Discrete Games Jonathan Williams 1 1 UNC - Chapel Hill
Mazzeo (RAND 2002) - Introduction As far back as Hotelling (1929), firms tradeoff intensity of competition against selection of product with strongest demand Proposes empirical model with joint entry and product-type choice Framework builds on Bresnahan and Reiss (1991) and Berry (1992), adds product-type choice Permits differential impact of competitors based on product-type choice
Mazzeo (RAND 2002) - Introduction Estimates model using data on motel quality choice along U.S. Highway system Inference done with a single cross-section of data and assumptions about entry process Performs sensitivity analysis of different equilibrium assumptions Finds differentiation represents strong effect on lessening competition Finds demand characteristics also have very strong effect in determining product-type choice
Mazzeo (RAND 2002) - Model Firms make their product choice by comparing payoffs to operating under each product type Collectively firms choose their product choice given competitors actions, and we have a system of discrete equations Results in dependent variable being a vector of number of each firm with product type Think of model as including two stages, product-type choice and then competition of some sort (quantity, prices, etc)
Mazzeo (RAND 2002) - Model Simple linear-index reduced-form profit function Firm type is indexed by T and markets by m Vector of number of each type of firm by N Note parameters indexed by T, so effect differs by firm type Error differs by type and market, not within type (iid)
Mazzeo (RAND 2002) - Model Considers two alternative (and largely un-testable) behavioral assumptions First is a Stackelberg game: sequential and irrevocable decisions about entry and product type Second is assuming that there are two stages, committed entry in first stage and then simultaneous product choice in second stage
Mazzeo (RAND 2002) - Model Stackelberg game is very simple Firms that move first anticipate that subsequent firms will have the opportunity to make decisions about entry and product after their committment The last firm of each type is profitable, but an additional firm of either type is not Nash equilibrium is given by the following equations:
Mazzeo (RAND 2002) - Model Two substage game, separates entry and product choice Firms enter with commitment and then choose product choice Number of firms that enter is the maximum (L+H) such that there is some (L,H) ordered pair such that both π L and π H are positive Proof of unique equilibrium under this assumption in Appendix
Mazzeo (RAND 2002) - Model Behavioral assumptions are not without cost, and do not predict same outcomes given same parameter values For example, assume the following inequalities hold: Stackelberg: L th low-type firm not enter despite third inequality, because once H + 1 th high-type firm follows, it will be unprofitable to operate as low-type, so outcome is (L 1, H + 1) Two-stage: only L + H enter, because unprofitable to poerate as low-type in (L, H + 1)
Mazzeo (RAND 2002) - Model Extending model beyond two types requires slight modification of second equilibrium concept No pure-strategy NE in two-stage version of game Solution is to break up the game into three stages (for L, M, H product choice) to ensure that a firm only has two options at each stage 1. Stage 1: firms choose to enter or not 2. Stage 2: firms choose whether to be L type or not 3. Stage 3: firms not choosing L type in stage 2 decide between M and H Notice this gets very messy with many firms
Mazzeo (RAND 2002) - Estimation Estimation proceeds using model that predicts equilibrium product-type configuration across markets Permits two product types and up to three firms of each type Thus, up to 15 possible values of dependent variable (L, H) Very similar to Ciliberto and Tamer in many ways, but here a firm must be categorized up front rather than data revealing the heterogeneity
Mazzeo (RAND 2002) - Estimation Consider one market with corresponding data (covariates entering profit function and outcome) Further consider particular values of the parameters of the profit function, and realization of the unobservable portion, (ɛ L, ɛ H ) Under model assumptions, two equilibrium assumptions assign one of 15 possible outcomes for this market, parameter values, and realization of (ɛ L, ɛ H ) Thus, the model provides an indicator function for each equilibrium outcome given these determinants By integrating these indicator functions across all possible realizations of (ɛ L, ɛ H ), df (ɛ L, ɛ H ), we get probabilities of each outcome
Mazzeo (RAND 2002) - Estimation The boundaries of each region are very complicated and correspond to the profit function inequalities that imply each outcome (different for two assumptions)
Mazzeo (RAND 2002) - Estimation Performing numerical integration over these complex regions (with only two types) is quite difficult and slow Alternative, used here, is to use a frequency simulator 1. Draw very large number of draws from the df (ɛ L, ɛ H ) distribution 2. Calculate probability of each equilibrium outcome given value of parameters 3. Construct econometric objective function from implied probabilities of each outcome observed in data 4. Repeat as necessary
Mazzeo (RAND 2002) - Estimation Estimation proceeds using MLE For each of the equilibrium outcomes observed in the data, and a given value of the parameters, the model predicts a probability of that outcome These simulated probabilities are used to construct a likelihood function: Caution on simulator: 1. Describe the exact sequence of steps for the algorithm you would use to perform the simulation and optimization? Order matters. 2. Why is frequency approach potentially problematic for MLE?
