1. Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure John B. Van Huyck, Raymond C. Battalio, Richard O. Beil The American Economic Review, Vol. 80, No. 1 (March 1990), 234-48.

2. Motivation Deductive equilibrium methods fail to determine unique equilibrium points Situations with multiple and non-interchangeable equilibria lead to strategic uncertainty Need to be able to identify self-enforcing equilibrium points and their expected outcomes

3. A Pure Coordination Game Let e 1,…e n be the actions of taken by players 1 through n. The period game A is defined by the payoff function where a > b > 0 and e i = min(e 1,…,e i-1,e i+1,…,e n ) and the action space for each player i = 1,…n.

4. A Pure Coordination Game (cont’d) Players assumed to have complete information about the payoff function and strategy space and to know that these are common knowledge Under explicit coordination, if a-b > 0, then each player should play ē Without explicit coordination and using the Nash equilibrium concept, player i chooses e i = e i, so any n-tuple (e,…,e), where e Є {1,…,ē} satisfies the best mutual response property

5. Two Coordination Problems Players incorrectly forecast the minimum e i, resulting in regret and outcomes without the mutual-best response property When equilibria can be Pareto-ranked, players can give best responses but the resulting equilibrium is Pareto dominated, i.e.,

6. Selection Principles: Payoff Dominance Equilibrium points are strictly Pareto ranked, resulting in selection of the strictly non-Pareto dominated point, (ē,…, ē) The game A, however, tests payoff dominance--consider the CDF for player j’s action F(e j ) and the CDF for the minimum F min (e) –F(e j ) = 1 for e j = ē and 0 otherwise –If e 1,…,e n are i.i.d., then F min (e) = 1-[1- F(e j )] n, which equals 1 for e j = ē and 0 otherwise –But, say, F(1) > 0, then F min (1) →1 as n→∞

7. Selection Principles: Security Equilibrium selected which supports the player’s maximin action In this case, each player ensures a payoff of a-b by choosing e i =1

8. Repeated Interaction In the repeated game A(T), n players play A for T periods –Payoff dominant equilibrium is the repeated play of the n- tuple (ē,…,ē) –Secure equilibrium is the repeated play of the of the n-tuple (1,…,1) Having t periods experience in A(T) gives some history for players to use for equilibrium selection in continuation game A(T-t) –Example: players give best response to the minimum observed in previous period

9. Experimental Design Subjects were undergraduates at Texas A&M and provided questionnaires to confirm understanding of instructions Description of game made as common knowledge at beginning No pre-play negotiation allowed Minimum action publicly announced after each repetition and subjects calculated earnings for that period In some experiments, design parameters altered in sequence of treatments (each preceded by new instructions) In all treatments, the feasible actions were the integers 1 through 7

10. Experimental Design (cont’d)

12. Results: Treatment A Initial outcome predicted by neither payoff dominance or security Repeated play makes it more likely that subjects obtain mutual best-response outcomes in the continuation game –Subjects determining the minimum in period one did not determine the minimum in the next period –Subjects playing above the minimum in period one reduced their choice of action –Some subjects play below the minimum of the preceding period In general, security predicts the stable outcome of the period game A

13. Results: Treatment B Each player’s individual action is not penalized in the payoff function π Players determining the minimum increase their action, those above the minimum do not In general, payoff dominance predicted the stable outcome of the game

14. Results: Treatment A’ After Treatment B, revert to treatment A Initial period shows bimodal distribution to actions and predictions (massed at the payoff-dominant and secure actions In general, security predicts the stable outcome of the game

15. Results: Treatment C For both fixed and random pairs –Initially, subjects generally increased their action (from end of A’) and close to half chose payoff dominant action –Subjects playing the minimum increased actions on average, subjects playing above the minimum decreased actions on average –No behavior where subjects play below the minimum In the fixed pair case, payoff dominance predicts the stable outcome, but not in random pair

16. Results: Treatment A with Monitoring At end of each period, report the distribution of actions, not just minimum action Reach a stable and mutual best response outcome (secure action) more quickly than without monitoring

17. Conclusions Convergence to inefficient secure outcome appears to be due to strategic uncertainty and the minimum rule With a large number of players, the secure equilibrium appears to describe coordination behavior Meanwhile, the payoff-dominant equilibrium appears unlikely as an initial play or in repeated interaction