3

Empirical Game Theoretic Models: Computational Issues OLIVIER ARMANTIER and JEAN-FRANÇOIS RICHARD Department of Economics, University of Pittsburgh, Forbes Quad, 4D11 Pittsburgh, PA 15260, U.S.A., E-mail: [email protected]

Abstract. This paper discusses computational issues raised by a generic solution and estimation methodology applicable to a broad range of empirical game theoretic models with incomplete information. By combining the use of Monte Carlo simulation techniques with that of smooth kernel estimation of empirical distribution functions, the authors develop a numerical algorithm of unparalleled performance and flexibility applicable, in particular, to models for which no operational solutions currently exist. An illustration to a set of procurement data from the French aerospace industry is used to illustrate the operation of this algorithm. Key words: game theory, Monte Carlo simulation, smooth kernel estimation

1. Introduction Game theory plays a central role in modern economic analysis. Rich data sets are available relative to such activities as auctions and procurements (Laffont, 1997). Nevertheless, the estimation of empirical game theoretic models remains a formidable task due, in particular, to the need to account properly for strategic behavior from participants. In two recent papers, Armantier et al. (1997) and Florens et al. (1997) – hereafter AFR and FPR, respectively – develop a general methodology for the analysis and estimation of a broad class of empirical game theoretic models. Its implementation raises non trivial numerical problems since it typically requires solving a pair of nested fixed-point problems under Monte Carlo (MC) simulations. The present paper addresses these numerical issues with the objective of enabling potential users to develop their own applications as the need arises. The paper is organized as follows: in Section 2 we briefly introduce the class of models under consideration, the methodology developed by AFR and FPR, and a flowchart of necessary computations. Section 3 discusses computational issues and constitutes the core of the paper. The computation of estimator standard errors is discussed in Section 4. An example is presented in Section 5, and Section 6 concludes.

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2. Solution and Estimation of Game Theoretic Models In order to keep notation simple, we shall not discuss the most general formulation of the class of models under consideration and, therefore, shall ignore such complications as data censoring, exogenous variables, endogenous participation decisions, and the like. A key advantage offered by our methodology arises because such extensions are typically to be accounted for in the initial selection of what will be referred to as an ‘unfeasible’ estimator, but do not require further modifications of the computational algorithm that leads to its transformation into a ‘feasible’ estimator. The class of games we are considering can be described as follows: players privately draw individual ‘types’ or ‘signals’ from a probability distribution F , which is assumed to be common knowledge to all. These unobservable signals are then transformed into observable actions by means of a transformation or ‘strategy’ that depends not only on one’s own signal but also on the distribution F itself. This dependence upon F captures the strategic implications of players’ understanding that the outcome of the game also depends upon their rivals’ actions, which are themselves transforms of random signals drawn from F . Game theory enters here by prescribing that players’ strategies be ‘mutual best responses’ relative to one another and, therefore, jointly correspond to an appropriate ‘equilibrium solution’ of the game under consideration. 2.1.

BASELINE MODEL

Consider the following set of assumptions: A.1 The unobservable types ξi ∈ Rk , for i : 1 → n, are identically independently distributed (i.i.d.) following a distribution F with support 4 ⊂ Rk , indexed by a parameter θ ∈ 2 ⊂ Rp . A.2 A type ξi is transformed into an observable action xi ∈ Rk by means of a transformation ϕi : 4 × 2 → Rk xi = ϕi (ξi ; θ),

i : 1 → n.

(1)

Note: If ϕi ≡ ϕ for all i, then the game and its solutions are said to be ‘symmetric’. Let Xiθ denote the support of the distribution Gi of xi as induced by transformation (1). A.3 The applications ϕi are continuously differentiable and are injective as functions of ξi , for any given θ ∈ 2. Let ϕi−1 : Xiθ × 2 → 4 denote the inverse transformation ξi = ϕi−1 (xi ; θ),

i : 1 → n.

(2)

Note: The index i often represents a pair (j, `), where j : 1 → J and ` : 1 → L denote plays of the game and players, respectively. Examples of relevant situations

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

5

are (1) a single play of a game (j = 1), where n players draw their types independently from one another; (2) n plays of a given game, where ξi denotes the types of the individual winners (in which case F denotes the distribution of the highest order statistic from L i.i.d. draws under the assumption that the game is ‘efficient’, i.e., that the winner holds the highest signal); or (3) J plays of the game with L players so that n = J · L, under the assumption that types are i.i.d. for all pairs (j, `). The solution concepts discussed below are defined in relation to a single play of the game, in which case i will be used as the players’ index for the ease of notation. Actual data sets often consist of repeated plays of the game and, as mentioned above, can be censored in various ways. 2.2.

ESTIMATION ( FPR )

Types are not observed. Furthermore, as discussed below, the ϕi ’s that result from equilibrium solutions often require fairly complex calculations and typically do not have a closed form solution. Therefore, the distribution Gθi of the observable actions rarely constitutes an operational basis for ‘direct’ estimation of θ (notwithstanding additional statistical complications originating from the fact that the support Xiθ of Gθi generally depends upon θ itself). The generic estimation principle introduced by FPR does not require derivation of Gθi and proceeds as follows: (1) Initially, one selects an ‘unfeasible’ estimator θ˜n , i.e., a function of the unobserved types, that would have desirable statistical properties if types could be observed. At a high level of generality θ˜n can be defined as the (unique) solution of an appropriate set of ‘moment’ conditions 1X h(ξi ; θ˜n ) = 0, n i=1 n

(3)

where h : 4 → Rp is a function such that θ is the unique solution of the moment condition Eξ |θ [h(ξ ; θ)] = 0.

