1 Some modern theoretical developments have underlined the importance of abandoning the assumption of price-taking agents . These developments require a different conception of the theoretical functioning of the market. This implies modeling a perfectly competitive market based on strategic behavior, rather than using the traditional Walrasian conception of individual behavior based only on preferences and technologies. In fact, the Walrasian assumption of price takers is based upon the idea that individuals choose their actions in isolation, without any consideration about the actions of others. While a few conjectures about prices and about the general state of the market are evoked, these are generally subsumed within the price-taker assumption. In fact, strategic behavior is essentially founded on agent interactions with each other and conjectures about other agents’ actions.
2 As has been stated by Arrow , if one identifies the price-taker assumption with perfect competition there is no other alternative than to introduce the Walrasian auctioneer. It seems to be a matter of fate linked with the whole Walrasian framework. This ‘pessimistic view’ of perfect competition pushes Arrow to postulate that in order to give a more realistic interpretation of economic reality (without the fiction of the centralizing auctioneer) we need to build imperfect competition models. But these models may only keep the optimality properties of the Walrasian competitive equilibrium as a normative reference, leaving aside the tâtonnement process as the representation of the competitive market mechanism.
3 This methodological suggestion has, however, entailed a progressive abandonment of the general equilibrium framework and the construction of imperfect competition models mainly built upon on a partial equilibrium framework. This is clearer regarding applied economics and in particular the theory of industrial organization . One of the main criticisms advanced since the 1960s against the Walrasian framework is its lack of an explicit treatment of decentralized exchange and price formation mechanisms. Arrow summarizes this idea as follows: “there exists a logical gap in the usual formulation of the theory of perfectly competitive economy, namely, that there is no place for a rational decision with respect to prices as there is with respect to quantities.” [Arrow 1959: 41]
4 The failures of the Walrasian theory of exchange and prices are thus identified with the presence of an auctioneer, according to Arrow’s idea. The absence of strategic behavior implied by the price-taker assumption seems to be inherently linked with the absence of an alternative to the auctioneer and thus to the Walrasian tâtonnement process. The auctioneer being linked with the absence of strategic behavior, and strategic behavior as the way to affect prices being the typical characteristic of imperfect competition, it is thus necessary to get rid of the auctioneer in order to explicitly treat imperfect competitive markets.
5 These arguments echo the radical opposition to Walras’s theory of prices addressed by Edgeworth among other of his contemporaries .The absence of an explicit treatment of agents’ strategic behavior and the centralized price mechanism were the main targets of Edgeworth’s poisoned darts against Walras.
6 The best known elements of Edgeworth’s disagreement with Walras are formulated in two concise articles reviewing Walras’ Elements of Pure Economics. In these texts, we find a recurring argument from Edgeworth concerning the unsatisfactory treatment of the dynamic process leading towards an equilibrium situation. Edgeworth’s weapons are constantly aimed at the tâtonnement process as a representation of a particular form of market and competition dynamics:
“If we compare the interplay of supply and demand in the market to a mass of water pouring down the slopes of a valley comprised of successive basins, as in the case considered by Mr Walras, there is indeed one definite position of equilibrium and he indicates it very well. However sometimes, in a thoroughly arbitrary fashion, he imagines the water as flowing into a particular bed, whereas the only thing we can say is that in one way or another it will arrive at a position of equilibrium.” [Edgeworth 1891: 317]
8 However, paradoxically enough, the 20th century rediscovered Edgeworth’s analyses and transformed it into the cornerstone upon which a new form of the Walrasian notion of perfect competition could be constructed. It seems generally accepted that the well-known Debreu-Scarf theorem (Debreu and Scarf 1963) on the convergence of the Core to the set of Walrasian equilibria is the final proof of the strong relation between Edgeworth’s and Walras’s views on perfect competition . For some historians of economics, this interpretation could be based upon Edgeworth’s rejection of Walras’s tâtonnement and his agreement with the latter’s definition of perfect equilibrium. Textual evidence has been advanced to support this claim:
“It is not easy to state in a few words all the various considerations that have led me to divide M. Walras’s theory of the market into two, accepting the static aspect and rejecting the dynamic. . .”[Edgeworth 1891: 313]
10 Following this interpretation, Edgeworth’s theory of recontracting is a suitable way to make explicit what is only implicit within the Walrasian framework: the very conditions under which a perfect competitive allocation emerges endogenously from agents’ decentralized interactions. Edgeworth’s approach, still following this interpretation, would show the process of negotiation between agents leading to a solution which in the Walrasian approach is left to the “auctioneer” fiction. This interpretation seems to lead us through a game of two players and two winners. Both authors were right. From the Walrasian point of view, the competitive price vector is seen as more than a spurious representation of an efficient allocation. It is the limit case of economic competition and it appears to be a very robust situation. From an Edgeworthian view, the Debreu-Scarf result has been seen as the demonstration of his genius conjecture, and a rigorous generalization of Edgeworth’s ideas. Aumann, whose contribution was essential to establish the idea of the Walras-Edgeworth convergence through the “great-number” of agents approach, summarized this way of thinking as follows:
“The definition of competitive equilibrium assumes that the traders allow market pressures to determine prices, and that they then trade in accordance with these prices, whereas that of core ignores the price mechanism and involves only direct trading between the participants. Intuitively, one feels that money and prices are no more than a device to simplify trading, and therefore the two concepts should lead to the same allocations. It is to be expected that this will not happen in finite markets, where the notion of competitive equilibrium is not really applicable; and, indeed, though every equilibrium allocation is always in the core, the core of a finite market usually contains points that are not equilibrium allocations. However, when the notion of perfect competition is built into the model, that is, in a continuous market, one may expect that the core equals the set of equilibrium allocations. That this is indeed the case is the main result of this paper. It has long been conjectured that some such theorem holds; the basic idea dates back at least to Edgeworth. The usual basic statement is that “the core approaches the set of equilibrium allocations as the number of traders tends to infinity.” Unfortunately, it is extremely difficult to lend precise meaning to this kind of statement, to say nothing of proving it. Very recently, Debreu and Scarf did succeed in stating and proving a theorem of this kind in a brilliant and elegant fashion.” [Aumann 1964: 40]
12 Following this idea, the Walrasian framework can be modified in order to incorporate an important solution concept from game theory, inspired from Edgeworth: the core. The convergence between Edgeworth and Walras is thus usually presented as the definitive proof of an ancient conjecture: perfect competition can be obtained as the limit situation of an economy with a great number of agents. The traditional interpretation finally joins the modern idea of convergence between both authors represented by the Debreu-Scarf theorem. This theorem, following that interpretation, makes less unacceptable the assumption of a price-taking behavior because it allows the establishing of a conjunction between this behavior and another assumption: the presence of a great number of agents. Following Rebeyrol [1999: 99]:
Les économistes contemporains assimilent la concurrence parfaite à un comportement de « preneurs de prix » de la part des agents. Cette conception est associée à l’idée qu’en concurrence les agents sont très nombreux, et donc atomisés, sans pouvoir sur le rapport d’échange global qui prévaudra au marché. Elle est plus ou moins renforcée par la reconnaissance du théorème limite d’Edgeworth.
