Retrading in market games - Science Direct

ARTICLE IN PRESS

Journal of Economic Theory 115 (2004) 151–181

Retrading in market games Sayantan Ghosala, and Massimo Morellib a

Department of Economics, University of Warwick, Coventry CV4 7AL, UK b Department of Economics, Ohio State University, OH, USA Received 6 December 2001; final version received 3 January 2003

Abstract When agents are not price takers, they typically cannot obtain an efficient real location of resources in one round of trade. This paper presents a non-cooperative model of imperfect competition where agents can retrade allocations, consistent with Edgeworth’s idea of recontracting. We show (a) there are Pareto optimal allocations, including competitive equilibrium allocations, that can be approximated arbitrarily closely when trade is myopic, i.e., when agents play a static Nash equilibrium at every round of retrading; (b) any converging sequence of allocations generated by myopic retrading can be supported along some retradeproof subgame perfect equilibrium path when traders anticipate future rounds of trading. r 2003 Elsevier Science (USA). All rights reserved. Keywords: Market games; Retrading; Myopic versus far-sighted behavior; Retrade-proofness

1. Introduction In [8], we find the following definition: ‘‘A final settlement is a settlement which cannot be varied by recontract within the field of competition’’. In this definition of a final settlement, the emphasis is on outcomes that are immune to recontracting. When individuals interact cooperatively, outcomes immune to recontracting are defined to lie in the core of an exchange economy [3]. In contrast, here our emphasis is on a noncooperative formulation of recontracting in a general equilibrium model characterized by imperfect competition. When the outcomes of trade are inefficient, traders must be allowed to reopen markets. The allocations from the previous round of trade are the initial endowments in any new round of trade, while the rules of exchange remain

Corresponding author. Fax: 1-44-2476-523032. E-mail address: [email protected] (S. Ghosal).

0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-0531(03)00102-9

ARTICLE IN PRESS 152

S. Ghosal, M. Morelli / Journal of Economic Theory 115 (2004) 151–181

constant. This generates an iterative process of retrading in which traders are able to reopen markets before they consume. We focus on the issue of whether retrading will allow traders to approximate allocations on the Pareto frontier. The non-cooperative game of exchange we use is the Shapley–Shubik [21] market game, where the rules of exchange allow all traders to influence prices by sending quantity signals. With a finite number of traders, Dubey and Rogawski [7] have shown, under some mild regularity assumptions on preferences, that the Nash equilibrium outcomes of the market game are Pareto optimal if and only if the initial endowments of traders are Pareto optimal as well. This result allows us to study the incentives traders have to reopen markets before they consume their final allocations even in trading environments characterized by complete information. In our model, traders can reopen markets a finite or infinite number of times before they consume. We think of the number of times traders can reopen markets as a way of capturing the frequency with which they can retrade. At each round of trade all commodities are exchanged at trading posts except for the numeraire commodity, in which bids for all other commodities have to be made. For each non-numeraire commodity, traders can submit bids for the commodity and make offers of a quantity of the commodity, at the relevant trading post. In any new round of trade, the endowments of individuals are their final allocations from the previous round of trade. Using these endowments, individuals now make bids and offers in the trading posts and obtain allocations determined by the same price formation rule and allocation rule. The cost of reopening trading posts in any new round of trade is measured by a common discount factor for all traders. We study the outcomes of myopic retrading as well as far-sighted retrading. A path of myopic retrading only requires that each period allocation be a Nash equilibrium outcome given the final allocation of the previous period. With farsighted retrading, traders anticipate that there will be retrading in future time periods. With myopic retrading, we show that there are allocations on the Pareto frontier that can be approximated arbitrarily closely along some equilibrium path of retrading, as the discount factor is close enough to perfect patience and the number of allowable retrading periods is large enough. We construct an example in which there is a unique path of myopic retrading, which approximates the Pareto frontier. In fact, typically, the competitive equilibrium allocation of the original economy can be approximated by such a retrading process. The same sequence of allocations that approximates a Pareto optimal allocation under myopic retrading can be sustained by a subgame perfect equilibrium profile under far-sighted retrading. The approximation result with far-sighted retrading is shown under two different information scenarios. In the first, each trader uses anonymous strategies where current bids and offers are conditioned only on the allocation obtained from the preceding round of trade and on the aggregate bids and offers in the preceding round of trade. With this restriction, deviations from the equilibrium path of play are punished by no trade. This is unsatisfactory as now (off the equilibrium path of play) traders may have an incentive to reopen trading posts. We, then show that when strategies are required to be retrade proof, both on and off the equilibrium path of

ARTICLE IN PRESS S. Ghosal, M. Morelli / Journal of Economic Theory 115 (2004) 151–181

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play, if all traders are able to observe the identity of the deviating trader, the approximation result still holds. However, we also show that, along any equilibrium path of finite retrading, with or without far-sighted behavior, no allocation on the Pareto frontier can be attained even when the cost of reopening trading posts is negligible. We are also able to demonstrate that any subgame perfect equilibrium that sustains a sequence of allocations that converges to some allocation on the Pareto frontier must have the property that it must look increasingly similar to the sequence of allocations generated by myopic retrading. Moreover, the set of allocations supported by subgame perfect equilibrium profiles is shown to expand as the cost of reopening trading posts falls. This weak monotonicity result holds with finite as well as infinite horizon. All the results just described are first proved under the simplifying assumption that traders can consume commodities (all tradeable) only after having stopped trading. However, we show that all results extend to the more general class of games where traders can decide to consume part of their current endowment at any time, while remaining on the market with the rest. Our model of retrading can also be derived as reduced form of a model where the tradeable goods are actually assets. The goods that agents consume can be simply viewed as derived from the flow yields of the currently owned stock of assets. With this interpretation in mind, our model of retrading can be thought of as providing a rationale for resale markets where assets (more generally, durable goods) are traded. Moreover, in this case the issue of consumption becomes irrelevant, since the assets owned by each individual at any given time cannot be consumed, they can only be kept or traded. Finally, although we focus on the possibility of eventually reaching an efficient allocation of resources (or assets) through retrading, we point out that a new type of market failure also arises in market games with retrading: there are ‘‘bad’’ subgame perfect equilibria where traders delay trade only because the other traders do the same. The rest of the paper is organized as follows. Section 1.1 compares our retrading model and our results with the related literature. Section 2 presents the economy and the basic models of non-cooperative trade that we study. Section 3 gives a simple example, in which the unique equilibrium path of retrading converges to the competitive equilibrium. Section 4 characterizes the equilibria of the benchmark retrading model with myopic players. Section 5 characterizes the subgame perfect equilibria and retrade-proof equilibria of the market game with far-sighted retrading. Section 6 contains the extensions to the case where each trader can always choose between consuming and trading any subset of her own commodities and to the case where the tradeable goods are assets. Some of the more technical material is relegated to Appendix A. 1.1. Related literature The model we study as well as the results we obtain are different from the body of related work that studies dynamic noncooperative games of exchange.



Gale [9–11] and McLennan and Sonnenschein [16] look at a model where traders are repeatedly pair-wise matched and bargain over the trades that they make with each other. With a continuum of traders, complete information, and endogenous replacement, there is a stationary equilibrium which converges to the competitive equilibrium as the discount factor converges to one. The following are the main differences from our model: (1) We have a finite number of traders; (2) Gale’s traders make direct transfers to each other in pairs, which are independent of the transfers made within other matched pairs at each round of trade (in contrast, in our model, trade is anonymous and each commodity is traded at a common price); (3) once a pair agree to trade, they exit and are replaced by identical copies. In this sense, in contrast to our model, the same set of traders never really agree to retrade with each other along the equilibrium path of play. In that framework, retrading refers to the fact that any type has a positive probability of being repeatedly matched with any given other type of trader. Moreover, in order to obtain convergence to efficient allocations, Gale needs traders to be far sighted. In contrast, we are able to obtain convergence when traders are myopic. Dubey et al. [5] is closer to our paper, as they also study retrading in market games. However, they have a model with a continuum of agents.1 Moreover, they do not allow for discounting of future consumption. They show that if equilibria in the one-shot market game fail to coincide with competitive equilibria due to the endowment constraints in the numeraire commodity binding for non-negligible subsets of traders, competitive equilibria can nevertheless be approximated arbitrarily when traders are allowed to reopen trading posts before they consume their final allocations. In our model, with a finite number of agents, the Nash equilibria of the market game are Pareto inefficient even when endowment constraints in the numeraire commodity do not bind for any individual trader. The process of myopic retrading that we study in Section 4 shares with the iterative processes studied by Dre´ze and de la Vallee Poussin [4a], Malinvaud [14] and Allen et al. [2], the property that reallocations can be Pareto improving at each step. Peck and Shell [18]2 study a model of a market game where traders can make arbitrarily large short sales, so that net trades are small relative to gross trades. Using this model they show that, at equilibrium, no individual action has a big effect on market prices, and therefore equilibrium allocations approximate competitive equilibrium allocations. Introducing the possibility of arbitrarily large short sales requires traders in their model to satisfy a budget constraint. They postulate some form of outside enforcement of the budget constraint via a bankruptcy rule. With these features, allowing for short sales has similar effects on imperfect competition as allowing for retrading (as they point out in footnote 6).

