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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. • A Proper Subtree is a subtree that is not the entire game. How many subtrees does this game tree have? A) 1 B) 2 C) 3 D) 4 E) 5 How many proper subtrees does the kidnapper game have? A) B) C) D) E) 1 2 3 4 5 What is a regular subtree 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 information set. • A proper, regular subtree is a regular subtree that is not the entire game tree. How many regular subtrees does this game tree have? A) 1 B) 2 C) 3 D) 4 E) 5 How many regular, proper,subtrees does this game tree have? A) 1 B) 2 C) 3 D) 4 E) 5 What is a subgame of a game? • A subgame is 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) How many proper subgames does this Game have? A) 1 B) 2 C) 3 D) 4 E) 5 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 strategy profile for an entire game induces a substrategy profile for each of its subgames. This substrategy profile specifies what each player will do at each of the player’s information sets in the subgame. 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. Find Subgame Perfect Nash equilibria Find Subgame Perfect Nash equilibria The subgame when Guy Kidnaps What are the substrategies for Guy? for Vivica? What are the Nash equilibria in this subgame? Subgame in Normal Form Vivica Pay Ransom Don’t Pay Ransom 4, 1 2, 2 5, 2 1, 4 Kill Guy Release What are the Nash equilibria in this subgame? What are the Nash equilibrium payoffs in this subgame? Find Subgame Perfect Nash equilibria Alice and Bob Play in the Dark How many proper subgames does this game have? Bob Go to A Go to B Alice Go to A 2 3 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 A) B) C) D) E) 0 1 2 3 More than 3 Is every Nash equilibrium for the previous game between Alice and Bob subgame perfect? A) Yes B) No Alice and Bob Play in the Dark How many subgame perfect Nash equilibria does this game have? Bob Go to A Go to B Alice Go to A 2 3 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 A) B) C) D) E) 0 1 2 3 4 Alice, Bob, and the outside option Bob Go to Movies Bob Go to A Go shoot pool Go to B Alice Go to A 2 3 2.5 1 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 How many proper subgames does this game have? A) 1 B) 2 C) 3 D) 4 E) 5 How would you play in this game if you were Bob? A) Go shoot pool B) Go to movie A C) Go to movie B How would you play in this game if you were Alice? A) Go to A B) Go to B Finding subgame perfect strategy profiles Bob Go to Movies Bob Go to A Go shoot pool Go to B Alice Go to A 2 3 2.5 1 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 Find Nash equilibria for the proper subgame. Look at the truncated game with equilibrium payoffs from subgame. Finding subgame perfect strategy profiles Bob Go to A Go to B Alice Go to A 2 3 2.5 1 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 Proper subgame has two N.E. Both go to A, Both go to B.We need to look at two possibilities. We may find more than one S P N E. Truncating the tree with both go to B in the subgame Bob Go to Movies Bob Go to A Go shoot pool Go to B Alice Go to A 2 3 2.5 1 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 If both go to B if Bob goes to the movies, then Bob will go to the movies rather than play pool. The profile, Bob goes to the movies and goes to B; Alice goes to B is a SPNE Truncating the tree with both go to A in subgame Bob Go to Movies Bob Go to A Go shoot pool Go to B Alice Go to A 2 3 2.5 1 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 If Alice’s strategy is Go to A, then Bob’s best response is Go shoot pool and Go to Movie A if he goes to the movies. This is a SPNE as well. Complete Information: Alice chooses first. Find SPNE Alice Movie A Movie B Bob Bpb Shoot pool 1 2.5 Movie A 3 2 Shoot pool Movie A Movie B Movie B 1 1 0 0 2 3 0 2.5 The Yule Ball Tale How many strategies are possible for Hermione? A) 2 B) 3 C) 4 D) 6 E) 8 The Yule Ball Tale How many strategies are possible for Ron ? A) 2 B) 3 C) 4 D) 6 E) 8 Dating Dilemma: Strategic Form Victor Asks Hermione Ron Y,Y,Y Y,Y,N Y,N,Y Y,N,N N,Y,Y N,Y,N N,N,Y N,N,N Ask 8,3,6 8,3,6 8,3,6 8,3,6 1,8*,8* 1,8*,8* 3,2,4 3,2,4 Don’t 7*,6*,5* 7*,6*,5* 7*,6*,5* 7*,6*,5* 2,5,3 2,5,3 2,5*,3 2,5*,3 N,Y,N N,N,Y N,N,N Victor Doesn’t Ask Hermione Ron Y,Y,Y Y,Y,N Y,N,Y Y,N,N N,Y,Y Ask 4,7*,7* 6,1,2 4,7*,7* 6,1,2 *4,7*,7* 6,1,2 *4,7*,7* 6,1,2 Don’t 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 The Yule Ball Tale How many proper subgames are there? A) 0 B) 2 C) 3 D) 6 E) 8 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. The Yule Ball Tale Victor Asks Hermione’s strategy Ron’s Strategy Yes to Victor No to Victor Ask 8,3,6 1,8*,8* Don’t Ask 7*,6*,5* 2,5,3 Victor Doesn’t Ask Hermione’s strategy Ron’s Strategy Yes to Victor No to Victor Ask 4,7*,7* 4*,7*,7* Don’t Ask 5,4,1* 5,4,1* Payoffs listed in order Victor, Ron, Hermoine Suppose Ron knows whether Victor asked Ron How many subgames are there now? A) 2 B) 3 C) 4 D) 6 E) 8 Suppose Ron knows whether Victor asked Ron What are N.E. in subgame where Victor Asks If Victor asks, then in remaining game, there are two things Hermoine can do, say Yes or No to Victor. There are two things, Ron can do. Ask Hermoine or Don’t ask her. What are the N.E. in this subgame? Strategic form if Victor asks: Ron Hermoine Ask Hermoine Don’t ask Hermoine Yes to Victor 6,3 (Victor 8) 5,6 (Victor 7) No to Victor 8, 8 (Victor 1) 3,5 (Victor 2) We have two Nash equilibria for the subgame between Hermoine and Ron starting at the node where Victor asks Hermoine. In one of them, Hermoine says Yes to Victor and Ron doesn’t ask. In the other, Hermoine says No to Victor and Ron asks. A SPNE in which Hermoine says Yes to Victor Ron A SPNE where Hermoine would say No to Victor Ron Valentine’s lesson: Subgame Perfection does not solve all of love’s quandries The Unpleasant Professor Problem Payoff information • Professor can give exam on Monday, Wednesday or Friday. • Students will study the night before exam if they know there will be an exam next day. • Professor prefers to have nobody prepared when exam is offered. • He also prefers earlier exam to later. Solve for SPNE by backward induction • Without drawing full tree, let’s try a shortcut. • If he doesn’t give exam on Monday, then he must either give it on Wednesday or on Friday. • If he doesn’t give it on Wednesday, students will know exam is Friday and will all study. • That is the worse for professor than giving it on Wednesday. • So he will not give exam on Friday. Working back… • So if he doesn’t give exam on Monday, he will give it on Wednesday. • Therefore if he doesn’t give exam on Monday, students will study on Tuesday. • If students will study on Tuesday if exam is not on Monday, professor would rather give exam on Monday. • Only subgame perfect Nash equilibrium has exam on Monday, students study on Sunday. Problem 8.16 Nick and Rachel divide 4 candy bars. They take turns choosing. Nick goes first. What should Nick choose first? Preferences are: For Nick Snickers Milky Way Kit Kat Baby Ruth For Rachel Milky Way Kit Kat Baby Ruth Snickers Hint: No matter what happens, Nick will get two bars. Rachel will never choose Snickers.