Mazzeo (RAND 2002) - Data Information from all motels operating in 492 oligopoly markets along interstates Caters almost exclusively to automobile travelers Differ substantially in service quality, which is ranked often by AAA and others AAA rating is used to characterize each motels product-type choice Sensitivity to apriori classification of types can be tested pretty easily For chains of motels, it is assumed that product-type choice is optimal in every market served Markets are defined by clustering of hotels at interstate highway exit, which permits demographic info to be pulled in
Mazzeo (RAND 2002) - Data Markets included in data
Market structure Mazzeo (RAND 2002) - Data
Mazzeo (RAND 2002) - Data Types by brand of motel
Market structure Mazzeo (RAND 2002) - Data
Mazzeo (RAND 2002) - Results Parameterization of g(θ T ; N)
Mazzeo (RAND 2002) - Results Parameterization of g(θ T ; N)
Mazzeo (RAND 2002) - Results Parameterization of g(θ T ; N)
Mazzeo (RAND 2002) - Robustness Tests Estimation of model permitting a third type, (L, M, H) Substantially complicates the two-step equilibrium as we saw above (third step to keep choice binary at each point) Estimates this model with less-rich specification
Mazzeo (RAND 2002) - Robustness Tests Implication of results are still very similar
Mazzeo (RAND 2002) - Conclusions Important firsts in this paper: 1. Incorporates other choice within entry game, opens door for modeling investment, product choice, etc 2. Firm heterogeneity and also heterogenous impact of competitors 3. Use of maximum simulated likelihood in game framework 4. Tests implications of equilibrium assumptions for estimates and model simulations
Seim (RAND 2006) - Introduction Introduces a model with endogenous product-type choices, similar to Mazzeo (2002) Choices are formalized as outcome of game with incomplete information and firm heterogeneity that implies nonuniform competitive effects Data comes from location choices of video retail industry Estimation proceeds with a nested fixed-point algorithm, substantially different from approaches studied thus far Performs many simulations to demonstrate tradeoffs between demand and intensified competition
Seim (RAND 2006) - Introduction Similar to what BLP did with demand, thinking of products as a point in characteristic space, you can think about a firm similarly This was Hotelling (1929) and Lancaster (1966, 1979), where the position of the firm was the source of differentiation Geography is important, particularly for high-frequency purchases, and people have idiosyncratic preferences over these locations Similar to the quality in Mazzeo (2002), this paper endogenizes location, which is another important strategic variable of firms
Seim (RAND 2006) - Introduction Presents an empirically tractable equilibrium model to study determinants of firms product positions. Characteristic space is now location rather than product quality as in Mazzeo Shows spatial differentiation used to shield from competition, distance decreases competitive effect Shows as market size grows, population fixed, firms gain market power (more scope for differentiation), but this is weighed against more dispersed population which lowers demand
Seim (RAND 2006) - Introduction Model fills a number of gaps in the literature 1. Model is tractable for a large number of locations, unlike Mazzeo (What happens to Mazzeo with many locations?) 2. Introduces idiosyncratic sources of incomplete information, now a Bayesian-Nash Equilibrium notion, so firms form expectations about others actions 3. Shows uniqueness of equilibrium in simple version of model
Seim (RAND 2006) - Model Model is static, firms choose location among set of discrete locations Firms are attracted to more favorable demand characteristics, but so are other firms, which lowers profits Entry and location decision determined by demand characteristics, expectation of competitors actions, and idiosyncratic shock Number of potential competitors is known to all firms and is greater than one
Seim (RAND 2006) - Model Number of potential entrants are indexed by f = 1...F, while possible locations are indexed by l = 0, 1...L m Firms location decision is denoted by d f, where d fl = 1 if firm l chooses location l and zero otherwise Profits of firm f in location l in market m is
Seim (RAND 2006) - Model Demand characteristics (population and income) specific to l are given by X m l Market specific profit shifter known to all firms ξ m, which includes demand and cost shifters Γ is an L m by L m matrix of competitive effects, where the competitive effects depend on distance between the locations within the market n m is the number of firms in each location Together Γ and n m give the competitive effect of other firms decisions on the profits for firm f ɛfl m gives the idiosyncratic part of profits, observed to firm f, but no one else
Seim (RAND 2006) - Model A few assumptions are necessary:
Seim (RAND 2006) - Model A1: Realizations of private information are iid, and provide no information about competitors realizations A2: Assumes effect of competitors location decisions on my profits are additively separable A3: Competitive effects differ by the distance band, since locations are only defined in the census data according to bands Resulting profit function in a market is:
Seim (RAND 2006) - Model Due to imperfect information about rivals profitability, firm can only form expectation of their optimal location choices Each firm will thus choose the location that maximizes its expected profit Expected profit in of firm f in location l is: The expected number of firms per distance band is
