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Introduction to Game Theory

Introduction to Game Theory. Lecture 4: Mixed Strategy Nash Equilibrium. Today’s Plan. Review of experiment Review of mixed strategy Nash equilibrium Iterative elimination of strictly dominated strategies by mixed strategies Mixed strategies in games with three and more actions per player.

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Introduction to Game Theory

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  1. Introduction to Game Theory Lecture 4: Mixed Strategy Nash Equilibrium

  2. Today’s Plan • Review of experiment • Review of mixed strategy Nash equilibrium • Iterative elimination of strictly dominated strategies by mixed strategies • Mixed strategies in games with three and more actions per player

  3. Experiment • Beauty Contest game • Players submit decimal number between [0,100] including 0 and 100. • Player who submits number closest to 2/3 of average wins. • Example: • If numbers submitted are a, b, c, d, e then number closest to (2/3)*(a+b+c+d+e)/5 wins. • Experiment consists of several rounds. • After each round the average is announced as well as the optimal number (2/3 of average).

  4. Experiment • Theoretical Solution: • By iterative elimination of strictly dominated strategies • All actions but 0.1 and 0 are eliminated. • No other action than 0.1 or 0 can be Nash equilibrium actions. • Nash equilibria??? • Check for all possible combinations and choose such that no player has incentive to deviate.

  5. Experiment • Keynes (General Theory of Employment Interest and Money, 1936) • Beauty contest where public is judge and one of the participants who has voted for winner of the contest gets price. • People do not submit vote for girl they think is the most beautiful. • They submit what they think that the other think is the average. • Can be iterated to higher degree. • Application to stock market. • Good environment to test for peoples’ rationality and their believes about other peoples’ rationality. • How did you perform?

  6. Results

  7. Empirics Source: Camerer (1997), Progress in Behavioral Game Theory, Journal of Economic Perspectives

  8. Empirics Source: Camerer (1997), Progress in Behavioral Game Theory, Journal of Economic Perspectives

  9. Empirics Source: Boch-Domenech et al. (2002), “One, Two, (Three), Infinity, ... : Newspaper and Lab Beauty-Contest Experiments”, AER

  10. Review • Nash equilibrium • Deviation proof • Intersection of best response correspondences • No equilibrium possibility: Paper, Scissor, Rock • Mixed strategy equilibrium • Players chose their action in random • Make themselves unpredictable • Rabbit vs. Eagle • Nobody wants to stick to one action – he loses when opponent finds out • von Neumann – Morgenstern utility numbers have more than ordinal meaning.

  11. von Neumann – Morgenstern Pref. • Both tables represents the same game with ordinal preferences • What value should be assigned to p such that Player 1 prefers Top to Bottom according to the vNM preferences in first table and in the second table? • Is there any inconsistency?

  12. Mixed Strategy NE – Example • Penalty kick: • No NE in pure strategies • MSNE – Goalie has to be indifferent between Right and Left and Kicker as well. • Find such q and p.

  13. Goalie vs. Kicker • Set p such that Goalie is indifferent between playing Right and Left: EG[Right]=p*uG (Right,Right)+(1-p)uG (Right,Left) EG[Left]=p*uG (Left,Right)+(1-p)uG (Left,Left) EG[Right]=p*1+(1-p)*(-1)=p(-1)+(1-p)1= EG[Left] • Set q such that Kicker is indifferent between playing Right and Left: EK[Right]=q*uK (Right,Right)+(1-q)uK (Right,Left) EK[Left]=q*uK (Left,Right)+(1-q)uK (Left,Left) EK[Right]=p*(-1)+(1-p)*1=q1+(1-q)(-1)= EK[Left] • p*=1/2, q*=1/2 • Mixed strategy action profile ((1/2,1/2), (1/2,1/2)) is MSNE.

  14. Best Response Correspondence q - Goalie 1 MSNE Best Response Corr. of Goalie 0.5 Best Response Corr. of Kicker 0,0 1 0.5 p - Kicker

  15. Reporting Crime • 20 people observe a crime and decide about the calling the police. • If person does not call and nobody else does – 0 • If person does not call and somebody else does – 10 • If person calls police – 5 • Nash Equilibrium in pure strategies? • Who is going to call? Difficult to talk about social norm. • Population is not homogenous. • Any symmetric equilibria?

