Uniqueness of Oblivious Equilibrium in Dynamic Discrete Games

Transcription

1 Unqueness of Oblvous Equlbrum n Dynamc Dscrete Games João Macera February 7, 2010 Abstract The recently ntroduced concept of oblvous equlbrum ams at approxmatng Markov-Perfect Nash Equlbra (MPNE) when ts calculaton s computatonally prohbtve n Ercson-Pakes-style games wth many players. Ths paper examnes and extends the oblvous equlbrum concept to dynamc dscrete choce games. We consder a set of assumptons commonly posed n the appled dynamc games lterature and establsh oblvous equlbrum propertes. In contrast to Ercson-Pakes models, we fnd that there s a unque oblvous equlbrum n the dynamc game for any number of players under the frequently posed assumpton of ndependence of state transtons across players. We demonstrate that the dstance between ths equlbrum and any MPNE converges n probablty to zero as the number of game players goes to nfnty. Unlke the orgnal work on oblvous equlbrum, our convergence result requres nether a "lght tal" condton nor the absence of aggregate state varables. Keywords: Dynamc dscrete games, oblvous equlbrum, equlbrum unqueness, convergence n probablty, approxmaton. JEL Codes: C61, C62, C73, L13. Emal address: joaom07@vt.edu. Department of Economcs, Vrgna Tech.

2 1 Introducton The ntroducton of the Markov-Perfect Nash Equlbrum (MPNE) concept by Maskn and Trole (1988a, 1988b) has resulted n a growng nterest on examnng dynamc games over the last two decades. To facltate the mappng between theoretcal results and concrete applcatons, several buldng-block frameworks have been ntroduced n the lterature. For example, Ercson and Pakes (1995) ntroduced what has snce become the buldng-block of modelng dynamc olgopoles. Agurregabra and Mra (2007), Pesendorfer and Schmdt-Dengler (2008) and Berry, Ostrovsky and Pakes (2007) developed estmable dynamc dscrete-choce games by buldng on earler advances n sngle-agent dynamc models (e.g. Rust 1987, Hotz and Mller 1993, Hotz et al. 1994). However, two man dffcultes persst on examnng equlbra n dynamc games. Frst, the calculaton of MPNE s computatonally unaffordable unless the dynamc game has a small number of players and a low-dmensonal state space. Concrete applcatons where ths problem s llustrated and dscussed nclude Pakes and McGure (1994, 2001), Gowrsankaran (1999) and Gowrsankaran and Town (1997). Second, a dynamc game typcally has multple Markov-Perfect equlbra, even when the researcher assumes that players restrct attenton to anonymous, symmetrc pure strateges (see Doraszelsk and Satterthwatte 2009 for an extensve dscusson). These dffcultes can be handled when the researcher s prmarly nterested on estmatng the parameters of the dynamc game (e.g. Agurregabra and Mra 2007, Pesendorfer and Schmdt-Dengler 2007, Pakes, Ostrovsky and Berry 2008, Bajar, Benkard and Levn 2007). However, unless the polcy experment of nterest to the researcher s spanned by estmaton output such as player polcy functons (e.g., Macera 2009a, Macera 2009b, Sweetng 2009), the researcher must compute the equlbra of the game. The lterature on dynamc equlbrum computaton has recently proposed approaches that deal wth some of the dffcultes on calculatng MPNE. One of those dffcultes s the curse of dmensonalty, defned as the fact that the state space grows exponentally wth the number of game players. Doraszelsk and Judd (2009) replace the usual dscrete-tme setup of dynamc games wth a contnuous-tme one and llustrate how ths dfference can mtgate the curse of dmensonalty. However, ther dynamc nvestment game wth entry and ext does not rule out multple equlbra. Doraszelsk, Besanko, Satherthwate and Kryukov (2009) propose a method of calculatng the set of possble equlbra n the context of a model of frm competton wth organzatonal learnng and forgettng. Nonetheless, even f ths model s approprate for the applcaton of nterest to the researcher, the problem of whch equlbrum to compute for polcy experments stll perssts. Agurregabra and Ho (2009b) and Agurregabra (2009) deal wth ths dffculty n dynamc dscrete games by calculatng a new equlbrum usng a lnear Taylor seres approxmaton around a gven equlbrum. Even though ths equlbrum selecton method leads to substantal computatonal savngs, t requres an ntal equlbrum to start wth. In addton, ths approach s vald only when the counterfactual experment nvolves small changes n game parameters. Abbrng and Campbell (2009) propose a dynamc verson of the model of 2

3 Bresnahan and Ress (1993) wth homogeneous frms that result n a unque symmetrc MPNE when players restrct attenton to what they name last-n-frst-out (LIFO) strateges. However, the unqueness result does not hold for more general settngs of nterest to appled researchers (see Abbrng, Campbell and Yang 2010). Fnally, Wentraub, Benkard and Van Roy (2008a) ntroduce the concept of oblvous equlbrum (OE) that ams at approxmatng MPNE n Ercson-Pakes models wth many players. However, ther approxmaton results apply only to settngs wthout aggregate shocks affectng all player s payoffs (e.g., market demand state). In addton, OE s not shown to be unque. Wentraub, Benkard and Van Roy (2009a) develop an extended oblvous equlbrum concept to accommodate aggregate shocks n ther model. However, the approxmaton theorem presented n Wentraub, Benkard and Van Roy (2008a) s not extended to ths setup despte ther smulaton results on OE approxmatons to MPNE. Ths paper ams at fllng these gaps by examnng the propertes of oblvous equlbrum n dynamc dscrete choce games. We buld on the setup of Agurregabra and Mra (2007) and extend the concept of oblvous equlbrum (OE) poneered by Wentraub, Benkard and Van Roy (2008) to that framework. We fnd that oblvous equlbrum s unque for the class of dynamc dscrete games where player-specfc state transtons do not depend on the actons of ts rvals. Ths s one of the most frequent setups consdered n concrete applcatons of dynamc games (e.g. Collard-Wexler 2009, Sweetng 2009, Ryan 2009, Beresteanu and Ellckson 2009, Agurregabra and Ho 2009a 2009b, among others 1 ). We also fnd that oblvous equlbrum approxmates MPNE n for games under a less restrctve set of assumptons than n Wentraub, Benkard and Van Roy (2008). In partcular, we fnd that the dstance between player value functons under MPNE and OE converges n probablty to zero as the number of players goes to nfnty. Our results apply to settngs where there exst aggregate state varables affectng all frm payoffs. In addton, we provde suffcent condtons under whch there exsts OE for nfntely many players. We llustrate our approach wth an example of frm competton market entry and ext borrowed from Agurregabra and Mra (2007). The concept of Oblvous Equlbrum (OE) s proposed by Wentraub, Benkard and Van Roy (2008a, 2009a) as an alternatve to MPNE for solvng dynamc games. It assumes that players restrct attenton only to ther ndvdual state varables (and aggregate states, f any) when makng a decson. Under ths soluton concept, players are oblvous by consderng the steady-state value of ther rvals states n leu of ther currently observed states. Ths nvolves the calculaton of the steady-state of a vector process contanng the number of players n each possble state, denoted s t. However, the drect calculaton of the steady-state of amarkovchan{s t,t Z + 0 } wth state transton Υ as the soluton to the fxed pont problem of s = Υs s computatonally unnafordable when the vector s t can assume many possble values. Even though Wentraub, Benkard and Van Roy (2008a, 2009a) avod ths problem by explotng the propertes of ther Ercson-Pakes- 1 For a comprehensve survey on dynamc dscrete games, see Agurregabra and Mra (2009) and Agurregabra and Nevo (2010) for a more comprehensve survey contemplatng dynamc olgopoles n general. 3

