Games with Sequential Moves Games with Sequential Moves

Games with Sequential Moves

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

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

ANN (2, 7, 4, 1) Up Down Branches (1, -2, 3, 0) 1 2 ANN Stop BOB DEB (10, 6, 1, 1) Nodes Go CHRIS Root (Initial Node) n Risky Safe High Low 3 NATURE (1. 3, 2, -11, 3) (0, -2. 718, 0, 0) Terminal Nodes Good 50% (6, 3, 4, 0) Bad 50% (2, 8, -1, 2) (3, 5, 3, 1) Game Tree (slightly different from the text)

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

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

Solving the Game Tree Backward Induction Rollback Equilibrium, Subgame Perfect Nash Equilibrium n 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 n

n Subgame ANN Ann’s move Down 1 2 ANN Stop BOB Go CHRIS (2, 7, 4, 1) Up Bob’s Move DEB Safe (1. 3, 2, -11, 3) High Low 3 Risky (1, -2, 3, 0) (10, 6, 1, 1) NATURE (0, -2. 718, 0, 0) Deb’s Move Good 50% (6, 3, 4, 0) Bad 50% (2, 8, -1, 2) (3, 5, 3, 1)

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 n It guarantees Chris can compare the payoff of 1. 5 by playing Risky move to that of 3 by playing Safe. n

ANN (2, 7, 4, 1) Up Down (1, -2, 3, 0) 1 2 ANN Stop BOB Go CHRIS Low 3 Risky Safe (1. 3, 2, -11, 3) (2, 7, 4, 1)High DEB (0, -2. 718, 0, 0) (10, 6, 1, 1) NATURE Good 50% (6, 3, 4, 0) Bad 50% (2, 8, -1, 2) (3, 5, 3, 1) Chris’ Move

n 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.

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

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

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

Remarks n n First-mover Advantage? -Not necessarily! Tic-tac-toe -9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1=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

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

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

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

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