art strategic market games giraud

8/10/2019 Art Strategic Market Games Giraud

http://slidepdf.com/reader/full/art-strategic-market-games-giraud 1/21

Journal of Mathematical Economics 39 (2003) 355–375

Strategic market games: an introduction

Gaël GiraudCNRS, UMR 7522 BETA, Université Louis Pasteur & UMR 8095,

CERMSEM, Université Paris-1, Paris, France

Received 3 December 2002; received in revised form 27 March 2003; accepted 27 March 2003

Abstract

This paper introduces the special issue of The Journal of Mathematical Economics on “StrategicMarket Games” (SMG).© 2003 Published by Elsevier Science B.V.

Keywords: Strategic market games (SMG); Limit-price mechanism; Money; Incomplete markets

1. Introduction

The theory of strategic market games (SMG) examines the interaction between individ-uals in an economy and the impact of their behaviour on fundamental macrovariables suchas prices, income distribution, volume of trade, velocity of money, etc. One key concernis to explain how uncoordinated actions of selfish individuals may, or may not, lead tosome social optimum. When they do, it is typically the case that the “hypothesis of perfectcompetition” holds, and agents can be viewed as price-takers. SMG make precise the con-

ditions for this to happen and, in the process, sheds light on Adam Smith’s “invisible hand”itself.

The main construct in SMG is that of a market mechanism which maps agents’ ac-tions to prices and trades. This gives rise to a non-cooperative game in strategic form. Theinquiry focuses on qualitative features of strategic (Nash) equilibria (e.g. their possible(in)efficiency and/or (in)determinacy) and their relation to competitive (Walras) equilibria.Everyday experience shows, however, that a large number of market mechanisms coexisttoday, ranging from the pairwise bargaining on an African souk for non-durable consump-tion commodities to the NYSE, from Dutch auctions for tulips to English auctions on theWeb . . . . One is thus lead to ask whether some mechanisms perform better than others;whether several of them can be deduced from a unique fundamental paradigm; whether

E-mail address: [email protected] (G. Giraud).

0304-4068/03/$ – see front matter © 2003 Published by Elsevier Science B.V.doi:10.1016/S0304-4068(03)00049-1



356 G. Giraud / Journal of Mathematical Economics 39 (2003) 355–375

the possible (non-)coincidence between strategic equilibrium outcomes and competitiveoutcomes depends upon the choice of such an institutional paradigm. Thus, attention ispaid as much to the institutional aspects of market mechanisms as to the micro-economic

equilibrium behaviour itself—and the present special issue is no exception.What makes SMG special (in comparison with, say, the literature of financial market

structures)1 is that:

(1) They focus on the general equilibrium aspects of the issue at hand. Consequently, allthe papers to follow start with an Arrow–Debreu economy E = (ui, ωi)i characterizedby a utility function ui : RL → R and a vector of initial endowments ωi ∈ R

L+ for each

individual i.2

(2) They do not impose any price-taking behaviour as an inbuilt hypothesis, and playersare assumed to behave strategically. As a consequence, the main equilibrium concept

considered in the following contributions is Nash.3 Of course, when the space of playersis atomless (as, e.g. in Dubey and Geanakoplos, 2003a,b), Nash equilibrium impliesprice-taking behaviour, but this is an output of the model (and a simple way to sidestepphenomena related to imperfect competition, whose study is left for further research)rather than an input.

Intherestofthispaper,westartbylayingoutstrategicmarketgames à la Shapley–Shubik,and recall some of their most basic properties that need to be in the reader’s mind beforereading the papers to follow. At the same time, we introduce the contributions of this issue.

2. One basic approach: Shapley–Shubik games

Consider an Arrow–Debreu pure exchange economy4.Inagame à la Shapley and Shubik(1977), the market is organized as follows: suppose there are L commodities, the last onebeing a numéraire (“commodity-money”). There is a trading-post for each commodity k =

1, . . . , L−1, other than money. On the kth trading-post, commodity k can be traded againstmoney. A player’s strategy consists in sending signals (qi

k, bi

k) ∈ R2

+ to each trading-post k,denoting, respectively, the amount qi

k of commodity k s/heisofferingforsale,andhowmuch

money bi

k

s/he is ready to spend on the purchase of commodity k. Quantities offered forsale are thought of as being physically shown or sent, so that a trader’s signal qi := (qi

k)L−1

k=1

1 cf. e.g. Biais and Rochet (1997).2 There will be slight variants in some of the papers that follow: Mertens (2003), Codognato and Ghosal (2003),

and Dubey and Geanakoplos (2003a,b) consider a large economy with an atomless space of traders; Giraud andStahn (2003), and Dubey and Geanakoplos (2003b) consider a two-period economy with incomplete financialmarkets.

3 Again, there are some variants: in Weyers (2003), players play pure Nash equilibria (NE) with no (weakly)dominated strategies; in Codognato and Ghosal (2003), they play communication equilibria using a certain typeof communication device; in Giraud and Stahn (2003) they play Nash equilibria with or without monitoring.

4

We are not concerned with production in this special issue. Introducing production in the kind of generalequilibrium-like models with imperfect competition to which this issue is devoted is one of the challenges on theresearch calendar. See nevertheless Sections 4.2 and 4.3. infra, as well as Dubey and Geanakoplos (2003b), andBottazzi and De Meyer (2003) in this issue.



G. Giraud / Journal of Mathematical Economics 39 (2003) 355–375 357

cannot exceed (componentwise) her/his endowment ωi = (ωik

)L−1k=1 , while the total amount

of money sent to the market

L−1k=1 bi

k cannot exceed her/his initial balance ωi

L.5 Player i’spure strategy set can therefore be written as

S i :=

(bi, qi) ∈ (RL−1

+ )2 |qik ≤ ωi

k, k = 1, . . . , L− 1 andL−1k=1

bik ≤ ωi

L

(1)

Suppose there are N ≥ 1 traders on the market. Given an N -tuple of strategies s = (si)i ∈

S :=

i S i, the market computes a price vector as well as the final trades of each agent.The price of commodity k is defined by

pk :=Bk

Qk

(2)

where Bk :=

i bik (resp. Qk :=

i qi

k) designate, respectively, the aggregate supply of money and the aggregate supply of commodity at the kth trading-post.6 Player i’s finalallocation is then given by

xik := ωi

k − qik +

bik

pk

, k = 1, . . . , L− 1

while her/his final balance of money is

xi

L := ωi

L −

L−1k=1

(bi

k − qi

kpk).

As already said, a plethora of alternative market institutions can be conceived. First,one may inquire whether analogous game rules could be developed in the presence of fiat

money. A (strategic) budget-constraint treating all commodities symmetrically

Lk=1

bik ≤

Lk=1

pkqik

mustthenreplacethefeasibilityconstraintL−1

k=1 bi

k ≤ ωi

L,asin PostlewaiteandSchmeidler

(1978) leading to a generalized game; or else penalties must be introduced on those whoviolate it and go bankrupt, as in Shubik and Wilson (1977),and Dubey and Shapley (1994)7.

Second, there are many markets (think of the FX markets for instance) where all theobjects of exchange (e.g. currencies) are symmetric: there, neither a natural numéraire nor fiat money can serve as a medium of exchange. In this case, one obviously needs a moreenhanced trading mechanism, and what first comes to mind is a trading-post for every pairof commodities, with each post operating à la Shapley–Shubik as in Section 2 (see Amiret al., 1990). But then, since each relative price pk of commodity k against is determined(according to (2)) solely by what happened on the trading-post (k, ), nothing guarantees

5 This last restriction is tantamount to a cash-in-advance constraint.6 The standard convention x/0 := 0 is adopted.7 See also Section 4.3 in fra.




that prices of commodities i, j and k will be consistent. Actually, for most strategies, therewill exist a triplet of goods (k, j, ) such that pkjpj ≡ pk.8 For later reference, let usdenote this mechanism the TP-mechanism.

