Nash Equilibrium. An obvious way to play? Player 1. Player 2. Player 2

Transcription

1 Nash Equilibrium An obvious way to play? In Joseph Heller s novel Catch 22, allied victory in WW2 is a foregone conclusion. Yossarian does not want to be one of the last ones to die. His commanding officer points out ``But suppose everyone on your side felt that way. Yossarian replies ``Then I would certainly be a damned fool to feel any other way, wouldn t I?. Yossarian others fight Not fight fight -5,25-5,0 Not fight 100,25 2,0 Strategy profile: a specification of one strategy for each player from that player s list of strategies. Nash Equilibrium: is a strategy profile such that no player, by changing his or her part of the strategy profile unilaterally, can improve his or her payoff. 1

2 L C R L C R U 1 0, 10 3, 3 U 1 0, 10 3, 3 M 2, 10 10, 2 M 2, 10 10, 2 D 3, 3 4, 6 6, 6 D 3, 3 4, 6 6, 6 Hawk and Dove Coordination game Hawk Dove Hawk 3, 1 1, 1 Dove 1, 3 2, 2 1, 1 2

3 Coordination problem Opera Movie A, B C, D Opera 2, 1 C, D A, B Movie 1, 2 A > C B > D QWERTY and DVORAK key boards Beta and VHS videos Mac and PC computers 2, 2 1, 1 Assumptions for Nash Equilibrium Each player is rational Each player believes all other players are rational The game correctly describes the utility/payoffs of the players. Players are flawless in execution Players have sufficient intelligence to deduce the solution Nash is not played Game of chicken- others may be irrational Prisoner's dilemma is not a dilemma if a player enjoys going to prison In chess players should be able to figure out the Nash strategy, but players make errors. Playing tic-tac-toe with a child 3

4 Is Nash equilibrium played: Economics Auctions Industrial Organization Organization design Biology-evolution Some obvious factors Players have a common understanding Games can be solved logically dominance Larger payoffs make you think harder, consultation before helps. Easier to play with smaller number of players Regular history of interaction. Dr Strangelove ( Muffley: What... what is it, what? DeSadeski: The fools... the mad fools. Muffley: What's happened? DeSadeski: The doomsday machine. Muffley: The doomsday machine? What is that? DeSadeski: A device which will destroy all human and animal life on earth. USSR USA Doomsday Peace Strangelove: Mr. President, it is not only possible, it is essential. That is the whole idea of this machine, you know. Deterrence is the art of producing in the mind of the enemy... the fear to attack. And so, because of the automated and irrevocable decision making process which rules out human meddling, the doomsday machine is terrifying. It's simple to understand. And completely Doomsday -, - credible, and convincing. Peace -, - 3, 3 -, - Nash Equilibrium is not unique. BP Narrow Wide Exxon Narrow 14 16, 2 Wide 2, 16 1, 1 Nash Equilibria ({Narrow,wide},{wide,Narrow}) If a strategy profile is a strictly dominant strategy equilibrium then it is the only Nash equilibrium as well. If a strategy profile is a weakly dominant strategy equilibrium then it is Nash equilibrium as well but may not be the only one. 4

5 Focal point and Co-related Equilibrium In case of coordination games Thomas Schelling suggested focal point equilibrium. pl1 pl2 5, 10-30,-20 An equilibrium which stands out. (Focal point) Finding a coordinating device. -2,-5 12,2 examples Consider a simple example: two people unable to communicate with each other are each shown a panel of four squares and asked to select one; if and only if they both select the same one, they will each receive a prize. Three of the squares are blue and one is red. It is likely both choose the red square. Of course, the red square is not in a sense a better square; they could win by both choosing any square. Social conventions. "Circle one of the following numbers: 7,100, 13, 261, 99, 555. You win if both you and your partner circle the same number." 87% of the people chose the first three numbers. "You are supposed to meet some one in New York City. You have not discussed where and when and you do not really know the other person and cannot communicate with the other person." More than half proposed a meeting at 12 noon at the information booth at Grand Central. "Write some positive number. You win if the number matches with the one chosen by your partner" More than 40% chose 1. Matching pennies Mixed strategy -1, 1 1, -1 1, -1-1, 1 is s best response to s strategy is s best response to s strategy is s best response to s strategy is s best response to s strategy Hence, NO Nash equilibrium Mixed Strategy: A mixed strategy of a player is a probability distribution over player s (pure) strategies. 5

