Taking Turns in the Dark Subgame perfection with

Taking Turns in the Dark (Subgame perfection with imperfect information) Econ 171

Homework from Chapter 8 Galileo and the Papal inquisition.

Strategic Form Three Players. —What are the strategies?

Payoffs if Pope Refers the Case to the Inquisition Galileo’s Strategy Confess Before Torture. Confess if tortured Inquisitor’s Torture Strategy Do not torture Confess before torture. Do not confess if tortured Do not confess before torture. Confess if tortured Do not confess before torture. Do not confess if tortured

Payoffs if Pope Does not refer the Case to the Inquisition Galileo’s Strategy Confess Before Torture and Confess if tortured Inquisitor’s Torture Strategy Do not torture Confess before torture but do not confess if tortured Do not confess before torture, do not confess if tortured

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.

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?

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.

Describe the subgame perfect N. E. • In subgame perfect Nash equilibrium, JS strategy is as follows: “Do not kidnap. If you kidnap, then kill if ransom paid, kill if no ransom is paid, and release if bounty is offered. ” JM’s strategy is “Offer a Bounty. ” Note that would happen out of equilibrium determines actions taken in equilibrium.

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.

Subgame Perfection with Imperfect Information Can the notion of subgame perfection help us if there is incomplete information?

What is a regular subtree of a game? • It is a tree starting from one of the nodes of the game such that the 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 subgame of the game, its substrategy profile is a Nash equilibrium.

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 will happen at every information set in the subgame.

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

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 What are the subgame perfect equilibria in this game?

The Yule Ball Story Page 283, in your text. How many proper subgames (subgames not equal to the whole game) does this game have? A) 0 B) 1 C) 2 D) 3 E) More than 3

Dating Dilemma Victor Asks Hermione Ron Y, Y, Y, N, Y Y, N, N N, Y, Y N, Y, N N, N, Y N, N, N Ask 8, 3, 6 1, 8*, 8* 3, 2, 4 Don’t 7*, 6*, 5* 2, 5, 3 2, 5*, 3 N, Y, N N, N, Y N, N, N Victor Doesn’t Ask Hermione Ron Y, Y, Y, N, Y Y, N, N N, Y, Y Ask 4, 7*, 7* 6, 1, 2 *4, 7*, 7* 6, 1, 2 Don’t 5, 4, 1 5, 4, 1

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 Yes to Victor No to Victor Ask 8, 3, 6 1, 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* 4*, 7* Don’t Ask 5, 4, 1*