1

A model of retrading with a continuum of agents corresponds to Edgeworth’s notion of recontracting in a field of perfect competition. In contrast, our model studies recontracting in a field of imperfect competition. 2 For a related liquidity based approximation result see also [17].


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2. The economy We study trade in pure exchange economies with a finite set of commodities L (indexed by l), a finite set of individuals I (indexed by i). Each individual’s consumption set is RLþ ; and his endowment is denoted by wi ARLþþ : The utility function is ui : RLþ -R: A pure exchange economy is E ¼ fL; ðui ; wi Þ: iAIg: An allocation x ¼ ðx1 ; y; xI Þ such that xi ARLþ for all iAI is feasible if, in addition, P P i i iAI x ¼ iAI w : A feasible allocation x is Pareto optimal if there is no other feasible allocation y such that ui ðyi ÞXui ðxi Þ for all iAI with ui ðyi Þ4ui ðxi Þ for some iAI: Throughout the paper, we keep the total endowments of each commodity fixed. Let P denote the set of Pareto optimal allocations and let IR denote the set of individually rational allocations x such that ui ðxÞXui ðwÞ for all iAI: Let F denote P P i i the set of feasible allocations, i.e., F fxARLI þ : iAI xl ¼ iAI wl ; l ¼ 1; y; Lg: Throughout the paper, we make the following assumption on the fundamentals of the exchange economy: Assumption 1. For each iAI; ui is strictly monotone, strictly concave, element of C r ; rXLI; and the closure of the indifference curves through wi are contained in RLþþ and remain bounded away from the boundary of the consumption set. 2.1. The one-shot market game In this section we describe the Shapley–Shubik [21] market game of noncooperative exchange. Each trader makes bids and offers of commodities at trading posts where commodities are exchanged; all bids are denoted in some numeraire commodity, which we set to be commodity 1. Traders are allowed to make offers in all the other commodities 2; y; L; each one traded on one of L 1 trading posts. A strategic action for a trader i is a vector si ¼ ðbi2 ; y; biL ; qi2 ; y; qiL Þ; where bil denotes the bid for commodity l while qil denotes the offer of commodity l; l ¼ 2; y; L: The corresponding set of strategic actions for each trader i is S i ðwi Þ ¼ P fðbi2 ; y; biL ; qi2 ; y; qiL Þ such that bil X0; iAI bil pwi1 ; 0pqil pwil ; l ¼ 2; y; Lg: All bids and offers have to be non-negative and the offer of a commodity made by a trader cannot exceed his endowment of that commodity. For each action profile P s ¼ ðs1 ; y; sI Þ; at the trading post for commodity l the aggregate bid is Bl ¼ iAI bil P Bl and the aggregate offer is Ql ¼ iAI qil : The corresponding price is pl ðsÞ ¼ Q if l Bl 40 and Ql 40; pl ðsÞ ¼ 0 otherwise. For each trader i; the allocation rule P determines commodity holdings as follows: If pl ðsÞa0; xi1 ðsÞ ¼ wi1 Ll¼2 bil þ PL bil i i i i i i l¼2 ql pl ðsÞ and xl ðsÞ ¼ wl ql þ p ðsÞ; l ¼ 2; y; L: If pl ¼ 0; xl ðsÞ ¼ wl ; for all iAI: l

Let vi ðsi ; s i Þ be the payoff associated with s: A Nash equilibrium profile is s such that vi ðsi ; s i ÞXvi ðsi ; s i Þ; for all si ASi ðwi Þ and iAI: A Nash equilibrium profile s



i such that bi l 40; ql 40 for all l ¼ 2; y; L and iAI is a non-trivial Nash equilibrium. PL i i i i A Nash equilibrium profile s such that bi l¼2 bl ow1 ; 0oql owl for all l 40; 0o l ¼ 2; y; L and iAI is an interior Nash equilibrium. Let NðwÞ denote the set of interior Nash equilibrium allocations of the market game. In the one-shot market game with variable offers, observe that the trivial Nash i equilibrium where bi l ¼ ql ¼ 0 for all l and iAI; always exists and yields the initial endowments as the final allocation. When wAP-RLI þþ ; there is also an interior Nash equilibrium at which individuals consume their initial endowments guaranteeing that NðwÞ-Paf; since w would be an element of such an intersection (see, for instance, [7]). What happens when weP? Consider the following three properties:

(P1) (Static inefficiency) If weP; then NðwÞ-P ¼ f: (P2) (Weak gains from trade) If weP; there exists xANðwÞ such that ui ðxi ÞXui ðwi Þ for all iAI; with ui ðxi Þ4ui ðwi Þ for some iAI: (P3) (Strong gains from trade) If weP; there exists xANðwÞ such that ui ðxi Þ4ui ðwi Þ for all iAI:

* *

*

Property (P1) requires that whenever the endowments in an exchange economy are Pareto suboptimal, there is no interior Nash equilibrium allocation that is also Pareto optimal. Property (P2) requires that whenever the endowments in an exchange economy are Pareto suboptimal, there is nevertheless some interior Nash equilibrium allocation that makes at east one trader better-off relative to his endowments. Property (P3) requires that whenever the endowments in an exchange economy are Pareto suboptimal, there is some interior Nash equilibrium allocation that makes all traders better-off relative to their endowments. Although when we state results in later sections we directly assume that one or all of (P1)–(P3) characterize NðwÞ whenever weP; it is worth pointing out that when preferences and endowments satisfy Assumption 1, Dubey and Rogawski [7] show that (P1)–(P3) characterize NðwÞ whenever weP (see also [19] for similar results in a related market game). Further, Dubey and Rogawski [7] also imply that if wANðwÞ; then wAP: We conclude this section with a result that characterizes the set of interior Nash equilibria. Consider two different endowment vectors w and w0 with the same set of feasible allocations (i.e., the aggregate amount of each commodity is the same at w and w0 ) but there is some individual %i who is better off at w0 relative to w: In the following proposition (see [20] for a similar argument in the case of two commodities and two individuals), we show that there exists an interior Nash equilibrium with endowments w0 at which individual %i is better off than at a different interior Nash equilibrium with endowments w: Note that the result stated below (and proved in the appendix) is a direct proof that (a) NðwÞ is non-empty whenever weP 3 and (b) NðwÞ is characterized by (P3) whenever weP: 3

Dubey and Shubik [6] show, under weaker assumptions, that the set of non-trivial Nash equilibria is non-empty.


157

Proposition 1. Suppose preferences and endowments of individuals satisfy Assumption P P 1. Consider two endowment vectors wc0 and w0 c0; w; w0 eP; Ii¼1 wil ¼ Ii¼1 w0il ; l ¼ 1; y; L such that u%i ðw%i Þou%i ðw0%i Þ for some %iAI: Then, there exists xANðwÞ and x0 ANðw0 Þ such that u%i ðx%i Þou%i ðx0%i Þ for the same individual %i: Proof. See Appendix A.

