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ECON E-1040: GAME THEORY & STRATEGIC GAMES Lecture #7 Zero-Sum Games Marion Laboure Fall 2018 Games with incomplete (asymmetric) information with complete (symmetric) information one-shot games sequential-move games repeated games simultaneous-move games simultaneous-move games sequential-move games simultaneous & sequential-moves Nash Equilibrium 2 use Do all players have a dominant strategy? NO Nash Equilibrium is the Nash Equilibrium YES Dominant strategy Equilibrium use Do players have strictly dominated strategies? YES NO Nash Equilibrium if you find an equilibrium Iteratively eliminate SDS it is the Nash Equilibrium use Do payoffs of both players add to the same constant in all outcomes? NO Nash Equilibrium finds the Nash Equilibrium YES Minimax Method Omnes viae Romam ducunt (All Roads lead to Rome) 3 Outline Constant-sum / zero-sum games Minimax method Midterm Exam & Revision 4 Von Neumann and Morgenstern John Von Neumann (19031957) Oskar Morgenstern (19021977) Princeton University Press, 1944 5 Games of pure conflict (strictly competitive games) Two-player constant-sum game: players’ payoff always add up to a constant. Examples? Zero-sum game: a special case of constant-sum game in which players’ payoff add to zero. In other words, payoffs of one player is the negative of the payoffs of the other player. 6 Maximin strategies (a.k.a. playing cautiously) You(Player1)areplayingazero-sumgamewithPlayer2. AssumethatPlayer2canseeintoyourhead(andthusknows yourstrategy). Whatisthebestthatyoucando? Remember,thisisazero-sumgame,soPlayer2willtryto minimize yourpayoff(i.e.,willtrytomaximize his). 7 Example: Penalty kicks “The strongest shooters can kick at speeds of up to 80 mph [~129 kmph]. This means that the ball reaches the goal line in 500 milliseconds.” Goalkeeper Left Right 0.2 0.8 0.7 0.3 1 0.5 0.5 Kicker popularmechanics.com Left Right 0 prob. of scoring a goal + prob. of not scoring a goal = 1 A constant-sum game (payoffs add to 1) 8 Example: Penalty kicks Transforming a constant-sum game to a zero-sum game Goalkeeper Kicker Left Left Right Right 0.2 0.7 1 0.5 Payoffs in each cell add to zero. 9 Example: Penalty kicks Goalkeeper Kicker Transforming a constant-sum game to a zero-sum game Left Right Left Right 0.2 -0.2 0.7 -0.7 1 0.5 -0.5 -1 Payoffs in each cell add to zero. Convention: write only Player 1’s (row player) payoffs 10 Example: Penalty kicks Goalkeeper Left Right 0.2 -0.2 0.7 -0.7 1 0.5 -0.5 Kicker Kicker’s perspective Left Right -1 The goalkeeper can read your mind! When you decide on your strategy, he will play his best-response. He will choose the strategy that maximizes his payoff (which minimizes yours). Play cautiously, and get the “better of the worse.” 11 Example: Penalty kicks Goalkeeper Left Right 0.2 -0.2 0.7 -0.7 min= 0.2 1 0.5 -0.5 min= 0.5 Kicker Kicker’s perspective Left Right -1 Maximin What is the minimum you can get from each strategy? Which of the minima is better? To guarantee yourself at least 0.5, you should play Right 12 Example: Penalty kicks Goalkeeper’s perspective Goalkeeper Kicker What is the minimum you can get from each strategy? Left Right Which of the minima is better? Left Right 0.2 -0.2 0.7 -0.7 1 -1 0.5 -0.5 min= -1 min= -0.7 Maximin To guarantee yourself at least -0.7, you should play Right 13 Example: Penalty kicks Kicker Goalkeeper Left Right Left Right 0.2 -0.2 0.7 -0.7 1 0.5 -0.5 -1 This method fails to find the equilibrium (0.5 and -0.7 are in different cells); 14 Pure-strategy Nash equilibrium of this game A. B. C. D. E. F. G. (Left, Left) (Right, Right) (Left, Right) (Right, Left) (Left, Left) and (Right, Right) (Left, Right) and (Right, Left) No pure-strategy Nash Equilibrium 15 Pure-strategy Nash equilibrium of this game A. B. C. D. E. F. G. (Left, Left) (Right, Right) (Left, Right) (Right, Left) (Left, Left) and (Right, Right) (Left, Right) and (Right, Left) No pure-strategy Nash Equilibrium 16 Why “Minimax method”? Kicker Goalkeeper Left Right Left Right 0.2 -0.2 0.7 -0.7 1 0.5 -0.5 -1 17 Why “Minimax method”? Goalkeeper Kicker Left Minimax Left Right max= 0.2 1 Right Minimax 0.7 Maximin 0.5 Maximin min= min= max= Minimax of the kicker is the Maximin of the goalkeeper 18 “Zero-sum games cannot have a Nash equilibrium in pure strategies.” A. True B. False 19 “Zero-sum games cannot have a Nash equilibrium in pure strategies.” A. True B. False 20 Can zero-sum games have equilibrium in pure strategies? Kicker A player with a very strong natural side (Payoffs: scoring prob. in %) Goalkeeper Left Right Left 38 65 Right 93 70 Maximin Minimax Right is dominant strategy for the kicker. N.E. in pure strategies (Right, Right) 21 Summing up Toolkit: Minimax method to find equilibrium in two-person zero-sum games. Minimax method will lead to a Nash Equilibrium in pure strategies (provided that it exists). This is true only for two-person zero-sum games. 22 Review for Midterm exam 23 Games withincomplete(asymmetric) information withcomplete(symmetric) information one-shot games sequential-movegames repeated games simultaneous-move games simultaneous-movegames sequential-movegames simultaneous&sequential-moves 24 1. Private Incentives to Control Pollution [30 points] Review for Midterm exam Suppose that a certain society consists of only two people and that each of these people drives a car. Suppose that each person could voluntarily choose to put a pollution control device on his or her car. Installing such a device on one car costs $100 and provides a benefit (in the form of cleaner air) that is worth $70 to each of the two people (both people benefit since both people breathe the cleaner air). In Each the following find allpossible strictlyactions dominant strictly person in thisgames, society has two – Installand or Not Install.dominated Assuming thispure strategies. Then solve one-time the game. game is a simultaneous, game, draw a payoff matrix showing the net (dollar) benefit (or cost) to each of the two people for all four possible outcomes of the game. How would describe this game? Find theyou equilibrium. 2. Dominant and Dominated Strategies [15 points] In the following games, find all strictly dominant and strictly dominated pure strategies. (a) Player 2 Player 1 Up Straight Down Left 0,1 5,9 7,5 Middle 9,0 7,3 10 , 10 Right 2,3 1,7 3,5 (b) Player 2 North Straight Up West 2,3 3,0 Center 8,2 4,5 East 10 , 6 6,4 25 device on his or her car. Installing such a device on one car costs $100 and provides a benefit (in the form of cleaner air) that is worth $70 to each of the two people (both people benefit since both people breathe the cleaner air). Review for Midterm exam Each person in this society has two possible actions – Install or Not Install. Assuming this game is a simultaneous, one-time game, draw a payoff matrix showing the net (dollar) benefit (or cost) to each of the two people for all four possible outcomes of the game. How you describe this game? In would the following games, find all strictly dominant and strictly dominated pure strategies. 2. Dominant and Dominated Strategies [15 points] In the following games, find all strictly dominant and strictly dominated pure strategies. Find the equilibrium. (a) Player 2 Player 1 Up Straight Down Left 0,1 5,9 7,5 Middle 9,0 7,3 10 , 10 Right 2,3 1,7 3,5 (b) Player 1 has a dominant strategy of “Down.” For player Player 2 1 the strategies “Straight” and “Up” are dominated by “Down.”Center West East North 2,3 8,2 10 , 6 Player 2 has neitherStraight a strictly Up dominant 3 , 0 nor a strictly 4 , 5 dominated6 ,strategy. 