Coordinating Expectations: Global Games with Strategic Substitutes

Transcription

1 : Global Games with Strategic Substitutes Stephen Morris Colin Clark lecture at the 2008 Australasian Meetings of the Econometric Society July 2008

2 Uniqueness versus Multiplicity in Economic Models I Economists have a love hate relationship with "unique" predictions I unique predictions = successful theory; multiple predictions = failure I multiple equilibria allow rich story telling, the role of self-ful lling beliefs,etc...

3 Uniqueness versus Multiplicity in Economic Models I Economists have a love hate relationship with "unique" predictions I unique predictions = successful theory; multiple predictions = failure I multiple equilibria allow rich story telling, the role of self-ful lling beliefs,etc... I This lecture: I I describe a rich family of strategic economic models where we can characterize when uniqueness does and does not arise... deliver insight into where uniqueness comes from...

4 Unifying Two Agendas What is this rich family? I Guesnerie (1992) and later work... I critizes "rational expectations hypothesis" for competitive economies; "expectational coordination" is assumed, not deduced I delivers a theory of when "uniqueness" = "expectational coordination" = "rational expectations hypothesis" should be expected to arise I Carlsson and van Damme (1993) and later "global games" literature... I complete information games have multiple equilibria I arbitrarily small noise about payo s ) unique equilibrium I CvD proved for all two player two action games I results generalize to games with strategic complementarities [Morris and Shin (2003), Frankel, Morris and Pauzner (2003)]

5 Global Games with Strategic Substitutes I Unify uniqueness conditions in Guesnerie s market expectation coordination and global games I This involves new, but special, results about global games with strategic subsitutes I Harrison (2005) analyzes global games with strategic substitutes and ex ante asymmetry and get di erent insights

6 Outline 1. Market Model in the spirit of Guesnerie (1992) 2. Normal examples unifying many literatures 3. Back entry into "global games"

7 Benchmark Model I Demand for a good D (p) = 1 p I A continuum of rms of mass 1; I rm i has cost x i of producing 1 unit I density of costs given by c.d.f. F I So supply curve: S (p) = F (p)

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12 Equilibrium Unique equilibrium price p solves D (p) 1 p = F (p) S (p) or b (p) 1 F (p) = p

13 Cobweb Dynamics Firms expect price from previous period to hold in the next period.

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19 Cobweb Dynamics Firms expect price from previous period to hold in the next period. I If everyone expected price p, supply would be F (p) implying price b (p) = 1 F (p) I observe that b 0 (p) = f (p) I Two possible limits of b k (p 0 ) I convergence to equilibrium, b k (p 0 )! p I two cycle convergence, p, p, p, p, p, p,. I Characterizing convergence to equilibrium I necc. and su. conn.: b 2 has unique xed point I necessary condition for eq. convergence: f (p ) 1 I su cient condition for eq. convergence: f (p) < 1 for all p 2 [0, 1]

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23 "Rationalizable" Prices I If everyone expected price p, supply would be F (p) implying price β (p) = 1 F (p). I Common knowledge that p 2 p, p ) supply 2 F p, F (p), market clearing price in b (p), b p. I kth level rationalizable prices (k even): b k ( ), b k ( ) I kth level rationalizable prices (k odd): b k ( ), b k ( ) I Since b is decreasing in p, rationalizable prices p, p, where p, p are smallest and largest xed points of b 2 I Recall that p is the unique xed point of b and thus a xed point of b 2. I Guesnerie extends these ideas in many directions I I "local expectational coordination" many markets, general stability conditions

24 "Rationalizable" Actions in Continuum Player Game I Continuum of players, mass 1. I Player i 2 [0, 1] has type x i (cdf F ) and chooses action a i 2 f0, 1g I Player i s payo is a i 0 Z i2[0,1] a i di I If player i expects other players to produce only if x j x, his best response is to produce only if x i 1 F (x ) = b (x ) I unique rationalizable outcome requires f (x ) = f (1 F (x )) = f (p ) 1 x i 1 C A

25 Normal Case I x i N θ, 1 β I θ common knowledge I unique rationalizable outcome for every θ if and only if β 2π I "private costs with no aggregate uncertainty"

26 Argument I b θ (p) = 1 F (p) = 1 Φ p β (p θ) I b 0 θ (p) = p βφ p β (p θ) I if β < 2π, jb 0 θ (p)j q I if β > 2π, let θ = 1 2 I b 12 1 = 0 2 I 12 b0 1 = 2 q β 2π > 1 β 2π < 1 for all p

27 Private Costs with Aggregate Uncertainty I x i N θ, 1 β I θ N θ, 1 α I unique rationalizable outcome for every y if and only if α + 2β β 2π α + β

28 Re-Parameterize I Takashi Ui I The x i are jointly normally distributed I Total ex ante variance of each x i : σ 2 = 1 α + 1 β I Correlation of x i and x j : ρ = I So α = 1 ρσ 2, β = 1 (1 ρ)σ 2 β α+β

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31 Common Costs with Noise I each rm s cost is θ I θ N y, 1 α I rm i observes x i = θ + ε i I ε i N 0, 1 β I unique rationalizable outcome for every y if and only if (α + β) (α + 2β) β 2π

