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1 – Introduction

1The study of what we refer today as ‘perfectly competitive markets’ has been the focus of economic theory throughout its history. Today economists have come to identify perfectly competitive markets with the so called ‘price taking’ behavior of participants in them, a term that signifies the inability (or lack of interest) of market participants to manipulate prices and thus influence transactions to their benefit. Of course, this is not the only ingredient of perfectly competitive markets, but it is certainly a crucial one. The framework developed by Walras [1874] naturally became the point of reference for the study of perfectly competitive markets and the Arrow-Debreu [1954] model of an exchange economy provided a widely accepted context of Walrasian ideas. It was soon recognized that there was a need for a justification of the price taking hypothesis.

2It is widely accepted among economists that in a system of markets where individual participants are small relative to the market size, individuals have a negligible effect on the determination of market outcomes, so they may be thought of as exhibiting a price taking behavior. This is the so called ‘price taking hypothesis.’ Of course, in order to make sense of this statement one has to attach a meaning to ‘a small individual relative to market size’. In this way it would be possible to distinguish when price taking is a reasonable assumption and when it is not. The significance of the price taking hypothesis in economics calls for a formal clarification of this point -a ‘theory of competition’ so to speak.

3The search for a justification of the price taking hypothesis has taken two paths. Aumann [1964] took the point of view that an atomless economy, that is an economy populated by an atomless measure space of individuals is the natural mathematical tool to model the idea of ‘negligibility’ of market participants. The interpretation of an atomless economy is that it constitutes an idealization of a ‘large finite economy’: an idealization of an Arrow-Debreu economy with a large number of individuals. Another more traditional approach of economic theory is the asymptotic study of equilibrium outcomes of finite economies, when the number of individuals increases without limit, i.e., the study of large finite economies themselves, rather than their idealization. This methodology starts by proposing a reasonable equilibrium notion when individuals are not price takers. The idea is that if we can identify conditions under which equilibrium outcomes of finite economies converge asymptotically (in some sense) to Walrasian ones, then we would have a context where individuals have negligible effect on market outcomes and hence when ‘price taking’ can be justified as a reasonable hypothesis.

4Of course those two methods to address the price taking hypothesis are not unrelated because the asymptotic study may be viewed as a link between ‘large finite’ and ‘atomless’ economies. In atomless economies ‘negligibility’ is built in the non atomicity of the measure of the space of agents. If a large finite economy is to be thought as a reasonable substitute of the idealized continuum model, it should be the case that equilibrium outcomes of a large finite economy are close to those of the atomless limit, i.e., the equilibrium outcomes of the former should asymptotically converge (in some sense) to those of the latter as the number of individuals increases.

5Asymptotic studies have been performed for a variety of equilibrium notions which have been developed in finite economies. For the most part asymptotic studies focus on the limits of the core [2] and Nash equilibria, mainly because those notions are associated with the traditional theories of Edgeworth [1881] and Cournot [1838], which are prevalent in economic theory.

6Here we study the asymptotic limits of Nash equilibria of strategic market games. Our interest in this is that these games feature an explicit physical description of the price formation and trade process. In this way, the ability of individuals to influence prices and trade volumes is explicit. This issue has been addressed by several authors, Dubey and Shubik [1978], Mas-Colell [1982], Peck and Shell [1989], Sahi and Yao [1989], Amir et al. [1990] among many others, albeit in the fragile context of sequences of economies obtained through replication. [3] Besides the particularity of this type of sequences (finite number of types of individual characteristics), the above results are shown only for the subset of ‘type symmetric’ Nash equilibria. Note that in replica sequences type symmetry is a property of the core (known as ‘equal treatment’), but not of Nash equilibria.

7By contrast, our results apply to more general sequences of economies with characteristics drawn from compact sets and do not depend on type symmetry. One of our results provides also a rate of convergence. In this way we address the issue of asymptotic convergence of Nash equilibria, at the same level of generality as some known core convergence results. Our approach is based on the idea developed in Koutsougeras [2009] of measuring individuals’ departure from price taking, via the wedge between the hyperplanes defined by the price vector and the supporting hyperplane of the indifference surface through the equilibrium bundle. We then demonstrate that under suitable assumptions on the distribution of individual characteristics [4] this wedge becomes arbitrarily small as the number of individuals converges to infinity.

2 – The model

8Let H, indexed by h = 1,2,…, N, be a finite set of agents. There are L, indexed by i = 1,2,…, L commodity types in the economy and the consumption set of each agent is identified with ℜL+. Each individual hH is characterized by a preference relation equation im1 and an initial endowment eh ∈ ℜℜL+{0}. We use the following assumption:

9Assumption 1 Preferences are continuous, convex, and strictly monotone.

