H and I please g) Take a first-order condition with respect to z. Find an expression for thesocially optimal level of z, according to the value judgement we made inquestion (f).h) What would have to equal, in order for Celia to choose the socially op-timal level of x in a Subgame Perfect Equilibrium? We are looking for amathematical expression, rather than a specific number.i) What is the intuition for the result you got in (h)?j) Finally, imagine a different value judgement, according to which the sociallyoptimal outcome would maximize the sum of all three human expected pay-offs (ie. including Luke's). Write down the optimisation problem that woulddefine this optimal outcome. You don't actually have to solve it, just writedown the problem. Consider the following steps1. Celia chooses how much care, z = [0, 1], to take in programming her robot.This effort costs her z²/2.2. Nature chooses whether the robot steps on Peter's pet salamander, leadingto emotional harm to Peter of H> 0 (with probability 1-x). If the robotdoes step on the salamander then there is a chance of that Celia will beidentified as the culprit.• If there is no accident (the salamander is not stepped on), then Celia'spayoff is V - r²/2. Peter and Luke both get zero.• If there is an accident, but Celia is not identified as the culprit, thenCelia gets V-²/2. Peter gets -H. Luke gets zero.• If there is an accident, and Celia is identified as the culprit, then Luke(the judge) decides a level of compensation D € R+ for Celia to payPeter. Celia gets V-2²/2-D. Peter gets D-H. Luke gets - (3H-D)².Now answer the following questions.a) Draw a game tree to represent this model.b) Write down Celia's expected payoff when she chooses x in step 1.c) What will Celia's first-order condition be in step 1? Your answer should bean expression that involves D.d) What is Luke's first-order condition? What decision rule will Luke follow instep 3?e) Now substitute the equilibrium expression for D that you obtained in ques-tion (d), into Celia's condition that you obtained in (c).f) Now we turn to social optimality rather than equilibrium. Consider thevalue judgement that the optimal level of care is the one that maximises theexpected sum of Celia's and Peter's payoffs. According to this approach, wecan ignore Luke's payoffs (as well as any payoff to the salamander or therobot) when considering social optimality. Write down this expected sum.

The socially optimum level of x (Celia's choice) is determined by maximizing the expected sum (ES)…

Consider the following steps 1. Celia chooses how much care, z = [0, 1], to take in programming her robot. This effort costs her z²/2. 2. Nature chooses whether the robot steps on Peter's pet salamander, leading to emotional harm to Peter of H> 0 (with probability 1-r). If the robot does step on the salamander then there is a chance of that Celia will be identified as the culprit. • If there is no accident (the salamander is not stepped on), then Celia's payoff is V - 1²/2. Peter and Luke both get zero. • If there is an accident, but Celia is not identified as the culprit, then Celia gets V-2²/2. Peter gets -H. Luke gets zero. • If there is an accident, and Celia is identified as the culprit, then Luke (the judge) decides a level of compensation DER for Celia to pay Peter. Celia gets V-2²/2-D. Peter gets D-H. Luke gets-(3H-D)². Now answer the following questions. a) Draw a game tree to represent this model. b) Write down Celia's expected payoff when she chooses z in step 1. c) What will Celia's first-order condition be in step 1? Your answer should be an expression that involves D. d) What is Luke's first-order condition? What decision rule will Luke follow in step 3? e) Now substitute the equilibrium expression for D that you obtained in ques- tion (d), into Celia's condition that you obtained in (c). f) Now we turn to social optimality rather than equilibrium. Consider the value judgement that the optimal level of care is the one that maximises the expected sum of Celia's and Peter's payoffs. According to this approach, we can ignore Luke's payoffs (as well as any payoff to the salamander or the robot) when considering social optimality. Write down this expected sum.

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.10P