{"title": "Oblivious Equilibrium: A Mean Field Approximation for Large-Scale Dynamic Games", "book": "Advances in Neural Information Processing Systems", "page_first": 1489, "page_last": 1496, "abstract": "", "full_text": "Oblivious Equilibrium: A Mean Field\n\nApproximation for Large-Scale Dynamic Games\n\nGabriel Y. Weintraub, Lanier Benkard, and Benjamin Van Roy\n\n{gweintra,lanierb,bvr}@stanford.edu\n\nStanford University\n\nAbstract\n\nWe propose a mean-\ufb01eld approximation that dramatically reduces the\ncomputational complexity of solving stochastic dynamic games. We pro-\nvide conditions that guarantee our method approximates an equilibrium\nas the number of agents grow. We then derive a performance bound to\nassess how well the approximation performs for any given number of\nagents. We apply our method to an important class of problems in ap-\nplied microeconomics. We show with numerical experiments that we are\nable to greatly expand the set of economic problems that can be analyzed\ncomputationally.\n\n1 Introduction\n\nIn this paper we consider a class of in\ufb01nite horizon non-zero sum stochastic dynamic\ngames. At each period of time, each agent has a given state and can make a decision.\nThese decisions together with random shocks determine the evolution of the agents\u2019 states.\nAdditionally, agents receive pro\ufb01ts depending on the current states and decisions. There\nis a literature on such models which focusses on computation of Markov perfect equilibria\n(MPE) using dynamic programming algorithms. A major shortcoming of, however, is the\ncomputational complexity associated with solving for the MPE. When there are more than\na few agents participating in the game and/or more than a few states per agent, the curse of\ndimensionality renders dynamic programming algorithms intractable.\nIn this paper we consider a class of stochastic dynamic games where the state of an agent\ncaptures its competitive advantage. Our main motivation is to consider an important class of\nmodels in applied economics, namely, dynamic industry models of imperfect competition.\nHowever, we believe our methods can be useful in other contexts as well. To clarify the\ntype of models we consider, let us describe a speci\ufb01c example of a dynamic industry model.\nConsider an industry where a group of \ufb01rms can invest to improve the quality of their\nproducts over time. The state of a given \ufb01rm represents its quality level. The evolution\nof quality is determined by investment and random shocks. Finally, at every period, given\ntheir qualities, \ufb01rms compete in the product market and receive pro\ufb01ts. Many real world\nindustries where, for example, \ufb01rms invest in R&D or advertising are well described by\nthis model.\nIn this context, we propose a mean-\ufb01eld approximation approach that dramatically sim-\npli\ufb01es the computational complexity of stochastic dynamic games. We propose a simple\n\n\falgorithm for computing an \u201coblivious\u201d equilibrium in which each agent is assumed to\nmake decisions based only on its own state and knowledge of the long run equilibrium\ndistribution of states, but where agents ignore current information about rivals\u2019 states. We\nprove that, if the distribution of agents obeys a certain \u201clight-tail\u201d condition, when the num-\nber of agents becomes large the oblivious equilibrium approximates a MPE. We then derive\nan error bound that is simple to compute to assess how well the approximation performs\nfor any given number of agents.\nWe apply our method to analyze dynamic industry models of imperfect competition. We\nconduct numerical experiments that show that our method works well when there are sev-\neral hundred \ufb01rms, and sometimes even tens of \ufb01rms. Our method, which uses simple code\nthat runs in a couple of minutes on a laptop computer, greatly expands the set of economic\nproblems that can be analyzed computationally.\n\n2 A Stochastic Dynamic Game\n\no\n\nx=0 s(x) = n\n\ns \u2208 N\u221e(cid:12)(cid:12)(cid:12)P\u221e\nn\n\nIn this section, we formulate a non-zero sum stochastic dynamic game. The system evolves\nover discrete time periods and an in\ufb01nite horizon. We index time periods with nonnegative\nintegers t \u2208 N (N = {0, 1, 2, . . .