ANALYSIS OF BIDDING BEHAVIOR OF CONTRACTORS IN VARIOUS ECONOMIC CONDITIONS USING UTILITY ASSESSMENT by ELIAS NICOLAS HANI B.S., American University of Beirut (1974) and YVES LESAGE Ingenieur Civil, Ecole Nationale des Ponts et Chaussees (1974) Submitted in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering at the Massachusetts Institute of Technology June, 1975 Signature of Coauthor Department of Civil Engineering, May 20, 1975 Signature of Coauthor Department of CilEngineering, May 20, 1975 Certified by / / I Thesis Supervisor Accepted by Chairman, Departmental Committee of Graduate Students of the Department of Civil Engineering tJUN 20 1975) ABSTRACT ANALYSIS OF FIbBING BEHAVIOR OF CO 'TRACTORS I) VARIOUS ECONOMIC CONDITIONS USING UTILITY ASSESSIhiENT by ELIAS NICOLAS HANI and YVES LESAGE Submitted to the Department of Civil Engineering on May 20, 1975 in partial fulfillment of the requirements for the de- gree of Waster of Science in Civil Engineering. Existing bidding strategies use expected monetary value as the decision criterion to account for the behavior of gene- ral contractors in risky situations. We show that this criterion has shortcomings and cannot ex- plain what is observed in real world bidding. The use of utility theory provides a better way of analysing the con- tractors' behavior. The results show that the economic position of a firm and the size of the project it bids on are two factors, amongst others, which are taken into consideration in bidding, fac- tors, that existing bidding strategies usually neglect. Thesis Supervisor Title Richard de Neufville Professor of Civil Engineering -2- ACKNOWLEDGEMENT We wish to thank our advisor, Professor Richard de Neufville, for his guidance and encouragements throughout this thesis. His many suggestions helped us to find our way through the difficulties of the subject and to bring this work to an early conclusion. We would like to extend our thanks to the general contractors of the Boston area we interviewed for their helpful cooperation. We are also grateful to the staff of the Bureau of Building Construction and the Depart- ment of Public Works who helped us in collecting the data. The authors wish to thank respectively the Lebanese National Council for Scientific Research and the French Ministry of Foreign Affairs for their financial support. Special thanks go to Mrs Christiane Hani for the numerous sacrifices she has made. -3-- TABLE OF CONTENTS Page Title page 1 Abstract 2 Acknowledgement 3 Table of contents 4 List of tables 8 List of figures 10 1. Introduction 1.1 Statement of the problem 12 1.2 Bidding strategies and utility theory 13 1.3 Organization of the study 15 2. A review of the literature of bidding strategies 17 2.1 Introduction 17 2.2 The general model 18 2.3 State of the art 22 2.3.1 Objectives 22 2.3.2 Profit 23 2.3.3 Probability of beating a given 25 competitor 2.3.4 Probability of beating a number of 27 competitors 2.3.5 Refinements 30 2.4 Other strategies 31 -It Page 2.5 Conclusion 33 3. Utility theory and assessment 38 3.1 Introduction 38 3.2 Utility 39 3.2.1 Expected utility 39 3.2.2 Selling and buying price of a lottery 40 3.2.3 Properties of utility functions 42 3.2.4 Risk premium and risk behavior 47 3.3 Utility assessment L9 3.4 Conclusion 51 4. Data analysis 53 4.1 Gathering data 53 4.2 Comparison of the number of bids per year 54 with the yearly allocated award 4.3 Number of bidders per project 58 4.4 Deviation from estimate 62 5. Utility assessment 67 5.1 Introduction 67 5.2 Development of the questionnaire 68 5.2.1 General considerations 68 5.2.2 Types of questions 69 5.2.3 Final form of the questionnaire 75 5.3 Utility assessment and results 77 -5- Page 5.3.1 From theory to practice 77 5.3.2 Treatment of data 79 5.3.3 Results 82 5.4 Conclusion 84 6. Comparison of results 85 6.1 Introduction 85 6.2 Interpretation of the results of the 85 utility assessment 6.2.1 Certain cost situation 85 6.2.1.1 Normal size of jobs 86 6.2.1.2 Large size of jobs 91 6.2.2 Uncertain cost situation 94 6.2.2.1 Normal size of jobs 96 6.2.2.2 Large size of jobs 97 6.2.2.3 Illustration 97 6.3 Comparison of results 100 6.3.1 Predicted behavior 100 6.3.2 Actual behavior 107 6.4 Conclusion 109 7. Conclusion 110 References 114 Appendix A Tables of data analysis 117 Appendix B Tables of utility assessment 127 -6- Page Appendix C Tables of comparison of results 131 Appendix D Questionnaire 140 -7 LIST OF TABLES Page 6.1 Variations in the degree of risk aversion, 92 certain cost case 6.2 Variations in the degree of risk aversion, 99 uncertain cost case Al BBC data, main features 118 A2 DPW data, main features 119 A5 Number of projects versus average number 120 of bidders per project - BBC - A4 Number of projects versus average number of 121 bidders per project - 3PW - A5 Deviation of bids from estimated cost for 123 BBC ( of estimated cost) A6 Deviation of bids from estimate for DPW 124 (f of estimated cost) AT Deviation of bids from estimate for DPW 125 (/ of estimated cost) AS Coefficients a and b found by linear 126 regression B1 Assessed utility function 128 B2 Minimum rate of return 130 for the various contractors interviewed (% return) C1 Optimum markup for contractor 1 on normal 132 size of jobs, uncertain cost situation C2 Optimum markup for contractor 1 on large 133 size of jobs, uncertain cost situation -8- C3 Cumulative distribution function (C.C.D.F) 134 for 3 competitors - markup in percentage - c4 Cumulative distribution function (C.C.D.F) 135 for 4 competitors - marKup in percentage - C5 Cumulative distribution function (C.C.D.F) 136 -for 5 competitors - markup in percentage - C6 Cumulative distribution function (C.C.D.F) 137 for 6 competitors - markup in percentage - C7 Optimum markups for contractor 1, 138 certain cost situation, using expected utility and monetary values C8 Optimum markups for contractor 1, 139 certain cost situation using expected utility and monetary values -9- LIST OF FIGURES Page 2.1 Probability distribution of the ratio of 20 competitors bids over estimated cost 2.2 Expected profits 21 2.3 Optimum markup versus number of competitors 36 3.1 Example of utility function of a contractor 45 on a $1,000,000 job - 0 U(x) 1 - 3.2 Example of utility function of a contractor 46 on a $1,000,000 job - 5 < V(x) < 15 - 3.3 Characteristic shapes of utility curves 48 4.1 Amount of money awarded and number of 56 projects awarded per year versus year (BBC) 4.2 Amount of money awarded and number of 57 projects awarded per year versus year (DPW) 4.3 Average number of bidders per project and 59 number of projects versus year (BBC) 4.4 Average number of bidders per project and 60 number of projects versus year construction projects (DPW) 4.5 Average number of bidders per project and 61 number of projects versus year maintenance projects (DPW) 4.6 Deviation from estimAte versus number of 63 bidders for good and bad years (BBC) 4.7 Deviation from estimate versus number of 66 bidders for 1967, 1969, 1971, 1973 6.1 Utility functions given certain cost and 87 normal size of job 6.2 Plot of the ratio, w(x), of utilities in bad 89 and good times 6.3 Expected utility function in good times 90 -10- 6.4 Utility functions given certain cost and 93 large size of job 6.5 Complementary cumulative distribution 101 function for 3 competitors 6.6 Complementary cumulative distribution 102 function for 4 competitors 6.7 Complementary cumulative distribution 103 function for 5 competitors 6.8 Complementary cumulative distribution 104 function for 6 competitors 6.9 Optimum markups for contractor 1, certain 106 cost situation, using expected utilities and monetary values, cost = 0.95 EE 6.1o Optimum markups for contractor 1, certain 108 cost situation, using expected utilities and monetary values, cost = 0.97 EE - 11 - CHAPTER 1 INTRODUCTION 1.1. Statementof the problem All firms in the construction industry have had to resort to closed competitive bidding as a means for getting work. A contractor is given a set of plans and specifications and is requested to submit in a sealed envelope his bid, or the price for which he agrees to do the work. The contractor takes many factors into account before submitting his bid: he considers the competition on the job, the type of work involved, his need for work, his financial status, the re- quirements of the job in men, materials and equipment, the availability of those, the size of the job, its risk, the overhead involved, etc... He then relies on his judgment and experience to submit his final bid. This bid will be based on his cost estimate increased by a certain percen- tage, called the markup, which allows for profit, overhead and contingencies. If his markup is too low, he will get the job but he will either lose money on the job, or get a very low return on investment and thus he will barely be able to survive. Usually the only difference between a flourishing company and another on the edge of bankruptcy lies in the management decisions taken at critical points - 12 - in time. One of the contractor's most important decisions is what optimum markup will insure maximum profits in the long run together with a good level of return on investment. This decision problem which faces the contractor can therefore be described as follows. After the contractor estimates the cost of the job he has to decide on the markup to be added, which can be any value between two extreme limits (usually 2% to 15%); then he has to observe what we call the state of nature, or the lowest bid among his competitors. If his bid is higher he will not get the job and will have lost the money spent on estimating the work which usually ranges between 0.5 to 2.0 percent of the total project cost. On the other hand, if his bid is the lowest, he will get the job and will end up after the execution of the work with a profit (or loss) equal to his markup minus job overhead minus variation in cost from the estimate. At the same time he suffers an opportunity loss which is the difference between his bid and the second lowest bid; this is referred to as the money left on the table, or the spread. 1.2. Bidding strategies and utility theory To help the contractor determine the optimum markup, compe- titive bidding strategies have been developed by several -13- investigators. A strategy can be defined in a number of different ways, but perhaps the most suitable definition for our purposes is: A bidding strategy is the science and art of meeting competition under the most advantageous condi- tions possible in order to achieve a certain goal. The problem we face can be stated in the following manner: Are these mathematical formulations of a fairly complex situation realistic enough to be applied in practice? Are the assumptions on which they are based justified? Is the actual behavior of contractors in real situations consis- tent with what is predicted by these strategies? In fact all bidding models considered that the contractors have expected monetary preferences, and that the number of bid- ders and the probability of winning are the major factors influencing the bid price. These factors can be repre- sented schematically as follows: NUMBER OF BIDDERS C.C.D.F BID PRICE where C.C.D.F is the complementary distribution function of the density distribution of markups: it gives the probabi- lity of winning given a certain markup. However we feel in this study that the decision-makers have expected utility preferences, and that some other factors, like the relative size of the project and the economic situation ("good" or "bad" times), are also of major influ- ence. We suggest the alternative representation: ECONOMIC CONDITIONS SIZE OF PROJECT [NUIMBER OF BIDDERSUTILITYJ C.D.F BID PRIC 1.3. Organization of the study The second chaper of this study reviews the state of the art in 1975 concerning bidding strategies. It first intro- duces the basic idea common to all the models, and then discusses the sometimes contradicting refinements or - 15 - assumptions considered in the different strategies. The expected monetary value criterion which constitutes the basic principle of all the models is criticized in chapter 3 and the invalidity of such a criterion is shown. The same chapter also introduces the utility theory which was deve- loped to deal more efficiently with uncertain situations. The next step is then to analyse actual behavior in bidding situations, in order to verify the validity of the expected utility criterion. The results of the analysis of public bids collected from two state agencies are presented in chapter 4. Chapter 5 is concerned with the development of a questionnaire for assessing the utility functions of some contractors in the Boston area. The factors influencing the utility functions are first determined, then the ques- tions used and their interpretation in terms of the utility functions are considered. We finally discuss the diffi- culties encountered in the actual assessment, as well as the results obtained. Chapter 6 presents the interpretation of the assessed utility functions, followed by a correlation between the predicted behavior, obtained through the utility functions, and the measured behavior from public bids. This exposes the deficiency in the existing bidding strategies. The conclusion suggests some possible extensions of this study for future research. - 16 - CHAPTER 2 A REVIEW OF THE LITERATURE OF BIDDING STRATEGIES 2.1. Introduction Numerous bidding strategies have been developed to apply to different kinds of situations. They can be classified according to whether they are concerned with: (1) Determining the optimal bid for a single job. (2) Sequential bidding, that is, simultaneous bidding on more than one contract. (3) Situations involving lump-sum bids. (4) Unbalanced bidding on unit price contracts. Although most of the strategies deal with bidding against competitors, some of them also deal with bidding against the owner. This study focuses on non-sequential closed competitive bid- ding on lump-sum contracts, since most other bidding situa- tions can be considered as extensions of this basic model. Almost all of the existing bidding strategies more or less follow the first bidding model, developed by Friedman (1956) for the oil industry. But many of the assumptions change, sometimes significantly, between different models. Fur- thermore each model considers different factors that, - 17 - according to the author, significantly affect the optimum bid, factors that are not considered or even are rejected by other models. The general contractor who is the poten- tial user of such models should be aware of the weaknesses (present in the assumptions) involved in any specific model before he decides to use it. We review and present the different bidding strategies cur- rently available for use in the construction industry as follows: first the basic general model common to almost all strategies; secondly, and in some detail, all known devia- tions ar elaborations around this model; finally, some of the models which differ significantly from the basic idea of the general model. This part is based on the study of dif- ferent bidding models developed by Friedman (1956), Park (1966), Gates (1967), Morin and Clough (1969), Shaffer and Micheau (1971), which generally represent the state of the art in 1975. 