Question

Consider a game with \(n\) players. Simultaneously and independently, the players choose between \(\mathrm{X}\) and \(\mathrm{Y}\). That is, the strategy space for each player \(i\) is \(S_{i}=\\{\mathrm{X}, \mathrm{Y}\\}\). The payoff of each player who selects \(\mathrm{X}\) is \(2 m_{x}-m_{x}^{2}+3\), where \(m_{x}\) is the number of players who choose X. The payoff of each player who selects \(\mathrm{Y}\) is \(4-m_{y}\), where \(m_{y}\) is the number of players who choose \(\mathrm{Y}\). Note that \(m_{x}+m_{y}=n\). (a) For the case of \(n=2\), represent this game in the normal form and find the pure-strategy Nash equilibria (if any). (b) Suppose that \(n=3\). How many Nash equilibria does this game have? (Note: you are looking for pure-strategy equilibria here.) If your answer is more than zero, describe a Nash equilibrium. (c) Continue to assume that \(n=3\). Determine whether this game has a symmetric mixed-strategy Nash equilibrium in which each player selects \(\mathrm{X}\) with probability \(p\). If you can find such an equilibrium, what is \(p\) ?

Short answer

(n = 2): Pure Nash Equilibrium at (X, X). (n = 3): Multiple Nash Equilibria with 2X, 1Y combinations. Mixed-strategy equilibrium with X probability p = 1/2.

Step-by-step solution

Setting Up the Game for n=2

First, let's determine the payoff matrix for each player when \(n=2\). For this, we use the given payoff functions. If Player 1 and Player 2 both choose \(\mathrm{X}\), then \(m_x = 2\) and each player's payoff is \( 2(2) - 2^2 + 3 = 7\). If Player 1 chooses \(\mathrm{X}\) and Player 2 chooses \(\mathrm{Y}\), then Player 1's payoff is \( 2(1) - 1^2 + 3 = 4\) and Player 2's payoff is \(4 - 1 = 3\). When both choose \(\mathrm{Y}\), the payoff for each is \(4 - 2 = 2\). The complete payoff matrix becomes:\[\begin{array}{c|c|c} & \mathrm{X} & \mathrm{Y} \\hline\mathrm{X} & (7,7) & (4,3) \\hline\mathrm{Y} & (3,4) & (2,2) \\end{array}\]

Finding Pure-Strategy Nash Equilibria for n=2

Next, we identify any pure-strategy Nash equilibria by examining the payoff matrix. A Nash equilibrium occurs when each player's strategy is a best response to the other player's strategy. From the matrix:- When both players choose \(\mathrm{X}\), neither benefits from deviating, so \((\mathrm{X},\mathrm{X})\) is a Nash equilibrium.Thus, the pure-strategy Nash equilibrium for \(n=2\) is \((\mathrm{X},\mathrm{X})\).

Setting Up the Game for n=3

For \(n=3\), we consider cases where players choose different combinations of \(\mathrm{X}\) and \(\mathrm{Y}\), using similar calculations as for \(n=2\). When all players choose \(\mathrm{X}\), \(m_x = 3\), and each payoff is \( 2(3) - 3^2 + 3 = 3\). If two players choose \(\mathrm{X}\) and one chooses \(\mathrm{Y}\), those choosing \(\mathrm{X}\) get \( 2(2) - 2^2 + 3 = 7\), and the one choosing \(\mathrm{Y}\) gets \( 4 - 1 = 3\). When one chooses \(\mathrm{X}\) and two choose \(\mathrm{Y}\), the payoff for \(\mathrm{X}\) is \( 4\) and for \(\mathrm{Y}\) is \( 4 - 2 = 2\). Finally, all \(\mathrm{Y}\) is payout \( 2\).

Identifying Nash Equilibria for n=3

Check all combinations for Nash equilibria:- \((\mathrm{X},\mathrm{X},\mathrm{X})\): Each has \(3\), and switching to \(\mathrm{Y}\) doesn't improve payoffs beyond this group strategy.- \((\mathrm{X}, \mathrm{X}, \mathrm{Y})\) and permutations: Two get 7, and one gets 3; sole \(\mathrm{Y}\)-player switching improves payoff, balancing to \(3\), forming a Nash equilibrium when players swap.Thus, Nash equilibria occur at combinations with exactly two \(\mathrm{X}\) and one \(\mathrm{Y}\).

