discrete games - jonathan w williams: unc...

Mazzeo (RAND 2002) Seim (RAND 2006) Grieco (RAND 2014)

Discrete Games

Jonathan Williams 1

1UNC - Chapel Hill


Mazzeo (RAND 2002) - Introduction

• As far back as Hotelling (1929), firms tradeoff intensity ofcompetition against selection of product with strongestdemand

• Proposes empirical model with joint entry and product-typechoice

• Framework builds on Bresnahan and Reiss (1991) and Berry(1992), adds product-type choice

• Permits differential impact of competitors based onproduct-type choice


Mazzeo (RAND 2002) - Introduction

• Estimates model using data on motel quality choice alongU.S. Highway system

• Inference done with a single cross-section of data andassumptions about entry process

• Performs sensitivity analysis of different equilibriumassumptions

• Finds differentiation represents strong effect on lesseningcompetition

• Finds demand characteristics also have very strong effect indetermining product-type choice


Mazzeo (RAND 2002) - Model

• Firms make their product choice by comparing payoffs tooperating under each product type

• Collectively firms choose their product choice givencompetitors actions, and we have a system of discreteequations

• Results in dependent variable being a vector of number ofeach firm with product type

• Think of model as including two stages, product-type choiceand then competition of some sort (quantity, prices, etc)



• 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)



• Considers two alternative (and largely un-testable) behavioralassumptions

• First is a Stackelberg game: sequential and irrevocabledecisions about entry and product type

• Second is assuming that there are two stages, committedentry in first stage and then simultaneous product choice insecond stage



• Stackelberg game is very simple

• Firms that move first anticipate that subsequent firms willhave the opportunity to make decisions about entry andproduct after their committment

• The last firm of each type is profitable, but an additional firmof either type is not

• Nash equilibrium is given by the following equations:



• 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 thatthere is some (L,H) ordered pair such that both πL and πH

are positive

• Proof of unique equilibrium under this assumption in Appendix



• Behavioral assumptions are not without cost, and do notpredict same outcomes given same parameter values

• For example, assume the following inequalities hold:

• Stackelberg: Lth low-type firm not enter despite thirdinequality, because once H + 1th high-type firm follows, it willbe unprofitable to operate as low-type, so outcome is(L− 1,H + 1)

• Two-stage: only L+H enter, because unprofitable to poerateas low-type in (L,H + 1)



• Extending model beyond two types requires slightmodification 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 optionsat each stage

1. Stage 1: firms choose to enter or not2. Stage 2: firms choose whether to be L type or not3. 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 equilibriumproduct-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 afirm must be categorized up front rather than data revealingthe heterogeneity



• Consider one market with corresponding data (covariatesentering profit function and outcome)

• Further consider particular values of the parameters of theprofit function, and realization of the unobservable portion,(εL, εH)

• Under model assumptions, two equilibrium assumptions assignone of 15 possible outcomes for this market, parameter values,and realization of (εL, εH)

• Thus, the model provides an indicator function for eachequilibrium outcome given these determinants

• By integrating these indicator functions across all possiblerealizations of (εL, εH), dF (εL, εH), we get probabilities ofeach outcome


Mazzeo (RAND 2002) - Estimation• The boundaries of each region are very complicated and

correspond to the profit function inequalities that imply eachoutcome (different for two assumptions)



• 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 (εL, εH )distribution

2. Calculate probability of each equilibrium outcome given valueof parameters

3. Construct econometric objective function from impliedprobabilities of each outcome observed in data

4. Repeat as necessary



• Estimation proceeds using MLE

• For each of the equilibrium outcomes observed in the data,and a given value of the parameters, the model predicts aprobability of that outcome

• These simulated probabilities are used to construct alikelihood function:

• Caution on simulator:

1. Describe the exact sequence of steps for the algorithm youwould use to perform the simulation and optimization? Ordermatters.

2. Why is frequency approach potentially problematic for MLE?


Mazzeo (RAND 2002) - Data

• Information from all motels operating in 492 oligopolymarkets along interstates

• Caters almost exclusively to automobile travelers

• Differ substantially in service quality, which is ranked often byAAA and others

• AAA rating is used to characterize each motels product-typechoice

• Sensitivity to apriori classification of types can be testedpretty easily

• For chains of motels, it is assumed that product-type choice isoptimal in every market served

• Markets are defined by clustering of hotels at interstatehighway exit, which permits demographic info to be pulled in


Mazzeo (RAND 2002) - Data

• Markets included in data


Mazzeo (RAND 2002) - Data• Market structure


Mazzeo (RAND 2002) - Data• Types by brand of motel


Mazzeo (RAND 2002) - Data• Market structure


Mazzeo (RAND 2002) - Results

• Parameterization of g(θT ;N)


Mazzeo (RAND 2002) - Robustness Tests

• Estimation of model permitting a third type, (L,M,H)

• Substantially complicates the two-step equilibrium as we sawabove (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 formodeling investment, product choice, etc