&

Proposition 1, together with Dubey and Rogawski [7] result that if wAP-RLI þþ then wANðwÞ; guarantees the non-emptiness of NðwÞ whenever weP and Assumption 1 is satisfied. Although this conclusion is of independent interest, here the main relevance of Proposition 1 is in the section on far-sighted retrading when it is used to characterize retrade-proof subgame perfect equilibria. 2.2. The market game with retrading From the results discussed in the preceding subsection, it follows that the gains from trade are never exhausted after a one-period exchange. Therefore there are always incentives to retrade. In this section we describe an exchange mechanism that takes into account these incentives. Trading posts can reopen over a sequence of finite or infinite periods, t ¼ 0; 1; y; T: At each t an action for trader i is a vector sit : The corresponding set of strategic actions at t for each trader i is Sti ðxit 1 Þ; since the endowments for the traders at time t are the allocations obtained from trading in the previous period. Start from si 1 ¼ ð0; y; 0Þ for all iAI and xil; 1 ¼ wil ; for all l ¼ 1; y; L and for all iAI: For each strategic action profile st ; in the trading post for commodity l; the aggregate bid is Bl;t ; the aggregate offer is Ql;t ; with the corresponding price pl ðst Þ; defined as in the static game. For each trader i; the allocations xit ðst Þ are also defined as before. Along a sequence of action profiles s ¼ fs0 ; y; st ; yg; we say that player i stops trading after period T˜ i iff bil;t0 ¼ qil;t0 ¼ 0 for all t0 XT˜ i ; l ¼ 2; y; L and bi 0 a0; qi 0 a0 for all t0 oT˜ i ; l ¼ 2; y; L: Even though l;t

l;t

traders can stop trading at different times, it is convenient not to complicate notation by explicitly keeping track of traders who drop out. We can do so, without loss of generality, as the bids and offers of a trader can be zero at any round of trade and hence a trader i who stops trading at some period T˜ i can be counted as a market player who makes zero bids and offers in all periods including and subsequent to T˜ i : In what follows, we shall consider two models of retrading, labelled as myopic and far sighted. Case 1 (Myopic retrading): When retrading is myopic, at each new round of retrading traders behave in a very simple way: at each new round of retrading, they choose a vector of bids and offers that constitutes a static Nash equilibrium to the final allocation obtained from the previous round of trade. In the notation developed before, at each t; the strategy profile chosen, st ; satisfies the condition that



xt ðst ÞANðxt 1 Þ: Traders consume when they stop trading.4 As the utility function of each trader is continuous and the set of feasible allocations compact, we remark that even when T˜ i ¼ N; the payoff to any player i remains well defined. Myopic traders can be seen as traders who do not expect that trading posts can be reopened, so they play their best responses as if the current trading round were the last. Consistent with this, we will study myopic retrading without discounting, even though the results extend to the case where discounting occurs. Case 2 (Far-sighted retrading): When retrading is far sighted, all traders know that future play will, in general, be conditioned on the outcomes of the current round of trade. Here, as before, we assume that an individual trader consumes only when she has stopped trading. However, now we endow each trader i with a common discount factor d: When T is finite, d lies in ½0; 1 : When T ¼ N; d lies in ½0; 1Þ:5 Trader i’s payoff, once she has stopped trading in period T˜ i ; is ˜i

dT ui ðxiT˜i Þ: A history of play at period t is ht ¼ fs0 ; y; st 1 g: The corresponding set of histories is denoted by Ht : A pure strategy for trader i is a sequence si ¼ fsi0 ; y; sit ; yg with sit : Ht -Sti for all t: Denote by si jht the restriction of si to the subgame from period t after history ht : A pure strategy profile s ¼ ðs1 ; y; sI Þ is a subgame perfect equilibrium (SPE henceforth) if, for every ht ; the restriction si jht for all traders iAI is a Nash equilibrium in the subgame from period t: Let ˜ w; TÞ denote the set of SPE allocations of the market game with far-sighted Xðd; retrading.

3. An example In this section, we analyze retrading in an example. There are two commodities and two individuals, with quasi-linear utility functions uk ðxÞ ¼ xk1 þ f k ðxk2 Þ; k ¼ i; j: We assume that f k ð:Þ is strictly monotone, strictly concave, twice-continuously differentiable and satisfies the boundary condition that limx2 -0 @f k ðx2 Þ ¼ N for k ¼ i; j: Further, for simplicity, we choose the units in which commodities are P measured so that k wk2 ¼ 1: We focus on retrading in the ‘‘sell-all’’ market game. The ‘‘sell-all’’ version of the Shapley–Shubik market game is obviously simpler than the variable offers version: At each time t where trader i is still active, his offer is assumed to equal xi2;t 1 ; which is the endowment of commodity 2 inherited from the trades of the previous period. Other than for this simplification, the strategies, aggregate variables, and the allocation rules are identical to the more general variable offers model described before.6 In this case there is a unique Nash 4 Note that the assumption that each trader consumes when he stops trading is only made for simplicity, and it is not crucial for the results, as discussed in Section 6. 5 We interpret d as a measure of the cost of reopening trading posts in any new round of trade. 6 One additional difference would be in the precise definition of what it means to stop trading in the sellall market game. We avoid the formal definitions since they are not relevant for this example, but the


159

equilibrium with trade in the one-shot market game.7 This means that finitely repeated trade would not add anything, whereas we now show that finite retrading leads the traders towards the competitive allocation even if they are myopic. It will be convenient to refer to wi2 ¼ ai0 Að0; 1Þ as individual i’s initial share of commodity 2 and ait as individual i’s share at the end of round t 1 of retrading. For the moment, we simply assume that at any round of trade, all traders have enough of the numeraire commodity to ensure existence of an interior one-shot Nash equilibrium in any one round of trade.8 Using the allocation rule, we obtain that at any round of retrading t; t ¼ 0; 1; y; if j ; bi2;t Þ; player i’s objective function at time t is the current profile of actions is st ¼ ðb2;t i b xi1;t bit þ ait Bt þ f i t ; Bt bi

j : Using the fact that the ratio Btt ¼ aitþ1 ; if the current profile of where Bt ¼ bi2;t þ b2;t actions st is an interior Nash equilibrium, we can rewrite the first-order conditions of traders to obtain the dynamical system that characterizes the evolution of the sequence of allocations generated by myopic retrading:

@f i ðaitþ1 Þð1 aitþ1 Þ ð1 ait Þ ¼ : ait @f j ð1 aitþ1 Þaitþ1 Evidently, a stationary point of the preceding map is an interior allocation on the Pareto frontier. Moreover, as both individuals have quasi-linear utility functions, the allocations of commodity 2 is uniquely determined at an interior Pareto optimum. Let a% i denote individual i’s share of commodity 2 at the interior Pareto optimum. Suppose ai0 oa% i : Then, as f k ð:Þ is strictly concave, we must have that Moreover,

ð1 ai0 Þ ai0

i

aÞ 4ð1 % : For all t40 such that a% i

there exists tˆ such that

@f i ðaiˆÞ t @f j ð1 aiˆÞ t

o1;

ð1 %ai Þ 4 a% i

@f i ðai0 Þ 41: @f j ð1 ai0 Þ

i @f i ðait Þ ai Þ ð1 ait Þ ð1 at 1 Þ 41; ð1 % o ai o ai : a% i @f j ð1 ait Þ t t 1

ð1 aiˆÞ t aiˆ t

If

and as long as for all t4tˆ;

(footnote continued) intuitive feature of any such definition is that traders must bid the exact amounts that give them back the endowments obtained with their last real trade. 7 The existence of an equilibrium with trade follows from Dubey ([4b], Remark 2). Further, using Remark 5 in [7], it also follows that if weP then Nsell-all ðwÞ satisfies (P1)–(P3). 8 By rewriting the first-order conditions for individual i at an interior Nash equilibrium, at period t we obtain the equation bit ¼ gi ðait ; aitþ1 Þ where gi ðait ; aitþ1 Þ ¼ aitþ1 @f i ðaitþ1 Þ i

g

ðait ; aitþ1 Þpxi1;t

1 aitþ1 1 ait

: We require that at period t;

for i ¼ 1; 2: Starting from t ¼ 1 and applying the above equality and inequality

recursively, we obtain that at each T 0 pT; the required inequality is wi1 XKTi 0 where KTi 0 ¼ PT 0 1 t¼1

ð1 aitþ1 Þgi ðait ;aitþ1 Þ ait g j ðait ;aitþ1 Þ k

þ gi ðaiT 0 ; aiT 0 þ1 Þ þ T 0 1; ait

iaj;

i; j ¼ 1; 2:

Without

loss

of

generality,

as

limx2 -0 @f ðx2 Þ ¼ N for k ¼ i; j; at each t; lies in a compact set bounded away from 0 and 1. If follows that maxT 0 pT KTi 0 ; i ¼ 1; 2; is finite and therefore, if wi1 XmaxT 0 pT KTi 0 ; i ¼ 1; 2; an interior Nash equilibrium exists at each t:


