Inertial Coordination Games

Abstract

Coordination lies at the heart of many economic phenomena. A well-known example is currency crises, in which traders decide whether to launch a speculative attack. On one hand, shocks to the currency's fundamentals can propagate: as more traders attack, the central bank's foreign reserves are depleted, which in turn encourages further attacks as traders seek to exploit a weakening currency. On the other hand, shocks can also fizzle out: traders may quickly learn that the central bank's balance sheet is strong, causing pessimism to dissipate and attacks to subside. When do shocks propagate, and when do they fizzle out? In particular, how do these outcomes depend on the speed at which traders learn about the fundamental? Motivated by these questions, we propose a model of inertial coordination games—dynamic coordination games where players repeatedly decide whether to take a risky action. The payoff from this risky action depends on (i) a persistent fundamental; and (ii) an endogenous component that depends on others' past play. Players receive private signals about the persistent fundamental over time and form beliefs about the current state. Notably, the current state depends on past play, which in turn depends on past beliefs about play yet farther back into the past. Thus, expectations about histories shape behavior in the present which, in turn, drives the evolution of future states and future play. Our main result develops a tight connection between the speed of learning and limit aggregate play: the risk-dominant action is played in the limit if and only if posterior precisions grow sub-quadratically. This has sharp implications for the long-run propagation of shocks. With slow (sub-quadratic) learning, limit play exhibits history independence: initial shocks have no lasting effect, and limit play is determined solely by fundamentals. By contrast, with fast (super-quadratic) learning, limit play is history dependent: initial shocks can be self-fulfilling, and whether they propagate depends jointly on fundamentals, the size of the shock, and the speed of learning. Our results offer a novel perspective on whether 'history' or 'expectations' shape long-run coordination outcomes: in our model, expectations about histories is what matters for whether self-fulfilling spirals occur. Finally, we show that the speed of learning also shapes the path of play, focusing on the case of sub-quadratic learning. When signals are precise, aggregate play exhibits a sudden transition from nearly all players choosing the non-risk-dominant action to nearly all players choosing the risk-dominant action. In contrast, when signals are noisy, the transition is gradual, with the share of players choosing the risk-dominant action increasing gradually over time. This suggests that "spikes" in aggregate behavior (such as a sudden and massive sell-off of a currency) can be consistent with transition to limit equilibrium play, and need not indicate an "equilibrium shift."