Subgame perfection

Consider the following steps 1. Celia chooses how much care, it: E [0, 1], to take in programming her robot. This effort costs her 32/2. 2. Nature chooses whether the robot steps on Peter's pet salamander, leading to emotional harm to Peter of H > 0 (with probability 1 x). If the robot does step on the salamander then there is a chance of 1r that Celia will be identied as the culprit. o If there is no accident (the salamander is not stepped on), then Celia's payoff is V m2/ 2. Peter and Luke both get zero. 0 If there is an accident, but Celia is not identied as the culprit, then Celia gets V neg/2. Peter gets H. Luke gets zero. a If there is an accident, and Celia is identied as the culprit, then Luke (the judge) decides a level of compensation D E R... for Celia to pay Peter. Celia gets VmZ/2D. Peter gets DH. Luke gets (,8HD)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 m in step 1. c) What will Celia's rst-order condition be in step 1? Your answer should be an expression that involves D. d) What is Luke's rst-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 ((1), 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. g) Take a rstorder condition with respect to 1:. Find an expression for the socially optimal level of 3;, according to the value judgement we made in question (f ) h) What would have to equal, in order for Celia to choose the socially op timal level of :c in a Subgame Perfect Equilibrium? We are looking for a mathematical expression, rather than a specic number. i) What is the intuition for the result you got in (h)? j) Finally, imagine a different value judgement, according to which the socially optimal outcome would maximize the sum of all three human expected pay offs (ie. including Luke's). Write down the optimisation problem that would dene this optimal outcome. You don't actually have to solve it, just write down the