160 @f i ðaiˆÞ

o1; we must have that

t @f j ð1 aiˆÞ t

that

i

ð1 %ai Þ ð1 ait Þ ð1 at 1 Þ 4 ai 4 ai : a% i t

Suppose there exists t˜4tˆ such

t 1

@f i ðai˜Þ t

41: @f j ð1 ai˜Þ t

Consider the ratio

@f i ðait˜Þy@f i ðaitˆÞ : @f j ð1 ait˜Þy@f j ð1 aitˆÞ Notice that if the above ratio is equal to one we must be on the Pareto frontier. On the other hand, the above ratio must be strictly greater than one as otherwise 1 ai˜ ai˜ t

t

i

aÞ 4ð1 % a contradiction. Therefore, a% i i

ð1 ait Þ ait

t

which implies that

t

i

ð1 %ai Þ 1 at˜ ð1 atˆ 1 Þ o ai o ai : a% i ˜ t

@f i ðai˜Þy@f i ðaiˆÞ t t 41; @f j ð1 ai˜Þy@f j ð1 aiˆÞ

By repeating the above argument from t˜; it follows that the ratio

tˆ 1

ð1 %ai Þ and therefore, ait to a% i : A symmetric argument establishes a% i when ai0 4a% i : An immediate consequence is that the sequence of

converges to

convergence allocations generated by myopic retrading must converge to the Pareto frontier. Moreover, note that from the equations that determine final allocations, we also obtain that individuals consumption of commodity 1 is identical to that at the competitive equilibrium. What about far-sighted retrading? We show that the sequence of allocations generated by myopic retrading can be supported as SPE outcomes when T is very large but finite. Consider the sequence of allocations y1 ; y; yt ; y; with y0 ¼ w; associated with myopic retrading. Note that yt ¼ Nf ðyt 1 Þ;9 t ¼ 1; y; with the associated sequence of payoffs uðy1 Þ; y; uðyt Þ; y in utility space R2 : Consider the following strategy profile s: * For tpT; play s˜t such that yit ¼ xi ð˜st Þ (and u˜ it ¼ ui ðxi ð˜st ÞÞ) as long as h˜t ¼ f˜s0 ; y; s˜t 1 g; otherwise, if there has been a deviation, play s0t such that xðs0t Þ ¼ Nf ðxt 1 Þ: We need to show that s* is a SPE. By construction, after any deviation, both players continue to choose bids according to one-shot Nash equilibria. As all sequences of allocations generated by one-shot Nash equilibria converge to the same allocation for both commodities, no player has an incentive to deviate when T is large. 4. Myopic retrading We start the general analysis with myopic retrading. We show that, starting from an arbitrary configuration of initial endowments, traders are able to converge to some allocation in the Pareto set. Nevertheless, convergence cannot take place in a finite number of rounds of myopic retrading. We state the results only for the market game with variable offers described in Section 2.2, but we note that the same results also obtain for the ‘‘sell-all’’ market game (as one could guess from the previous section). 9

The subscript f refers to ‘‘fixed’’ offers, since in this example offers are not strategic.


161

The following definition identifies the sequences of allocations that are consistent with myopic retrading. Definition of myopic retrading. A sequence of allocations fxt g; t ¼ 1; y is generated by myopic retrading if and only if it satisfies the inclusion xt ANðxt 1 Þ; for all tX1:10 An allocation y is stationary with myopic retrading if and only if y ¼ NðyÞ: Some notation is needed before proving the main result of this section. For any allocation y; let uðyÞ ¼ ðu1 ðy1 Þ; y; uI ðyI ÞÞ: For any KCRLI ; let uðKÞ ¼ fuðyÞ: yAKg: Observe that uðKÞCRI ; for all K: Let jj:jj denote the Euclidean norm. Then, we define the distance between a vector y and a set K as dðuðyÞ; uðKÞÞ ˆ inf uAuðKÞ jjuðyÞ ujj: ˆ Proposition 2. Consider wARLI þþ ; suppose that NðwÞ satisfies (P1) and (P2) whenever weP: Then, for any w ¼ y0 ARLI ˜ t g; t ¼ þþ ; there exists a sequence of allocations fy 0; 1; y; y˜ t ANðy˜ t 1 Þ for all tX1; such that, for any E40; there is a T40 with dðuðy˜ t Þ; uðP-IRÞÞoE for all t4T: Proof. If wAP; then w ¼ NðwÞ and we are done. Therefore assume that weP: Consider the sequence of sets N1 ; y; Nt ; y; with y0 ¼ w; and Nt ¼ fx: xANðyÞ; for some yANt 1 g; t ¼ 1; y; with the associated sequence of sets uðN1 Þ; y; uðNt Þ; y in utility space RI : By (P2) there exists a sequences of sets, denoted by fU˜ t g; where at t; and u˜ t AU˜ t ; t ¼ 0; 1; y; u˜ t AuðNt Þ for all t and u˜ tþ1 4u˜ t at each t: Denote by fY˜ t g (and, respectively, by y˜ t ; t ¼ 0; 1; y its generic element) the associated sequence of sets of allocations generated by myopic retrading and satisfying P2 (all starting from y0 ¼ w). Note that for each iAI; any sequence u˜ it ; t ¼ 0; 1; y; in fU˜ it g is bounded above, as the utility of each individual is continuous and the set of feasible allocations is compact. Let u% i denote the supremum of the sequence u˜ it : As every increasing sequence converges to the supremum, it follows that the sequence u˜ t converges to u% ¼ ðu% 1 ; y; u% I Þ; the component-wise supremum of u˜ t ¼ ðu˜ 1t ; y; u˜ It Þ; t ¼ 0; 1; y . Moreover, it also follows that the associated sequence of allocations y˜ t ; t ¼ 0; 1; y has a limit point y% such that uðyÞ % ¼ u: % By considering every sequence of utilities in fU˜ t g and the corresponding sequences in fY˜ t g; we obtain a set of allocations Y% w which consists of the limit allocations of each of those sequences of allocations y˜ t ; t ¼ 0; 1; y . If we show that there exists some sequence of allocations y˜ t ; t ¼ 0; 1; y in fY˜ t g that converges to yA % Y% w such that y% is stationary under myopic retrading, we are done: in fact, as Dubey and Rogawski [7] have shown that if wANðwÞ then wAP; if y% ¼ NðyÞ; % yAP: % To this end, define the binary relation ! on F as follows: Given two feasible allocations x and y; y!x if either (a) xANðyÞ and ui ðxÞXui ðyÞ for all iAI and ui ðxÞ4ui ðyÞ for some iAI or (b) x is a limit point of a sequence of allocations fxt g; 10

x0 is obviously the initial endowment.



t ¼ 1; y with xt ANðxt 1 Þ; where x0 ¼ y: Remark that ! is transitive: if y!x and x!z; then by (b) y!z: Therefore, ! is a partial order on F [13, p. 13]. Consider any sequence of allocations y˜ t ; t ¼ 0; 1; y in fY˜ t g (i.e., in the set of sequences generated by myopic retrading satisfying (P2)). Note that either y˜ t !y˜ t0 or y˜ t0 !y˜ t for all tat0 ; moreover, if y˜ t !y˜ t0 and y˜ t0 !y˜ t ; y˜ t ¼ y˜ t0 : Therefore, ! is a linear ordering [13, p. 14] and hence any sequence in fY˜ t g is a chain given ! [13, p. 15]. By Kuratowski’s lemma [13, p. 33], each chain in a partially ordered set is contained in a maximal chain. Moreover, any chain in F under !; and hence even the maximal chain, is a subset of some sequence of allocations in fY˜ t g: Given that every subset of F is partially ordered under ! and every linearly ordered subset of F (and hence, every chain in F ) has an upper bound, Zorn’s lemma [13, p. 33] guarantees that F has a maximal element under !: We have already shown that every sequence in fY˜ t g has a limit point y: % In any chain (and hence, in a, maximal chain), y˜ t ; t ¼ 0; 1; y; y˜ t !y% for all t; and hence y% is an upper bound [13, p. 13]. A maximal element of a maximal chain in F cannot be preceded by any other element of F under !: Therefore, if y% is an upper bound of the maximal chain, y˜ t ; t ¼ 0; 1; y; y% ¼ NðyÞ; % and therefore yAP: & % The above proposition demonstrates that starting from an arbitrary Pareto suboptimal vector of initial endowments, there is some sequence of allocations, generated by myopic retrading, that converges to an allocation that is stationary under myopic retrading, and hence to some allocation on the Pareto set. Note that each profile of actions along the myopic retrading sequence constitutes a static Nash equilibrium to the allocation inherited from the preceding round of trade. By (P2), for every configuration of Pareto suboptimal endowments, there is a static Nash equilibrium at which allocations are such that every trader is at least as well-off and some trader(s) strictly better-off relative to their initial endowments. This implies that the sequence of utility profiles associated with the sequence of allocations generated by myopic retrading is an increasing sequence. But then, along each dimension, corresponding to a specific individual, this sequence of utilities must converge to its supremum, which in turn determines a limit point of the sequence of allocations generated by myopic retrading. To complete the proof, it suffices to show that some limit allocation must be stationary under myopic retrading. To this end, we define a binary relation (on the set of feasible allocations) which is a suborder of Pareto dominance. Each sequence of allocations generated by myopic retrading satisfying (P2) is a linearly ordered chain under this binary relation, and under this binary relation, any linearly ordered chain is a subset of some sequence of allocations in the set of all sequences of allocations generated by myopic retrading satisfying (P2). By Kuratowski’s lemma, this set must contain a maximally ordered chain under this binary relation. As each sequence of allocations generated by myopic retrading has an upper bound (as the set of feasible allocations is compact), by Zorn’s lemma some sequence of allocations in the set of all sequences of allocations generated by myopic retrading satisfying (P2) has a maximal element. If a sequence of allocations generated by myopic retrading satisfying (P2) has a maximal element, then the