2.2. The general model The general model supposes that the objective of the com- pany is to maximize the total expected profits, given by the general formula: E(P) = p(P) x P where E(P) is the expected profit - 18 - P is the profit (or markup) included in the bid price p(P) is the probability of being the low bidder given a profit P. Based on past bidding data the probability p (P) of beating competitor i given P can be determined. This is done by developing bidding patterns for the different competitors. Given competitor i's bids on past projects, a frequency distribution of the ratio of com1 etitor's bids as a percent of estimated cost can be plotted and a probability distri- bution function can be fitted (see figure 2.1). Then, for any given P, pi(P) can be found: it is the complementary distribution function (C.C.D.F) Pi(P)= 1 -J p dy If there are n competitors the probabilities of beating each one individually are combined in a certain way (discussed later in this study) to give the probability p(P) of beating them all given a certain profit level P. E(P) is then plotted versus P (expressed as a percentage of the estimated cost of the work) and the optimum profit P0 can be deter- mined. (see figure 2.2) However there are significant deviations between the dif- ferent strategies on the following points: 1) What is to be considered as profit. - 19 - >- C 0 so too 105 It'O 's COMPETITOR'S BIDS xlOO ESTIMATED COST FIGURE 2.1 PROBABIMITY DISTRIBUTION OF THE RATIO Of COMPETITORS BIDS OVER ESTIMATED COST. - 20- tlog L La VI Ll FIGURE 2.2 EXPECTED PROFITS. - 21 - 2) How to obtain the probability of beating a given compe- titor. 3) How to combine the n individual probabilities in order to obtain the probability of beating the n competitors. 4) How to treat the individual competitors when their iden- tity is either known or unknown. 5) How to deal with the case when the number of competitors is unknown. In addition to the above mentionned points, which can be considered to have a direct relation to the general model, some other factors are introduced by different models and have a direct influence on the optimum markup. These in- clude the following: the class of the work, its size and contingencies. 2.3. State of the art 2.3.1. Objectives The assumptions and deviations between the different stra- tegies will be presented in the same order as mentionned above. It is obvious that the optimum markup depends pri- marly on the objectives of the company. In the case of the construction industry there are many possible objectives: 1) Maximize total expected profit - 22 - 2) Maintain a prescribed level of return on investment. 3) Minimize expected losses (during idle periods). 4) Minimize competitors' profits to maintain competitive position. 5) Keep a certain share of the market. 6) Increase volume of work as much as possible. 7) Increase volume of work to keep up with inflation only. 8) Try to maintain labor force at work at any cost. Almost all of the bidding strategies developed until now have the objective of maximizing total expected profit (which is most common). But, for example, if a company's objective is to maintain its labor force, its optimum markup can well be negative. The user of bidding models should be aware of this basic assumption of maximization of expected profits which is the foundation of all the strategies. 2.3.2. Profit The profit P on a certain job can be considered to be equal to B - Ca - OH where B is the bid price, Ca the actual cost of contruction and OH the overhead. Both Park (1966) and Gates (1967) assume in their models that profit is equal to the bid price (B) minus the estimated cost of construction (Ce). No consideration is given to overhead which is as- sumed to be included in the cost estimate and no correction is introduced to adjust for the fact that in actual cases -23- the cost of construction is different from the cost estimate. Gates recognizes this difference and contends that it is small enough to be neglected. He also suggested a correc- tion for bias in a brief explanation of break-even analysis. Similarly, Friedman (1956) assumed that overhead is included in the cost estimate but he considered profit to be the dif- ference between the estimated cost of fulfilling the con- tract corrected for bias and the bid amount. He suggests that one can develop, through a study of past data on esti- mates and actual costs, a probability distribution of the true cost as a fraction of the estimated cost. Letting S be the ratio of true cost to estimated cost (0 ), B the amount bid for the contract, p(B) the probability that a bid B will be the lowest and winn, then the expected profit suggested by Friedman will be: E(P) = p(B) B - S C3 h(S) dS where h(S) is the probability of a certain S. Alternatively E(P) = p(B) [B - C'] ca where C' jS h(S) dS and is the expected actual cost. On the other hand, iorin and Clough (1969) assert that the expected actual cost will equal the estimated cost. They found, from the study of data for a certain company, that the ratio of actual to estimated cost approached a symme- trical distribution with a median of 1.0 and a standard deviation of less than 2%. But, unlike the previous models, they recognized the fact that overhead actually differs significantly from one company to another. So the profit to be maximize is the net profit which is the difference between the markup (MP) and the overhead (OH): E(P) = (MP - OH) p(MP) where p(WLP) is the probability of being the low bidder with a markup of MP. 2.3.3. Probability of beating a given competitor To determine the probability of beating a known competitor, all models suggest that one should study past bidding data and develop (in different ways) a probability function for the ratio of the competitor's bid to our estimated cost. Friedman (1956) and Park (1966) suggest fitting a continuous probability function to the available data; Casey and Shaffer (1964), a normal distribution function; Gates (1967) uses on the other hand a statistical linear regression to determine this probability and expresses the profit P as: P = a logpi- b where pi is the probability of beating competitor i given - 25 - a profit P. a and b are determined from the plot of P versus log pi for all past bidding data. When the identity of the competitor is unknown or when insufficient data are available to develop the probability functions, all the above models combine past data to get a general bidding pattern for a typical competitor. The OPBID model of Morin and Clough varies significantly from the other models. First they felt that recent data should receive more weight than older data because the-com- petitive situation changes with time, they suggested using an exponential weighting scheme; secondly they classified competitors as being either key or average, depending on the percentage of past biddings they participated in. If this percentage is greater than a certain specified ratio (0.4 to 0.5 was suggested) the competitor is classified as key com- petitor, if it is lower the competitor is considered as ave- rage. The advantage of this classification is that it is a weighting scheme in which companies competing on a relati- vely large number of jobs are given much more attention than occasional competitors. Finally in order to assess the pro- bability of winning with a given bid, the OPBID model uses a discrete probability distribution which has the following three advantages: 1) It eliminates errors introduced by fitting smooth - 26- probability curves to existing data. 2) It permits fast calcutions by a digital computer 3) It provides a general model that can be used by dif- ferent contractors. 2.344. Probability of beating a number of competitors Given that the number n of competitors on a certain project is known and given the probability pi(B), all models follow more or less one of two methods to determine the probability of beating the n competitors. Some of the authors (Clough, Friedman, Park) assume that the probability of beating any given competitor is independent of that of beating any other competitor, and therefore the probability of beating them all is the product of the indi- vidual probabilities. Pn(B) = iT P(B) pn (B) is the probability of winning over the n competitors given a bid price B. In the.case of Clough's model this probability is: N' pn(B) = [TTN p(B )I a(B) av key j~ v where n can be subdivided in Nkey competitors and N' ave- rage competitors. - 27 - When the identity of the competitors is not known the proba- bility of winning becomes: Pn(B) = Pav(B)]" The second method, developed by Gates, rejects the idea of the probabilities of beating individual competitors being statiscally independent, and contends that when you beat a certain competitor you will revise your estimate of beating the others. The probability of winning becomes: Pn(B)= ; ;B) F. I- )i () +1 n pi(B)I When the identity of the competitors is not known is sim- plifies to the following: PnB)=n p (B) n (1 -p ay(B)) + 1 Broemser (1968) developed and used another method in his model. He dissociates the probability of winning from the number of competitors on a given job and contends that it is sufficient to beat the lowest bidder. So, using past data, he determines the distribution of ra, the ratio of the lowest competitor's bid to the estimated cost, by a linear regression which attempts to explain the behavior of the -28- low competitor by certain requirements of the particular job. These include estimates of the percent of cost not subcontracted, the duration of the job, the ratio of job duration to estimated cost, and the cost. This model tries to go around the problem of determining the probability of beating n competitors by implicitly assuming a certain rela- tion between the number of competitors and the characte- ristics of the job. When the number of competitors is unknown the different strategies use different ways to predict this number. In his original work, Friedman suggested the use of one of two methods: 1) The number of competitors might be given by a Poisson distribution whose parameters are determined from tests of past data. 2) A linear regression is used on past data of number of bidders against the cost estimate of the contract. He assumes that the larger the size of the contract, the higher the number of competitors. Similarly Park assumes that the number of competitors is a function of the job size, and that this number increases until a certain limit job size then decreases for larger sizes. On the other hand, Gates as well as Morin and Clough - 29 - suggests using the mean of the number of competitors encoun- tered on previous biddings,since, using real world data they could not come up with a relation similar to that suggested by Friedman and Park. 2.3.5. Refinements Some other refinements were introduced by some bidding models. These concern the class and size of the work as well as contingencies. Morin and Clough suggested the clas- sification of historical data in different "classes of work" since the chances of winning with a certain markup on one class of work might be different from the chances of winning with the same markup on a different class of work due to the variability of risks involved or the time required for exe- cution. Park suggested an interesting extension for his bidding model. He recognized that the optimum mark up de- pends on the size of the job, the larger the job, the lower the optimum markup, and suggests the following relationship: x[Coil MP 2 where Co is the estimated direct cost on job 1, and LP1 the optimum markup on job 1. The exponent x can be determined from the bidding history and he proposes a value of 0.2 in his article. - 30 - As far as contingencies are concerned Gates (1971) catego- rized contingencies into four groups: mistakes, subjective uncertainties, ojective uncertainties, and chance variations. He dealt with such problems as mistakes of omission, pro- blem of natural events, production rates etc... and he used analytical methods based on statistics and probability to obtain for each case likelihood of occurence, costs and fi- nally expectation values. 