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33 Strategic Complementarities I let payo s be instead of I a i = 1 "Invest" a i a i I a i = 0 "Not Invest" Z i2[0,1] Z i2[0,1] a i di a i di x i x i 1 C A 1 C A

34 Private Cost with no Aggregate Uncertainty (SC) I x i N θ, 1 β I θ common knowledge I unique rationalizable outcome for every θ if and only if β 2π I Quantal Response Equilibria [McKelvey and Palfrey 1995)]; Herrendorf, Valentinyi and Waldmann (2000), Baliga and Sjostrom (2004)

35 Private Costs with Aggregate Uncertainty (SC) I x i N θ, 1 β I θ N y, 1 α I unique rationalizable outcome for every y if and only if α 2 β (α + β) (α + 2β) 2π I Carlsson and van Damme (1993) Appendix B, Morris and Shin (2005) "Heterogeneity and Uniqueness" I as β! 0 or β!, uniqueness. but if α = and β!, multiplicity but puri cation

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37 Common Costs with Noise (SC) I each rm s cost is θ I θ N y, 1 α I rm i observes x i = θ + ε i I ε i N 0, 1 β I leading "linear" example in global games applications, e.g., Morris and Shin (2000, 2003) I unique rationalizable outcome for every y if and only if α 2 α + β 2π β α + 2β

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39 Global Games I let players be nite or continuum I asymmetric payo s u i (a i, a,θ) I I increasing di erences limit dominance I each rm s cost is θ = y + 1 p α η, η g () I rm i observes x i = θ + p 1 ε i, ε i f () β I unique rationalizable outcome for every y for high enough β I Frankel, Morris and Pauzner (2003)

40 Summary p.v. p.v. with agg. unc c.v. + noise SS β α+2β (α+β)(α+2β) 2π β α+β 2π β 2π α SC β 2π 2 β (α+β)(α+2β) 2π α 2 α+β β α+2β 2π

41 Summary with Re-Parameterization p.v. p.v. with agg. unc c.v. + noise SS σ 2 1 2π σ ρ 2π 1 ρ σ 2 1 2π SC σ 2 1 2π σ 2 1 2π 1 ρ 1+ρ σ 2 1 2π 1+ρ 1 ρ 1 ρ 1+ρ 1 ρ 2 1 ρ 2

42 Summary I Heterogeneity (increasing σ) favors uniqueness I Correlation favors uniqueness under strategic complementarities but not strategic substitutability I Global games small noise case correlation goes to 1 "faster" than heterogeneity goes to zero I Strategic Uncertainty intuition

43 Heuristic Uni ed Derivation I Consider game where i s payo is a i (1 + cba (qθ + (1 q) x i )) I c = 1 SC; c = 1 SS; q = 1 CC; q = 0 PC I 1 Agent i thinks x j N ρx i + (1 ρ) y, σ 2 (1 ρ) 2 I If agent i expects all others to "invest" if x j x, his expected payo is 8 < v (x i, x x 1 + cφ ρx i (1 ρ)y ) = p σ 1 ρ : 2 (1 q (1 ρ)) x i q (1 ρ) y 9 = ;

44 Heuristic Uni ed Derivation v (x i, x ) = 8 < : 1 + cφ x ρx i (1 ρ)y σ p 1 ρ 2 (1 q (1 ρ)) x i q (1 ρ) y v ρ = cφ () x i σ p 1 ρ 2 (1 q (1 ρ)) v 1 x = cφ () σ p 1 ρ 2 now if b (x ) solves v (x i, x ) = 0, cφ () p 1 b 0 (x σ 1 ρ ) = 2 1 q (1 ρ) + cφ () ρ σ p 1 ρ 2 9 = ;

45 Heuristic Uni ed Derivation, Algebraic Intuition I setting y = 1 + c 2 and x = 1 + c 2, b (x ) = 1 + c 2 I at this point b 0 (x ) = c 1 σ p 2π p 1 ρ 2 1 q (1 ρ) + c ρ σ p 2π p 1 ρ 2 cφ () p 1 σ 1 ρ 2 1 q (1 ρ) + cφ () ρ σ p 1 ρ 2 1 σ 2 1 2π (1 sn(c)ρ) 2 (1 ρ 2 ) (1 q (1 ρ)) 2

46 Ex Ante Asymmetry Harrison (2005) I let each rm s total cost be x i = θ + η i I η i g () I each rm observes z i = θ + p 1 ε i β I ε i f (), θ h () I each rm s type is pair (η i, z i ) I as β!, unique rationalizable outcome

47 Conclusion I Guesnerie has observed that "strong rational expectations equilibria" (unique rationalizable outcomes) arise only if supply is su cient elastic compared to demand I heterogeneity in costs ) inelastic supply I heterogeneity ) unique rationalizable outcomes in general by smoothing best replies I but in global games, small heterogeneity smooths best replies a lot I examined role of heterogeneity in generating uniqueness I in symmetric environments, correlation helps generate uniqueness under SC, not SS I general analysis of global games with SS much more complex