163

maximal element must be its upper bound and therefore its limit allocation. But then this limit allocation is stationary under myopic retrading and hence a Pareto optimal allocation.11; 12 Evidently, the preceding proposition goes through with the stronger requirement that NðwÞ satisfies (P3) whenever weP: The result (as well as the next one) can also be extended to do1: The reason for dealing only with d ¼ 1 in the propositions of this section is that this is the case where the assumptions of myopic retrading make the most sense intuitively: in fact, a myopic player who does not discount the future can be assumed to believe he will consume right away. So myopia here means that traders cannot predict that after trading they will change their mind, trading again instead of consuming. Remark 1. At each stage of myopic retrading, the final allocation from the preceding round of trade defines the distribution of endowments for a ‘‘new’’ economy. As the sequence of allocations converge to some allocation on the Pareto frontier, in the limit we obtain an economy with Pareto optimal endowments. As no trade is the only outcome at the competitive equilibrium of an economy with Pareto optimal endowments, in this sense the converging sequence of allocations associated with myopic retrading converges to competitive equilibria of the limit economy as well. Remark 2. Proposition 2 holds even when we limit attention to sequences of Pareto undominated Nash equilibrium allocations in the paths of myopic retrading. In other words, consider the sequences of sets N˜ 1 ; y; N˜ t ; y; where N˜ t are the Pareto undominated allocations in Nt ; t ¼ 1; y . Note that as the set of interior Nash equilibrium allocations Nt is a closed subset of the set of feasible allocations, it is also compact, and therefore N˜ t is non-empty.13 Remark 3. Let CðwÞ denote the set of competitive equilibrium allocations associated with some initial endowment vector w: For any allocation x ACðwÞ; let p denote the associated set of competitive equilibrium prices (with the normalization rule that the price of commodity one is set equal to one). Then, Lemma A.1 (see Section A.1) shows that there exists E% w 40 such that for any feasible allocation x with (a) jjx 11 For a similar argument see [2] (see observation 8) who study a convergent iterative process where at each step individuals exchange assets to share risks inherent in the multiplicity of competitive equilibria. 12 Note that it is not possible to show that all limit allocations are stationary points under myopic retrading, because some sequences of allocations generated by myopic retrading satisfying (P2) may have a limit allocation where the Nash equilibrium correspondence fails to be continuous. 13 In market games like the one in [19], the presence of a budget constraint and the bankruptcy rules to insure feasibility imply that the equilibrium of the sell-all model is also an equilibrium of the more general variable offers model, and under some conditions (see for instance [12a]) such equilibria are undominated. However, in the Shapley–Shubik model that we use it is not true that the equilibrium of the sell-all model is always an equilibrium of the more general variable offers model, and even when it is there is no reason why it should be better than any other equilibrium. Hence, since in the Shapley–Shubik model there is no salient type of equilibrium that always belongs to N˜ t ; the analysis does not benefit from limiting attention to the undominated Nash equilibria.



wjjoE%; (b) ui ðxi ÞXui ðwi Þ for all iAI (with strict inequality for individuals j with xj aw j 14), (c) p xi ¼ p wi ; then xANðwÞ: Using this result and modifying the proof of the preceding convergence result appropriately, it follows that starting with y0 ¼ w; we can generate a sequence of allocations fy˜ t g; t ¼ 0; 1; y; y˜ t ANðy˜ t 1 Þ and p yit ¼ p wi for all tX1: It follows, by adapting the proof of Proposition A.1 in Appendix A, in a dense set of economies, we can show that the sequence of allocations fy˜ t g; t ¼ 0; 1; y; converges to x : Although Proposition 2 demonstrates that traders will obtain allocations in the vicinity of the Pareto set, it still leaves open the question of whether traders are able to converge to an allocation on the Pareto frontier after a finite number of rounds of myopic retrading. Proposition 3. If weP and NðwÞ satisfies (P1), there is no ToN; and no sequence of allocations fyt g; t ¼ 1; y; yt ANðyt 1 Þ; with y0 ¼ w and t ¼ 0; y; T; such that yT AP: Proof. If yT AP; then yT ¼ NðyT Þ: Moreover, as weP; there must be some T 0 oT such that the allocation obtained at T 0 1; yT 0 1 ; is not in P; while for all tXT 0 yt AP: Then we must have that yT 0 ANðyT 0 1 Þ-P; a contradiction. & The intuition behind this result is simple. If trade concludes after some finite length of time, at some finite stage in the game it must be the case that while the traders’ inherited allocation from the previous period is Pareto suboptimal, the final allocation they obtain after reopening trading posts is both (a) a Pareto optimal allocation, and (b) satisfies the inequalities for a Nash equilibrium allocation for the one-shot market game with the traders’ inherited allocation as the endowment. But by (P1), with Pareto suboptimal endowments no Nash equilibrium allocation of the one-shot market game can ever be Pareto optimal. This guarantees that no allocation on the Pareto set will be attained by traders after a finite number of rounds of myopic retrading. Without discounting, this implies that trading posts will always be reopened. This makes the assumption of myopic traders hard to swallow, but the next section shows that not only the results above extend to far-sighted behavior, but also that far-sighted behavior becomes indistinguishable from myopic behavior over the process of retrading.

5. Far-sighted retrading Let us now allow traders to be far sighted. The approximation result of Proposition 2 is confirmed, and we show that the set of SPE paths ‘‘converges’’ to the set of myopic retrading paths as traders keep retrading. Far-sighted retrading can lead the economy to Pareto improvements ‘‘faster’’ than myopic retrading, but does 14

Remark that this set is non-empty as weP:


165

not have to. We will see that it is possible to have a new kind of market failure with far-sighted retrading: traders may delay trade along a SPE path merely because they expect other traders to do the same. We will also see that anonymous strategies, i.e., strategies that can only be functions of prices and aggregate demand for each commodity, are sufficient to guarantee the existence of SPE profiles such that the final allocation converges to the Pareto set. On the other hand, the appealing property of renegotiation-proofness, appropriately defined for our context, can only be proved by allowing for non-anonymous punishment strategies. The next proposition and its corollary provide a negative result, in line with Proposition 3. ˜ w; TÞ-P ¼ f; for all dA½0; 1 ; Proposition 4. If weP; and NðwÞ satisfies (P1), Xðd; and all ToN: % Proof. Let TpT be the first period at which an allocation xT% AP is obtained along some SPE path. Given that trade cannot take place after reaching the Pareto set, it % must be the case that traders stop trading at some TpT; i.e., bil;t ¼ qil;t ¼ % Moreover, since T% is the first period where p is reached, xT 1 0 8i; 8l; 8t4T: % eP: As xT 1 eP; by (P1), Nðx Þ-P ¼ f: This is a contradiction, since, at the last % % T 1 round of trade, any SPE profile requires the final allocation to be in the set of Nash equilibrium allocations with respect to the inherited allocation. & ˜ w; NÞ-P ¼ f; for all dA½0; 1Þ: Corollary 1. If weP; and NðwÞ satisfies (P1), Xðd; Proof. When dA½0; 1Þ; any trader gets a payoff of zero if he trades indefinitely. Therefore, along any SPE path, all traders will stop trading after some finite length % % of time, implying that there exists a ToN such that bil;t ¼ qil;t ¼ 0 for all tXT; l ¼ 2; y; L: Trade stops before T% 0 ¼ inf T fT: bil;t ¼ qil;t ¼ 0 for all tXT; l ¼ 2; y; L; iAIg: Then, the proof immediately follows from Proposition 4. & Even far-sighted traders cannot obtain allocations on the Pareto set. As trade always concludes after some finite length of time, at some finite stage in the game, it must be the case that both the traders’ inherited allocation from the previous period and the final one are Pareto suboptimal, otherwise there would be a contradiction with (P1). We now extend the approximation result obtained under myopic retrading to this world of far-sighted players. Proposition 5. If NðwÞ satisfies (P1)–(P3) whenever weP-RLI þþ ; then, for every E40; ˜ w; TÞ such that dðuðyÞ; uðPÞÞoE for all dAðd; 1Þ: there is a T and d and yAXðd; % % % % Proof. Using Proposition 2, for d close to 1 we obtain that whenever weP-RLI þþ ; if NðwÞ satisfies (P1)–(P3) whenever weP; for any w ¼ y0 ARLI ; there exists a þþ sequence of allocations fy˜ t g; t ¼ 0; 1; y; y˜ t ANðy˜ t 1 Þ for all tX1; such that, for any



