IDENTIFICATION AND ESTIMATION OF A DISCRETE GAME OF COMPLETE INFORMATION P. Bajari, H. Hong, S. P. Ryan Econometrica 2010

28 February 2012

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Outline of the Talk

1

Introduction

2

The Model

3

Simulation

4

Estimation

5

Identification

6

Conclusion

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Introduction

Discrete games: games with a finite number of players, moves, events, outcomes, etc. Examples: labor force participation, entry, technology choice, advertising, analyst stock recommendations, etc. Usual representation in the normal form ⇒ For a given set of payoffs, (quite possibly despite refinements) multiple Nash equilibria. ⇒ Problem with Identification

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Introduction Approaches to deal with multiple Nash equilibria: 1 Equilibrium selection mechanism in which equilibrium is selected as part of an econometric model; e.g. random or extremal eqm selection (Bjorn and Vuong (1984), Jia (2008)) 2 Restricting attention to a particular class of games and identifying of payoff parameters even with multiple equilibria; e.g. entry games (Bresnahan and Reiss (1990,1991), Berry (1992)) 3 Bounds estimation of best-response functions; e.g. pure Nash eqm games (Tamer (2002), Jia (2005), Berry and Reiss (2007) and so on). Most papers in the literature are on entry games. Econometrically, a discrete game is naturally modeled by conditional logit or multinomial probit models.

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Introduction

This paper studies identification and estimation of discrete games, allows for multiple and mixed strategy equilibria, proposes a simulation-based estimator. Model primitives: players’ utilities an equilibrium selection mechanism for the probability of an equilibrium being played.

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Introduction

This approach, applicable to any normal form game with complete information (mixed strategies included), explicitly models and estimates equilibrium selection mechanism, proposes conditions for semiparametric identification of structural parameters underlying payoff functions and the equilibrium selection mechanism: Identification at infinity (Index restriction): Rich support for covariates assuming structural utility parameters can be indexed. Identification by exclusion restriction: Covariates that affect one player but not the other locally identify payoffs and equilibrium selection mechanism, e.g. firm-specific cost shifters.

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The Model Setup

For i = 1, 2, ..., N players, the finite action set is given by Ai A = Xi Ai with a = (a1 , ..., aN ) The vNM utility function, ui : A → R, is given by ui (a, x, θ1 , i ) = fi (x, a; θ1 ) + i (a) where x are covariates, θ1 parameters, a is the action profile and i (a) a random preference shock. A Nash eqm is a vector π = (π1 , ..., πN ) s.t. each agent’s mixed strategy is a best response.

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The Model Setup

i (a) reflects info about the utility (that is common knowledge to players), but unobserved by the econometrician i (a) are iid Q with Q the density gi (i (a)|θ2 ) and joint distribution g (|θ2 ) = i a∈A gi (i (a)|θ2 ). Let u = (u1 , ..., uN ). Then E (u) denotes the set of Nash equilibria, λ(π; E (u), β) denotes the probability that π ∈ E (u) is selected, and β is the vector of parameters

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The Model

Including λ allows for estimating how equilibrium is selected, rather than imposing an ad hoc rule For λ to generate a well-defined distribution, ∀u, β, X λ(π; E (u), β) = 1. π∈E (u)

A parsimonious, parametric model of λ, λ(π; E (u), β) = P

exp(β · y (π, u)) 0 π 0 ∈E (u) exp(β · y (π , u))

By estimating β, one can find which equilibrium best matches the outcomes observed in the data. (⇒ Sort of a rationality measure.)

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The Model

Combining everything together, the probability of observing a certain action profile in one play:  ! Z  N  X Y g ()d. P(a|x, f , λ) = λ(π; E (u(f , ), x), x) π(ai )   i=1

π∈E (u(f ,),x)

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Simulation Goal: Simulating the model to get the method of simulated moments (MSM) estimator. Simulation of the probability of observing a certain action profile in one play: P(a|x, θ, β) =

 Z  

X

λ(π; E (u(x, θ1 , )), β)

π∈E (u(x,θ,))

N Y i=1

!  πi (ai ) g (|θ2 )d. 

