Chapter 3. Dynamic discrete games and auctions: an introduction

Chapter 3. Dynamic discrete games and auctions: an introduction Joan Llull Structural Micro. IDEA PhD Program I. Dynamic Discrete Games with Imperfect Information A. Motivating example: firm entry and exit The estimation of oligopolistic discrete games is very popular in industrial organization (IO). As a motivating example, we consider firms decision of entry/continuation/exit from a market. Markets are small and isolated, and we focus on the retail sector. Each active firm operates at most in one location or store. We observe a random sample of markets m = 1, 2,..., M. In each market, there are N potentially active and infinitely lived firms which decide simultaneously whether to operate or not. The distinction here between firms and markets is very important. We have N M firms (we assume that firms at most operate in one market) whose payoffs are affected by the decisions of other players in the same market. This is the main difference with what we have seen so far. In our example, the profit function is: Π it = θ RS ln S m(i)t θ RN ln 1 +, (1) {j:j m(i),j i} where d jt = 1 if firm j is active in marked m(j) and S m(i)t is the market size. This profit function can be interpreted as the outcome of a static symmetric game. In a competitive setup, there are so many players that individual firm s decisions do not affect decisions of other firms, as the measure of firms operating in the market is unaffected. However, in an oligopolistic market, the number of firms operating in the market is a best-response function with respect to competitor s choices. The fixed cost paid every year by a firm active on a market is: F it = ω F i + ε 1it, (2) where ε 1it represents an idiosyncratic shock to firm i s fixed cost. The entry cost, paid only when the firm was not active in the previous period, is given by: E it = (1 d it 1 )ω Ei. (3) Hence, the total current profits of an active firm with observable state variables x it (S m(i)t, d i, d it 1 ), where d i is the vector of choices for all firms in the d jt 1

same market as i except i itself, are: U it (1, x it, ω i, ε it ) = (4) θ RS ln S m(i)t θ RN ln 1 + ω F i (1 d it 1 )ω Ei + ε 1it. {j:j m(i),j i} The profit function of a non-active firm is given by the value of the best outside option, that we assume to be: U it (0, x it, ω i, ε it ) = ω Ni + ε 0it, (5) where ω Ni is normalized to zero for identification purposes. The unobserved state vector ε it is assumed to be private information of firm i, unknown by other players in the market. We assume that it is normally distributed, i.i.d. across firms and markets, and over time, with zero mean. x it and ω i are common knowledge. The researcher observes x it but not ε it and ω i. Note that this model embeds Rust s framework with unobserved heterogeneity whenever θ RN = 0. However, if θ RN 0, d it(x it, ω i, ε it ) is a best response function. The actual decisions are given by the solution of the Nash equilibrium. Full solution methods are often unfeasible in this context. Aguirregabiria and Mira (2007) propose a Hotz-Miller based estimation method for dynamic discrete games. A potential complication of this may be the existence of multiple equilibria, but it can also be handled in estimation. B. General setup One of the costly aspects of dynamic discrete games is the need for solving for the equilibrium in each period. If in single agent problems solving for the value functions implies a nested fixed points, in the games context there is a double nesting because on top of solving for the individual s problem, one also needs to solve for the equilibrium conditions. CCP estimations has turned particularly important in developing techniques that allowed us to advance in the possibilities of estimating such problems. To generalize the setup from the example above, let d it denote the own action of individual i, and let d it (d 1t,..., d i 1,t, d i+1,t,..., d It ) denote the actions of all other I 1 players. The flow utility of individual i choosing alternative j is thus: u ij (x t, d it ) + ε ijt, (6) where ε it (ε i1t,..., ε ijt ) is an i.i.d. random variable privately observed by individual i but not by the others. The vector x t, on the contrary, is observed by 2 d jt