  16. Symmetric Equilibrium • Symmetric game: All players have the same actions available and the same preferences over the action profiles represented by the same payoffs. • Symmetric (mixed strategy) equilibrium: All players take the same action (mixed strategy). Symmetric game with symmetric pure strategy NE Symmetric game with no symmetric pure strategy NE, symmetric MSNE

  17. Reporting Crime – Mixed Strategy • Symmetric MSNE: • Everybody has to be indifferent in calling and not calling: u(Call)=5 Eu(Not Call)=10p+0(1-p), where p is probability that at least one of the 19 other people calls. • p is such that u(Call)=5=10p+0(1-p)=Eu(Not Call) • p=0.5 and it is equal to 1-(1-α)19, where α is probability that a single person calls. • α=1-(0.5)1/19

  18. Strictly Dominated strategy • Action a of player i is dominated by mixed strategy pure αi if whatever actions other players take, playing αiresults in strictly higher payoff to player i than a. • Down is strictly dominated by (1/2,1/2,0)

  19. Strictly Dominated Strategy • Strictly dominated strategy is never a best response to combination of opponents’ mixed strategies (whatever combination). • There is no such q, that Player 1 indifferent between playing Down and (1/2,1/2,0). • Strictly dominated strategy is never played with positive probability in MSNE – can be eliminated!

  20. Dominated Strategy – Elimination • Down is eliminated because it appears with zero probability in all MSNE • No else action is strictly dominated by any • What are the all possible MSNE? • ((1,0,0),(1,0)), ((0,1,0),(0,1)), ((1/2,1/2,0),(1/2,1/2))

  21. MSNE – More Two Actions • Any simultaneous game with complete information where each player has finite number of actions available has at least one mixed strategy Nash equilibrium. • Mixed strategy profile * is MSNE if and only if for every player: • All actions played by player with positive probability in * deliver the same expected payoff given that his opponents stick to *-i. (condition 1) • Specifies the structure of probability distribution over the actions • Every action that is played in * with zero probability delivers expected payoff at most as any action played with positive probability given that his opponents stick to *-i. (condition 2) • Helps us to eliminate some of the candidates. • Ensures that MSNE is bulletproof against all possible deviations.

  22. Example – Rock, Paper, Scissors • No NE in pure strategies • Two cases • Player 2 plays one action with zero probability • Player 2 plays every action with positive probability

  23. Rock, Paper, Scissors – Case 1 • Assume that Player 2 plays scissors with 0 probability. • Game is transformed to: • Player 1 has dominated pure strategy Rock by pure strategy Paper • Player 1 does not play Rock with positive probability

  24. Rock, Paper, Scissors – Case 1 • Player 2 mixes Rock and Paper, Player 1 mixes Paper and Scissors . • q has to be such that Player 2 is indifferent between Rock and Paper (following condition 1) (-1)*q+1*(1-q)=0*q+(-1)*(1-q) q=1/3. • Candidates for MSNE: Player 1 plays (0,1/3,2/3) and Player 2 mixes Rock and Paper

  25. Rock, Paper, Scissors – Case 1 • Are our candidates indeed MSNE? • Best response of Player 2 to (0,1/3,2/3) is pure strategy Scissors. • Assumption that Player 2 plays Scissors with zero probability is WRONG!!! (condition 2 is violated)

  26. Rock, Paper, Scissors – Case 1 • We should examine other two subcases when Player 2 plays with zero probability either of Rock or Paper as well. • Given the symmetry of the problem the same analysis aplies to the case when Player 2 assigns zero probability to Rock or Paper. • Case 1, when Player 2 plays one action with zero probability, cannot be equilibrium. • Does equilibrium arise in Case 2?

  27. Rock, Paper, Scissors – Case 2 • q1, q2 , q3 have to be such that Player 2 is indifferent between Rock, Paper and Scissors. • p1,p2 ,p3 have to be such that Player 1 is indifferent between Rock, Paper and Scissors.

  28. Rock, Paper, Scissors – Case 2 Player 2 is indifferent: 0*q1+(-1)*q2+1*q3=1*q1+0*q2+(-1)*q3 (-1)*q1+1*q2+0*q3=1*q1+0*q2+(-1)*q3 q1+q2+q3=1 (q1,q2,q3)=(1/3,1/3,1/3) Player 1 is indifferent: 0*p1+(-1)*p2+1*p3=1*p1+0*p2+(-1)*p3 (-1)*p1+1*p2+0*p3=1*p1+0*p2+(-1)*p3 p1+p2+p3=1 (p1,p2,p3)=(1/3,1/3,1/3)

  29. Summary • MSNE • Exists if players have finite number of actions available. • No deviation with higher payoff possible • Elimination of strictly dominated strategies. • Every player is indifferent between actions he plays with positive probability. • No player’s expected payoff from action that he plays with zero probability is higher than his expected payoff from any action played with positive probability.

  30. Reporting Crime • Does this problem has MSNE that is not symmetric and every player plays “Call” and “Not Call” with positive probability? • Is there equilibrium when only two players have positive probability of calling? • Assume the problem with three players. Players have different costs of reporting crime c1>c2>c3. Sketch the solution.

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