Checking for Mixed-Strategy Equilibria for n=3

For a symmetric mixed-strategy equilibrium where each player chooses \(\mathrm{X}\) with probability \(p\), the expected payoffs from choosing \(\mathrm{X}\) and \(\mathrm{Y}\) equations are equivalent. Calculate expected payoffs assuming the same strategy: for \(\mathrm{X}\), its expectation is based on potential opponents' combinations, and \(\mathrm{Y}\): direct losses reduce value. Solving expected payoffs equality yields \(p = \frac{1}{2}\) each, balancing between both strategic choices as equilibrium.

Key concepts

Game Theory

Game Theory is a field of study that examines the way individuals make strategic decisions in settings where the outcome depends not only on their own actions but also on the actions of others. It is used to analyze situations where players—such as individuals, companies, or countries—make decisions that together form an overall outcome. This process involves determining the best strategies or moves given the potential decisions of other players. In understanding Game Theory:

• Players: These are the decision-makers in the game. In our scenario, each decision-maker can choose between two actions, X or Y.
• Strategies: These are the plans of action players may take, which involve choosing X or Y based on the situation.
• Payoffs: These are the rewards or outcomes the players receive based on the actions chosen by each player.

Game Theory helps us predict the behavior of players by considering their preferences, strategies, and the likely decisions of others. This makes it a powerful tool in economics, politics, and business, ensuring one can anticipate competitive or cooperative behaviors.

Strategy Space

Strategy Space is the set of all possible strategies that a player can choose from in a game. For every player, the strategy space dictates their potential actions. In our exercise, each player can select between two actions: X and Y. These are the sole elements of the strategy space for each player. Understanding the Strategy Space involves:

• Finite Choices: Often, as in this example, the number of strategies available is limited, simplifying analysis.
• Assessing Outcomes: A player needs to consider the possible outcomes for each strategy in their space to make an informed decision.
• Player Independence: Each player's strategy space is independent of the others, even though outcomes do depend on all players' choices.

By clearly defining the strategy space, it becomes easier to analyze potential outcomes and strategies each player might take. It's a core part of structuring any game scenario in Game Theory, as it helps clarify the scope of analysis.

Mixed-Strategy Equilibrium

Mixed-Strategy Equilibrium is a concept in Game Theory where players randomize over possible strategies. In mixed-strategy equilibrium, each player selects different strategies according to a specific probability distribution. This approach ensures that players are indifferent between strategies since their expected payoffs are equal.In the exercise discussed, we explored finding a mixed-strategy equilibrium for three players:

• Probability: Each player selects X with probability \( p = \frac{1}{2} \).
• Balance: The mixed-strategy equilibrium achieves balance by making the expected payoffs of choosing X and Y equal. This means no player can increase their payoff by unilaterally changing their strategy.
• Symmetry: Since each player employs the same probability for their choices, the equilibrium is symmetric.

Mixed-strategy equilibria are particularly useful when no pure-strategy equilibrium exists or when players benefit from randomness in their strategy choice. It reflects the complex decision-making process when uncertainties or incomplete information are involved.

Payoff Matrix

The Payoff Matrix is a table used in Game Theory to illustrate the payoffs for each player given their strategic choices. It lays out the rewards for each combination of strategies chosen by players. In a two-player scenario, this matrix showcases the interdependencies of strategies and outcomes between the players. In our example with two players:

• Structure: The matrix provides a clear visualization of potential outcomes and payoffs for each decision combination (X, Y).
• Analysis: By examining the matrix, one identifies Nash equilibria where no player can unilaterally improve their payoff by changing their strategy.
• Outcomes: Each cell in the matrix represents the payoff for each player with specific strategy choices, enhancing clarity about which strategies yield the best outcomes.

Payoff Matrices are essential tools to analyze games systematically, helping identify equilibrium points and evaluate the best responses. They are widely used in strategic planning and prediction in various fields.