Question

Can we say that If a game has a unique nash equilibrium, it is also a dominant strategy equilibrium. How can we prove this statement? This is just a statement. we should find whether it is true or not by making proof.

Answer 1

Answer) Nash equilibrium - Strategy chosen by a player such that she does best given what other player has chosen.

Dominant strategy - A strategy chosen by a player irrespective of what other player does.

In case of prisoner's dilemma game itvis observed that the best setbod strategy may not be choosen as a equilibrium strategy. This is because under prisoner's dilemma sub optimum set results in convergence of nash strategies.

In the above example Prisoner 1 chooses confess if prisoner 2 chooses confess because othereise one will spend ten years in jail while confessing results in 5 years. Similarly if prisoner 2 chooses not confess the best strategy for 1 is to confess because it results in no imprisonment. Therefore nash strategies for 1 in both cases is confess which means confess becomes Dominant strategy for 1. In the same way confess is also dominant strategy for 2 because no matter what one chooses best strategy for 2 is always to confess when the players will strategise to make their optimum choice they realise that no matter what they choose dominant strategy for other is always confess given the other player chooses confess their best strategy is to confess. The intersection of two strategies is nash equilibrium. Given player 1 has choosen the best strategy for 2 is to choose confess & given two has choosen confess best strategy for 1 is to choose confess. This makes (confess, confess) nash equilibrium for this game.

It is called sub optimum because if both had choosen not confess then their imprisonment will be for lesser time period .

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