Question

Consider the following \(n\)-player game. Simultaneously and independently, the players each select either X, Y, or Z. The payoffs are defined as follows. Each player who selects X obtains a payoff equal to \(\gamma\), where \(\gamma\) is the number of players who select Z. Each player who selects Y obtains a payoff of \(2 \alpha\), where \(\alpha\) is the number of players who select X. Each player who selects \(Z\) obtains a payoff of \(3 \beta\), where \(\beta\) is the number of players who select Y. Note that \(\alpha+\beta+\gamma=n\). (a) Suppose \(n=2\). Represent this game in the normal form by drawing the appropriate matrix. (b) In the case of \(n=2\), does this game have a Nash equilibrium? If so, describe it. (c) Suppose \(n=11\). Does this game have a Nash equilibrium? If so, describe an equilibrium and explain how many Nash equilibria there are.

Short answer

For n=2, Nash equilibria are (Y, Z) and (Z, Y). For n=11, multiple equilibria exist with all players choosing the same strategy (X, Y, or Z).

Step-by-step solution

Game Formulation for n=2

For two players, the strategies can be X, Y, or Z for each player. We need to create a 3x3 payoff matrix where rows represent Player 1's strategies, and columns represent Player 2's strategies. Players' payoffs are calculated based on their strategy choice — selecting X gives them a payoff equal to the number of Z choices, Y gives twice the number of X choices, and Z gives three times the number of Y choices.

Construct the Payoff Matrix for n=2

For Player 1 and Player 2, the payoff matrix is constructed as: | | X | Y | Z | |------|---|---|---| | **X** |(0,0)|(2,2)|(2,0)| | **Y** |(0,2)|(0,0)|(0,3)| | **Z** |(3,0)|(0,3)|(0,0)| Each cell represents the payoffs (Player 1, Player 2) given their chosen strategies. The payoff is calculated based on how many other players choose the relevant strategies.

Evaluating Nash Equilibria for n=2

A Nash equilibrium occurs when neither player can benefit by unilaterally changing their strategy. From the matrix: - Cell (Y, Z) and (Z, Y) with payoffs (0, 3) for Player 1 and (3, 0) for Player 2 demonstrate that switching strategies doesn't improve their payoff unilaterally.

Analyzing n=11 Scenario

Now consider n=11. A Nash equilibrium occurs when each player’s choice remains the best response to the others' strategies if no one changes unilaterally. Let's check if all select the same strategy. We evaluate uniform strategies and possible mixed strategies. One clear equilibrium is where all players choose the same option due to symmetry.

Determine Multiple Nash Equilibria for n=11

Given the symmetrical nature of payoffs, each player choosing the same strategy ensures a balance where no player gains by switching. Hence, there are equilibria at X, X,...,X; Y, Y,...,Y; Z, Z,...,Z. By symmetry, as any mixed strategy distribution allowing players to optimize their payoffs against others' consistent choices would also be a Nash equilibrium.

Key concepts

Nash Equilibrium

In game theory, a Nash Equilibrium represents a situation where no player can gain by changing their strategy unilaterally, assuming the strategies of other players remain unchanged. Imagine a two-player game where each player has to choose a strategy (X, Y, or Z) without knowing the other player’s choice. The Nash Equilibrium is found when players settle on strategies where neither would benefit from deviating while the other's strategy stays the same.
For example, in a game of two players, if Player 1 and Player 2 both choose (Y, Z) or (Z, Y), they reach a Nash Equilibrium. In these combinations, any switch in strategy by one player only decreases their payoff, making this choice stable. The players' satisfaction with their choices, knowing what the other chooses, defines this state of equilibrium. This way, Nash Equilibrium is crucial for predicting likely outcomes in strategic interactions.

Payoff Matrix

A payoff matrix is a table that describes the payoffs players receive based on the strategies chosen by themselves and their opponents. It is a key element in analyzing decisions in a normal form game. Let’s consider two players with choices of X, Y, or Z.
In our scenario, the payoff matrix for n=2 players might look like this:

• If both choose X, the payoff is (0,0) as no Z is picked by anyone.
• If one chooses X and the other Y, the payoff is (0,2) for Player 1 and (2,2) for Player 2, since the number of X's chosen benefits Y players.
• Similarly, if Player 1 chooses Z and Player 2 chooses Y, Player 1 receives a payoff of 0, while Player 2 benefits with a higher payoff of 3.

This matrix helps visualize all potential outcomes and determine optimal strategies. It’s an essential tool for calculating Nash Equilibria, highlighting possible gains or losses from each strategy combination.

Normal Form Game

A normal form game is a framework used in game theory to present strategic interactions between multiple players simultaneously choosing their actions. Each row and column represents a player's strategy, and the corresponding cell details the payoff for that choice combination. This helps in understanding how different strategies impact the payoffs among players.
Consider our example with two players where each can choose X, Y, or Z. The normal form game here is showcased through a 3x3 matrix for clarity. Each cell contains the payoffs for both players given their chosen strategies. By analyzing this matrix, players can evaluate potential strategies. These decisions help predict the outcomes in strategic situations, making it easier to deduce Nash Equilibria and optimal strategy choices.

Strategy Choice

Strategy choice in game theory involves selecting an optimal action based on anticipating other players' decisions. It requires an understanding of both one’s own incentives and how they may align or conflict with opponents'.
In games where players choose between X, Y, or Z, the payoff dictates their choice. For example, a player may choose X if they believe many will opt for Z—as X's payoff grows with the number of Zs. Alternatively, if many choose X, Y becomes attractive due to its payoff multiplier. These choices depend on personal payoff maximization while predicting competitors' moves.
When n=11, strategy choice involves comprehensive analysis. Even a collective strategy, such as all choosing X, can become a Nash Equilibrium if every player perceives it as the best response to others doing the same, preserving payoff balance. Hence, informed strategy selection drives the dynamics in multiplayer games.