Question

Consider the following game played between 100 people. Each person \(i\) chooses a number \(s_{i}\) between 20 and 60 (inclusive). Let \(a_{-i}\) be defined as the average selection of the players other than player \(i\); that is, \(a_{-i}=\left(\Sigma_{j \neq i} s_{j}\right) / 99\). Person i's payoff is \(u_{i}(s)=100-\left(s_{i}-\frac{3}{2} a_{-i}\right)^{2}\). For instance, if the average of the \(-i\) players' choices is 40 and player \(i\) chose 56 , then player \(i\) would receive a payoff of \(100-\left(56-\frac{3}{2} \cdot 40\right)^{2}=100-4^{2}=84\). (a) Find an expression for player \(i\) 's best response to her belief about the other players' strategies as a function of the expected value of \(a_{-i}\), which we can denote \(\bar{a}_{-i} .\) What is the best response to \(\bar{a}_{-i}=40\) ? (b) Use your answer to part (a) to determine the set of undominated strategies for each player. Note that dominated strategies are those that are not best responses (across all beliefs). (c) Find the set of rationalizable strategies. Show what strategies are removed in each round of the deletion procedure. (d) If there were just two players rather than 100 , but the definition of payoffs remained the same, would your answers to parts (a) and (b) change? (e) Suppose that instead of picking numbers between 20 and 60 , the players can select numbers between \(y\) and 60 , where \(y\) is a fixed parameter between 0 and 20 . Calculate the set of rationalizable strategies for any fixed \(y\). Note how the result depends on whether \(y>0\) or \(y=0\).

Short answer

Best response is \(s_i = \frac{3}{2} \bar{a}_{-i}\); for \(\bar{a}_{-i}=40\), choose 60. Undominated are between \(40/3\) and 60; rationalizable converge near 60.

Step-by-step solution

Define the Best Response Function

For player \(i\), the goal is to maximize their payoff \(u_i(s) = 100 - \left(s_i - \frac{3}{2} \bar{a}_{-i}\right)^2\). To find the best response \(BR_i(\bar{a}_{-i})\), differentiate the payoff function with respect to \(s_i\) and set the derivative to zero: \[\frac{d}{ds_i} \left(100 - \left(s_i - \frac{3}{2} \bar{a}_{-i}\right)^2\right) = 0.\] This gives \(s_i = \frac{3}{2} \bar{a}_{-i}\).

Evaluate the Best Response to \(\bar{a}_{-i}=40\)

Substitute \(\bar{a}_{-i}=40\) into the best response function from Step 1 to find the specific strategy: \[s_i = \frac{3}{2} \times 40 = 60.\] Thus, the best response to \(\bar{a}_{-i} = 40\) is \(s_i = 60\).

Identify Undominated Strategies

Undominated strategies are the strategies resulting from the best response function for any \(\bar{a}_{-i}\). Since \(s_i = \frac{3}{2} \bar{a}_{-i}\), \(s_i\) must be chosen between 20 and 60. Check cases: - If \(\bar{a}_{-i} \leq \frac{40}{3}\), then \(s_i\) would be less than 20, hence the effective lower bound is \(40/3\) to ensure \(s_i\geq 20\). - Maintain \(s_i \leq 60\).Final set of undominated strategies is \(\frac{40}{3} \leq \bar{a}_{-i} \leq 40\).

Finding Rationalizable Strategies

Start with possible strategies \(s_i\) between 20 and 60, the range \(\frac{40}{3} \leq \bar{a}_{-i} \leq 40\) offers solutions: - Round 1: If all players choose above \(\frac{40}{3}\), it favors convergence towards \(\bar{a}_{-i} = 40 \). - Round 2: Remove strategies resulting in lower payoffs further from 60. The rationalizable strategies are consistent with undominated strategies pathway to remaining choices near \(s_i=60\).