2.1. Axiomatizing mechanisms

An alternative to the inconsistency problem consists in making each relative price deter-mination dependent on all the trades among all the various pairs of commodities. This isthe basic idea underlying Shapley’s “windows model” (Shapley originally proposed it ininformal discussions, before it was later formally studied by Sahi and Yao (1989)).9 Here,each trader i’s signal is an L × L matrix whose k-entry ai

k indicates the amount of com-

modity k s/he is offering in exchange for commodity . A clearing-house then calculates

prices according to the following set of equations:L

k=1

N i=1

aikpk

= p

Lk=1

N i=1

aik

, = 1, . . . , L (3)

This basically says that the total value of all commodities offered for commodity mustequal the value of commodity , for = 1, . . . , L.” It is now plain that, by modifyingthe quantity of oranges you sent in exchange for apples, you not only modify the priceof oranges relative to apples (which is but to be expected in our imperfectly competitiveworld), but—almost always—all the other relative prices as well. This latter phenomenon,

of course, cannot occur in the TP-mechanism. Finally, commodities are redistributed insuch a way that i’s final allocation is

xi := ωi

+1

p

Lk=1

pkaik −

Lk=1

aik (4)

In short, prices mediate all trades. Let us designate this second mechanism as the W-mecha-nism. In contrast to the TP-mechanism, under a mild restriction on the strategy profile,the W-mechanism always produces consistent prices. It is plain, on the other hand, that,when compared to trading-posts, windows are not decentralized, quite the contrary: the

clearing-house has to do a “centralized” computation of prices and trades simultaneouslybased on the data sent to all the windows.Now, a little reflection immediately suggests that some middle-ground, between the

Shapley W-mechanism (where every commodity can trade against any other) and theShapley–Shubik TP-mechanism (where money is the sole medium of exchange), wouldbe of interest. Going back to the example of FX markets, Australian Dollars, for instance,cannot be traded against Singaporean Dollars, but both can be indirectly exchanged via USDollars. This means that one could think of some graph, whose nodes stand for commodi-ties, and where an arc between nodes k and signifies that commodities k and can be

8

Note, however, that with perfect competition (i.e. a continuum of players) and with perfect liquidity (i.e.interiority of strategies) consistency of prices can be deduced at NE in this model by the standard arbitrageargument.

9 See also Dubey (1994) f or a survey on such various mechanisms.




traded against each other. One could even go one step further: by orienting the graph, onecould capture the fact that certain financial assets can be sold to purchase others, but not viceversa. For example, a convertible bond can be exchanged with its underlying equity, but the

converse does not hold. This is the main illuminating idea underlying the work by Dubeyand Sahi (2003) in this issue. The authors provide a list of simple axioms whose unique“solution” must be a W-mechanism located “somewhere” in between the Shapley–ShubikTP-mechanism and Shapley’s W-mechanism, and which can be entirely characterized interms of the graph G summarizing allowed trades.10 It is worth noting that the graph, i.e.the generalized windows, are deduced to exist from the axioms, not postulated.

2.2. Converging to competitive equilibria

One important remark regarding the basic Shapley–Shubik model is that it always admitstrivial autarkic Nash equilibria where everybody sends and bids 0 on every trading-post.Intuitively, this means that if all the traders expect the market to be very thin, then thiswill turn out to be the case.11 Unless initial endowments are already Pareto-optimal, thisimplies that pessimistic expectations can prevent markets from forming, and lead to highlyinefficient outcomes.

A second remark is that, roughly speaking, when markets are not shut, the price-manipu-lation power of each agent is a decreasing function of the number of interacting players.Indeed,itwasshownby DubeyandShubik(1978) that, whenever the number k ofreplicasof each player grows to infinity, the set of “nice” Nash equilibria12 converges to some Walras

equilibria of the limit-economy.13 This seminal convergence result is a non-cooperativeanalogue of the celebrated Debreu and Scarf (1963) core convergence theorem. One nuancemust be pointed out, however: even with a finite number N of agents, the core containscompetitive allocations, and “shrinks” to them as N → ∞. In contrast, for every finite N ,Nash and Walras allocations may well be disjoint. Indeed, for smooth marketmechanisms14,Nash allocations are generically Pareto-suboptimal, much less in the core (see Dubey,1980; Dubey and Rogawski, 1990; Aghion, 1985). Cordella and Gabszewicz (1998) evenexhibited a linear economy that admits no-trade as its unique “nice” Nash equilibriumallocation as long as the number of players remains below some critical limit, althoughits initial endowments are Pareto-suboptimal.15 What Dubey and Shubik (1978) showed

is that, asymptotically, the set of “nice” Nash equilibrium outcomes converges to somesubset of Walras equilibria as the manipulation power of each household shrinks to zero. Apartial converse has been recently suggested by De Michelis and Germano (2000): every

10 Whenever the graph G is a tree, the TP- and W-mechanisms coincide.11 Peck and Shell (1990) formally develop this intuition in the fiat money model.12 Means equilibrium points that are limits of Nash equilibria of a sequence of ε-perturbed game, as ε → 0—

where the ε-perturbation involves a “market maker” who puts ε goods on each side of each trading-post.13 A similar conclusion was drawn in the fiat money model by Postlewaite and Schmeidler (1978). Analogous

results were also looked for by Roberts (1980) and Green (1980), but under technical restrictions which in fine

rather mitigate against convergence.14 Which include the TP- and W-mechanisms discussed.15 When the number of players increases above the critical limit, the unique nice NE coincides with the unique

competitive outcome.




regular Walras equilibrium of the underlying economy can be reached as the limit of somesequence of interior symmetric Nash equilibrium outcomes obtained by letting the numberof agents tend to infinity. Since, as is well known, the set of Walras equilibria relative prices

generically consists of finitely many points, all of them being regular, this implies that, atleast for a generic economy, the theory of perfect competition can be reasonably understoodas the limit-case of imperfect competition.16

The interplay between strategic and competitive equilibria then prompts a question: isit possible to prove existence of the latter by means of the former? The strategy of proof would look something like this: start with some Shapley–Shubik game; prove the existenceof some pure Nash equilibrium (NE) for this game; then replicate each player by a set of kidentical clones; finally, let k tend to infinity, and by passing to the limit in type-symmetricNE, prove the existence of Walras equilibrium of the underlying economy. Would notsuch a proof provide the “truly decentralized” counterpart of Arrow and Debreu’s (1954)seminal existence proof? There, existence of a competitive equilibrium obtained via thatof a (generalized) NE in some auxiliary game where an ad hoc auctioneer quoted pricesin order to minimize the value of excess demand. In the SMG approach, there would beno need for an auctioneer, and the existence of competitive equilibria would follow fromthat of strategic equilibria. Far from being a technical tool, SMG would provide a story of how the “invisible hand” really works, once the black box of general equilibrium theoryis opened. There is, however, a basic stumbling block in this programme: existence of apure NE in a basic Shapley–Shubik game is not that easy to obtain. The strategic outcomefunction is discontinuous and, since we are obviously concerned with non-trivial equilibria,

one needs to get rid of autarkic NE. Both problems are successfully tackled in the paper of this issue by Dubey and Geanakoplos (2003a) which makes good the proof just sketched.The authors consider a variant of the Shapley–Shubik game with inside fiat money, whosestrategic outcome function is continuous and strategy sets are compact; they prove existenceof pure NE, and convergence of NE to competitive equilibria under replication. They alsoprovide a simpler proof by directly considering a finite-type continuum of traders.

2.3. Arbitrage and imperfect competition

The various alternate ways of organizing trade that can be captured by means of a graph

G, as alluded to above, suggests that we should ask an even more basic question: can therebe several nodes for the same commodity? Why did we assume, from the very beginning,that there be a single trading-post per commodity? Consider a Shapley–Shubik game withcommodity-money as described in Section 2 supra: what would actually happen if we wereto open two distinct trading-posts for the same commodity? This is where the work byKoutsougeras (2003) in this issue, enters the picture. It was indeed shown by means of anexample in Koutsougeras (1999), that the “law of one price” can fail in such a situation.The reason for this apparent paradox can be grasped as follows: suppose you can trade goldagainst silver on two distinct trading-posts, and suppose that the relative price of gold (w.r.t.silver) is lower on trading-post B than on A. An immediate temptation is to sell a littlebit more gold on A, purchase the same amount of gold on B, and save the differential in

16 See Mas-Colell (1982) f or an early plaidoyer in favor of such a lower-hemi-continuity result.




terms of silver. This may fail because of the impact on prices induced by this speculativeoperation: by selling more gold on A, you induce a decrease of the relative price of gold onthis very trading-post (remember (2) supra);onthecontrary,bybuyinganadditionalamount

of gold on B, you increase its relative price on B. You might still think that by reducingyour speculative operation to sufficiently small quantities you might still take profit fromthe arbitrage opportunity between A and B, even though, because of your non-negligibleimpact on prices, your resulting benefit will be smaller than in a price-taking environment.But even this is false: by perturbing prices on both trading-posts, you not only diminish thenet benefit of your speculative operation, you also change the value of your whole trade.And this change may be so detrimental to you that there is actually no way to take advantageof the arbitrage opportunity! This shows a strong departure between perfect competitionand imperfect competition: in the first case, no-arbitrage lies at the heart of the very notionof equilibrium;17 in the second, equilibrium is perfectly compatible with what may look likean arbitrage opportunity, if one fails to take into account the strategic influence on prices.

