Games with Sequential Moves

1. Games with Sequential Moves

2. Games with Sequential Moves • Games where players move one after another. • Possible to combine with simultaneous moves. (But not considered in this chapter) • Players, when makes moves, have to consider what the opponents may do. • Game Trees are commonly used to specify all possible moves by all players and all possible outcome and payoffs.

3. Games in extensive (tree) form. • Games with perfect and complete information

4. Game Tree (slightly different from the text) (2, 7, 4, 1) Up ANN Down Branches (1, -2, 3, 0) 1 (1.3, 2, -11, 3) High DEB 2 Low (0, -2.718, 0, 0) Stop BOB 3 ANN (10, 6, 1, 1) Terminal Nodes Nodes (6, 3, 4, 0) Good 50% Go NATURE Risky CHRIS (2, 8, -1, 2) Bad 50% Root (Initial Node) Safe (3, 5, 3, 1)

5. v.s. Decision Tree • Nodes Places where players make moves. -Root -Terminal nodes • Branches Possible choices of players

6. Strategy vs. Moves • Payoffs -(A, B, C, D) -Comparison • Nature Uncertainty

7. Solving the Game Tree • Backward Induction Rollback • Rollback Equilibrium, Subgame Perfect Nash Equilibrium • Subgame the part of a game where the subsequent nodes after the starting nodes can separate from other nodes not after the starting node of the subgame

8. Subgame (2, 7, 4, 1) Up ANN Ann’s move Down (1, -2, 3, 0) 1 Bob’s Move (1.3, 2, -11, 3) High DEB 2 Low (0, -2.718, 0, 0) Stop Deb’s Move BOB 3 ANN (10, 6, 1, 1) (6, 3, 4, 0) Good 50% Go NATURE Risky CHRIS (2, 8, -1, 2) Bad 50% Safe (3, 5, 3, 1)

9. Solving the Game Tree • Expected Utility Theorem (von Neumann and Morgenstern) When taking Risky move, Chris expects to obtain 50% X 4 + 50% X (-1)= 1.5 • It guarantees Chris can compare the payoff of 1.5 by playing Risky move to that of 3 by playing Safe.

10. (2, 7, 4, 1) (2, 7, 4, 1) Up ANN Down (1, -2, 3, 0) 1 (1.3, 2, -11, 3) High DEB 2 Low (0, -2.718, 0, 0) Stop BOB 3 ANN (10, 6, 1, 1) (6, 3, 4, 0) Good 50% Go NATURE Risky CHRIS (2, 8, -1, 2) Bad 50% Safe Chris’ Move (3, 5, 3, 1)

11. In equilibrium, A chooses “Go” in the beginning, and “Up” if she has the chance to go after B . B chooses “1” C chooses “Safe” D chooses “High” The payoff is 3 to A, 5 to B, 3 to C and 1 to D.

12. The Secret Garden Game (3, 3, 3) TALIA C C D NINA (3, 3, 4) D C (3, 4, 3) D C TALIA EMILY D (1, 2, 2) D TALIA C (4, 3, 3) C NINA D (2, 1, 2) D C (2, 2, 1) TALIA D (2, 2, 2)

13. In equilibrium, Emily chooses D, Nina follows C, and then Talia chooses C. • Equilibrium Path (Subgame Perfect Equilibrium (SPNE)) -Reinhard Selten, 1994 Nobel Laureate

14. Strategies Emily {C, D} 2 strategies Nina {CC, CD, DC, DD } 4 strategies Talia {CCCC, CCCD, CCDC, CCDD, …..} 16 strategies for Talia • Nash Equilibrium (NE) is not necessarily a SPNE, but SPNE must be a NE.

15. Remarks • First-mover Advantage? -Not necessarily! • Tic-tac-toe -9x8x7x6x5x4x3x2x1=362,880 terminal nodes • Chess? • Existence of the equilibrium? Zermelo-Theorem: A finite game of perfect information has (at least) one pure-strategy Nash equilibrium

16. Theory vs. Evidence • A simple bargaining problem • Traveler’s Dilemma

17. The Centipede Game B Pass A B A Pass Pass Pass Pass Pass B A 90, 90 Take Dime Take Dime Take Dime Take Dime Take Dime Take Dime 10, 0 0, 20 30, 0 0, 40 90, 0 0, 100

18. The Survivor A constant-sum game. • Players Rich, Rudy, Kelly • Every 3 days, a person will be voted off if not the immunity winner.

19. Homework question 2, 3, 5, and 10.