Advanced Artificial Intelligence

Lecture 3B: Game theory

2

Outline

• Decisions with multiple agents: game theory(Text book: 17.5 )

Game Theory

• For games where two players move simultaneously (or, without knowledge of each other)

• Often, single-move games• Two problems are studied– Agent design– Mechanism design

Prisoner’s Dilemma

• Dominant Strategy: testify(does better no matter what other player does)

• Pareto Optimal: no other outcome that all players would prefer

• Equilibrium: no player can benefit from switching (assuming other players stay the same)Every game has at least one (Nash)

Game Console Game

• Dominant Strategy?• No• Equilibrium?• blu,blu; dvd,dvd• Pareto Optimal?• Blu,blu: +9, +9

Sample Game

• Two finger Morra: players E and O show 1 or 2 fingers; total f. E wins f if even; O if odd.

• What is best strategy for each player (solution)?– Pure strategy: single move– Mixed strategy: probability distribution on moves

-3 ≤ UE ≤ 2

--1/12 ≤ UE ≤ -1/12

Poker

Deck: KKAADeal: 1 card eachRounds:(1) raise/check(2) call/fold

Sequential game;Extensive form

Convert to Normal Form

Problem: number of strategies exponential in number of information setsTexas Hold’em: 1018 states.

Extensive Games

• Sequence form (Koller): Up to 25,000 states• Abstraction– Suits; Hi/mid/lo; bet amounts; only some deals

• Can handle– PO, multi-agent, stochastic, sequential, dynamic

• Can’t handle (very well)– Unknown actions; continuous actions– Irrational opponents– Unknown utilities

Mechanism Design

• Game Theory:– Given a game, find rational policy

• Mechanism Design:– Given utility functions Ui, design a game such that

the rational strategies maximize ∑ Ui

• Examples:– Google ads, airplane tickets, radio spectra, TCP

packets, dating, doctor internships

Auction• Auction is globally better with more bidders• Easier if bidders have dominant strategy– Strategy-proof; truth-revealing; incentive compatible

• Sealed bid auction– Strategy-proof?

• Second-price (Vickrey) auction– Payoff for bid b with value v, best other bid c:

U = (v – c) if (b > c) else 0• Optimal bid:– Bid v