4 lke game, ths dffculty apples to dynamc dscrete-choce games wth many players. We deal wth dffculty by buldng on the Multnomal Markov Dstrbuton framework of Wang and Yang (1995). By dervng the probablty-generatng functon of sums of ndvdual statonary Markov Chans, the framework of Wang and Yang (1995) allows us to redefne the fxed-pont problem s = Υs by replacng ts rght-hand sde wth a vector of partal dervatves of the probablty-generatng functon. Ths results n a powerful smplfcaton of steady-state calculatons and eases our proofs on OE propertes. Even though ths paper focuses on the relatonshp between OE and MPNE, our approach to steady-state representaton n dynamc dscrete-choce games s applcable to other equlbrum concepts, provded they can be represented by choce probablty profles. The man contrbuton of Wentraub, Benkard and Van Roy (2008a) s a theorem that shows that a sequence of OE satsfes what they defne as the Asymptotc Markovan Equlbrum (AME) property. That s,asthenumberofplayersgoestonfnty, the expected net present value gan of devatng from an OE to a best-response prescrbed by a Markovan strategy converges to zero. Ths result s used to clam that OE approxmates MPNE well when the game has many players, snce by defnton an MPNE mples no net present values from equlbrum devatons for any number of players. Wentraub, Benkard and Van Roy (2009a) provde smulaton results that agree wth ther clam, yet there are two mportant concerns that matter for practcal applcatons. Frst, the expectaton measurng value functon dfferences s taken wth respect to the long-run dstrbuton under an oblvous strategy. Ths expected dfference metrc allows player payoffs todffer consderably across states under Markovan and oblvous strateges whle "cancelng out" when the expectaton s taken, resultng n a value close to zero. Second, the theorem of Wentraub, Benkard and Van Roy (2008a) requres a "lght tal" condton on the flow payoff of players. Ths condton ams at rulng out equlbra where the shares of game payoffs are concentrated n a few players. In ths paper we deal wth these ssues n the context of dynamc dscrete-choce games. We consder convergence n probablty as our metrc for approxmaton between net present values under dfferent strateges. Ths type of convergence concept consders dstance n absolute value and enjoys mportant propertes under functon contnuty. Ths allows a drect mappng between net present values and choce probabltes, ensurng convergence at every possble state of the game. In addton, we do not requre a "lght tal" condton snce OE s proven to be unque n our framework. Most of dynamc models followng the framework of Ercson and Pakes (1995) nvolve a state varable observable to all players that affects each agents payoff (e.g. number of potental consumers, state of demand). However, the theorem of Wentraub, Benkard and Van Roy (2008a) does not extend to ths mportant settng. Although Wentraub, Benkard and Van Roy (2009a) provde smulaton evdence ndcatng that the presence of aggregate states does not compromse the approxmaton power of OE, t s convenent to have a concrete proof agreeng wth the computatonal evdence. Our paper also contrbutes n ths 4

5 dmenson for the case of dynamc dscrete-choce games. Our approxmaton result based on an asymptotc theorem also apples to models wth aggregate states. In addton, OE s also unque n that stuaton. Ths latter result s grounded on the fact that OE n dynamc dscrete games wth ndependent transtons across players essentally replcate Markov decson models smlar to the one Agurregabra and Mra (2002). Our unqueness proof therefore bulds on steps n the sprt of Agurregabra and Mra (2002) wth proper modfcatons to the presence of a steady-state vector for s t. Ths paper s organzed as follows. Secton 2 descrbes the dynamc dscrete-choce model and the two equlbrum concepts consdered for solvng the game. Secton 3 presents the propertes of oblvous equlbrum and applcable asymptotc results descrbng approxmaton to MPNE for games wth many players. Secton 4 outlnes our man conclusons and dscusses possble extensons to our framework. All proofs not dsplayed n the man sectons of the paper are presented n the Appendx. 2 The Dynamc Model Our framework bulds on the dynamc dscrete choce games of Agurregabra and Mra (2007) but wth some dfferences. We consder flow payoff functons whch do not depend on the actons of rvals. That s, rval actons only nterfere wth future player payoffs through ts stochastc mpact on the state vector evoluton. Ths s the case most frequently consdered n the dynamc games lterature and t s a basc feature of all work nvolvng the oblvous equlbrum concept (e.g., Wentraub, Benkard and van Roy 2006, 2008a, 2008b, 2009a, 2009b, Xu 2008). In addton, we adapt the notaton of Agurregabra and Mra (2007) to the one used by Wentraub, Benkard and Van Roy (2008a) to facltate the defnton of oblvous equlbrum n dynamc dscrete games. 2.1 Model and Assumptons Tme s assumed dscrete and ndexed by t =0, 1,...,. In each perod, there are N players ndexed by =1,..., N who smultaneously decde whch acton to take out of a set of dscrete and fnte optons, denoted A. We represent player s acton by a t A and we let a t =(a 1t,..., a Nt ) represent the vector of player choces at perod t. In addton, we assume that all payoff-relevant features for each player can be encoded nto a state vector. At the begnnng of each perod, a player s characterzed by two state vectors: a vector of varables that are common-knowledge to all players, denoted x t X, and a vector ε t Σ that s prvate nformaton of player. In what follows, ε t =(ε,t,..., ε N,t ) represents a vector contanng prvate nformaton of all N players at perod t.we assume that the set X has a dscrete and fnte support. In addton, we defne the ndustry state s t as the vector over player-specfc observed states that specfes how many players out of N are n each state x t X. 5

