Manipulated Nash equilibrium

MAPNASH
MAPNASH
Solution concept in game theory
Relationship
Subset of	Nash equilibrium, Subgame perfect equilibrium
Significance
Proposed by	A. Amershi, A. Sadanand, and V. Sadanand
Used for	Dynamic games of imperfect information
Example	Battle of the sexes

In game theory, a Manipulated Nash equilibrium or MAPNASH is a refinement of subgame perfect equilibrium used in dynamic games of imperfect information. Informally, a strategy set is a MAPNASH of a game if it would be a subgame perfect equilibrium of the game if the game had perfect information. MAPNASH were first suggested by Amershi, Sadanand, and Sadanand (1988) and has been discussed in several papers since. It is a solution concept based on how players think about other players' thought processes.

Formal definition and an example

Consider a dynamic game of imperfect information, G. Based on G, construct a game, PG, which has the same strategies, payoffs, and order of moves as G except PG is a game of perfect information (every player in PG is aware of the strategies chosen by those players who moved before). A strategy, S, in G is a MAPNASH of G if and only if S is a Nash equilibrium of G and S is a subgame perfect equilibrium of PG.

As an example, consider a sequential version of Battle of the sexes (pictured above on the left). This game has three Nash equilibrium: (O, o), (F, f), and one mixed equilibrium. We can construct a perfect information version (pictured above on the right). This game has only one subgame perfect equilibrium (O, Oo) If the first player chooses O, the second will choose Oo because 2 is better than 0. If the first player chooses F, the second will choose Ff because 3 is better than 0. So, player 1 is choosing between 3 if she chooses O and 2 if she chooses F. As a result, player 1 will choose O and player 2 will choose Oo.

In the imperfect information Battle of the sexes (G) the only MAPNASH is (O, o). Effectively, by moving first, player 1 can force the other player to choose her preferred equilibrium, hence the name "manipulated."

Significance

In traditional game theory the order of moves was only relevant if there was asymmetric information. In the case of battle of the sexes discussed above, the imperfect information game is equivalent to a game where player 2 moves first and a game where both players move simultaneously. If players follow MAPNASH, the order of moves is relevant even if it does not introduce asymmetries in information. Experimental evidence seems to suggest that actual players are influenced by the order of moves even if the order does not provide players with additional information.

Cooper et al. (1993) study a version of battle of the sexes and find that when one player moves before the other, the first player tends to choose his favorite equilibrium more often and the second player chooses her less favored equilibrium more often. This is a reversal for the second player compared to the same game where both players chooses simultaneously. Similar results are observed in public good games by Budescu, Au, and Chen (1997) and Rapoport (1997).

All of these games are coordination games where equilibrium selection is an important problem. In these games one player has a preferred equilibrium, and one might suppose that the order of moves introduces an asymmetry that solves the coordination problem. In order to resolve this problem Weber, Camerer, and Knez (2004) study a coordination game where no player prefers one equilibrium over another. They find that in this game introducing order results in different equilibria being selected, and they conclude that MAPNASH may be an important predictive tool.

References

Amershi, A. A. Sadanand, and V. Sadanand (1989) "Manipulated Nash equilibria I: Forward induction and thought process dynamics in extensive form." University of Minnesota Discussion Paper 1989–4.
Budescu, D.V., W.T. Au, and X.-P. Chen (1997) "Effects of protocol of play and social orientation on behavior in sequential resource dilemmas." Organizational Behavior and Human Decision Processes.' 69(3), 179–193.
Cooper, R., D. DeJong, R. Forsythe, and T. Ross (1993) "Forward Induction in the Battle-of-the-sexes game." The American Economics Review. 83: 1303–1316.
Rapoport, A. (1997) "Order of play in strategically equivalent games in extensive form." International Journal of Game theory. 26(1), 113–136.
Weber, R. C. Camerer, and M. Knez (2004) "Timing and virtual observability in ultimatum bargaining and 'weak link' coordination games." Experimental Economics 7: 25–48.

v t e Topics of game theory
Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game Mechanism design
Equilibrium concepts	Bayes correlated equilibrium Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Coalition-proof Nash equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand equilibrium
Strategies	Appeasement Backward induction Bid shading Collusion Cheap talk De-escalation Deterrence Escalation Forward induction Grim trigger Markov strategy Pairing strategy Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat
Classes of games	Auction Bargaining problem Global game Intransitive game Mean-field game n-player game Perfect information Large Poisson game Potential game Repeated game Screening game Signaling game Strictly determined game Stochastic game Symmetric game Zero-sum game
Games	Go Chess Infinite chess Checkers All-pay auction Prisoner's dilemma Gift-exchange game Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Lewis signaling game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Electronic mail game Rock paper scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Bertrand competition Cournot competition Stackelberg competition Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Induction puzzles Trust game Princess and monster game Rendezvous problem
Theorems	Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
Search optimizations	Alpha–beta pruning Aspiration window Principal variation search max^n algorithm Paranoid algorithm Lazy SMP
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