Nash Equilibrium Topics

• strategy nash equilibria
• strategy choices
• nxn matrix
• credible threats
• credible threat
• player one
• payoffs
• set in stone
• businessmen
• cartel
• equality
• cells
• profits
• classified
• game
• columns
• games

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Nash Equilibrium, Nash Equilibrium

If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. In other words, to be in a Nash equilibrium, each player must answer negatively to the question: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?" Stated simply, Amy and Bill are in Nash equilibrium if Amy is making the best decision she can, taking into account Bill's decision, and Bill is making the best decision he can, taking into account Amy's decision.

However, Nash equilibrium does not necessarily mean the best cumulative payoff for all the players involved; in many cases all the players might improve their payoffs if they could somehow agree on strategies different from the Nash equilibrium (e.g. competing businessmen forming a cartel in order to increase their profits).

This is because it may happen that a Nash equilibrium is not pareto optimal.

For such games the Subgame perfect Nash equilibrium may be more meaningful as a tool of analysis.

If instead, for some player, there is exact equality between x^*_i and some other strategy in the set S , then the equilibrium is classified as a weak Nash equilibrium.

If these conditions are met, the cell represents a Nash Equilibrium. Check all columns this way to find all NE cells. An NxN matrix may have between 0 and NxN pure strategy Nash equilibria.

The subgame perfect equilibrium in addition to the Nash Equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game. This eliminates all non-credible threats, that is, strategies that contain non-rational moves in order to make the counter-player change his strategy.

If player one goes right the rational player two would de facto be kind to him in that subgame. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) nash equilibrium. Therefore, if rational behavior can be expected by both parties the subgame perfect Nash equilibrium may be a more meaningful solution concept when such dynamic inconsistencies arise.

But this is a clear contradiction,so all the gains must indeed be zero. Therefore \sigma^* is a Nash Equilibrium for G as needed.

If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays his strictly dominant strategy.

Source: Wikipedia > Nash Equilibrium