6 In what follows, t s convenent to defne what Wentraub, Benkard and Van Roy (2008, 2009) call the state of compettors of player. Ths state vector, denoted s,t,sdefned by s,t (x) =s t (x) 1 f x t = x and s,t (x) =s t (x) otherwse. Fnally, we assume that there s an aggregate shock state ω t Ω whch s common-knowledge to all players. Ths aggregate shocks affects each players payoff functon, denoted π (a t,ω t,x t, s,t,ε t ). Here we dffer from Wentraub, Benkard and Van Roy (2008, 2009) by consderng a more general payoff functon. Our payoff functon depends on player acton a t and prvate nformaton ε t, whle Wentraub, Benkard and Van Roy (2008, 2009) restrct attenton to flow payoffs that depend only on player choce and observed states. We assume that each frm decdes ts acton to maxmze expected dscounted sum of payoffs " # X E β τ t π (a t,ω t,x t, s,t,ε t ) ωt,x t, s,t,ε t τ=t where β [0, 1) s a dscount factor common to all players. The condtonal expectaton s taken over current and future values of state varables and prvate shocks consderng the state transton probablty p(ω t+1,x,t+1, s,t+1, ε t+1 ω t,x t,,s,t, ε t, a t ). As n Agurregabra and Mra (2007), we pose the followng assumptons: (2.1) Assumpton 1 (Addtve separablty) The flow payoff functon of each player s lnear n prvate nformaton for each a t A. That s, π (a t,ω t,x t, s,t,ε t )=π (a t,ω t,x t, s,t )+ε t (a t ), where ε t (a t ) denotes the prvate nformaton value that player obtans f he chooses acton a t at tme t. Assumpton 2 (Condtonal Independence) The state transton functon satsfes the condton p(ω t+1,x,t+1, s,t+1, ε t+1 ω t,x t,,s,t, ε t, a t ) = p ε (ε t+1 ω t+1,x,t+1, s,t+1 ) (2.2) p x (x,t+1 ω t,x t,,s,t,a t ) p s (s,t+1 ω t, a t,x t,,s,t ) p ω (ω t+1 ω t,x t,,s,t ) Assumpton 3 (Independent Prvate Values) The prvate nformaton vector ε t s ndependently dstrbuted across players gven observed states (ω t+1,x,t+1, s,t+1 ). Its factors as NY p ε (ε t+1 ω t+1,x,t+1, s,t+1 )= g (ε,t+1 ω t+1,x,t+1, s,t+1 ) (2.3) =1 where, for each player =1,..., N, g (..) s an absolutely contnuous densty functon measurable wth respect to the Lebesgue measure. 6

7 The frst assumpton s posed for techncal convenence and s shared by nearly all the dynamc dscrete choce models. The second assumpton mposes several restrctons on the probablty transton functon. Frst, the prvate nformaton shocks are ndependent over tme. Even though ths s an mportant lmtaton, t s shared by most theoretcal and emprcal work on dynamc dscrete games for yeld model tractablty. Second, the evoluton of each player s ndvdual future state value, x,t+1, s a stochastc functon of observed states and own player s acton, a t. Even though other dynamc dscrete game frameworks also consder the dependence on other player actons (e.g., Agurregabra and Mra 2007, Pesendorfer and Schmdt-Dengler 2008), most practcal applcatons of dynamc games consder dependence only on own player s acton (e.g. Collard-Wexler 2009, Ryan 2009, among others). The thrd assumpton shares smlartes wth most dynamc dscrete game frameworks and s also motvated by tractablty. It s not as strong as n other manstream works on dynamc dscrete games, n whch the condtonal densty g (ε,t+1 ω t+1,x,t+1, s,t+1 ) s replaced by ts uncondtonal analog g (ε,t+1 ) (e.g. Agurregabra and Mra 2007, Pesendorfer and Schmdt-Dengler 2007). In order to provde some ntuton and fx an nterpretaton for our framework, we provde an example whch conssts of a modfed verson of Example 1 n Agurregabra and Mra (2007, page 6) that satsfes the above-lsted assumptons: Example 1 (Market Entry and Ext) There are N players n the game consstng of frms decdng on operatng or not n a gven market. At each moment t, eachplayer canbentwostuatonsrepresented bythesatevarablex,t ; ether they are "potental entrants" (denoted x,t =0) or they are "ncumbents" (denoted x,t =1). Thus the set of possble ndvdual states s X {0, 1}. The ndustry state s t s a vector whch dsplays how many players are n each state n X. In our setup, f the market has n ncumbents at tme t, thens t =(N n, n). The compettor s state vector, s,t,s(n n 1,n) f player s potental entrant and t s (N n, n 1) otherwse. We also assume that there s an aggregate shock varable ω t representng market demand at tme t. An example of the set of possble values for ω t s Ω = {a, b} where a, b R + and a<b.intutvely, ths would correspond to a model of "Low" and "Hgh" demand states quantfed by the values a and b, respectvely. We assume that at perod t each player must decde whether or not to become an ncumbent for the next perod. We denote player s choce on operatng n the market at perod t +1 by a t =1and the decson of not operatng by a t =0. In our model, ths mples an acton set A {0, 1} for every player. Thevector of prvate nformaton shocks for chosng each alternatve s ε t =(ε t (0),ε t (1)). An example that satsfes our condtons on the dstrbuton of prvate values s assumpton that ε t (0) and ε t (1) are dentcally and ndependently dstrbuted accordng to a Type I Extreme Value dstrbuton. The flow payoff functon s assumed to take the form 7