}). All random variables are de\ufb01ned on a probability space\n(\u2126,F,P) equipped with a \ufb01ltration {Ft : t \u2265 0}. We adopt a convention of indexing by t\nvariables that are Ft-measurable.\nThere are n agents indexed by S = {1, ..., n}. The state of each agent captures its ability\nto compete in the environment. At time t, the state of agent i \u2208 S is denoted by xit \u2208 N.\nWe de\ufb01ne the system state st to be a vector over individual states that speci\ufb01es, for each\nstate x \u2208 N, the number of agents at state x in period t. We de\ufb01ne the state space S =\n. For each i \u2208 S, we de\ufb01ne s\u2212i,t \u2208 S to be the state of the\ncompetitors of agent i; that is, s\u2212i,t(x) = st(x) \u2212 1 if xit = x, and s\u2212i,t(x) = st(x),\notherwise.\nIn each period, each agent earns pro\ufb01ts. An agent\u2019s single period expected pro\ufb01t\n\u03c0m(xit, s\u2212i,t) depends on its state xit, its competitors\u2019 state s\u2212i,t and a parameter m \u2208\n<+. For example, in the context of an industry model, m could represent the total number\nof consumers, that is, the size of the pie to be divided among all agents. We assume that for\nall x \u2208 N, s \u2208 S, m \u2208 <+, \u03c0m(x, s) > 0 and is increasing in x. Hence, agents in larger\nstates earn more pro\ufb01ts.\nIn each period, each agent makes a decision. We interpret this decision as an investment\nto improve the state at the next period. If an agent invests \u00b5it \u2208 <+, then the agent\u2019s state\nat time t + 1 is given by, xi,t+1 = xit + w(\u00b5it, \u03b6i,t+1), where the function w captures\nthe impact of investment on the state and \u03b6i,t+1 re\ufb02ects uncertainty in the outcome of\ninvestment. For example, in the context of an industry model, uncertainty may arise due to\nthe risk associated with a research endeavor or a marketing campaign. We assume that for\nall \u03b6, w(\u00b5, \u03b6) is nondecreasing in \u00b5. Hence, if the amount invested is larger it is more likely\nthe agent will transit next period to a better state. The random variables {\u03b6it|t \u2265 0, i \u2265 1}\nare i.i.d.. We denote the unit cost of investment by d.\nEach agent aims to maximize expected net present value. The interest rate is assumed to\nbe positive and constant over time, resulting in a constant discount factor of \u03b2 \u2208 (0, 1) per\ntime period. The equilibrium concept we will use builds on the notion of a Markov perfect\nequilibrium (MPE), in the sense of [3]. We further assume that equilibrium is symmetric,\nsuch that all agents use a common stationary strategy. In particular, there is a function \u00b5\nsuch that at each time t, each agent i \u2208 S makes a decision \u00b5it = \u00b5(xit, s\u2212i,t). Let M\ndenote the set of strategies such that an element \u00b5 \u2208 M is a function \u00b5 : N \u00d7 S \u2192 <+.\n\n\fWe de\ufb01ne the value function V (x, s|\u00b50, \u00b5) to be the expected net present value for an agent\nat state x when its competitors\u2019 state is s, given that its competitors each follows a common\nstrategy \u00b5 \u2208 M, and the agent itself follows strategy \u00b50 \u2208 M. In particular,\n\nV (x, s|\u00b50, \u00b5) = E\u00b50,\u00b5\n\n\u03b2k\u2212t (\u03c0(xik, s\u2212i,k) \u2212 d\u03b9ik)\n\n#\n(cid:12)(cid:12)(cid:12)xit = x, s\u2212i,t = s\n\n,\n\n\" \u221eX\n\nk=t\n\nwhere i is taken to be the index of an agent at state x at time t, and the subscripts of\nthe expectation indicate the strategy followed by agent i and the strategy followed by its\ncompetitors. In an abuse of notation, we will use the shorthand, V (x, s|\u00b5) \u2261 V (x, s|\u00b5, \u00b5),\nto refer to the expected discounted value of pro\ufb01ts when agent i follows the same strategy\n\u00b5 as its competitors.\nAn equilibrium to our model comprises a strategy \u00b5 \u2208 M that satisfy the following condi-\ntion:\n(2.1)\n\nV (x, s|\u00b50, \u00b5) = V (x, s|\u00b5)\n\n\u2200x \u2208 N, \u2200s \u2208 S.