14 On a different ground, historians of economic thought are well aware of the confrontation between these two important figures of late 19th century economics. However, this confrontation has been interpreted as a methodological misunderstanding . Most scholars credit this misunderstanding to a difference in their conceptions of the dynamic process leading towards an equilibrium position. Making use of the authors’ public and private opinions  (as in their correspondence), this debate has been well studied. The common point of view of historians of economics points mainly  to the idea of the possibility of an effective out-of-equilibrium exchange during the process of convergence to a general equilibrium situation. As Walker  puts it, abridging Edgeworth’s position :
“. . .that is why Edgeworth rejected the tâtonnement approach; his alternative was an advance over the Walrasian conception, an early step in a new direction towards a line of investigation that is still being slowly worked out—a dynamic non tâtonnement theory of exchange, production and consumption.” (p. 147)
16 From this interpretation of Edgeworth’s analysis of the recontracting process, it follows that the main difference between his and Walras’s theory of perfect competition can only be observed if we take into account the out-of-equilibrium process. However, in what follows, we will try to show that there is a more fundamental divergence between both authors; namely, their conception of the strategic behavior of agents.
17 In this paper we try to challenge this traditional interpretation of the Walras-Edgeworth convergence on perfect competition. Following modern literature on the Debreu-Scarf theorem, we show that this convergence can be questioned from two main points of view. Firstly, a main challenge for the Walrasian approach is revealed by the theoretical framework within which the notion of core and recontracting has been built by Debreu and Scarf. The Walrasian General Equilibrium model is based on an assumption about individual behavior and the institutional structure of markets that avoids a lot of questions which become unavoidable once the Edgeworthian recontracting process is considered. Secondly, the interpretation of Edgeworth’s theory of the recontracting process is misleading because it withdraws central and interesting elements of his conception about the market mechanism; the most significant being the strategic behavior behind Edgeworth’s conception of market competition that vanishes in the cooperative games framework of the Debreu-Scarf tradition. We claim that Edgeworth’s visions on the dynamic process of the market appeal for a strategic-game way of thinking.
18 This article tries thus to reconstruct Walras’s and Edgeworth’s conceptions of perfect competition to evaluate the well-founded notion of their convergence on the properties of the final situation (perfect competitive allocations) inspired by the Debreu-Scarf theorem. We will show that it is useful to reconstruct a historical debate using modern economic analysis. This allows not only the shedding of new light on to the historians’ debate but it also opens up new questions for modern economic theory.
19 In what follows, we begin (section 1) by presenting the main theoretical assumptions that Walras has to introduce in order to ensure the price-taker assumption. We show that, contrary to what modern economics does, it is not enough to assume price-taking agents without discussing the institutional framework within which this assumption can be sustainable. We show that Walras’s entire institutional framework converges to a particular conception of individuals: nonstrategic agents.
20 Here again, the notion of strategic behavior concerns agents’ decision theory based on conjecture about other agents’ behavior and on their strategies facing the former actions, and so on. We shall show that Walras’s conception of perfect competition is based on a systematic discarding or neutralization of any situation that allowed the consideration of the consequences of agents exploiting strategic behavior. In this sense, the Walrasian analysis is hard to be translated into a modern strategic non-cooperative game theoretical framework (the Nash program). This allows us to explain why the cooperative-game theoretical approach is more suitable to develop a Walrasian tradition. It is exactly in this direction that the Debreu-Scarf theorem and the ‘Core’ as an equilibrium solution tried to establish a convergence between Edgeworth and Walras. Next (section 2), we discuss this interpretation of Edgeworth as the forerunner of the cooperative-game tradition. We conclude that the main elements of the Debreu-Scarf theorem are extracted from a particular example given by Edgeworth that obscures the general idea this author wanted to convey. Following recent works on the analysis of the ‘Core’ as an equilibrium solution [Vind, 1995] it is possible to recover the originality of Edgeworth’s analysis and the interest to go further in an alternative path to the cooperative-game framework. We finally present some concluding remarks on the consequences of this analysis for modern “imperfect competition economics.”
1 - Walras’s theory: perfect competition and the neutrality of pure economics
21 As stated by Jaffé [1980: 530], Walras’s pure economics is an integral part of a larger system of social philosophy. Walras tried to build a pure theory of economics as a neutral system, with respect to the criteria of distributional justice. His pure theory of market, prices and money must satisfy commutative justice in order to describe production and exchange as free individual decisions isolated from distributional considerations. Agents must be free to exchange and produce within the limits of feasible allocations that respect individuals’ budget constraints. These requirements led Walras to impose a regime of free competition as the main (and sole) framework of his pure theory of economics.
22 We shall show that the price-taker behavior is complemented by the “no-exchange-out-of-equilibrium” assumption as the main characteristics of Walras’s perfect competition framework. This construction is founded on two elements: the numéraire and the fiction of a tâtonnement process. The former allows us to avoid the treatment of strategic behaviors within a general equilibrium system, and establishes a dichotomy between the exchange process and price formation. The latter prevents pure theory of prices from dealing with distributional effects of out-of-equilibrium exchanges.
23 In order to accomplish a coherent description of the institutional framework that characterizes a perfect competitive market, Walras introduces some apparently pure metaphorical figures as brokers and the well-known auctioneer. We shall show that these figures are more than metaphors and that their presence reveals the way some possible behaviors of agents “threaten” the distributional neutrality of pure economics and thus the perfect competition ideal.
24 Furthermore, Walras relegates the treatment of the exchange process to the last “flats” of his pure theoretical construction; the analysis of money as a medium of exchange. This last characteristic of Walras’s framework allows us to reinforce the idea that, within pure economics, everything that may possibly open the door to strategic considerations must be neutralized or removed altogether.
1.1 - From the Tâtonnement Process to Price-taking Behavior
25 So far, we have only evoked a central question concerning the close relation between the dynamic question on price formation and the institutional assumption allowing this process to take place within a competitive market. It is now necessary to distinguish between these two theoretical aspects. In order to do so, we must first underline that the second element, namely the notion of perfect competition, can be dealt without any reference to dynamics. In fact, the evolution of Walras’s main work, Elements of Pure Economics (henceforth EEPP ), shows the need to distinguish these questions that were unsuitably mixed in the first editions of the book.
