Question

Players \(A\) and \(B\) are engaged in a coin-matching game. Each shows a coin as either heads or tails. If the coins match, \(B\) pays A \(\$ 1 .\) 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: There is no Nash equilibrium in the coin-matching game. Since there is no outcome where neither player A nor player B has a reason to change their strategy, players might choose their strategies based on personal play style, risk tolerance, or other factors. One possible approach is to use randomized strategies, where each player chooses heads or tails based on a certain probability, making game outcomes unpredictable and challenging for the other player to determine the best strategy.

Step-by-step solution

Understanding the Game

In this coin-matching game, player A and player B each show a coin with one of two outcomes, heads (H) or tails (T). If the coins match, player B pays player A \(1, otherwise, player A pays player B \)1.

Creating the Payoff Matrix

To construct the payoff matrix, utilize the four possible outcomes in this game: 1. Player A shows H and Player B shows H. 2. Player A shows H and Player B shows T. 3. Player A shows T and Player B shows H. 4. Player A shows T and Player B shows T. The payout for each outcome will be displayed as (A's payout, B's payout). The arrangement would look like this: | | B_H | B_T | |---|-----|-----| | A_H | (1, -1) | (-1, 1) | | A_T | (-1, 1) | (1, -1) |

Identify Nash Equilibria

To find a Nash equilibrium, neither A nor B should have a reason to change their strategy, given the other player's strategy. For outcome (A_H, B_H): If A switches to T, the payout becomes (-1, 1), which is worse for A. If B switches to T, the payout becomes (-1, 1), which is better for B. So, (A_H, B_H) is not a Nash equilibrium. For outcome (A_H, B_T): If A switches to T, the payout becomes (1, -1), which is better for A. If B switches to H, the payout becomes (1, -1), which is worse for B. So, (A_H, B_T) is not a Nash equilibrium. For outcome (A_T, B_H): If A switches to H, the payout becomes (1, -1), which is better for A. If B switches to T, the payout becomes (1, -1), which is worse for B. So, (A_T, B_H) is not a Nash equilibrium. For outcome (A_T, B_T): If A switches to H, the payout becomes (-1, 1), which is worse for A. If B switches to H, the payout becomes (-1, 1), which is better for B. So, (A_T, B_T) is not a Nash equilibrium. Since there is no outcome where neither A nor B have a reason to change their strategy, there is no Nash equilibrium in this game.

Choosing Strategies

Since this game does not have a Nash equilibrium, the players might choose their strategies based on their personal play style, risk tolerance, or other factors. One possible approach is to use randomized strategies, where each player chooses heads or tails with a certain probability, such as flipping their coin and echoing the outcome. This could lead to unpredictable game outcomes and make it more challenging for the other player to determine the best strategy.

Key concepts

Payoff Matrix

When playing a game, it is crucial to understand the potential outcomes and rewards for each participant based on their chosen strategies. This is where the payoff matrix comes into play. In simple terms, a payoff matrix is a table that describes the gains or losses (payoffs) for each player, depending on the combination of actions or strategies both players employ. For our coin-matching game, there are two players, each with two possible moves: showing heads (H) or tails (T).

Given these possibilities, we can devise a payoff matrix that reflects four scenarios:

• If both players show H, player A gains \(1, and player B loses \)1.
• If player A shows H and player B shows T, player A loses \(1, and player B gains \)1.
• If player A shows T and player B shows H, the reverse payouts happen.
• If both show T, the payouts are similar to the first scenario but reversed.

By constructing this matrix, we provide a visual representation of the potential outcomes, which is invaluable for players to strategize their next move. It is worth noting that the matrix is symmetric owing to the game's nature, where players' fortunes are directly opposed.

An improved understanding of the game comes when one tries to identify the best move for each player based on the matrix, a challenge given the absence of a Nash equilibrium in this particular setup.

Game Theory

Game theory is a fascinating field of mathematics and economics that examines strategic interactions between rational decision-makers. In its essence, game theory aims to understand and predict players' behavior in strategic situations where their success depends on the choices of others. One of the foundational concepts in game theory is Nash equilibrium. This occurs when players have selected strategies such that no player can benefit by changing their strategy while the others keep theirs unchanged. In other words, it's like coming to an unstated agreement on the strategies where everyone is in a state of mutual contentment.

The coin-matching game is a perfect exercise in game theory because it involves simple, opposing strategies and immediate payoffs. However, as the solution reveals, not every game has a Nash equilibrium. Sometimes, the best move for each player changes based on what the other player does, leading to a strategic shuffle without that settled arrangement. In games without a clear Nash equilibrium, understanding players' motivations, risk aversions, and even psychological factors can be just as crucial as calculating payoffs.

Randomized Strategies

In games where a clear cut winning strategy is not apparent, like our coin-matching game without a Nash equilibrium, players often resort to randomized strategies. A randomized strategy doesn't follow a predictable pattern but instead uses randomness to add an element of surprise, thus making it harder for the opponent to anticipate and counteract one's moves.

In a practical sense, this can mean that a player chooses to show heads or tails based on the flip of their coin, not following any discernible sequence. This randomness has the potential to make both players' decisions more unpredictable and, therefore, the game more balanced. Randomized strategies are a sophisticated method to combat the issue of predictability in competitive situations, allowing for more dynamic and less exploitable gameplay.

Using randomized strategies can be an advantageous approach, particularly in repeated matches where patterns could otherwise be detected and exploited. Yet, it's worth considering that while randomization introduces uncertainty, it does not guarantee a win, as the nature of probability ensures both victory and defeat will occur over time.