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## Problems from Chapter 8

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**Describe strategies in the subgame perfect equilibrium.**What does pope do? What does Galileo do at each of his decision nodes? What does the inquisitor do?**Strategic Form**Three Players.—What are the strategies?**Strategic form if Pope refers case**Payoffs if Pope Refers the Case to the Inquisition Galileo’s Strategy Inquisitor’s Strategy Payoffs listed x,y,z means x for Pope, y for inquisitor, z for Galileo**Strategic form if Pope doesn’t refer case**Payoffs if Pope Does not refer the Case to the Inquisition Galileo’s Strategy Inquisitor’s Strategy Payoffs listed x,y,z means x for Pope, y for inquisitor, z for Galileo**Some Nash equilibria**• Pope refers, Galileo will confess before torture and will confess if tortured, Inquisitor will torture if Galileo doesn’t confess beforehand. • Pope refers, Galileo confess before torture, would not confess if tortured, Inquisitor will torture. • Pope doesn’t refer, Galileo will not confess before torture, wouldn’t would confess if tortured, Inquisitor would torture if G doesn’t confess.**More Nash equilibria**• Pope doesn’t refer the case. Galileo would not confess either before or after torture. Inquisitor would torture. • Pope doesn’t refer the case. Galileo would not confess either before or after torture. Inquisitor would not torture.**Piquant facts for fans of the waterboard.**• Galileo would rather confess before being tortured than be tortured. • But if he is tortured, he would rather not confess. • Pope would like Galileo to confess without being tortured. • Pope would also be happy if Galileo is tortured and confesses. • But Pope would rather not refer the case if Galileo would be tortured and not confess. • So Galileo is not brought before the Inquisition.**Goblins Gold Problem**• Seven goblins, A,B,E,G,K,R, and U, divide 100 gold pieces • A proposes an allocation of the gold. If at least half vote yes, allocation is accepted • If not, A is sent away. Then B proposes and the remaining Goblins vote. • This process continues down the list until either a • proposal is accepted or only U is left in which case • U gets all the gold.**Working backwards**If only two goblins are left, R and U, then it is R’s turn to propose. R can make anything pass by voting for it. So he will choose 100 for R and 0 for U. Suppose there are 3 goblins left, K, R, and U, then it is K’s turn to propose. K’s best strategy is to offer 1 to U, 0 to R and 99 to himself. Why?**What’s the pattern?**• What if there were 100 goblins? • How about 200? • How about 300? • What does this problem have to do with subgame perfection?**Taking Turns in the Dark:(Subgame perfection with incomplete**information ) Econ 171**Subgame Perfection with Imperfect Information**How can the notion of subgame perfection help us if there is incomplete information? Look back at kidnapper game**What is a subtree of a game?**• It is a non-terminal node, together with all of the nodes that could be reached from this node. • Incidentally, a Proper Subtreeis a subtree that is not the entire game.**What is a regularsubtree of a game?**• It is a subtree starting from one of the nodes of the game such that this subtree contains an entire information set if it contains at least one node from that set. • A subgame is defined to be a regular subtree together with the associated payoffs. • A proper subgame of a game is a subgame that does not contain the entire game. (by analogy to a proper subset of a set)**Subgame perfection**• In a game with imperfect information, a strategy profile is a subgame perfect Nash equilibrium if for every proper subgame of the game, its substrategy profile is a Nash equilibrium. • That is, the actions taken in the proper subgame are a Nash equilibrium for the game that consists of just that subgame.**What is a substrategy profile?**• A strategy profile for a game specifies what a player will do at every information set in the game and specifies the payoffs at the end of the game. • A substrategy profile of the original strategy profile specifies what each player will do at every information set in the subgame.**Alice and Bob Play in the Dark**Bob How many proper subgames does this game have? 0 1 2 3 More than 3 Go to A Go to B Alice Alice Go to A Go to B Go to A Go to B 2 3 0 0 1 1 3 2**Alice and Bob Play in the Dark**Bob How many subgame perfect Nash equilibria does this game have? 0 1 2 3 4 Go to A Go to B Alice Alice Go to A Go to B Go to A Go to B 2 3 0 0 1 1 3 2**Alice, Bob, and the outside option**Bob Go to Movies Bob Go shoot pool Go to A Go to B 2.5 1 Alice Alice Go to A Go to B Go to A Go to B 2 3 0 0 1 1 3 2 What are the subgame perfect equilibria in this game?**What if they know if you have invested?**• What is the subgame perfect equilibrium?**The Yule Ball Story**Page 283, in your text. How many proper subgames (subgames not equal to the whole game) does this game have? • 0 • 1 • 2 • 3 • More than 3**Dating Dilemma**Victor Asks Hermione Ron Victor Doesn’t Ask Hermione Ron**Simplifying the Game**If Hermione ever reaches either of the two nodes where Ron gets to ask her, she would say Yes. So a subgame perfect equilibrium must be a Nash equilbrium for the simpler game in which Hermione always says “yes” to Ron if she hasn’t accepted a date from Victor.**Victor Asks**Hermione’s strategy Ron’s Strategy Victor Doesn’t Ask Hermione’s strategy Ron’s Strategy**One lesson:Subgame Perfection does noteliminate all of**love’s quandries**When does a lawmaker want a moderate law?**• Pick numbers so that a gentle law is enforced but not obeyed, and a severe law is neither enforced nor obeyed, but a moderate law is enforced and obeyed. • Make b=f=j>4 (Violate law and not convicted is better than obeying the law) • Make c>g>k. (Judge doesn’t like to punish) • Make a > 4 (Weak law not obeyed if enforced) • Make i<e<4 (Moderate and strong laws obeyed if enforced) • Make g>8>k. (judge will enforce moderate law but not strong law. • Make d=h=l Try g=9, c=10, k=7, ,d=f=g=6, a=5