2Payoff informationProfessor 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.
3Solve 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.
4Working 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.
5Problem 8.16Nick and Rachel divide 4 candy bars. They take turns choosing.Nick goes first. What should Nick choose first?Preferences are: For Nick For RachelSnickers Milky WayMilky Way Kit KatKit Kat Baby RuthBaby Ruth SnickersHint: No matter what happens,Nick will get two bars. Rachel will never choose Snickers.
6Taking Turns in the Dark: (Subgame perfection with incomplete information ) Econ 171
7Subgame Perfection with Imperfect Information How can the notion of subgame perfection help us if there is incomplete information? Look back at kidnapper game
9What 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.
10How many subtrees does this game tree have? A) 1 B) 2 C) 3 D) 4 E) 5
11How many proper subtrees does the kidnapper game have? 12345
12What 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.
13How many regular subtrees does this game tree have?A) B) 2 C) 3 D) E) 5
14How many regular, proper,subtrees does this game tree have?A) B) 2 C) 3 D) E) 5
15What 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)
16How many proper subgames does this Game have?A) B) 2 C) 3 D) E) 5
17What 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.
18Subgame perfectionIn 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.
19Alice and Bob Play in the Dark How many proper subgames does this game have?123More than 3Go to AGo to BAliceAliceGo to BGo to AGo to AGo to B23132
20Alice and Bob Play in the Dark How many subgame perfect Nash equilibria does this game have?1234Go to AGo to BAliceAliceGo to BGo to AGo to AGo to B23132
21Alice, Bob, and the outside option Go to MoviesBobGo shoot poolGo to AGo to B2.5 1AliceAliceGo to BGo to AGo to AGo to B23132How many proper subgames does this game have?A) 1 B) 2 C) 3 D) 4 E) 5
22How would you play in this game if you were Bob? Go shoot poolGo to movie AGo to movie B
23How would you play in this game if you were Alice? Go to AGo to B
24Finding subgame perfect strategy profiles BobGo to MoviesBobGo shoot poolGo to AGo to B2.5 1AliceAliceGo to BGo to AGo to AGo to B23132Find Nash equilibria for the proper subgame.Look at the truncated game with equilibriumpayoffs from subgame.
25Finding subgame perfect strategy profiles BobGo to AGo to B2.5 1AliceAliceGo to BGo to AGo to AGo to B23132Proper 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.
26Truncating the tree with both go to B in the subgame BobGo to MoviesBobGo shoot poolGo to AGo to B2.5 1AliceAliceGo to BGo to AGo to AGo to B23132If both go to B if Bob goes to the movies, then Bob will go tothe movies rather than play pool. The profile, Bob goes to themovies and goes to B; Alice goes to B is a SPNE
27Truncating the tree with both go to A in subgame BobGo to MoviesBobGo shoot poolGo to AGo to B2.5 1AliceAliceGo to BGo to AGo to AGo to B23132If 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.
28Complete Information: Alice chooses first. Find SPNE Movie AMovie BBpbBobShoot poolShoot pool2.512.5Movie AMovie AMovie BMovie B32321
29The Yule Ball Tale How many strategies are possible for Hermione? A) 2 B) 3 C) 4 D) 6 E) 8
30The Yule Ball Tale How many strategies are possible for Ron ? A) 2 B) 3 C) 4 D) 6 E) 8
31Dating Dilemma: Strategic Form Victor AsksHermioneY,Y,YY,Y,NY,N,YY,N,NN,Y,YN,Y,NN,N,YN,N,NAsk8,3,61,8*,8*3,2,4Don’t7*,6*,5*2,5,32,5*,3RonVictor Doesn’t AskHermioneY,Y,YY,Y,NY,N,YY,N,NN,Y,YN,Y,NN,N,YN,N,NAsk4,7*,7*6,1,2*4,7*,7*Don’t5,4,1Ron
32The Yule Ball Tale How many proper subgames does this game have? A) 0 B) 2 C) 3 D) 6 E) 8
33Simplifying the GameIf 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.
34Victor AsksHermione’s strategyYes to VictorNo to VictorAsk8,3,61,8*,8*Don’t Ask7*,6*,5*2,5,3Ron’s StrategyVictor Doesn’t AskHermione’s strategyYes to VictorNo to VictorAsk4,7*,7*4*,7*,7*Don’t Ask5,4,1*Ron’s StrategyPayoffs listed in order Victor, Ron, Hermoine
38What 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?
39Strategic form if Victor asks: RonAsk HermoineDon’t ask HermoineYes to Victor6, (Victor 8)5,6 (Victor 7)No to Victor8, 8 (Victor 1)3,5 (Victor 2)HermoineWe have two Nash equilibria for the subgame between Hermoineand 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.