Seim (RAND 2006) - Model Calculating the equilibrium conditional on a vector of parameters is more complicated than in other static models we ve studied Start by considering the location decision of a firm (representative since problem is symmetric) for given number of entrants Probability that competitor g chooses location l, p gl is Thus, the number of competitors that firm f expects in location l is (E 1)p gl and number of firms entering each distance band b is
Seim (RAND 2006) - Model Assume that unobserved shocks are from iid type 1 extreme-value distribution Provides a closed form for firms choice probabilities The equilibrium is a symmetric Bayesian-Nash notion that has each firms optimal response maximizing their expected profits such that This system of L equations defines location probabilities as a fixed point of the mapping from firm s conjecture of its rivals strategies into its rivals conjecture of the firm s own strategy
Seim (RAND 2006) - Model Equilibrium above characterizes the probability of a particular location The probability of entry at any location (versus alternative of not entering, which has a mean normalized to zero) is The number of expected entrants is then
Seim (RAND 2006) - Estimation The equilibrium system of equations is highly nonlinear and difficult to numerically solve Takes an approach similar to BLP (1995) by finding the market effect that is just large enough so expected number of entrants equals number observed in the data More precisely, the market level effect can be solved for given all parameters and observables as It is assumed that these market effects are normally distributed, and the mean and variance are estimated along with the other model parameters
Seim (RAND 2006) - Estimation Estimation is done via MLE Each market is treated as an independent location game Model gives predictions over the discrete location of each firm, so that dependent variable is vector of each firm s observed location choice stacked across firms and markets Likelihood function is then given by First part of likelihood gives probability of actions conditional on market effect, and second gives probability of that realization of market effect
Seim (RAND 2006) - Data Models entry and product-type choices in video retail industry Main form of differentiation is location (others include contract terms, price, inventory, etc) Average customer travels 3.2 miles for a round trip to video store Selects cities with population between 40,000 and 150,000, average of 74,367 Total of 151 cities drawn from most states in US Locations of stores are defined at the population-weighted centroid of their Census tract
Markets and Locations Seim (RAND 2006) - Data
Seim (RAND 2006) - Data Demand Shifters: Population and Income
Seim (RAND 2006) - Data Store Choice Locations
Seim (RAND 2006) - Results Prediction errors in choice probabilities (why right skewed?)
Main Results Seim (RAND 2006) - Results
Main Results Seim (RAND 2006) - Results
Seim (RAND 2006) - Results Market-level Normality assumption
Seim (RAND 2006) - Conclusion Many meaningful contributions 1. Incorporating imperfect information permits possibility ex-post location-choice regret, and seems like step towards reality 2. Discrete choice beyond entry much richer than any previous models, essentially continuous, as framework permits any form of differentiation Big steps towards dynamic games we will consider with imperfect information and both discrete (entry/exit/etc) and continuous choices (investment/price/etc)
Grieco (RAND 2014) - Introduction Most empirical research focuses on estimating payoff functions Informational assumptions tend to be strong, and made for convenience Provides framework to relax the informational assumptions in discrete games Applies the method to data on the entry and exit patterns of grocery stores Approach provides bounds on equilibrium outcomes, which provides point identification of payoff parameters with rich support condition on private information Demonstrates bias resulting from incorrect informational assumptions in counterfactual simulations
Grieco (RAND 2014) - Introduction Contributions 1. Unifies two empirical literatures: complete information (starting with Bresnahan and Reiss 1990) and incomplete information (Seim 2006) 2. Provides identification results to show payoff parameters can be point identified even if information structure is only set identified 3. Application that looks at super center s impact on local grocery stores
Grieco (RAND 2014) - Model Consider small market with two entrants, i = 1, 2 Binary entry decision of each firm is, y 1, y 2 Researcher and players observe covariates that could be firm or market specific about each firm, x 1, x 2 All firms observe public information about payoffs, ɛ 1, ɛ 2, researcher does not Only each firm knows its own private information about payoffs, ν 1, ν 2 Payoffs are given by
Grieco (RAND 2014) - Model Players choose actions using equilibrium strategies, focused on Bayesian Nash equilibria Game is one of incomplete information so long as ν has positive variance Equilibrium provides a mapping from a firm s type, ν to an action profile y = 0, 1 Optimal strategies are a cutoff in ν, since ν increases profits, you can t have a firm enter for a low ν and not enter for a high ν Denote the optimal strategies (s) by
Grieco (RAND 2014) - Model Makes assumption that ν is normally distributed for each player (and independent) Equilibrium beliefs are then Cutoffs are such that each player best responds given their beliefs about other players cutoff. At cutoff, each player is indifferent about action. Optimal cutoff as function of beliefs is One-to-one beliefs and cutoffs, so equilibrium can be described either way
Grieco (RAND 2014) - Model Given the assumptions made, the equilibrium set (not unique) for the incomplete information game is the solution set to the system of nonlinear equations Assumes there is an equilibrium selection mechanism (or coordination device) that may depend on (ɛ, x, θ), but is independent of ν (similar to Bajari et al 2010).