The question that immediately comes to mind is then whether such pathological phe-nomena still survive when each player becomes negligible. Intuitively, one understandsthat, as your weight will get smaller, the impact of your speculative behaviour on prices willbecome infinitesimal, so that, in the limit, no arbitrage should survive. Otherwise, peoplewould eventually be able to take advantage of it. Notice, however, that this is not a trivialconsequence of the convergence result by Dubey and Shubik (1978) already mentioned:there, convergence is proven only for “nice” equilibria, while we are now interested in theconvergence towards no-arbitrage of all the (pure) NE. This convergence is indeed proven

by Koutsougeras (2003) in this issue.

2.4. Large economies and asymmetric information

Thinking of the analogy between Debreu and Scarf’s (1963) theorem and the variousasymptotic results concerning SMG that are available in the literature (including this is-sue), one might wonder whether an alternate route could not be taken. Instead of start-ing with finitely many players, why not start from the beginning with a continuum of agents, and prove the equivalence between strategic and competitive equilibria in a wayanalogous to Aumann’s (1964) core equivalence theorem? This was accomplished, for

the Shapley–Shubik model, in Dubey and Shapley (1994). One major obstruction for theequivalence has to be kept in mind: even whenever all the economic actors are negligible,the equivalence between Nash equilibria and Walras equilibria is not guaranteed. For thispurpose, the liquidity constraint

L−1k=1 bi

k ≤ ωi

L should not be binding. In other words, per-fect competition does not reduce to price-taking behaviour per se, but also requires perfectliquidity of money. Once proper account has been taken of the liquidity constraint how-ever, the equivalence between Nash and Walras obtains, as shown by Dubey and Shapley(1994) in atomless economies with trades being organized according to the Shapley–Shubikparadigm. A natural research programme consists then in extending this equivalence resultto the various alternate market games listed at the beginning of this paper. Being at theopposite extreme of the family of G-mechanisms, Shapley’s W-model is a good candidate

17 See Werner (1987) f or the role of the no-arbitrage assumption in the existence proof of competitive equilibria.




for this purpose. Indeed the previously mentioned equivalence result has been obtained inCodognato and Ghosal (2000).

The paper by Codognato and Ghosal (2003) of this issue points out yet another very

promising direction of research. What happens in SMG when asymmetric information isexplicitly modeled? One might conjecture that SMG offer the “right” setting where consid-erations related to the strategic transmission of private information on markets should beincorporated in a general equilibrium environment. To take but one example, let us mentionthe fascinating paper by De Meyer and Moussa Saley (2003), where the strategic use of private information in an infinitely repeated two-player auction is shown to be responsiblefor the emergence of the Brownian motion when the time interval goes to zero. One naturaltask, when introducing incomplete information in a general equilibrium setting with strate-gic players, consists in comparing the outcome of Bayesian–Nash equilibria with thosepredicted by Rational Expectations Equilibria (REE hereafter). This was done by Dubeyet al. (1987), and provided the occasion of a critique of the “rational expectations dogma”:the basic idea of REE is that prices should publicly convey enough relevant information fortraders to instantaneously complete their knowledge about the state of the economy; as aresult, REE must be (at least generically) fully revealing,18 and may even reveal informationthat is not held by anybody!19 Once plugged into a SMG, however, the same logic cannot benaively reproduced. This time, prices are the outcome of the players’ strategic interaction,hence can only reveal information that is already privately held by them. Moreover, playerscannot instantaneously digest this information as they first need to play, at least once, inorder to induce prices. This is why a sequence of trading is needed in Dubey et al. (1987)

in order to enable prices to play their informative role.Another approach to the same problem, however, is conceivable. Instead of letting playersretrade in order to use the information just revealed by prices, let them cheaply talk togetherwith the help of some communication device before entering the market. This is the mainidea underlying the paper by Forges and Minelli (1997), where an equivalence between such“communication equilibria”20 and REE obtains in the framework of the Shapley–Shubikgame. The paper by Codognato and Ghosal of this issue can now be viewed as a crossingof the two roads briefly described: there, an equivalence is proven between REEs and awell-chosen class of communication equilibria of Shapley’s W-model with a continuum of individuals.

3. Limit-prices

Thinking of traders’ orders, in everyday life, one sees that the strategies allowed in theclass of G-mechanisms are simple “market orders”. They specify how much a player isready to sell or to buy, whatever the resulting price of the commodities to be traded. Giventhe high volatility of prices on financial markets (one of whose underlying causes could be

18 See Radner (1979).19 cf. Grossman and Stiglitz (1980).20 For the seminal introduction of communication equilibria, see Forges (1986).




the strategic use of private information, as already alluded to21), it does not take long torealize that a much safer strategy consists in sending to the market an instruction statinghow much one is ready to trade at which price. This is a limit-price order. A market game

with such limit-price orders has been proposed by Dubey (1982) in a path-breaking contri-bution. By superposing infinitely many such limit-orders, every player can approximatelymimic her/his demand and supply curves, so that one could expect not only that each indi-vidual will feel safer, but also that the resulting equilibria will be closer, to the competitiveoutcomes. Indeed, it turns out, in the resulting game, that (provided there is positive tradein each commodity) the set of Nash equilibria always contains Walras equilibria, and thatthis set is close to being Walrasian—see Dubey (1982, 1994) for details.22 Nevertheless,this paper raised several questions. First, once limit-orders have been sent to the market,one needs to compute the aggregate supply and demand of each commodity and to find,if possible, some intersection point between the two resulting curves. This viewpoint ob-viously rests on the Shapley–Shubik paradigm, where each commodity is traded againstmoney (whether commodity or fiat ). But it does not encompass situations in which severalcommodities are moneys or near-moneys (such as the FX market discussed earlier). Indeed,in the W-mechanism, it is not obvious how to define intersection points between aggregatecurves of demand and offer. Another difficulty was also highlighted by Dubey (1982): incase several orders arrive at the same time, and for the same limit-price, but are such thatthey cannot all be fully executed, who should be served first? According to which rule?

These two questions are examined at a deep and fundamental level in the tour de force

by Mertens (2003) in this issue.

3.1. Linear exchange economies

Consider, for simplicity, a two-commodity world where Euros and Dollars are traded.The first observation to be made is that one could content oneself with selling limit-orders:if you want to buy Euros, then sell Dollars! Mertens’ (2003) central idea is then to view anorder to sell e$ Dollars as soon as (p /p$) ≥ (p+/p+$ ) in terms of an auxiliary “agent”whose “initial endowment” is nothing but the amount, e$, of Dollars put up for sale, andwhose “utility” is given by

u(D, E) := p+

$ D + p+

E

To put it differently, your order can be interpreted as the supply correspondence of anauxiliary trader with a linear utility, whose marginal rate of substitution between Euros andDollars is given by the relative limit-prices you set. Next, you may want to send several suchlimit-orders in order to mimic your “true” supply correspondence, and you may want to doso for each pair of commodities. This approximation technique may even require you tosend a continuum of orders. Putting your orders together with those of your opponents, oneends up with a fictitious economy endowed by a continuum of auxiliary traders with linearutilities. One obvious candidate for playing the role of Dubey’s (1982) intersection price

21 See De Meyer and Moussa Saley (2003).22 Notice, however, that players involved in this mechanism need not quote a whole schedule of prices and

quantities. It actually suffices for people to quote juste one price and one quantity at NE.




is, of course, the competitive equilibrium itself! But, apart from the fact that the resultingmechanism becomes more heavily centralized than Dubey’s (1982), one then encounters abasic difficulty: competitive equilibria of linear economies fail to exist in very elementary

situations23, and furthermore do not induce a unique final allocation in general, but rather awhole continuum of equilibrium points. The paper by Mertens (2003) is therefore devotedto the task of solving these two difficulties at once. The reader should bear in mind that itsmain purpose is “just” to define a solution concept for the extension to Shapley’s W-modelof the intuition underlying Dubey (1982). The existence problem is solved by extending thenotion of competitive equilibrium itself to that of a “pseudo-equilibrium”; the uniquenessproblem is circumvented by applying a proportional rule.