(4)

(2) The corresponding ‘feasible’ estimator θˆn is then defined as a solution of the transformed moment conditions 1X h(ϕi−1 (xi ; θˆn ); θˆn ) = 0. n i=1 n

(5)

Let ξˆn = (ξˆi,n ; i : 1 → n), where ξˆi,n = ϕi−1 (xi ; θˆn ), denote the corresponding estimator of the unobserved types. In practice, as discussed further below, (ξˆn , θˆn )

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OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

generally obtains as a (fixed-point) solution to the system consisting of Equations (2) and (3). FPR provide sufficient conditions for the (statistical) identification of θ from the sole observation of the xi , as well as for the (local) existence of a unique √ fixed-point solution (ξˆn , θˆn ). They also provide an asymptotic distribution for n · θˆn . 2.3.

CONSTRAINED STRATEGIC EQUILIBRIUM SOLUTIONS ( AFP )

We now discuss equilibrium solutions and consider a single play of the game, letting i : 1 → n denote players.? Game solution concepts require the ϕi to be mutual best responses to one another in the sense that no single player has an incentive to ‘deviate’ on his/her own from the prescribed decision rule. Unconstrained Nash Equilibrium (NE) solutions essentially impose no restrictions upon the class of strategies under consideration (except for standard technical conditions). There are not many games for which analytical NE solutions exist. Example: First price Independent Private Value (IPV) auction. Let n = N+1 players draw their type ξi > 0 independently from a common univariate distribution with cumulative distribution function (c.d.f.) F (·|θ) and submit non negative sealed bids. The highest bidder wins the item and pays the winning bid. The symmetric NE bid function is given by Riley and Samuelson (1981) Rξ ϕNE (ξ ; θ) = ξ −

0

F N (u|θ)du . F N (ξ |θ)

Note that, if 4 = R+ , then X θ = [0, x(θ)] ¯ with Z ∞ x(θ) ¯ = [1 − F N (u|θ)]du .

(6)

(7)

0

 In more general cases, even though it might be possible to derive first-order conditions for a NE solution, often in the form of a (set of) differential equation(s), there might not exist operational numerical procedures to compute such a solution. Marshall et al. (1994) illustrate the numerical difficulties associated with a simple asymmetric auction game with only two (classes of) players. In view of the conceptual as well as numerical complications often associated with unconstrained NE solutions, one is lead to question their validity as a basis for inference in real-life situations. See, for example, Binmore (1987), Simon (1995), Rosenthal (1993a,b) as well as the rapidly expanding literature on ‘bounded rationality’. ? The MC simulation framework which is introduced below also (conceptually) applies to more

complex ‘repeated game’ situations (learning, . . . ), though at the cost of additional notation and, possibly, of significant increases in computational complexity.

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

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These are the main reasons that led AFR to propose an operational concept of Constrained Strategic Equilibrium (CSE) solution that can be evaluated by MC simulations for a broad class of single play games – and can also serve to produce numerical approximations to the more complex unconstrained NE solutions. Briefly, one initially assigns to each player a constrained set of strategies, typically characterized by a simple preassigned functional expression depending upon a small number of auxiliary parameters, say xi = ψi (ξi ; di ),

di ∈ Di .

(8)

As illustrated below, low-order polynomials constrained to be monotone on 4 can usefully be considered. Piecewise linear strategies, as in Levin et al. (1997), can also be considered. In specific applications these can usefully be interpreted as ‘rules of thumb’ in the sense of Rosenthal (1993a,b). When applicable, symmetry obtains under the additional requirement that ψi ≡ ψ and di ≡ d for all i. Next, for any given θ ∈ 2, one can solve the game in ‘strategic form’, which, as discussed below, can be done by MC simulation, to obtain a set of mutually best response parameters di∗ (θ). The corresponding CSE solution of the game is then given by xi = ψi (ξi ; di∗ (θ)) =say ϕCS (ξi ; θ),

(9)

for i : 1 → n. More specifically, let the indices i and −i denote an arbitrary player and his/her n − 1 rivals, respectively. Up to an obvious permutation of indices, the vectors regrouping auxiliary parameters, types, and actions are factored conformably with one another into       di ξi xi d= ξ= x= , (10) d−i ξ−i x−i respectively. Let ψ(ξ ; d) denote the stacked individual strategies ψ(ξ ; d) = (ψi (ξi ; di ); i : 1 → n) = (ψi (ξi ; di ), ψ−i (ξ−i ; d−i )),

(11)

and let Ui (x; ξ ) denote the payoff function of player i, which may (ex-post) depend on types as well as on actions. The ‘strategic form’ of the game requires evaluating expected payoffs under arbitrary assignments of strategies. Under the present formulation, these expected payoffs are defined as Vi (d; θ) = Eξ |θ [Ui (ψ(ξ ; d); ξ )].

(12)

Let dˆi (d−i ; θ) denote player’s i best response to an arbitrary choice of parameters by rival players; dˆi (d−i ; θ) = ArgMaxVi (d; θ). di ∈Di

(13)

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OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

A CSE solution for the game is then defined as a fixed-point solution to the problem, whereby ∗ di∗ (θ) = dˆi (d−i (θ); θ),

i : 1 → N.

(14)

Example (continuation): First Price IPV. Let ψ denote a symmetric affine ‘rule of thumb’ ψ(ξ ; d) = d · ξ,

0 ≤ d ≤ 1.

(15)

As shown in AFR, the unique CSE solution is given by d ∗ (θ) = with

Z A(θ) = n

∞

A(θ) , A(θ) + B(θ)

ξ 2 F n−1 (ξ |θ)f 2 (ξ |θ)dξ ,

(16)

(17)

0

Z B(θ) =

∞

ξ F n (ξ |θ)f (ξ |θ)dξ ,

(18)

0

where f denotes the density function associated with F . If ξ is uniformly distributed on [0,1], so F (ξ ) = ξ , then the unconstrained NE solution (6) coincides with n the CSE solution (15), both being given by n+1 · ξ. 2.4.

