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Advanced Artificial Intelligence
Lecture 3B: Game theory
Outline
• Decisions with multiple agents: game
theory(Text book: 17.5 )
2
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
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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: KKAA
Deal: 1 card each
Rounds:
(1) raise/check
(2) call/fold
Sequential game;
Extensive form
Convert to Normal Form
Problem: number of strategies exponential in number of information sets
Texas 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