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**Unformatted text preview: **Analysis of Sta.c Se1ngs •
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• 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}. •
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• 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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