\n\nsup\n\u00b50\u2208M\n\nUnder some technical conditions, one can establish existence of an equilibrium in pure\nstrategies [4]. With respect to uniqueness, in general we presume that our model may\nhave multiple equilibria. Dynamic programming algorithms can be used to optimize agent\nstrategies, and equilibria to our model can be computed via their iterative application. How-\never, these algorithms require compute time and memory that grow proportionately with the\nnumber of relevant system states, which is often intractable in contexts of practical interest.\nThis dif\ufb01culty motivates our alternative approach.\n\n3 Oblivious Equilibrium\n\nWe will propose a method for approximating MPE based on the idea that when there are\na large number of agents, simultaneous changes in individual agent states can average out\nbecause of a law of large numbers such that the normalized system state remains roughly\nconstant over time. In this setting, each agent can potentially make near-optimal decisions\nbased only on its own state and the long run average system state. With this motivation,\nwe consider restricting agent strategies so that each agent\u2019s decisions depend only on the\nagent\u2019s state. We call such restricted strategies oblivious since they involve decisions made\nwithout full knowledge of the circumstances \u2014 in particular, the state of the system. Let\n\u02dcM \u2282 M denote the set of oblivious strategies. Since each strategy \u00b5 \u2208 \u02dcM generates\ndecisions \u00b5(x, s) that do not depend on s, with some abuse of notation, we will often drop\nthe second argument and write \u00b5(x).\nLet \u02dcs\u00b5 be the long-run expected system state when all agents use an oblivious strategy\n\u00b5 \u2208 \u02dcM. For an oblivious strategy \u00b5 \u2208 \u02dcM we de\ufb01ne an oblivious value function\n\n\u02dcV (x|\u00b50, \u00b5) = E\u00b50\n\n\u03b2k\u2212t (\u03c0(xik, \u02dcs\u00b5) \u2212 d\u03b9ik)\n\n#\n(cid:12)(cid:12)(cid:12)xit = x\n\n.\n\n\" \u221eX\n\nk=t\n\nThis value function should be interpreted as the expected net present value of an agent that\nis at state x and follows oblivious strategy \u00b50, under the assumption that its competitors\u2019\nstate will be \u02dcs\u00b5 for all time. Again, we abuse notation by using \u02dcV (x|\u00b5) \u2261 \u02dcV (x|\u00b5, \u00b5)\nto refer to the oblivious value function when agent i follows the same strategy \u00b5 as its\ncompetitors.\nWe now de\ufb01ne a new solution concept: an oblivious equilibrium consists of a strategy\n\u00b5 \u2208 \u02dcM that satisfy the following condition:\n(3.1)\n\n\u02dcV (x|\u00b50, \u00b5) = \u02dcV (x|\u00b5),\n\n\u2200x \u2208 N.\n\nsup\n\u00b50\u2208 \u02dcM\n\n\fIn an oblivious equilibrium \ufb01rms optimize an oblivious value function assuming that its\ncompetitors\u2019 state will be \u02dcs\u00b5 for all time. The optimal strategy obtained must be \u00b5. It\nis straightforward to show that an oblivious equilibrium exists under mild technical condi-\ntions. With respect to uniqueness, we have been unable to \ufb01nd multiple oblivious equilibria\nin any of the applied problems we have considered, but similarly with the case of MPE, we\nhave no reason to believe that in general there is a unique oblivious equilibrium.\n\n4 Asymptotic Results\n\nIn this section, we establish asymptotic results that provide conditions under which obliv-\nious equilibria offer close approximations to MPE as the number of agents, n, grow. We\nconsider a sequence of systems indexed by the one period pro\ufb01t parameter m and we as-\nsume that the number of agents in system m is given by n(m) = am, for some a > 0.\nRecall that m represents, for example, the total pie to be divided by the agents so it is\nreasonable to increase n(m) and m at the same rate.\nWe index functions and random variables associated with system m with a superscript\n(m). From this point onward we let \u02dc\u00b5(m) denote an oblivious equilibrium for system m.