COMPUTATIONS : FLOWCHART

Consider now an empirical situation for which a set of observed actions X = (xj ` ) is available, appropriate constrained strategy sets have been assigned (or, in simple cases, first-order conditions for an unconstrained NE have been derived), and an unfeasible estimator θ˜n (ξ ) has been selected. A general recursive algorithm for computing the corresponding feasible point estimate θˆn , and also ξˆn , if required, is outlined in the flowchart in Figure 1. 3. Computational Issues The implementation of the algorithm just outlined raises a number of significant computational issues which we discuss now. The main difficulties to be addressed are those relating to (a) the NE inversion module, (b) a more general problem of occasional non-invertibility, and (c) the CSE simulation module. Other less critical issues are discussed briefly at the end of this section.

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

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Figure 1. Flowchart.

3.1.

NE INVERSION MODULE

In practice the distribution F may depend on exogenous variables that are specific to each play of the game under consideration. As above, to keep notation simple, we discuss here inversion only within a specific play of the game, using a single index i to denote participants. The calculations described here are repeated across all plays of the game for which data are available. NE strategies typically are solutions of a (set of) differential equation(s). Regardless of whether these solutions obtain analytically or numerically, they represent complicated transformations of the types, and their (numerical) inversion is

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no trivial operation. It essentially requires solving (with respect to ξ ) non linear equations of the form φi (ξi ; xi , θ) = ϕi (ξi ; θ) − xi ,

i : 1 → n.

(19)

We found that the solution routines included in GAUSS or IMSL can reliably produce accurate solutions to (19) if one makes good use of all the information relative to the NE strategies themselves. Note, in particular, that the ϕ’s are generally monotone in ξi , that their derivatives w.r.t. ξi obtain as a direct byproduct of the associated differential equation(s), and, most importantly in view of our own experience, that good initial guesses and fairly ‘tight’ brackets are generally available for the types. Let us illustrate the last point in the context of the example introduced in Section 2.3 (first price IPV auction). Let ξ¯ and x(θ) ¯ denote the upper bounds of 4 and X(θ), respectively. In cases where 4 is unbounded it generally proves helpful to assign to ξ¯ a sensibly large value, pending subsequent verification that it does not affect the final estimation results. In view of the monotonicity of the ϕi , we can assume without loss of generality that observed actions and unobserved types have been ranked in decreasing order, i.e., that x1 ≥ . . . ≥ xm and ξ1 ≥ . . . ≥ ξm , where xi = ϕi (ξi ; θ). In view of frequent data censoring, m represents here the number of players whose actions are actually being recorded (for example, m = 1 if only the winning bid is recorded) and is not to be confused with n = N + 1, the number of active participants that enters Equation (6) and is assumed to be common knowledge. We temporarily assume that x1 ≤ x(θ), ¯ a critical condition that will be discussed further in Section 3.2 below. Finally, let ξi0 denote an initial guess for ξi . Note: Careful examination of the problem under consideration and, in particular, of the economic issues at stake often suggests ‘sensible’ guesses for the types. Furthermore, in view of the recursive nature of our algorithm, beyond the initial step we can use the type estimates obtained at step r − 1 as guesses for step r. Actually, types are generally easier to guess than the parameters themselves, which is precisely why our algorithm is initialized on types. We then invert individual bid functions in the order associated with the above ranking. To avoid searches outside the relevant intervals, which, in our experience, often results in subsequent failure of the entire algorithm, we find it useful to adopt a logistic transformation of the form ξi = xi +

ξi−1 − xi , 1 + bi exp(−ki yi )

yi ∈ R, i : 1 → n ,

(20)

with ξ0 = ξ¯ and where bi and ki are chosen in such a way that the search ‘centers’ on an appropriate bracket for ξi . In particular, we can take bi = (ξi−1 − ξi(r−1))/(ξi(r−1) − xi ), so that (i) ξi = ξi(r−1) for yi = 0, and (ii) searching for yi ∈ R is equivalent to searching for ξi ∈ [xi , ξi−1 ]. Actually, ki is a scale para-

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

11

meter that must be assigned in reasonably sophisticated solution routines, though default values are available. Another numerical problem may arise when ϕ has an analytical expression that depends on an integral that must be numerically evaluated, as in the case in Equation (6). It is then essential to use an integration routine powerful enough to produce accurate results across the relevant range of values for ξ and θ. Note, in particular, that for N large, the integrand in Equation (6) can peak quite sharply over a small subinterval of the range of integration. The subroutines QDAG and QDAGS in IMSL have proved sufficiently flexible and reliable for our purpose. The same cannot be said of the procedure INTQUAD1 in GAUSS, which lacks flexibility and often produces inaccurate results for integrals like that in Equation (6), even under the maximum number of points allowed. Common sense also dictates avoiding potential sources of rounding errors by judicious use of integrations by part. For example, Equation (6) can be advantageously replaced by the equivalent formulation Rξ udF N (u|θ) ϕNE (ξ ; θ) = 0 N , (21) F (ξ |θ) except for values of ξ (very) close to zero, for which we can use the approximation ϕNE (ξ ; θ) ∼ = ξ. 3.2.

NON - INVERTIBILITY PROBLEM

The fact that the support X(θ) of the distribution of the actions generally depends on θ can occasionally prevent inversion for some observed actions. If, in particular, an action xi lies outside of X(θˆn(r−1) ), then the inversion module will fail on inversion of ϕi at step r. In auction problems such failures typically occur for high bids at early steps of the iterative algorithm, especially if ‘poor’ initial guesses of the types were to produce values of the parameters inherently incompatible with observed actions. Failures at near-convergence might flag out model misspecification or, at minimum, ‘anomalies’ in the corresponding play(s) of the game. We have experimented with three methods to eliminate this non-invertibility problem. (1) At step r, we temporarily delete any action that cannot be inverted and compute θˆn(r) solely on the basis of the nr < n actions that can. Though this procedure is easy to implement, it often produces subsequent parameter estimates that remain incompatible with the deleted observations and, therefore, converges toward an ’invalid’ final estimate. This is due to the fact that the observations that cannot be inverted often turn out to be the most ‘influential’ ones (e.g., they are high bids in an auction problem). Deleting such observations can irretrievably bias the estimation of θ. (2) Somewhat more reliably though less ‘generically’, we have found that noninvertibility can be greatly reduced – and often completely eliminated – by judi-