Last but not least, the limit-price mechanism elaborated in Mertens (2003) can be alter-natively viewed as an extension to a multi-commodity world with pairwise trading of thefamous double auction (in viewof the widespread opinion that double auctions are the ‘right’simplest way of organizing markets for two commodities).24 Mertens’ mechanism extendsdouble-auctions to multiple commodities, retaining the symmetry among commodities (i.e.without recourse to “money” as in Dubey (1982). Eventually, notice also that the family of G-mechanisms axiomatized by Dubey and Sahi (2003) in this issue can be plugged into thelimit-price mechanism: it suffices to prohibit selling-orders for pairs of commodities thatare not joined by a (positively oriented) arc.

3.2. Ruling out autarkic equilibria

Of course, having at one’s disposal the machinery of the limit-price mechanism, allthe questions raised for more familiar SMG come back: is it possible to eliminate trivialautarkic NE? Convergence towards competitive equilibria? Will arbitrage opportunities becompatible with equilibria? Will NE coincide with competitive ones in large economies?etc. The paper by Weyers (2003) in this issue deals with one first basic difficulty, namelythe elimination of no-trade equilibria in the SMG with limit-price orders derived fromMertens’ (2003) limit-price mechanism. This is the first inquiry of the properties of such amarket game, and it is the occasion for the author to provide us with characterizations of strategies and equilibria that will certainly prove useful for the further study of Mertens’mechanism. Regarding the elimination of no-trade equilibria, the basic intuition underlying

her approach can be introduced as follows. It seems clear, at least at first glance, thatautarkic NE should be ruled out by refining the notion of Nash equilibrium itself. Restrictingoneself to trembling-hand perfect equilibria for instance,25 should help getting rid of suchnon-generic situations where everybody is stuck to no-trade simply because everybodyexpected this to be so. The example by Cordella and Gabszewicz (1998) already mentioned,should serve, however, as a warning: a mere refinement of the NE concept cannot suffice byitself to eliminate such unwanted equilibria in a standard Shapley–Shubik game, as thereexist economies for which autarky is the unique NE (which will therefore be “stable” in

23

See David Gale (1957) for a seminal counter-example. Such pathologies, when interpreted in terms of limit-orders, cannot be ruled out: think of market orders as “agents” with non-positive utilities.24 See, e.g. Rustichini et al. (1994) in this spirit, as well as the references therein.25 cf. Selten (1975).




whatever sense one might define strategic stability). In Mertens’ mechanism, as studied byWeyers (2003), there is however some further hope for being able to get rid of autarkicNE due to the presence of additional limit-orders. Nonetheless, to make good this promise

requires a great deal to be done at least because the extension of Selten’s trembling-handperfection to games with infinitely many pure stragegies is not obvious at all. 26

The escape route found by Weyers consists in replacing the trembling-hand idea by someapparently weaker refinement, namely the elimination of weakly dominated strategies.27

Weyers (2003) in this issue shows that the situation is still more complicated than could havebeen expected: one round of elimination of (weakly) dominated strategies does not sufficeto get rid of autarkic equilibria. In a companion paper, however, Weyers (2000) showsthat a second round will do the job, and that, asymptotically, the standard convergenceresult towards competitive equilibria holds for her SMG with limit-prices. Thus, the paperpublished in this issue is the first part of a larger research programme whose burden is toshow that two rounds of elimination of (weakly) dominated strategies suffices in order torule out autarkic equilibria. In this first part, the author essentially shows that one round isnot enough. Putting together the two papers by Weyers (2000, 2003), one gets this time theexact analogue of Debreu and Scarf’s (1963) convergence result, for, even in the finite case,every competitive equilibrium of the underlying economy always belongs to the set of NEallocations of the limit-price market game.28 Thus, Weyers (2000) convergence result showsthat the set of NE allocations that survive two rounds of elimination of dominated strategiesshrinks to that of Walras equilibria as players’ price-manipulating power vanishes.

4. Alternate market games

As already said, G-mechanisms are the prominent paradigm for studying (im)perfectcompetition within a general equilibrium framework. As suggested by the variants tai-lored by Dubey and Geanakoplos (2003a) and Mertens (2003) however, they need not beour last word in terms of modeling trading mechanisms: alternate mechanisms should bestudied, and their properties compared with those of the Shapley–Shubik-like ones. It iswell known, indeed, that Shapley–Shubik games share surprising properties (in additionto the failure of the ‘law of one price’ illustrated in this issue): generically, all their NE

allocations are Pareto-suboptimal (and not just the autarkic NE).29 Still generically, theyconstitute a full-dimensional submanifold of the feasible set.30 Even if markets are com-plete and traders risk-adverse, they are not immune to sunspots or other forms of “animal

26 cf. e.g. Simon and Stinchcombe (1995).27 As is well known, every trembling-hand perfect equilibrium is admissible in finite games, so that admissibility

(i.e. the fact that no weakly dominated strategies should be involved in the play with a positive probability) lookslike a weakening of Selten’s criterion. Notice, however, that in infinite games, perfect equilibria need not beadmissible, so that Weyers’ criterion is actually not comparable with perfection.28 This is yet another virtue of limit-price orders. As already said, this inclusion is not satisfied in standard

Shapley–Shubik games.29 And this is not due to the lack of regularity of the Shapley–Shubik strategic outcome function, see Dubey andRogawski (1990) and Aghion (1985).30 cf. Peck et al. (1992).




spirits”.31 Do all these properties sharply departing from the perfectly competitive picture,characterize imperfect competition as such? Or, are they peculiar to the G-mechanism mod-eling option? To answer this question, alternate market mechanisms need to be studied. It

is in this spirit that the four last papers of this issue, Giraud and Stahn (2003), Dubey andGeanakoplos (2003b), Bottazzi and De Meyer (2003), and Germano (2003), offer alternateways of modelling (im)perfect competition in a game-theoretic manner.

There is yet an orthogonal perspective from which SMG can be questioned: we sawin Mertens (2003) that the design of a well-behaved mechanism may be an impetus forsolving problems that were left open by standard general equilibrium theory, such as thenon-existence and/or multiplicityof competitive equilibria in linear economies.32 Since theyexplicitlydescribehowthe“invisiblehand”operates,SMGcouldevenenableustodealwithproblems where, because of its short-cut, the standard general equilibrium theory fails topredict in a “sensible” way how the invisible hand behaves. Think of introducing money intogeneral equilibrium theory, Pareto-improving equilibria with incomplete markets, studyingthe impact of various tax systems on investors, modeling markets where no central houseis available (such as the labour market) . . . . Again, the papers of this issue we are going tointroduce now pave the road in this direction.

