Matching pennies

This article is about the two-person game. For the confidence trick, see coin-matching game.

	Heads	Tails
Heads	+1, −1	−1, +1
Tails	−1, +1	+1, −1
Matching pennies

Matching pennies is the name for a simple example game used in game theory. It is the two-strategy equivalent of Rock, Paper, Scissors. Matching pennies is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium.^[1]

The game is played between two players, Player A and Player B. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails) Player A keeps both pennies, so wins one from Player B (+1 for A, −1 for B). If the pennies do not match (one heads and one tails) Player B keeps both pennies, so receives one from Player A (−1 for A, +1 for B). This is an example of a zero-sum game, where one player's gain is exactly equal to the other player's loss.

The game can be written in a payoff matrix (pictured right). Each cell of the matrix shows the two players' payoffs, with Player A's payoffs listed first.

This game has no pure strategy Nash equilibrium since there is no pure strategy (heads or tails) that is a best response to a best response. In other words, there is no pair of pure strategies such that neither player would want to switch if told what the other would do. Instead, the unique Nash equilibrium of this game is in mixed strategies: each player chooses heads or tails with equal probability.^[2] In this way, each player makes the other indifferent between choosing heads or tails, so neither player has an incentive to try another strategy. The best response functions for mixed strategies are depicted on the figure 1 below:

Figure 1. Best response correspondences for players in the matching pennies game. The leftmost mapping is for the coordinating player, the middle shows the mapping for the discoordinating player. The sole Nash equilibrium is shown in the right hand graph. x is a probability of playing heads by discoordinating player, y is a probability of playing heads by coordinating player. The unique intersection is the only point where mys strategy of first player is the best response on the strategy of second and vice versa.

The matching pennies game is mathematically equivalent to the games "Morra" or "odds and evens", where two players simultaneously display one or two fingers, with the winner determined by whether or not the number of fingers match. Again, the only strategy for these games to avoid being exploited is to play the equilibrium.

Of course, human players might not faithfully apply the equilibrium strategy, especially if matching pennies is played repeatedly. In a repeated game, if one is sufficiently adept at psychology, it may be possible to predict the opponent's move and choose accordingly, in the same manner as expert Rock, Paper, Scissors players. In this way, a positive expected payoff might be attainable, whereas when either player plays the equilibrium, everyone's expected payoff is zero.

Nonetheless, statistical analysis of penalty kicks in soccer—a high-stakes real-world situation that closely resembles the matching pennies game—has shown that the decisions of kickers and goalies resemble a mixed strategy equilibrium.^[3]^[4]

References

↑ Gibbons, Robert (1992). Game Theory for Applied Economists. Princeton University Press. pp. 29–33. ISBN 0-691-00395-5.
↑ "Matching Pennies". GameTheory.net.
↑ Chiappori, P.; Levitt, S.; Groseclose, T. (2002). "Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer" (PDF). American Economic Review 92 (4): 1138–1151. doi:10.1257/00028280260344678. JSTOR 3083302.
↑ Palacios-Huerta, I. (2003). "Professionals Play Minimax". Review of Economic Studies 70 (2): 395–415. doi:10.1111/1467-937X.00249.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Screening game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Tragedy of the commons Tyranny of small decisions

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