6 Mixed strategy: example Matching pennies has two pure strategies: H and T ( Prob 1 (H)=0.5, Prob 1 (T)=0.5 ) is a Mixed strategy. That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively. ( Prob 1 (H)=0.3, Prob 1 (T)=0.7 ) is another Mixed strategy. That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively. Solving matching pennies p 1-p (q) -1, 1 1, -1 (1-q) 1, -1-1, 1 Randomize your strategies to surprise the rival chooses and with probabilities p and 1-p, respectively. chooses and with probabilities q and 1-q, respectively. Solving matching pennies Mixed Strategy: Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities. Probabilities are chosen such that the players are indifferent to playing the available strategies Expected pay off of player 1 if she plays heads -1(q)+1(1-q) Expected pay off of player 1 if she plays tails 1(q)-1(1-q) In order to randomize player 1 will choose between playing heads and tails -1(q)+1(1-q)=1(q)-1(1-q) Expected pay off of player 2 if she plays heads 1(p)-1(1-p) Expected pay off of player 2 if she plays tails -1(p)+1(1-p) In order to randomize player 2 will choose between playing heads and tails 1(p)-1(1-p)=-1(p)+1(1-p) Therefore at equilibrium -1(q)+1(1-q)=1(q)-1(1-q) 1(p)-1(1-p)=-1(p)+1(1-p) q*=(1/2),(1-q)*=(1/2) p*=(1/2),(1-p)*=(1/2) 6

7 Mixed strategy: example Player 1 T (3/4) M (0) B (1/4) L (0) 0, 2 4, 0 3, 4 C (1/3) 3, 3 0, 4 5, 1 R (2/3) 1, 1 2, 3 0, 7 : (3/4, 0, ¼) is a mixed strategy. That is, Prob 1 (T)=3/4, Prob 1 (M)=0 and Prob 1 (B)=1/4. : (0, 1/3, 2/3) is a mixed strategy. That is, Prob 2 (L)=0, Prob 2 (C)=1/3 and Prob 2 (R)=2/3. Rock, Paper and Scissors pl1 rock paper scissor s pl2 rock 0,0 1,-1-1,1 paper -1,1 0,0 1,-1 scissors 1,-1-1,1 0,0 `Takashi Hashiyama, president of Maspro Denkoh Corporation, an electronics company based outside of Nagoya, Japan, could not decide whether Christie's or Sotheby's should sell the company's art collection, which is worth more than $20 million, at next week's auctions in New York. `Instead, he resorted to an ancient method of decision-making that has been time-tested on playgrounds around the world: rock breaks scissors, scissors cuts paper, paper smothers rock. - Child's Play Wins Auction House an Art Sale-By CAROL VOGEL, Published: April 29, The New York Times. Bart and Lisa playing RPS Lisa: Look, there's only one way to settle this. Rock-paper-scissors. Lisa's brain: Poor predictable Bart. Always takes `rock'. Bart's brain: Good ol' `rock'. Nuthin' beats that! Bart: Rock! Lisa: Paper. Bart: D'oh! 2007 World Rock Paper Scissors Champion Arnold Farina R R R R R S P R <--- Arnold wins set 1 R R S S R P S S R P <--- Farina wins set 2 P S P P R R R P <--- Farina wins championship 7

8 Two pure strategy equilibria [(right, right);(left,left)] pl1 right left pl2 right 5, 10-2,-5 left -30,-20 12, 2 Expected pay off of player 1 if she plays right 5(q)-30(1-q) Expected pay off of player 1 if she plays left -2(q)+12(1-q) In order to randomize player 1 will choose between playing left and right 5(q)-30(1-q)=-2(q)+12(1-q) Expected pay off of player 2 if she plays right 10(p)-5(1-p) Expected pay off of player 2 if she plays left -20(p)+2(1-p) In order to randomize player 2 will choose between playing left and right 10(p)-5(1-p)=-20(p)+2(1-p) Therefore at equilibrium 5(q)-30(1-q)=-2(q)+12(1-q) 10(p)-5(1-p)=-20(p)+2(1-p) q*=(6/7),(1-q)*=(1/7) p*=(7/37),(1-p)*=(30/37) Employee Monitoring Employee Monitoring Employees can work hard or shirk Salary: 100K unless caught shirking Cost of effort: 50K Managers can monitor or not Value of employee output: 200K Profit if employee doesn t work: 0 Cost of monitoring: 10K Employee Work ( p ) Shirk (1-p ) Monitor ( q ) 50, 90 0, -10 Manager Not Monitor (1-q) 50, ,

9 Expected pay off of employee if she works 50 Expected pay off of employee if she shirks 100(1-q) In order to randomize employee will choose between working and shirking 50= 100(1-q) Expected pay off of manager if she monitors 100p-10 Expected pay off of manager if she does not monitor 200p-100 In order to randomize manager will choose her probabilities such that she is indifferent between monitoring and not monitoring 100p-10 = 200p-100 q*=1/2 p*=9/10 9