8 π (a t,ω t,x t, s,t,ε t )=x t à θ M! ω t (2 + s,t (2)) 2 θ FC + θ X (1 a t ) θ E (1 x t )a t + ε t (a t ) where s,t (2) denotes the second cell of s,t (.e., the number of player s rval ncumbents). In ths formulaton, a potental entrant bears no market profts at tme t, and must pay an entry cost θ E n case she decdes to operate at perod t +1. An ncumbent receves market varable profts θ M ω t / (2 + s,t (2)) 2 net of fxed costs θ FC. The expresson for varable market profts could be obtaned from two well-known models: a Cournot equlbrum wth same margnal costs across frms, and the Salop (1979) crcle cty model of frm competton n prces. The ncumbent also ncurs nto an extra payoff of θ X n case she decdes not to operate the market n perod t +1. The parameter θ X could be ether postve or negatve, dependng on whether a market leave decson yelds a scrap value or ext costs, respectvely. One example of observed state transtons satsfyng our assumptons would be the case where {ω t,t 0} s an ergodc Markov chan and where x,t+1 = a t. That s, the decson of operatng fully determnes the player s state n the followng perod. The mpled transton probabltes for every possble value of x,t+1 and s,t+1 gven observed states and actons are defned by 1 f x,t+1 = a t p x (x,t+1 ω t,x t,,s,t,a t )= 0 otherwse 1 f s,t+1 = N 1 T a t (1 a t ), 1 T a a t p s (s,t+1 ω t, a t,x t,,s,t )= 0 otherwse where 1 T s a 1 N matrx of ones. 2.2 Equlbra In ths secton we provde a convenent representaton of each agent s problem and the equlbrum concepts consdered to solve the dynamc game. We assume that all players restrct attentone to statonary Markovan strateges. That s, player actons are determnstc functons of observed states and prvate nformaton. We omt tme subndexes and denote next perod observed states, prvate nformaton and compettor s state by ω 0,x 0, ε0 and s0, respectvely. We denote player s choce functon by σ (ω, x, s,ε ), whch are formally a mappng σ : Ω X S Σ A. Inwhatfollows,tsconvenenttodefne the condtonal choce probabltes assocated wth the profle of strategy functons σ = {σ (ω, x, s,ε )} N =1 as Z P σ (a ω, x, s ) Pr(σ (ω, x, s,ε )=a ω, x, s )= 1(σ (ω, x, s,ε )=a )g (dε ω, x, s ) (2.4) 8

9 As n Agurregabra and Mra (2007), the set of probabltes {P σ(a ω, x, s ):a A } N =1 represents expected behavor from player from the perspectve of the remanng players when player abdes by hs strategy n the set σ. We use ths fact to provde a recursve representaton of the player s problem 2.1. Under a strategy profle σ, player s problem can be represented by the Bellman equaton ˆV σ (ω, x, s,ε ) = max {π (a,ω,x, s )+ε (a ) (2.5) a A +β X X X ω 0 Ω x 0 X s S Z ˆV σ (ω 0,x 0, s 0,ε 0 )g (dε 0 ω 0,x 0, s 0 ) o p ω (ω 0 ω, x, s ) p σ s (s 0 ω, x, s,a ) p x (x 0 ω, x,,s,a ) where p σ s (s 0 ω,x, s,a ) s the transton probablty of player s compettors state gven that they follow the strategy profle σ and player chooses a. Takng condtonal expectatons of 2.5 wth respect to ε allows us to relate value functons wth choce probabltes. The resultng expresson s the ntegrated Bellman Equaton V σ (ω, x, s ) = X P σ (a ω, x, s ) {[π (a,ω,x, s )+E(ε (a ) a,ω,x, s )] (2.6) a A +β X X X (ω 0,x 0, s 0 )p x (x 0 ω, x,,s,a ) V σ ω 0 Ω x 0 X s S o p ω (ω 0 ω, x, s ) p σ s (s 0 ω,x, s,a ) Markov-Perfect Nash Equlbrum The Markov-Perfect Nash Equlbrum (MPNE) poneered by Maskn and Trole (1988a, 1988b) has been the most frequently used soluton concept for dynamc games. The followng defnton of MPNE n pure strateges bulds on the work of Agurregabra and Mra (2007) wth dfferent notaton adapted to our framework: Defnton 1 Aprofle of Markovan strateges σ s a Markov-Perfect Nash Equlbrum f for every player =1,.., N and any (ω, x, s,ε ) Ω X S Σ, σ (ω,x, s,ε ) s a soluton for the Bellman Equaton 2.5 when all rvals of player choose accordng to ther strategy functon descrbed n σ. As n Agurregabra and Mra (2007), the MPNE just defned can be represented n probablty space by followng the framework of Mlgrom and Weber (1985). Workng n probablty space s partcularly 9

10 convenent for several reasons. Frst, t facltates equlbrum exstence proofs by transformng the strategy set nto a bounded contnuous set. Second, the rght-hand sde of the ntegrated Bellman Equaton 2.6 conssts of a contracton mappng operator that s contnuous n choce probabltes. Ths property s os central mportance for the proofs outlned below. Fnally, ntegratng out prvate nformaton from 2.5 allows appled researchers to map the structural model to concrete emprcal settngs (see for example, Agurregabra and Mra 2007, Pesendorfer and Schmdt-Dengler 2007, Beresteanu and Ellckson 2009). Let P =(P 1,..., P N ) denote a profle of choce probablty functons assocated wth a Markovan strategy profle σ and let (V1 P,..., V N P ) denote the players value functon vectors assocated wth P. Then we can rewrte 2.6 wth slght changes n notaton as V P (ω, x, s ) = X P (a ω,x, s ) π (a,ω,x, s )+e P (a,ω,x, s ) (2.7) a A +β X X X (ω 0,x 0, s 0 )p x (x 0 ω, x,,s,a ) V P ω 0 Ω x 0 X s S o p ω (ω 0 ω,x, s ) p P s (s 0 ω,x, s,a ) where e P (a,ω,x, s ) corresponds to the expected value of prvate nformaton gven observed states and the acton chosen by player equals σ (ω, x, s,ε ). The fact that e P (a,ω,x, s ) s a functon only of player s choce probablty vector P and the condtonal prvate nformaton densty g (..) s establshed by followng the same steps as n Agurregabra and Mra (2007, pp 9-10). We can explctly wrte the fxed pont problem that defnes equlbrum probabltes P by takng the followng two steps. Frst, we stack the state-specfc elements of 2.7 for each player and defne a contnuous matrx operator on P, denoted Γ (P), as s the unque soluton for V P gven a profle P. Second, we combne the Defnton 1 wth 2.4 to represent MPNE n probablty space as the soluton to the fxed pont problem nvolvng Γ(P), where Γ(P) =(Γ 1 (P),...,Γ N (P)). The system representaton of 2.7 followng the stackng procedure descrbed for the frst step s V P = X a A P (a ) π (a )+e P (a )+βf P (a )V P (2.8) where P (a ),π (a ) and e P (a ) are column vectors of dmenson R = Ω X S stackng state and acton-specfc elements of choce probablty, payoff and condtonal expected prvate nformaton, respectvely. The R R matrx F P (a ) contans the transton probabltes condtonal on player chosng acton a, whle represents the Hadamard element-by-element matrx product. As n Agurregabra and Mra 10