2. Firm heterogeneity and also heterogenous impact ofcompetitors

3. Use of maximum simulated likelihood in game framework4. 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 incompleteinformation and firm heterogeneity that implies nonuniformcompetitive 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 betweendemand and intensified competition



• Similar to what BLP did with demand, thinking of products asa point in characteristic space, you can think about a firmsimilarly

• This was Hotelling (1929) and Lancaster (1966, 1979), wherethe position of the firm was the source of differentiation

• Geography is important, particularly for high-frequencypurchases, and people have idiosyncratic preferences overthese locations

• Similar to the quality in Mazzeo (2002), this paperendogenizes location, which is another important strategicvariable of firms



• Presents an empirically tractable equilibrium model to studydeterminants of firms’ product positions.

• Characteristic space is now location rather than productquality as in Mazzeo

• Shows spatial differentiation used to shield from competition,distance decreases competitive effect

• Shows as market size grows, population fixed, firms gainmarket power (more scope for differentiation), but this isweighed against more dispersed population which lowersdemand



• Model fills a number of gaps in the literature

1. Model is tractable for a large number of locations, unlikeMazzeo (What happens to Mazzeo with many locations?)

2. Introduces idiosyncratic sources of incomplete information,now a Bayesian-Nash Equilibrium notion, so firms formexpectations 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 discretelocations

• Firms are attracted to more favorable demand characteristics,but so are other firms, which lowers profits

• Entry and location decision determined by demandcharacteristics, expectation of competitors’ actions, andidiosyncratic shock

• Number of potential competitors is known to all firms and isgreater than one



• Number of potential entrants are indexed by f = 1....F , whilepossible locations are indexed by l = 0, 1....Lm

• Firms location decision is denoted by df , where dfl = 1 if firml chooses location l and zero otherwise

• Profits of firm f in location l in market m is



• Demand characteristics (population and income) specific to lare given by Xm

l

• Market specific profit shifter known to all firms ξm, whichincludes demand and cost shifters

• Γ is an Lm by Lm matrix of competitive effects, where thecompetitive effects depend on distance between the locationswithin the market

• nm is the number of firms in each location

• Together Γ and nm give the competitive effect of other firms’decisions on the profits for firm f

• εmfl gives the idiosyncratic part of profits, observed to firm f ,

but no one else



• A few assumptions are necessary:



• A1: Realizations of private information are iid, and provide noinformation about competitors’ realizations

• A2: Assumes effect of competitors’ location decisions on myprofits are additively separable

• A3: Competitive effects differ by the distance band, sincelocations are only defined in the census data according tobands

• Resulting profit function in a market is:



• Due to imperfect information about rivals’ profitability, firmcan only form expectation of their optimal location choices

• Each firm will thus choose the location that maximizes itsexpected profit

• Expected profit in of firm f in location l is:

• The expected number of firms per distance band is



• Calculating the equilibrium conditional on a vector ofparameters is more complicated than in other static modelswe’ve studied

• Start by considering the location decision of a firm(representative since problem is symmetric) for given numberof entrants

• Probability that competitor g chooses location l , pgl is

• Thus, the number of competitors that firm f expects inlocation l is (E − 1)pgl and number of firms entering eachdistance 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 haseach firms optimal response maximizing their expected profitssuch that

• This system of L equations defines location probabilities as afixed point of the mapping from firm’s conjecture of its rivals’strategies into its rivals’ conjecture of the firm’s own strategy



• Equilibrium above characterizes the probability of a particularlocation

• The probability of entry at any location (versus alternative ofnot 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 anddifficult to numerically solve

• Takes an approach similar to BLP (1995) by finding themarket effect that is just large enough so expected number ofentrants equals number observed in the data

• More precisely, the market level effect can be solved for givenall parameters and observables as

• It is assumed that these market effects are normallydistributed, and the mean and variance are estimated alongwith 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 eachfirm, so that dependent variable is vector of each firm’sobserved location choice stacked across firms and markets

• Likelihood function is then given by

• First part of likelihood gives probability of actions conditionalon market effect, and second gives probability of thatrealization of market effect


Seim (RAND 2006) - Data

• Models entry and product-type choices in video retail industry

• Main form of differentiation is location (others includecontract terms, price, inventory, etc)

• Average customer travels 3.2 miles for a round trip to videostore

• 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-weightedcentroid of their Census tract



• Markets and Locations



• Demand Shifters: Population and Income



• Store Choice Locations


Seim (RAND 2006) - Results

• Prediction errors in choice probabilities (why right skewed?)