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OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

cious selection of initial values. As already discussed, this is often easier to achieve with types than with parameters, hence the structure of our recursive algorithm. So, when our first method fails immediately to reintegrate the deleted observation(s), we suggest restarting the algorithm with new initial values for types and, in particular, for those associated with the ‘critical’ observations. In an auction problem, for example, one could raise the initial guesses associated with those observations in an obvious attempt to produce parameter estimates that can accommodate them better. (3) Finally, a ‘brute force’ procedure for solving cases where the simpler remedies fail consists of bypassing the inversion problem by conflating the three central modules of our algorithm into a single joint optimization module of either of the following forms (depending upon the actual definition of θ˜n ): min

ξ ∈4n

n X

[xi − ψ(ξi ; d ∗ (θ˜n (ξ )))]2 ,

(22)

i=1

or minn

(ξ,θ)∈4 ×2

n X

{[xi − ψi (ξi ; d ∗ (θ))]2 + ci [hi (ξi ; θ)]2 } ,

(23)

i=1

where the ci ’s are appropriate scaling factors. Note that, if a fixed-point solution (ξˆn , θˆn ) exists, then it will obtain from either of these optimization problems whose objective function at convergence equals zero (up to the usual errors associated with stopping rules). In our experience, this brute force algorithm has proved highly reliable. Since, however, it can be CPU intensive for moderate to large sample sizes, it should be used only as a last resort. A more sophisticated implementation, which we propose to develop and test in the future, consists in replacing any step that fails on inversion by a single step of the joint optimization problem, immediately reverting to the baseline algorithm. 3.3.

CSE SIMULATION MODULE

By using a CSE approach instead of the NE approach just discussed, we generally eliminate all numerical problems associated with inversion per se (but not the ‘noninvertibility’ problem itself) since, by definition, CSE rules typically have simple (monotone) functional forms. On the other hand, we have to address a new set of numerical problems associated with the search for a CSE fixed-point solution as defined by Equations (12)–(14). These problems originate largely from the fact that, in many games, payoff functions actually are discontinuous functions of the di . For example, most auctions are ‘winner take all’ games in that infinitesimal changes in the di may produce a different winner and, therefore, discrete jumps in some of the resulting payoff functions. This would not be a problem if we could obtain analytical expressions

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

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for the expected payoff functions themselves. For any given values of d and θ, the integrand in (12) includes an indicator function delineating the region in 4n over which player i is the winner. The boundaries of that region are continuous functions of d and θ. Therefore, expected payoffs themselves are continuous in d and θ. In most problems, however, we cannot expect to produce analytical expressions for expected payoffs. It is natural, then, to consider estimating them by MC simulation techniques, since games played under preassigned CSE rules can trivially be ‘played’ under simulated types. Expected payoffs are then approximated by arithmetic means of the actual payoffs that result from a sufficiently large number of MC replications of the game. Our experience unequivocally suggests that damping out the effect of payoff discontinuities up to a point where the CSE fixed-point problem can actually be solved numerically requires prohibitively large numbers of MC draws (in the hundreds of thousands if not in the millions!). Actually, continuity w.r.t. θ trivially results if, in addition to the kernel implementation discussed below, we apply the technique known as ‘Common Random Numbers’ (CRN), whereby random draws {ξ˜(t )(θ); t : 1 → T } from F (·|θ) for any arbitrary value of θ are obtained by transforming a fixed (common) set of ‘canonical’ random draws {u˜ (t ) ; t : 1 → T } whose distribution does not depend upon θ, say ξ˜(t )(θ) = g(u˜ (t ) ; θ),

t : 1 → T.

(24)

See e.g. Richard (1996, Section 23.2.3) for a discussion of the CRN technique. Within a univariate framework, a common implementation of CRN consists in transforming uniform random draws by the inverse of the distribution F. As discussed (Devroye, 1986), there exist more efficient techniques for generating random draws from most distributions, such as composition and rejection. It is, however, easy to verify that such techniques can produce draws that are discontinuous in θ even when common seeds are reused. Note also that faster transformations may be available for specific distributions. For example, draws from an arbitrary Normal distribution are most efficiently produced by a linear transformation of univariate standardized Normal draws, which is based upon a Cholesky decomposition of its covariance matrix. Under CRN draws, MC estimates of expected payoffs are given by T 1X V¯i,T (d; θ) = Ui (ψ(ξ˜(t )(θ); d); ξ˜(t ) (θ)), T t =1

(25)

which, as already discussed, often are discontinuous w.r.t. d. As introduced by AFR, two key modifications of this ‘naive’ MC implementation not only produce estimates of the Vi that are continuous w.r.t. d and θ, but also lead to a CSE module that performs better than one consisting of implementing Equations (13) and (14). The first modification consists in replacing the arithmetic means in Equations (25) by smooth ‘kernel’ estimates thereof. The second one

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restates the fixed-point problem in a way that is partially analytical and also avoids actually solving the optimization problem in Equation (13) at each step of the iterative search for a CSE solution to Equation (14). In view of their importance, these two modifications are discussed next in separate subsections. 3.3.1. Kernel Estimates To keep notation simple, we illustrate the use of kernel estimates in the determination of CSE’s with reference to a fairly common auction scenario in which a player’s payoff depends only on one’s own type, own bid, and highest rival bid,? say Ui (ψ(ξ ; d); ξ ) = Ui (ψi (ξi ; di ), βi (ξ−i ; d−i ); ξi ),

(26)

βi (ξ−i ; d−i ) = max ψj (ξj ; dj ).