11 (2007), the unque soluton for the player-specfc value functon vector V P 2.8 gven P s defned by the lnear system n ϕ (P) = ( I βf P 1 X P (a ) π (a )+e P (a ) ) (2.9) a A where F P = P a A P (a ) F P (a ). Usng the defnton of choce probablty n 2.4 to the partcular case of a strategy that solves player s problem 2.5, an operator mappng the choce probabltes profle P onto the the set of player-specfc choce probabltes Ψ (P) Ψ (a 1 ω,x, s ; P),..., Ψ (a A ω, x, s ; P) s defned as Ψ (a ω, x, s ; P) = Z 1(a = σ (ω, x, s,ε ))g (dε ω, x, s ) (2.10) Z ½ 1 a =argmax{π (a,ω,x, s )+ε (a ) (2.11) a A +β X X X ϕ (ω 0,x 0, s 0 ; P)p x (x 0 ω, x,,s,a ) ω 0 Ω x 0 X s S o p ω (ω 0 ω, x,,s ) p P s (s 0 ω, x, s,a ) g (dε ω, x, s ) where ϕ (ω 0,x 0, s0 ; P) regards the value functon cell of ϕ (P) assocated wth the combnaton of states (ω 0,x 0, s0 ) computed usng choce probablty profle P. An MPNE n choce probabltes space conssts of any profle P that s a soluton the fxed pont problem P = Ψ(P), whereψ(p) =(Ψ 1 (P),..., Ψ N (P)). Exstence of P such that P = Ψ(P ) follows drectly from Brower s fxed pont theorem. However, the set of solutons for ths fxed-pont problem s rarely a sngleton; Doraszelsk and Satterthwate (2009) dscuss and llustrate multplcty of MPNE for closely-related, general dynamc games. In an attempt to deal wth multplcty of equlbra whle allowng good approxmatons to MPNE, we defne the concept of oblvous equlbrum for the model outlned above n the next secton Oblvous Equlbrum So far we have outlned a model smlar to Agurregabra and Mra (2007) wth some modfcatons n notaton and assumptons. We now defne the concept of oblvous equlbrum poneered by Wentraub, Benkard and Van Roy (2008a) n the context of the model outlned above. As ther orgnal work was amed at examnng models smlar to the one of Ercson and Pakes (1995), we modfy some aspects of equlbrum defnton whle keepng ts ntuton. 11

12 The concept of oblvous equlbrum was motvated by the fact that computaton of MPNE s not computatonally affordable when a dynamc game has more than a few players. Ths problem s due to two facts. Frst, ncreasng the number of players ncreases the number of dynamc programmng problems requred to obtan at least one (out of possbly many) equlbra. Second, the dmensonalty of the state space grows exponentally wth the number of players. Ths curse of dmensonalty mposes substantal computatonal and memory storage problems, whch cannot be handled wth computatonal resources commonly avalable to researchers. To deal wth ths problem, Wentraub, Benkard and Van Roy (2008) proposed the concept of oblvous equlbrum. In ths equlbrum concept, each agent solves ts dynamc problem consderng ts own state, x, whle treatng the compettors state, s, as a constant. Ths constant s the steady-state value of the ndustry state vector computed under the assumpton that all players choose optmal strateges n the fashon just descrbed. Ths approach transforms the mult-agent problem nto a sngle-agent problem by treatng other players states as constant, leadng to substantal computatonal gans. For an Ercson-Pakes game wth no aggregate shocks, Wentraub, Benkard and Van Roy (2008) demonstrate that the expected dfference between value functons computed under MPNE and oblvous strateges converges to zero as the number of game players goes to nfnty 2. Ther proof does not extend to a game wth aggregate shocks affectng all player payoffs, although smulaton results ndcate that oblvous equlbrum may stll provde good approxmatons to MPNE n that stuaton (for a dscusson Wentraub, Benkard and Van Roy 2009). The computatonal challenges on calculatng MPNEs n models wth many players also apply to dynamc dscrete games smlar to the ones of Agurregabra and Mra (2007) and Pesendorfer and Schmdt-Dengler (2007). Ths fact motvates a defnton of oblvous equlbrum n the sprt of Wentraub, Benkard and Van Roy (2009). Let s P (ω) denote the steady-state ndustry vector gven a profle of player choce probabltes P and the current value of the aggregate shock ω. In what follows, t s convenent to follow the probabltygeneratng functon framework of Wang and Yang (1995) to concsely defne the steady-state ndustry vector. The probablty-generatng functon of a dscrete random varable Y s defned as G(z) =E(z Y ). It s by defnton a power seres whose coeffcents are postve snce E(z Y )= P y=0 Pr(Y = y)zy. The computaton of probabltes and expectatons usng G(z) s straghtforward snce Pr(Y = k) =G (k) (0)/k! and E(Y )= lm z 1 G (1) (z), where G (l) (z) denotes the l th dervatve of G wth respect to z. By defnton, we have s P (ω) Nη P (ω), where η P (ω) s a column vector contanng the fracton of players that, n the steady-state, wll be n each state consdered n X. Under our assumpton that the value of player s ndvdual state at perod t +1s not stochastcally dependent on ts rvals actons, the transton matrx for x,t+1 gven the player s choce probabltes P andobservedstatess 2 To be exact, the lmt s takng by lettng one of the game parameters - the market sze - go to nfnty. The result s establshed usng the property that the equlbrum number of players s ncreasng n market sze. 12

13 1 x,t. J x,t+1 1 J p 1,1 (ω, s ) p 1,J (ω, s ).. p J,1 (ω, s ) p J,J (ω, s ) (2.12) where p l,j (ω, s )= P a A p x (x 0 = j ω,x = l, s,a )P (a ω, x = l, s ). Usng Theorem 1 n Wang and Yang (1995), the probablty-generatng functon for the ndustry state vector s gven that the game has N players s Ĝ N (z; η P (ω),ω,s) = η P (ω) T (1,z 2,.., z J ) Q N 1 (z; ω,s)1 (2.13) where the cells of Q(z; ω, s) are defned by [p l,j (ω, s )z j ] l,j {1,...,J},, Q k (.) represents the k th power of matrx Q(z; ω, s), and 1 s a J 1 vector of ones. Wthout loss of generalty, we consder the probablty generatng functon G N (z; s P (ω),ω,s) by merely multplyng and dvdng η P (ω) by N. Usng the expectaton formula for probablty-generatng functons, the steady-state ndustry vector s P (ω) s formally defned as the soluton to the fxed-pont problem on the vector s s = T ( s P (ω),ω,s) (2.14) where lm z1 1 G N ((z 1, 0,..., 0); s P (ω),ω,s) z 1 T ( s P (ω),ω,s) =. lm zj 1 G N ((0,..., 0,z J ); s P (ω),ω,s) z J (2.15) The column vector T ( s P (ω),ω,s) mplctly assumes that there exsts at least one steady-state vector s P (ω) by ncludng t as one of ts arguments. As our current focus s on the defnton of oblvous equlbrum for our dynamc dscrete game, we postpone the dscusson about exstence and unqueness of a soluton for 2.14 to the next secton. Our exposton proceeds by assumng the exstence of a soluton s P (ω) for ndustry steady-state gven current aggregate shock value ω and a choce probablty profle P. As n Wentraub, Benkard and Van Roy (2009), we assume that agents take s P (ω) as exogenously gven n ther dynamc problem by neglectng the mpact of ther ndvdual choces on the computaton of ndustry 13