26 The evolution throughout the different chapters and editions of the EEPP of Walras’s theory of price formation explained by a tâtonnement process has important consequences for the final version of his views on perfect competition. Because of Walras’s attachment to the internal coherence of his pure economics, the theory of stability and dynamics is subordinated to the notion of neutrality of the exchange process. This implies that the process of price formation in perfect competition is finally presented as an instantaneous adjustment of all markets towards equilibrium .
27 The well-known discussion of the Walrasian tâtonnement appears in the first editions of the EEPP as a description of an adjustment process that actually takes place in the market. Here, Walras underlines the role that competition plays in this process. The literary description of the process, which will prevail until the last edition of the EEPP, sustains the idea that during the formation of equilibrium prices agents have an active role and that competition is the result of individual decisions and interactions:
28 La valeur d’échange laissée à elle-même se produit naturellement sur le marché sous l’empire de la concurrence. Comme acheteurs, les échangeurs demandent à l’enchère, comme vendeurs, ils offrent au rabais, et leur concours amène ainsi une certaine valeur d’échange des marchandises tantôt ascendante, tantôt descendante et tantôt stationnaire. Selon que cette concurrence fonctionne plus ou moins bien, la valeur d’échange se produit d’une manière plus ou moins rigoureuse.[Walras 1988: 70]
29 One question arises after reading this passage: What does a more or less rigorous competition mean? Walras answers in the same paragraph:
Les marchés les mieux organisés sous le rapport de la concurrence sont ceux où les ventes et achats se font à la criée, par l’intermédiaire d’agents tels qu’agents de change, courtiers de commerce, crieurs, qui les centralisent, de telle sorte qu’aucun échange n’ait lieu sans que les conditions en soient annoncées et connues et sans que les vendeurs puissent aller au rabais et les acheteurs à l’enchère.[Walras 1988: 70]
31 That is, if the process of price formation is to take place under perfect competition, without interfering with distributive justice, agents cannot exchange at prices different from those of equilibrium. In order to guarantee this result, market intermediaries centralize supplies and demands thus avoiding exchanges out of equilibrium. These intermediaries are the auctioneer and stockbrokers.
32 The need to introduce an agent playing the role of an auctioneer (crieur de prix), other than those participating as exchangers, is a consequence of a conception of perfect competition as a situation where no one can manipulate prices. Commutative justice is the main ideal of this social arrangement. Price taking is not only a rhetoric figure; it is also an active assumption within this theoretical construction. As mentioned above (see the Introduction), Arrow clearly identifies the need for an auctioneer when agents are price takers. However, it is possible that even Arrow and certainly most modern economists do not realize the deep implications of this conception of perfect competition.
33 The presence of the brokers (courtiers) is even more interesting and less known in modern economic theory. These agents are introduced by Walras in a passage of EEPP where he recognizes that, even in the presence of an auctioneer and price-taker’s there is still a place for strategic behavior. Once an auctioneer is doing his job, shouting out prices, one needs to ensure agents are transmitting their “real” demand and supply plans in order to avoid the possibility they being manipulating the auction process with a speculative intention. This is a similar problem to the one tackled by modern auctions’ theory and mechanism design. Walras in fact refers to an apparently anecdotal situation where agents fall ill and need to give their “real” demand and supply plans to a third agent who could be a broker. It is interesting to note that Edgeworth understood very clearly Walras’s intentions introducing this apparently anodyne story. Let’s give Edgeworth himself the privilege of describing it:
“To avoid all misunderstanding, I beg leave to employ a very convenient supposition that M. Walras has set out with his accustomed clarity. Suppose that in a certain market a certain quantity of commodity A is to be exchanged for a certain quantity of commodity B. Each holder of A, finding it impossible to come in person, sends an agent whom he informs of his readiness to trade, that is to say how much B he is ready to buy and how much A he is ready to sell at each price quoted. Each holder of B does the same. When all these agents meet together, the market price will be determined.” [Edgeworth 1891: 312]
35 Therefore, in spite of the initial assertion, according to which agents are active during the price formation process, Walras ends up reducing perfect competition to a situation where agents are price-takers because the active part of the market is left to other metaphorical characters: the crieur and the courtiers.
36 This description of tâtonnement, in the presence of an auctioneer and brokers, which seems to appeal to an empirical argument in the text presented above, is clearer in Walras’s lecture at the Académie de sciences morales et politiques de Paris read in 1873 [Walras, 1993]. Here, Walras explains that to avoid any possible ambiguity and to guarantee coherence with the aim of his pure theory he assumes tâtonnement takes place through the use of an automatic calculator (nowadays we could say a computer) that determines the quantities each individual offers and demands, and can finally calculate general equilibrium prices [Walras, 1993]. Walras thus avoids the problem of the consequences of disequilibrium on expectations (if the process takes place under an auctioneer) or on distributive effects (if exchanges take place in disequilibrium). As most of Walras’s scholars, starting with Jaffé’s works [1976, 1977 and 1980], have remarked, the tâtonnement process is one during which no exchanges take place and can be interpreted (see [Bridel 1997, chapter 4; 2002]) as a method of solving a system of simultaneous equations whose roots are the equilibrium prices. As Edgeworth clearly realizes–as shown in the quotation above–once the “real” plans of supply and demand are known or revealed during the tâtonnement process, a market-clearing price vector could mathematically exist.
37 Finally, an important reason that completes and reinforces the requirements of commutative justice and explains why Walras focuses on the equilibrium situation is that it is optimal (i.e. Pareto optimal). If exchanges were to take place at disequilibrium prices, there would be unsatisfied exchange possibilities that could induce an agent aware of this situation to propose a different price that would better his condition and would not worsen that of the person willing to exchange with him. The normative  properties of general equilibrium are then implicitly founded on a pure theoretical assumption of price models under perfect competition. This is why the existence of a general equilibrium price vector and the welfare theorems can be dealt with independently from dynamics. These theorems allow establishing a theoretical reference with regard to Walras’s idea of commutative justice. Therefore, without any need to demonstrate or make reference to the stability of equilibrium Walras may assert, just as contemporary applied economics does, in his Studies of applied political economy  that:
Les éléments du système économique sont des services qui, sous le régime de la libre concurrence, tendent naturellement à se combiner en produits de la nature et de la quantité propres à donner la plus grande satisfaction possible des besoins dans les limites de cette double condition que chaque service comme chaque produit n’ait qu’un seul prix sur le marché et que le prix de vente de chaque produit soit égal à son prix de revient en services.([Walras 1992: 77], our emphasis)
39 This reasoning may be easily understood by looking at what is nowadays known as recontracting, in a model à la Edgeworth. The famous debate between Walras and Edgeworth regarding the tâtonnement process leads to the conclusion that, although Walras does not accept the terms of Edgeworth’s argument, he accepts that at least for the pure theory of prices the criteria of commutative justice and distributive neutrality imply that “il est parfaitement juste de proposer immédiatement le prix unique” [Walras, 1896: 184]. We shall show in a later section that many theoretical differences between both authors are present behind this apparent methodological misunderstanding .