⇒ Computationally burdensome! Changing the variable of integration from  to u, with the density of u|θ, x being h(u|θ, x) and i ∼ iiN(0, 1), the density becomes YY h(u|θ, x) = φ(ui (a) − fi (x, a; θ1 )|0, σ) i

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a∈A

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Simulation

With the change of variable from  to u, prob. of a certain action profile in one play:  ! Z  X N  Y P(a|x, θ, β) = λ(π; E (u), β) πi (ai ) h(u|θ, x)du   i=1

π∈E (u)

With importance sampling, where q(u|x) is the importance density, P(a|x, θ, β) =

 Z  X

λ(π; E (u), β)



i=1

π∈E (u)

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N Y

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!  h(u|θ, x) πi (ai ) q(u|x)du  q(u|x)

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Simulation

(s)

(s)

For a given value of x, a pseudorandom sequence u (s) = (u1 , ..., uN ), s = 1, ..., S, is drawn from q(u|x). Then P(a|x, θ, β) is simulated as  ! N S  X  h(u (s) |θ, x) Y X 1 ˆ P(a|x, θ, β) = λ(π; E (u (s) ), β) πi (ai )  q(u (s) |x)  S (s) s=1

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π∈E (u

i=1

)

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Simulation

Advantages of this simulation approach: ˆ P(a|x, θ, β) is an unbiased estimator of P(a|x, θ, β). Using u (s) , θ and β do not enter into E (u (s) ). ⇒ No need to recompute the eqm set with parameter changes Simulator function is smooth in underlying parameters. ⇒ MSM objective function will be well behaved.

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The Estimator

Maximum simulated likelihood estimator is asymptotically biased unless √ST → ∞.

√

T −consistent but

Maximum simulated moments estimator is consistent and unbiased for any value of S. To form the estimator, enumerate elements of A from k = 1, ..., #A ⇒ #A − 1 linearly independent conditional moments (sum of #A probabilities is 1). ωk (x) : vector of weight functions

At true parameters of the DGP, E [1(at = k) − P(k|x, θ, β)]ωk (x) = 0.

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The Estimator The sample counterpart of the moment for the k−th element of a vector of #A − 1 moments is T 1 X mk,T (θ, β) = [1(at = k) − P(k|xt , θ, β)]ωk (xt ). T t=1

So, the simulated analog is T 1 X ˆ m ˆ k,T (θ, β) = [1(at = k) − P(k|x t , θ, β)]ωk (xt ). T t=1

Then, for positive-definite weighting matrix WT , the MSM estimator is ˆ β) ˆ = arg min m (θ, ˆ T (θ, β)0 × WT × m ˆ T (θ, β). (θ,β)

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Identification

Unknown model parameters to be identified: f (a, x), entering into the utility function λ(x), probability of a certain equilibrium being played Two approaches to identification: 1 Sufficient conditions to identify payoffs and the eqm selection mechanism as covariates’ support gets larger 2 Identification based on agent-specific payoff shifters

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Scale and Location Normalizations

Assumption The payoffs of one action for each agent are fixed at a known constant. Adding a constant to all deterministic payoffs does not perturb the set of equilibria. ⇒ Location normalization.

Assumption The joint distribution,  = (i (a)) is independent and known to all agents and the econometrician. Multiplying all deterministic payoffs by a positive constant does not alter the set of equilibria. ⇒ Scale normalization

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Difficulty of Nonparametric Identification

P(a|x, f , λ) =

 Z 

X

λ(π; E (u(f , ), x))



N Y i=1

π∈E (u(f ,),x)

!  g ()d πi (ai ) 

Holding x fixed, model primitives to be identified: f (a, x) and λ(x) Denote P(a|x) = H(f (a, x), λ(x)) where H is the map defined above.

Assumption H(.) is continuously differentiable with the Jacobian denoted by DHf ,λ (x)

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Difficulty of Nonparametric Identification

Global identification: For f 0 (a, x) and λ0 (x) satisfying the P(a|x) equation, if no other pairs of f (a, x) and λ(x) also satisfy it, then (f 0 (a, x), λ0 (x)) is globally identified. Local identification: ∃ Nx (an open neigborhood of (f 0 (a, x), λ0 (x))) s.t. there’s no other ˜ ˜ vector (f˜(a, x), λ(x))) ∈ Nx with (f 0 (a, x), λ0 (x))) 6= (f˜(a, x), λ(x))) that also satisfies P(a|x).

