Analysis of Static Settings

Analysis of Static Settings - Analysis of Sta.c Se1ngs...

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Unformatted text preview: Analysis of Sta.c Se1ngs •  •  •  •  •  •  •  •  Dominance and Best Response Efficiency First Strategic Tension Iterated Dominance and Ra.onalizable Strategies Second Strategic Tension Nash Equilibrium Third Strategic Tension Mixed Strategy Nash Equilibria Dominance Pigs S D PD WD PS 4, 2 6, –1 WS 2, 3 0, 0 Consider the decision of the submissive pig (S):  ­ If the dominant pig plays PD, what’s best for S?  ­ If the dominant pig plays WD, what’s best for S? Since WS has a strictly higher payoff for S than PS, regardless of D’s strategy, we say PS is dominated by WS 1 Dominance Pigs S D PD WD PS 4, 2 6, –1 WS 2, 3 0, 0 Consider the decision of the dominant pig (D):  ­ If the submissive pig plays PS, what’s best for D?  ­ If the submissive pig plays WS, what’s best for D? PD is some.mes preferred to WD, so WD is not dominated by PD WD is some.mes preferred to PD, WD is not dominated by PD Dominance 2 A player is thought to be ra+onal if (1)  through some cogni.ve process, he forms a belief about the strategies of others, and (2)  given this belief, he selects a strategy to maximize his expected payoff. A ra+onal player will never play a dominated strategy. A dominated strategy has a lower payoff than the strategy that dominates it, for any strategy other players play. A ra.onal player would always select the strategy that dominates the dominated strategy. Dominance by Mixed Strategies 2 1 X Y Z L 1, 2 3, 5 0, 1 R 1, 3 0, 4 3, 2 Consider player 1’s decision:  ­ Neither Y or Z dominates X.  ­ What about a mixture of Y and Z? The mixed strategy shown here dominates X. 3 Best Response Best Response Best choices given your beliefs Strategies that give you weakly higher payoffs than any alterna.ve strategy given your beliefs 4 Considera.on of Dominance Y and Z are not dominated. Is X? •  If so, not by Y or Z. (X is some.mes beaer, some.mes worse than either.) •  What about a mixed strategy (0, p, 1 ­p) = σ1? Yes, e.g. X is dominated by σ1=(0, 1/2, 1/2) Considera.on of Best Response •  M is a best response to every belief of player 2 about player 1. •  Y is a best response to the simple belief (1,0) •  Z is a best response to the simple belief (0,1) 5 Considera.on of Best Response Is X a best response to some belief? NO. X is never the best strategy. Considera.on of Dominance 6 Considera.on of Best Response •  M and Q are best responses to simple beliefs (1, 0, 0) and (0, 1, 0). Y and Z are best responses to simple beliefs(1, 0) and (0, 1). •  Is X the best response to some belief? YES Best Response / Dominance Result Result: 7 Procedure for finding B=UD 1.  Look for strategies that are best responses to simplest beliefs: they’re obviously in B (=UD). 2.  Look for strategies that are dominated by other pure strategies: they’re obviously not in UD (=B). 3.  Test each remaining strategy to see if it’s dominated by a mixed strategy. Efficiency •  If a strategy profile is efficient, we cannot make any player better off without making another person worse off 8 Efficiency Example 2 1 L X 0, 2 Y 3, 6 M 2, 8 4, 1 R 3, 1 5, 5 Which strategy profiles are Pareto efficient?  ­ Player 1’s (unique) best outcome is 5, YR.  ­ Player 2’s (unique) best outcome is 8, XM. YR and XM are both Pareto Efficient. YR is more efficient than XL, XR and YM.  ­ XL, XR and YM are not Pareto Efficient. YL is also Pareto Efficient. Efficiency in Prisoner’s Dilemma Prisoner’s Dilemma 2 1 C D C 2, 2 3, 0 D 0, 3 1, 1 CC is more efficient than DC. CC, CD, DC, are efficient. 9 First Strategic Tension Prisoner’s Dilemma 2 1 C D C 2, 2 3, 0 D 0, 3 1, 1 First Strategic Tension The conflict between individual and group incen.ves In the Prisoner’s Dilemma, the group is beaer off at (C,C). But players ac.ng individually and ra.onally will each want to play D. 10 Efficiency in Prisoner’s Dilemma Prisoner’s Dilemma 2 1 C D C 2, 2 3, 0 D 0, 3 1, 1 Common Knowledge of Ra2onality All players are ra.onal All players know that all are ra.onal All players know that all know that all are ra.onal All players know that all know that all know that all are ra.onal . . and so on to infinity 11 Ra.onalizability •  Three behavioral assump.ons: –  Players have beliefs about each other’s strategy choices. –  Each player selects a strategy that is a best ­response to his belief. –  The above two assump.ons are common knowledge to all the players. •  The ra.onalizable set of strategy profiles, R, consists of all profiles consistent with our three assump.ons. Iterated Dominance •  Our simple theory from before was that no ra.onal player would play a dominated strategy. –  If a ra.onal player knows she’s playing a game with another ra.onal player it makes sense that she won’t expect the other player to play a dominated strategy. –  Her knowledge of the other player’s ra.onality helps her to refine her beliefs about the other player’s strategy. –  If the other player has similar knowledge about her, the other player may refine his beliefs. 