E40; there is a T40 with dðuðy˜ t Þ; uðP-IRÞÞoE for all t4T: Now construct the following strategy profile s: * For tpT þ 1; play s˜t such that yit ¼ xi ð˜st Þ (and % u˜ it ¼ ui ðxi ð˜st ÞÞ) as long as h˜t ¼ f˜s0 ; y; s˜t 1 g; otherwise, if there has been a deviation, play bit% ¼ qit% ¼ 0; iAI; for all t%4t: Finally, when t4T þ 1; play bit ¼ qit ¼ 0: To % complete the proof, we need to show that s* is a SPE. By construction, observe that no player has an incentive to deviate after T þ 1 or in any subgame following a % deviation from the SPE path. It remains to check that no player has an incentive to deviate at any tpT þ 1: Indeed, consider player i who deviates at t choosing some % action s0it : As bit0 ¼ qit0 ¼ 0; iAI; for all t0 4t; denote i’s maximum payoff from such a deviation by dtþ1 vi ðs0it ; s˜ i;t Þ; where xi ðs0it ; s˜ i;t Þ is the resulting allocation for i when i chooses s0it while all other players choose according to s: * On the other hand, his * i payoff from continuing to choose according to s* is dT ðsÞ u ðyÞ: As yt ANðyt 1 Þ; we i 0i i i Þ: Consider must have v ðst ; s˜ i;t Þ less than or equal to u ðyt Þoui ðyiT i ðsÞ * i

* i i * dT ðsÞ u ðyT i ðsÞ Þ dtþ1 vi ðs0it ; s˜ i;t Þ ¼ dtþ1 ½dT ðsÞ t 1 ui ðyiT i ðsÞ Þ vi ðs0it ; s˜ i;t Þ : Let dtþ1 be i * * i

i

* such that ½dT ðsÞ t 1 ui ðyiT i ðsÞ Þ vi ðs0it ; s˜ i;t Þ ¼ 0: Set d ¼ inf i;0ptpT i ðsÞ 1 dtþ1 * i : It * % follows that for all dA½d; 1Þ; s* is a SPE strategy profile. & % i

The proof of Proposition 5 shows that the sequence of allocations generated by myopic retrading that converges to the Pareto set can be supported as SPE outcome with far-sighted retrading when traders are sufficiently patient. To understand the main idea of the proof, consider first the case where d ¼ 1: The strategy profile is constructed so that traders continue to choose the bids and offers that implement the sequence of allocations generated by myopic trade. If a trader deviates at some round of trade, in all subsequent rounds of trade all traders make null bids and offers at the trading post for each commodity, thus ensuring that no trade is the outcome. In the no trade phase, no individual trader has an incentive to deviate. This is because the bids and offers at any round of trade constitute a static Nash equilibrium to the final allocation from the previous round of trade, which implies no individual trader can gain by deviating, as a deviation will be followed by no trade in all subsequent rounds. Under (P3), all traders strictly gain in utility along some sequence of allocations generated by myopic retrading. This implies that if traders are sufficiently patient, they will prefer to retrade over consuming their current allocation. Remark that the above argument would also go through if at each t; along the equilibrium path of play, traders were required to choose strategies according to some undominated myopic Nash equilibrium. The following corollary shows that Proposition 5 goes through even when strategies are conditioned only on a subset of the entire history of play namely, the aggregate bids and offers at each trading post in the preceding round of trade. Denote by sM an anonymous strategy profile where each player i conditions his choice of bids and offers in period t, ðbit ; qit Þ; only on Bl;t 1 and Ql;t 1 ; l ¼ 2; y; L (and therefore on the preceding period’s market price vector pt 1 ðst 1 Þ), and on her own individual allocation xit 1 ðst 1 Þ with SM the set of


167

anonymous strategy profiles. Let X˜ M ðd; w; TÞ the set of SPE allocations for strategy profiles in SM : Corollary 2. For every E40; there is a T and d and yAX˜ M ðd; w; TÞ such that % % % dðuðyÞ; uðP-IRÞÞoE for all dAðd; 1Þ: % Proof. It is sufficient to observe that the sequence of allocations along the SPE path, y0 ; y; yt ; y; used in the proof of Proposition 4 can also be supported by a strategy profile s* M specified as follows. For tpT; play s˜t as long as Bt 1 ¼ B˜ t 1 and Qt 1 ¼ % Q˜ t 1 ; otherwise, if there has been a deviation, play bit% ¼ qit% ¼ 0; iAI; for all t%4t: Finally, when t4T; play bit ¼ qit ¼ 0: It is immediate that sM ASM is also a SPE % strategy profile. & Although anonymous strategy profiles minimize the amount of information used to sustain a sequence of myopic Nash equilibria along the equilibrium path of a SPE strategy profile, Proposition 5 is problematic because deviations from the equilibrium path of play are punished by no trade, but if the allocation reached is suboptimal no trade is not retrade-proof. This leads us to define retrade-proof strategy profiles. Consider a sequence of feasible allocations xt ; t ¼ 0; 1; y such that there exists some finite time period T such that xt0 ¼ xT for all t0 4T: For a given value of d; we say the sequence of allocations is retrade proof if there is no xANðxT Þ such that ui ðxT Þodui ðxÞ for all iAI: Under the strategy profile s; let TðsÞ be the set of all time periods at which trade stops under the strategy profile s both on or off the equilibrium path of play. Under the strategy profile s; let X ðsÞ be the set of sequences of allocations generated by s both on and off the equilibrium path of play. It follows that for each sequence of allocations xt ; t ¼ 0; 1; y in X ðsÞ there exists ˆ ˆ Then, s is retrade-proof if every sequence of TATðsÞ such that xt0 ¼ xTˆ for all t0 4T: allocations in X ðsÞ is retrade-proof. Let X˜ R ðd; w; TÞ denote the set of allocations supported by retrade proof SPE. We now extend the approximation result obtained with anonymous strategy profiles to the case where we require retrade-proof SPE. Proposition 6. If NðwÞ satisfies (P1)–(P3) whenever weP-RLI þþ ; then, for R ˜ every E40; there is a T and d and yAX ðd; w; TÞ such that dðuðyÞ; uðPÞÞoE for % % % all dAðd; 1Þ: % Proof. See the appendix.

&

Suppose all traders are sufficiently patient and that there are at least three active traders on each side of a trading post. Consider a SPE strategy profile where deviations off the equilibrium path of play are not punished by no trade. What prevents a trader from deviating from the equilibrium path of play profile of bids and offers? By definition, the current profile of bids and offers is a one-shot Nash equilibrium. It follows that a trader will deviate if she anticipates that in the



continuation subgame, there is an equilibrium which supports an allocation where she is better off relative to the allocation obtained as the limit of the sequence of allocations along the SPE path of play. To prevent such a contingency from occurring, the other traders punish the deviating trader by coordinating at each subsequent round of trade on a interior myopic Nash equilibrium at which the deviating trader is worse-off. Now, each individual trader’s strategies is conditioned on both the aggregate bids and offers and on the identity of the deviating trader. Proposition 1 guarantees the existence of such a strategy profile. When traders use such a strategy profile, following a deviation, bids and offers in the continuation subgame are conditioned on the identity of the deviator. Moreover, given d; off the equilibrium path of play, by construction such a strategy profile allows traders to reopen trading posts as long as there are incentives to retrade. Are there other SPE strategy profiles, which do not require players to implement allocations generated by myopic retrading but which, nevertheless, support some sequence of allocations that approximate the Pareto frontier? While the answer is generally yes, the next proposition shows that any SPE strategy profile must, after some length of time, begin to look like a strategy profile that implements allocations generated by myopic retrading. Formally, for T ¼ N; for any SPE strategy profile s; let y1 ðsÞ; y; yt ðsÞ; y; yTs ðsÞ (where yt ¼ xðst ðsÞÞ) denote the allocations generated along the equilibrium path of play associated with s and Ts denotes the last period ˜ ˜ with trade under s: For ToT s ; let y1 ðsÞ; y; yt ðsÞ; y; yT˜ ðsÞ denote a T truncation of Ts : We say that a SPE strategy profile s approximates the Pareto frontier if ˜ for E40 there exists TpT s and yT˜ ðsÞ such that dðuðyT˜ ðsÞÞ; uðPÞÞoE: Moreover, for LI any e40; and wARþþ ; let Ne ðwÞ denote the set of non-trivial e-Nash equilibrium allocations.15 Proposition 7. For any SPE strategy profile s that approximates the Pareto frontier, ˜ ˜ for every e; there is a ToT s 1 such that for each t4T; yt ðsÞANe;t ðyt 1 ðsÞÞ: Proof. Consider the sequence of strategies along the equilibrium path of play of s; s1 ðsÞ; y; st ðsÞ; ysTs ðsÞ: At any t such that yt ðsÞeNe ðyt 1 ðsÞÞ; there is some player i whose maximum payoff from a deviation, denoted by vi ðsit ; s i;t ðsÞÞ; where xi ðsit ; s˜ i;t ðsÞÞ is the resulting allocation for i when she chooses sit while all other players choose according to s; is such that vi ðsit ; s i;t ðsÞÞ ui ðyit ðsÞÞ40: By choosing bit0 ¼ qit0 ¼ 0; for all t0 4t; player i can obtain a payoff dtþ1 vi ðsit ; s i;t ðsÞÞ: As s is SPE, ˜ # 1 and t0 4t and it follows that dtþ1 vi ðsi ; s i;t ðsÞÞpdTþ1 ui ðyi 0 ðsÞÞ for all dA½d; t