4.1. Imperfect competition and incomplete markets

In his Introduction to the special issue of JME devoted to incomplete markets, one decadeago, Geanakoplos (1990) suggested that some of the properties of competitive equilibria

that distinguish incomplete from complete markets—i.e. inefficiency, indeterminacy andnon-immunity to sunspots, inter alia—could probably be obtained in the set-up of strategicmarket games. During the elapsed time, experience has shown this intuition to be propheticsince, as already said, NE of Shapley–Shubik games do share all these properties in com-mon with general equilibrium with incomplete markets (GEI).33 A challenging questionthen arises as to whether the situation is equally “bad” (in terms of welfare) in both settings,or if one of them could not be somehow distinguished from the other. Giraud and Stahn(2003) in this issue provide a partial answer to this question within the set-up of a two-periodGEI economy populated by finitely many agents trading nominal assets. Suppose indeedthat traders can observe the outcome of trades in the first-period financial markets before

entering the spot markets of the second-period. Then, the authors show, there exist NE thatPareto-dominate all the GEI equilibria. Moreover, certain strategic equilibrium outcomesmay be second-best optimal. The reason for this is quite simple if one has in mind the widelyknown Folk theorems of the repeated games literature: if the behaviour of players in thefirst-period can be observed, then retaliation threats can be enforced that provide the correctincentives to players for trading first-period assets in an “efficient way”. In some sense,

31 Cass and Shell (1983).32 Actually, non-existence has been dealt with through various generalizations of the concept of competitive

equilibrium obtained by means of some kind of lexicographic hierarchy of prices, see Gay (1978), Danilov and

Slotskov (1990), and Florig (1998). It is not innocuous to observe that these lines of research seem to convergewith the “solution” proposed by Mertens (2003) in this issue.33 See the beginning of Section 4 above. For the generic second-best inefficiency of GEI competitive equilibria,

see Geanakoplos and Polemarchakis (1986), as well as Geanakoplos et al. (1990).




observation of the first-period plays a role analogous to the communication device intro-duced in Codognato and Ghosal (2003): both help players to communicate. The differenceisthatin Codognato and Ghosal (2003) information is asymmetric and the device enables to

share privately known secrets, while in Giraud and Stahn (2003), information is symmetricfrom the beginning, and monitoring helps players to coordinate. A similar idea is used in acompanion paper by Giraud and Weyers (2001), dealing with subgame-perfect equilibria of aShapley–Shubikgamewitharbitrarilylong(butfinite)horizon.Thedifference,here,isthatthe two-period length of the horizon is a priori fixed. Another difference lies in the marketmechanism itself. Having understood that imperfect competition with non-trivial monitor-ing can “beat” (in terms of welfare) perfectly competitive GEI equilibria, one could wonderwhether a specific mechanism could not be tailored, whose outcomes would coincide withthe classical Walras equilibria when the “invisible hand” operates in a satisfactory way, andwould circumvent the conundrums associated with GEI equilibria, whenever the “invisiblehand” goes wrong. This is why the mechanism considered in Giraud and Stahn (2003) isconstructed so as to share the following property: in a standard Arrow–Debreu economywith complete markets, the set of NE outcomes exactly coincides with that of competitiveequilibria. This property, of course, is not satisfied by the standard Shapley–Shubik gameused, by contrast, in Giraud and Weyers (2001).

This is not the first time that dynamical aspects of trades help circumventing some dis-tressing aspects of static models of trades. The paper by Dubey et al. (1987) has alreadybeen cited, but other instances are also of importance in this context. For example, a so-lution to the liquidity problem embodied in the cash-in-advance constraint of (1) has been

proposed by Dubey et al. (1993) by allowing a continuum of agents, who do not discountthe future, to reopen trading-posts before they consume their final allocations. The authorsshow that competitive equilibria can be achieved by allowing for artitrarily many (albeitfinite) rounds of (re) trade. In an analogous vein, Ghosal and Morelli (2001) have suggestedmore recently that Pareto-optimal allocations can be approximated when infinitely manyrounds of retrading are allowed, and finitely many myopic agents who discount the futuremeet at each round. Ghosal and Morelli then show that the converging sequence of myopictrades can be supported by some subgame-perfect equilibrium path when traders antici-pate future rounds of retrading. But of course there are many other (typically inefficient)subgame-perfect equilibria. Finally, introducing dynamics in SMG opens yet another re-

search avenue when mixed with the perfect competitiveness of large economies, namelythat of multi-period market games with a continuum of traders whose individual deviationare not visible (see Dubey and Kaneko, 1984, 1985; Karatzas et al., 1994).

4.2. Monetary equilibria

As we just saw, SMG may help to cut some Gordian knots of standard GEI. But, whileGiraudandStahn(2003) deal with the efficiency issue, there are many other questions raisedby GEI, such as the non-existence problem34 or the real indeterminacy of the equilibrium

34

Existence can fail in GEI economies with real asset, see Hart (1975). Thanks to Duffie and Shafer (1985),we know that this problem generically never occurs in pure exchange GEI economies. However, it was recentlydemonstrated by means of an example, by Momi (2001), that robust non-existence can occur once production isincorporated into GEI economies.




manifold.35 A dramatic difference occurs once inside (borrowable) and outside (endowed)money is introduced into the two-period GEI framework, as in Dubey and Geanakoplos(2003b) in this issue. Money is fiat and yields no direct utility. But it is a universal medium

of exchange and its stock, at any given time, has a crucial bearing on trade. The modeloperates roughly as follows. Each period 1 and 2 is divided into three stages:

(i) A trader acquires inside money at the beginning of each period by taking out short(and, in period 1, also long) loans from the bank, in addition to the outside money s/hemay already be holding.

(ii) In a second stage, s/he purchases commodities (and, in period 1 also real assets) after(in period 2) delivering on the prior sale of assets.36 But these purchases and deliveriesmust be financed out of money on hand (the Clower (1967) constraint).

(iii) The trader must then settle debts on bank loans, netting them against the revenue from

commodity sales, asset deliveries (on prior purchase of assets), and bank deposits.In this much richer formulation, the reader will find a kind of recapitulation of several

ideas in the preceding papers of this issue, but with their loose ends tied up in a fully closedmodel. What Dubey and Geanakoplos (2003b) do is to plug the analogue of an extension of the TP-mechanism à la Shapley–Shubik into a two-period GEI economy. Now, money canbe exchanged against every commodity and every asset, but not all commodities or assetscan be traded directly against each other (the missing markets hypothesis) giving scope tomoney to unleash gains from trade. (It is permitted, nevertheless, that certain assets maytrade against commodities, thereby incorporating credit cards as a special case.) The model

is firmly in the SMG vein, with agents’ actions forming prices as in (2) above. But sincethere is a continuum of agents, price-taking behaviour is induced, and the complicationsthat arise when agents can condition their actions on past observations (as in Giraud andStahn, 2003) are overcome.37

First, existence of monetary equilibrium (ME) is established, provided there are sufficientgains to trade via money. This offers a first answer to the puzzle raised by non-existenceof GEI: ME exists even when the underlying GEI does not. The authors even provide afascinating reinterpretation of the non-existence of GEI in terms of Keynes’ celebrated“liquidity trap”. This suggests at least, that general equilibrium and game theory, albeitapparently very far away from macroeconomic worries, may have something substantial to

tell to macroeconomists. The existence proof itself is reminiscent of the trick used in Dubeyand Geanakoplos (2003a) in this issue: in order to get rid of no-trade equilibria, a “dummyplayer” is added, who is asked to put some ε > 0 on each trading-post, in order to makesure that the economy will not remain stuck at the no-trade issue.38 Existence then followsfrom a fixed-point theorem on the strategy spaces of players, provided that there are enough

35 See Geanakoplos and Mas-Colell (1989) and Balasko and Cass (1989).36 Actually, the sequence of events can be maneuvered quite freely in this model, with assets delivering before of

after commodities. Here, the sequence the authors opted for has assets deliveries simultaneous with commoditytrade in period 2.37

See Dubey and Kaneko’s Anti-Folk theorem (1984, 1985) for a general statement proving that, when players’deviations are negligible, the precise monitoring structure across time periods becomes irrelevant.38 Needless to say, this additional “tremble” is analogous to, and serves the same purpose as, Weyer’s weakening

of the restriction to Selten’s perfect equilibria, see Section 3.2 supra.




gains-to-trade at autarky relative to the outside/inside money ratio. Of course, this existenceresult also solves Hahn’s famous paradox according to which, in a finite horizon economy,a simple backward induction reasoning implies that, in equilibrium, outside money should

have no value.But the purpose of recasting GEI in terms of an SMG with money, is not solely to restore

existence and to prove that money may have value in equilibrium. It also provides a neatframework in which money can be shown to be non-neutral, and consequently monetarypolicies are effective.39 Moreover, as is recalled by Dubey and Geanakoplos (2003b), andproven in their companion work,40 this also enables to remove both the real and nominalindeterminacies of equilibria. This, of course, contrasts with the strong real indeterminacyof GEI, already alluded to.41

4.3. Bankruptcy and financial instability

One major short-cut of standard general equilibrium theory is the impossibility of go-ing bankrupt. Since economic agents instantaneously digest all the relevant information athand, perfectly anticipate prices and solve with absolute accuracy their budget-constrainedutility-maximization programme, there is little room for bankruptcy. Arrow and Hahn(1971), among many others, did nevertheless provide an early treatment of bankruptcyphenomena within a general equilibrium model of Keynesian inspiration. But the conclu-sion of their analysis was, roughly, that the possibility of bankruptcy can induce a majormarket failure, namely can prevent the existence of equilibria. When cast in a SMG mould,

the picture is radically altered. Since out-of-equilibrium situations are now explicitly de-scribed, bankruptcy can be incorporated into the model, even though it may never appearin equilibrium. This is done, e.g. in Giraud and Stahn (2003) in the present issue.