2.4. Other strategies This section presents two models which significantly vary from the fundamental idea of the general bidding model. The first is the "least bidding strategy" developed by Gates and the second is the strategy proposed by Shaffer and Micheau (1971). The "least bidding strategy" is based on Gates' finding that the average spread ( Ba) is a function of the size (C) of av the low bid. Specifically he came up with the following expression: Bay = 1.08 CO.734 He also found in his original study that 67P" ' Ba- 1-(p) - 31 - where P' is the amount to be added to the completed bid and (p) the probability of still being the low bidder after in- creasing the bid by P'. Using these relations and assuming that you were 100o cer- tain (relatively) that for your initial bid B you would be the low bidder, Gates determined the optimum amount P' to add to your bid to maximize your expected profit. This is given by: P',= 0.80 C 0 '34 - 0.50 B It is interesting to note at this point that a similar rela- tion was found by Park relating the percent average spread to the number of bidders and he suggests a linear relation: the higher the number of bidders the smaller the average spread, but he did not pursue his idea further. All the models presented until this point rely only on ma- thematicals expressions using past bidding data, and the contractor had only to input his information into the model to come up with an optimum bid. An extension of previous models was presented by Shaffer and Micheau (1971) who reco- gnized that too many variables influence actually the selec- tion of an optimum bid, that no model took all of these into consideration, and that judgment based on experience can be of great value in arriving at a final choice of a bid. So they suggested a combination of formal and informal means, - 32- with the formal means being competitive strategy models and the informal means being personal judgment of the contractor. The whole object was to determine with the formal means an upper and a lower bound to provide the contractor with a focus- for his judgment. The actual selection of the bid would be obtained solely by informal means, the lower bound of the range would be the bid that would give the contractor the greatest chance of submitting the low bid for the pro- ject and the upper bound would be the bid that would give him the greatest chance of submitting the second low-bid. Shaffer and lMicheau also recognized the fact that different bidding models apply best to different situations and that a person cannot a priori decide that this particular bidding strategy will apply best to all companies or to a particular company. So they recommend to select a few strategies and to test these using past bidding data and then the models to be used to determine the upper and lower bound will be those whose results have had the greatest success on historical data of yielding the low and second low bids respectively. 2.5. Conclusion Although most of the strategies follow the same basic idea presented in the general model, apparent large differences exist in many of the fundamental points considered. These -33- points of disagreement can be restricted to two major ones: 1) What is profit. 2) How to get the combined probability of winning. As far as profit is concerned, some models assumed the ac- tual cost to be equal to the estimated cost and most of the models did not consider overhead which affects obviously the probability of winning. However this disagreement is mostly superficial. As regards the correction for bias of the es- timated cost, it is safe to assume that the estimator, based on a knowledge of his past performance, will continuously adjust his method for computing the estimate to account for the bias that has occured in the past; there will then be no need to correct for bias. Consider on the other hand the overhead; if it is accounted for in the cost estimate (as assumed in most strategies) and is high, the probability of winning for a given markup will be lower than when the over- head is low for the same markup. However if the overhead of a certain company did not fluctuate much in past bids and is still at the same level, one can affirm that this overhead would have influenced the "bidding patterns" developed for the competitors and no other correction is required. As far as the probability of winning against a number of competitors is concerned, numerous articles and discussions were written to support either of the two assumptions pre- sented by Friedman and Gates - Stark (1968), Baumgarten 34 (1970), Benjamin (1970), Naykki (1973), Dixie (1974) - while other articles tried to reconcile them - Rosenshine (1972) - All the strategies agreed, however, on the fact that the higher the number of competitors, the lower the probability of winning and therefore the lower, the optimum markup (see figure 2.3). Nearly all of the models presented developed or introduced certain relations between the different variables, and cer- tain factors which they considered would have an effect on the optimum bid. But as suggested by Benjamin Neal (1972) the weaknesses and the disagreements between the different models may be due to the fact that they use single variable statistical techniques to deal with a situation which is ac- tually much more complex. Although there is much controversy between the different bidding models studied in this part, all agreed on the basic assumption of maximization of total expected profits. They all implicitly assume that people behave in a specific way, behavior which may not be observed in real world situations. To consider that people actually decide according to expec- ted monetary value may be a major weakness. Although this is the subject of the next chapter, a simple example illus- trates this point: if a person behaved consistently with the above assumption then he should be indifferent between participating to a lottery offering an equal chance of - 35 - 0! 0 HUM13ER OF COMPETITORS. FIGURE Z.. 3 opTimum MARKUP VER.Sus NUMBER OF COMPETITORS . -36- gaining or losing $10,000, or not participating at all. But in fact most, if not all, people will not even consider such a lottery. The fact that all models do not account for the effect of various economic conditions on the optimum bid is one aspect of this weakness. All available bidding stra- tegies indicate that the contractor should add the same markup for a given project, regardless of the economic con- ditions. In fact, one might expect that in bad economic conditions the bids will be lower: the contractors are much more interested in winning the contract due to the fact that they are more in need for work than in normal times. Other aspects of the weakness of the models will be discussed later on, in this study. The next step is therefore to introduce a criterion, expec- ted utility value, that would do away with the weakness of the expected monetary value criterion. We then determine through an analysis of actual bidding data the behavior of the contractors in various economic conditions. The aim is to test the validity of the existing models and of the ex- pected utility value criterion to determine if its use gives consistent results with observed real world behavior. - 37m CHAPTER 3 UTILITY THEORY AND ASSESSMENT 3.1. Introduction The review of various models of bidding strategies in the literature shows that all of them deal with expected mone- tary value as a decision criterion. This makes implicit assumptions which may bias the evaluation of a bid, and lead to erroneous conclusions. The first assumption is that a unit of loss has the same value as a unit of gain, i.e. that a contractor should be indifferent between staying in his present situation and having a lottery where he can lose $A with a probability of .5 or win $A with a probability of .5. That may be true if A equals $1 or $10 but if the amount of money goes to $10,000 or $100,000 the contractor may think he is better off away from that gamble. The fact is that people generally give more weight to losses than to gains, and taking expected monetary value does not take into account the spread of the outcomes. The size of the loss that one can afford mainly depends upon one's assets. For instance a small contractor might not be willing to bid on a risky $10,000,000 job, because if it fails, he may go bankrupt. On the other hand a bigger - 38- company may take it, for, in the worst case it may run into a cash problem. However, if the small company is at the point where it needs a high gain to avoid bankruptcy, it might take the project on the grounds that if it fails the bankruptcy will only be a little worse, whereas if it suc- ceeds, it will not go bankrupt at all. All these factors are not taken into account by the expected monetary value criterion, for it makes the assumption that everyone shares the same values for all items at all times. These various difficulties encountered in using this criterion can be cleared up by the use of a recently developed tool of deci- sion analysis : utility theory. 3.2. Utility 3.2.1. Expected utility Given a certetit number of axioms, which we will discuss later, a utility function can be defined as the represen- tation of a set of numbers which are generated for each possible outcome of a decision, numbers which can be used to order all choices according to their desirability to the decision-maker. An outcome can be represented by a vector of attributes. In this study we will only be concerned with utility func- tions with one attribute. Such functions scale preferences - 39 - for the attribute - in this case, the percentage of markup on a project. Instead of working with expected monetary values we will use expected utility value. This criterion enables us to compare two possible decisions, knowing the various values of the attribute which result from the deci- sions. We proceed in the following manner : letting U(x) be the utility function of a contractor, the expected value of the utility EUV of a decision is EUV = En U(x) where x. is the value of the attribute corresponding to one of the n possible outcomes of the decision, and p. the pro- bability of occurence of this outcome. To compare two deci- sions, one compares their two expected utility values, and chooses the decision with the maximum expected utility. 3.2.2. Selling and buying price of a lotters Suppose that you own a lottery ticket which gives you a 50-50 chance of winning $1000 or $0, and that we ask you to sell that ticket for $300. Must you accept ? Such a deci- sion can be represented by (Raiffa 1968) - 4o- $300 $1000 decision node chance node $0 lottery If you are indifferent between the two choices, $300 is said to be the certainty equivalent of the lottery ($1000, .5 ; $0, .5). This can be represented by $1000 $300 - 'too < $0 If two choices are equivalent their respective expected utility values are equal. We can write 1 x U($300) = .5 x U($1000) + .5 x U($0) -h41 - $300 is said to be the selling price of such a lottery. We can consider another type of lottery corresponding to the following question : how much would you pay to have a 50-50 chance of winning $1000 ? This can be represented by $1000 - b 0 $0 .4 - b where b is the amount you would pay. b is said to be the buying price of the lottery. 3.2.3. Properties of utility functions Very often a utility function is monotonic. For instance if the attribute is the profit on a project, more is better the utility function is monotonically increasing. If the utility of a decision is twice the utility of another decision, it does not mean that the results of the former decision will be twice as good. The function has ordinal properties and not cardinal ones, it works exactly as temperature scaling. You can compare two temperatures and say that one day is hotter ot colder than another one ; nevertheless, one cannot say that a given day with a tempe- rature of 60"F is twice as hot as another with a 30F reading, - 42 - You can take a linear transformation of the utility function U(x), which transforms U(x) into V(x) = a U(x) +b (a>0), without changing the relative ranking of the decisions ; that is exactly what happens when temperature is expressed in degrees centigrade instead of degrees Farenheit. In that case U(x) and V(x) are said to be strategically equivalent. A utility function is defined on a certain range which is meaningful to the person whose utility function is assessed. For instance it is meaningless for a general contractor to consider a 50% profit on a project, because he could never reach that. Therefore the range of definition of a utility function should cover all the values of the attribute which can be considered in any decision-making process. Then two utility values are assigned to two different outcomes (as 326F and 212*F are respectively assigned to freezing point and boiling point of water). Usually the values 0 and 1 are respectively given to the lowest and highest value in the range considered. In the following example, which illus- trates what we have said about utility functions until now, we consider a range of profit going from -5% to 10%. We assume that the worst thing which can happen on the project is a loss of 5% and the best thing is a profit of 10%. - 43- In figure 3.1 we can see the following correspondance x -5% -1% 2% 10% U(X) 0 .5.75 1 In figure 3.2 we have another correspondance x -5% -1% 2% 10% V(x) 5 10 12.5 15 It is easy to see that V(x) = 10 U(x) + 5 Using these curves, let us find the certainty equivalent of the following lottery : -1% profit $0 10% profit Let us first compute the expected utility value U(x) U(x) = .5 U(-1%) + .5 u(i0%) = .5 x .5 + .5 x 1 = .75 400 4 4. 'C 5 I I., - - -- - - Ir -g I IF I I I I I I I I I I I I I 1 I I * - 4 e~6 4 10 MARKUP C V. OF COST) FIGURE 3~1 EAMPLL OF UTILITY FUNCTION OF A CONTRACTOR ON A $ IPOQOOO JOB - 0 ~ V(fl141 "i MAKU( OFCOT 5IUE3 XML F TLT UCINO 10 OTATO NA 1 sOoOOJB.5 M 5 We conclude from figure 3.1 that x equals 2%. Let us compute the expected utility value V(x) : V(x) = .5 V(-i%) + .5 V(10%) .5 x 10 + .5 x 15 = 12.5 From figure 3.2, we conclude that x equals 2%. As expected the answer obtained with V(x) is the same as the one obtained with U(x), since the two are strategically equivalent. 