Question

Fudenberg and Tirole (1992) develop a game of stag-hunting based on an observation originally made by Rousseau. The two players in the game may either cooperate in catching a stag or each may set out on his own to catch a hare. The payoff matrix for this game is given by a Describe the Nash equilibria in this game. b. Suppose \(B\) believes that \(A\) will use a mixed strategy in choosing how to hunt. How will 5 's optimal strategy choice depend on the probability \(A\) will play stag? c. Suppose this game is expanded to \(n\) players (the game Rousseau had in mind) and that all \(n\) must cooperate in order for a stag to be caught. Assuming that the payoffs for one specific player, say \(B\), remain the same and that all the other \(n-1\) players will opt for mixed strategies, how will \(B\) 's optimal strategy depend on the probabilities with which each of the other players plays stag? Explain why cooperation seems less likely in this larger game.

Short answer

Answer: Player B's optimal strategy choice depends on the expected payoffs of playing Stag versus Hare. If the expected payoff for choosing Stag (2p) is greater than the expected payoff for choosing Hare (1), it is optimal for B to play Stag. If the expected payoff for choosing Stag (2p) is lower than the expected payoff for choosing Hare (1), it is optimal for B to play Hare. If the expected payoffs of both strategies are equal, B is indifferent between playing Stag or Hare.

Step-by-step solution

Part a: Identifying the Nash Equilibria

First, let's identify the Nash equilibria in this game. A Nash equilibrium occurs when no player has an incentive to change their strategy, given that the other players maintain their current strategies. In this case, the payoff matrix is as follows: A Stag Hare Stag (2,2) (0,1) B Hare (1,0) (1,1) There are two pure-strategy Nash equilibria: (Stag, Stag) and (Hare, Hare). In both equilibria, neither player has an incentive to change their strategy, since doing so would result in a lower payoff.

Part b: B's Optimal Strategy Choice if A Plays a Mixed Strategy

Suppose Player A adopts a mixed strategy, playing Stag with probability \(p\) and Hare with probability \(1-p\). Given A's strategy, B can choose to play Stag or Hare according to the following expected payoffs: Expected payoff for B if choosing Stag: \(2p + 0(1-p) = 2p\) Expected payoff for B if choosing Hare: \(1p + 1(1-p) = p + 1 - p = 1\) B's optimal strategy choice depends on the expected payoff of playing Stag versus Hare. If \(2p > 1\), it is optimal for B to play Stag. If \(2p < 1\), it is optimal for B to play Hare. If, however, \(2p=1\), B is indifferent between playing Stag or Hare since the expected payoffs of both strategies are equal.

Part c: B's Optimal Strategy Choice in an n-Player Game

Now, let's generalize the game to \(n\) players, where all \(n\) players need to cooperate for a stag to be caught. We can assume that the payoffs for B remain the same and all the other \(n-1\) players will opt for mixed strategies. Let \({q_i}\) represent the probability that player \(i\) (different from B) plays Stag. In this case, B's expected payoffs for each strategy are as follows: Expected payoff for B if choosing Stag: \(2(\Pi_{i=1}^{n-1} q_i)\) Expected payoff for B if choosing Hare: \(\displaystyle 1\) It is important to note that the expected payoff for B if they choose Hare is constant and does not depend on the other players' strategies. Comparing the expected payoffs, we find that B will choose Stag if their expected payoff for choosing Stag is greater than the expected payoff for choosing Hare, or when \(2(\Pi_{i=1}^{n-1} q_i) > 1\). As \(n\) gets larger, the likelihood that all the other players play Stag (\((\Pi_{i=1}^{n-1} q_i)\)) becomes smaller. This is because each player's decision to play Stag is multiplied together. As a result, it becomes increasingly difficult for the product to be greater than \(\frac{1}{2}\), meaning that cooperation is less likely in a larger game.

Key concepts

Nash Equilibrium

In game theory, a Nash equilibrium refers to a situation where each player, given the choices of others, has no incentive to change their strategy. This means that all players are making the most optimized decisions as long as the decisions of the others remain unchanged. There are two types of Nash equilibria: pure strategy and mixed strategy.
- In **pure strategy Nash equilibrium**, each player consistently chooses one strategy. In the stag hunt game, the pure strategy Nash equilibria are (Stag, Stag) and (Hare, Hare). Here, both players choose to either both hunt a stag or both hunt hares, receiving consistent payoffs without any incentive to deviate.
- Identifying a Nash equilibrium involves considering the payoff matrix where the strategic choice of each player is mapped. If neither player can benefit by unilaterally changing their strategy, equilibrium is reached. Understanding and identifying Nash equilibrium is crucial in analyzing strategic interactions in various scenarios.

Mixed Strategy

A mixed strategy involves players randomizing over possible moves, assigning a probability to each. This strategy is used when there isn't a dominant pure strategy equilibrium, or when a player wants to keep their opponents indifferent to its moves.
- In the stag hunt game, if player A adopts a mixed strategy by playing Stag with probability \( p \) and Hare with probability \( 1-p \), player B must decide their strategy based on expected payoffs. B calculates the benefits of each option by considering these probabilities.
- Key to understanding mixed strategies is the concept of expected payoff, which helps players evaluate potential benefits of different choices combined with the probability each occurs. By comparing expected payoffs, players can make informed decisions on their best strategy course.

Cooperative Game

In cooperative games, players work together to achieve better payoffs than playing independently. This contrasts with non-cooperative games where players make decisions selfishly. The core idea in cooperative games is forming coalitions that can enforce cooperative agreements.
- The stag hunt game is a classic example of a cooperative game. Players opt to hunt the stag collectively or go it alone hunting hares. The highest group payoffs come from cooperation, where both decide to hunt the stag.
- Cooperation can be threatened by mistrust or fear of one player defecting, leading to suboptimal outcomes like both opting for the less rewarding hare. Thus, creating a stable cooperative strategy involves ensuring all players trust each other to stick to the joint plan.

Stag Hunt Game

The stag hunt game is a classic model in game theory illustrating a mix between the dilemmas of coordination and cooperation among players.
- The origins trace back to Jean-Jacques Rousseau's description of two individuals going on a hunt. Collaborating for a stag yields higher rewards. However, hunting independently for hares provides a guaranteed, though smaller, payoff.
- The game's essence lies in the trust and assurance needed amongst players to collaborate for the greater benefit. The choice involves risk; if one cooperates and the other defaults, the cooperating player gets nothing. Hence, equilibrium points, whether pure or mixed strategies, drastically influence the decision-making process.
- Understanding the stag hunt game provides insight into strategic decisions in real-world scenarios where trust and cooperation are essential for optimal outcomes.