14 steady-state. That s, nstead of consderng the current value of compettor state vector s, agents take the ndustry steady-state functon s P (ω) as gven and replace s wth the mpled steady-state vector for compettor state, denoted s P (ω). Ths s the reason for namng ths equlbrum as oblvous, n the sense that each agent dsregards the true nformaton on ts rvals and replaces t wth a steady-state constant vector. The latter s a partcularly convenent value to consder, snce t allows the agent to smplfy consderably ts optmzaton problem by restrctng attenton to ts own state x and the aggregate shock ω as the relevant state varables to solve for her optmal polcy. Consequently, the sequental representaton of each player s objectve functon s now " X # E β τ t π (a t,ω t,x t, s P,t(ω t ),ε t ) ωt,x t,ε t τ=t (2.16) where s,t has been removed from the nformaton set snce ω t suffces to compute the vector s P,t (ω t). To solve ths dynamc problemy, each player no longer consders the set Θ Ω X S Σ to form a polcy functon; nstead, she restrcts attenton to the set Θ Ω X Σ and consders the mappng of oblvous polces σ : Θ A. We defne the oblvous choce probabltes s a fashon smlar to 2.4 wth the necessary modfcatons: Z P σ (a ω, x ) Pr( σ (ω, x,ε )=a ω, x )= 1( σ (ω, x,ε )=a )g (dε ω, x, s P (ω)) (2.17) where σ = { σ (ω,x,ε )} N =1 represents a profle of oblvous strategy functons and P s the choce probablty profle assocated wth σ. We assume that each player acknowledges that ts rvals restrct attenton to oblvous strateges. Consstent wth ths assumpton, the set of oblvous choce probabltes { P σ (a ω, x ):a A } N =1 represents expected behavor from player from the perspectve of the remanng players when player abdes by the oblvous strategy profle σ. The recursve representaton of the player s problem 2.16 gven oblvous strategy profle σ s represented by the Oblvous Bellman equaton n ˆV σ (ω, x, s P (ω),ε ) = max π (a,ω,x, s P (ω)) + ε (a ) (2.18) a A +β X X Z ˆV σ (ω 0,x 0, s P (ω 0 ),ε 0 )g (dε 0 ω 0,x 0,, s P (ω 0 )) ω 0 Ω x 0 X o p x (x 0 ω, x, s P (ω),a ) p ω (ω 0 ω, x,, s P (ω)) We now provde a defnton of oblvous equlbrum for the dynamc dscrete games: 14

15 Defnton 2 Aprofle of oblvous strateges σ s an Oblvous Equlbrum f for every player =1,..,N and any (ω, x,ε ) Θ, the followng condtons hold: () σ (ω, x,ε ) s a soluton for the Bellman Equaton 2.18 when all rvals of player choose accordng to ther strategy functon descrbed n σ ; () s P (ω) s a soluton to the steady-state problem 2.14 for the choce probablty profle P mpled by the oblvous strategy profle σ. There are two key dfferences between 2.5 and The frst s that the compettor state vector s merely a functon of ω n 2.18, not a tme-varyng vector governed by a transton as n 2.5. Secondly, the dscounted future payoff term n 2.18 does not contan a summaton over all possble values of s S. Ths follows from the assumpton that, from all players perspectve, the temporal evoluton of s s now governed by changes n the aggregate shock ω through the vector functon s P (ω). These two propertes result n two mportant smplfcatons of player s problem. Frst, 2.18 s now a sngle-agent problem, snce each player takes s P (ω) as exogenously gven and so her relevant state space conssts only of ts ndvdual states - x and ε and an exogenously-evolvng aggregate shock ω. Second, there s a consderable reducton on state space dmenson as a result of restrctng attenton to the compettors steady-state vector rather consderng the transton matrx for s. In a slght abuse of notaton, we defne the oblvous value functon by removng s P (ω) from ts argument set. Takng condtonal expectatons of 2.18 wth respect to ε, the ntegrated Oblvous Bellman Equaton becomes Ṽ σ (ω, x ) = X a A P σ (a ω,x ) +β X X Ṽ σ ω 0 Ω x 0 X ³ nhπ (a,ω,x, s P (ω)) + e P a,ω,x, s P (ω) (ω 0,x 0 ))p x (x 0 ω, x,, s P (ω),a ) p ω (ω 0 ω, x,, s P (ω)) (2.19) where e P ³a,ω,x, s P (ω) now corresponds to the expected value of prvate nformaton gven observed states (ω,x ), the compettor s steady-state functon s P (ω) and the acton chosen by player beng the oblvous strategy σ (ω, x,ε ). As n the case of Markovan strateges, e P ³a,ω,x, s P (ω) can be shown to be a functon only of player s choce probablty vector P and the condtonal prvate nformaton densty g (..) by mmckng the proof of Agurregabra and Mra (2007, pp 9-10). We now map the concept of oblvous equlbrum nto probablty space. We follow the same approach as n the defnton of MPNE n probablty space by stackng state-specfc elementsof?? for each player. The resultng system dennng oblvous ntegrated Bellman equatons gven an oblvous profle P s 15

