Two people enter a bus. Two adjacent cramped seats are free. Each person must decide whether to sit or stand. Sitting alone is more comfortable than sitting next to the other person, whihc is more comfortable than standing.
a.) Suppose that each person only cares about her own comfort. model the situation as a strategic game and find nash equilibria.
b.) Suppose that each person is altruistic, ranking the outcomes according to the other person's comfort, but, out of politeness, prefers to stand than to sit if the other person stands. Model the situation as a game.
How do I go about this? Thanks
a) To model the situation in part a) as a strategic game, we can use a normal form representation. In this case, we have 2 players, each with 2 possible strategies: Sit or Stand.
The payoffs for each player can be represented as follows:
Sit-Sit: (2, 2) - Both players sit and receive a payoff of 2 each.
Sit-Stand: (0, 3) - Player 1 sits (comfortable) while Player 2 stands (less comfortable). Player 1 receives a payoff of 0, and Player 2 receives a payoff of 3.
Stand-Sit: (3, 0) - Player 1 stands (less comfortable) while Player 2 sits (comfortable). Player 1 receives a payoff of 3, and Player 2 receives a payoff of 0.
Stand-Stand: (1, 1) - Both players stand and receive a payoff of 1 each.
To find the Nash equilibria, we need to look for strategy profiles where no player has an incentive to unilaterally deviate. In this case, the only Nash equilibrium is (Sit, Sit), where both players sit. If any player were to switch to standing, they would receive a lower payoff, so neither player has an incentive to deviate.
b) In part b), we consider that each person is altruistic and ranks the outcomes according to the other person's comfort.
We can represent the payoffs in terms of the other player's comfort:
Sit-Sit: (2, 2) - Both players sit comfortably, resulting in a payoff of 2 each.
Sit-Stand: (0, 4) - Player 1 sits (less comfortable) while Player 2 stands (more comfortable). Player 1 receives a payoff of 0, and Player 2 receives a payoff of 4.
Stand-Sit: (4, 0) - Player 1 stands (more comfortable) while Player 2 sits (less comfortable). Player 1 receives a payoff of 4, and Player 2 receives a payoff of 0.
Stand-Stand: (1, 1) - Both players stand and receive a payoff of 1 each.
In this case, we also have a unique Nash equilibrium, which is (Stand, Stand), where both players stand. If either player were to sit, they would increase the other player's comfort level but decrease their own, so neither player has an incentive to deviate.
Remember, even though the players might prefer more comfort for the other person, they still make decisions based on their own preferences and constraints.
a.) To model the situation as a strategic game, we need to identify the players, strategies, and payoffs.
Players: The two people entering the bus.
Strategies: Each person has two strategies - "Sit" and "Stand."
Payoffs: The payoffs represent the level of comfort for each player.
Let's denote the payoffs as follows:
- If both players sit, they will each receive a payoff of 3 (maximum comfort).
- If one player sits while the other stands, the sitter will receive a payoff of 2 (medium comfort), and the stander will receive a payoff of 1 (minimum comfort).
- If both players stand, they will each receive a payoff of 0 (no comfort).
To find Nash equilibria, we need to identify any combinations of strategies where neither player can improve their payoff by unilaterally changing their strategy. In this case, we can see that the combination of both players standing is a Nash equilibrium because neither player can increase their payoff by changing their strategy while the other player stays the same.
b.) To model the situation considering altruism, we need to consider the preferences of each person for the other person's comfort.
The possible outcomes can be ranked as follows, from best to worst:
1. Both players sit.
2. One player sits, and the other stands.
3. Both players stand.
Additionally, out of politeness, a person prefers to stand instead of sitting if the other person stands.
Let's denote the payoffs as follows:
- If both players sit, they will each receive a payoff of 3 (maximum comfort).
- If one player sits while the other stands, the sitter will receive a payoff of 2 (medium comfort), and the stander will receive a payoff of 0 (minimum comfort).
- If both players stand, they will each receive a payoff of 1 (low comfort).