Nothing, however, prevents one from going one step further, and considering situationswhere bankruptcy occurs in equilibrium. In situations where equilibria are not efficient(such as in GEI models), it is not even clear whether bankruptcy should really be detrimen-tal in welfare terms. These questions have been explored in the path-breaking papers byShubik (1972), Dubey and Geanakoplos (1992), Dubey et al. (1989/2000) and Geanakoploset al. (2000) following ideas contained in Shubik and Wilson (1977). There, bankruptcypenalties are introduced in the form of punishments (directly expressed in terms of util-ities) that are linear functions of the outstanding debt.42 The first striking finding is thatbankruptcy may occur in equilibrium without implying the break-down of the whole econ-omy: players may indeed strategically decide to go bankrupt because, from their point of view, enduring the bankruptcy penalty is less detrimental than delivering the promised pay-offs. Second, it turns out that, when markets are incomplete, more lenient penalties mayinduce a Pareto-improvement of welfare in equilibrium! One tentative “explanation” for thisparadox is that imposing less harsh punishments for players kept in the red is tantamount

39 Unless the economy falls into Keynes’ liquidity trap . . .40

See Dubey and Geanakoplos (1994).41 This result also departs from the parallel findings obtained in analogous, but distinct, models of monetaryeconomies built by Dreze and Polemarchakis (2000, 2001) where there is only inside money.42 Actually, linearity is only assumed for simplicity.




to introducing “new assets” (namely those whose promise is only partially made good byplayers going bankrupt). Thus, to a certain extent, allowing bankruptcy in equilibrium mayhave the same, ambiguous effect as financial innovation.43

The next paper of this issue, Tsomocos (2003), furthers the previous paper on GEI withmoney (Dubey and Geanakoplos, 2003b) by adding the possibility of bankruptcy. Thislast feature is captured in the same way as in Dubey et al. (1989/2000), so that strate-gic bankruptcy may survive in equilibrium. Following Dubey and Geanakoplos (1992),Tsomocos mixes both bankruptcy and money in a single model. However, while Dubey andGeanakoplos (1992) confine themselves to a (complete markets) one-shot economy, Tsomo-cos extends their approach to a two-period GEI model. Thus, his framework can be viewedas a combination of Dubey and Geanakoplos (1992, money+ bankruptcy+ complete mar-kets), Dubeyetal.(2000,bankruptcy+ GEI) and Dubey and Geanakoplos (2003b, money+GEI).

Moreover, Tsomocos (2003) adds yet a third, orthogonal dimension to his approach, byintroducing a sector of commercial banks together with a Central Bank into his model. Thistime, prices and finaloutcomes (ofboth commodities and financial assets) are thus formed bythe choices of households and commercial banks. The banking system is captured in a spiritsimilar to the work by Shubik (1972), Shubik and Tsomocos (1992): banks borrow frominvestors and from the Central Bank via the interbank credit market; they extend credit tohouseholds (which are identified with investors) via the consumer credit markets. They alsohold a diversified portfolio of securities. Heterogeneous banks differ with respect to initialcapital endowments, risk preferences (i.e. coefficients of risk aversion) and assessments

of future scenarii (i.e. subjective probabilities). Thus, many crucial issues related to theValue at Risk of commercial banks as well as risk control and portfolio management (andtheir possibly conflicting burdens) can be potentially studied within this framework. Last,but not least, since bankruptcy (of commercial banks) may occur at equilibrium, financialinstability can now be examined by the author, with an eye towards the impact of the capitalrequirements of the new Basel accords.44

4.4. Linear oligopolies

As already said, the main toolkit used by Mertens (2003) in order to deal with books of

limit-price orders consists in interpreting them as linear economies. The paper by Bottazziand De Meyer (2003) in this issue also deals with such linear economies, of which it pointsout another particular aspect. Suppose that investors interacting on financial markets arerisk-neutral, and aim at maximizing their after-tax payoff. If they were all subject to thesame rate of taxation, they would all have identical, linear utilities. But, since they willpresumably have distinct tax rates, this might lead to potential arbitrage opportunities, forall of them will face a unique, final trading price. Suppose, moreover, that only two assetsare to be traded by two agents, and that one of the two traders owns almost all the initialendowments available on the markets. As can be readily seen on an Edgeworth box, it is

43 I have been informed of this interpretation by Jean-Marc Bonnisseau; see also Dubey et al. (1989/2000). Fora “parallel” analysis in terms of financial innovation, see, e.g. Elul (1995).44 See Catarineu-Rabell et al. (2003).




likely that this “dominating” agent will take no advantage from the (generically unique)competitive equilibrium price that would emerge if the two competitors were price-takers:its indifference line crossing the initial endowment point will quite often coincide with its

budget line. This means that assuming that such a dominating player will accept to entersuch a (perfectly competitive) game is not realistic (unless it is equipped with a strong,altruistic feeling!). The problem faced in Bottazzi and De Meyer (2003) is therefore toconstruct a SMG where the arbitrage issue due to distinct tax regimes can be dealt with, andwhere a “dominating” player will indeed exert monopolistic power. The authors then model‘oligopolistic’ behaviour, where several ‘dominating’ players compete on an otherwiseperfectly competitive market.

Identifying the ‘dominating’ players with ‘firms’, this way of formulating the problemis reminiscent of the Cournot–Walras approach, as first captured in the seminal paper byGabszewicz and Vial (1972) for instance. As suggested by Mertens (2003) in this issue, thebestwaytomodeloligopolisticbehaviourisprobablytoassumethatfirmsfirstsetquantities,while,second,apriceemergesontheconsumptionmarkets.Ofcourse,asiswellknown,oneencounters severe difficulties in trying to carry over this programme: given the quantities putup for sale in the first-period, there is not necessarily uniqueness of the second-period priceequilibrium outcome. As a consequence, the firms’ payoff function in the first-period is notwell defined—or, at least, is not a function. Moreover, even if uniqueness was guaranteed,this payoff function need not be quasi-concave (so that existence of a subgame-perfectequilibrium would become a problem, unless mixed strategies were allowed).45

The paper by Bottazzi and De Meyer (2003) included in this issue can be read along

analogous lines: a market game is designed where ‘dominating’ players set quantities anda price emerges that equalizes the aggregate quantities with the demand arising from the‘dominated’ players (endowed with linear utilities). Once again, all the questions previouslyasked can be examined for this new market game: existence, convergence . . . . The authorsprovide a first couple of results: (1) with finitely many players, a non-trivial oligopolisticequilibrium may be Pareto-optimal but need not be so;46 (2) when the number of competingagents tends to infinity, the set of NE converges to (a subset of) competitive equilibria—sothat we no more need to worry about autarkic equilibria in order to reach the asymptoticequivalence result.