Grieco (RAND 2014) - Model The equilibrium selection mechanism completes the model Taken together, the two predict a unique probability distribution over actions and the model likelihood is well defined If we don t specify the selection mechanism, then model is only partially identified as discussed in Tamer (2003)
Grieco (RAND 2014) - Model Have model that nests the incomplete (σ ɛ = 0) and complete information (σ ν = 0) extremes Consider an even simpler model that drops the covariates to demonstrate the relationship between information structure and multiple equilibria
Grieco (RAND 2014) - Model Fixing the parameters of payoffs and σ ɛ, we can plot the region of multiple equilibria for different σ ν
Grieco (RAND 2014) - Model Varying σ ν reveals 1. Region of multiplicity shrinks as σ ν increases 2. Box when σ ν is small resembles that of Tamer (2003) 3. Multiplicity in the presence of uncertainty is more likely when two firms are similar in terms of their publicly observed propensity to enter (ɛ 1 ɛ 2 ), which captures strategic substitutability of players actions (reversed if δ parameter is positive and strategic complements)
Grieco (RAND 2014) - Model Existence of multiple equilibria can be seen by tracing out firm 1 s equilibrium probability of entering as function of σ ν Intuition is that as uncertainty becomes large, so that
Grieco (RAND 2014) - Model As uncertainty becomes large, probability that a firm s optimal action is independent of its expectations increases This forces equilibrium strategies to be more similar In limit as σ ν approaches infinity, the expectation of entry plays no role in its entry decision and ultimately its strategy Thus, if we have a rich support, σ ν, we eliminate multiple equilibria and this permits us to point identify the payoff function (well defined likelihood given unique predictions of outcomes)
Grieco (RAND 2014) - Estimation Inference is a technical mess Parameters are set identified (in most cases) and there is an infinite dimensional nuisance parameter, the selection mechanism λ (not of interest) Applies profiled sieve likelihood ratio approach of Chen, Tamer, and Torgovitsky (2011), which uses a weighted bootstrap procedure to perform estimation At the end of the day, it essentially relies on a flexible approximation of the equilibrium selection mechanism, which results in weighted log likelihood optimization problem where the weights come from the selection mechanism which is jointly estimated with the payoff parameters
Grieco (RAND 2014) - Application Focuses on store openings and closings in rural areas, and particularly the role of super centers by Walmart in this process Makes assumption that super center opening is independent of grocery stores decisions, so essentially a exogenous covariate from our perspective The game played by local grocery stores matches the game we just described
Grieco (RAND 2014) - Application Main source of data comes from annual extracts from the Trade Dimension TDLinx database of all grocery store locations in the US from 1995-2006. Demographic information comes from 2000 Decennial US Census Geographic markets are defined by zip codes (rural areas, so not too worried about spillovers between markets).
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Application
Grieco (RAND 2014) - Conclusion Provides a flexible framework to nest games of complete and incomplete information Shows power of incomplete information for mitigating issues associated with multiple equilibria Shows how to adapt Chen, Tamer, and Torgovitsky (2011) profiled sieve likelihood ratio approach to identify this model Demonstrates approach to show impact of super centers on grocery applications