3.2.4. Risk premium and risk behavior The risk premium is defined as the difference between the expected monetary value and the certainty equivalent. In the above example we found that the certainty equivalent was 2%o. The expected value is : .5 X 10% +.5 x -1% = 4.5% The risk premium r equals : r = 4.5% - 2% = 2.5% If the risk premium is positive, the decision-maker is said to be risk-averse. In case of a negative risk premium he is said to be risk-positive. If the risk premium is equal to zero, he is said to be an EMV'er. Figure 3.3 shows the three characteristic shapes of utility curves. Curve I rp I) RISK AVERSE Kr) x '.9 EMVir RISK POSITIVE MARKUP CZ OF COST) FIGURE 3.3 CHAACTERISTIC SHAPES OF UTILITY CURVES. represents the utility curve of a decision-maker who decides according to expected monetary value over the whole range. Curves II and III respectively represent risk-averse and risk-positive behavior over the whole range. It is often interesting to measure risk-aversion, and to be able to say that a decision-maker is more or less risk- averse than another. The risk premium is not a good mea- sure when the range of the utility function is changed. Pratt (1964) showed that if U(x) is the utility function, the ratio r(x) = - U"(x) / U'(x) gives a good measure of risk-aversion. It is easy to see that r(x) is identical for two strategically equivalent utility functions. As was said earlier, the existence and properties of uti- lity functions are based on a certain number of axioms. One of them deserves some special attention, because of the criticism it receives. It is the transitivity of prefe- rences : if you prefer outcome A to outcome B and B to C you must prefer A to C. Raiffa (1968) shows that somebody who has an intransitive behavior is bound to lose all his assets to someone who adequately uses this intransitivity. 3.3. Utility assessment To use the expected utility criterion in the bidding stra- tegy of a contractor, you need to assess his utility function. Such assessments have been made in other fields: Grayson (1960) made a study of oil wildcatters, Swalm (1966) and Spetzler (1968) of business executives, Lorange and Norman (1970) of shipowners, de Neufville and Keeney (1971) of executives of the Mexican Ministry of Public Works and Willenbrock (1973) of contractors. In all these studies, the interview method was used to assess utility functions; a discussion of the validity of such a procedure is made by Lorange and Norman (1970). The type o0 questions asked in any of the interviews is as follows: For what amount of money will you be indifferent between I and the following lottery? XX X2 In this lottery you can win X1 with probability p and X2 with probability 1-p. Answers to a certain number of these questions enable the interviewer to draw the utility func- tions of the decision-maker. Two procedures can be used to obtain these curves. One is to let X1 and X2 be fixed and to vary p; we call this the constant-attribute procedure. The other is to let p be fixed - generally equal to .5 - and vary X11 and X2; we call this the constant-probality - 50 - procedure. Grayson (1960) and Spetzler (1968) used the constant- attribute procedure. Swalm (1966), de Neufville and Keeney (1971) and #illenbrock (1973) used the constant-probability procedure. Lorange and Norman (1970) used both procedures. These studies show that the persons interviewed have pro- blems in dealing with probabilities different from .5. Grayson (1960) writes: "Probabilities created the greatest difficulty in the experiment. Some operators do not normally use numerical probabilities in their decisions, and they found it strange to try to reach a deci- sion on the basis of probabilities." In Lorange and Norman's study this problem was reflected by the fact that shipowners were significantly more risk-averse vis-a-vis constant-attribute gambles than versus constant- probability gambles. Both assessment procedures discussed above are valid, but the constant-probability one is more suitable because more easily understood by the decision-makers who are inter- viewed. 3.4. Conclusion One of the useful properties of utility functions is their - 51 - flexibility. The assessment made todEay can be revised two months later if economic conditions, liquidity position of the company or availability of work have changed. This allows comparison between various behaviors of a decision- maker according to his liquidity position - good or weak - or according to the environnement - normal or bad times - - 52 - CHAPTER 4 DATA ANALYSIS We found that the best way to understand contractors' beha- vior when bidding on projects was to gather data on bids of past projects, to analyze them and, as far as possible to find relationships between various factors. 4.1. Gatherin5 data Theoretically we had the choice of picking data of bids of either public or private projects. In fact we chose to gather data on public projects. Data on private projects are not easily available both because owners are reluctant to give out the information and because the information needed for a good sample size is scattered among many owners. On the other hand bids on public projects are public information, in the United States, and are all avai- lable in the same place. As we were mainly interested in building contractors we went first to the Massachusetts Bureau of Building Construction (B B C) where we gathered data of the years 1961 to 1974. We collected data on all projects of new construction which had a value of $100,000 or more. A description of various -53- features of these data is given in table Al (Appendix A). The total number of projects, over the 14 years, is 167. To obtain a larger sample size we went to the Massachusetts Department of Public Works (DPW) to get data of bids on highway projects. There we were able to collect data of highway projects of the years 1966 to 1974. These covered four types of project: construction, reconstruction, high- way work and resurfacing. A lower bound on the size of the projects was also fixed at $100,000. The main features of these data are given in table A2. The number of projects collected is greater than 650. Only the DPW data were ana- lyzed on a computer. Note that in this case the fiscal year was used instead of the calendar year, because the statistics of the annual awards were provided in that manner. In table Al and A2 of appendix A, one finds the total amount of money awarded to projects by the BBC and the DPW, as well as the corrected amount in 1974 dollars using the engineering news-record cost indices. 4.2. Comparison of the number of bids per year with the yearly allocated award Our first task in analyzing these data was to compare the number of projects awarded each year to the total amount of money expressed in 1974 dollars allocated yearly, and to see how they are related. If, for instance, the BBC was only awarding new construction projects, one would expect the two curves to be approximately parallel. As we can see in figure 4.1, that is not quite the case. As a matter of fact, the BBC also awards other types of work - renovation, utilities for instance - and there is sometimes an imba- lanced year - 1971 for example where many renovation pro- jects were awarded. In the case of the data of the DPW (figure 4.2), it appears that no relation exists between the two curves. Table A2 shows that when the construction award is sufficient, nu- merous construction projects are started at the expense of maintenance projects - highway work or resurfacing -, be- cause the former usually require a much greater investment. Conversely, when the money becomes scarce, maintenance pro- jects take priority: as they are less expansive than cons- truction projects, the total construction award can decrease whereas the total number of projects awarded increases. Therefore the total construction award cannot represent the real situation of the market of projects available to con- tractors bidding either on BBC work or on DPW work. In order to compare the results to the nature of the economic environnement, we needed a variable which represents the availability of projects for contractors. For the reasons - 55 - " � �• "' 0 a: C i 2.00 z 0 t- \.n \J :>°' 0: I ... "' z 0 u.100 NUM&EA 61 6 . ,. . A I I I I I ,"' I I \ \ j � MONE\' AW"RD£D 5' '' ' 68 6 C.ALENDAR YEARS. ' \ \ I ,_ _. 0 7f • 1 I I \ 20 \ \ ' ' 15 I l I g I � I 0 I 10.,, .. I / :a 0 ,,., c.. "' � 5 F\(;URE 4.1 "AMOUNT OF' MONEY AWARDED AND NUMBER OF PROJECTS AWARDED PER YE�R- VERSUS 'ua S. ^I U. 0lw v -bs- CHAPTER 5 UTILITY ASSESSMENT 5.1. Introduction After analyzing the bidding data on past projects, the next step was to develop a questionnaire to assess the utility functions of some general contractors in the Boston area. The purpose was to determine whether one can explain, through utility assessment, what is observed in real world situations, and subsequently to investigate the superiority of the expected utility criterion over the expected mone- tary criterion. The decision-maker, in any bidding situation, will usually consider a number of factors before deciding on a final bid. These factors, often interrelated, are: - Economic conditions prevailing in the construction in- dustry. - Assets position of the firm - Size, type, location and duration of the project - Percentage of project subcontracted - Identity of owner, architect and competitors - Number of competitors - Availability of materials - Possibility of price fluctuations and of labor strikes -67 - As far as utility assessment is concerned, one can expect the asset position of the firm and the size of project to be major influences on the shape of the utility functions. The attitude towards a loss (or a gain) of a $1000 will differ depending on whether the company is making a lot of money or is struggling to survive. Similarly a 2% gain on a small project is viewed differently from a 2% gain on a large pro- ject. The economic situation will also have an effect on the shape of the utility curves, but because of its strong interrelation with the asset position of the firm, only one of these two factors will be varied in the utility assess- ment. Most of the remaining factors affect the risk of the job and therefore the outcomes: they are handled through the incorporation of probability in the use of the expected utility criterion. 5.2. Development of the questionnaire 5.2.1. General considerations The utility assessment was conducted under varying assump- tions of size of project and economic conditions. As far as the size of the project was concerned, two sizes were consi- dered for each company interviewed; one corresponded to the normal size of projects handled by the firm; the second was roughly half-way between the normal size and the maximum - 68 - size ever undertaken by this particular company. The eco- nomic conditions were described as normal (or "good" times), and actual (in 1975 this meant "bad" times). All companies agreed that normal times corresponded to the years 1968, 1969, and also that actual times are bad. Therefore the utility assessment was conducted under four general condi- tions: good times and normal project, good times and large project, bad times and normal project, and finally bad times and large project. To determine the range over which the utility assessment was to be conducted, a utility value of 1 was arbitrarily as- signed to a 15% markup (profit and overhead). This was thought to be the maximum amount that any firm, in any con- ditions, can expect to make on a project. The value of 0 was assigned to the minimum rate of return (MRR) acceptable by the company. This minimum rate was determined by of- fering the decision-maker a project on a cost plus fixed fee basis, and determining the minimum fee for which he would consider accepting the contract. 5.2.2. Types of questions The general type of questions used for utility assessment is the following: the decision-maker is offered two projects, one involving a fixed outcome, the second two outcomes with some chances of occurence, and he is asked to give his - 69 - preferences. The purpose is to arrive at an indifference state between the two choices. As stated in a preceeding chapter, there are two major ways that can be used to arrive at the indifference point, the first being to vary the pro- babilities of occurence of the various outcomes, the second to vary the value of the outcomes. The second procedure was followed in our assessment, since it was felt that the contractors would be much more sensi- tive to variations in monetary outcomes rather than to va- riations in probabilities. It also permits the elimination of a subjective interpretation of probabilities that could distort the shape of the utility function and give incorrect results. Futhermore the questions should permit the assess- ment of utilities for points within the range minimum rate of return and 15%, and also for points that are below this range. Since the contractor may be faced, in some risky situations, with the possibility of outcomes below his mini- mum acceptable return. Another very important characteristic for the questions is that they should be phrased in a certain way that makes sense to the general contractor. They should represent specific situations that the contractor is likely to en- counter in his every day work. The first type of questions used for the utility assessment - 70 - is as follows: the contractor is offered two contracts. Contract A involves a job under a unit price arrangement with two equally likely monetary outcomes. These outcomes are the minimum rate of return, determined in a previous question, and 15% of the total cost of the project. Project A was represented by 15% x size of job *o Minimum rate of return x size of job Contract B involves a job on a cost plus fixed fee basis, and the contractor is guaranteed a certain monetary return. The contractor was then asked to decide whether he would choose contract A or contract B, if he had the opportunity to take only one. The fixed fee of project B was varied until the point of indifference between projects A and B was reached. To determine the utilities of markups below the minimum rate of return, the same kind of questions could be used. Con- tract A would have its outcomes replaced by one which is smaller than the minimum rate of return (S) and one which is larger (L). Then, the fixed fee in contract B would be varied until, for a certain value, the indifference point between the two contracts is reached. Knowing the utilities - 71 - of the fixed fee (in contract B) and L, the utility of S could be found (it would be negative). L 10 'S. However the disadvantage of such type of questions soon became apparent. When the first contractor interviewed was offered contract A with the possibility of an outcome smaller than the minimum acceptable to him, he refused to consider such a project. Another type of question was therefore devised in order to go around this difficulty. A job with two equally likely final costs was presented to the contractors, and they were asked what their minimum bid would be. If the final costs of the job are denoted by C1 and C2 , the bid price by B and the present situation without the project by P.S. then we would have: B - C - cost of preparing bid P.S.u B - C 2 - cost of preparing bid I.S. - cost of preparing bid -72- Neglecting the cost of preparing the bid, which can be con- sidered to be accounted for in the overhead of the company, we have: x 1000/0 C B - C2 S. 2 C P.S. The outcomes have been replaced by the percentage profit for consistency. Let f, represent a very small positive value, then one could say that: P.S. ^.