Question

Imagine a game in which players 1 and 2 simultaneously and independently select \(\mathrm{A}\) or \(\mathrm{B}\). If they both select \(\mathrm{A}\), then the game ends and the payoff vector is \((5,5)\). If they both select B, then the game ends with the payoff vector \((-1,-1)\). If one of the players chooses \(\mathrm{A}\) while the other selects \(\mathrm{B}\), then the game continues and the players are required simultaneously and independently to select positive numbers. After these decisions, the game ends and each player receives the payoff \(\left(x_{1}+x_{2}\right) /\left(1+x_{1}+x_{2}\right)\), where \(x_{1}\) is the positive number chosen by player 1 and \(x_{2}\) is the positive number chosen by player 2 . (a) Describe the strategy spaces of the players. (b) Compute the Nash equilibria of this game. (c) Determine the subgame perfect equilibria.

Short answer

(A, A) is both Nash and subgame perfect equilibrium.

Step-by-step solution

Understanding the Strategy Spaces

The strategy space for each player can be described as a first decision to choose either A or B. If both players end up in the scenario where one chooses A and the other chooses B, there is an additional decision to choose a positive number in continuation. So, the strategy space for each player is {Choose A, Choose B, and if reached, choose a positive number}.

Identifying Payoffs for Initial Moves

If both players choose A, the payoff vector is (5,5). If both choose B, the payoff vector is (-1,-1). If one chooses A and the other B, the game transitions to the next stage where they choose positive numbers to determine the payoff.

Finding Nash Equilibria in Initial Moves

For a pair (A, A), both players secure a payoff of 5 which is optimal against each other, leading to a Nash Equilibrium. Any deviation leads to a payoff less than 5. Both (B, B) is not a Nash Equilibrium as any player deviating to A can potentially earn more in the subsequent stage by choosing optimal numbers.

Analyzing the Continuation Game for Equilibrium

In the continuation stage, each player aims to maximize the payoff \[ \frac{x_{1}+x_{2}}{1+x_{1}+x_{2}} \]. Best responses typically involve choosing large numbers, reducing the other's possible gain, effectively creating a zero-sum perception. However, the payoff remains restricted since a large x makes \(x_1 + x_2 \) approach 1.

Determining Subgame Perfect Equilibria

A subgame perfect equilibrium involves players choosing strategies that are optimal based on continued play. Therefore, (A, A) is subgame perfect because it avoids the difficult-to-optimize second stage altogether. In any (A, B) or (B, A) scenario leading to the continuation game, rational players will select large numbers greedily enhancing complexity, but (A, A) assures maximum immediate payoff making it subgame perfect.

Key concepts

Strategy Spaces

In game theory, a player's strategy space is a comprehensive set of all possible actions they can take. In the given exercise, players 1 and 2 have two stages of decision-making, which defines their strategy spaces. Initially, they both must choose between two options: A or B. This decision is crucial as it determines whether the game ends immediately or continues to a subsequent stage.

• If both players choose A, the game concludes with excellent payoffs: 5 for both players.
• If both choose B, the game also ends but with negative payoffs, -1 each.
• However, if one chooses A while the other opts for B, the game moves to a second stage.

In this second stage, the strategy space expands. Each player now faces the challenge of selecting a positive number. This additional decision significantly affects the final payoffs. Players must consider their initial choice's impact on this continuation. Thus, the complete strategy space for each player includes choosing A or B initially, plus deciding on a positive number if they end up in the game's second stage.

Nash Equilibrium

Nash Equilibrium is a concept in game theory where each player, given the strategies of others, has no incentive to only alter their strategy. In our exercise, the Nash Equilibrium was found during the game's first stage, where both players deciding on (A, A) etablishes this equilibrium. Here's why:

• When both choose A, each receives a payoff of 5, which cannot be improved by unilaterally changing their strategy.
• If a player switches to B while the opponent chooses A, it leads to a continuation game with complex dynamics and potentially lower payoffs.
• Therefore, sticking with A as both players ensures they do not deviate to a worse-off outcome.

In contrast, (B, B) isn’t a Nash Equilibrium. Players choosing B can move to A to improve their payoff in the secondary game, where better outcomes might be possible. Thus, the equilibrium is best depicted by the (A, A) choice in the initial round, defining a mutual benefit ensuring no player wants to deviate.

Subgame Perfect Equilibrium

A subgame perfect equilibrium refines the idea of Nash Equilibrium by considering optimal strategies in every possible point throughout the game. In our game, the selection of (A, A) by both players results in an optimal outcome, qualifying as subgame perfect. This is due to its strategic path:

• By choosing A, players instantly receive the maximum possible payoff without the game entering the complexity of a second stage.
• The subgame beginning with one player choosing A and the other B leads to a continuation stage that becomes less advantageous due to its challenging payoff calculation.
• Therefore, rational players recognize that choosing (A, A) preemptively eliminates the intricate and suboptimal scenarios of the continuation game.

By opting for strategies that avert the second stage, the players ensure immediate gratification of maximum gains. Thus, the (A, A) outcome is not only a Nash Equilibrium but also the subgame perfect equilibrium, as it addresses the entirety of the game's decision points with optimal decisions.