\nLet V (m) and \u02dcV (m) represent the value function and oblivious value function, respectively,\nwhen the system is m. To further abbreviate notation we denote the expected system state\nassociated with \u02dc\u00b5(m) by \u02dcs(m) \u2261 \u02dcs\u02dc\u00b5(m). The random variable s(m)\ndenotes the system state\nat time t when every agent uses strategy \u02dc\u00b5(m). We denote the invariant distribution of\n{s(m)\n: t \u2265 0} by q(m). In order to simplify our analysis, we assume that the initial system\nstate s(m)\nis distributed\naccording to q(m) for all t \u2265 0. It will be helpful to decompose s(m)\nt =\nt n(m), where f (m)\nf (m)\nis the random vector that represents the fraction of agents in each\nstate. Similarly, let \u02dcf (m) \u2261 E[f (m)\n] denote the expected fraction of agents in each state.\nWith some abuse of notation, we de\ufb01ne \u03c0m(xit, f\u2212i,t, n) \u2261 \u03c0m(xit, n \u00b7 f\u2212i,t). We assume\nthat for all x \u2208 N, f \u2208 S1, \u03c0m(x, f, n(m)) = \u0398(1), where S1 = {f \u2208 <\u221e\nx\u2208N f(x) =\n1}. If m and n(m) grow at the same rate, one period pro\ufb01ts remain positive and bounded.\nOur aim is to establish that, under certain conditions, oblivious equilibria well-approximate\nMPE as m grows. We de\ufb01ne the following concept to formalize the sense in which this\napproximation becomes exact.\nDe\ufb01nition 4.1. A sequence \u02dc\u00b5(m) \u2208 M possesses the asymptotic Markov equilibrium\n(AME) property if for all x \u2208 N,\n\nis sampled from q(m). Hence, s(m)\n\nis a stationary process; s(m)\n\n+ |P\n\naccording to s(m)\n\nt\n\nt\n\n0\n\nt\n\nt\n\nt\n\nt\n\nt\n\n(cid:20)\n\n(cid:21)\n\nlim\nm\u2192\u221e E\u02dc\u00b5(m)\n\nsup\n\u00b50\u2208M\n\nV (m)(x, s(m)\n\nt\n\n|\u00b50, \u02dc\u00b5(m)) \u2212 V (m)(x, s(m)\n\nt\n\n|\u02dc\u00b5(m))\n\n= 0 .\n\nThe de\ufb01nition of AME assesses approximation error at each agent state x in terms of the\namount by which an agent at state x can increase its expected net present value by de-\nviating from the oblivious equilibrium strategy \u02dc\u00b5(m), and instead following an optimal\n(non-oblivious) best response that keeps track of the true system state. The system states\nare averaged according to the invariant distribution.\nIt may seem that the AME property is always obtained because n(m) is growing to in\ufb01nity.\nHowever, recall that each agent state re\ufb02ects its competitive advantage and if there are\nagents that are too \u201cdominant\u201d this is not necessarily the case. To make this idea more\nconcrete, let us go back to our industry example where \ufb01rms invest in quality. Even when\nthere are a large number of \ufb01rms, if the market tends to be concentrated \u2014 for example,\nif the market is usually dominated by a single \ufb01rm with a an extremely high quality \u2014\n\n\fthe AME property is unlikely to hold. To ensure the AME property, we need to impose a\n\u201clight-tail\u201d condition that rules out this kind of domination.\n\nNote that d ln \u03c0m(y,f,n)\nof agents in state x. We de\ufb01ne the maximal absolute semi-elasticity function:\n\nis the semi-elasticity of one period pro\ufb01ts with respect to the fraction\n\ndf (x)\n\ng(x) =\n\nmax\n\nm\u2208<+,y\u2208N,f\u2208S1,n\u2208N\n\n(cid:12)(cid:12)(cid:12)(cid:12) d ln \u03c0m(y, f, n)\n\ndf(x)\n\n(cid:12)(cid:12)(cid:12)(cid:12) .\n\nFor each x, g(x) is the maximum rate of relative change of any agent\u2019s single-period pro\ufb01t\nthat could result from a small change in the fraction of agents at state x. Since larger\ncompetitors tend to have greater in\ufb02uence on agent pro\ufb01ts, g(x) typically increases with x,\nand can be unbounded.\nFinally, we introduce our light-tail condition. For each m, let \u02dcx(m) \u223c \u02dcf (m), that is, \u02dcx(m)\nis a random variable with probability mass function \u02dcf (m). \u02dcx(m) can be interpreted as the\nstate of an agent that is randomly sampled from among all agents while the system state is\ndistributed according to its invariant distribution.\nAssumption 4.1. For all states x, g(x) < \u221e. For all \u0001 > 0, there exists a state z such that\n\nh\n\ni \u2264 \u0001,\n\nfor all m.