- RR-a since, theoretically, the contractor would refuse the job if the return was slightly smaller than the minimum rate of return. He would therefore be indifferent between his present situation and a return of MRR - E. Since e is a very small value the continuity of the function U(x) implies U(MrR - a) ~'U(MRR) Therefore P.S. a MRR Using the axiom of transitivity (see chapter 3) we obtain: - 73 - B - .* C' B- C, MRR 100% x 100%4 2 Equating the utilities on both sides: U(MRR) = Pr(win) U(job) + (1-Pr(win)) U(MRR) where U(MRR) is the utility of MRR Pr(win) is the probability of winning the bid We therefore obtain U(MRR) = U(job) and B -C 11 ox 100 IARR Po B-C 2 x 100% C2 This is equivalent to saying that the contractor is indif- ferent between a project with the minimum rate of return and the uncertain final cost project with his minimum bid, since if the minimum bid or the minimum rate of return are de- creased by a, both projects will be unacceptable to him. From U(MRR) = 0.5 U(k2 1 x 100) + 0.5 U(L 2 x 100) 1, 2 it is evident that: U( 2x 100) < U(MRR) < U( L 1 x 100) 021 and because of the assumption of monotonicity of the utili- ty function: B-C2 x 100 < MRR < 1 x1000 2 C 1 The utility of -1 x 100 can be obtained from the utility 1 function assessed, using the first type of questions, for the range MRR to 15%. The utility of MRR being known (equal to zero), one can therefore obtain the utility of 2 x 100 which lies below the MRR. C2 5.2.3. Final form of the questionnaire One of the major constraints that determined the final form of the questionnaire was the time needed by the contractor to go over and answer all parts. A limit of 40 minutes was arbitrarily set to be the maximum amount not to be exceeded. The introduction to the questionnaire consisted of a rela- tively ehort part describing the work that had already been done and the final aim of this study. The purpose was to explain to the contractor the usefulness of the question- naire, and also to motivate him in order to obtain answers -75- that really represented his preferences. The first series of questions are intended to give an idea of the present status of the company, type of work in which it is involved and the goals of the firm. No specific ans- wers were required but rather the contractor was asked to give answers of the type: normal, lower than normal or higher than normal, since it was felt that most of the con- tractors would be reluctant to give exact figures which, moreover, would be of no use for the purpose of this study. The remaining part of the questionnaire consists of three basic questions repeated under different situations. These are; normal size job and good times, large size job and good times, normal size job and bad times, and finally large size job and bad times. The three types of questions were dis- cussed in a previous part of this chapter: the first was used to determine the minimum rate of return; the second to determine the utility of one point within the range DJUuII aa. 15%; and the third to determine the utility of one point be- low this range. In this question the variation in the costs was taken equal to 15%, since it was felt that a relatively large difference should exist in order to obtain a better reaction to the risk of the job and therefore a better as- sessment. However, it turned out that this difference was in fact too large and is unlikely to occur on any - 76 - type of job. A smaller difference in the order of 5% to 7% would have been preferable. The utilities of at least two other points, in .ie range considered, were in fact needed to obtain a reasonably ac- curate utility curve. However due to the time constraint we were limited to only three questions per situation and fur- thermore the exact shape of the utility curves was of no direct interest in this study since we were mainly concerned with obtaining a measure of the variation in risk-aversion. A form of the questionnaire with certain fictitious sizes of jobs and answers appears in appendix D. 5.3. Utility assessment and results 5.3.1. From theory to practice The first problem encountered was how to get to know general contractors interested enough to give us 40 minutes of their time. The Massachusetts Chapter of the Associated General Contractors (AGC) provided us with the name of two general contractors who, in their turn, introduced us to other con- tractors in the Boston area. The utility assessment was conducted for the decision-makers in five firms representing a fairly good cross-section of the industry in terms of sizes. The questionnaire could not be sent by mail to general contractors because the nature of the questions - 77 - involved necessitated the presence of at least one of us to direct the decision-maker and explain what is exactly needed. The most common problem encountered in the utility assess- ment was that many decision-makers considered the questions to be too hypothetical. For example when some of the gene- ral contractors were asked for the minimum acceptable rate of return on a certain project given certain economic con- ditions, they answered that it depenaed on the idendity of the owner or the number of jobs undertaken by the company: we had to make reasonable assumptions in each case. Simi- larly some of the contractors interviewed used subjective probabilities in their answers for certain questions. When they were asked to bid on the job with uncertain final costs they gave amazingly low bids; when we explained once more that the final costs had equal chance of occurence and that this was totally independent of their control, the bids were much higher. In fact when the decision-makers were faced with the uncertain costs most of them were confident that they were capable of bringing the final cost near the lower bound and they gave their bids accordingly. Another common type of pitfall encountered was that when the contractor gave us his minimum acceptable rate of return, - 78 - say 5%, and was then asked if he would accept 4.5% generally the answer was yes. In fact the contractor gave us the rate of return he would like to get, not the minimum accept able to him. So we kept asking the contractors, in all questions for their attitudes towards different values below the first answer they gave, therefore making sure that the final answer really represented the minimum rate of return, bid or fee acceptable to them. Some other problems encountered were that two of the compa- nies interviewed had enough work actually going that they felt they are personnally operating in good conditions. Therefore they said that their answers given for actual times will be the same as those for good times: the four situations were hence reduced to two. Furthermore these contractors considered the question of uncertain cost situa- tion in a specific way: they argued that since they are in relatively good operating conditions, they will bid by adding the minimum rate of return to the higher cost, ne- glecting completely the possibility of occurence of the lower cost. 5.3.2. Treatment of data All the answers to the questions on the utility assessment, whether given in percentage or gross monetary values, were dealt with as a percentage of the cost of the project. - 79 - The minimum rate of return and the 15% markup were respec- tively given utility values of 0 and 1. The markup having the 0.5 utility was obtained through the second type of questions discussed earlier. Finally a point with a nega- tive utility value was obtained. In order to interpret the results, constant risk aversion was assumed throughout the range of assessment. This kind of behavior is represented by utility curves of the form: U(x) = a - b e-CX a,b,c > 0 In our case: x is the markup, percent of total cost U(x) is the utility of a markup x The risk aversion function r(x) is the appropriate measure of the degree of risk aversion. For the exponential func- tion given above: r(x) = - =+1. - 2 -cx = c U, ) _ - cexp(-c-x) where U"(x) is the second derivative of U(x) with respect to x and U'(x) the first derivative with.respect to x. The assumption of constant risk aversion was resorted to for two major reasons: the first is that due to the variation in the ranges of the different utility assessments, the risk premium could not be utilized for comparing degrees of risk aversion and no other practical criteria could be used for this purpose; the second reason is that since the shape of -80- the utility curve is of no concern to us, the exponential form was assumed because it provides an easy measure of the degree of risk aversion. Although the two types of questions utilized for the uti- lity assessment are complementary, the results obtained from these different types of questions were treated separately. The reason is that it was not possible to fit a curve of the exponential form through the four points obtained (with one point having negative utility). Since the major concern in this study is to determine the change in risk aversion given different conditions, the data was treated in the fol- lowing ways: first a utility curve of the exponential form was fitted through the points MRR, 15% markup and the point with a 0.5 utility value; then the results obtained from the type of questions involving uncertain final costs were trea- ted separately,- and a constant risk aversion type of curve was also used. Subsequently these two situations will be referred to respectively as the certain cost situation and the uncertain cost situation. In the uncertain cost situation, utility values of 0 and 1 were arbitrarily assigned to the two uncertain monetary outcomes, C1-and Cz2, and the utility of the MRR was t e02 therefore 0.5. 5.3.3. Results The utility functions obtained for the 5 general contractors are presented in table B1 of appendix B. The utility assess ment in the uncertain cost situation could not be obtained for contractors 1,2 and 3. All three neglected completely the possibility of occurence of the low cost and gave their bids considering the high cost only. They considered the project to be too risky, and this for various reasons; con- tractors 2 and 3 are operating in relatively good conditions while contractor 1 is dealing with negociated types of con- tracts and felt that if he had to bid on such a risky pro- ject he will be on "the safe side". Furthermore contractors 2 and 3 could not consider the bad times situations because their companies were never, in these last 15 years, in bad working conditions. Finally the answers of contractor 5 on the uncertain cost and good times situation were not con- sidered because they were obviously inconsistent. The results concerning the minimum assessed rate of return are presented in table B2. The minimum rate of return appeared to be significantly higher in good times than in bad times regardless of the size of the job. It was also higher for the normal size jobs than for the large size jobs given the same economic conditions. As far as variations in the degree of risk aversion are - 82 - concerned, the certain cost situation and the uncertain cost situation will be considered separately. All the contractors, in the certain cost situation, appeared to be much more risk averse for the normal size jobs in bad times than in normal times: the risk aversion c (table B1) varied from below 10 values in good times to the 20's and 30's in bad times. For the large size jobs, the contractors appeared to be much more risk averse than for the normal size jobs. Consequently the risk aversion was much less affected by good and bad times. One can therefore expect the bids, for normal size jobs, to be much lower in bad times than in good times, since higher risk aversion means that lower outcomes are given relatively more value. However the bids, for large size jobs, will vary much less with changing economic conditions. In the uncertain cost situation, the risk aversion was very high given good times, regardless of the size of the job. This risk aversion diminished drastically for the normal size jobs in bad times but did not almost vary for the large size jobs. One can expect therefore the contractors to take much more risk and bid much lower in normal size jobs in bad times than in good times. - 83 - 5.4. Conclusion We were mainly concerned in this chapter with the assess- ment of the utility functions for some general contractors in the Boston area. The assumptions used and difficulties encountered in both the development stage of the question- naire and the actual assessment were discussed in some detail. Then the results of the assessment were presented, their detailed interpretation and relation to observed behavior being the object of our next chapter. CHAPTER 6 COMPARISON OF RESULTS 6.1. Introduction The object of this chapter is to interpret the results ob- tained from the utility assessment in terms of what is to be observed in practice, we then correlate the expected be- havior and the actual behavior obtained from the analysis of public bids in order to investigate the superiority of the expected utility value criterion over the expected monetary value criterion used in all bidding strategies. Finally we use, in an example, the utility functions assessed for one of the contractors and the expected monetary value approach to determine variations in the optimum markup given diffe- rent situations. 