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