12 Iterated Dominance Pigs S D PD WD PS 4, 2 6, –1 WS 2, 3 0, 0 For the submissive pig (S):  ­ WS dominates PS . A ra.onal S will never play PS. The dominant pig (D) doesn’t have a dominant strategy.  ­ What should D do if it knows S is ra.onal? uD(PD, WS) = 2 > uD(WD, WS) = 0 => Play PD. Iterated Dominance Example 2 1 L U 2, 6 C 8, 6 D 6, 0 M 2, 2 2, 8 4, 2 R 10, 0 6, 4 8, 0 •  M dominates R for player 2. –  If player 1 knows R won’t be played: •  D dominates U. •  Then M dominates L. •  Then D dominates C. –  The outcome will be D, M. 13 Iterated Dominance •  Procedure: –  First delete all dominated strategies for each player. •  This leaves us with a “smaller” game. –  Restrict aaen.on to the smaller game, and delete all dominated strategies in this smaller game. –  Con.nue un.l there are no dominated strategies in the “smallest” game. •  If all players have common knowledge of ra.onality, the structure of the game, etc., each will play a strategy that’s leq in the “smallest” game. –  In general we won’t be leq with a unique solu.on. Ra.onalizability •  We can go through a similar process with beliefs. –  Start with the idea that player i could play any strategy from Bi. •  The best response to some belief, θj. –  Player j knows the above and now plays the best response to some belief over this smaller set of beliefs. 14 2 1 L M R U 3, 9 3, 3 15, 0 C 12, 9 3, 12 9, 6 D 9, 0 6, 3 12, 0 Iterated Dominance/Ra.onalizability •  For two player, finite games, the set of strategy profiles that survive iterated dominance are iden.cal to R. –  Not necessarily the case for other games. –  Oqen easier to use iterated dominance to find R. 15 A Loca.on Game •  Consider a street with 5 evenly spaced apartments. –  Each apartment has 100 residents. •  Two food trucks decide where to locate. –  Si = {1, 2, 3, 4, 5}. –  Each consumer goes to whichever stand is closer. –  Each stand makes $1 in profit per customer. –  The trucks can locate in the same loca.on. –  Half the consumers go to each stand when the distance is the same. 1 2 3 4 5 (100) (100) (100) (100) (100) A B •  Suppose firm A locates at 1 and firm B at 3. –  Firm A will have 150 customers. uA(1, 3) = 150 •  All the customers at 1 and half of the customers at 2. –  Firm B will have 350 customers. uB(1, 3) = 350 •  All the customers at 3, 4 and 5 plus half of the customers at 2. •  Suppose firm A locates at 2 and firm B at 3. –  Firm A will have 200 customers. uA(2, 3) = 200 –  Firm B will have 300 customers. uB(2, 3) = 300 16 1 2 3 4 5 (100) (100) (100) (100) (100) A, B •  Find R for this loca.on game. •  Consider Player A. –  Does loca.ng at 3 dominate 2? –  uA(3, sB) versus uA(2, sB) for all sB in {1, 2, 3, 4, 5}. •  •  •  •  uA(3, 1) is NOT > uA(2, 1) (350 vs 400) uA(3, 2) is NOT > uA(2, 2) (300 vs 350) sA = 2 is not dominated by sA = 3 Loca.ng at 2 is beaer than loca.ng at 3 when Player B locates at 1 or 2. –  But with common knowledge of ra.onality we can perform iterated dele.on of dominated strategies. 1 2 3 4 5 (100) (100) (100) (100) (100) A, B •  Consider Player A. –  Does loca.ng at 2 dominate 1? –  uA(1, sB) versus uA(2, sB) for all sB in {1, 2, 3, 4, 5}. •  sA = 1 is dominated by sA = 2. –  Similarly 4 dominates 5. –  Similarly 1 and 5 are dominated for player B. •  Applying common knowledge of ra.onality –  Player A knows Player B will not ra.onally play 1 or 5 –  Aqer 1 and 5 are eliminated, now 2 and 4 will be dominated by 3. –  R = {3} x {3} 17 Calcula2ng the Ra2onalizable Set In simple finite games just repeat procedure for finding B=UD. Games with con.nuous ac.on spaces: do the same thing, but some details are different. Start with en.re strategy space. Itera.vely remove dominated strategies. Ra.onalizability: Guess .7 of the average game Ri0 = Si = [0,100] R1 = [0, 70] i Ri2 = [0, 49] . . Ra.onal players know that ra.onal players will not play above 70 and will not play above 49 Ri = [0,1] € Ra.onal players will not play above 70 Common knowledge of ra.onality Ra.onal players know… will