6.2. Interpretation of the results of the utility assessment 6.2.1. Certain Cost Situation This refers to the situation in which the contractor is sure, beforehand, that his estimated cost will be equal to his actual cost after execution of the work. Most of the bid- ding strategies in the literature considered the certain - 85 - cost situation, and some of the researchers asserted that the ratio of actual to estimated cost is represented by a normal probability density of mean 1 and standard deviation smaller than 2%. In fact there are several situations in which this may be true; the most common ones are when the contractor is dealing with jobs that are of a type well known to him; or when he is involved in cost plus fixed fee contracts on regular types of constructions. 6.2.1.1. Normal size of jobs The general shapes of the assessed utility function in good and bad times are represented in figure 6.1. In good times, the interviewed contractors appear to have a higher accep- table minimum rate of return and a smaller degree of risk aversion than in bad times. One can expect therefore lower optimum markups in bad times than in good times, because increased risk aversion means in this case that the contrac- tor will prefer lower markups with an increased chance of winning, to higher markups. The contractor is much more con- cerned with the possibility of losing the job and will con- sequently submit a low.bid. This effect of risk aversion on the level of the bids will be proven below for the con- tractors interviewed. Let w(x) be the ratio of the utility function of a contrac- tor in bad times, U(x), over his utility function in good - 86 - BAD TIM 4 I I / I I %j - iiiilllk___ -I m MRS6 MRS3 MARKUP C% OF TOTAL COST). is LEGEND: MRRb =MINIMUM RATE OF RETURN IN BAD Ti MES MR 3 =MINIMUM RATE OF RETURN IN GOOD TIMES FIGURE 6.1 UTILITY FUNCTIONS GIVEN CERTAIN COST AND NORMAL SILE OF JOB. - 87 - -II 'C 'me >"' a-i I- a 0 000 60- ES - GODTIE times, V(x). We find that, for the contractors interviewed, w(x) is decreasing with increasing values of x. The general shape of the curve obtained is shown in figure 6.2. The optimum markup in good times, xo, is the number that maxi- mizes the expected utility value: E(V(x)) = p(x) V(x) where p(x) is the previously defined probability of winning given a markup x. p(x) is decreasing with increasing values of x because it represents the complementary cumulative dis- tribution function of the repartition of bids (C.C.D.F), a function which is decreasing by definition. Plotting the expected utility function for good times we get the curve shown in figure 6.3. We will now show that, for the same number of competitors and therefore the same probability distribution of winning, the optimum markup in bad times, x0 , is always lower than or equal to x'.0 If we assume that x, - c, with ca> xA, we should have: 0 E(U(c)) > E(U(x')), since c is optimuim in bad times. The expected utility of x in bad times is: E(U(x)) = p(x) U(x) = p(x) V(x) = p(x) V(x) w(x) = E(V(x)) w(x) - 88 - coI LAI LL, 0 2 MARKUP k%). RIGURE 6 I. PLOT OF THE RNTIOW(x, OF. UTILITIES IN BAD AND GOOD "TIMES. - 89- IP :2 w Loi MRR FIGURE 6.. 3 YO MARKUP EXPECTED UTILITY FUNCT10N IN GOOD TIMES. - 90 - Considering now the ratio of expected utilities: E(U(c)) E(V(c)) w(c) E(U(x)) E(V(x')) w(x')o o Since x' is the optimum markup in good times we have:0 E(V(c)) I E(V(x')) Furthermore since we have assumed c > x', we obtain from figure 6.2 w(c) < w(x'), and so the ratio w(c)/w(x) is less than 1. Therefore: E(U(c)) E(U(x'))0 and E(U(c)) P -Io 0 - =;imavw I Furthermore one can expect that, in good times, the markups on the large jobs will be lower than those on the smaller jobs. 6.2.2. Uncertain cost situation The decision-maker is often faced with the problem of esti- mating work with uncertainties as to the final cost. Until now no specific tools were available to the contractor for dealing rigorously and consistently witr such situations. The uncertainty may be related to the type of job or it may be caused by external environmental factors. The contractor may, for example, be bidding for the first time on a certain type of job and there is a possibility that he will not con- sider certain costly aspects of the work. The job may also be of such a nature that one cannot determine in advance the construction method which is best suited and the contractor will have to resort to a trial approach in the field; this is especially the case for underwater or soil foundations types of work. Another characteristic type of uncertainty is when the specifications recommend a specific type of ma- terial, which is costly, and leave the possibility to the contractor of using an equivalent type of material condi- tional on obtaining the agreement of the owner. This usual- ly happens on governmental jobs, and the contractor is un- certain whether to base his estimations on the relatively - 94 - cheap material which may be rejected by the owner, or con- sider only the specified type and risk losing the job be- cause of his high bid. The uncertainty resulting from external environmental factors may be caused for example by weather conditions, possible strikes of the labor force, fluctuations in prices of materials, irregular and late payments by the owner or faulty plans by the architect. An uncertain cost bidding situation can be represented by the following: B-C B-C 2 P.S. where C is the maximum cost of the work, C2 the minimum cost, p the probability of occurence of C1 , B the bid price and p(B) the probability of winning given B. It is assumed for simplicity that there are only two possible costs, but one can easily extend the problem for an infinity of pos- sible costs knowing the probability distribution function of their ocourence. Furthermore, the cost of preparing the bid is not accounted for since most of the time it is considered as being part of the company overhead. - 95 - The above tree representation can be simplified to the following: B-C B -C 2 The probability p(B) of being the low bidder cannot be ob- tained through the use of statistical methods presented in a previous chapter. The contractor should assess and use his judgmental probabilities of chances of winning. As Raiffa (1968) said: "As far as action is concerned ... you should feel free to use these judgmental probabilities just as if they were the real thing ... " 6.2.2.1. Normal size of jobs Given the same uncertain situation the contractor appears much more risk averse in good times than in bad times. We shall see that this behavior implies lower level of markups in bad times than in good times, similarly to the certain cost situation. Increasing risk aversion, in the uncertain cost case, means that the smaller outcomes that are below the contractor's minimumrate of return are given much more importance or weight in arriving to the final decision. The - 96 - contractor will therefore increase his bid in order to ac- count for the possible occurence of these outcomes with negative utility. 6.2.2.2. Large size of job The assessed utility functions for contractor 4 showed that the degree of risk aversion is very little affected by bad times. In fact c decreases by 9% in bad times for the large size of jobs as compared to a 90% decrease for normal sizes. One can therefore expect the variation in optimum markups with changing economic conditions to be very small for the large projects. 6.2.2.3. Illustration An example will help to clarify the above points. Let us consider a project with two equally likely uncertain final costs, C and C2. It can be represented by: C< $1,000,000 .60 C2 $900,000 Assuming a certain p(B) for different levels of the bid B and using the utility functions of a normal job assessed for contractor 4 (table B1), we can determine the optimum bids - 97 - in the following situations: 1) Using expected monetary value criterion. 2) Using expected utility for good times. 3) Using expected utility for bad times. The results of the computation are shown in table C1 of ap- pendix C, where: V(x) = 1.0 - 522 e-.69 x good times U(x) = 1.5 - 1.67 e-.0 7 x bad times An optimum bid price of $1,100,000 was obtained given good times, as compared to a $1,050,000 given bad times. Fur- thermore the bid price obtained using the expected monetary value criterion was also $1,050,000 but this is due to the fact that the increment between two successive bids was too large for a good sensitivity in the results. The optimum bid in good times was higher than the one in bad times, which is the expected result. Moreover the optimum bid sug- gested by the expected monetary value criterion is unac- ceptable, in good times, by the decision-maker since a nega- tive utility value is associated with such a bid. Let us consider the same example and assume that it is a relatively large job as far as contractor 4 is concerned. Using his utility functions assessed for a large project we can determine his optimum bid; the calculations are shown in table C2. The optimum bid obtained in good times was - 98 - $1,100,000 compared to $1,075,000 in bad times. The bid in bad times is, as expected, lower than the one in good times. The level of the bid appears also to be less affected by changing economic conditions, the larger the size of the job. Furthermore the level of the bid is proportionally higher, in good times, the larger the relative size of the project. Table 6.2 summarizes the variations in the degree of risk aversion in the uncertain cost case, given varying sizes of projects and economic conditions. Table 6.2 Variations in the degree of risk aversion, uncertain cost case. Times Good Bad Size of job Normal Large Sall(c 0.69) (c 0.07) Large(Large Large(c 0.69) (c 0.63) We can therefore conclude that, for the contractors inter- viewed, we observe the same behavior in both certain and un- certain cost cases. The level of the bids in good times is - 99 - higher than in bad times, this variation in the level being much smaller the larger the relative size of the job. Fi- nally the markup, in good times, on large jobs is higher than the markup on small jobs. 6.3. Comparison of results 6.3.1. Predicted behavior Using the collected public bidding data to generate probabi- lities, and the assessed utility functions for normal sizes of jobs, we will determine the optimum markups of contrac- tor 1, in the certain cost situation, as a function of the number of bidders and chnnging economic conditions. We will then verify whether the actual behavior obtained from the analysis of the data is similar to the predicted behavior. Discrete probabilities of winning for various levels of mark- ups and different numbers of competitors were calculated on a computer utilizing the data of the DPW; these data were prefered to those of the BBC because of their larger size. The C.C.D.F obtained appear in figure 6.5, 6.6, 6.7, 6.8 and tables 03, C4, C5, C6. The sizes of projects considered ranged between $100,000 and $1,000,000, the upper limit being imposed to eliminate all construction types of pro- jects and restrict the analysis to maintenance projects. The bidding data of all years (1966 to 1975) were consi- -100- .... 0 .... 1 O.f I&.: ci u u -J o s "10 MAR\ .. s FIGURE. 6. 5 COMPLENl£NTM'( CUMULP..TIVE D1STR\8UTl0N FUNCTION FOR 3 C.OMPETITORS. .. I OS -5 5 MARKUP 00 10 FIGURE G_ 6 COMPLEM£NTAR't CUMULATNE DISTRIBUTION F"UNCTION FOR 4 C ON\PET\TORS. o., - 0 vi 0 -to MARl<\JP ("J flGUR[ b .. 7 COMPLEN\ENTAR'f CUMULATIV£ OISTR\BUT\ON f"UNCTION FOR 5 COM PE.Tl TORS. OS -s 0 10 MARKUP C%l ti GURE 6. 8 COMPLEMENTARY CUM\JLATlVE DISTRIBUTION FUNCTION F'OR 6 COMPETITORS. dered in order to have a sufficient number of bids for each specified number of competitors. Furthermore the markup was taken to be the difference between the bid price and the cost estimate (EC) divided by the cost estimate: B - EC EC This cost estimate was defined to be equal to the engineer's estimate (EE) minus 5%, the 5% being the arbitrarily assumed allowance for overhead and profit present in the agency's estimate. The optimum markups were then respectively com- puted using the expected monetary value criterion and the utility functions assessed for contractor 1 given a normal size of job and varying economic conditions. The general shape of the curves showing the optimum markups as a func- tion of the number of competitors appear in figure 6.9, with the exact values given in table C7. It appears that, con- sistently with what is predicted by the bidding strategies, the level of the markups is decreasing the higher the number of competitors. Furthermore the optimum markups in bad times and using expected monetary values are coinciding, but this is due to the fact that discrete probability functions are used; these are not sensitive enough to show the diffe- rences that exist in this specific case. In fact the opti- mum markups were calculated for another form of the C.C.D.F, - 105 - - °' I --. • s.._-----------------.�-----.....-- 4- s 6 NUMBER OF BIDDERS FIGUR[ G_ 9 OPTIMUM MARKUPS F'OR C.ONTRACTOR 1, CERTAIN COST SITU""T\ON, USING E�PEC.1'ED UT\LlTI ES AND MONETAR"< VA.LUES. _COST: 0.95 EE - form corresponding to a different assumption of the percen- tage of overhead and profit included in the engineer's esti- mate on the public bids. Figure 6.10 shows the curve ob- tained for a 7% allowance, the values appear in table C8. One can see that the optimum markup using expected monetary values are lower than those obtained using expected utili- ties in bad times. Most importantly, optimum markups, in good times, are in both cases higher than the optimum markups in bad times. 