16 Ṽ P = X h P (a ) π (a )+e P a A (a )+β f P (a )Ṽ P (2.20) where P (a ), π (a ) and e P (a ) are column vectors of dmenson G = Ω X stackng state and actonspecfc elements of choce probablty, oblvous payoff and condtonal expected prvate nformaton, respectvely. Player consders the probablty matrx f P (a ), whch represents state transtons of (ω, x ) gven that acton a s chosen and compettors state vector s evaluated at s P (ω). Note that G<Rsnce player now restrcts attenton to (ω, x ) and uses ths nformaton and the probablty profle to compute s P (ω). In addton to allevatng the curse of dmensonalty through state space reducton, ths smplfcaton transforms the mult-agent dynamc game nto a sngle-agent Markov decson process smlar to the one of Agurregabra and Mra (2002). Ths fact wll play a key role on showng unqueness of oblvous equlbrum. The value functon operator as a functon of the oblvous choce probablty s ϕ ( P) = ( ³ 1 X h P I β F P (a ) π (a )+e P (a ) ) (2.21) a A where F P = P a A P (a ) f P (a ). We complete our defnton of oblvous equlbrum (OE) n probablty space by representng t as a fxed pont n P. As n the probablstc representaton of MPNE, we use the defnton of choce probablty n 2.4 to form an operator mappng the space of probablty profles nto the space of ndvdual choce probabltes. Ths operator conssts of the probablty that a gven acton solves the Oblvous Bellman equaton Snce players restrct attenton to oblvous strateges to solve ths equaton, each component of the operator s defned by Ψ (a ω, x ; P) = = Z 1(a = σ (ω, x,ε ))g (dε ω, x, s P (ω)) (2.22) Z n 1(a =argmax π (a,ω,x, s P (ω)) + ε (a ) (2.23) a A +β X X ϕ (ω 0,x 0 ; P)p x (x 0 ω, x,, s P (ω),a ) ω 0 Ω x 0 X p ω (ω 0 ω, x, s P (ω)) g (dε ω, x, s P (ω)) where ϕ (ω 0,x 0 ; P) regards the value functon cell of ϕ ( P) assocated wth the combnaton of states (ω 0,x 0 ) computed gven choce probablty profle P. 16

17 Lettng Ψ ³ ( P) Ψ (a 1 ω,x ; P),..., Ψ (a A ω, x ; P), an OE representaton n choce probabltes space conssts of any profle P that s a soluton the fxed pont problem P = Ψ( P), where Ψ( P) = ( Ψ 1 ( P),..., Ψ N ( P)). We use ths representaton of OE to examne the propertes of OE and ts relatonshp wth MPNE n the next secton. 3 Results on Oblvous Equlbrum In ths secton we examne two topcs that are at the core of our contrbutons. Frst, we dscuss the exstence and unqueness of oblvous equlbrum. Second, we examne the approxmaton propertes of oblvous equlbrum to MPNE when the number of players goes to nfnty. The steady-state ndustry vector s a key ngredent n the defnton of oblvous equlbrum whose exstence we have taken as gven. For ths reason, we start by dscussng the exstence and unqueness of a steady-state soluton for 2.14 for any oblvous choce probablty profle P. Lemma 1 For any oblvous choce probablty profle P, there exsts an unque steady-state vector of compettor s states. Proof. Note that under an oblvous profle P the steady-state operator T ( s P (ω),ω,s) defnedn2.15 smplfes to T ( s P (ω),ω) snce the ndustry state s s not an argument of oblvous strateges. From assumptons 2.2 and 2.3, t follows that the transton matrx for the ndustry state s mplct to 2.13 defnes an rreducble Markov Chan for any oblvous profle P snce every state of the ndustry vector s reaches other states wth strctly postve probablty. In addton, the transton matrx for s mplct to 2.13 s aperodc by constructon snce all ts dagonal elements are strctly postve 3. Unqueness of ndustry state vector follows from the well-known result that an rreducble and aperodc Markov Chan has an unque steady-state dstrbuton. A drect mplcaton of ths lemma s that for any oblvous profle P there s only one steady-state vector of player s compettors state, denoted s P (ω). Ths smplfes consderably the analyss of OE exstence and unqueness, as t mples that every player consders at most one constant vector n leu of s. 3.1 Exstence and Unqueness As n the above dscusson about MPNE, we establsh the exstence of oblvous equlbrum by drect applcaton of Brower s fxed pont theorem to the system P = Ψ( P). A more complcated yet nterestng 3 AMarkovchan{Z t,t 0} s sad to be aperodc f the greatest common dvsor of the sequence {n :Pr(Z n = Z 0 = )} s 1 for every state. 17

18 ssue s the unqueness of oblvous equlbrum. The followng theorem addresses ths ssue: Theorem 1 There s an unque oblvous equlbrum for the dynamc dscrete game. We dscuss the ntuton behnd ths theorem s proof, whch we present n the Appendx. Frst, the fact that there exsts an unque steady-state for any oblvous choce probablty profle P mples that the functon s P (ω) s unque regardless of whether P s an equlbrum or not. Snce each agent takes s P (ω) as exogenously gven, t s treated as a vector functon of the aggregate shock ω. Thus, each agent s solvng a Markov decson process smlar to the one of Agurregabra and Mra (2002) snce ϕ ( P) only depends on P, not the whole profle P. Thus, the oblvous profle P consstng of the collecton of ndvdual choce probabltes solvng each player s Markov decson process gven s P (ω) must be an equlbrum. Agurregabra and Mra (2002) show that the soluton to ther Markov decson process s unque by explotng the unqueness of fxed pont for the value functon operator. Unqueness of the oblvous equlbrum P s proven s establshed n a smlar fashon by usng the unqueness of fxed ponts for both the value functon and steady-state operators. 3.2 Asymptotc Results The ntroducton of the oblvous equlbrum concept n the lterature by Benkard, Wentraub and Van Roy (2008) s motvated by ts ablty to approxmate MPNE when the number of players n the game s large. Here we address ths ssue n the context of the dynamc dscrete game descrbed above. We demonstrate that the dstance between the value functon vectors of MPNE and OE converges n probablty to zero as the number of players goes to nfnty. There are three key dfferences between our results and the ones of Benkard, Wentraub and Van Roy (2008). Frst, the metrc consdered n the convergence result of Benkard, Wentraub and Van Roy (2008) s the expected dfference between value functons under MPNE and OE, where expectatons are taken consderng the steady-state dstrbuton under OE. Ths metrc does not rule out the possblty that value functons dffer consderably under the two equlbrum concepts whle havng the same expected value. The metrc consdered n our work s convergence n probablty, whch by defnton rules out that caveat. Second, Benkard, Wentraub and Van Roy (2008) consder a game wthout aggregate shocks ω. Ths condton s requred to valdate ther convergence proof. Even though smulaton results presented n Benkard, Wentraub and Van Roy (2009) ndcate that the presence of aggregate shocks does not compromse the qualty of OE approxmatons to MPNE, no concrete proof s presented. In our famework we explctly allow for aggregate states wthout compromsng our covergence proofs. Fnally, Benkard, Wentraub and Van Roy (2008) requre what they name as a "lght-tal" condton to proof ther asymptotc result. In the Ercson-Pakes model of 18

