Games with Sequential Moves

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## Games with Sequential Moves

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**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.**Games in extensive (tree) form.**• Games with perfect and complete information**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)**v.s. Decision Tree**• Nodes Places where players make moves. -Root -Terminal nodes • Branches Possible choices of players**Strategy vs. Moves**• Payoffs -(A, B, C, D) -Comparison • Nature Uncertainty**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**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)**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.**(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)**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.**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)**In equilibrium, Emily chooses D, Nina follows C, and then**Talia chooses C. • Equilibrium Path (Subgame Perfect Equilibrium (SPNE)) -Reinhard Selten, 1994 Nobel Laureate**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.**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**Theory vs. Evidence**• A simple bargaining problem • Traveler’s Dilemma**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**The Survivor**A constant-sum game. • Players Rich, Rudy, Kelly • Every 3 days, a person will be voted off if not the immunity winner.**Homework**question 2, 3, 5, and 10.