(27)

where j 6 =i

The point to note here is that, though the distribution of βi is all but impossible to derive analytically, it is trivial to produce random draws from it by MC simulation of the actual game for any arbitrary values of d−i and θ. AFR propose to use such draws to construct kernel estimates of the distribution and/or density function of βi , as required for the evaluation of expected payoffs. To keep presentation simple, we also assume that types are independent.?? Let Gi (b|d−i , θ) and gi (b|d−i , θ) denote the distribution and density functions of βi , respectively and let bi (d−i , θ) denote the lower bound of the support of that distribution. The expected payoff of player i is then given by "Z # Z ψi (ξi ;di ) Vi (d; θ) = fi (ξi |θ) Ui (ψi (ξi ; di ), b; ξi )gi (b|d−i , θ)db dξi , (28) 4

bi (d−i ,θ)

where, in view of the bounds of integration for b, Ui can actually be interpreted as player’s i payoff conditionally on winning. We now use S MC plays of the game to construct S draws from βi , say {β˜i,s (d−i , θ); s : 1 → S}. Let K(·) and k(·) denote a standardized univariate ‘kernel’ (distribution) function and its derivative (density), respectively (Hardle, 1990). The standardized Normal c.d.f. is commonly used for this purpose, but additional simplifications (and reductions in computing time) result if we use an analytical c.d.f. instead, such as that of the logistic distribution. Kernel estimates of gi and, if required, of Gi are then given by ! S 1 X b − β˜i,s (d−i ; θ) gˆi,S (b|d−i , θ) = k , (29) hS s=1 h ? As discussed in AFR and illustrated by the application in section 5 below, multivariate bids can be replaced by univariate ‘scores’ when bids are ranked according to a scoring rule. ?? Extensions to more general cases essentially require bivariate kernel estimation of the joint distribution of (ξi ; βi ). AFR also briefly discuss extensions to ‘affiliated’ types.

15

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES S 1X ˆ Gi,S (b|d−i , θ) = K S s=1

b − β˜i,s (d−i ; θ) h

! ,

(30)

where h denotes a ‘bandwith’ controlling the smoothness of the kernel estimates.? ˆ i,S are continuous functions of d−i and θ (under CRN), in Note that both gˆi,S and G sharp contrast with the frequency (step function) estimate of Gi . Substituting gˆi,S for gi in Equation (28), we can now go one step further to produce an analytical expression for the inner integral in b, taking full advantage of the known properties of k and/or K and of the fact that Ui typically is a (very) simple function of b. For example, Ui does not depend upon b for firstprice auctions and is linear in b for ‘risk neutral’ bidders in second-price auctions. Note that the inner integral in Equation (28) represents player i’s expected payoff, conditionally on his/her own type ξi . Most importantly it is continuous in d and θ by our combined usage of kernel estimation and CRN. A MC estimate of player i’s unconditional expected payoff result from conventional MC simulation of ξi and is denoted Vˆ i,S,T . It immediately follows from our discussion that (under CRN) Vˆ i,S,T is itself continuous in d and θ. Example (continuation): First price IPV. The winner receives an item with private value ξi and pays the amount of the winning bid. Hence, player i’s payoff is ξi −xi , conditionally on winning, and zero, otherwise. Equation (28) then simplifies to Z ˆ i,S (ψi (ξi ; di )|d−i ; θ) · fi (ξi |θ)dξi . Vˆi,S (d; θ) = [ξi − ψi (ξi ; di )] · G (31) 4

The corresponding MC estimate is given by T 1 X (t ) ˆ i,S (ψi (ξ˜i(t )(θ); di )|d−i ; θ). Vˆ i,S,T (d; θ) = [ξ˜ (θ) − ψi (ξ˜i(t )(θ); di )] · G T t =1 i

(32)  We conclude this section by noting that all of the above calculations have to be repeated for each (class of) player(s) in ’asymmetric’ games though, obviously, all kernel and MC estimates can be based upon a single set of S MC plays of the game. In symmetric games, all calculations are run only once for a ‘typical’ (index free) player. It is worth emphasizing here the fact that, under CSE’s, asymmetric games are conceptually as simple to solve as symmetric ones, in sharp contrast with situa? The selection of a bandwith is extensively discussed in the literature on nonparametric estimation – see e.g. Hardle (1990). Here, however, our objective is simply that of controlling the ˆ i,S by an appropriate combination of h and S. Visual inspection under smoothness of gˆ i,S and G trial values proves extremely useful in that respect.

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OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

tion where one tries to derive unconstrained NE solutions (whether analytically or numerically). 3.3.2. Reformulation of the CSE Fixed-Point Problem One possible approach for solving the CSE fixed-point problem defined by Equations (13) and (14) consists in rewriting it in the form of the following optimization problem n X min [di − dˆi (d−i ; θ)]2 , d

(33)

i=1

where dˆi denotes player’s i best response to d−i . This procedure offers two major problems. (1) It is extremely CPU intensive. For each tentative value of d, we have to solve n optimization problems of the form given in Equation (13) in order to evaluate the objective function to be minimized. (2) The properties of the objective function are not well understood and convergence of the optimization algorithm is by no means guaranteed. As discussed by AFR, there exists a far more efficient way to solve the CSE fixed-point problem. It amounts to solving the following system of non linear equations, which consists of the first-order conditions for the optimization problem in Equation (14): ωi (d; θ) =

∂ Vi (d; θ) = 0. ∂di

(34)

A key advantage resulting from this reformulation comes from the fact that the derivative of Vi w.r.t. di is trivially derived from Equation (28) and is given by " Z ∂ ωi (d; θ) = fi (ξi |θ) · ψi (ξi ; di ) · Ui (xi , xi ; ξi ) · gi (xi |d−i , θ) ∂di 4 # (35) Z xi ∂ + Ui (xi , b; ξi ) · gi (b|d−i , θ)db dξi . bi ∂xi xi =ψi (ξi ;di )