10Denote by Pcm the set of preferences that satisfy (1) endowed with the topology of closed convergence. Let TPcm × ℜℜL+. An economy is defined as a mapping E : HT.

11We now turn to describe a strategic market game, which proposes an explicit model of how exchange in the economy takes place.

2.1 – Trade using inside money

12We will develop our results for the strategic market game version appearing in Postlewaite and Schmeidler [1978] and in Peck et al. [1992] which is described below.

13Trade in the economy is organized via a system of trading posts where individuals offer commodities for sale and place bids for purchases of commodities. Bids are placed in terms of a unit of account. The strategy set of each agent is equation im2. Given a strategy profile (b, q) ∈ ∏hHSh let Bi = ∑hHbih and Qi = ∑hHqih denote aggregate bids and offers for each i = l, 2, …, L. Also for each agent h denote Bi−h = ∑khbik, Qi−h = ∑khqik. For a given a strategy profile, the consumption of consumer hH is determined by xh = eh + zh(b, q), where for i = 1, 2, …, L :

15and it is postulated that whenever the term 0/0 appears in the expressions above it is defined to equal zero. When BiQi ≠ 0, the fraction equation im4 has a natural interpretation as the (average) market clearing ‘price’. The relation equation im5 is a ‘bookkeeping’ restriction which ensures that units of account remain at zero net supply (inside money). The interpretation of this allocation mechanism is that commodities (money) are distributed among non-bankrupt consumers in proportion to their bids (offers), while the purchases of bankrupt consumers are confiscated.

16An equilibrium is defined as a strategy profile (b, q) ∈ ∏hHSh that forms a Nash equilibrium in the ensuing game with strategic outcome function given by (1). Let N(E) ⊂ ∏hHSh denote the set of Nash equilibrium strategy profiles of the strategic market game and N(E) ⊂ ℜLN+ the set of consumption allocations corresponding to the elements of N(E).

17The following notation and familiar facts will be useful in the sequel. Fix (bh, qh) ∈ ∏khSk and let [5] equation im6. The set of allocations which an individual hH can achieve via the strategic outcome function is given by the convex set

19Conversely, equation im8. Thus, due to the bankruptcy rule, at an equilibrium with nonzero bids and offers we have: equation im9 if and only if:

21We say that equation im11 is fully active if, for the corresponding equation im12, we have equation im13, i.e., there is trade in all commodities. In the sequel we will focus on such equilibria. [6]

2.2 – Strategic vs. price taking behavior

22The methodology that we will follow uses the fact that price taking is characterized by a ‘tangency condition’: the Walrasian price vector defines a hyperplane that supports the set of bundles which are at least as good as the Walrasian one for each individual and separates that set from the budget set. Given a Nash equilibrium, the idea is to find a ‘surrogate’ vector that separates the ‘budget set’ from the set of bundles which are at least as good as the Nash equilibrium one for each individual and compare that to the strategic market game price vector. The asymptotic exercise becomes then to study under what conditions the difference between the two vectors becomes arbitrarily small as the number of individuals becomes arbitrarily large.

23Formally, let us fix a fully active equation im14 corresponding to a strategy profile equation im15. Consider one hH and denote equation im16.

24The monotonicity of preferences implies that equation im17, i.e., equation im18 lies on the boundary of the convex set ch, which is C2. Since preferences are also convex, by the separating hyperplane theorem there is a ph ∈ ℜL+, specifically equation im19, where Dgh (·) denotes the gradient of gh(·), such that

26Using the definition of ch we have

28Now observe that if for some λh > 0, equation im22 then the behavior of such an individual would be identical to price taking at the market clearing prices equation im23. To see this notice that because equation im24 (equation im25 is active) there is a cheaper point, i.e., w ∈ ℜL+ with equation im26. Since furthermore preferences are continuous and convex, the first part of (4) implies equation im27. Finally, equation im28.

29Therefore, the measurement

31serves as an indicator of ‘how far’ the strategic behavior of individual h falls from price taking. [7] Clearly, for each agent h we have equation im30 and equation im31 is Walrasian if (and only if) equation im32 for each agent h. Therefore, a sequence of market game price-allocation pairs tends to become a price taking one, if (and only if) the above indicator tends to zero (in an appropriate sense) for all individuals.

32We are ready now to proceed with the results of this paper.

3 – Results

33For the results that follow we consider a sequence {En}nN of economies En : HnPcm × [0, r]L, where #Hn → ∞, equation im33 and associated xnN(En), for each nN which are fully active. Let (bn, qn) ∈ N(En) corresponding strategies, and zn,h = xn,heh the corresponding net trades for each hH.