6.3.2. Actual behavior The bidding strategies using the expected monetary value criterion suggest the same markup given the same probability distribution of winning and regardless of the economic con- dition of the firm. On the other hand the expected utility criterion suggests that, for the contractors interviewed, the optimum markups in good times are higher than those in bad times. If the expected monetary value criterion reflec- ted the true behavior of the contracting companies, one should observe more or less the same level of bids for the same number of competitors regardless of the year. But if the expected utility criterion is the correct one to use, then the markups in good years will be higher than those in bad years for the same number of competitors. The analysis of public bids showed in figure 4.6 that this latter beha- - 107 - 15' a & %0 IL n GOOD TIMES m m 0O co BAD TMMES xN. NUMBER OF BIDDERS FIGURE 6.10 OPTIMUM MARKUPS FOR C.ONTRACTOR I CERTAIN COST SITUATIONUS, IGLXPECTED UTILITIES AND MONETAAY VALUES. . COST= 0.93 EE . vior is actually observed; this showed that general contrac- tors behave in a way as to maximize their expected utilities, and furthermore that the general shape of the utility furic- tions found for the contractors interviewed, are represen- tative of the preferences in the industry as a whole. 6.4. Conclusion This chapter explains in a first step the results of the uti- lity assessment, in both certain and uncertain cost cases. We find that the contractors will bid consistently lower in bad times than in good times. The second step correlates the actual behavior obtained through the analysis of public bids in chapter 4, with the predicted behavior obtained by using the utility functions of one of the contractors inter- viewed. The expected utility value criterion appears to be a more realistic criterion to use than the expected monetary value. - 109 - CHAPTER 7 CONCLUSION As we have shown in the preceeding chapters, some factors which are important in bidding behavior are not considered in the existing bidding strategies. In chapter 2 we have seen that the following diagram is used in all cases. NUMBER OF BIDDERS C.C.D.F AB I D PRICE where C.C.D.F is a complementary distribution function G(x) which gives the probability of winning the bid, for any level of markup. This function is generally generated from past data. By considering the data of actual bidding we showed that the number of bidders depends on the number of projects avai- lable to the bidders (see chapter 4). On the other hand we showed that a contractor determines his markup according to the utility of tvis markup to him (see chapter 3). The utility assessments that we made on contractors of the - 110 - Boston area show that at least two factors influence the utility function of a contractor; these are the size of the project and the economic condition of his firm at the time he bids, this condition generally being related to the eco- nomic condition of the construction industry as a whole (see chapter 5). Therefore we suggest that a diagram which is more closely representative of real world bidding is as follows: ECONOMIC CONDITIONS SIZE OF kNUMBER OF BIDDERS!UTILITY C.C.D.F i BID PRICE This diagram may not be complete but it gives a better idea of what should be taken into account by bidding strategies in order for them to be more useful. In that respect we think there are at least three possible extensions of this study which could lead to the development of better bidding - 111 - PROJECT] strategies. It would be interesting to assess the utility of one con- tractor very precisely and use his past bids to compare the predictions given by the expected monetary value model and those given by the expected utility value model. Another line of study would be to verify which of the sta- tistical methods of the literature can best be used for predictive purposes. It might even be interesting to use multivariate statistical analysis to determine how different variables which have an influence on the bid are correlated. In our study we have been concerned by utility functions with one attribute only. During our interviews we saw that other objectives than monetary outcomes are taken into con- sideration. Some of them are: expand as much as possible, expand to keep up with inflation, prestige of the work. In the case of a contractor who wants to expand as much as pos- sible, he will take a much lower profit to have more jobs. In that case a second attribute would be the volume of work on hand. In the case of the prestige attribute, a good example is the case of a contractor who accepted a $250,000 worth job in the White House for a nominal markup of $1. The use of a one attribute utility function will have short- comings in this case. Consequently we think that a multi- attribute utility function could be defined and that its - 112 - combination with multivariate statistical analysis may lead to the development of good and hopefully more useful bidding strategies. - 113 - REFERE',CES 1. Baumgarten, R.M. (1970) "Discussion" Journal of the Construction Division ASCE, 96, NoC01 June, 2. Benjamin, Neal B.H. (1970) "Discussion" Journal of the Construction Division ASCE, 96, NoC01 June, 3. Benjamin, Neal B.H. (1972) "Competitive Bidding: the probability of winning" Journal of the Construction Division ASCE, 98, NoC02 Sept, pp. 313-330 4. Broemser, G.M. (1968) "Competitive Bidding in the Construction Industry" PhD dissertation, Department of Civil Engineering, Stanford University (unpublished) 5. Casey, B.J. and L.R. Shaffer (1964) "An evaluation of some competitive bid strategy models for contractors" Construction Research Series Report 4, Department of Civil Engineering, University of Illinois 6. de Neufville, R. and R.L. Keeney (1972) "Multiattri- bute Preference Analysis : the Iexico City Airport" Transportation Research, ,April, pp. 63-75 7. Dixie, J.M. (1974) "Bidding IUodels. The final reso- lution of a controversy" Journal of the Construction Division ASCE, 100, NoC03 Sept, pp. 265-271 8. Friedman, L. (1956) "A Competitive Bidding Strategy" Operations Research, 4, pp. 104-112 9. Gates, M. (1961) "Statistical and economic analysis of a bidding trend" Transactions ASCE, 126, pp..601-623 10. Gates, TM. (1967) "Bidding strategies and probabili- ties" Journal of the Construction Division ASCE, 93, NoC01 June, pp. 75-107 11. Gates, M. (1970) "Closure of biacting strategies and probabilities" Journal of the Construction tivision ASCE, 96, NoC01 June, pp. 77--78 - 114- 12. Gates, TA. (1971) "bidding contingencies and proba- bilities" Journal of the Construction Division ASCE, 97, NoCO2 Nov., pp. 277-303 13. Grayson, C.J. (1960) Decisions under Uncertainty, Drillin Decisions by Oil andGasfOperators. Harvard Business School, Division of Research, Boston. 14. Lorange, P. and V.D. Norman (1970) Risk Preference Patterns among Scandinavian Tankship Owners, Institute for Shipping Research.Bergen 15. Morin, T.L. and R.H. Clough (1969) "OPBID: Competitive bidding strategy model" Journal of the Construction Division ASCE, 95, NoC01 July, pp. 85-106 16. Naykki, P. "Discussion" Journal of the Construction Division ASCE, 99, NoC012July,pp. 224-225 17. Park, W.R. (1962) "How low to bid to get both job and profit" Engineering News-Record, April 19, pp. 38-39 18. Park, W.R. (1963) "Let's have less bidding for bigger profits" Engineering ews-Records, Feb 14,, p. 41 19. Park, W.R. (1966) "The strategy of contracting for profit" Prentice-Hall, N.J. 20. Park, W.R. (1968) "Bidders and job size determine your optimum markup" Engineering News-Record, June 20, pp.122-123 21. Pratt,John W. (1964) "risk Aversion in the Small and in the Large" Econometrica pp. 122-136 22. Raiffa, Howard (1968) Decision Analysis - Introductory Lectures on Choices under Uncertainty, Addison Wesley Publishing Company, Reading, Mass. 23. Rosenshine, M. (1972) "Bidding Models Resolution of a Controversy" Journal of the Construction Division ASCE, 98, NoC01l Mar., pp. 143-148 24. Shaffer, L.R. and T.W. Micheau (1971) "Bidding with competitive strategy models" Journal of the Construc- Sion Division ASCE, 97, NoC01 Mar. pp. 113-12o - 115 - 25. Spetzler, Carl S. (1968) "The Development of a Corpo- rate Risk Policy for Capital Investment Decisions" ITEE Transactions on System Science and Cybernetics Vol SSC-4 No3 September pp. 279-299 26. Stark, R.TA. (1968) "Discussion" Journal of the Cons- truction Division ASCE, 91 , No C01 Jan. pp.110-112 27. Swalm, R.O. (1966) "Utility Theory Insights into Risk-Taking" Harvard Business Review Vol )44 No 6 Nov. Dec. pp. 123-136 28. Willenbrock, J.H. (1973) "Utility function determi- nation" Journal of the Construction Division ASCE, 99, No C01, July, pp. 155-153 - 116 - APPENDIX A TABLES OF DATA ANALYSIS - 117 - Table Al: BBC data, main features Calendar Total Corrected Corre. Number of construction value projects year award ($1974) factor collected 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 28,437,260 21,226,812 32,106,696 42,225,777 44,361,143 32,069,899 34,196,479 41,685,853 59,928,186 86,254,671 247,745,497 157,327,847 59,015,706 17,975,973 60,571,4oo 44,576,300 64,534,500 86,883,800 89,165,900 61,157,300 61,895,600 68,781,700 92,888,700 124,206,726 312,159,326 184,073,581 63,146,800 17,975,973 2.13 2.10 2.01 2.01 2.01 1 .91 1.81 1.65 1.55 1 .41 1.26 1.17 1.07 1.00 - 118 - 4 4 6 10 11 15 12 13 21 16 16 21 8 10 Table A2 : DPW data, main features Total construction award Corrected value ($1974) Number of projects t 1* 4 4 83,101,064 Unavailable 102,931,319 95,2L9,553 125,852,673 61,035,346 106,453,506 132,016,067 Unavailable Unavailable 163,709,096 178,0719,182 153,351,780 188,779,009 81,047,717 125,615,137 141,257,719 a 1 .1 __________ * In 1966 the number of projects is taken from January 1966 to June 1966, in 1975 from July 1974 to December 1974. - 119 - Const. llaint. Fiscal year 1966* 1967 1968 1969 1970 1971 1972 1973 1974 1975* 1 15 19 15 16 10 6 11 8 2 12 41 53 56 48 71 78 103 79 36 Table A3 : Number of projects versus average number of bidders per project - BBC - Calendar Number Average number year of projects of bidders 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 4 4 6 10 11 15 12 13 21 16 16 21 8 10 10.75 10.00 7.50 7.50 7.18 6.40 6.25 4.23 4.71 5.75 5.69 8.85 10.50 9.60 _____ I. ________ I ________ - 120 - Table A4 : Number of projects versus average number of bidders per project - DPW - N.P : Number of projects A.N.B : Average number of bidders per project - 121- Fiscal Construction Maintenance year N.P A.N.B N.P A.N.B 1966 12 6.60 12 4.58 1967 15 5.00 hi 4.64 1968. 19 4.45 53 3.71 1969 15 5.36 56 3.96 1970 16 5.80 48 3.84 1971 10 6.10 71 4.84 1972 6 8.40 78 4.71 1973 11 5.67 103 4.60 1974 8 5.12 79 4.79 1975 2 7.50 36 5.94 Table A4: Number of projects versus average number of bidders per project - DPW - (continued) N.P : Number of projects A.N.B : Average number of bidders per project - 122 - Fiscal Reconstruction Highway work Resurfacing year N.P A.N.B N.P A.N.B Na.P A.N.B 1966 3 5.30 4 5.00 5 3.80 1967 13 4.30 20 5.21 8 3.75 1968 10 3.10 22 4.08 21 3.50 1969 30 3.93 6 5.50 20 3.50 1970 29 4.04 9 3.40 10 3.60 1971 29 5.62 22 4.90 20 3.75 1972 27 5.63 15 5.87 36 3.61 1973 62 5.02 0 - 41 4.02 1974 30 5.70 1 9.00 48 4.25 1975 18 7.10 0 - 18 5.29 Table A5 : Deviation of bids from estimated cost for BBC (% of estimated cost) Good years : 66, 67, 68, 69, 70, 71 Bad years : 61, 62, 63, 64, 65, 72, 73, 74 - 123 - Number of bidders Good years Bad years 1-2-3 8.76 5.03 4-5-6 4.71 1.32 7-8-9 -2.22 -4.31 10-11-12 -3.03 -5.32 Table A6 Deviation of bids from estimate for DPW (o of estimate cost) D.E : Deviation from estimate N.P : Number of projects (B - E)/E x 100 -124- Fiscal year 1967 1968 1969 1970 Number of bidders D.E N.P D.E 1 N.P D.E N.P D.E N.P 1 - 0 - 0 51.34 3 17.66 1 2 6.81 5 13.37 10 9.82 11 6.88 8 3 2.83 9 3.00 17 9.38 11 7.47 13 4 -2.34 8 -0.62 9 4.65 16 5.04 14 5 -3.25 7 0.43 13 6.34 4 -3.34 5 6 -5.33 5 1.01 3 -4.02 4 -4.23 4 7 -4.56 4 4.17 1 3.29 3 -3.23 1 8 -5.74 1 - 0 -5.36 2 - 0 9 -4.53 1 - 0 -5055 1 1.48 2 10 -2.40 1 - 0-4.26 1 - 0 Table A7 : Deviation of bids from estimate for DPW (o of estimate cost) D.E : Deviation from estimate N.P : Number of projects (B - E)/E x 100 - 125 - Fiscal year 1971 1972 1973 1974 Number of bidders D.E N.P D.E N.P D.nE N.P D.E N.P 1 9.55 2 -0.01 3 12.42 5 -1.60 1 2 11.13 3 -2.07 9 3.00 12 5.87 10 3 8.55 15 -4.12 13 -1.22 13 7.15 17 4 2.40 16 -0.72 16 -6.40 24 5.91 16 5 -0.70 10 -4.00 11 -6.00 15 4.24 13 6 -7.26 7 -11.9 11 -11.9 12 5.60 10 7 -6.20 9 -1.56 5 -6.02 7 11.10 2 8 0.65 3 -4.25 4 -5.61 8 -6.28 3 9 -4.73 5 -3.29 4 0.92 5 4.13 5 10 -3.13 1 -i0.4 1 -7.27 1 0.11 1 Table A8 : Coefficients a and b found by linear regression - 126 - Fiscal year a b r2 (fit) 1966--- 1967 -2.77 11.50 0.85 1968 -5.79 24.31 0.73 1969 -3.18 17.49 0.81 1970 -3.93 ~ 18.06 0.84 1971 -4.30 20.86 0.91 1972 -3.84 10.82 0.51 1973 -3.69 10.25 0.94 1974 -1.25 10.74 0.28 1975 - - APPENDIX B TABLES OF UTILITY ASSESSMENT - 127 - Table B1 : Assessed utility functions A- Certain cost situation a- Normal size job Contractor Good times Bad times 1 2.53 - 4.42 eO.0 8 x 1.10 - 3.70 e-O.2 4x 2 x* 3 2.73 - 3.06 e-.04% 5 2.70 - 5.40 e- 0 8 x 1.08 - 8.52 e0.31x b- Large size job Contractor Good times Bad times 1 1.01 - 5.03 e-AOX 1.01 - 2.80-34X 2 1.25 - 2.25 e-0.15x 3 1.25 - 1.81 eO1 3X - 5 1.16 - 11.0 e-0. 2 8 x 1.4 - 5.36 e-O.33x * : Expected monetary valuer, r(x) = 0 - 128 - Table B1 : Assessed utility functions (continued) B- Uncertain cost situation a- Normal size job Contractor Good times Bad times 4 1.00 - 512. jO.69x 1.45 - 1.67 e-G'O 7 5 -1-43 - 1.53 e-0.08x b- Large size job Contractor Good times Bad times 4 1.00 - 256. e-0 6 9x 1.00 - 41.2 e-o.63x 5 -101 - 1.82 e-O.25x - 129 - Table B2 : Minimum rate of return for the various contractors interviewed (/ return) Normal size job Large size job Good Bad Good Bad Contractor times times times times 1 8. 5. 4. . 