Question

Explain the difference between a dominant strategy and a Nash equilibrium.

Short answer

Answer: The main differences between a dominant strategy and a Nash equilibrium are: 1. A dominant strategy is always the best choice for a player, regardless of other players' strategies, while a Nash equilibrium considers the best strategy for each player, given the strategies chosen by the other players. 2. A game can have multiple Nash equilibria but can only have at most one dominant strategy for each player. 3. A dominant strategy is a stronger concept since a player will always choose it without considering the choices made by other players, while a Nash equilibrium is based on a player's response to the choices made by others. 4. A game with a dominant strategy for all players will always result in a Nash equilibrium, but not all Nash equilibria involve dominant strategies.

Step-by-step solution

Definition of Dominant Strategy

A dominant strategy is a strategy that is the best choice for a player, regardless of the strategies chosen by the other players in the game. In other words, a player with a dominant strategy will always choose this strategy because it gives them the best payoff, regardless of what the other players do.

Example of Dominant Strategy

Consider a simple game between two players, A and B. They each can choose either action X or action Y. The payoffs for each player are as follows: || X | Y | |--|--|--| |X|(2,3)|(0,4)| |Y|(1,1)|(1,2)| In this game, player A should always choose action X, because action X gives him a higher payoff, regardless of the action chosen by player B. The payoffs for player A are (2,0) for action X and (1,1) for action Y. Since 2 > 1 in both cases, action X is the dominant strategy for player A.

Definition of Nash Equilibrium

A Nash equilibrium is a situation in which each player chooses the best strategy, given the strategies chosen by the other players in the game. In a Nash equilibrium, no player has an incentive to deviate from their current strategy.

Example of Nash Equilibrium

Using the same example as before: || X | Y | |--|--|--| |X|(2,3)|(0,4)| |Y|(1,1)|(1,2)| There is a Nash equilibrium when player A chooses action X and player B chooses action Y. When player A chooses X and player B chooses Y, each player is playing their best possible strategy, given the strategy chosen by the other player. Player A, given B's strategy Y, prefers X (payoff 0) over Y (payoff 1), and player B, given A's strategy X, prefers Y (payoff 4) over X (payoff 3). No player has an incentive to deviate from their current strategy.

Differences between Dominant Strategy and Nash Equilibrium

The main differences between a dominant strategy and a Nash equilibrium are: 1. A dominant strategy is a single strategy that is always the best choice for a player, while a Nash equilibrium considers the best strategy for each player, given the strategies chosen by the other players. 2. A game can have multiple Nash equilibria but can only have at most one dominant strategy for each player. 3. A dominant strategy is a stronger concept since a player will always choose it without considering the choices made by other players, while a Nash equilibrium is based on a player's response to the choices made by others. 4. A game with a dominant strategy for all players will always result in a Nash equilibrium, but not all Nash equilibria involve dominant strategies.