34The following result is shown in Koutsougeras [2007] and its proof applies unchanged here.

35Theorem 1. For each ε > 0, there is an nεN so that for all n > nε

37Or, equivalently

39The above theorem asserts some kind of convergence (in measure) for our indicator. Its strength is that it requires no assumptions, so it applies to all sequences of active Nash equilibria. It is noteworthy that some sort of convergence already obtains only as a consequence of the number of individuals without any assumptions on the distribution of their characteristics. This fact reveals that a large number of market participants goes already a long way towards price taking.

40On the other hand it hardly fits the bill: we still need to show some convergence of the Nash equilibrium allocations themselves. Furthermore, we need to ensure that ‘most’ of the commodities are consumed by ‘most’ of the individuals who exhibit ‘almost’ a price taking behavior, as the above theorem asserts. We proceed with two lemmas which will be useful to us in pursuing this end. We skip the proofs which can be found in Koutsougeras [2009].

41Lemma 1. Let r be an upper bound for the endowments in the economy. Define equation im36 where equation im37. Then equation im38.

42The above lemma asserts that the proportion of individuals whose net trades remain bounded above along a sequence is uniformly bounded below.

43Lemma 2. Suppose that equation im39. There is a subsequence (still indexed by n) and ε > 0, so that for each i = 1, 2, …, L, we have: equation im40.

44This lemma claims that the proportion of individuals whose consumption remains bounded away from zero along a sequence is uniformly bounded below.

45We now turn to develop an asymptotic convergence theorem, by introducing appropriate assumptions on the distribution of characteristics along a sequence of economies. In particular, consider a sequence of economies En : HnT where TPcm × [0, r]L is compact. For such sequences the set of Nash equilibrium allocations is uniformly bounded as the following result shows.

46Proposition 1. Let {En}nN be a sequence of economies, En : HnT, where #Hn → ∞ and let xnN(En) for each nN be fully active. There is B ⊂ ℜL+, which is bounded and depends only on T, such that for all nN, xn,hB for each hHn, i.e., the set of Nash equilibrium allocations remains uniformly bounded along a sequence of economies with characteristics drawn from T.

47We can now prove the following result.

48Theorem 2. Consider a sequence of economies{En}nN, where En : HnT, #Hn → ∞, and TPcm × [0, s]L is compact. Let(bn, qn) ∈ N(En) and suppose that for some β > 0, equation im41 for n large enough, so that the corresponding xnN(En) is fully active. Then given any ε > 0, there is N so that if # Hn > N, then δh(xn) < ε, ∀hHn.

49Remark 1. In the theorem above, the set of commodities is held fixed along the sequence. However, the number of commodities could vary as well. In that case, the solution is still valid as long as equation im42, i.e., the number of agents increases faster than the number of commodities.[8]

50The theorem above is the desired result, which asserts that along a sequence of economies with a distribution of characteristics drawn from a compact set, strategic behavior becomes asymptotically nearly price taking. The interpretation is that in an economy with a large number of individuals none of whom has exceptional characteristics, strategic behavior is approximately the same as price taking, the approximation becoming finer the larger the number of individuals. Notice that the proof suggests also a rate of convergence, although it cannot be argued that this rate is ‘tight’.

4 – Purely competitive sequences of economies

51The results of the previous section can become more transparent by considering sequences of economies converging to a limit. To this end in this section we will consider ‘purely competitive’ sequences of economies [Hildenbrand, 1974:138] which are defined as follows.

52Let TPcm × ℜL+ be compact. Consider a sequence {En}nN, where En : HnT such that:

  1. #Hn → ∞.
  2. The sequence of distributions of characteristics (μn) converges weakly on T.
  3. If μ = lim μn, then ∫ e dμn → ∫ e dμ.
  4. e dμ > 0.
Let {En}nN be such a sequence and consider a sequence of fully active Nash equilibria xnN(En). Since T is compact, it follows by proposition (1) that {xn}nN is uniformly bounded, so we can extract a subsequence (still indexed by n) which converges in distribution, i.e., by defining for each B ∈ ℜLequation im43 we have that λnλ weakly.

53Denote now by τn the joint distribution of (En, xn) : HnT × ℜL. The sequence (τn)nN is tight since the sequences of its marginal distributions are tight, so we may assume, by passing to a subsequence if necessary, that τnτ weakly. Hence, this sequence of economies and associated allocations admits a continuous representation [Hildenbrand, 1974, proposition 2: 139]: there is an atomless measure space (H, H, ν), (E, x) : HT × ℜL and measurable functions an : HHn, so that equation im44 in H and the respective distributions of equation im45 and (E, x) are τn and τ respectively.