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Difficulty of Nonparametric Identification

Necessary condition for identification: Full-rank DHf ,λ (x) Sufficient conditions for local identification: ”The rank of DHf ,λ (x) equaling to the number of parameters in (f (a, x), λ(x))) at (f 0 (a, x), λ0 (x))).” (Rothenberg (1971), Thm 6) global identification: ”∃ a square submatrix W of DHf ,λ (x) with dimension (f (a, x), λ(x))) s.t. W has a positive determinant and W + W 0 is positive definite throughout the parameter space.” (Gale and Nikaido, (1965)) ⇒ Checking these conditions in games with multiple players and strategies can be difficult.

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Identification at Infinity

Covariates have full support Mean utilities are defined by a linear index of the covariates ⇒ fi (a, x) = xia βia for all ai . Two-step identification procedure: Identify mean utilities with covariate values that give a unique eqm with prob close to 1 → Perturb the covariates locally to identify the utility parameters Under invariance assumption of eqm selection mechanism, identify the eqm selection probabilities from observable choice probabilities

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Identification at Infinity Assumption k

For any i = 1, ..., N and action profile k−i ∈ A−i , there exists a set τ−i−i of covariates x s.t. lim

k

P(a−i = k−i |x) = 1.

kxk→∞,x∈τ−i−i

For each player i, covariates x can be shifted along a dimension s.t. k−i is a dominant strategy.

Assumption ∀ i, a ∈ A s.t. mean utility fi (a, x) is not normalized, ∃ L0 > 0 s.t. a

inf min eigE [xia xia |x ∈ τ−i−i , kxk ≥ L] > 0.

L≥L0

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Identification at Infinity

Theorem Under those assumptions, βia and hence fi (a, x) is identified up to the ex ante normalization for all i and all a for fixed x. Now, an invariance property to identify the eqm selection probabilites:

Assumption Equilibrium selection probabilities depend only on latent utility indices, ρ(x, ) = ρ(u(a, x, )) and are scale invariant w.r.t the latent utility indices, i.e. ρ is homogeneous of degree 0.

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Identification at Infinity Finally, to ensure total utility indices to be approximated arbitrarily well:

Assumption There exists a set τ s.t. for all δ > 0,   fi (a, x) lim min P − 1 < δ = 1. ui (a, x, ) |x|→∞,x∈τ i,a

Theorem Under all assumptions, equilibrium selection probabilities ρ(u(x, )) are all identified from observed choice probabilities whenever the cardinality of E (u(x, )) is less than or equal to #A − 1.

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Exclusion Restrictions

Search for variables that influence one equation but not the other, as in sample selection models. Here, covariates that shift the utility of agent i but not uj , for any j 6= i, or the eqm selection mechanism λ.

Assumption For each i, there exists some covariate xi that enters ui and not the others. Furthermore, ρ(u(α, x, )) depends on u(α, x, ) only through a set of sufficient statistics. λ cannot depend freely on utility indices.

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Exclusion Restrictions

Theorem Under these assumptions, if #xi are sufficiently large, the necessary order condition is satisfied. ⇒ Identification is possible if we have covariates that are indexed by agent’s identity, i. Examples of imposing exclusion restrictions in discrete games: Strategic entry games: Airline operations (a possible shifter: number of connecting routes) (Holmes (2008), Jia (2008)) Technology adoption in the presence of network effects: Adoption of video-conference systems by in-firm employees (possible shifter: level of employees)

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Application

The application presented in the paper: California Highway Procurement Auctions: Strategic entry by bidders Estimation of the probability of selecting a particular eqm, λ All strategic possibilities allowed Two stages of the games: Bidding and Entry. A possible equilibrium is decided based on the expected entry profits. λ depends on whether joint payoffs are maximized (efficiency).

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Conclusion

1 2 3 4

Estimation of both utilities and equilibrium selection mechanisms Generally applicable toolbox for normal form games In-depth discussion of identification Theory overlapping with a real-life example

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Conclusion

Possible directions for future research: Identification and estimation of network games: For example, identification of peer effects on learning, estimating the diffusion of information in general networks, etc. Extensions to continuous games (e.g. Cournot competition) via set-identification tools Discrete games with incomplete information using Bayesian updating

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