Note, furthermore, that the derivatives ∂ψi /∂di and ∂Ui /∂xi typically have simple functional forms. In other words, ωi can be computed without recourse to evaluating numerical derivatives. The CRN and kernel techniques described in Section 3.3.1 can be applied directly to the ωi ’s so that we solve a system of non linear equations of the form ωˆ i,S,T (d; θ) = 0,

i : 1 → n,

(36)

whose evaluation now is very efficient. The IMSL subroutine DNEQNF has proved very effective for this purpose. In the symmetric case, all di ’s are set equal and the

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

17

system (36) reduces to a system of p equations in p unknowns, where p denotes the dimension of an arbitrary di . In cases where players are regrouped into m classes, so that the game is symmetric within classes and asymmetric across, we have to solve a system of P equations in P unknowns, where P denotes the total number of auxiliary parameters used to characterize m distinct constrained strategy sets. Though the system (36) is better behaved than the initial CSE formulation in Equations (13) and (14), it, nevertheless, remains a system of non linear equations with all the usual caveats. Here again it is important to select a good initial guess. In particular, it is essential to impose fairly tight constraints on the di s, using all theory and factual information . Examples in auctions are monotonicity, boundary conditions, any heuristic estimates of bidders’ ‘markdowns’, or interviews with actual participants. We can also produce second-order derivatives for the Vi ’s ωii (d; θ) =

∂Vi , ∂di ∂di

(37)

that allow us to verify that the second-order conditions are satisfied at convergence. Cross-derivatives, however, cannot be obtained in such convenient way, since d−i appears in the conditioning sets for gi and Gi and for which we have no operational expressions – which is why they have to be estimated by kernels.

3.4.

ADDITIONAL NUMERICAL CONSIDERATIONS

The estimation module itself raises none of the numerical problems specific to game theoretic models since it considers (hypothetical) estimation assuming types are directly observed. Numerical problems in estimating econometric models are well documented and have spawned software developments that can be incorporated in our estimation module. We note in passing that the definition of the unfeasible estimator in terms of the moment conditions given by Equation (3) proves exceptionally useful for the statistical analysis of the properties of the corresponding feasible estimator. See FPR for details. From a computational viewpoint, however, it often proves more convenient to obtain the unfeasible estimator as the solution of an optimization problem (Minimum distance, maximum likelihood, . . . ). We mention here that FPR also discuss a Generalized Method of Moments (GMM) extension of Equation (3) that enables one to construct overidentification tests of a model in the sense of Hansen (1982) and also to estimate additional parameters that might be specific to the bid functions themselves but do not affect the distribution F . An example of the latter situation is found in Marshall and Raiff (1997).

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OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

As for the convergence criterion, we have experimented with several variations and are currently using a relative difference criterion of the form (r−1) (r−1) k ˆ (r) n ˆ (r) ˆ ˆ X X θ ξ − θ − ξ n,j n,j n,j n,j (38) +c· < ε, θˆ (r−1) ξˆ (r−1) j =1

n,j

j i=1

n,j

where c is a scaling coefficient. For applications where N is relatively small (say ≤ 50) we generally set c = 1 and ε = 10−8 . As usual, it is important to make sure that all θ’s and ξ ’s be of similar order of magnitude by appropriate rescaling. 4. Statistical Standard Deviations In order to assess the statistical significance of the point estimates θˆn and/or ξˆn , it is important to compute the corresponding (estimated) √ standard √ deviations. Expressions for the asymptotic covariance matrices of n · θˆn and n · ξˆn are derived in FPR. They are both transformations of 6θ , the asymptotic covariance matrix of the √ unfeasible estimator n · θ˜n . If, for example, ϕ ≡ ϕ, for i : 1 → n, then? √ A Var( nθˆn ) = Tθ−1 6θ T "−1 (39) θ , where Tθ denotes the matrix "    ∂h −1 Tθ = Ik − Eξ |θ · Eξ |θ ∂θ 0

∂h · ∂ξ 0



∂ϕ ∂ξ 0

−1

∂ϕ · 0 ∂θ

!# .

(40)

In practice θ is not known and, as usual, 6θ and Tθ are evaluated at θ = θˆn . Furthermore, expectations in the expression of Tθ are generally replaced by arithmetic means over estimated types. For example, we have   n X ∂h ∂h(ξ , θ) 1 i ∼ Eξ |θ . (41) = 0 0 ∂θ θ=θˆn n ∂θ i=1

θ=θˆn ,ξ =ξˆn

As discussed in FPR, the non-singularity of Tθ is sufficient for the identification of θ from the sole observation of the x’s. Therefore, in situations where the estimation algorithm fails to converge, one should verify whether this failure originates from a fundamental lack of identification, which would result in a (near) singular Tθ matrix. ? Equation (40) is derived under the assumption that observations are i.i.d. and that ϕ ≡ ϕ. i

Following our discussion of indexation in Section 2.4., the index i may refer to J · L observations relative to L players over J plays of a game, in which case the condition ϕi ≡ ϕ implies symmetry. In other cases, it might refer to plays of the game, in which case ϕ represents a vector of individual bid functions. The condition ϕi ≡ ϕ then reads across games but does not preclude asymmetry within games.

