Chapter 15: Problem 4

Consider a game with two players who cannot communicate, and in which each player is asked a question. The players can answer honestly or lie. If both answer honestly, each receives $\$ 100$. If one player answers honestly and the other lies, the liar receives $\$ 500$ and the honest player gets nothing. If both lie, then each receives $\$ 50$. a. Describe the strategies and the payoffs. b. Construct the payoff matrix. c. What is the equilibrium of this game? d. Compare this game to the prisoners' dilemma. Are the games similar or different? Explain.

Short Answer

Expert verified

Strategies: H or L. Payoff Matrix: Player 2: H L; Player 1 H: (100,100) (0,500); Player 1 L: (500,0) (50,50). Nash Equilibrium: (L, L). Similar to Prisoners' Dilemma.

Step by step solution

Define the Strategies and Payoffs

For each player, there are two possible strategies: answer honestly (H) or lie (L). We need to determine the payoffs for all possible outcomes:1. Both answer honestly (H, H): Each receives $100.2. One answers honestly and the other lies (H, L or L, H): The honest player receives $0, the liar receives $500.3. Both lie (L, L): Each receives $50.

Construct the Payoff Matrix

Create a 2x2 matrix to represent the payoffs for the possible combinations of strategies:\[\begin{array}{c|c|c} & H & L \ \hlineH & (100, 100) & (0, 500) \ \hlineL & (500, 0) & (50, 50) \ \end{array}\]

Identify the Nash Equilibrium

A Nash Equilibrium occurs when neither player can unilaterally improve their payoff by changing their strategy. Analyze the payoff matrix:1. If Player 1 chooses H, Player 2's best response is L (500 > 100).2. If Player 1 chooses L, Player 2's best response is L (50 > 0).3. If Player 2 chooses H, Player 1's best response is L (500 > 100).4. If Player 2 chooses L, Player 1's best response is L (50 > 0).Thus, (L, L) is a Nash Equilibrium.

Compare with the Prisoners' Dilemma

The structure of the game is similar to the Prisoners' Dilemma.1. In both games, mutual cooperation (H, H) provides a moderate payoff for both players, while mutual defection (L, L) provides a lower payoff.2. In the given game, lying (defection) strictly dominates answering honestly (cooperation) as it results in a higher individual payoff.Both games illustrate the tension between individual rationality (defection) and collective rationality (cooperation).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nash Equilibrium

In our game, Nash Equilibrium occurs when neither player can improve their situation by changing their strategy alone. Let's walk through this again. If Player 1 decides to always lie, Player 2's best move is to also lie because getting $50 is better than getting $0. The same goes vice versa. Therefore, if both players choose to lie (L, L), that's our Nash Equilibrium. This is the point at which both players are content with their choices, given the choice of the other player. In this scenario, lying is the best response for both, so no one gains by switching their strategy alone.

Payoff Matrix

A payoff matrix is a handy tool used in game theory to show the possible outcomes and rewards for each player based on their chosen strategies. For our game:

If both answer honestly (H, H), they each get \(100.
If one lies and the other answers honestly (H, L or L, H), the liar gets \)500 and the honest player gets \(0.
If both lie (L, L), they each get \)50.

We organize these outcomes into a 2x2 matrix to visualize possible payoffs. This makes it easier to analyze and find the Nash Equilibrium. The matrix lets us see all potential outcomes and understand how each player's choice affects both players.

Prisoners' Dilemma

This game mirrors the classic Prisoners' Dilemma. In both games:

Cooperating (H, H) offers a moderate reward for both players.
Defecting (L, L) results in a lower, but still positive, payoff.

In the Prisoners' Dilemma, two prisoners are better off if they both stay silent (cooperate), but they end up betraying each other (defecting) because they aim for the higher individual reward. Similarly, in our game, answering honestly (cooperation) is better for both collectively, but lying (defection) provides a higher individual reward, leading both to lie ultimately. This showcases how individual rationality conflicts with collective well-being.

Strategies and Payoffs

Understanding strategies and payoffs is critical in game theory. In our game, each player has two strategies: answer honestly (H) or lie (L). The payoffs associated with these strategies are:

Both answer honestly: each gets \(100.
One answers honestly, the other lies: honest gets \)0, liar gets \(500.
Both lie: each gets \)50.

These payoffs define how the game unfolds and influence each player’s strategy. Players choose strategies based on expected payoffs, leading to different outcomes. Recognizing these outcomes helps players understand optimal choices and reach equilibria that reflect strategic decision-making.