To determine the presence of dominant strategies for both players in the game, we need to analyze the given payoff matrix. A dominant strategy is a strategy that results in a better outcome for a player, regardless of the opponent's actions.
Here is the payoff matrix:
[tex]\[
\begin{array}{|c|c|c|}
\hline
& \text{Left} & \text{Right} \\
\hline
\text{Up} & \text{R: 10, C: 8} & \text{R: 4, C: 3} \\
\hline
\text{Down} & \text{R: 11, C: 15} & \text{R: 5, C: 7} \\
\hline
\end{array}
\][/tex]
We'll analyze the strategies for both players, considering Roland first:
1. Roland's strategies: "Up" vs. "Down"
- When Colleen chooses "Left":
- If Roland chooses "Up", his payoff is 10.
- If Roland chooses "Down", his payoff is 11.
- In this case, "Down" is better for Roland (11 > 10).
- When Colleen chooses "Right":
- If Roland chooses "Up", his payoff is 4.
- If Roland chooses "Down", his payoff is 5.
- In this case, "Down" is better for Roland (5 > 4).
Since the strategy "Down" yields a higher payoff for Roland regardless of whether Colleen chooses "Left" or "Right", "Down" is a dominant strategy for Roland.
Next, we analyze Colleen's strategies:
2. Colleen's strategies: "Left" vs. "Right"
- When Roland chooses "Up":
- If Colleen chooses "Left", her payoff is 8.
- If Colleen chooses "Right", her payoff is 3.
- In this case, "Left" is better for Colleen (8 > 3).
- When Roland chooses "Down":
- If Colleen chooses "Left", her payoff is 15.
- If Colleen chooses "Right", her payoff is 7.
- In this case, "Left" is better for Colleen (15 > 7).
Since the strategy "Left" yields a higher payoff for Colleen regardless of whether Roland chooses "Up" or "Down", "Left" is a dominant strategy for Colleen.
Thus, both Roland and Colleen have dominant strategies in the given game.
So, the correct answer is:
[tex]$\boxed{\text{both Roland and Colleen have}}$[/tex]