t

therefore, ui ðyit0 ðsÞÞ4vi ðsit ; s i;t ðsÞÞ4ui ðyit ðsÞÞ: As s approximates the Pareto frontier, for every e40; there exists T˜ such that if t4T˜ and t0 4t; ui ðyit0 ðsÞÞ ui ðyit ðsÞÞoe and therefore, vi ðsit ; s i;t ðsÞÞ ui ðyit ðsÞÞoe which implies that ytþ1 ðsÞANet ðyt ðsÞÞ: & 15

A non-trivial e-Nash equilibrium allocation x satisfies the condition that there is a profile s0 with x ¼ xi ðs0 Þ such that for all iAI; ui ðxi ðs0 ÞÞXui ðxi ðsi ; s0 i ÞÞ e for all si ASi ðwÞ: i


169

When at some t players do not choose bids and offers according to myopic retrading, along they obtain an allocation yt eNðyt 1 Þ (where yt 1 is the allocation obtained from t 1). This implies that there must be some individual i who would have incentive to deviate from the SPE strategy profile at t and then choose bit0 ¼ qit0 ¼ 0 for all t0 4t: Therefore, if fyt : tX0g is generated along some SPE path of play, it must be the case that the gain in utility for i in the continuation game along the SPE path of play from t þ 1 outweighs the gain in utility from deviating at t: As we approach the Pareto frontier along a SPE, an individual’s gain in the continuation game along the SPE path becomes smaller, and so must the gain in utility by deviating from the equilibrium path of play. Note that a similar result goes through for when T is large but finite (simply substitute T for Ts throughout). It is intuitive that far-sighted retrading can lead the economy within a given neighborhood of the Pareto frontier faster than myopic retrading. To see this, consider do1 and T such that dðuðyT Þ; UðP-IRÞÞoe; where yT is the allocation obtained through the myopic retrading path y0 ; y; yt ; y; yT : The same path can be sustained as a SPE path of far-sighted retrading, hence far-sighted traders can always do at least as well as myopic traders. They can also do strictly better: Suppose that T43; then it is possible to construct a SPE profile where at the first round of trade the obtained allocation is directly yT 1 as long as yT 1 can be attained by some combination of bids and offers with the initial endowments w: The threat of no trade in the last round can make deviations from this profile not attractive, if d is high enough and the game satisfies (P3). Does this mean that far-sighted retrading always leads to greater gains in efficiency? The answer is no, and the following remark shows that there are also SPE of the game that make traders worse off than with just one round of trade.16 A new type of market failure can also arise: there are SPE where traders will delay trade only because all other traders do the same. Remark 4. By Proposition 1, we know that there always exists a static Nash equilibrium in the one-shot market game where all traders gain relative to the notrade equilibrium. Denote the bid-offer profile that constitutes a Nash equilibrium with trade s ¼ ðb ; q Þ: Now suppose that traders are allowed to retrade in an extra round of trade. Consider the following strategy profile s: * (1) for all iAI; play si0;l ¼ i i i i ðb0;l ; q0;l Þ ¼ ð0; 0Þ for all l ¼ 2; y; L; (2) if s0;l ¼ ðb0;l ; qi0;l Þ ¼ ð0; 0Þ for all l ¼ 2; y; L; and all iAI; play s ¼ ðb ; q Þ next round; otherwise, play bi1 ¼ qi1 ¼ 0; for all %i

%i

i

i

Þ Þ iAI: Then, for each dAðu%iuðxðw i ¼ arg maxiAI fuiuðxðw * is a SPE. %i ðs ÞÞ; 1 ; where % i ðs ÞÞg; s %i

%i

Þ However, observe that for dAðu%iuðxðw * all traders obtain payoffs which are %i ðs ÞÞ; 1Þ; at s;

Pareto dominated by their payoffs corresponding to the static Nash equilibrium. The next result shows that the set of SPE allocations with far-sighted retrading expands as d becomes larger. 16

Note also that the Pareto set can be approximated only in terms of final allocation, whereas discounting makes the convergence process itself ‘‘inefficiently long’’ in terms of utility.



Proposition 8. Consider d0 ; d00 A½0; 1 such that d0 pd00 : For each ToN; then, ˜ 00 ; w; TÞ: ˜ 0 ; w; TÞDXðd Xðd Any allocation that satisfies the inequalities that characterize the sequence of allocations along the SPE path for a specific d must continue to do so as d becomes larger. The proof is Appendix A, and the result can be easily extended to the case where traders can retrade infinitely often.

6. Discussion on consumption and asset trading Throughout the paper we have made the simplifying assumption that consumption by trader i may occur only after he has stopped trading. A natural question to ask is whether therefore our results hold when individual traders can decide otherwise, i.e., when they can opt to consume part of their current endowment instead of using it all for trading purposes. We will divide the analysis of this issue in two parts. First, we give a direct answer to this question, keeping the assumption that all tradeable goods are also consumable. The second part of the analysis makes an argument that in fact one of the best interpretations of our model ought to be the case where the tradeable goods on the trading posts are assets, which are long-lived, yield consumption indirectly, but are not directly consumable themselves. In the second case, of course, the consumption issue becomes irrelevant. Let us start by keeping the assumption that all tradeable are consumables. Clearly, when the discount factor is equal to one it cannot make a difference, and the SPE profiles that approximate points on the Pareto set remain SPE profiles even when consumption is in principle allowed at any time. On the other hand, when the discount factor is in the open interval ð0; 1Þ; individuals will typically have an incentive to consume (part of) their endowments even before leaving the market. An important observation is that along a SPE path, the bids and offers typically do not exhaust the endowments at any round of trade. In other words, considering a sequence of actions s0 ; y; st ; y that constitutes a SPE path of far-sighted retrading, it is typically the case that qit oxit 1 at all times.17 Therefore, individuals can consume a small fraction of their current endowments at each new round of trade and not necessarily affect the SPE profile of retrading. Hence, it is obvious that allowing traders to consume whenever they want the final utilities must be higher. But what matters here is that if d is high enough the same path s0 ; y; st ; y can remain an equilibrium path of retrading. The equilibrium path used in the approximation results is such that everybody is made better off by each successive round of trade, and hence, for d high enough, the utility difference can always compensate for the longer wait to consume. A deviation to consume current endowments that affects the feasibility of bids and offers at the current or subsequent rounds of trade will not be profitable. 17

Recall that we are looking at environments in which no trader has a shortage of tradeable goods.