all individuals, and, therefore, includes the state variables of all individuals. The dependence of u ij ( ) on i is a reflection of the possibility of different state variables affecting payoffs differently (e.g. own state variables vs other individuals ); the absence of t in it reflects that we are in a stationary environment. Choices are taken simultaneously in each period. We concentrate on rational stationary Markov perfect equilibria, which imply that, given that ε it is i.i.d. across individuals, individual i expects other agents to make choices d it with probabilities: Pr(d it x t ) = Pr(d jt x t ). (7) j i These CCPs represent the best-response probability functions. In this type of models, an equilibrium exists, but uniqueness is rather unlikely to hold. However, these CCPs uniquely identify the beliefs of agents. Taking expectations of u ij (x t, d it ) over d it, we obtain: ũ ij (x t ) = Pr(d it x t )u ij (x t, d it ). (8) d it D I 1 These concentrated payoffs resemble the standard payoffs of the single-agent models seen so far. Now we additionally need to construct the continuation values. The difficulty on that front is that the state variables, which follow a process given by F x (x t+1 x t, d it, d it ), depend on the unobserved choices by the other individuals. Following a similar idea, we can obtain the concentrated transition functions as: F i (x t+1 x t, d it ) = Pr(d it x t )F x (x t+1 x t, d it, d it ). (9) d it D I 1 Given all this, the conditional value functions can be expressed as: v ij (x t ) = ũ ij (x t ) + β V (x t+1 )d F i (x t+1 x t, d it ), (10) and then V (x t+1 ) can be replaced the standard CCP representation, i.e.: v ij (x t ) = ũ ij (x t ) + β [v ik (x t ) + ψ k (p i (x t ))] d F i (x t+1 x t, d it ). (11) Once the model is transformed this way, estimation follows standard CCP procedures, which range from likelihood or GMM versions of Hotz and Miller (1993) to the iterative Nested Pseudo-Likelihood algorithm in Aguirregabiria and Mira (2002). The latter is the approach adopted in the seminal paper by Aguirregabiria and Mira (2007). 3

Note that the use of CCP estimation methods made the estimation of these models feasible. Full solution maximum likelihood approaches would not be tractable, because it would require solving for the equilibrium of the game (on top of solving for the dynamic problem). Furthermore, the estimation is based on conditions that are satisfied by every Markov perfect equilibrium, which skips the complication generated by the existence of multiple equilibria. II. Auctions A. Introduction Auctions are often used as mechanisms for allocation of resources to bidders in an as efficient as possible way. There are many interesting real-world applications that generate interesting data and a great deal of opportunities for empirical investigation, raising interesting issues. As a result, there is an important and growing literature in empirical micro that aims at structurally estimating the parameters of the bidders valuations. This is of fundamental importance for auction design, because they allow the simulation of efficiency results under different types of auction mechanism. It allows to determine the optimal design, and the parameters of this design (e.g. the reserve prices). Since the aim of this section is only to provide a brief introduction to the structural estimation of auctions, we focus on two particular types of auctions: first and second price sealed bid auctions for one good, N players, and independent valuations v i for i = 1,..., N. In the first price sealed bid auctions, each player simultaneously submits a bid, and the object is assigned to the highest bidder, who pays the price she bid. In the second price sealed bid auctions, the highest bidder wins, but pays the bid of the second highest bidder. The literature has studied other types of auctions. For example, English auctions are similar to first price sealed bid auctions except that bidders sequentially call ascending prices, and other players observe bids. In Dutch auctions, the price is reduced until a player accepts the offer, so only the winning bid is ever observed. In Japanese auctions, players exit as the auctioneer raises the price, and the winner pays the price at which the only other remaining bidder exits. While Dutch auctions are strategically equivalent to first price sealed bid auctions, Japanese auctions are not necessarily strategically equivalent to second price sealed bid auctions, because players update their information sets as the auction evolves. As noted above, we focus on single object first and second price sealed bid auctions. There are N risk-neutral bidders (indexed by i = 1,..., N) and they have valuations v i, independently drawn from a common distribution F ( ). The 4