1.2 - The numéraire and the market organization
40 Let us now concentrate on another element of the décor of Walras’s masterpiece: the numéraire. Frequently regarded at as a simple mathematical consequence of the degrees of freedom in the determination of the competitive price vector, or as a reference to a common measure of the value problem, Walras introduces the presence of a common unit of account in order to solve a more complicated problem. This problem is related to the need to establish a coherent system of prices. In other terms, the numéraire appears to be the central element in the generalization of the model from two commodities to three and more. This quest is equivalent to demonstrating the existence of a clearing market price vector, where every commodity has a unique price quoted in terms of a common unit of account.
41 In the EEPP Walras presents the foundations of his theory of prices and competition in a two commodities pure-exchange framework (section II, 4th edition). Any medium of exchange is thus absent from this analysis (it shall remain absent until section VI in the 4th edition). Within this framework, the two commodities are directly exchanged in a common market and consequently there exists a unique rate of exchange (relative price). Even in the simplest case of exchange, Walras introduces the assumption of price taking. In order to fulfil the commutative justice requirements, agents exchange only at the equilibrium price of the market. As underlined by Rebeyrol [1999: 100]: “[…] l’absence de transaction en déséquilibre n’est une hypothèse exorbitante que si l’on admet l’absence de conscience du déséquilibre.” This assumption is the reduced form of the assumption of common knowledge of all exchange opportunities. Even if Walras does not explicitly describe the details of the organization of this two-commodities market, the reference to the stock-market structure leads to the conclusion that this is not a disorganized bilateral exchange process.
42 The generalization of the price theory to the n-commodities case strengthens the requirements of a well-organized market process . In section III of his EEPP, 4th edition, Walras attempts to demonstrate a “true theorem of general equilibrium.” A simple extension of the two-commodities framework leads to a system of one independent market for every couple of commodities (i.e. n (n-1)/2 markets for n commodities). However, if nothing is said about the agents’ level of information, this can result in an “incoherent” system of prices. A price system is incoherent if different exchange strategies lead to different final allocations for an agent; in other words, if some arbitrage gains of indirect exchange are still possible.
43 The condition of a general equilibrium system of prices is resumed by Walras [1988: 161–3] in EEPP: “L’équilibre parfait ou général du marché n’a lieu que si le prix de deux marchandises quelconques l’une en l’autre est égal au rapport des prix de l’une et l’autre en une troisième quelconque.”
44 This condition is, yet again, a requirement of neutrality of the perfect competition solution. If there are unexploited possibilities of arbitrage some agents lose exactly what other agents win by exchanging at those prices. The value of an individual’s budget constraint is not independent of the order s/he follows in order to obtain her/his desired final allocation. Furthermore, a generalized direct barter exchange within a system of isolated two-commodities markets may not necessarily result in a perfect general equilibrium situation [Walras 1988: 163]. However, if those possibilities of gains are common knowledge, the arbitrage strategies may be neutralized. Walras’s solution to this problem is the introduction of a common unit of measure of the individuals’ budget constraints, namely the common numéraire. If prices are all quoted in terms of a common numéraire, agents can verify the coherence of the actual price system. Due to the common knowledge assumption on prices, arbitrages are neutralized. As a consequence, Walras concludes that « …si on a crié des prix en numéraire, la condition d’équilibre général a été remplie ipso facto. » [1988: 200]
45 The numéraire is thus more than an assumption aiming at simplifying the computation of a price system. The existence of a common measure of prices is accompanied by a particular conception of the organization of markets. The numéraire is the necessary language for the common knowledge assumption. The consequence of these assumptions is a market system where strategic behaviors are absent (because they have already been neutralized!) .
1.3 - Money and decentralized exchange
46 Walras considers that the perfect competitive market should be the general frame for a pure economic theory. If there is to be a pure theory of money, it must be adapted to the frame of competitive general equilibrium. Besides being static with passive agents as regards prices and only focusing on equilibrium situations, this framework leaves no place for a theory of the organization of the exchange process without having to take into account the perturbations of the equilibrium so induced. However, Walras tried, through the different editions of his EEPP and other writings (in particular Walras 1992 ), to introduce a monetary theory aimed at demonstrating the neutrality of the monetary exchange vis-à-vis the equilibrium of real prices.
47 From the first versions of his pure theory of prices, Walras intends to distinguish perfect competition from monetary exchange. In the following well known quotation from his 1873 lectures at the Académie de sciences morales et politiques of Paris, Walras presents money as a pure practical economic device, excluded from the pure theory of competition:
Nous allons étudier le phénomène des prix se produisant dans ces conditions de concurrence supposées rigoureuses, en faisant abstraction des petites circonstances perturbatrices […] Seulement, nous écarterons aussi l’intervention de la monnaie. Pour avoir une idée exacte du mécanisme de la concurrence, il nous a bien fallu l’emprunter à l’un de ces marchés où se font des ventes et des achats de marchandises contre or et argent; mais il est clair que l’intervention de la monnaie, qui est une simplification pratique, est une complication théorique qui doit être écartée.[Walras 1993 (1874): 33]
49 After the first edition of EEPP , and probably as a consequence of Bertrand’s and Edgeworth’s criticisms, Walras makes more explicit his assumption of absence of exchange out of equilibrium. In spite of this textual evidence, Walker [1990a and 1990b] considers, contrary to most interpretations, that it is wrong to assume that this assumption means Walras does not study the disequilibrium exchanges. Be this as it may, it is certain that as long as Walras’s monetary theory is concerned, disequilibrium exchanges are absent. In lesson 29 of the EEPP 4th edition, “circulation” (exchange process) begins after equilibrium prices are quoted (see Rebeyrol 1999).