In this case, to find Nash equilibria, we need to identify any combinations of strategies where neither player can improve their payoff by unilaterally changing their strategy. Here, we can see that the combination of both players sitting is a Nash equilibrium because if any player were to deviate and stand instead of sit, their payoff would decrease from 3 to 1 or 0.
Remember that this analysis assumes that both players are rational decision-makers, always aiming to maximize their own comfort (in the first case) or consider the other person's comfort (in the second case).
To model the situation as a strategic game and find the Nash equilibria, we can follow these steps:
a.) Consider each person's decision to be a player in the game. Let's denote the players as Player 1 and Player 2.
1. Identify the strategies: In this case, the strategies for each player are:
- Stand: Player chooses to stand.
- Sit Alone: Player chooses to sit alone.
- Sit Next to the Other Person: Player chooses to sit next to the other person.
2. Determine the payoffs: Assign payoffs to each player for each combination of strategies. Given the preference order, we can assign arbitrary numbers indicating the comfort level. Let's use the following payoffs, where higher numbers represent higher comfort:
- If both players choose to Stand: Player 1 payoff: 0, Player 2 payoff: 0.
- If both players choose to Sit Alone: Player 1 payoff: 2, Player 2 payoff: 2.
- If one player Sits Alone and the other Sits Next to the Other Person: Player 1 payoff: 1, Player 2 payoff: 1.
3. Construct the payoff matrix: Combine the payoffs for each player into a matrix. The matrix will have rows representing Player 1's strategies and columns representing Player 2's strategies. The entries in the matrix will correspond to the payoffs for each player.
| Stand | Sit Alone | Sit Next to Other |
---------------------------------------------
Stand | 0, 0 | 0, 0 | 0, 0 |
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Sit Alone| 2, 2 | 2, 2 | 1, 1 |
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Sit Next | 1, 1 | 1, 1 | 1, 1 |
4. Find the Nash equilibria: A Nash equilibrium is a combination of strategies where no player has an incentive to unilaterally deviate from their chosen strategy. To find the Nash equilibria, look for any combinations where both players' strategies are the best responses to each other. In this case, we can see that all three strategies satisfy this condition, resulting in three Nash equilibria:
- Both players Stand,
- Both players Sit Alone, and
- Both players Sit Next to the Other Person.
b.) To model the situation as a game where each person is altruistic but prefers to stand than to sit if the other person stands, we can modify the payoffs and follow the same steps as in part a.
1. Modify the payoffs: Since both players now prioritize the other person's comfort, the previous payoffs need adjustment. Let's use the following modified payoffs:
- If both players choose to Stand: Player 1 payoff: 1, Player 2 payoff: 1.
- If both players choose to Sit Alone or both Sit Next to the Other Person: Player 1 payoff: 2, Player 2 payoff: 2.
- If one player Sits Alone and the other Stands: Player 1 payoff: 3, Player 2 payoff: 0.
2. Construct the new payoff matrix with the modified payoffs following the same pattern as in part a.
| Stand | Sit Alone | Sit Next to Other |
---------------------------------------------
Stand | 1, 1 | 3, 0 | 3, 0 |
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Sit Alone| 0, 3 | 2, 2 | 2, 2 |
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Sit Next | 0, 3 | 2, 2 | 2, 2 |
3. Find the Nash equilibria: Using the new payoff matrix, analyze the strategies where each player's choice is their best response to the other player's choice. In this case:
- If Player 1 chooses to Stand, Player 2's best response is also to Stand.
- If Player 1 chooses to Sit Alone or Sit Next to the Other Person, Player 2's best response is to Sit Alone or Sit Next to the Other Person.
- If Player 2 chooses to Stand, Player 1's best response is also to Stand.
- If Player 2 chooses to Sit Alone or Sit Next to the Other Person, Player 1's best response is to Sit Alone or Sit Next to the Other Person.
Thus, there are four Nash equilibria in this case:
- Both players Stand,
- Both players Sit Alone,
- Both players Sit Next to the Other Person, or
- Player 1 Sits Alone, and Player 2 Stands.
Follow these steps, and you should be able to model the situations as strategic games and identify the Nash equilibria for both cases.