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Network Theory and Dynamic Systems Game Theory: Mixed Strategies

Network Theory and Dynamic Systems Game Theory: Mixed Strategies. Prof. Dr. Steffen Staab. Randomization. Rationale to randomize Matching pennies do not result in a Nash equilibrium , because opponent would switch if he knew your choice

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Network Theory and Dynamic Systems Game Theory: Mixed Strategies

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  1. Network Theory and Dynamic SystemsGame Theory: Mixed Strategies Prof. Dr. Steffen Staab

  2. Randomization • Rationale torandomize • Matchingpennies do not result in a Nash equilibrium,becauseopponentwouldswitchif he knewyourchoice → Randomize your choice such that your opponent cannot determine the best response → Mixed strategy

  3. Nash Equilibrium Definition: A Nash equilibriumfor a mixed-strategygameis a pair ofstrategies (S,T), where S and T arebothprobabilitydistributionsover individual strategiesavailabletoplayer 1 and 2, respectively, such thateachis a bestresponsetotheother. Can a pure strategyleadto a Nash equilibriumforthepenniesgame?

  4. Mixed Strategy • Eachplayerchooses a probabilityforeachstrategy • Player 1 chooses Head with p andTailwith 1-p • Player 2 chooses Head with q andTailwith 1-q • Rewardforplayer 1: • pq*1 + (1-p)(1-q) * 1 + p(1-q) * (-1) + (1-p)q * (-1) = pq + 1 + pq - p - q - p + pq - q + pq = 1 – 2 (p+q) + 4pq

  5. Mixed Strategy • Rewardforplayer 1: • pq*1 + (1-p)(1-q) * 1 + p(1-q) * (-1) + (1-p)q * (-1) = pq + 1 + pq - p - q - p + pq - q + pq = 1 – 2 (p+q) + 4pq • Rewardforplayer2: Multiplywith -1: -1 + 2 (p+q) - 4pq • Chooseoptimum: differentiateforboth variables • d(1 – 2 (p+q) + 4pq)/dp = -2 + 4q = 0 • d(1 – 2 (p+q) + 4pq)/dq= -2 + 4p = 0 (also forplayer 2) Nobetterresultachievablefor p=q=0.5 Nash Equilibrium for p=q=0.5 !

  6. Note • Whilethereneed not be a Nash equilibriumforthe pure strategies, thereisalwaysat least one Nash equilibriumformixedstrategies.

  7. Howtosolvethis a bitmoreeasily? Considermatchingpenniesagain: • If Player 2 chooses a probabilityof q, and Player 1 chooses Head theexpectedpayoffto Player 1 is: • (-1) (q) + 1 (1-q) = 1-2q • If Player 1 choosesTail: • 1 (q) + (-1) (1-q) = 2q-1 • Note 1: no pure strategycanbepartof a Nash equilibrium • Note 2a: If 1-2q > 2q-1 then Head isthebeststrategyfor P1 • Note 2b: If 1-2q <2q-1 thenTailisthebeststrategyforP1 • Note 2c: Pure strategies do not leadto Nash equilibrium, thus Player 2 must choose a pointwhere1-2q = 2q-1 • Note 2d: Likewisefor Player 1

  8. DerivedPrinciple • A mixedequilibriumariseswhentheprobabilitiesusedbyeachplayermakehisopponent indifferent betweenhistwooptions.

  9. Applyingthe „indifferenceprinciple“ toanotherexample • „American Football“ • Assumedefensechooses a probabilityof q fordefendingagainstthe pass: • Thenpayoffforoffensefrompassingis • 0 q + 10 (1-q) = 10-10q • Payofffromrunningis • 5q + 0 (1-q) = 5q • 10-10q=5q issolvedfor q=2/3 • Likewise p=1/3

  10. Multiple Nash equilibria • Howmany Nash equilibria in thiscase? Unbalancedcoordinationgame

  11. Multiple Nash equilibria • Mixed-strategyequilibrium • 1 q + 0 (1-q) = 0q + 2 (1-q) → q=2/3 Also, becauseofsymmetry → p=2/3 Unbalancedcoordinationgame

  12. Zero Sum Games andLinear Optimization

  13. Linear Programming • Linear optimizationtask • Maximize or minimize a linear objective function in multiple variables • Linear constraints • Givenwith linear (in-)equalities

