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Sagot :
To find the Nash equilibrium in the given game, we can follow a systematic process. Let's consider the payoffs for both players, Roland and Colleen, and determine their best responses to the strategies of each other.
Here is the payoff matrix again for better clarity:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{} & \text{Left} & \text{Right} \\ \hline \text{Up} & (8, 2) & (7, 4) \\ \hline \text{Down} & (4, 5) & (3, 1) \\ \hline \end{array} \][/tex]
### Step 1: Identify Best Responses
First, we need to find the best responses for each player to the strategy of the other player.
#### Roland's Best Response
If Colleen plays Left:
- Up: Roland gets 8
- Down: Roland gets 4
- Best response: Up (since 8 > 4)
If Colleen plays Right:
- Up: Roland gets 7
- Down: Roland gets 3
- Best response: Up (since 7 > 3)
#### Colleen's Best Response
If Roland plays Up:
- Left: Colleen gets 2
- Right: Colleen gets 4
- Best response: Right (since 4 > 2)
If Roland plays Down:
- Left: Colleen gets 5
- Right: Colleen gets 1
- Best response: Left (since 5 > 1)
### Step 2: Find the Nash Equilibrium
A Nash equilibrium occurs when both players are playing best responses to each other’s strategies simultaneously.
Let’s check each possible combination to see if it qualifies as a Nash equilibrium:
1. Up, Left:
- Roland’s payoff: 8 (best response to Left)
- Colleen’s payoff: 2 (not best response to Up; best response would be Right with payoff 4)
- Not a Nash equilibrium since Colleen would prefer to switch to Right.
2. Up, Right:
- Roland’s payoff: 7 (best response to Right)
- Colleen’s payoff: 4 (best response to Up)
- This is a Nash equilibrium since neither Roland nor Colleen would benefit by switching their strategies.
3. Down, Left:
- Roland’s payoff: 4 (not best response to Left; best response would be Up with payoff 8)
- Colleen’s payoff: 5 (best response to Down)
- Not a Nash equilibrium since Roland would prefer to switch to Up.
4. Down, Right:
- Roland’s payoff: 3 (not best response to Right; best response would be Up with payoff 7)
- Colleen’s payoff: 1 (not best response to Down; best response would be Left with payoff 5)
- Not a Nash equilibrium since both would prefer to switch their strategies.
### Conclusion
The Nash equilibrium for this game is Up, Right. Neither player has an incentive to deviate unilaterally from this strategy combination. So, the result is:
Up, Right
Here is the payoff matrix again for better clarity:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{} & \text{Left} & \text{Right} \\ \hline \text{Up} & (8, 2) & (7, 4) \\ \hline \text{Down} & (4, 5) & (3, 1) \\ \hline \end{array} \][/tex]
### Step 1: Identify Best Responses
First, we need to find the best responses for each player to the strategy of the other player.
#### Roland's Best Response
If Colleen plays Left:
- Up: Roland gets 8
- Down: Roland gets 4
- Best response: Up (since 8 > 4)
If Colleen plays Right:
- Up: Roland gets 7
- Down: Roland gets 3
- Best response: Up (since 7 > 3)
#### Colleen's Best Response
If Roland plays Up:
- Left: Colleen gets 2
- Right: Colleen gets 4
- Best response: Right (since 4 > 2)
If Roland plays Down:
- Left: Colleen gets 5
- Right: Colleen gets 1
- Best response: Left (since 5 > 1)
### Step 2: Find the Nash Equilibrium
A Nash equilibrium occurs when both players are playing best responses to each other’s strategies simultaneously.
Let’s check each possible combination to see if it qualifies as a Nash equilibrium:
1. Up, Left:
- Roland’s payoff: 8 (best response to Left)
- Colleen’s payoff: 2 (not best response to Up; best response would be Right with payoff 4)
- Not a Nash equilibrium since Colleen would prefer to switch to Right.
2. Up, Right:
- Roland’s payoff: 7 (best response to Right)
- Colleen’s payoff: 4 (best response to Up)
- This is a Nash equilibrium since neither Roland nor Colleen would benefit by switching their strategies.
3. Down, Left:
- Roland’s payoff: 4 (not best response to Left; best response would be Up with payoff 8)
- Colleen’s payoff: 5 (best response to Down)
- Not a Nash equilibrium since Roland would prefer to switch to Up.
4. Down, Right:
- Roland’s payoff: 3 (not best response to Right; best response would be Up with payoff 7)
- Colleen’s payoff: 1 (not best response to Down; best response would be Left with payoff 5)
- Not a Nash equilibrium since both would prefer to switch their strategies.
### Conclusion
The Nash equilibrium for this game is Up, Right. Neither player has an incentive to deviate unilaterally from this strategy combination. So, the result is:
Up, Right
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