\n\nE\n\ng(\u02dcx(m))1{\u02dcx(m)>z}\n\nPut simply, the light tail condition requires that states where a small change in the fraction\nof agents has a large impact on the pro\ufb01ts of other agents, must have a small probability\nunder the invariant distribution. In the previous example of an industry where \ufb01rms invest\nin quality this typically means that very large \ufb01rms (and hence high concentration) rarely\noccur under the invariant distribution.\nTheorem 4.1. Under Assumption 4.1 and some other regularity conditions1, the sequence\n\u02dc\u00b5(m) of oblivious equilibria possesses the AME property.\n\n5 Error Bounds\n\nWhile the asymptotic results from Section 4 provide conditions under which the approxi-\nmation will work well as the number of agents grows, in practice one would also like to\nknow how the approximation performs for a particular system. For that purpose we derive\nperformance bounds on the approximation error that are simple to compute via simula-\ntion and can be used to asses the accuracy of the approximation for a particular problem\ninstance.\nWe consider a system m and to simplify notation we suppress the index m. Consider an\noblivious strategy \u02dc\u00b5. We will quantify approximation error at each agent state x \u2208 N by\n\nE(cid:2)sup\u00b50\u2208M V (x, st|\u00b50, \u02dc\u00b5) \u2212 V (x, st|\u02dc\u00b5)(cid:3) . The expectation is over the invariant distribu-\n\ntion of st. The next theorem provides a bound on the approximation error. Recall that \u02dcs\nis the long run expected state in oblivious equilibrium (E[st]). Let ax(y) be the expected\ndiscounted sum of an indicator of visits to state y for an agent starting at state x that uses\nstrategy \u02dc\u00b5 .\nTheorem 5.1. For any oblivious equilibrium \u02dc\u00b5 and state x \u2208 N,\n\n(5.1)\n\nE [\u2206V ] \u2264 1\n1 \u2212 \u03b2\n\nax(y) (\u03c0(y, \u02dcs) \u2212 E [\u03c0(y, st)]) ,\n\nE[\u2206\u03c0(st)] +X\n\ny\u2208N\n\n1In particular, we require that the single period pro\ufb01t function is \u201csmooth\u201d as a function of its\n\narguments. See [5] for details.\n\n\f=\n\nsup\u00b50\u2208M V (x, st|\u00b50, \u02dc\u00b5) \u2212 V (x, st|\u02dc\u00b5)\n\nwhere \u2206V\nmaxy\u2208N (\u03c0(y, s) \u2212 \u03c0(y, \u02dcs)).\nThe error bound can be easily estimated via simulation algorithms. In particular, note that\nthe bound is not a function of the true MPE or even of the optimal non-oblivious best\nresponse strategy.\n\nand \u2206\u03c0(s)\n\n=\n\n6 Application: Industry Dynamics\n\nMany problems in applied economics are dynamic in nature. For example, models involv-\ning the entry and exit of \ufb01rms, collusion among \ufb01rms, mergers, advertising, investment in\nR&D or capacity, network effects, durable goods, consumer learning, learning by doing,\nand transaction or adjustment costs are inherently dynamic. [1] (hereafter EP) introduced\nan approach to modeling industry dynamics. See [6] for an overview. Computational com-\nplexity has been a limiting factor in the use of this modeling approach. In this section we\nuse our method to expand the set of dynamic industries that can be analyzed computation-\nally.\nEven though our results apply to more general models where for example \ufb01rms make exit\nand entry decisions, here we consider a particular case of an EP model which itself is a\nparticular case of the model introduced in Section 2. We consider a model of a single-good\nindustry with quality differentiation. The agents are \ufb01rms that can invest to improve the\nquality of their product over time. In particular xit is the quality level of \ufb01rm i at time t.\n\u00b5it represents represents the amount of money invested by \ufb01rm i at time t to improve its\nquality. We assume the one period pro\ufb01t function is derived from a logit demand system\nand where \ufb01rms compete setting prices. In this case, m represents the market size. See [5]\nfor more details about the model.\n\n6.1 Computational Experiments\n\nIn this section, we discuss computational results that demonstrate how our approximation\nmethod signi\ufb01cantly expands the range of relevant EP-type models like the one previously\nintroduced that can be studied computationally.