Question

In two-player games, what do we mean when we have a Nash equilibrium? What is a dominant strategy? Using a payoff matrix, illustrate a two-player, two strategy game where the Nash equilibrium is not the social optimum.

Answer 1

Under nash equilibrium , the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice. Overall, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies. A game may have multiple Nash equilibria or none at all.In the Nash equilibrium, each player's strategy is optimal when considering the decisions of other players.

In game theory, a dominant strategy is the course of action that results in the highest payoff for a player regardless of what the other player does. Not all players in all games have dominant strategies; but when they do, they can blindly follow them.

The prisoner's dilemma is a paradox in decision analysis in which two individuals acting in their own self-interests do not produce the optimal outcome. The typical prisoner's dilemma is set up in such a way that both parties choose to protect themselves at the expense of the other participant. As a result, both participants find themselves in a worse state than if they had cooperated with each other in the decision-making process.

Example:

each prisoner will analyse their best strategy given the other prisoner’s possible strategies. Prisoner 1 (P1) has to build a belief about what choice P2 is going to make, in order to choose the best strategy. If P2 confesses (P2C), he will get either -8 or 0, and if he lies (P2L) he will get either -10 or -1. It can be easily seen that P2 will choose to confess, since he will be better off. Therefore, P1 must choose the best strategy given that P2 will choose to confess: P1 can either confess (P1C, which pays -8) or lie (P1L, which pays -10). The rational thing to do for P1 is to confess. Proceeding inversely, we analyse the beliefs of P2 about P1’s strategies, which gets us to the same point: the rational thing to do for P2 is to confess. Therefore, “to confess” is the dominant strategy. P1C, P2C is the Nash equilibrium in this game, since it is the set of strategies that maximise each prisoner’s utility given the other prisoner’s strategy.

Nash equilibriums can be used to predict the outcome of finite games, whenever such equilibrium exists. On the downside, we find the issue that arises when dealing with a Nash equilibrium that is neither social nor ethical, and where efficiency may be subjective, which is the case in the prisoner’s dilemma, where the Nash equilibrium does not meet the criteria for being Pareto optimal