page_title: Manipulated Nash equilibrium
section_title: Formal definition and an example
hierarchical_section_title: Manipulated Nash equilibrium / Formal definition and an example
caption_attribution_description: An extensive form representation of a battle of the sexes game with imperfect information
context_page_description: 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 and has been discussed in several papers since. It is a solution concept based on how players think about other players' thought processes.
context_section_description: 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."
caption: Battle of the Sexes with imperfect information (G)
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