\nFirst, we propose an algorithm to compute oblivious equilibrium [5]. Whether this al-\ngorithm is guaranteed to terminate in a \ufb01nite number of iterations remains an open issue.\nHowever, in over 90% of the numerical experiments we present in this section, it converged\nin less than \ufb01ve minutes (and often much less than this). In the rest, it converged in less\nthan \ufb01fteen minutes.\nOur \ufb01rst set of results investigate the behavior of the approximation error bound under\nseveral different model speci\ufb01cations. A wide range of parameters for our model could\nreasonably represent different real world industries of interest. In practice the parameters\nwould either be estimated using data from a particular industry or chosen to re\ufb02ect an\nindustry under study. We begin by investigating a particular set of representative parameter\nvalues. See [5] for the speci\ufb01cations.\nFor each set of parameters, we use the approximation error bound to compute an upper\nbound on the percentage error in the value function, E[sup\u00b50\u2208M V (x,s|\u00b50,\u02dc\u00b5)\u2212V (x,s|\u02dc\u00b5)]\n, where\n\u02dc\u00b5 is the OE strategy and the expectations are taken with respect to s. We estimate the\nexpectations using simulation. We compute the previously mentioned percentage approxi-\nmation error bound for different market sizes m and number of \ufb01rms n(m). As the market\nsize increases, the number of \ufb01rms increases and the approximation error bound decreases.\nIn our computational experiments we found that the most important parameter affecting\n\nE[V (x,s|\u02dc\u00b5)]]\n\n\fthe approximation error bounds was the degree of vertical product differentiation, which\nindicates the importance consumers assign to product quality. In Figure 1 we present our\nresults. When the parameter that measures the level of vertical differentiation is low the\napproximation error bound is less than 0.5% with just 5 \ufb01rms, while when the parameter is\nhigh it is 5% for 5 \ufb01rms, less than 3% with 40 \ufb01rms, and less than 1% with 400 \ufb01rms.\n\nFigure 1: Percentage approximation error bound for \ufb01xed number of \ufb01rms.\n\nMost economic applications would involve from less than ten to several hundred \ufb01rms.\nThese results show that the approximation error bound may sometimes be small (<2%) in\nthese cases, though this would depend on the model and parameter values for the industry\nunder study.\nHaving gained some insight into what features of the model lead to low values of the\napproximation error bound, the question arises as to what value of the error bounds is\nrequired to obtain a good approximation. To shed light on this issue we compare long-run\nstatistics for the same industry primitives under oblivious equilibrium and MPE strategies.\nA major constraint on this exercise is that it requires the ability to actually compute the\nMPE, so to keep computation manageable we use four \ufb01rms here. We compare the average\nvalues of several economic statistics of interest under the oblivious equilibrium and the\nMPE invariant distributions. The quantities compared are: average investment, average\nproducer surplus, average consumer surplus, average share of the largest \ufb01rm, and average\nshare of the largest two \ufb01rms. We also computed the actual bene\ufb01t from deviating and\nkeeping track of the industry state (the actual difference E[sup\u00b50\u2208M V (x,s|\u00b50,\u02dc\u00b5)\u2212V (x,s|\u02dc\u00b5)]\n).\nNote that the the latter quantity should always be smaller than the approximation error\nbound.\nFrom the computational experiments we conclude the following (see [5] for a table with\nthe results):\n\nE[V (x,s|\u02dc\u00b5)]]\n\n1. When the bound is less than 1% the long-run quantities estimated under oblivious\n\nequilibrium and MPE strategies are very close.