• Main Results



• Market-level Normality assumption


Seim (RAND 2006) - Conclusion

• Many meaningful contributions

1. Incorporating imperfect information permits possibility ex-postlocation-choice regret, and seems like step towards reality

2. Discrete choice beyond entry much richer than any previousmodels, essentially continuous, as framework permits any formof differentiation

• Big steps towards dynamic games we will consider withimperfect information and both discrete (entry/exit/etc) andcontinuous choices (investment/price/etc)


Grieco (RAND 2014) - Introduction

• Most empirical research focuses on estimating payoff functions

• Informational assumptions tend to be strong, and made forconvenience

• Provides framework to relax the informational assumptions indiscrete games

• Applies the method to data on the entry and exit patterns ofgrocery stores

• Approach provides bounds on equilibrium outcomes, whichprovides point identification of payoff parameters with “richsupport” condition on private information

• Demonstrates bias resulting from incorrect informationalassumptions in counterfactual simulations


Grieco (RAND 2014) - Introduction

• Contributions

1. Unifies two empirical literatures: complete information(starting with Bresnahan and Reiss 1990) and incompleteinformation (Seim 2006)

2. Provides identification results to show payoff parameters canbe point identified even if information structure is only setidentified

3. Application that looks at super center’s impact on localgrocery stores


Grieco (RAND 2014) - Model

• Consider small market with two entrants, i = 1, 2

• Binary entry decision of each firm is, y1, y2

• Researcher and players observe covariates that could be firmor market specific about each firm, x1, x2

• All firms observe public information about payoffs, ε1, ε2,researcher does not

• Only each firm knows its own private information aboutpayoffs, ν1, ν2

• Payoffs are given by



• Players choose actions using equilibrium strategies, focused onBayesian Nash equilibria

• Game is one of incomplete information so long as ν haspositive variance

• Equilibrium provides a mapping from a firm’s type, ν to anaction 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 ahigh ν

• Denote the optimal strategies (s) by



• Makes assumption that ν is normally distributed for eachplayer (and independent)

• Equilibrium beliefs are then

• Cutoffs are such that each player best responds given theirbeliefs about other players cutoff. At cutoff, each player isindifferent about action.

• Optimal cutoff as function of beliefs is

• One-to-one beliefs and cutoffs, so equilibrium can bedescribed either way



• Given the assumptions made, the equilibrium set (not unique)for the incomplete information game is the solution set to thesystem of nonlinear equations

• Assumes there is an equilibrium selection mechanism (orcoordination device) that may depend on (ε, x , θ), but isindependent of ν (similar to Bajari et al 2010).



• The equilibrium selection mechanism “completes” the model

• Taken together, the two predict a unique probabilitydistribution over actions and the model likelihood is welldefined

• If we don’t specify the selection mechanism, then model isonly partially identified as discussed in Tamer (2003)



• Have model that nests the incomplete (σε = 0) and completeinformation (σν = 0) extremes

• Consider an even simpler model that drops the covariates todemonstrate the relationship between information structureand multiple equilibria


Grieco (RAND 2014) - Model• Fixing the parameters of payoffs and σε, we can plot the

region of multiple equilibria for different σν



• Varying σν reveals

1. Region of multiplicity shrinks as σν increases2. Box when σν 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 observedpropensity to enter (ε1 ε2), which captures strategicsubstitutability of players’ actions (reversed if δ parameter ispositive 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 σν

• Intuition is that as uncertainty becomes large, so thatprobability that a firm’s optimal action is independent of itsexpectations increases, so the set of rational



• As uncertainty becomes large, probability that a firm’s optimalaction is independent of its expectations increases

• This forces equilibrium strategies to be more similar

• In limit as σν approaches infinity, the expectation of entryplays no role in its entry decision and ultimately its strategy

• Thus, if we have a rich support, σν, we eliminate multipleequilibria and this permits us to point identify the payofffunction (well defined likelihood given unique predictions ofoutcomes)


Grieco (RAND 2014) - Estimation

• Inference is a technical mess

• Parameters are set identified (in most cases) and there is aninfinite dimensional nuisance parameter, the selectionmechanism λ (not of interest)

• Applies profiled sieve likelihood ratio approach of Chen,Tamer, and Torgovitsky (2011), which uses a weightedbootstrap procedure to perform estimation

• At the end of the day, it essentially relies on a flexibleapproximation of the equilibrium selection mechanism, whichresults in weighted log likelihood optimization problem wherethe weights come from the selection mechanism which isjointly estimated with the payoff parameters


Grieco (RAND 2014) - Application

• Focuses on store openings and closings in rural areas, andparticularly the role of super centers by Walmart in thisprocess

• Makes assumption that super center opening is independentof grocery stores decisions, so essentially a exogenouscovariate from our perspective

• The game played by local grocery stores matches the game wejust described



• Main source of data comes from annual extracts from theTrade Dimension TDLinx database of all grocery storelocations in the US from 1995-2006.

• Demographic information comes from 2000 Decennial USCensus

• Geographic markets are defined by zip codes (rural areas, sonot too worried about spillovers between markets).


Grieco (RAND 2014) - Conclusion

• Provides a flexible framework to nest games of complete andincomplete information

• Shows power of incomplete information for mitigating issuesassociated 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 ongrocery applications

discrete games - jonathan w williams: unc...

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