50 This clearly shows the strict coherence of Walras’s conception of perfect competition. Even when he deals with the problem of exchange and money, the whole structure of a competitive market is unaltered. Agents are price takers, equilibrium prices have already been found and quoted by the auctioneer, and there is no possible strategic behavior related with any of the forms studied so far. There still remains a problem with this conception of money as a medium of exchange: it needs to be imposed as an ad hoc assumption taking the form of the nowadays well-known cash-in-advance constraint . This is the price Walras paid for his praise of a coherent theory of perfect competition.
2 - Edgeworth’s theory of perfect competitive field: the limit case of a decentralized exchange economy
51 Edgeworth’s intellectual project of constructing an economic theory is very different to Walras’s. The English economist is attached to a utilitarian tradition, originating from Bentham and including Jevons and, in some aspects, also Marshall. Edgeworth’s main concern is the scientific description of economic behavior and the possibility of merging psychological theories with a scientific moral philosophy. This leads him to study the market and, in general, economic problems as a branch of a more general behavioral science. According to his views, perfect competition is thus a result of agents’ interactions under special circumstances. Perfect competition is thus a special case of a general framework, namely a limit case.
52 Edgeworth does not need to impose the price-taking assumption or the presence of auctioneers and brokers in order to explain the functioning of a market. In some sense, his is a theory constructed in the opposite sense in comparison to the Walrasian edifice. As we shall try to show, Edgeworth’s conception of perfect competition is built upon strategic behaviors and the causes of their being neutralized under some circumstances. In opposition to Walras’s permanent attempt to erase any strategic conjectures, as we have shown so far, Edgeworth makes of this kind of behavior the rule and the engine of a competitive market.
2.1 - Edgeworth’s “final settlement” and his equilibrium solution
53 Edgeworth’s market description is presented as a battleground where agents are constantly bargaining over contracts of exchange that could be destroyed if a better contract is established with another partner. Within this apparently chaotic interaction of agents there may exist some general states of “peace.” A peaceful state is characterized by the absence of any possibility to better at least two agents by breaking their actual contracts and sign a new one among them. Edgeworth defines the set of those feasible peaceful situations as the “final settlement.”
54 Following the developments of General Equilibrium Theory during the 1970’s it seems generally accepted that the concept of the “core” as an equilibrium solution is the mathematical representation of Edgeworth’s “final settlement.” Hildebrand says:
“Since an economy with a finite number of agents is not perfectly competitive, Edgeworth introduced a new concept of equilibrium; the ‘final settlement’ or in today’s language, the core. The ‘Limit Theorem’ expresses that under ‘perfect competition’ these two concepts coincide.” 
56 In the present section we shall show how this, now common, interpretation of Edgeworth’s theory is not only misleading but it also contributes to erroneously reinforce the idea of the convergence of the price-taking assumption and the recontracting analysis. It is necessary to revisit Edgeworth’s texts  in order to show that the source of this confusion comes from his own works.
57 The main textual evidence generally quoted or referred to in modern economics is based on an example of what we nowadays call a ‘replica economy'. This was a simple illustration of Edgeworth’s mechanism of recontracting. The convergence of an economy with a great number of agents to a perfect competitive situation has been developed in the 20th century economics as the general Edgeworthian framework. It is precisely within this framework that the idea of a Walras-Edgeworth convergence, and the consequent disregard of their difference and the founding of the price-taker assumption as the sole requirement for perfect competition, takes root.
58 There are two main consequences derived from the traditional interpretation of Edgeworth, as exposed in the above quotation from Hildenbrand. One related is related to the idea of coalitions as used in modern cooperative game theory; a concept alien to Edgeworth’s general framework. The second concerns the conception of prices implicitly forming during the recontracting process and the presence of a de facto rule of unique price undermining the notion of competition as a battleground. Both issues point in the direction of the interpretation of Edgeworth as a theorist in the line of cooperative-games and the forerunner of the notion of the core as a solution concept. We will discuss these notions from Edgeworth’s texts. This allows us to plead in favour of Edgeworth’s originality and of the need to rescue his theory of perfect competition.
2.2 - Blocking coalitions and combinations
59 The first consequence, mentioned above, refers to the interpretation of Edgeworth’s “final settlement” as necessarily linked with the formation of ‘blocking-coalitions.' This idea has been developed within the framework of cooperative games. In a nutshell, the core is the set of contracts that cannot be improved–from the point of view of an individual–by an alternative arrangement with a different group of agents. This notion thus leads to a representation of Edgeworth’s ‘recontracting’ process as a permanent formation and dissolution of coalitions. In this way, the set of actions available to individuals are replaced by the actions of a coalition. Furthermore, the very notion of cooperative games implies that the coalition may be interpreted as a collective decision body. Within this collective body, individual strategies are subject to the distributional rule imposed within the coalition. However, this interpretation does not really follow from Edgeworth’s texts as we shall now show.
60 Two textual elements allow us to assert that the language of cooperative game theory, and the subsequent modern theory of the core, does not correspond to Edgeworth’s theory. (i) The modern notion of coalition seems to be a misinterpretation of Edgeworth’s concept of combination and of cooperative associations. However, these two notions are deeply different from the modern notion of coalition when one examines them more carefully.
61 Edgeworth clearly establishes a difference between a group of agents entering voluntarily in a series of bilateral contract relations and a collective decision body that he calls a combination. Whereas the latter is a collective body, the logic of the former is purely individual, in the sense of an agent accepting bilateral contracts within a new group, rather than the bilateral contracts this same agent has formerly agreed to within another group. In fact, within the new improving group, all contracts continue to be bilaterally established and, more importantly, these kinds of association do not necessary imply giving up all previous exchanges outside this group.
62 In opposition to the purely individual logic of these bilateral arrangements, a combination relies on a collective logic, as the distributional rules established among their members impose a form of collectivization of private property. It is, in fact, when Edgeworth actually discusses the extension of his ‘theorem of the perfect competitive field,' that he presents the notion of a combination as a source of imperfect competition ([Edgeworth, 1881], in Newman (ed.) [2003: 44]): “Combinations tend to introduce or increase indeterminateness; and the final settlements thereby added are more favourable to combiners than the previously existing ones.”
63 Furthermore, Edgeworth goes on to present a cooperative association, a particular form of combination, as a source of imperfection. This type of collective arrangement is exactly what the cooperative game theory introduces as a coalition. Let’s follow Edgeworth’s text to clearly appreciate this:
“The fourth imperfection would seem likely to operate in the case of cooperative associations up to the time when the competitive field shall contain a practically infinite number of such bodies; that is, perhaps for a long time. To illustrate the idea, let us imagine associations of capitalist-workmen, consisting each of 100 members, 50 contributing chiefly capital, and 50 chiefly labor. Let the field of competition consist of 1000 individuals. The point here is that, notwithstanding the numerical size of the field, contract will not be more determinate than if the field consisted of 10 individuals.” Edgeworth, 1881 in Newman (ed.) [2003: 44]
65 From this, Edgeworth draws a completely different conclusion than that implied by the coalitional formation presented in the cooperative game theoretical framework. The core of an economy, in this framework, is simply reduced by the presence of an increasing number of agents forming this kind of coalitions. From the Edgeworthian point of view, the number of agents does not guarantee increasing competition per se when this kind of cooperative arrangement is the rule.