Two Players Scenario Analysis

With two players, each calculates the other's choice, simplifying to \(\bar{a}_{-i} = s_{-i}\). The best response and dominations concept remain unchanged per player individually maximizing the same payoff form, experiencing convergence of strategy at \(s_i = 60\). Thus answers to parts (a) and (b) do not change.

Impact of Varying Lower Bound \(y\)

Extend the range to \(y\leq s_i\leq 60\). Redefine possible \(\bar{a}_{-i}\) based on \(\frac{3y}{2} \leq s_i \leq \min(60, \frac{3}{2}60)\), driven by tighter boundaries. - If \(y = 0\), the strategy set includes full spectrum until dominated near 60. - If \(0 < y < 20\), lower limit \(y\) bounds affect for different \(\bar{a}_{-i}\) narrowing rationalizable trajectory closer towards the upper bound \(60\).

Key concepts

Best Response Function

In game theory, the best response function describes how a player should choose their strategy based on their beliefs about other players' strategies. For each player, the aim is to maximize their payoff by adjusting their strategy optimally according to what they expect others to do.

• In this exercise, each player's payoff is defined by the function: \(u_i(s) = 100 - \left(s_i - \frac{3}{2} \bar{a}_{-i}\right)^2\).
• To find the optimal response, we differentiate this payoff with respect to the chosen strategy \(s_i\). Setting the derivative to zero isolates \(s_i\) where the payoff is maximized: \(s_i = \frac{3}{2} \bar{a}_{-i}\).

This means that players adjust their choice \(s_i\) relative to their belief about the average choice of others, \(\bar{a}_{-i}\). A specific example given is when \(\bar{a}_{-i} = 40\), leading to the best response \(s_i = 60\). This method helps players act in their best interest given the expected actions of others, thereby aligning with the broader game dynamics.

Dominated Strategies

Dominated strategies are those that a player will never choose, because there are better options available regardless of what the others do. In our exercise, determining dominated strategies involves understanding the range of \(s_i\) based on the best response function.

• The optimal \(s_i = \frac{3}{2} \bar{a}_{-i}\) dictates that \(s_i\) depends directly on \(\bar{a}_{-i}\). To avoid any dominated strategies, \(s_i\) must lie within the feasible bounds of the game, which are 20 to 60.
• Specific conditions arise: if \(\bar{a}_{-i} \leq \frac{40}{3}\), \(s_i\) falls below 20, violating the allowable range.

Hence, the effective undominated strategy set is \(\frac{40}{3} \leq \bar{a}_{-i} \leq 40\). Any strategy outside this set is dominated because it cannot be an optimal response under any belief about others' actions. Players naturally avoid such strategies to ensure maximized payoffs.

Rationalizable Strategies

Rationalizable strategies are those that can be expected to be chosen based on mutual reasonable assumptions about rational behavior, given that all players are trying to maximize their payoffs.

• We start with all potential choices between 20 and 60. As the players iteratively update their beliefs and eliminate dominated strategies, the range narrows.
• In this exercise, the remaining strategies from the previous round (the undominated ones) automatically become the rationalizable ones — they support the leading payoff, aligning closely around \(s_i=60\).

In essence, players focus on choices that rational counterparts might make, reinforcing their position on decisions that are consistent among rational thinkers, usually converging towards maximizing strategies such as \(s_i=60\), in synergy with the best response path.

Payoff Maximization

Payoff maximization is a player's primary objective in game theory, where each participant tries to adopt strategies that ensure the highest possible returns under the circumstances.

• In our example, the payoff is expressed through a quadratic loss function: \(u_i(s) = 100 - (s_i - \frac{3}{2} \bar{a}_{-i})^2\). Here, maximizing it means minimizing the squared term, which occurs when the term inside equals zero.
• By choosing \(s_i = \frac{3}{2} \bar{a}_{-i}\), each player aligns their strategy with this condition, leading directly to payoff maximization.

Achieving maximum payoff involves strategic decision-making that is consistent with the best response function. Players constantly adjust to others, seeking the highest rewards while maintaining rational behavior across all potential scenarios offered by the game.