: Global Games with Strategic Substitutes Stephen Morris Colin Clark lecture at the 2008 Australasian Meetings of the Econometric Society July 2008
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...
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...
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)]
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
Outline 1. Market Model in the spirit of Guesnerie (1992) 2. Normal examples unifying many literatures 3. Back entry into "global games"
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)
Equilibrium Unique equilibrium price p solves D (p) 1 p = F (p) S (p) or b (p) 1 F (p) = p
Cobweb Dynamics Firms expect price from previous period to hold in the next period.
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]
"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
"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 B @1 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
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"
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
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
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
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
Strategic Complementarities I let payo s be instead of I a i = 1 "Invest" a i a i I a i = 0 "Not Invest" 0 B @1 + 0 B @1 Z i2[0,1] Z i2[0,1] a i di a i di x i x i 1 C A 1 C A
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)
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
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
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)
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
Summary with Re-Parameterization p.v. p.v. with agg. unc c.v. + noise SS σ 2 1 2π σ 2 1 1+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
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
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 = ;
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 = ;
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
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
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