66 The particular case, which is typically presented as the Edgeworthian archetype of the recontracting process (see Hildenbrand ),where agents systematically renounce their old contracts in order to recontract within a new group, is only an example presented by Edgeworth. It is the well-known example of a ‘replica economy,' where a great number of only two types of agents are gradually introduced in the market. Edgeworth intended to formulate a general concept, with multiple types and unequal number of agents [Edgeworth, 1881] in Newman (ed.) [2003: 43].
2.3 - The example of a replica economy and the price-taker assumption
67 The second reason that confirms the distance between the cooperative game notion of coalition and the Edgeworthian pairwise arrangement within a subset of agents is that the former implies the presence of a unique price rule within the coalition and the latter does not. In fact, when Edgeworth presents his example of a replica economy, gradually introducing the same number to both sides of a two-type of agents’ economy, he clearly states that contracts continue to be bilateral and the implicit exchange rates could be different. In fact, the settlements (contracts) prevailing within a group of, let’s say, two agents of type A and three agents of type B are not the same. Exchange rates may be bilaterally defined and they could be different. However, when we consider a group composed by agents with the same characteristics and we suppose perfect fluidity of information, it is impossible to establish this difference. However, the important question that arises from this example is not whether the bilateral contracts between any A and any B could differ from another similar contract, but whether those contracts can be stable in the presence of imperfect information . The real advantage of the Edgeworthian framework is the possibility of discussing the presence of imperfections and therefore not being confined to a rule imposing a unique price within a coalition or for the whole economy.
68 In fact, Edgeworth compares this example, where the same price prevails among all bilateral contracts within a subset of agents, with the result obtained by Jevons and Walras. This comparison is important because it allows Edgeworth to show the particularity of his example of a replica economy, compared with his broader notion of recontracting:
“The dealing of an individual in an open market, in which there prevails what may be called the law of price, the relation between the individual’s requirements and that quantity collectively-demanded-at-a-price, usually designated by the term Demand, between little d and big D in M. Walras’s terminology, is elegantly exhibited by that author.
Here it is attempted to proceed without postulating the phenomenon of uniformity of price by the longer route of contract-curve. When we suppose plurality of natures as well as of persons, we have to suppose a plurality of contract-curves.” Edgeworth, 1881 in Newman [2003: 40]
70 Edgeworth constantly criticized the use Jevons and Walras make of the unique-price assumption. In fact, as we have forewarned in the introduction, the main point of contention between Edgeworth and Walras refers to the presence of this assumption. As Bridel & Huck [2002: 531] have clearly stated:
“Well beyond the static or dynamic nature of the tâtonnement, the opposition between the recontracting principle and the notion of supply-and-demand at given prices, between ‘fields of competition’ and ‘competitive markets’ is at the heart of the controversy between Edgeworth and Walras.”
73 The richness of Edgeworth’s theoretical goal has been obscured in modern theory as a consequence of the enthusiasm generated by the Debreu-Scarf theorem and the reconciliation of those great fathers of modern economic analysis. Contrary to the hope of this line of reasoning, if there exists a convergence in terms of the result (namely the final allocations) of the perfect competitive situation between Edgeworth and Walras, there is no possible reconciliation in terms of the exchange mechanism that allows obtaining such results. In brief: Edgeworth’s theory of recontracting cannot give content to the elusive notion of tâtonnement as an actual economic process. We think it is much more interesting for modern economic theory to take seriously the confrontation well stated by Edgeworth himself:
“Equilibrium is attained when the existing contracts can neither be varied without recontracting with the consent of the existing parties, nor by recontracting within the field of competition. The advantage of this general method is that it is applicable to the particular cases of imperfect competition: where the conceptions of demand and supply at a price are no longer appropriate.” Edgeworth, 1881 in Newman [2003: 31]
75 In this sense, Negishi’s interpretation of the convergence of the prefect competitive situation in both authors gives more credit to Edgeworth’s claims but needs to be reassessed. Negishi, as well as Walker , relies on the non-tâtonnement theory as the correct interpretation of Edgeworth’s message. However, he accepts that the convergence of the final outcome of both authors justifies the price-taker assumption:
“The assumptions made in the Cournot-Walras approach may not be realistic, unless there is an auctioneer, as in the case of well-organized markets. Edgeworth’s equivalent theorem justifies Walrasian assumption, however, since it is not the assumption but the outcome that matters for a theory, and we can assume that the traders behave as though they were price takers, even though traders are not actually price takers, provided we have the same outcome as assured by Edgeworth.” [Negishi 1989: 338]
77 Contrary to this view, we think that the price-taker assumption has to be taken for what it is: an assumption. This implies not only a different conception of an agent’s behavior, as we have shown in our interpretation of Walras’s theory, but also a different institutional framework in order to neutralize all possible sources of strategic behavior. In other words, it is not possible to assume Walras’s price-taker assumption as a consequence of a situation where there exists a huge number of individuals neutralizing all the possible strategic (i.e. arbitrage) actions each other might take. The main result of our analysis of Walras’s pure economics is that the auctioneer, the numéraire and the presence of brokers are institutions giving content to the price-taker assumption; the result of the tâtonnement (namely the general equilibrium price vector) and the non-trade-out-of-equilibrium notion are assured by this institutional framework.
78 In Edgeworth’s theory, the institutional framework and individual behavior are completely different. However, the interpretation given in terms of the cooperative-game approach leaves no place for a different view. This framework contributes to reduce individual behavior to a nonstrategic one, as can be clearly seen from the opposition between the two main approaches on game theory. As Gabszewicz [1999: 26] recognizes:
“. . .when a small number of sellers are conscious of the interactive strategic context created by their simultaneous selection of product supply, this selection can be viewed as a non-cooperative equilibrium when sellers do not coordinate their choices, and as cooperative equilibrium when sellers enter into collusion to select their aggregate supply and share the production of it among themselves.”
80 We come now to the second implication of the modern interpretation of Edgeworth as a forerunner of cooperative-game theory. This has to do with the conception of the theory of the core, and thus Edgeworth’s theory of ‘final settlement', as a theoretical representation of the market with the simplest or even no institutional framework.
