The game proposed in Chapter3constitutes a dynamic game of incomplete information.
Lee and Fong (2013) propose a two-step procedure to estimate the parameters in this type of problem when actions are networked in a non-trivial way.
The computational difficulties associated to the estimation of dynamic problems are well understood. Until recently, the origin of these difficulties has been the need for solving the underlying dynamic programming problem: continuation values needed to be generated by finding a fixed point in the value function for each player, repeating this process for different parameter candidates to search for the one that mimicked the
observed behaviour best 1. The burden imposed by this type of procedure naturally increases with the number of players, making the computational problem particularly severe in the context of strategic games. The literature in this area, in the last decade, has then focussed on alleviating the computational costs imposed by the fixed point procedure.
The developments that are more relevant to the work I present here exploit theInvert- ibility Result proved inHotz and Miller (1993) in the context of a single-agent dynamic discrete choice problem: under relatively general assumptions, it can be shown that there exists a one-to-one mapping between the choice specific value functions and the condi- tional choice probabilities induced by the dynamic programming problem, in a fashion that is similar to static discrete choice problems. Moreover, the invertibility of that mapping allowed for estimating non-parametrically the continuation values, using the choice probabilities in the data, without computing the fixed point problem described above 2. Early generalisations of this original idea were presented in Hotz et al.(1994) offering an extension, based of forward simulation techniques, of the findings inHotz and Miller(1993) to problems with no terminal state. Rust’s chapter in the 1994 Handbook offered a survey of these methods under a unifying framework (Rust,1994).
Different adaptations of these techniques to games with strategic interaction were pro- posed (see Aguirregabiria and Mira (2007); Bajari et al. (2007); Pakes et al. (2007) as salient, but not exclusive, examples) and the reader is referred toAckerberg et al.(2007) and Aguirregabiria and Mira(2010) for comprehensive surveys on the literature. The immediate additional complication associated with moving from single agent to multiple-agent problems is the (potential) multiplicity of equilibria. As pointed out in
Pakes et al.(2007), non-uniqueness in games of strategic interaction implies the impossi- bility of working out the probability distribution over possible outcomes, conditional on the parameters and the set of observable variables. This, in turn makes most standard estimators unsuitable for settings in which multiple equilibria are possible, aggravated when the relevant dimension of heterogeneity across agents grows and, with it, the scope for multiplicity as it tends to be the case in network formation games.
The type of game I propose belong to the class in Ericson and Pakes’s general frame- work for Markov industry dynamics (1995). Their first and second Theorems state the conditions that underlie the “one-MPE-data” assumption, that most of the economet- ric approaches relevant to my work make. Proposition 1 in Pakes et al. (2007) states that, under certain assumptions, each equilibrium of the game generates a finite chain
1
These complications are not exclusive to the multiple-agent setting. The single agent problem in
Rust(1987) already evidenced this.
2
of actions and states that depend only on their current and immediately observable re- alisations. This (once more finite) chain defines a recurrent class of states that are the only visited states. Then, given a data generating process consistent with a given re- current class, the policies of the players that correspond to that data generating process need to be the same across all equilibria. In other terms, given the current state, the distribution of future states can be computed and policies are well defined functions of the parameters and observables (Ericson and Pakes,1995;Pakes et al.,2007).
A number of alternative two-step estimators were developed, making use in one way or another of the “one-MPE-data” assumption and some form of non-parametric cir- cumvention of the explicit computation of continuation values. The general structure of the estimators is similar across all these alternatives. They all share a first stage in which transition probabilities over states and players’ conditional choice probabilities are obtained from the data. Asecond stage then searches for the parameters that best match the observed behaviour, using the conditions of a Markov Perfect Equilibrium in the corresponding game. This requires no actual computation of en equilibrium. The first step in this direction was, of course, that taken in Hotz and Miller (1993). The improvements on their framework have largely been devoted to i) introducing more engaged interactions between individual current profits and rivals’ actions and ii) re- ducing the sample bias induced by the fact that continuation values are estimated in a first stage and “fed into” a second stage from which the parameters are recovered with, potentially, sample bias 3.
It is in the tradition of these two-step estimators that I exploit the advances in Lee and Fong(2013) to recover the structural parameters of the game I am interested in. In its most general form, the estimation procedure suggested inLee and Fong (2013) imposes some additional challenges when implemented in my setting. The first one is that, like in many other applications, the state space is indeed large and exponentially growing with the number of players. Even when all states of the world were visited with equal proba- bility, stepping in each of these at least once would require an unrealistically long panel. Moreover, as it will be seen below, the type of data I work with, shows high recurrence in linking choices, generating observations compatible with a distribution over states that never visits some of its nodes. The second one is that in my setting I do observe the prices of interactions that take place in the data, so the fixed point that recovers prices needs to be internally consistent not only with the computed value functions, but also with the prices in the data. Finally, the computer times are prohibitively long even
3
The severity of this depends, among other things, on the size (relative to the data) of the state space and the second stage method, especially if the estimating objective function exhibits non linearities over the estimated values, as in the context of likelihood-based techniques. SeePakes et al.(2007) and
when no full equilibria of the game is computed. Actually, the algorithm in Lee and Fong’s approach requires computing equilibrium CCPs for deviations of the parameters, which makes the computation almost as taxing as solving for the equilibrium fully. The following section presents the algorithm as developed by Lee and Fong, with the variations that correspond to my setting.