data consists of the outcomes observed across independent auctions k = 1,..., K that follow the same paradigm. In this introductory review, we focus on the case where individual i observes her own valuation v i, but not other players valuations. The literature also analyzes the case in which the individual, instead, has a signal x i v i, but does not observe v i, and also the case in which different players have a (partially) common valuation. Players submit a single bid b i IR + and do not observe other players bids. The econometrician only observes bids, either all of them, or only the winning bid or price. The structural estimation of auction models usually rely on the equilibrium bid functions and on distributional assumptions regarding F ( ), which often is specified parametrically. The literature focuses on Perfect Bayesian Equilibria in weakly undominated pure strategies. A bidding strategy is a function that maps valuations into bids. The bidding strategy is the equilibrium solution of a expected utility maximization problem. B. Equilibrium responses in second and first price sealed bid auctions In second price sealed bid auctions, it is a weakly dominant strategy for every individual to bid her expected valuation, i.e. b i = v i. Intuitively, bidding more implies winning some auctions that yield negative expected value but leaves unchanged the expected value of any other auction that would be won, whereas bidding less implies losing some auctions that yield positive expected value but leaves unchanged the expected value of any other auction that she would win. In a first price sealed bid auction, best responses are slightly more complicated, because, unlike in the second price counterpart, changing the bid not only affects the probability of winning, but also the price to pay. Let p(b) denote the probability of winning the auction with bid b, defined as: p i (b) Pr(max{b j } j i b). (12) Then, b i solves: b i = arg max(v i b)p i (b). (13) b The resulting b i is the best response to other player s expected actions. The first order condition yields: (v i b i )p (b i ) p(b i ) = 0. (14) 5

Totally differentiating this expression with respect to b and v, we obtain: db i dv i = p (b i ) (v i b i )p (b i ) 2p (b i ) > 0. (15) The last inequality is obtained from observing that the denominator is the second order condition (and, hence, it is negative), and that the winning probability is increasing in the bid. Therefore, if players are in pure strategy equilibrium with an interior solution, then b i is increasing in v i. This ensures invertibility, and, hence, identification. C. Identification Let F (v) denote the distribution of valuations. In a second price sealed bid auction, the distribution of valuations is trivially identified if all bids are observed because, as noted before, all players bid their valuation. The case in which only the winning price is observed (in each of K auctions in which the same equilibrium is played) is a bit more convoluted. If only the winning price is observed, which equals to the second highest bid, then the probability distribution of the second highest valuation, denoted by F N 1,N (v) is identified trivially from the distribution of paid prices. Let f N 1,N (v) denote the corresponding density. Given symmetry and independence, this density equals: f N 1,N (v) = N(N 1)F (v) N 2 (1 F (v))f(v). (16) Intuitively, the N(N 1) comes from the combinatorial possibilities in which v is the second highest valuation, F (v) N 2 is the probability that N 2 valuations are lower than v, and 1 F (v) is the probability that one of them is higher. Given a boundary condition F N 1,N (v) = F (v) = 0, and noting that f(v) > 0 above the boundary condition, the identification of F (v) comes from solving the differential equation above. In first price sealed bid auctions, identification comes from the first order condition above. If all bids are observed, then p(b) is trivially identified. Hence, v i is identified from (14): v i = b i + p(b i) p (b i ) = b G(b i ) i + (N 1)g(b i ), (17) where G( ) and g( ) are respectively the cdf and pdf of observed bids, and the latter equality is obtained from noting that p(b) = G(b) N 1. Therefore, the probability distribution of (v 1,..., v N ) is identified off the bidding distribution G(b). 6

If only the winning bid is recorded, the distribution of winning bids, H(b) is identified from the outcomes observed in the different auctions. Since the winning bid is defined as the highest one, H(b) is just the probability that all the bids in all the auctions are less than or equal to b, such that: H(b) = Pr(b k i b i = 1,..., N) = G(b) N. (18) Therefore: G(b) = H(b) 1 N, (19) and: g(b) = 1 N H(b) 1 N 1 h(b), (20) where h(b) is the density of winning bids. Replacing these two in (17), this shows that the bidding distribution is identified off the winning bid distribution. D. Estimation Estimation strategies range from minimum distance to maximum likelihood. In minimum distance estimation, once the distribution of bids is derived, up to parameter values, these parameters are estimated comparing sample moments of the observed bids against theoretical moments of the G( ) distribution for each parameter value. Sometimes, this moments are trivial functions of the parameters, and standard minimum distance methods are easy to implement. Other times, it is too costly to derive the theoretical moments from the distribution, and we proceed with simulated method of moments, using Monte Carlo approaches. In particular, for a given set of parameters, valuations are drawn for all players. This valuations correspond to bids, given the equilibrium bidding strategies of each player. Keeping the seed fixed, iterate over parameters to minimize the distance between simulated and data moments. Maximum likelihood approaches are also feasible, but have the complication that the upper bound of the support often depends on parameter values, which lead to estimates that, while consistent, are not asymptotically normal. 7