2.4 - The ‘Core’ as a free-of-institutions framework and Edgeworth’s equilibrium exchange
81 We now turn our attention towards an epistemological consequence derived from the traditional interpretation of Edgeworth’s equilibrium analysis. We shall show that the Debreu-Scarf’s cooperative-game approach forced an interpretation of Edgeworth’s market views not only as compatible with, but even complementary of, Walrasian general equilibrium. We shall show that this tour de force is based on the idea that both representations of the market are based on the idea of an economic system liberated from any institutional framework. This contributed to the overlooking of Edgeworth’s most interesting insights about the functioning of a competitive market and the related notion of competition. Both ideas could allow economics to address an equilibrium approach in an alternative way where the Walrasian auctioneer and the price-taking assumption are absent from the scene.
82 During the 1980s, the evolution of the cooperative-game theoretical approach led economists to believe that the ‘Core’ as an equilibrium solution was very attractive because of its simplified Walrasian-like framework. The simplicity came from the fact that, as in the Walrasian-General-Equilibrium approach, the ‘Core’ is a solution concept built with little data: tastes, technologies and endowments, without any institutional structure. As Mas-Colell [1982: 16] states:
83 For the Cournot type analysis, one needs a specification of transaction institutions. This may be a cumbersome and non-trivial task. In contrast, the ‘Core’ determines final allocations from the basic economic data: tastes, technologies, endowments. This is the feature of the ‘Core’ which makes it so attractive, but the level of abstraction at which it moves has an unfortunate consequence. It is this: in the absence of an institutionally explicit model, the only sensible feasibility constraint for a blocking coalition is that the proposed reallocation affect can be attainable for all commodities simultaneously considered. This, of course, means that competition works by the setting of entirely separated subeconomies!
84 As it arises from this quotation, the main characteristic of the ‘Core’ approach, and thus of Edgeworth’s, is that the economy seems to be completely described by this simple data. This is exactly the same conception of a Walrasian economy as presented in the Arrow-Debreu model. This reveals how the Debreu-Scarf tradition constructs the Edgeworth-Walras convergence upon a common view on what is a complete description of an economy. Both representations are misleading. To interpret Walras’s pure economics as a free-of-institutions framework is equal to ignoring the fact that an auctioneer, the numéraire, non-out-of-equilibrium-exchange, etc. are all institutional assumptions needed to enforce the price-taker assumption and thus the perfect competitive economy.
85 This misleading interpretation of Walras’s and Edgeworth’s theories, prevailing in modern economic theory, gives place to the mistreatment of the informational context and the exchange technology. It is commonplace to say that a Walrasian economy explicitly assumes perfect information and a nonfrictional exchange technology. Both ideas are far from being accurate. De Vroey (2003) has shown that Walras’s theory presumes less informational requirements than Marshall’s, and we can say now, than an Edgeworthian framework; the reason for this is simple, and has to do with the price formation and technology of exchange those authors assume. As we have shown above for Walras, the presence of the auctioneer implies that agents only need the price-vector information in order to make decisions. Furthermore, agents do not make conjectures on others’ decisions; they have no strategic behavior.
86 When strategic behavior is introduced and there is no centralized price-formation system (i.e. no auctioneer)–and this is certainly the case for Edgeworth–the informational flow becomes an important issue. Furthermore, the fluidity of information is the main issue in order to assure a perfect competitive field in terms of Edgeworth  as in Newman (ed) [2003: 18]: There is free communication throughout a normal competitive field. You may suppose the constituent individuals are collected in one place, or connected by telephones.
87 Concerning the exchange technology and price-formation mechanism there is also a very different, but explicit, presence of institutions for both authors. Walras’s has been described in the second section of this paper. In the case of Edgeworth, it has been mentioned that the recontracting process supposes a decentralized price pairwise framework. The implicit exchange rates of the recontracting process are fixed through a bilateral bargaining game. This is not a cooperative game as the strategies of all parties are (i) interdependent, and (ii) the context within which the decisions to accept or not a definitive exchange is explicitly modelled. In fact, the most interesting characteristic of Edgeworth’s theory of perfect competition is that it is developed as a limit case of an imperfect competition framework. The notion of indeterminateness and the recurrent criticism Edgeworth addresses to Cournot show how important the analysis of monopolies and all other forms of imperfect competition is for him. In fact, most of Mathematical Psychics is devoted to the analysis of “imperfect competition” and a scientific analysis of the economic policies to be adopted for solving potential problems of imperfect competition . Briefly, following Vind [1995: 1740]:
“Edgeworth’s definition of competition and exchange equilibrium did not share the norms and goals of the later general equilibrium theorists. His notion of competition only makes sense in a society with institutions, and would not have regarded the core as an interesting solution concept even if it was ‘institution free.’”
3 - Concluding remarks: imperfect competition… On which notion of perfection do we stand?
89 It is obvious, and almost goes without saying, that imperfect competition is a concept built upon a conception of what is a perfect competitive situation. However, this simple reasoning is not always adopted in economics. In fact, imperfect competition has rarely been constructed in relation to a clear-cut concept of perfect competition. It is usual to find a mere rhetorical argument explaining why the model violates the so-called Walrasian-competitive assumptions by giving agents the power to establish prices. Once the price-taker assumption is left aside, Pandora’s box is opened for theoreticians. It is the box of the institutional framework where the exchanges take place. However, ‘Walrasian allocations’ are always kept as the benchmark. The way from the imperfect situation to the so-called competitive situation is, however, not explicit. This is, in fact, one of the reasons why the convergence between Edgeworth and Walras was a hopeful result for general equilibrium theoreticians. It seems to demonstrate that the central point may be the number of agents, without giving importance to other, more important, differences between those authors.
