Question

Consider a guessing game with ten players, numbered 1 through \(10 .\) Simultaneously and independently, the players select integers between 0 and 10. Thus player \(i\) 's strategy space is \(S_{i}=\\{0,1,2,3,4,5,6,7,8,9,10\\}\), for \(i=1,2, \ldots, 10\). The payoffs are determined as follows: First, the average of the players' selections is calculated and denoted \(a\). That is, $$ a=\frac{s_{1}+s_{2}+\cdots+s_{10}}{10} $$ where \(s_{i}\) denotes player \(i\) 's selection, for \(i=1,2, \ldots, 10\). Then, player \(i\) 's payoff is given by \(u_{i}=(a-i-1) s_{i}\). What is the set of rationalizable strategies for each player in this game?

Short answer

The rationalizable strategies are influenced by the predicted average, likely ranging near the center of the interval, such as 5, contingent on players' estimates of others' choices.

Step-by-step solution

Understanding the Payoff

Each player receives a payoff calculated by the formula \( u_i = (a - i - 1)s_i \), where \( a \) is the average selection from all players, and \( s_i \) is player \( i\)'s selection. The term \( a - i - 1 \) indicates that each player aims to optimize their own score given all other players' choices.

Defining Rationalizable Strategies

A rationalizable strategy is one where it's the best response to some belief about what others are doing. In this context, a player's best response would depend on their prediction of the average \( a \). If a player believes others will pick low numbers (thus lowering \( a \)), they may benefit more from selecting a higher number to influence the average, balancing the term \((a - i - 1)\).

Exploring Strategy Extremes

There are extreme cases in this game. If every player selects 0, the average \( a = 0 \), making the payoff formula \( u_i = - (i + 1) imes 0 = 0 \). Conversely, if every player picks 10, \( a = 10 \) leading to \( u_i = (10 - i - 1) imes 10 \).

Analyzing Influence of Average on Strategy

Consider what happens if a player selects a number greater than the average. For \( s_i > a \), if \( a - i - 1 \) is negative or zero, the payoff is not maximized. Last, note \( s_i < a \) might be advantageous only when \( a \) itself is optimal based on beliefs about other players.

Iterating Reasonable Assumptions

Given the symmetry and the range of choices, any guess to skew \( a \) prioritizes middle ground approaches, aligning with majority predictions. Rationalizability narrows to strategy profiles where players reasonably guess average which leads to similar choices or patterns.

Key concepts

Rationalizable Strategies

In the guessing game, determining rationalizable strategies involves focusing on what is logically justifiable for each player based on their expectations of others' actions. A rationalizable strategy is a choice that makes sense when considering how others might play. Each player forms a belief about the others’ choices, and rationalizability means choosing a strategy that is the best response to these beliefs.

• Rationalizability encourages strategic thinking – formulating a plan based on expectations of how others will behave.
• If a player believes others will choose low numbers, selecting a higher number may serve as a rational strategy.
• Conversely, if higher numbers are anticipated, choosing a smaller value might maximize relative gain in regard to the average.

Therefore, the rationalizable strategy is dynamic, adjusting based on perceived tendencies of opponents’ choices.

Payoff Calculation

Understanding how payoffs work in this game is crucial to determining the best strategy. Each player's payoff is calculated using the formula:\[u_i = (a - i - 1) s_i\]Where:- \(u_i\) is the payoff for player \(i\).- \(a\) is the average of all selected numbers.- \(s_i\) is the number chosen by player \(i\).

• Players need to understand how their choice impacts both the average \(a\) and their payoff.
• The calculation emphasizes the importance of the average \(a\), as it directly affects the payoff.
• Choices that lower \(a\) may increase the chance of a higher payoff, depending on the position of player \(i\).

Hence, each player’s objective is to optimize their number to either influence or follow the average logically.

Best Response

A player's "best response" is the strategy that maximizes their payoff given their beliefs about the other players' strategies. This is central to deciding on rationalizable strategies.

• Identifying the best response involves predicting others’ average choices (\(a\)) and selecting \(s_i\) to maximize their payoff.
• Often, the best response requires balancing your number with expected averages to leverage the payoff formula effectively.
• If a player's predicted average differs greatly from others', their optimal number may diverge to adjust to this difference.

By consistently choosing the best response, players aim to secure the highest possible payoff given the existing dynamics of the game.

Average Selection

The concept of average selection plays a pivotal role in the game, as it forms the baseline for calculating payoffs. Each player's choice impacts and is impacted by this average. - The average \(a\) is the sum of all selections divided by the number of players:\[a = \frac{s_1 + s_2 + \cdots + s_{10}}{10}\]

• Choosing numbers influences \(a\), subsequently affecting the individual payoffs of the players.
• Predicting the average accurately allows players to make informed strategic choices.
• A common strategy is to aim for numbers that influence \(a\) to a favorable range for oneself.

Ultimately, mastering the concept of average selection enables a player to strategically determine their best response within the game's framework.