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?
Did you see the suggestions I did for you earlier today?
Sra
To formulate the situation as a strategic game, we need to define the players, their possible strategies, and the payoffs associated with each combination of strategies.
In this particular scenario, we have two players, representing the two animals. Let's call them Player 1 and Player 2. The possible strategies for each player are being "passive" or "aggressive".
To determine the payoffs, we need to consider the preferences of each player. You have mentioned that each player prefers the outcome in which its opponent is passive to that in which its opponent is aggressive. We can assign a higher payoff to this preferred outcome and a lower payoff to the less preferred outcome.
Let's say the payoff to each player in the preferred outcome (opponent is passive) is 1, and the payoff in the less preferred outcome (opponent is aggressive) is 0.
Now, we can create a payoff matrix to represent the game:
```
Player 2
Passive Aggressive
Player 1 ______________________
Passive | 1, 1 | 0, 0 |
|__________|__________|
Aggressive | 0, 0 | 1, 1 |
|__________|__________|
```
In this matrix, the first number in each cell represents the payoff to Player 1, and the second number represents the payoff to Player 2.
To find the Nash Equilibrium, we need to identify the stable outcome where neither player has an incentive to unilaterally change their strategy. It is generally a combination of strategies where neither player can improve their payoff by switching strategies.
In this case, we can observe that both players have an incentive to be aggressive if their opponent is passive, and passive if their opponent is aggressive. Therefore, there are two Nash Equilibria: (Aggressive, Passive) and (Passive, Aggressive).
To summarize the process:
1. Identify the players and possible strategies.
2. Determine the payoffs based on the preferences of the players.
3. Create a payoff matrix representing the game.
4. Find the stable outcomes where no player has an incentive to unilaterally change strategies. These are the Nash Equilibria.