90 The cooperative-game approach has thus constructed an ideal framework in which it is possible to obtain the main results of the Arrow-Debreu model and, more importantly, to finally get rid of the annoying auctioneer. However, all the properties and shortcomings of the centralized Arrow-Debreu framework reappeared within this new land of freedom. Even the wishes of the ‘founding fathers’ of Game Theory were betrayed by cooperative-game theory:
“Let us look more closely at the type of economy which is represented by the “Robinson Crusoe” model, that is, an economy of an isolated single person or otherwise organized under a single will . . .Thus Crusoe faces an ordinary maximum problem the difficulties of which are of a purely technical–and not conceptual–nature …
Consider now a participant in a social exchange economy. His problem has, of course, many elements in common with a maximum problem. But also it contains some very essential elements of an entirely different nature. He too tries to obtain an optimum result. But in order to achieve this he must enter into relations with others. . . Thus each person tries to maximize a function. . . of which he does not control all variables. This certainly is no maximum problem but a peculiar and disconcerting mixture of several conflicting maximum problems.” [Von Neuman & Morgenstern 1944: 10]
92 This is exactly what the ‘Core’ as a solution does not tackle. The centralized framework of Walras’s theory translated into a game-theoretic approach, which produced nonstrategic behavior; at least for the sole case that interested the Debreu-Scarf limit theorem, which is the perfect competitive allocation. There is a complete lack of a notion of competitiveness, at least as common sense and economic rhetoric conceive of it. Let us give the last word to an expert on the ‘Core’ theory in order to express this point more clearly:
“. . .the present game theoretical treatment of General Equilibrium is also guilty of this lack of competitiveness. While the process leading to the establishment of the core is indeed a process of rivalry, the eventual solution is purely cooperative and can be approximated by a set of competitive prices. This is no surprise, in the sense that if we give two correct theories the same set of assumptions, they will arrive at the same conclusions.” [Schotter 1973: 307]
94 The paradoxical conception of perfect competition as the absence of any form of competition relies on another paradox of modern neo-Walrasian macroeconomics. The representative agent, or the more literary figure of a Robinson Crusoe, is a way to express the confusion between perfect competition and optimality. In modern dynamic macroeconomics, prices are purely rhetorical as the only exchange rate is the one established between nature, mediated by a production function, and human labor. Within this Robinsonian world, the efficient allocation of resources is considered to be a representation of a more complex coordination mechanism: the price system. Furthermore, the so-called prices implicit in the efficient allocation are considered as the result of a competitive market. The last paradox appears here: the central question about the capacity of a competitive market to coordinate decentralized decisions of a multitude of agents is reduced to the optimizing choice of a sole agent. ‘Great numbers’ seems thus to be merely another rhetorical idea.
This work was supported by the Belgian French-speaking Community (Grant ARC 03/08-302) and the Belgian Federal Government (Grant PAI P5/10) through a Post-doctoral Fellowship given to A. Alvarez and by the National University of Colombia (Convocatoria de Investigación 2006). I thank all the participants of the “First Workshop on the Interface between Economic Theory and History of Economics” organized at the Université Catholique de Louvain. In particular, I am in intellectual debt to Michel De Vroey for his constant encouragement during my visit to Louvain-la-Neuve and Claire Pignol for her enlightening comments on an earlier version of this paper. I am particular indebted to two anonymous referees whose remarks and sharp criticisms allowed me to improve considerably this article. All remaining errors are my sole responsibility.
Contact: Andrés Alvarez, Department of Economics, Universidad de los Andes, Calle 19a, No. 1-37 Este, Bloque W, oficina W-816, Bogotá - Colombia. Email: email@example.com
See Makowski and Ostroy  and Gale  for two essential references on the recent literature about the reestablishment of the perfect competition framework within a strategic behavior approach as an alternative view regarding the traditional price-taker assumption. These alternative approaches aim at founding perfect competition not on an ex-ante assumption (i. e., price takers) but as the result; as the limit case of imperfect competition.
To be clear, it is important to acknowledge we are not considering here relevant theoretical attempts at building general equilibrium models of imperfect competition (see Gabszewicz  for a general discussion on this). However, we only try to underline a general trait of standard economics moving out of the general equilibrium framework, not only by embracing a neo-Marshallian framework, but also a partial equilibrium approach based on a game theoretical framework, as per the Laffont & Tirole tradition.
The main goal of this article being to compare the treatment of perfect competition in Edgeworth’s and Walras’s theories, we do not consider necessary to discuss a third alternative such as for example Cournot’s theory. However, as we will refer to it in the concluding remarks, this paper pleads for considering new approaches founded upon multiple authors, but authors with essentially compatible frameworks.
To mention only some few relevant authors, we can relate this way of thinking with Robert Aumann Hildenbrand and Mas-Colell. But a most significant reference is the treatment of this question in the now classic advanced textbook on “Microeconomic Theory” by Mas-Colell et al.
See Walker , Rebeyrol [1999, p. 78-100], Berta  and Bridel & Huck .
Edgeworth [1889a, 1889b, 1891] as in Newman (ed.)  and Walras [1988, 1965]. It is also important to include the intervention of Bortkiewicz  on behalf of Walras.
With the important exception of Bridel & Huck  that we follow very closely concerning the interpretation of this debate, though we go further in the analytical interpretation of the implications for modern economics.
This idea is mainly developed in Negishi’s interpretation of Edgeworth [Negishi 1985, 1987]. For an interesting analysis of this idea see Pignol . We study different modern interpretations of Edgeworth’s recontracting theory in another paper.
From the original title in French: Éléments d’économie politique pure.
For an extensive discussion see P. Bridel [1997 and 2002].
It is important to underline that here we are referring to the modern concept of “Pareto optimality” which has nothing to say about distributive justice neither about first-best results. We accept thus the critical point made by Jaffé  against Hicks and Baumol concerning Walras’s extremely optimistic view of a free market issue in terms of a social welfare criterion regarding first best allocation or distributive justice.
For an in-depth discussion of this point see Rebeyrol [1999: 90-100] and Bridel & Huck . For an opposite point of view see Walker .
A well-organized market process refers here to the idea Walras constantly evokes of the commodities or financial stock markets as a paradigm of the conditions under which agents are forced by the institutional context to behave as if they were price-takers. Markets with less structured systems of informational flow contain the possibility of exploiting strategic actions in order to take advantage of any informational asymmetry or bargaining power.
Costa [1988 and 2002] also analyses the problems Walras faces in trying to avoid the arbitrage questions. He concludes that the analytical difficulties associated with those problems lead Walras to avoid an explicit treatment of competition and exchange. Following Rebeyrol , we rather argue that it was not a matter of difficulty but a more profound attachment to a particular conception of pure economics.
For a detailed discussion of this question see Álvarez & Bignon .
All references to and quotations from Edgeworth’s texts are taken from the recent edition of his Economic Works by Newman (ed.) . This contains also the translation of some articles originally published by Edgeworth in French.
“It is not necessary to resolve analytically the composite mechanism of a competitive field. It will suffice to proceed synthetically, observing in a simple typical case the effect of continually introducing into the field additional competitors.” [Edgeworth, 1881] in Newman [2003: 40]
We refer here to the authors mentioned in the introduction of this paper: Gale, Makowski & Ostroy, etc.
See pages 34 to 56 of Edgeworth  as in Newman (ed) .