54Using this continuous representation, our indicator can be extended in a natural way on H, by equation im46. The meaning of theorem (2) can be made more transparent as follows:

55Lemme 3. equation im47 in measure.

56The following proposition establishes that the allocation x is Walrasian for the economy E, provided that the associated sequence of strategic prices does not converge to the boundary of ℜL+.

57Proposition 2. Let xnN(En), for each nN be fully active and suppose that the sequence of associated strategic market game prices {πn}nN are such that no subsequence converges to the boundary ofL+. Then δh (x) = 0, ae in H.

Concluding remarks

58Notice that the proof of theorem (2) provides a rate of convergence which depends on the set of characteristics T (the constants C and s), but it also depends on the sequence itself (the constant ξ, which in turn depends on β —the uniform lower bound on offers). This is sensible because in strategic market games there is no parameter β that works for all possible sequences: it is possible that some sequences of prices converge to the boundary of the simplex, irrespectively of the set of characteristics. In that case the corresponding sequence of active equilibria converges to one where some markets are inactive, which typically will not be Walrasian. For the same reason a similar qualification on the sequence of Nash equilibria was needed in proposition (2). Hence, the results in this paper must be understood as asserting that the limit of all sequences of Nash equilibria which remain active in the limit is Walrasian.

59We have established here a generalization of the idea that Nash equilibria converge to Walrasian ones as the number of individuals increases without limit. We showed that this conclusion holds for all Nash equilibria which are fully active in the limit and this conclusion holds for general sequences of economies with characteristics drawn from a compact set. In this way we have developed an asymptotic argument for the emergence of price taking behavior based on non-cooperative games, in the same level of generality as the analogous results based on cooperative games.

60By now the sufficient conditions for the emergence of price taking behavior are well understood. There are two directions where we believe that would be worthwhile to pursue. First, the extension of these results in the asymmetric information context. When information becomes one of the private characteristics of individuals a number of issues arise which make a straightforward extension of the existing results impossible. Second, we believe that it is worthwhile to explore the possibility of formulating necessary conditions for the emergence of price taking. It is unclear whether the sufficient conditions that we have developed for the emergence of competition are also necessary, i.e., whether or not competition may arise in a richer set of circumstances e.g. with small numbers of individuals.


  • [1]
    School of Social Sciences, University of Manchester. E-mail:
    Acknowledgment: I am grateful to the organizers of the Paris X Workshop on Perfect Competition in January 2011 for the opportunity to present this paper. I also thank the participants and discussant in that workshop for valuable comments. My thanks are also extended to a referee of this issue for thoughtful comments. The standard disclaimer applies for any shortcomings.
  • [2]
    See Anderson [1992] for a survey of core equivalence results and references.
  • [3]
    In some related papers but somewhat distinct in scope, Peck and K. Shell [1990] features an asymptotic exercise where the number of agents remains finite but the volume of trade increases without limit, while Postlewaite and Schmeidler [1981] shows that Nash equilibria are Walrasian for a properly defined ‘nearby economy.’
  • [4]
    Remarkably these assumptions are the same as in the case of core convergence.
  • [5]
    In order to save on notation we omit the dependency on (b-h, q-h). In the results the values of those variables will be fixed so no confusion should arise.
  • [6]
    Alternatively we could consider the subset of commodities L for which there is active trade.
  • [7]
    In the case of C2 preferences, the indicator δh(·) coincides with γh(·) in Koutsougeras [2007].
  • [8]
    We thank an anonymous referee for pointing this out to us.


We address the asymptotic convergence of active Nash equilibria of strategic market games to Walrasian ones for general sequences of economies whose distribution of characteristics has compact support.


  • strategic behavior
  • price taking
  • convergence

Le price-taking comme limite asymptotique de comportements stratégiques


Dans cet article, nous considérons un modèle de jeu stratégique de marché. Nous étudions la convergence asymptotique d’équilibres de Nash non triviaux vers les équilibres walrassiens. Cette étude utilise des séquences dont la distribution des caractéristiques est à support compact.
JEL classification: C72, D43, D50


  • comportement stratégique
  • comportement concurrentiel
  • convergence


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Leonidas C. Koutsougeras [1]
  • [1]
    School of Social Sciences, University of Manchester. E-mail:
    Acknowledgment: I am grateful to the organizers of the Paris X Workshop on Perfect Competition in January 2011 for the opportunity to present this paper. I also thank the participants and discussant in that workshop for valuable comments. My thanks are also extended to a referee of this issue for thoughtful comments. The standard disclaimer applies for any shortcomings.
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