1. Advanced Artificial Intelligence Lecture 3B: Game theory

2. 2 Outline Decisions with multiple agents: game theory(Text book: 17.5 )

3. 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

4. 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)

5. Game Console Game Dominant Strategy? No Equilibrium? blu,blu; dvd,dvd Pareto Optimal? Blu,blu: +9, +9

6. 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

7. -3 ≤ U E ≤ 2

9. - -1/12 ≤ U E ≤ -1/12

10. Poker Deck: KKAA Deal: 1 card each Rounds: (1) raise/check (2) call/fold Sequential game; Extensive form

11. Convert to Normal Form Problem: number of strategies exponential in number of information sets Texas Hold’em: 10 18 states.

12. 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

13. Mechanism Design Game Theory: – Given a game, find rational policy Mechanism Design: – Given utility functions U i, design a game such that the rational strategies maximize ∑ U i Examples: – Google ads, airplane tickets, radio spectra, TCP packets, dating, doctor internships

14. 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