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Nash Equilibria
“Nash Equilibria” is a concept in game theory named after the mathematician and Nobel laureate John Nash. In the context of strategic interactions, a Nash equilibrium is a situation in which each participant in a game makes the best decision they can, taking into account the decisions of others. In other words, at a Nash equilibrium, no player has an incentive to unilaterally change their strategy given the strategies chosen by the other players.
In a Nash equilibrium, each player’s strategy is a best response to the strategies chosen by the other players. It represents a state of mutual consistency where no player can improve their own outcome by changing their strategy, assuming the other players’ strategies remain unchanged.
Nash equilibria are important in understanding strategic interactions in various fields, such as economics, political science, and evolutionary biology. They provide a foundation for predicting stable outcomes in situations where multiple decision-makers interact and influence each other’s choices.
Question:
Compute the set of Nash equilibria of the bimatrix game (A, B) = ([4, 3, 2, 5, 1, 0], [0, 3, 3, 0, 3, 4]). Also report the best response correspondences.
Answer:
let’s find the Nash equilibria of the bimatrix game defined by matrices A and B.
Best Response Correspondences
For Player 1, the best response to Player 2’s strategies can be found by identifying the maximum payoff for each of Player 1’s strategies. We can create a best response table:
| Player 2 chooses 1 | Player 2 chooses 2 | Player 2 chooses 3 | |
|---|---|---|---|
| 1 | 4 | 3 | 2 |
| 4 | 5 | 1 | 0 |
For Player 2, the best response to Player 1’s strategies can be found similarly:
| Player 1 chooses 1 | Player 1 chooses 4 | |
|---|---|---|
| 1 | 0 | 3 |
| 2 | 3 | 0 |
| 3 | 3 | 3 |
We determine the best responses for each player based on the strategies chosen by the other player. For Player 1, we look at each of Player 2’s strategies and find the maximum payoff for Player 1. Similarly, for Player 2, we find the maximum payoff for each of Player 2’s strategies given Player 1’s choices. This gives us insight into the strategies that are optimal for each player depending on the actions of the other.
Identify Pure Nash Equilibria
Now, we need to find the intersections where both players are playing a best response to each other. In this case, the pair (1, 3) is a pure Nash equilibrium.
We find the intersections where both players are playing a best response to each other. A pure Nash equilibrium occurs when no player has an incentive to unilaterally deviate from their chosen strategy. We identify these points by looking for mutual best responses, where both players are making optimal decisions based on the other player’s choices.
Report the Nash Equilibria
The set of Nash equilibria for this bimatrix game is {(1, 3)}.
In this equilibrium, Player 1 chooses strategy 1, and Player 2 chooses strategy 3. At this point, neither player has an incentive to unilaterally deviate from their chosen strategy, as both are playing their best responses given the other’s choice.
This is the Nash equilibrium for the given bimatrix game.
Once we’ve identified the points where both players are playing best responses to each other, we report these as the Nash equilibria of the game. In this context, a Nash equilibrium represents a stable state where neither player has an incentive to change their strategy, given the strategy chosen by the other player. The set of Nash equilibria provides insight into the potential outcomes where the players’ choices align optimally. In this specific example, the Nash equilibrium is given by the strategy pair (1, 3), meaning Player 1 chooses strategy 1, and Player 2 chooses strategy 3.
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