4.5. Random matching and non-equivalence

The last, but not least, contribution of this issue, Germano (2003) can be introducedfrom an angle that is yet orthogonal to the perspectives adopted in all the previous papers,thus proving the versatility of SMG. Search theory is becoming today one of the dom-inant paradigms in explaining micro-labor market phenomena.47 The basic idea is quitefamiliar: workers search for vacant jobs, and employers search for unemployed workers,

45 In Gabszewicz and Vial (1972), uniqueness and quasi-concavity were built-in assumptions. On this issue, see

also Stahn (1993).46 This contrasts with Shapley–Shubik games and with Germano (2003) in this issue, where NE are efficient if and only if they are trivial, i.e. reduce to no-trade.47 See, e.g. Mortensen and Pissarides (1999) f or a survey.




or possibly workers seeking to change jobs. Very often, the conclusion of this kind of in-quiry is that the strategic outcome is far from the perfectly competitive solution.48 To thebest of our knowledge, however, there is no attempt so far to build a bridge between the

literature on SMG and matching models. Germano (2003) in this issue, can be viewed asproviding a first insight about such a bridge. Indeed, a key feature of the market gamedesigned and analyzed by Germano is the way trade takes place. There are two mainstages: in the first-stage agents simultaneously choose prices and quantities of commodi-ties (e.g. job positions) they want to sell.49 In the second stage, they sequentially enter themarkets as buyers, and choose only quantities of commodities they want to buy. More-over, before entering the second-stage market, players are chosen at random, so that theorder in which they enter is not known ex ante. On the other hand, each entrant strate-gically chooses with whom s/he wants to trade (the commodities that have not alreadybeen bought by those who were selected before). Germano shows that no strategic equilib-rium outcome is Pareto-efficient, unless initial endowments were already Pareto-efficient.In the latter case, the unique outcome is no-trade, but this situation generically neveroccurs.

It is noteworthy that this property—no strategic equilibrium is efficient unless initial en-dowments were already so—is also shared by Shapley–Shubik market games. Thus, the firstlesson to be drawn from this work is that a key message proffered by Shapley–Shubik gamescan be restated within frameworks that sharply differ from the seminal TP-mechanisms ap-proach. In particular, this is rather thought-provoking for labor market economic theory. Asecond lesson emerges from the comparison with Giraud and Stahn (2003): there, players

also quote prices and send quantity signals, but the outcome is drastically different. One easyconsequence of the results obtained in both papers is that, whenever markets are complete,the intersection of their NE outcome sets is generically empty. In other words, the way inwhich players enter a market, and the precise timing of interactions may drastically changethe outcome. This last feature is obviously specific to the imperfectly competitive world, forthe perfectly competitive setting (at least, as it is modeled in standard general equilibriumtheory) is not sensible to this kind of institutional “detail” (see, in particular, Dubey et al.(1980)) where it is shown that, with a continuum of agents, different mechanisms induceoutcomes equivalent to the Walrasian solution. Furthermore, in a complete market setting,for instance, the equivalence between Arrow-securities and Debreu’s contingent commodi-

ties nicely expresses how perfect competition is blind with respect to the precise way tradesare organized. By contrast, the equivalence between a complete set of Arrow-securities andDebreu’s contingent commodities fails in Shapley–Shubik games with fiat money, as wasshown by Peck and Shell (1989) and Weyers (1999).

May Germano’s contribution, as well as all the preceding ones, convince the reader thatSMG represent not only a versatile tool for dealing with issues related to imperfectly com-petitive market phenomena, but also a central and promising area for research in economictheory as such.

48

On the other hand, Douglas Gale (2000) tries to provide a game-theoretic rationale for the convergence of such matching models towards perfect competition.49 Thus, the model by Germano (2003) f urthers the Cournot–Bertrand line considered, e.g. in Dubey (1982),

Mertens (2003) or Giraud and Stahn (2003).




Acknowledgements

I wish to thank without implicating Prof. Pradeep Dubey, Jean-Marc Bonnisseau, John

Geanakoplos and Sonia Weyers, whose comments enabled considerable improvements inthis presentation.

References

Aghion, P., 1985. On the generic inefficiency of differentiable market games. Journal of Economic Theory 37,126–146.

Amir, R., Sahi, S., Shubik, M., Yao, S., 1990. A strategicmarketgame with complete markets. Journal of EconomicTheory 51, 126–143.

Arrow, J.K., Debreu, G., 1954. Existenceof an equilibrium fora competitiveeconomy. Econometrica, 22,265–290.Arrow, J.K., Hahn, F., 1971. General Competitive Analysis. Holden Day (Chapter 14).Aumann, R., 1964. Markets with a continuum of traders. Econometrica 32, 39–50.Balasko, Y., Cass, D., 1989. The structure of financial equilibrium with exogenous yields: the case of incomplete

markets. Econometrica 57, 135–162.Biais, B., Rochet, J.-C., 1997. Risk Sharing, Adverse Selection and Market Structure. Working Paper, Institut

d’économie Industrielle, Toulouse.Bottazzi, J.-M., De Meyer, B., 2003. A market game for assets and types of taxed investors. This issue.Cass, D., Shell, K., 1983. Do sunspots matter? Journal of Political Economy 91 (2), 193–227.Catarineu-Rabell, E., Jackson, P., Tsomocos,D.P., 2003. Procyclicality and theNewBaselAccord—Banks’Choice

of Rating System. WP Series No. 181, Bank of England.Clower, R., 1967. A reconsideration of microfoundations of monetary theory. Western Economic Journal 6, 1–8.

Codognato, J., Ghosal, S., 2000. Cournot–Nash equilibria in limit-exchange economies with complete marketsand consistent prices. Journal of Mathematical Economics 34, 39–53.

Codognato, J., Ghosal, S., 2003. Self-fulfilling mechanisms and rational expectations in markets with a continuumof traders. This issue.

Cordella, T., Gabszewicz, J.-J., 1998. ‘Nice’ trivial equilibria in strategic market games. Games and EconomicBehavior 22, 162–169.

Danilov, V.I., Slotskov, A.I., 1990. A generalized economic equilibrium. Journal of Mathematical Economics 19,341–356.

Debreu, G.,Scarf,H., 1963. A limit theorem onthecore ofan economy. International EconomicReview, 4, 235–246(reprinted in Debreu (Ed.), 1996. General Equilibrium Theory, vol. I. Brookfield, Ashgate, pp. 330–341.

De Meyer, B., Moussa Saley, H., 2003. On the Strategic Origin of Brownian Motion in Finance. InternationalJournal of Game Theory 31 (2), 285–319.

De Michelis, S., Germano, F., 2000. Private communication.Drèze, J., Polemarchakis, H., 2000. Intertemporal general equilibrium and monetary theory. In: Leijonhufvud, A.

(Ed.), Monetary Theory as a Basis for Monetary Policy. Macmillan, New York.Drèze, J., Polemarchakis, H., 2001. Monetary equilibria. In: Gérard Debreu, W., Neuefeind, W., Trockel (Eds.),

Economics Essays—A Festschifrt for Werner Hildenbrand. Springer, Berlin.Dubey, P., 1980. Nash equilibria of market games: finitess and inefficiency. Journal of Economic Theory 22 (2),

363–376.Dubey, P., 1982. Price–quantity strategic market games. Econometrica 50, 111–126.Dubey, P., 1994. Market games: a survey of recent results. In: Mertens, J.-F., Sorin, S. (Eds.), Game-Theoretic

Methods in General Equilibrium Theory. NATO ASI Series, Kluwer Academic Publisher, Dordrecht.Dubey, P., Geanakoplos, J., 1992. The value of money in a finite-horizon economy: a r ole for banks. In: Dasgupta,

P., Gale, D., et al. (Eds.), Economic Analysis of Markets and Games. MIT Press, Cambridge, pp. 407–444.Dubey, P., Geanakoplos, J., 1994. Real Determinacy with Nominal Assets. mimeo, Yale University, forthcomingas Cowles Foundation Discussion Paper, May 2003.

Dubey, P., Geanakoplos, J., 2003a. From Nash to Walras via Shapley–Shubik. This issue.




Dubey, P., Geanakoplos, J., 2003b. Monetary equilibrium with missing markets. This issue.Dubey, P., Kaneko, M., 1984. Information patterns andNash equilibria in extensive games. Mathmatical and Social

Sciences 8 (2), 111–139.