2 5. - - 3 3. -2.8 - 4 ~10. 8. 9. 7. 5 9. 6.7 8. 5. - 130 - I APPENDIX C TABLES OF COMPARISON OF RESULTS - 131 - Table C1 : Optimum markup for contractor 1 on normal size of jobs, uncertain cost situation. B P(B) B - C1 B - C2 x11x 2 ($1000) ($1000) ($1000) (%) (%) 950 0.95 -50 50 -5.0 5.5 1,000 0.65 0 100 0 11.1 1,025 0.55 25 125 2.5 13.88 1,050 0.45 50 150 5.0 16..66 1,075 0.35 75 175 7.5 19.4 1,100 0.20 100 200 10.0 22.2 1,125 0.10 125 225 12.5 25.0 B EMV E(V(x)) E(U(x)) ($1000) 950 0 -7937 -0.57 1,000 32.5 -169 0.156 1,025 41.25 -24.8 0.24 1,050 45 -3.2 0.27 1,075 43.75 -0.15 0.26 1,100 30 0.149 0.17 1,125 17.5 0.095 0.095 - 132 - Table C2 : Optimum markup for contractor 1 on large size of jobs, uncertain cost situation. - 133 - B EMV E(V(x)) E(U(x)) ($1000) 950 0 -3893 -61.26 1,000 32.5 - 82.6 -12.74 1,025 41.25 -11.9 -1.79 1,050 45 -1.35 0.053 1,075 43.75 0.103 0.285 1,100 30 0.175 0.192 1,125 17.5 0.097 0.099 riviarkup CoC*D.F lvlarkul3 CoCe D-mF Markup C.C.D.F l4e86 OoO15 lOo23 09359 3o22 Oo687 l4e67 OeO31 10*05 0*375 3e14 09703 l4e57 oo47 9080 OV,391 3s13 09719 l4o37 o.o62 9.69 o,,406 3*11 Oo734 14.17 OoO78 8s8l o.422 2.25 Oo 750 13,o84 o.o94 8000 o.437 2,001 0*765 l3o'74 o.log 7*89 o.453 2eOO Oo781 13*57 Oe125 7*77 o*469 lm75 Oo797 13,9 o.141 7o54 o.484 lo64 09812 13,o4i Oe156 7*53 09500 1,510 Oo828 l3e35 Oel'72 6-.94 Oo516 0982 008 l3eO9 Oo167 6,91s Oo531 o.67 0*859 13oO8 Oo2O3 6oll 0*547 -Oo22 Oo8 75 12-m57 09219 5,*62 Os562 0.64 00891 l2o48 0.234 5e56 09578 0690 0*906 12o42 Oo250 5o))lTV Oe594 o,.94 Oo922 l2o12 0*266 5o16 0*609 1*03 0*937 llo45 0*281 4e4q Oe625 -le 34 Oe953 11019 Oo297 4o34 Oo641 -w-1 .37 Oo969 11*15 Oe oz 12 .-A 3993 Oo656 2,973 Oo984 l0e50 Oe328 3.64 Oe672 wm2995 10000 l0w36 Ov3 Table C3 Cumulative distribution function (CoCeD.F) for 3 competitors, aw markup in percentage 134 Table C4 : Cumulative distribution function (C.C.D.F) for 4 competitors. - markup in percentage - Markup C.C.D.F Markup C.C.D.F Markup C.C.D.F 14.74 0.012 9.02 0.345 1.29 0.678 14.72 0.024 9.00 0.357 1.24 0.690 14.57 0.035 7.84 0.369 1.23 0.702 14.46 0.047 7.73 0.381 1.22 0.714 14.39 0.059 7.61 0.393 1.08 0.726 13.91 0.071 7.11 0.405 1.02 0.738 13.41 0.083 6.95 0.416 0.87 0.750 13.4o 0.095 6.79 0.428 0.84 0.762 12.94 0.107 6.72 0.440 0.77 0.774 12.90 0.119 6.52 0.452 0.76 0.786 12.85 0.131 6.14 o.464 -o.14 0.797 12.71 0.143 5.74 o.476 -o.43 0.809 12.56 0.155 5.56 o.488 -o.48 0.821 12.22 0.166 5.04 0.500 -0.71 0.833 12.00 0.178 4.83 0.512 -0.98 0.845 11.74 0.190 4.71 0.523 -1.01 0.857 11.32 0.202 4.66 0.536 -1.30 0.869 11.07 0.214 4.57 0.548 -1.56 0.881 10.68 0.226 4.08 0.559 -2.03 0.893 10.16 0.238 3.92 0.571 -2.04 0.904 9.99 0.250 3.68 0.583 -2.46 0.916 9.97 0.262 3.36 0.595 -2.53 0.928 9.93 0.274 3.29 0.607 -2.77 0.940 9.82 0.286 3.01 0.619 -3.65 0.952 9.59 0.298 2.91 0.631 -3.66 0.964 9.36 0.309 2.14 0.643 -4.58 0.976 9.24 0.321 2.06 0.655 -4.71 0.988 9.11 0.333 1.45 0.667 -4.75 1.000 - 135 - MarIcap C.CoDoF Markup C.CoDeF liarkup C.C.DoF 14,m74 OeO17 4943 Oo351 1*15 0 o 6134 l4e7O OwO35 4*36 09368 le06 09702 14e47 OeO53 4o23 Ow386 00 Oo719 13,o52 OwO70 4ol7 o,,403 o,.42 Oo737 13,946 OoO88 3w95 O's )a 1 0,032 Ow754 l3o16 OolO5 3.85 o,,438 Oo24 09772 12o78 Oo123 3*71 Oe456 Om13 Oo789 12*06 o.14o 3,m47 0,*474 OeV Oe8O7 llo89 Oe158 3o43 o.491 o.41 0,*824 ilo16 Oe175 3,m34 0*509 --0*56 o.842 9091 Oo193 2o69 0*526 MM1930 Oe859 9o55 Oo210 2w39 Oo5 -1-32 Oe877 8oB2 Oo228 2o36 0,9561 -2*21 09895 8,,6o 0*245 2o26 0*579 2eO Oe912 6o29 Oo263 2oo25 Oo596 w-2o72 0*930 5998 0*281 2.20 0.614 -2*80 o.947 5o89 Oo298 lv95 09631 3.64 o.965 5oO7 0,*316 lo37 0*649 4,938 Oe982 L 4o78 0*333 lo20 Oe667 -4o49 10000 Table C5 Cumulative distribution function (C.C.D.F) for 5 competitors. 4=M markup in percentage 136 Markup C*C.D.F Markup C.C.DoF Markup C.C. D.F 14.20 OeO23 4o16 Os372 0.93 o,.698 l2o39 oo46 3953 0*395 --lo62 Oo721 11,943 OoO70 3o17 o.419 2s2O 0*7 10,974 O-wO93 lo67 00 --2o36 0*767 10*07 0,9116 Oe88 Oo465 2o39 0*791 10905 00139" 0.61 o,,488 2959 09814 9o76 Oo163 Ov32 00512 3*36 Om837 8o73 0.186 Oe3l Oe535 -4.2o Os860 8*35 Oo2O9 0001 Oe558 .4.24 oe8s4 7939 Oo232 0,*30 09581 ,4o 46 Oo9O7 6993 Oo256 0931 0.6o4 4949 Oo930 6o23 Oo279 Oe53 Oo628 49 62 Oe953 5oO6 0*302 -Os68 o.651 4.63 Oo977 4o64 Oo325 -o,.84 0.674 4.69 10000 4o28 09349 1 1 Table C6 Cumulative distribution function (C.C.D.F) for 6 competitors. markup in percentage 137 Table C7 : Optimum markups for contractor 1, certain cost situation, using expected utility and monetary values. Cost = 0.95 EE Optimum markups (o) using expected Number of Monetary Utility value competitors value Bad times Good times 3 9.69 9.69 12.12 4 9.00 9.00 11.74 S 5 8.60 8.60 11.89 6 6.93 6.93 10.06 - 138 - Table C8 : Optimum markups for contractor 1, certain cost situation, using expected utility and monetary values. Cost = 0.93 EE Optimum markups (%) using expected Number of Monetary Utility value competitors value Bad times Good times 3 7.42 9.84 12.05 4 6.81 8.81 11.35 5 5.56 8.17 10.94 6 5.39 8.01 10.62 - 139 - APPENDIX D QUESTIONNAIRE - 1ho - RISK PREFERENCE OF CONTRACTORS The object of our study is to understand the behavior of the contractor when he bids on projects with different sizes and also when he bids in different overall economic condi- tions. The study is to be conducted on two levels. We collected data on bids from the Bureau of Building Cons- truction (we took all bids on new construction for projects of $100,000 or more covering the years 1961 to 1974). We went also to the Public Works Department and collected all highway work, construction and resurfacing bids from 1966 to 1974. These data were studied and the following results were apparent: 1- When the number of projects available per year is low, the number of bidders on a project is high. 2- When the number of bidders on a project is high, the lowest bid is below the estimate, whilst when the number of bidders on a project is low the lowest bid is well above the estimate. Now the data is being analyzed in a computer for all pos- sible trends. We are now conducting interviews with general contractors in order to determine their attitudes in risky situations and how these change with different economic conditions. A similar study on highway contractors was conducted in 1972 and results were published in the "Journal of the Construction Division", ASCE, - July 1973 - The utility of such studies is that they are attempts to understand how a decision-maker arrives at decisions and what his attitude is towards risk in order to help him make more consistent decisions in his work and possibly also to enable him to delegate responsability to subordinates while being sure that they will behave similarly to him. II - 142 - QUESTIONNAIRE We shall give you a series of questions regarding your deci- sions in several hypothetical investment choice situations. As far as possible you should answer each question as if it were a real world situation and feel free to ask questions anything seems unclear. 1- What is the usual size of projects you bid on? (Give a range) Answer: $ 200,000 To AP2.000,000 2- How would you describe your company's present economic conditions: * Liquidity position (this concerns the availability of funds for you and the cost of such funds.) Better than normal CDNormal [6 Lower than normal El * Availability of work: Do you think that the average number of contractors bidding on a project is: Higher than normal R Normal M Lower than normal C Do you think that the number of projects available is: Higher than normal M Normal ElLower than normal Ef - 143- 3- What is the type of work in which you are involved? Answer: BUILDIt CONSTRUCTION It is available today? Yes oENo Have you decided or started to switch to another type of work? Answer: NO L- What are the goals of your firm: - To expand as much as possible E] - To keep a fixed share of the market - To expand only to keep up with inflation 126 - To reduce the amount of work - Others 5- What kind of profit goals are you following for the company: - Long term - Intermediate term - Short tern Has it changed lately? Yes 6- Assuming actual times, suppose that you are offered a $400,000 job on a cost plus fixed fee basis, what would be the minimum profit and home office overhead for which you would consider accepting the contract. - Answer: $ 20,000 - 145 - 7- Now, you have to decide between two projects: (you can only take one) a- A job under a unit price arrangement where the final monetary return is uncertain. b- A job on a cost plus fixed fee basis where a certain profit level is guaranteed. Assume first that you have a $400,000 job with two uncertain outcomes: $60,000 markup or $20,000 with a 50-50 chance of occurence (this is similar to the flip of a coin: heads you get $60,000, tails you get $20,000 ) and a similar cost plus fixed fee contract for a $400,000 job (assume the same resources: capital, equipment, manpower are required for either jobs) where you would be guaranteed a fixed profit. The unit price arrangement is represented for conve- nience by $60,000 $ 20,000 What would be the minimum profit and home office over- head for which you would prefer the cost plus fixed fee contract? - Answer: $&1,000 - 146 - 8- Assuming actual times, suppose that you are offered a $1,000,000 job on a cost plus fixed fee basis, what would be the minimum profit and home office overhead for which you would consider accepting the contract. - Answer: $ 35,000 - 147 - 9- Now, you have to decide between two projects: (you can only take one) a- A job under a unit price arrangement where the final monetary return is uncertain. b- A job on a cost plus fixed fee basis where a cer- tain profit level is guaranteed. Assume first that you have a $1,000,000 job with two uncertain outcomes: $150,000 markup or $ 35,000 with a 50-50 chance of occurence (this is similar to the flip of a coin: heads you get $150,000, tails you get $ 35,000 ) and a similar cost plus fixed fee contract for a $1,000,000 job (assume the same resources: capital, equipment, manpower are required for either jobs) where you would be guaranteed a fixed profit. The unit price arrangement is represented for conve- nience by $150,000 $ 35,000 What would be the minimum profit and home office overhead for which you would prefer the cost plus fixed fee contract? - Answer: $ 50,000 - 1ha- 10- Assuming actual times: (you have eight to ten compe- titors on a project) You have t.o bid on a project with uncertain final cost: the final cost cannot exceed $430,000 and cannot go below $370,000. You have an equal chance of spending $430,000 or $370,000 on it. We will represent it by $430,000 0 to $370,000 What would you bid on this project. Include profit and home office overhead. Keep in mind that if you bid too high you will lose the contract. - Bid: $ ++0,000 - 149 - 11- Similarly you have to bid on a project with two un- certain final costs: $1,075,000 and $925,000 (maximum and minimum costs). There is an equal chance of spending $1,075,000 or $925,000. $1,075,000 $925,000 What would you bid on this project. - Bid: $ 1;103,000 - 150 - 12- Assuming normal times, suppose that you are offered a $400,000 job on a cost plus fixed fee basis, what would be the minimum profit and home office overhead for which you would consider accepting the contract. - Answer: $ 30,000 - 151 - 13- Now, you have to decide between two projects: (you can only take one) a- A job under a unit price arrangement where the final monetary return is uncertain. b- A job on a cost plus fixed fee basis where a cer- tain level of profit is guaranteed. Assume first that you have a $400,000 job with two uncertain outcomes: $60,000 markup or $30,000 with a 50-50 chance of occurence (this is similar to the flip of a coin: heads you get $60,000, tails you get 4 30,000) and a similar cost plus fixed fee contract for a $400,000 job (assume the same resources: capital, equipment, manpower are required for either jobs) where you would be guaranteed a fixed profit. The unit price arrangement is represented for conve- nience by $60,000 $30,oOO What would be the minimum profit and home office overhead for which you would prefer the cost plus fixed fee contract? - Answer: 4 40,000 - 152 - 14- Assuming normal times, suppose that you are ofiered a $1,000,000 job on a cost plus fixed fee basis, what would be the minimum profit and home office overhead for which you would consider accepting the contract. - Answer: $ 45,000 - 153 - 15- Now, you have to decide between two projects: (you can only take one) a- A job under a unit price arrangement where the final monetary return is uncertain. b- A job on a cost plus fixed fee basis where a cer- tain profit level is guaranteed. Assume first that you have a $1,000,000 job with two uncertain outcomes: $150,000 markup or $ 45,000 with a 50-50 chance of occurence (this is similar to the flip of a coin: heads you get $150,000, tails you get $ 4,000 ) and a similar cost plus fixed fee contract for a $1,000,000 job (assume the same resources: ca- pital, equipment, manpower are required for either jobs) where you would be guaranteed a fixed profit. The unit price arrangement is represented for conve- nience by $150,000 0 $ 45,000 What would be the minimum profit and home office overhead for which you would prefer the cost plus fixed fee contract? - Answer: $ 59,000 - 154 - 16- Assuming normal times: (you have three to four compe- titors on a project) You have to bid on a project with uncertain final cost: the final cost cannot exceed $430,000 and cannot go below $370,000. You have an equal chance of spending $430,000 or $370,000 on it. We will represent it by $430,000 $370,000 What would you bid on this project. Include profit and home office overhead. Keep in mind that if you bid too high you will losE the contract. - Bid: $+55,000 - 155 - 17- Similarly you have to bid on a project with two un- certain final costs: $1,075,000 and $925,000 (maximum and minimum costs). There is an equal chance of spending $1,075,000 or $925,000. $1,075,000 $925,000 What would you bid on this project. - Bid: $1,44,000 - 156 -