19 nvestment n qualty, the "lght-tal" condton conssts of a restrcton on the players proft functon amed at rulng out equlbra where a small porton of players holds a hgh fracton of total market profts. Ths would be the case of markets where a few frms have most of market share (e.g., software ndustry). In these stuatons, OE could result n a very poor approxmaton to MPNE despte the large number of players snce the dstance between steady-state and current ndustry vector s large due to the presence of domnant frms. Ths type of assumpton s not requred for provng that OE approxmates well MPNE when the number of players s large. There are two ntutve reasons for ths fact. Frst, we have shown that OE s unque for any number of game players. So mposng a "lght-tal" condton wll not lead to any mprovement of the approxmaton propertes of OE. Second, the observed state space s dscrete and fnte. Consequently, the fracton of players who are at a gven state converges n probablty to a constant by nvokng a sutable law of large numbers. Ths fact avods the need of any "lght-tal" condton to ensure convergence under a gven metrc. However, we need an auxlary assumpton on the flow payoff functon to ensure that the concept of convergence as N s well-defned. The followng assumpton ensures that property: Assumpton 4 The flow payoff functon π (a t,ω t,x t, s,t ) can be wrtten as a functon contnuous n η t, denoted π,n (a t,ω t,x t, η t ),whereη t s the relatve frequency of each state of X for a game wth N players. Ths assumpton s farly weak and apples to most appled work on dynamc dscrete games, ncludng the market entry and ext example descrbed above. We are now ready to establsh our man asymptotc result: Theorem 2 The dstance between player value functons usng Markovan and Oblvous strateges converges to zero n probablty as the number of game players goes to nfnty. Here we dscuss the proof man steps and ntuton whle outlnng the complete verson n the Appendx. Frst, we nvoke the unform law of large numbers of Vapnk and Chervonenks (1971) to clam that the relatve frequency vector for X, (.e. η t ), converges under any Markovan or Oblvous strategy profles tothesamevectorasn. That s, the lmtng steady-state dstrbuton of players under MPNE and OE s the same. Second, we note that strategy functons defned over the Markovan state space Θ Ω X S Σ do not mprove upon strateges defned over the oblvous space set Θ Ω X Σ for solvng the dynamc programng problem of an oblvous player descrbed n Ths s a feature of oblvous strateges that our framework shares wth the one of Wentraub, Benkard and Van Roy (2008). We use the trangular nequalty to bound the dfference between value functons by the sum of two components whch are shown to converge to zero n probablty. The convergence n probablty of the latter to zero follows from our assumpton on flow payoff and the unform convergence n probablty of η t. 19

20 Corollary 1 The dstance between an MPNE and the unque OE converges n probablty to zero as N. Proof. Ths results follows from the convergence of state dstrbutons η t establshed usng the unform law of large numbers of Vapnk and Chervonenks (1971) and the contnuty of the choce probablty operators Λ (.) and Λ (.). By the unform law of large numbers of the unform law of large numbers of Vapnk and Chervonenks (1971) the relatve frequency vector η t (ω) converges n probablty unformly to a constant vector η(ω) as N.Usng the trangular nequalty, we have Λ (V P,N) Λ (Ṽ P,N ) Λ (V P,N) Λ (V,N) P + Λ (V,N) P Λ (Ṽ P,N ) (3.1) where V P s player 0 s value functon under MPNE profle P when the compettor s state s calculated usng the lmt η(ω). It follows from Assumpton 4 that value functon vectors are contnuous n η t. In addton, Λ (.) and Λ (.) are contnuous functonals mappng value functon vectors nto choce probabltes. Therefore, n o t follows from the Mann-Wald Contnuous Mappng theorem that lm N Pr Λ (V,N P ) Λ (V,N P ) n >ε Λ 0 and that lm N Pr (V,N P ) Λ o (Ṽ P,N ) >ε =0, ε >0. Snce each parcel of the rght-hand sde of 3.1 converges n probablty, the proof s complete. = The ntuton behnd ths result reles on the fact that, as N, η t (ω) converges to a constant vector. Ths means that s P (ω) becomes approxmately constant when the game has many players. Ths mples that, for each player, ther rval state vector s provdes approxmately the same nformaton as the steadystate computed usng OE. Snce the key dfference between MPNE and OE s that the former consders compettors state vector S n the strategy set, t comes at no surprse that, that the choce probabltes converge. Ths argument s confrmed usng the contnuty of value functon and choce probablty operators and the Contnuous Mappng theorem. 4 Concluson Ths paper bulds upon and extends the oblvous equlbrum concept of Benkard, Wentraub and Van Roy (2008a, 2009a) to Agurregabra-Mra-style dynamc dscrete games. We fnd that under assumptons commonly posed n both emprcal and theoretcal lterature on dynamc games, the resultng oblvous equlbrum s unque and can approxmate Markov-Prefect equlbra n games wth many players. Our approxmaton metrc s stronger than the one consdered by Wentraub and Van Roy (2008). Unlke ths latter poneer work, our proofs do not requre absence of aggregate shocks as state varables. These results allow researchers to compute equlbra for polcy experments wthout facng equlbrum multplcty ssues frequently present at dynamc games. 20

21 The calculaton of the unque oblvous equlbrum s computatonally affordable even for large state space dmensons for two reasons. Frst, the oblvous equlbrum computaton essentally requres the researcher to compute an unque fxed pont n probablty space representng a sngle-agent problem. The computatonal costs of ths type of fxed-pont have been mnmzed courtesy of mproved computng resources and advances n equlbrum calculaton n probablty spaces (e.g. Agurregabra and Mra 2002). Second, the oblvous equlbrum also nvolves computng a fxed pont for ndustry steady-state vector, whose dmenson s equal to the player state space X. The probablty-generatng functon approach of Wang and Yang (1995) for sums of Markov chans avods the cumbersome and possbly prohbtve task of dervng the transton matrx for the ndustry state vector. The fact that the dstance between value functons of MPNE and OE shrnks as N by no means guarantees that OE s well-defned for nfntely many players. Wentraub, Benkard and Van Roy (2009b) examne ths ssue n the context of ther Ercson-Pakes-lke model. In our context, the exstence and propertes of OE when N = depends on the concrete assumptons on game prmtves. In partcular, t depends on whch (and how many) of the cells of the ndustry steady-state vector s P (ω) converge to a constant as N. In ths latter case, the asymptotc dstrbuton results of Wang and Yang (1995) can be used to show that the OE n dynamc dscrete games has well-defned equlbrum choce probabltes when N =, where the cells of s P (ω) convergng to constants as N are gven by the Compound Posson dstrbuton parameters.snce the fulfllment of those condtons requres a more extensve examnaton of the model prmtves at hand, further work on ths drecton s necessary and s left for future research. 21