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

19

Except for simple cases, some of the derivatives in the expression of Tθ may not have analytical expressions. Note, in particular, that if we use CSE rules, then ϕ may be a simple function of the auxiliary parameter d, but the latter is an implicit function of θ, obtained by the MC fixed-point algorithm discussed earlier. The computation of ∂ϕ/∂θ 0 would then require evaluating numerical derivatives, a non trivial task which would itself require high accuracy computation of d(θ). Actually, we might often find it more convenient (and/or statistically more relevant) to compute ‘finite sample’ standard deviations based upon MC reruns of the entire estimation algorithm. While this might prove to be a CPU intensive operation, it requires little additional programming. The corresponding flowchart is outlined in Figure 2. Once we have produced S draws from the feasible estimator (conditional on θ = θˆn ), we can compute its finite sample mean and covariance matrix in the usual way, say, θˆ n,S =

S 1X θˆn,s , S s=1

S 1X ˆ ˆ V n,S = (θn,s − θˆ n,S )(θˆn,s − θˆ n,S )0 . S s=1

(42)

(43)

5. An Example The following application is taken from AFR to which the reader is referred for additional derivations and results. It is presented here briefly in order to emphasize the organization of the computations, their inherent simplicity, and excellent performance. The available data are relative to tenders for pieces of satellite equipment in the French aerospace industry. Firms submit technical proposals together with financial plans. These are evaluated by an independent committee and summarized in a quality grade Qi ∈ [0, 1] and a price Pi , which can be ex-post standardized across tenders. Tenders with a quality below a preannounced threshold Q0 are eliminated. Among the qualified firms (Qi ≥ Q0 ) the one with the highest quality-price ratio is awarded the contract. If only one firm qualifies, it receives a prenegotiated price, whence the strategic component of the game is conditional upon two or more firms qualifying. Note that the latter condition is trivial to replicate within a MC simulation framework, whether by deletion of draws which are below the threshold or by truncated draws, obtained by ‘inversion’, when an operational expression is available for the c.d.f. of quality. It turns out that firms have far more to lose than to gain by misrepresenting their quality, so we can assume that they announce their ‘true’ quality. Within this

20

OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

Figure 2. Finite sample moments of θ˜n .

framework, types are cost-quality pairs, ξi = (ci , qi ), and actions are price-quality pairs,? xi = (pi , qi ) with pi = ϕi (ci , qi ). The payoff of firm i, conditionally on winning, is given by   pi ci Ui (ci , qi , pi ) = pi − ci = qi . (44) − qi qi Additional assumptions are (1) the IPV framework applies; (2) the pairs (ci , qi ) ¯ × [0, 1];?? and (3) the ϕ’s satisfy the are i.i.d. with c.d.f. F and support [0, C] ¯ Q0 ) = C. ¯ boundary condition ϕi (C, ? The general estimation framework introduced by FPR allows for partial observability of types. The reason for including quality in ξi and xi , rather than treating it as an ‘exogenous’ variable lies in the fact that the (univariate) scoring rule depends on both price and quality. ?? The actual expression for F is found in AFR but is irrelevant to the discussion here.

21

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

As shown by AFR, the symmetric NE bid function for a qualifying firm (Qi ≥ Q0 ) is given by   ci pi = ϕ(ci , qi ) = qi · λ , (45) qi with 1 λ(z) = z + · 1(z; Q0 ) 

Z

1(u; Q0 ) = 1 −

Z

¯ 0 C/Q

1(u; Q0 )du ,

(46)

z

N

1

fQ (ν) · FC|Q (uν|ν)dν

− [FQ (Q0 )]N ,

(47)

Q0

where N = n − 1 denotes the number of rival participants (of which only a random subset of at least 1 firm will actually qualify), FQ (fQ ) the marginal c.d.f. (density) of Qi , and FC|Q the conditional c.d.f. of Ci conditional on Qi = Q. As an alternative, consider a CSE bid function of the form?   ci , (48) pi = qi · `i qi where `(z) is constrained to be a second order polynomial `i (z) = αi + βi z + γi z , 2

  C¯ . z ∈ 0, Q0

(49)

Leaving `(z) completely unrestricted – searching for CSE parameter values in R3 (or R3n in asymmetric scenarios) – proves far too demanding for our CSE algorithm and generally results in either failure or ‘false’ convergence toward unacceptable bid functions (non-monotone or pathological in other respects). Therefore, AFR impose the following ‘natural’ restrictions on the CSE bid function given by Equation (48): (1) It is monotone increasing in c and q over the support of F ; ¯ Q0 ). (2) It equals C¯ and its slope equals 1 at the boundary (C, These conditions imply the following restrictions on the parameters of `(z):  ¯ 2 C αi = γ i , Q0

 ¯  C , βi = 1 − 2γi Q0

0 ≤ γi ≤

Q0 . 2C¯

(50)

? In this formulation, we assume that firms essentially select a price-quality ratio as a function of

their (private) cost-quality ratio. This ‘assumption’ naturally follows from the payoff function given in Equation (44) and is also verified by the NE bid function in Equation (45).

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OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

Therefore, the final version of the constrained bid function is given by  pi = ψi ((ci , qi ); di ) = qi `i

ci ; di qi



 = ci + di qi

  Q0 di ∈ Di = 0, 2C¯

2 ci C¯ − , qi Q0

(51)

where γi is replaced by di for conformability with the general notation introduced in Equation (8). Finally, the condition under which firm i wins is given by   qi qj (52) ≥ βi (ξ−i ; d−i ) = max |qj ≥ Q0 j 6 =i pi pj and can be rewritten as   −1 ci `i ; di ≥ βi (ξ−i ; d−i ). qi

(53)

It follows that the inner integral in the expected payoff Equation (28) is univariate. Since, furthermore, the procurements are first price, the Ui (·) term does not depend upon βi and can be taken outside the inner integral. The resulting expression for expected payoff of firm i is given by     ZZ ci ci Vi0 (d; θ) = fi0 (ci , qi |θ) · qi `i ; di − qi qi 4(Q0 ) (54) !   −1 c i ·G0i `i ; di ; d−i , θ dci dqi , qi where the superscript 0 is used to denote distributions and results that are condi¯ × [Q0 , 1]. The function ωi (d; θ) is tional upon Qi ≥ Q0 and 4(Q0 ) = [0, C] trivially obtained by derivation of Vi0 w.r.t. the scalar parameter di , which only appears as an argument in `i (·). Straightforward MC simulation of {(ci , qi ); i : ˆ 0i,S and ωˆ i,S needed to solve the CSE 1 → n} on 4n (Q0 ) produces the estimates G module per Figure 1. The search for a symmetric CSE solution sets di ≡ d in the expression of ωi (after derivation w.r.t. di ) and, therefore, only requires solving a single non linear equation in a single parameter d. If one considers instead m (classes of) heterogeneous players and searches for an asymmetric CSE solution, then one has to solve a system of m equations in m parameters.? ? AFR illustrate the unparallelled flexibility of the CSE approach by considering different sce-

narios for which m = 2. These will not be discussed here but do provide operational solutions for cases where the derivation of unconstrained NE solutions would prove close to impossible or, at a minimum, would require non-trivial extensions of the numerical algorithm developed by Marshall et al. (1994).