  14. Zero sumgame Players x and y chooseelectiontopicstogathervotes. Depending on thechosentopicstheygainorloosevotes. Whatisthebestmixedstrategy? y x

  15. Zero Sum Game Player x knowsthatplayer y triestominimizehislossbyminimizingthefollowingtwovalues: While x triestomaximizethisvalue Dual argumentforplayer y:

  16. Write zerosumgameas linear programs Spieler x Spieler y -2y1+y2 -2

  17. Dual Linear Programs Every linear programming problem, referred to as a primal problem, can be converted into a dual problem: • Given primal problem:Maximize cTx subject to Ax ≤ b, x ≥ 0; • Has same value of the objective function as the corresponding dual problem: Minimize bTy subject to ATy ≥ c, y ≥ 0.

  18. Write zerosumgameas linear programs Spieler x Spieler y -2y1+y2 -2 Observation: The twoproblemsare dual toeachother, thustheyhavethe same objectivefunctionvalue: Equilibrium

  19. MinimaxTheorem • Minimax-Theorem: • There are mixed strategies that are optimal for both players.

  20. Zero Sum Game Whatarethebeststrategiesnow? • Player x choosesstrategieswith • Player y choosesstrategieswith • Bothachieve an objectivefunctionvalueof

  21. Pareto Optimality & SocialOptimality

  22. Optimality – 1 • Nash Equilibrium • Individual optimum (-4,-4) • Not a global optimum Prisoners‘ Dilemma

  23. SocialOptimality Definition: A choiceofstrategies – onebyeachplayer – is a socialwelfaremaximizer(orsocially optimal) ifitmaximizesthesumoftheplayers‘ payoffs. However: Addingpayoffsdoes not alwaysmake sense! Prisoners‘ Dilemma

  24. Pareto Optimality Definition: A choiceofstrategies – onebyeachplayer – isPareto-optimal ifthereisnootherchoiceofstrategies in which all playersreceivepayoffsat least as high, andat least oneplayerreceives a strictlyhigherpayoff. • Core here: bindingagreement • (not-confess,not-confess) is Pareto-optimal, but not a Nash equilibrium • Nash equilibriumisonly Non-Pareto optimal choice!

  25. Multiplayer Games

  26. Multiplayer Games Structure: • n players • eachplayerhassetofpossiblestrategies • eachplayerhas a payofffunction Pi such thatforreachoutcomeconsistingofstrategieschosenbytheplayers(S1, S2,…Sn), thereis a payoff Pi(S1, S2,…Sn) toplayer i. Best response: A strategy Siis a bestresponseby Player i to a choiceofstrategies(S1, S2,…Si-1, Si+1,…,Sn), if Pi(S1, …Si-1, Si, Si+1,…,Sn) ≥ Pi(S1, …Si-1, S'i, Si+1,…,Sn) for all otherpossiblestrategiesS'iavailabletoplayer i.

  27. Multiplayer Games Structure: • n players • eachplayerhassetofpossiblestrategies • eachplayerhas a payofffunction Pi such thatforreachoutcomeconsistingofstrategieschosenbytheplayers(S1, S2,…Sn), thereis a payoff Pi(S1, S2,…Sn) toplayer i. Nash equilibrium: A choiceofstrategies(S1, S2,…Sn) is a Nash equilibriumifeachstrategyitcontainsis a bestresponseto all others.

  28. DominatedStrategies

  29. DominatedStrategies Definition: A strategyisstrictlydominatedifthereissomeotherstrategyavailabletothe same playerthatproduces a strictlyhigherpayoff in responsetoeverychoiceofstrategiesbytheotherplayers. (notionisonlyusefuliftherearemorethantwostrategies)

  30. Facility Location Game (cf Hotelling‘slaw) • Firm 1: Strategy A isdominatedby C • Firm 2: Strategy F isdominatedby D • Reasonaboutthestructureanddeletethesestrategies

  31. Facility Location Game (cf Hotelling‘slaw) • Firm 1: Strategy Eisdominatedby C • Firm 2: StrategyB isdominatedbyD • Reasonaboutthestructureanddeletethesestrategies

  32. Facility Location Game (cf Hotelling‘slaw) • One pair ofstrategiesremaining • The unique Nash equilibrium • Generalizesfrom 6 to an arbitrarynumberofnodes

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