\n\n2. Performance of the approximation depends on the richness of the equilibrium in-\nvestment process. Industries with a relatively low cost of investment tend to have\na symmetric average distribution over quality levels re\ufb02ecting a rich investment\nprocess. In this cases, when the bound is between 1-20%, the long-run quantities\nestimated under oblivious equilibrium and MPE strategies are still quite close. In\nindustries with high investment cost the industry (system) state tends to be skewed,\nre\ufb02ecting low levels of investment. When the bound is above 1% and there is little\ninvestment, the long-run quantities can be quite different on a percentage basis\n(5% to 20%), but still remain fairly close in absolute terms.\n\n\f3. The performance bound is not tight. For a wide range of parameters the perfor-\nmance bound is as much as 10 to 20 times larger than the actual bene\ufb01t from\ndeviating.\n\nThe previous results suggest that MPE dynamics are well-approximated by oblivious equi-\nlibrium strategies when the approximation error bound is small (less than 1-2% and in some\ncases even up to 20 %). Our results demonstrate that the oblivious equilibrium approxima-\ntion signi\ufb01cantly expands the range of applied problems that can be analyzed computation-\nally.\n\n7 Conclusions and Future Research\n\nThe goal of this paper has been to increase the set of applied problems that can be addressed\nusing stochastic dynamic games. Due to the curse of dimensionality, the applicability of\nthese models has been severely limited. As an alternative, we proposed a method for ap-\nproximating MPE behavior using an oblivious equilibrium, where agents make decisions\nonly based on their own state and the long run average system state. We began by show-\ning that the approximation works well asymptotically, where asymptotics were taken in\nthe number of agents. We also introduced a simple algorithm to compute an oblivious\nequilibrium.\nTo facilitate using oblivious equilibrium in practice, we derived approximation error\nbounds that indicate how good the approximation is in any particular problem under study.\nThese approximation error bounds are quite general and thus can be used in a wide class of\nmodels. We use our methods to analyze dynamic industry models of imperfect competition\nand showed that oblivious equilibrium often yields a good approximation of MPE behavior\nfor industries with a couple hundred \ufb01rms, and sometimes even with just tens of \ufb01rms.\nWe have considered very simple strategies that are functions only of an agent\u2019s own state\nand the long run average system state. While our results show that these simple strategies\nwork well in many cases, there remains a set of problems where exact computation is not\npossible and yet our approximation will not work well either. For such cases, our hope\nis that our methods will serve as a basis for developing better approximations that use\nadditional information, such as the states of the dominant agents. Solving for equilibria\nof this type would be more dif\ufb01cult than solving for oblivious equilibria, but is still likely\nto be computationally feasible. Since showing that such an approach would provide a\ngood approximation is not a simple extension of our results, this will be a subject of future\nresearch.\n\nReferences\n[1] R. Ericson and A. Pakes. Markov-perfect industry dynamics: A framework for empir-\n\nical work. Review of Economic Studies, 62(1):53 \u2013 82, 1995.\n\n[2] R. L. Goettler, C. A. Parlour, and U. Rajan. Equilibrium in a dynamic limit order\n\nmarket. Forthcoming, Journal of Finance, 2004.\n\n[3] E. Maskin and J. Tirole. A theory of dynamic oligopoly, I and II. Econometrica,\n\n56(3):549 \u2013 570, 1988.\n\n[4] U. Doraszelski and M. Satterthwaite. Foundations of Markov-perfect industry dynam-\nics: Existence, puri\ufb01cation, and multiplicity. Working Paper, Hoover Institution, 2003.\n[5] G. Y. Weintraub, C. L. Benkard, and B. Van Roy. Markov perfect industry dynamics\n\nwith many \ufb01rms. Submitted ofr publication, 2005.\n\n[6] A. Pakes. A framework for applied dynamic analysis in i.o. NBER Working Paper\n\n8024, 2000.\n\n\f", "award": [], "sourceid": 2786, "authors": [{"given_name": "Gabriel", "family_name": "Weintraub", "institution": null}, {"given_name": "Lanier", "family_name": "Benkard", "institution": null}, {"given_name": "Benjamin", "family_name": "Van Roy", "institution": null}]}