EMPIRICAL GAME THEORETIC MODELS: COMPUTATIONAL ISSUES

23

Table I. Estimated moments. Method

Cost

Quality

Profit margin

NE

Expectation St. dev.

0.905 0.437

0.500 0.147

0.494 0.195

CSE

Expectation St. dev.

0.921 0.454

0.500 0.147

0.478 0.202

The actual econometric model in AFR is completed by selecting of a bivariate distribution for (ci , qi ) and an appropriate (censored) maximum Likelihood unfeasible estimator θ˜n . The data consist of 15 procurements for which a total of 80 firms were consulted but only 50 qualified. Accurate point estimates of θ and finite sample standard deviations? were produced under both NE and CSE solutions. The corresponding point estimates for the means and standard deviations of the cost, quality, and profit margin distributions are reproduced in Table I. The results for both approaches are quite similar, indicating that CSE bid functions of second order are close approximations to NE bid functions [as indicated in AFR, results are nearly identical with fourth-order CSE bid functions]. Most interestingly, a complete estimation run (using the algorithm outlined in Figure 1) requires on the order of 17 minutes of CPU time on a 6 year old DEC 5000/240 workstation for CSE estimates and nearly ten times as much for NE estimates. This highly significant difference, which shows up in spite of the fact that CSE requires solving a fixed-point problem under MC simulation, largely results from the fact that the algorithm under CSE converges much faster than under NE.

6. Conclusions Based on our current experience, we believe that the CSE algorithm presented here provides a fully operational method of unparalleled flexibility for solving and estimating a broad range of empirical game theoretic models. The object of the present paper is to provide potential users with a resolution of the main numerical problems to be addressed in order to take full advantage of the CSE technique. The combination of MC simulation techniques, kernel estimation of the distribution of the ‘closest’ rival action, and careful assessment of constrained strategic sets and initial guesses for unobserved types produces an effective and numerically robust algorithm. Additional auxiliary results, such as finite sample standard deviations for the parameter estimates, results as a direct (though somewhat CPU intensive) byproduct of the main algorithm. ? Standard deviations were produced by auxiliary MC simulations as described in Section 4.

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OLIVIER ARMANTIER AND JEAN-FRANÇOIS RICHARD

Even in the few simple cases where NE solutions can be numerical evaluated, CSE solutions can provide close approximations at a fraction of the computational cost. Last, but not least, once a model has been solved and estimated, modifications such as the selection of an alternative unfeasible estimator or another distribution for the unobserved types generally require only minor modifications of specific modules within the general algorithm. For example, once we had the CSE algorithm running for the symmetric CSE application described in Section 5, it required only a couple of hours to run asymmetric extensions thereof, as described in AFR. A similar effort with NE strategies might take several weeks to design and implement, if possible at all, along the lines of the analysis in Marshall et al. (1994). Finally we find the combination of a programming language like FORTRAN 77 and a large library of subroutines like IMSL to be especially useful for implementing specific applications of the CSE algorithm. Acknowledgements We acknowledge financial support from the National Science Foundation under grant SBR-9601220. We are grateful to Jean-Pierre Florens, Marie-Anne Hugo, John Kagel, Robert Marshall, Matthew Raiff, Philip Reny and Alvin Roth for helpful discussions. All remaining errors are ours. References Armantier, O., Florens, J.P. and Richard, J.F. (1997). Empirical game theoretic models: Constrained equilibrium and simulation. Mimeo, University of Pittsburgh. Binmore, K. (1987). Modeling rational players: Part 1 and 2. Economics and Philosophy, 3, 4. Devroye, L. (1986). Non-Uniform Random Variable Generation. Springer Verlag, New York. Florens, J.P., Protopopescu, C. and Richard, J.F. (1997). Identification and estimation of a class of game theoretic models. Mimeo, University of Pittsburgh. Hansen, L.P. (1982). Large sample properties of generalized methods of moments estimators. Econometrica, 50, 1029–1054. Hardle, W. (1990). Applied Non Parametric Regression. Econometric Society Monographs # 19. Laffont, J-J. (1997). Game theory and empirical economics: The case of auction data. European Economic Review, 41, 1–36. Levin, D., Kagel, J. and Richard, J.F. (1996). Revenue effects and information processing in English common value auctions. The American Economic Review, 86, 442–460. Marshall, R.C. and Raiff, M. (1997). The impact of synergies on bidding in the Georgia school milk market. Mimeo, Pennsylvania State University. Marshall, R.C., Meuer, M., Richard, J.F. and Stromquist, W. (1994). Numerical analysis of asymmetric first price auction. Game and Economic Behavior, 7, 193–220. Richard, J.F. (1996). Simulation techniques. In Econometrics of Panel Data, a Handbook of the Theory with Applications, Ch. 23, 2nd Edition, Kluwer, Boston. Riley, J.G. and Samuelson, W.F. (1981). Optimal auctions. American Economic Review, 71, 381–392. Rosenthal, R. (1993a). Rules of thumb in games. Journal of Economic Behavior and Organization, 22, 1–13. Rosenthal, R. (1993b). Bargaining rules of thumb. Journal of Economic Behavior and Organization, 22, 15–24. Simon, H. (1995). A behavioral model of rational choice. Quarterly Journal of Economics 69, 99– 118.