91Ó°ÊÓ

The concept of a payoff matrix is fundamental in game theory. It visually represents the outcomes for each player depending on the strategies they choose. In this game, if both players select the same number, player 2 pays $1 to player 1. If they choose different numbers, no exchange takes place. This matrix helps in understanding the consequences of each player's choices in different scenarios. Visualizing these outcomes through the matrix is crucial as it simplifies the understanding of payoffs and helps in analyzing the strategies in mixed strategy Nash equilibrium and other equilibrium conditions.

In game theory, a mixed strategy refers to a scenario where players randomize over possible moves rather than sticking to a single pure strategy. Here, each player randomizes their integer choices from 1 to K. Each choice is made with an equal probability of 1/K. Mixed strategies allow for more complex strategic thinking since players must consider not just specific actions but probabilities across all potential actions. This broadens the strategic landscape and requires players to optimize based on expected payouts rather than fixed outcomes.

The expected payoff is a crucial concept when dealing with mixed strategies. It represents the average outcome a player can anticipate, given the probabilities of different strategies. For player 1, the expected payoff is calculated as: \ \( E(U_1) = \frac{1}{K} \left( \sum_{k=1}^{K} 1 \cdot \frac{1}{K} \right) \) \ This simplifies to \ \( E(U_1) = \frac{1}{K} \).\ For player 2, the expected payoff becomes: \ \( E(U_2) = -\frac{1}{K} \left( \sum_{k=1}^{K} 1 \cdot \frac{1}{K} \right) \) \ which simplifies to \ \( E(U_2) = -\frac{1}{K} \).\ These calculations show the balance achieved through mixed strategies, where each player’s choice yields an expected payoff that reflects the fairness and balance of the game structure.

For a strategy to be in Nash equilibrium, it must be the best response to the other player's strategy. For mixed strategy Nash equilibrium, this means no player can improve their expected payoff by changing their strategy unilaterally. In our game, if each player chooses each integer from 1 to K with equal probability 1/K, neither player can alter their strategy to increase their expected payoff. This mutual best response condition ensures that the players' strategies are indeed in equilibrium.

The uniqueness of the mixed strategy Nash equilibrium in this game can be shown by examining any deviations. Assume player 1 assigns a positive probability to some action k. Player 2 must also assign a positive probability to k, or player 1 could alter their strategy to improve their payoff by not choosing k. This interplay enforces that both players must assign equal probability to each action. Any deviation from this equal probability would result in a worse expected outcome for at least one player, thus proving that no other mixed strategy Nash equilibria exist beyond the one where each choice is made with probability 1/K.