not play above… 18 Cournot Duopoly Example Demand: P = 1 ­Q , Q=q1+q2 Zero produc.on cost Normal Form n=2 S1 = S2 = [0, ∞ ) Denote i‘s strategy qi Payoff func.ons: ui(qi,qj)=(1 ­qi ­qj)qi Cournot Duopoly Example Suppose player i has the belief θj about the strategy of player j. •  We think of qj as a random variable distributed according to according to θj. •  The expected payoff of qi: 19 Cournot Duopoly Example Cournot Duopoly Example 20 Cournot Duopoly Example Cournot Duopoly Example 21 MODIFIED Cournot Duopoly Example Now suppose: n=2 S1 = S2 = [0,∞) ui(qi,qj)=(1 ­qi+qj)qi BR(qj) = (1+qj)/2 Modified Cournot Duopoly Ra.onalizable Strategies q2 BR1 BR2 q1 R = [1, ∞) × [1, ∞ ) € 22 Second Strategic Tension Ra.onalizable set oqen contains many strategy profiles. •  We call this “strategic uncertainty.” •  Ra.onalizability only requires beliefs and behavior be consistent with common knowledge of ra.onality—beliefs can be wrong! Second Strategic Tension Stag Hunt 2 1 S H S 5, 5 4, 0 H 0, 4 4, 4 23 Second Strategic Tension Stag Hunt game: Stag Hare Stag 5,5 0,4 Hare 4,0 4,4 When only 1 other player’s coopera.on was required for the high payoff, 1/3 of the class played stag. When 100 other players coopera.on was required for the high payoff, 1/5 of the class played stag. 47 Nash Equilibrium 24 Notes on Nash Equilibrium A case in which all strategic uncertainty has been removed, so players behave exactly as others believe they will behave  ­  Self ­fulfilling, mutually ­confirming beliefs A strategy profile from which no player, unilaterally, can profitably deviate  ­  Self ­enforcing agreement  ­  “No regrets”  ­  Mutual best response Nash Equilibria in Matrix Games •  We can find pure strategy Nash equilibrium by iden.fying the best response(s) to each of opponent’s pure strategies. 2 1 P L S P 1, 2 0, 0 0,  ­10 L 0, 0 2, 1 0,  ­10 S  ­10, 0  ­10, 0  ­9,  ­9 •  The Nash equilibria are (P, P) and (L, L). 25 NE in 4 Player Nash Pickup Game Scene from “A Beau.ful Mind” movie Calcula2ng Nash Equilibrium 2. Games with an infinite number of pure strategies: (a) If calculus can be used (if payoff func.ons are differen.able and first ­order condi.ons characterize the best ­response func.ons): For each player i, calculate (b) If calculus does not apply: Rule out strategy profiles in which a player is obviously not best responding. Check others. 26 Calcula2ng Nash Equilibrium (when Calculus applies) General procedure 1.  Find best response func.ons  ­ Take account of any “edges” (where the func.on runs into the edge of the strategy space) 2.  Solve best response func.ons as a system of equa.ons. Nash Equilibrium q2 BR1 BR2 q1 Nash equilibrium = Intersec.on of BR1 and BR2 27 Calcula2ng Nash Equilibrium (when Calculus doesn’t apply) Bertrand Duopoly / Price Compe..on: n=2 S1 = S2 = [0, ) Q = 1000 ­p p=min(pi,pj) Marginal Cost=100 (cost per addi.onal unit) ui(pi,pj)=(1000 ­pi)Q ­100Q if pi<pj =(1000 ­pi)(pi ­100) if pi<pj 0 if pi>pj Third Strategic Tension •  Inefficient coordina.on. –  The players may coordinate on an inefficient Nash equilibrium. •  Even if individual incen.ves don’t go against the group’s. •  And if there is no strategic uncertainty. –  Examples: •  Keyboard layout. •  Betamax. 28 Pareto Coordina.on 2 1 A A 2, 2 B 0, 0 B 0, 0 1, 1 •  AA and BB are both NE •  AA is more efficient than BB 29 Mixed Strategy Nash Equilibrium If player i picks a mixed strategy she can’t get higher payoffs from any pure strategy, not even ones over which she’s mixing •  In fact, she must be indifferent between the pure strategies to which she assigns posi.ve probability. •  How do we make her indifferent to those strategies? With Player j’s (mixed) strategy Finding Mixed ­Strategy Equilibria For simple two ­player games: 1.  Calculate the set of ra.onalizable strategies by performing the iterated dominance procedure. 2.  Restric.ng aaen.on to ra.onalizable strategies, write equa.ons for each player to characterize mixing probabili.es that make the other player indifferent between the relevant pure strategies. 3.  Solve these equa.ons to determine the equilibrium mixing probabili.es. 30 Mixed Strategy Nash Equilibrium •  All pure strategies that are being played in a mixed ­strategy Nash equilibrium must yield the same expected payoff. •  All strategies that are not being played cannot yield a higher payoff. •  Every finite game has at least one Nash equilibrium (in pure or mixed strategies). –  Finite number of players and finite strategy space. 31 ...
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