Complex Decision-Making & Game Theory Concepts

1. Chapter 17: Making Complex Decisions April 1, 2004

2. 17.6 Decisions With Multiple Agents: Game Theory • Assume that agents make simultaneous moves • Assume that the game is a single move game.

3. Uses • Agent Design (2 finger Morra) • Mechanism Design

4. Game Components • Players • Actions • Payoff Matrix e.g. rock-paper-scissors

5. Terminology • Pure Strategy – deterministic policy • Mixed Strategy – randomized policy, [p: a; (1-p): b] • Outcome – result of game • Solution: player adopts a strategy profile that is a rational strategy

6. Prisoner’s Dilemna

7. Terminology • (testify, testify) is a dominant strategy • s strongly dominates s’ – s is better than s’ for all other player strategies • s weakly dominates s’ – s is better than s’ for one other strategy and is at least as good as all the rest

8. Terminology • An outcome is Pareto optimal if there is no other outcome that all players would prefer • An equilibrium is a strategy profile where no player benefits by switching strategies given that no other player may switch strategies • Nash showed that every game has an equilibrium • Prisoner’s Dilemna!

9. Example: Two Nash Equilibria

10. Von Neumann’s Maximin • zero sum game • E maximizer (2 finger Morra) • O minimizer (2 finger Morra) • U(E = 1, O = 1) = 2 • U(E = 1, O = 2) = -3 • U(E = 2, O = 1) = -3 • U(E = 2, O = 2) = 4

11. Maximin • E reveals strategy, moves first • [p: one; 1-p: two] • O chooses based on p • one: 2p -3(1-p) • two: -3p + 4(1-p) • p = 7/12 • UE,O = -1/12

12. Maximin • O reveals strategy, moves first • [q: one; 1-q: two] • E chooses based on q • one: 2q -3(1-q) • two: -3q + 4(1-q) • q = 7/12 • UO,E = -1/12

13. Maximin • [7/12: one, 5/12: two] is the Maximin equilibrium or Nash equilibrium • Always exists for mixed strategies! • The value is a maximin for both players.

14. Repeated Move Games • Application: packet collision in an Ethernet network • Prisoner’s Dilemna – fixed number of rounds – no change! • Prisoner’s Dilemna – variable number of rounds (e.g. 99% chance of meeting again) • perpetual punishment • tit for tat

15. Repeated Move Games • Partial Information Games – games that occur in a partially observable environment such as blackjack

16. 17.7 Mechanism Design • Given rational agents, what game should we design • Tragedy of the Commons

17. Auctions • Single Item • Bidderi has a utility vi for the item • vi is only known to Bidderi • English Auction • Sealed Bid Auction • Sealed Bid Second Price or “Vickrey” auction (no communication, no knowledge of others)