Question

Players \(A\) and \(B\) are engaged in a coin-matching game. Fach shows a coin as either heads or tails. If the coins match, \(B\) pays \(A\) S1. If they differ, \(A\) pays \(B \$ 1\) a. Write down the payoff matrix for this game, and show that it does not contain a Nash equilibrium. b. How might the players choose their strategies in this case?

Short answer

Answer: In the absence of a Nash equilibrium, players A and B may choose to adopt a mixed strategy, where they play each strategy (heads or tails) with a certain probability, making it harder for the opponent to predict the player's strategy and thus minimizing the risk of making a losing move. For example, both players might decide to play heads and tails with equal probability, so that each strategy is played half of the time.

Step-by-step solution

Define possible strategies for players A and B

In the coin-matching game, each player can choose between two strategies: showing heads (H) or tails (T).

Create the payoff matrix

We can now create the payoff matrix for this game. The rows will represent player A's strategies, and the columns will represent player B's strategies. The entries of the matrix will show the payoff for both players, where the first value represents player A's payoff, and the second value represents player B's payoff. Payoff matrix: ``` B H T A H (1,-1) (-1,1) T (-1,1) (1,-1) ```

Analyze the payoff matrix for Nash equilibria

A Nash equilibrium is a situation where neither the row player (A) nor the column player (B) would want to unilaterally change their chosen strategy if they knew the other player's strategy. To find a Nash equilibrium, we will look for the best response for each player for each of the other player's strategies. If there is an intersection of best responses between A and B, that would be a Nash equilibrium. Best responses for player A: - If B plays H, A should play H (1 > -1). - If B plays T, A should play T (1 > -1). Best responses for player B: - If A plays H, B should play T (1 > -1). - If A plays T, B should play H (1 > -1). There is no intersection between the best responses of A and B, so there is no Nash equilibrium in this game.

Strategy selection in the absence of a Nash equilibrium

Without a Nash equilibrium, players A and B do not have any dominant strategy that always leads to the best outcome for them. In this case, the players may choose to adopt a mixed strategy, where they play each strategy (H or T) with a certain probability. For instance, players A and B could both decide to play heads and tails with equal probability, so that each strategy is played half of the time. This would make it harder for the opponent to predict the player's strategy, thus minimizing the risk of making a losing move.

Key concepts

Nash Equilibrium

In game theory, a Nash Equilibrium represents a stable state in which no player can benefit by changing their strategy while the other player's strategy remains unchanged. It's a point where each player's chosen strategy yields the highest payoff given the strategies of the others.

In the coin-matching game described, we investigated the potential for a Nash Equilibrium using the provided payoff matrix. A Nash Equilibrium would exist if there was an intersection of best responses between players A and B. However, analysis shows that:

• Player A's best response to Player B showing Heads (H) is to also show H, and similarly, respond with T to B's T.
• In contrast, Player B finds the best responses to A's strategies inversely, preferring to choose the opposite of what A presents.

As a result, we observe no overlapping best responses between players A and B, indicating that no stable Nash Equilibrium exists for this particular payoff matrix.

Mixed Strategy

Without a dominant strategy that ensures the best outcome in every scenario, players may consider using a mixed strategy. In mixed strategies, players do not stick to a single choice; instead, they vary their moves according to certain probabilities.

For instance, in our game, both players can decide to display heads and tails with equal probability. This means:

• Player A would show Heads and Tails each with a 50% chance.
• Player B would do the same, showing Heads and Tails with a probability of 50%.

By doing this, each player minimizes predictability, thus reducing the risk of exploitation by the opponent. A mixed strategy in this context doesn't guarantee a win every round but allows both players to fare better over time by eliminating any potential predictable patterns in their choices.

Payoff Matrix

A payoff matrix is a visual representation of the possible outcomes of a game for each combination of strategies chosen by the players. It helps in analyzing the interactions between players and understanding the potential payoffs or losses associated with each outcome.

In our example, the payoff matrix was a simple 2x2 grid due to the binary choices available to both players A and B. Here's what our payoff matrix looked like for the coin-matching game:

• When both A and B choose Heads (H), A receives a payoff of 1, and B gets -1.
• If A shows Heads and B shows Tails, A's payoff is -1, and B's is 1.
• The opposite scenarios of Tails for A and Tails for B reflect similarly.

Each cell in the matrix provides insights into how the players' strategies interact, aiding in the decision-making process by clearly laying out the consequences for each possible combination of moves.