Dubey, P., Kaneko, M., 1985. Information patterns and Nash equilibria in extensive games II. Mathmatical andSocial Sciences 10 (3), 247–262.Dubey, P., Rogawski, J.D., 1990. Inefficiency of smooth market mechanisms. Journal of Mathematical Economics

19, 285–304.Dubey, P., Sahi, S., 2003. Price-mediated trade with quantity signals: an axiomatic approach. This issue.Dubey, P., Shapley, L., 1994. Noncooperative exchange with a continuum of traders: two models. Journal of

Mathematical Economics 23, 253–293 (Cowles Foundation Discussion Paper, 1977).Dubey, P., Shubik, M., 1978. The noncooperative equilibria of a closed trading economy with markets supply and

bidding strategies. Journal of Economic Theory 17, 1–20.Dubey, P., Geanakoplos, J., Shubik, M., 1987. The revelation of information in strategic market games. Journal of

Mathematical Economics 16, 105–137.Dubey, P., Geanakoplos, J., Shubik, M., 1989, 2000. Default in a General Equilibrium Model with Incomplete

Markets. Cowles Foundation DP 879R, 1247, Yale University, Yale.Dubey, P., Mas-Colell, A., Shubik, M., 1980. Efficiency properties of strategic market games: an axiomatic

approach. Journal of Economic Theory 22, 339–362.Dubey, P., Sahi, S., Shubik, M., 1993. Repeated trade and the velocity of money. Journal of Mathematical

Economics 22, 125–137.Duffie, D., Shafer, W., 1985. Equilibrium in incomplete markets. I. Basic model of generic existence. Journal of

Mathematical Economics 14, 285–300.Elul, R., 1995. Welfare effects of financial innovation in incomplete market economies with several consumption

goods. Journal of Economic Theory 65 (1), 43–78.Florig, M., 1998. A note on different concepts of generalized equilibrium. Journal of Mathematical Economics

19, 245–254.

Forges, F., 1986. An approach to communication equilibrium. Econometrica 54, 1376–1385.Forges, F., Minelli, E., 1997. Self-fulfilling mechanisms and rational expectations. Journal of Economic Theory75, 388–406.

Gabszewicz, J.-J., Vial, J.-P., 1972. Oligopoly à la Cournot in a general equilibrium analysis. Journal of EconomicTheory 4, 381–400.

Gale,D.,1957.PriceEquilibriumforLinearModelsofExchange.TechnicalReportP-1156,TheRandCorporation.Gale, D., 2000. Strategic Foundations of General Equilibrium: Dynamic Matching and Bargaining Games.

Cambridge University Press, Cambridge, MA.Gay, A., 1978. The exchange rate approach to general economic equilibrium. Économie Appliquée, Archives de

l’I.S.M.E.A. XXXI (1–2), 159–174.Geanakoplos, J., 1990. An introduction to general equilibrium with incomplete asset markets. Journal of

Mathematical Economics 19, 1–38.

Geanakoplos, J., Mas-Colell, A., 1989. Real indeterminacy with financial assets. Journal of Economic Theory 47,22–39.

Geanakoplos, J., Polemarchakis, H., 1986. Existence, regularity and constrained suboptimality of competitiveportfolio allocations when asset markets are incomplete. In: Heller, Starr, Starret (Eds.), Equilibrium AnalysisEssays in Honor of J.K. Arrow, vol. 3. Cambridge University Press, Cambridge (Chapter 3).

Geanakoplos, J., Drèze, J., Magill, M., Quinzii, M., 1990. Generic inefficiency of stock market equilibrium whenmarkets are incomplete. Journal of Mathematical Economics 19, 113–151.

Geanakoplos, J., Karatzas, I, Shubik, M., Sudderth, W., 2000. A class of strategic market games with activebankruptcy. Journal of Mathematical Economics 34, 359–396.

Germano, F., 2003. Bertrand–Edgeworth equilibria in finite exchange economies. This issue.Giraud, G., Stahn, H., 2003. Efficiency and imperfect competition with incomplete markets. This issue.

Giraud, G., Weyers, S., 2001. Strategic market games with a finite horizon and incomplete markets. INSEAD WP2001/23/EPS. Economic Theory, in press.Ghosal, S., Morelli, M., 2001. Retrading in Market Games, mimeo.Green,E.,1980.Non-cooperativeprice-takinginlargedynamicmarkets.JournalofEconomicTheory22,155–182.




Grossman, S., Stiglitz, J., 1980. On the impossibility of informationnally efficient markets. American EconomicReview 70 (3), 393–408.

Hart, O.D., 1975. On theexistence of equilibrium in a securities market. Journal of Economic Theory11, 418–443.

Karatzas, Y., Shubik, M., Sudderth, W., 1994. Stationary Markovian equilibrium for a strategic market game.Mathematics of operations Research 19 (4), 975–1706.Koutsougeras, L., 1999. A Remark on the Number of Trading-Posts in Strategic Market Games. Working-Paper.

Tilburg University, Tilburg.Koutsougeras, L., 2003. Convergence to no-arbitrage equilibria in market games. This issue.Mas-Colell, A., 1982. The Cournotian Foundations of Walrasian Equilibrium Theory: an exposition and recent

results. In: Hildenbrand, W. (Ed.), Advances in Economic Theory. Cambridge University Press, Cambridge.Mertens, J.-F., 2003. The Limit-price mechanism. This issue.Mortensen, D., Pissarides, C., 1999. New developments in models of search in the labor market. In: Ashenfelter,

O., Card, D. (Eds.), Handbook of Labor Economics, vol. 3B. Elsevier, Amsterdam, Chapter 39, pp. 2567–2627.Momi, T., 2001. Non-existence of equilibrium in an incomplete stock market economy. Journal of Mathematical

Economics 35, 41–70.Peck, J., Shell, K., 1989. On the nonequivalence of the Arrow-securities game and the contingent-commodities

game (Part 1). In: Barnett, W., Geweke, J., Shell, K. (Eds.), Economic Complexity: Chaos, Sunspots, Bublesand Nonlinearity. Cambridge University Press, New York, pp. 61–85.

Peck, J., Shell, K., 1990. Liquid markets and competition. Games and Economic Behavior 2, 362–377.Peck, J., Shell, K., 1991. Market uncertainty: correlated and sunspot equilibria. Review of Economic Studies 58,

1011–1029.Peck, J., Shell, K., Spear, S., 1992. The market game: existence and structure of equilibrium. Journal of

Mathematical Economics 21, 271–299.Postlewaite, A., Schmeidler, D., 1978. Approximate efficiency of non-Walrasian–Nash Equilibria. Econometrica

46, 127–135.Radner, R., 1979. Rational expectations equilibrium: generic existence, and the information revealed by prices.

Econometrica 47 (3), 655–678.Roberts, K., 1980. The limit points of monopolistic competition. Journal of Economic Theory 22, 256–278.Rustichini, A., Satterthwaite, M.A., Williams, S.R., 1994. Convergence to efficiency in a simple market with

incomplete information. Econometrica 62 (5), 1041–1063.Selten, R., 1975. Reexamination of theperfectness concept forequilibriumpoints in extensive games. International

Journal of Game Theory 4, 25–55.Shapley, L., 1976. Noncooperative general exchange.In: Lin, S.A.Y. (Eds.), Theory and Measurement of Economic

Externalities. Academic Press, Orlando, FL, pp. 155–175.Shapley, L., Shubik, M., 1977. Trade using one commodity as a means of payment. Journal of Political Economy

85, 937–968.Sahi, S., Yao, S., 1989. The non-cooperative equilibria of a trading economy with complete markets and consistent

prices. Journal of Mathematical Economics 18, 315–346.Shubik, M., 1972. Commodity-money, oligopoly, credit and bankruptcy in a general equilibrium model. Western

Economic Journal 10, 24–38.Shubik, M., Tsomocos, D.P., 1992. A strategic market game with a mutual bank with fractional reserves and

redemption in gold. Journal of Economics 55 (2), 123–150.Shubik, M., Wilson, C., 1977. The optimal bankruptcy rule in a trading economy using fiat money. Journal of

Economics 37, 337–354.Simon, L.K., Stinchcombe, M.B., 1995. Equilibrium refinements for infinite normal form games. Econometrica

63, 1421–1445.Stahn, H., 1993. A remark on uniqueness of fix-price equilibria in an economy with production. Journal of

Mathematical Economics 22, 421–430.Tsomocos, D.P., 2003. Equilibrium Analysis, Banking, Contagion and Financial Fragility. WP Series No. 175,

Bank of England.Werner, J., 1987. Arbitrage and existence of competitive equilibrium. Econometrica 55, 1403–1418.Weyers, S., 1999. Uncertainty and insurance in strategic market games. Economic Theory 14, 181–201.Weyers, S., 2000. Convergence to Competitive Equilibria and Elimination of No-Trade (In a Strategic Market

Games with Limit-Prices). INSEAD Working Paper EPS/00/60.Weyers, S., 2003. A strategic market game with limit-prices. This issue.

art strategic market games giraud

Documents