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Game Theory Definitions June 11, 2007

Posted by jyu in Qualifying.
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Nash Equilibrium (NE) (after John Nash):

A kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally.  In other words, if no player has incentive to deviate from his strategy given that the other players do not deviate.

Subgame Perfect Equilibrium (SPE): 

  1. a refinement of a NE used in dynamic games
  2. a strategy set is a SPE if it represents a NE of every subgame of the original game
  3. the players palyed at any smaller game that consisted of only one part of the larger game and their behavior represents a NE of that smaller game, then their behavior is a SPE of the larger game
  4. Common method for determining SPE is backward induction

Perfect Bayesian Equilibrium (PBE):

A PBE is a strategy combination and a set of beliefs such that a strategy combination consisting of the best responses given that equilibrium beliefs follow Bayes’s rule and out-of-equalibrium beliefs follow a specific pattern that does not contradict Bayes’s rule. 

Folk Theorem: 

A class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players’ minimax conditions are satisfied.

The minimax condition states that a player will minimize the maximum possible loss which they could face in the game

Pure Strategy:

A pure strategy maps each of a player’s possible information sets to one action.

Mixed Strategy:

A mixed strategy maps each of a player’s possible information sets to a probability distribution over actions.

Incentive Compactibility Constraint (IC):

The IC constraint takes account of the fact that the agents move second, so the contract must induce him to voluntarily pick the desired effort

Individual Rationality (participation constraint) IR:

The agents prefer the contract to his reservation utility.

Incomplete Information:

The utility payoffs of each player remain private ifnormation of each player.  Games with incomplete information require the players form beliefs about their opponents’ private information

Minimax Punishment:

Minmax strategy is defined as the most severe sanction possible if the offender does not cooperate in his own punishment.  The set of strategies is a set of minimax strategies chosen by all the players except i to keep i’s payoff as low as possible, no matter how he responds

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