4 Player 1 Down 5,4 6,1 2,5 South 4,5 2,3 5,2 26 benefit (or cost) to each of the two people for all four possible outcomes of the game. How would you describe this game? Review for Midterm exam 2. Dominant and Dominated Strategies [15 points] In the following games, find all strictly dominant and strictly dominated pure strategies. (a) In the following games, find all strictly dominant and strictly dominated pure Player 2 strategies. Find the equilibrium. Player 1 Left 0,1 5,9 7,5 Up Straight Down Middle 9,0 7,3 10 , 10 Right 2,3 1,7 3,5 (b) Player 2 Player 1 West 2,3 3,0 5,4 4,5 North Straight Up Down South Center 8,2 4,5 6,1 2,3 East 10 , 6 6,4 2,5 5,2 (c) Player 2 A A 5,5 B 0,6 27 would you describe this game? Review for Midterm exam 2. Dominant and Dominated Strategies [15 points] In the following games, find all strictly dominant and strictly dominated pure strategies. (a) In the following games, find all strictly dominant Player and strictly dominated pure 2 strategies. Find the1equilibrium. Player Left 0,1 5,9 7,5 Up Straight Down Middle 9,0 7,3 10 , 10 Right 2,3 1,7 3,5 (b) Player 2 West 2,3 3,0 5,4 4,5 North Straight Up Down South Player 1 Center 8,2 4,5 6,1 2,3 East 10 , 6 6,4 2,5 5,2 (c) There are no strictly dominant strategies or strictly dominated strategies, in pure strategies. Player 2 Player 1 A B A 5,5 8,4 B 0,6 3,1 28 Player 1 Up Straight Down 0,1 5,9 7,5 9,0 7,3 10 , 10 Review for Midterm exam 2,3 1,7 3,5 (b) Player 2 In the following games, find all strictly dominant and strictly dominated pure West Center East strategies. North 2,3 8,2 10 , 6 Straight Up 3,0 4,5 6,4 Find the1 equilibrium. Player Down 5,4 6,1 2,5 South 4,5 2,3 5,2 (c) Player 2 Player 1 A B C A 5,5 8,4 4,5 B 0,6 3,1 5,3 29 Down (b) 7,5 10 , 10 3,5 Review for Midterm exam Player 2 West Center East North 2,3 8 , 2 dominated 10 ,6 In the following games, find all strictly dominant and strictly pure Straight Up 3,0 4,5 6,4 strategies. Player 1 Down 5,4 6,1 2,5 South 4,5 2,3 5,2 Find the equilibrium. (c) Player 2 Player 1 A B C A 5,5 8,4 4,5 B 0,6 3,1 5,3 There are no strictly dominant strategies. For Player 1 “A” is strictly dominated by “B.” 30 Review for Midterm exam “If a player has a strictly dominant strategy in a simultaneous-move game, then she is sure to get her best possible outcome.” True, or false? Explain, and give an example of a game that illustrates your answer. 31 Review for midterm exam “In a sequential-move game, the player who moves first is sure to win.” Is this statement true or false? State the reason for your answer in a few brief sentences, and give an example of a game that illustrates your answer. [10 points] 32 Review for midterm exam 33 Review for Midterm exam: Sequential-Moves Game 34 Review for Midterm exam: Sequential-Moves Game 35 Review for Midterm exam: Sequential-Moves Game 36 Review for Midterm exam: Sequential-Moves Game 37 Review for Midterm exam: Sequential-Moves Game 38 Review for Midterm exam: Sequential-Moves Game 39 Review for Midterm exam: Sequential-Moves Game 40 Review for Midterm exam: Sequential-Moves Game 41 Review for Midterm exam: Sequential-Moves Game 42 Review for Midterm exam: Sequential-Moves Game 43 Review for Midterm exam: Sequential-Moves Game 44 Review for Midterm exam: Sequential-Moves Game 45 Review for Midterm exam: Sequential-Moves Game 46 Review for Midterm exam: Sequential-Moves Game 47 Review for Midterm exam: Sequential-Moves Game 48