For anyone with experience working with game theories...
1.) Two animals are fighting over some prey. Each can be passive or aggressive. Each prefers to be aggressive if its opponent is passive, and passive if its opponent is aggressive; given its own stance, it prefers the outcome in which its opponent is passive to that in which its opponent is aggressive. Formulate this situation as a strategic game and find its Nash Equilibrium? How should i go about doing this?
To formulate the situation as a strategic game and find its Nash Equilibrium, you can follow these steps:
1. Identify the players: In this scenario, the two players are the two animals involved in the fight.
2. Determine the set of strategies: Each animal can choose to be either passive or aggressive. Thus, there are two possible strategies for each player.
3. Construct the payoff matrix: The payoff matrix represents the preferences or rewards for each player based on their strategies. In this case, we can assume a simple payoff matrix as follows:
| Player 2 - Passive | Player 2 - Aggressive |
Player 1 - Passive | 0, 0 | -1, +1 |
Player 1 - Aggressive | +1, -1 | -2, -2 |
The numbers in the matrix represent the payoffs for Player 1 and Player 2, respectively.
4. Find the Nash Equilibrium: A Nash Equilibrium occurs when neither player has an incentive to change their strategy given the other player's strategy. In other words, it is a stable outcome where no player can unilaterally improve their payoff.
To find the Nash Equilibrium, we analyze each player's strategy:
- Player 1's strategy: If Player 2 is passive, Player 1 prefers to be aggressive, as it results in a payoff of +1. If Player 2 is aggressive, Player 1 prefers to be passive, as it results in a payoff of 0.
- Player 2's strategy: If Player 1 is passive, Player 2 prefers to be aggressive, as it results in a payoff of +1. If Player 1 is aggressive, Player 2 prefers to be passive, as it results in a payoff of 0.
Considering these preferences, we can see that there is no incentive for either player to change their strategy:
- (Player 1 - Aggressive, Player 2 - Passive) is a Nash Equilibrium because Player 1 gets a payoff of +1, and Player 2 gets a payoff of -1.
- (Player 1 - Passive, Player 2 - Aggressive) is also a Nash Equilibrium because Player 1 gets a payoff of -1, and Player 2 gets a payoff of +1.
So, the Nash Equilibria in this game are (Player 1 - Aggressive, Player 2 - Passive) and (Player 1 - Passive, Player 2 - Aggressive). Both are stable outcomes where neither player has an incentive to deviate from their chosen strategy.
By following these steps, you can formulate the situation as a strategic game and find the Nash Equilibrium.