Game Theory: Dominant Strategies, Prisoners’ Dilemma, and Nash Equilibrium

Game Theory in Microeconomics

Introduction to Game Representation

Game theory analyzes strategic interactions where the outcome for each participant depends on the actions of all. Games are often represented using a payoff matrix, which displays the payoffs for each player under every possible combination of strategies.

• Payoff Matrix: A table that shows the payoffs for each player for every possible combination of strategies.

• Players: The decision-makers in the game.

• Strategies: The possible actions each player can take.

Dominant Strategies and Equilibrium

Dominant Strategy

A dominant strategy is one that yields the highest payoff for a player, regardless of what the other players do. If a player has a dominant strategy, it is always optimal to play it.

• Definition: A strategy is dominant if, for every possible strategy of the other player(s), it provides a higher payoff than any other strategy.

• Example: In the Prisoners’ Dilemma, both players have a dominant strategy to confess.

Equilibrium in Dominant Strategies

An equilibrium in dominant strategies occurs when each player plays their dominant strategy. This is a natural prediction for the outcome of a game, but not all games have such an equilibrium.

• Not every player or game has a dominant strategy (e.g., Rock-Paper-Scissors).

• Not every game has an equilibrium in dominant strategies.

The Prisoners’ Dilemma

Payoff Matrix and Analysis

The Prisoners’ Dilemma is a classic example in game theory, illustrating the conflict between individual rationality and collective welfare. The payoff matrix is as follows:

	Prisoner 2: Confess	Prisoner 2: Deny
Prisoner 1: Confess	-5, -5	0, -20
Prisoner 1: Deny	-20, 0	-1, -1

• Both prisoners have a dominant strategy: Confess.

• The equilibrium in dominant strategies is (Confess, Confess).

Pareto Domination

An outcome Pareto dominates another if at least one player is better off and no player is worse off. In the Prisoners’ Dilemma, (Deny, Deny) Pareto dominates (Confess, Confess), but rational self-interest leads both to confess.

• Pareto Dominated: An outcome is Pareto dominated if another outcome makes at least one player better off without making anyone worse off.

• Example: (Deny, Deny) with payoffs (-1, -1) is better for both than (Confess, Confess) with (-5, -5).

Key Features of the Prisoners’ Dilemma

• There is an equilibrium in dominant strategies.

• The equilibrium is Pareto dominated by another feasible outcome.

Practice Problems: Identifying Prisoners’ Dilemma Features

Consider the following payoff matrices:

	Wife: Clean	Wife: Don’t Clean
Husband: Clean	10, 10	0, 15
Husband: Don’t Clean	15, 0	2, 2

• Both have dominant strategies (Don’t Clean).

• The equilibrium (Don’t Clean, Don’t Clean) is Pareto dominated by (Clean, Clean).

In a modified game where (Don’t Clean, Don’t Clean) yields (12, 12), the equilibrium is not Pareto dominated, so it is not a true Prisoners’ Dilemma.

Nash Equilibrium

Definition and Best Response

A Nash equilibrium is a set of strategies where each player is playing their best response to the other’s strategy. No player can improve their payoff by unilaterally changing their strategy.

• Best Response: The strategy that yields the highest payoff given the other player’s choice.

• Mathematical Definition: In a two-player game, (S1, S2) is a Nash equilibrium if:

$S_1 \text{ is a best response to } S_2$ $S_2 \text{ is a best response to } S_1$

The Pig and Piglet Game

Payoff Matrix and Dominant Strategies

This game illustrates the concepts of dominant strategy and Nash equilibrium. The payoff matrix is:

	Piglet: Wait	Piglet: Press
Pig: Wait	0, 0	10, -1
Pig: Press	1, 8	6, 2

• The Piglet has a dominant strategy: Wait.

• The Pig does not have a dominant strategy; its best response depends on the Piglet’s choice.

• There is no equilibrium in dominant strategies.

Nash Equilibrium in the Pig and Piglet Game

To find the Nash equilibrium, identify the best responses:

• If the Piglet chooses Wait, the Pig’s best response is Press.

• If the Pig chooses Press, the Piglet’s best response is Wait.

Thus, (Press, Wait) is a Nash equilibrium, as both are playing their best responses to each other.

Summary Table: Nash Equilibrium Verification

Strategy Profile	Nash Equilibrium?	Reason
(Press, Wait)	Yes	Both are playing best responses
(Wait, Wait)	No	Pig should Press if Piglet Waits
(Press, Press)	No	Piglet should Wait if Pig Presses
(Wait, Press)	No	Piglet should Wait if Pig Waits

Conclusion

Game theory provides tools to analyze strategic interactions, including the concepts of dominant strategies, Pareto domination, and Nash equilibrium. Not all games have dominant strategies or equilibria in dominant strategies, but Nash equilibrium offers a broader prediction of outcomes in strategic settings.