1. Consider a team with n individuals, n > 2. Each individual must choose whether to put in effort (e) or shirk (s), i.e. must choose from {e, s}. The team succeeds if and only if all n individuals put in effort. The cost of effort for each individual, c; is random, and is drawn uniformly from the interval [0, 1]. That is, each c; is independently drawn, and uniformly distributed on [0, 1]. An individual observes his own cost realization, but not the cost realizations of anyone else in the team. Shirking has zero cost. Each individual gets a benefit of v > 1 if the team succeeds, and zero if it fails. The overall payoff to an individual equals the benefit from the success or failure of the team, minus the cost of effort (if she puts in effort). An equilibrium is symmetric if every player chooses the same strategy. a) A player is said to be pivotal, in this context, if his effort decision determines the public outcome, i.e. whether the team succeeds or not. When is player i pivotal in this game? (That is, set out the profile of action choices for the other players where player i is pivotal.) a) Solve for a symmetric Bayes Nash equilibrium of this game where each player chooses e if and only if his cost is below some threshold c, where 0 < c < 1. b) Are there any other symmetric Bayes Nash equilibria? c) Does the game have an asymmetric Bayes Nash equilibrium, where some players choose s irrespective of their cost, while others choose e irrespective of their cost? d) Consider a more general situation where the team succeeds if and only if at least k of the n individuals put in effort, where 1 ≤ k

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Hi, just some homework help. Please don't use the responses found from chegg, they are not thorough and are incorrect.
The first a) point is just part of the question, doesn't need to be answered.

Thank you!

$**1. Consider a team with $ n $ individuals, $ n > 2 $.** Each individual must choose whether to put in effort ($ e $) or shirk ($ s $), i.e., must choose from $\{e, s\}$. The team succeeds if and only if all $ n $ individuals put in effort. The cost of effort for each individual, $ c_i $, is random, and is drawn uniformly from the interval $[0, 1]$. That is, each $ c_i $ is independently drawn, and uniformly distributed on $[0, 1]$. An individual observes his own cost realization, but not the cost realizations of anyone else in the team. Shirking has zero cost. Each individual gets a benefit of $ v > 1 $ if the team succeeds, and zero if it fails. The overall payoff to an individual equals the benefit from the success or failure of the team, minus the cost of effort (if he puts in effort). An equilibrium is **symmetric** if every player chooses the same strategy. a) A player is said to be **pivotal**, in this context, if his effort decision determines the public outcome, i.e., whether the team succeeds or not. When is player $ i $ pivotal in this game? (That is, set out the profile of action choices for the other players where player $ i $ is pivotal.) a) Solve for a symmetric Bayes Nash equilibrium of this game where each player chooses $ e $ if and only if his cost is below some threshold $ c^* $, where $ 0 < c^* < 1 $. b) Are there any other symmetric Bayes Nash equilibria? c) Does the game have an asymmetric Bayes Nash equilibrium, where some players choose $ s $ irrespective of their cost, while others choose $ e $ irrespective of their cost? d) Consider a more general situation where the team succeeds if and only if at least $ k $ of the $ n $ individuals put in effort, where $ 1 \leq k < n $. When is player $ i $ pivotal in this game?$

Transcribed Image Text:**1. Consider a team with $ n $ individuals, $ n > 2 $.** Each individual must choose whether to put in effort ($ e $) or shirk ($ s $), i.e., must choose from $\{e, s\}$. The team succeeds if and only if all $ n $ individuals put in effort. The cost of effort for each individual, $ c_i $, is random, and is drawn uniformly from the interval $[0, 1]$. That is, each $ c_i $ is independently drawn, and uniformly distributed on $[0, 1]$. An individual observes his own cost realization, but not the cost realizations of anyone else in the team. Shirking has zero cost. Each individual gets a benefit of $ v > 1 $ if the team succeeds, and zero if it fails. The overall payoff to an individual equals the benefit from the success or failure of the team, minus the cost of effort (if he puts in effort). An equilibrium is **symmetric** if every player chooses the same strategy. a) A player is said to be **pivotal**, in this context, if his effort decision determines the public outcome, i.e., whether the team succeeds or not. When is player $ i $ pivotal in this game? (That is, set out the profile of action choices for the other players where player $ i $ is pivotal.) a) Solve for a symmetric Bayes Nash equilibrium of this game where each player chooses $ e $ if and only if his cost is below some threshold $ c^* $, where $ 0 < c^* < 1 $. b) Are there any other symmetric Bayes Nash equilibria? c) Does the game have an asymmetric Bayes Nash equilibrium, where some players choose $ s $ irrespective of their cost, while others choose $ e $ irrespective of their cost? d) Consider a more general situation where the team succeeds if and only if at least $ k $ of the $ n $ individuals put in effort, where $ 1 \leq k < n $. When is player $ i $ pivotal in this game?

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