Question

Consider a version of the "social unrest" game analyzed in Chapter 8 (including Exercise 8) with incomplete information. Two people (players 1 and 2 ) have to simultaneously choose whether to protest \((\mathrm{P})\) or stay home (H). A player who stays home gets a payoff of 0 . Player \(i\) ' \(s\) payoff of protesting is determined by this player's protest value \(x_{i}\) and whether the other player also protests. Specifically, if player \(i\) decides to protest, then her payoff is \(x_{i}-\frac{1}{3}\) if the other player also protests, whereas her payoff is \(x_{i}-\frac{2}{3}\) if the other player stays home. Each player knows her own protest value but does not observe that of the other player. Thus, \(x_{i}\) is player \(i\) 's type. Assume that \(x_{1}\) and \(x_{2}\) are independently drawn from the uniform distribution on \([0,1]\). (a) Calculate the Bayesian Nash equilibrium of this game. (Hint: Note that each player's strategy is a function from her type to \(\\{\mathrm{P}, \mathrm{H}\\}\). Consider cutoff strategies, where player \(i\) will protest if and only if \(x_{i} \geq y_{i}\), for some constant \(y_{i}\).) Document your analysis and report the equilibrium cutoffs \(y_{1}^{*}\) and \(y_{2}^{*}\). (b) What is the set of rationalizable strategies in this game? Without providing mathematical details, try to give reasoning based on considering (i) a lower bound on the types that protest regardless of their beliefs and (ii) an upper bound on the types that stay home regardless of their beliefs. (c) Consider an \(n\)-player version of the game in which the payoff of protesting is \(x_{i}-\frac{1}{3}\) if at least \(m\) of the other players also protests, and it is \(x_{i}-\frac{2}{3}\) if fewer than \(m\) of the other players also protests. Can you find values of \(n\) and \(m\) for which this game has multiple Nash equilibria?

Short answer

The Bayesian Nash equilibrium cutoffs are \( y_1^* = y_2^* = \frac{1}{2} \). Rationalizable strategies start at type \( \frac{2}{3} \). In n-player, multiple equilibria depend if \( n = 3 \) and \( m = 2 \).

Step-by-step solution

Define the Players' Payoffs

Player 1 and Player 2 have the option to either protest (P) or stay home (H). If they protest, their payoff depends on whether the other player also protests. Each player gets a payoff of 0 for staying home. Player i's protest payoff is \( x_i - \frac{1}{3} \) if the other player protests, and \( x_i - \frac{2}{3} \) if the other player stays home.

Define the Strategy and Type

Each player's decision to protest is based on their type \( x_i \), which is drawn from a uniform distribution on \([0, 1]\). Therefore, we consider strategies based on cutoffs, where Player i protests if \( x_i \geq y_i \), with \( y_i \) being the cutoff for Player i.

Derive the Best Response

Given the strategy of Player 2 using cutoff \( y_2 \), the best response for Player 1 is to choose P if \( x_1 - \frac{1}{3} \Pr(x_2 \geq y_2) \geq -\frac{2}{3}(1 - \Pr(x_2 \geq y_2)) \). Similarly, we derive the best response for Player 2.

Solve for the Bayesian Nash Equilibrium

Equilibrium occurs when strategies are mutual best responses. Assuming cutoff strategies, set \( x_1 \) equal to the critical cut point where Player 1 is indifferent, solving \( x_1 - \frac{1}{3} \times (1-y_2) = x_1 - \frac{2}{3} \times y_2 \). Solving yields \( y_1 = y_2 = \frac{1}{2} \). Correspondingly, do this reasoning for Player 2 and verify consistency.

Identify Rationalizable Strategies

With payoff differences of \( \frac{1}{3} \), only individuals with types \( \geq \frac{2}{3} \) will protest without relying on others based on expectations, ensuring payoff gain. Conversely, risk-averse individuals of types \( < \frac{1}{3} \) will prefer staying home regardless of others due to high protest perceived risk.

Analyze the N-Player Game

For an \( n \)-player frame, consider if several equilibria exist due to different dynamics. For example, if \( n = 3 \) and \( m = 2 \), a player benefits if a crucial mass protests. Different \( y_n \) such as all or few thresholds leads to multiple equilibria as \( m \) manipulates consensus thresholds.

Key concepts

Game Theory

In the realm of advanced mathematics and economics, Game Theory serves as a powerful tool to analyze strategic interactions. Imagine a strategic game where players must make decisions without knowing the choices of others. This is the core of Game Theory, a framework that models scenarios where players' outcomes depend on their strategies and those of others.
In this particular exercise, we see two players in the 'social unrest' game. They can choose to protest or stay home. Their decision isn't just about personal gain; it hinges on what they anticipate the other will do.

• Players: Individuals making choices (here, protest or stay home).
• Payoffs: Outcomes associated with each strategy chosen by the players.
• Strategies: Plans of action prepared by the players based on expected actions of others.

These elements allow players to try and maximize their payoffs, based on rational predictions of others' behavior. The goal is to understand and predict these interactions logically and strategically.

Incomplete Information

The concept of Incomplete Information in Game Theory is fascinating. It reflects real-world scenarios where individuals lack full knowledge about others’ traits or intentions. In our social unrest model, players know their own willingness to protest, called the 'type,' but they don't know the type of their counterpart.
Let's break this down:

• Type: Each player's protest value, represented as a variable, which is unknown to the opponent.
• Information Symmetry: While players know their own type, they have no insight into the other's type. Here, symmetry lies in the fact both players have similar knowledge deficits.
• Uniform Distribution: The protest types are drawn from a uniform distribution, meaning every protest value between 0 and 1 is equally likely.

Using this framework, players form strategies despite these uncertainties, making predictions purely from available information.

Rationalizable Strategies

Rationalizable strategies are key in making sense of decisions in environments with uncertainty. They represent strategies that remain logical under common knowledge assumptions about others' rationality. Each player considers the most rational approach based on the expectations of the other's behavior.
In our game:

• Lower Bound: Types that will protest regardless of beliefs. For types above \( rac{2}{3} \), the advantage of protesting secures a gain without concern for the others’ actions.


• Upper Bound: Types that will stay home without regard to beliefs. Typically, those below \( rac{1}{3} \), to avoid the risk of losing payoff by protesting unnecessarily.

Rationalizability provides a spectrum: from risk-taking behavior to conservative awareness, guiding players in decision-making amidst unknowns.

Payoff Calculation

Calculating payoffs is central to determining the best course of action in any game theoretic model. Payoffs indicate the success of a strategy chosen by each player.Understanding the exercise:

• If a player stays home, the payoff is zero.
• If a player protests, the payoff depends on the opponent's choice:
• • If both protest, payoffs align with \( x_i - rac{1}{3} \).
  • If the player protests alone, the payoff reduces to \( x_i - rac{2}{3} \).

The task then becomes calculating when each choice yields the highest payoff, by comparing these scenarios and evaluating potential rivals’ choices. Mastering these calculations ensures that players are utilizing their types strategically, deriving maximum benefit from their actions in the game.