171

Even though it should be clear from the above discussion that our assumption of ‘‘consumption at the end’’ is irrelevant for the main results, it is also worth noting that this consumption issue would not even be raised if the trading posts were just markets for assets. Let x ¼ ðx1 ; y; xL Þ be reinterpreted as an allocation of assets. For any xi ; let yi ðxi Þ ¼ ðyi1 ðxi Þ; y; yiM ðxi ÞÞ be the associated allocation of commodities.18 Let vî ðyðxi ÞÞ represent trader i’s preferences over the commodity bundle y: Traders are endowed with assets but not commodities. A feasible allocation of assets generates a feasible allocation of commodities. An allocation of assets is Pareto optimal if and only if the associated allocation of commodities is Pareto optimal. Traders trade assets 2; y; L using asset l ¼ 1 as numeraire. For simplicity, we assume that traders cannot trade commodities directly. They can only trade commodities indirectly, by trading assets. The retrading process, both myopic and far sighted, is as in the previous sections. The difference is that now at each round of trade, if xit is trader i’s current allocation of assets, then yit is trader i’s current commodity bundle, which he consumes to obtain a current utility of vi ðyit Þ: P Then, trader i’s total utility from retrading will be Tt¼0 dt vî ðyit ðxi ÞÞ: With this specification, all our previous results apply by appropriately rephrasing the propositions and proofs. After all players have stopped trading, the final allocation of assets will keep giving the same consumption bundle to all traders thereafter every period. If we extended the model to allow for stochastic yields of assets, then asset trading could continue forever, since every shock on the productivity of assets may change the incentives (or needs) of traders to readjust their asset portfolio. Issues related to uncertainty and/or asymmetric information are however beyond the objective of this paper.

7. Conclusion The main result of this paper has been to show that allowing retrading in markets where the one-shot allocations are inefficient allows traders to approximate allocations on the Pareto frontier arbitrarily closely. This ‘‘approximation’’ result, however, needs to be qualified on the following grounds: (1) allocations on the Pareto frontier are never attained in finite time by retrading; (2) getting to an allocation close to the Pareto frontier may take several rounds of retrading and therefore, when traders discount future consumption heavily, in payoff space traders may still be far away from the Pareto frontier of utilities; (3) there is a huge multiplicity of equilibria with retrading, and therefore not all subgame perfect equilibrium allocations with retrading are close to the Pareto frontier; (4) in other contexts (see for instance Jehiel and Moldovanu [12b]), where there are externalities in consumption and traders use trading mechanisms which 18 As a metaphor, think of the allocation of assets as being allocation of trees, and the vector y would be the corresponding allocation of fruits. People consume fruits, not trees, but trade trees only in this interpretation of the model.



allow some subset of traders to be excluded from the market, retrading may not approximate allocations on the Pareto frontier. Beside the issue of efficiency, this paper has also demonstrated some interesting ‘‘behavioral’’ properties of retrading processes. In particular, we have shown that the set of equilibrium paths of retrading that converge to the Pareto frontier when agents are forward looking shrinks towards the converging path of myopic retrading. We have also shown by example that convergence holds even when there is a unique Nash equilibrium in the one-shot game, i.e., in a context where finitely repeated trade could not have efficient equilibrium outcomes. The properties of retrading that we have studied seem therefore to be quite general, and independent on the assumptions made on the rationality of traders.

Acknowledgments We thank M. Cripps, B. Dutta, D. Easley, G. Giraud, R. Guesnerie, M. Jackson, M. Machina, J. Peck, H. Polemarchakis, A. Postlewaite, S. Spear, S. Weyers, and the participants of various workshops for their comments. Massimo Morelli thank Deutsche bank for sponsoring his membership at the Institute for Advanced study. The usual disclaimer applies.

Appendix A A.1. Proof of Proposition 1 In order to prove Proposition 1, it is convenient to associate to the original exchange economy and market game, a pseudo-exchange economy and pseudomarket game. In the original exchange economy, there are L commodities and individual i’s is characterized by his consumption set RLþ ; endowments wARLþþ and utility function u : RLþ -R: In the pseudo-exchange economy, commodity 1 is replaced by L 1 copies of itself i.e. commodity 1 is replaced by commodities labelled by the pair ð1; lÞ; l ¼ 2; y; L implying that there are 2ðL 1Þ commodities. A commodity bundle in the original exchange economy is xARL : A commodity bundle in the pseudo-exchange economy is xˆ ¼ ðxˆ 1;2 ; y; xˆ 1;L ; xˆ 2 ; y; xˆ L ÞAR2ðL 1Þ P with Ll¼2 xˆ 1;l ¼ x1 and xˆ l ¼ xl ; l ¼ 2; y; L: The consumption set of individual i in 2ðL 1Þ

the pseudo-exchange economy is Rþ : We abuse notation slightly to denote individual’s preferences over the new set of commodities by the utility function P PL ˆ 1;l ¼ x1 and xˆ l ¼ xl ; l ¼ ui ð Ll¼2 xˆ 1;l ; xˆ 2 ; y; xˆ L Þ ¼ ui ðx1 ; x2 ; y; xL Þ; where l¼2 x 2; y; L with endowments wˆ i ¼ ðwˆ i1;2 ; y; wˆ i1;L ; wˆ i2 ; y; wˆ iL Þ; where wˆ i1;l 40; l ¼ PL ˆ i1;l ¼ wi1 and wˆ il ¼ wil ; l ¼ 2; y; L: The allocation 2; y; L; l¼2 w ðxˆ 1;2 ; y; xˆ 1;L ; xˆ 2 ; y; xˆ L Þ is feasible in the pseudo-exchange economy if and only if


173

P PI P ˆ il ¼ Ii¼1 wˆ il ; l ¼ 2; y; L: Remark that if the xˆ i1;l ¼ Ii¼1 wi1;l and i¼1 x allocation ðxˆ 1;2 ; y; xˆ 1;L ; xˆ 2 ; y; xˆ L Þ is feasible in the pseudo-exchange economy, the P allocation ð Ll¼2 xˆ 1;l ¼ x1 ; x2 ; y; xL Þ where xˆ l ¼ xl ; l ¼ 2; y; L is also feasible in the original exchange economy. Moreover, if the allocation ðx1 ; x2 ; y; xL Þ is feasible in the original exchange economy, the allocation ðxˆ 1;2 ; y; xˆ 1;L ; xˆ 2 ; y; xˆ L Þ P where Ll¼2 xˆ 1;l ¼ x1 and xˆ l ¼ xl ; l ¼ 2; y; L may not be feasible in the pseudoexchange economy. Therefore, the set of feasible utility profiles in the pseudo-exchange economy is contained with the set of feasible utility profiles in the original exchange economy. Consider the allocation x in the original economy where uðxÞcuðwÞ: Consider the allocation ðxˆ 1;2 ; y; xˆ 1;L ; x2 ; y; xL Þ P in the pseudo-exchange economy where Ll¼2 xˆ 1;l ¼ x1 and xˆ l ¼ xl ; l ¼ 2; y; L: It P follows that ui ð Ll¼2 xˆ 1;l ; xˆ i2 ; y; xˆ iL Þ ¼ ui ðxi1 ; xi2 ; y; xiL Þ4ui ðwi1 ; wi2 ; y; wiL Þ ¼ P ui ð Ll¼2 wˆ i1;l ; wˆ i2 ; y; wˆ iL Þ: Conversely, if xˆ is an allocation in the pseudo-exchange P ˆ ˆ then the allocation x ¼ ð Ll¼2 xˆ 1;l ¼ x1 ; x2 ; y; xL Þ economy such that uðxÞcuð wÞ; xˆ l ¼ xl ; l ¼ 2; y; L in the original exchange economy is such that uðxÞcuðwÞ: Next, we turn to specification of the pseudo-market game. In the pseudo-market game, bids for commodity l are denoted in commodity ð1; lÞ: A strategy for player i is therefore ðbî ; qˆ i Þ ¼ ðbî2 ; y; bîL ; qˆ i2 ; y; qˆ iL Þ such that 0pbîl pwˆ i1;l and 0pqˆ il pwˆ il ; l ¼ 2; y; L: The price formation rules and allocation rules remain unchanged. At a strategy profile where pl 40; l ¼ 2; y; L; the payoff function for individual i is

PI

i¼1

u

i

L X l¼2

ðwˆ i1;l

bî1;l

þ

pl qˆ il Þ; wˆ i2

qˆ i2

! bî2 bîL i i þ ; y; wˆ L qˆ L þ : p2 pL

Remark that an interior Nash equilibrium of the pseudo-market game where 0obîl owˆ i1;l and 0oqˆ il owˆ il ; l ¼ 2; y; L; i ¼ 1; y; I which yields allocation xˆ is also an interior Nash equilibrium of the original market game, with bil ¼ bîl ; P and qil ¼ qˆ il ; l ¼ 2; y; L; which yields allocation x where x ¼ ð Ll¼2 xˆ 1;l ¼ x1 ; xˆ 2 ; y; xˆ L Þ: To see why, note that the first-order conditions that characterize an interior Nash equilibrium in the pseudo-market game coincide with the first-order conditions that characterize an interior Nash equilibrium. Moreover, it also follows that by appropriately choosing ðwˆ i1;2 ; y; wˆ i1;L Þ so that wˆ i1;l 4bil ; for all l ¼ 2; ::; L and i ¼ 1; y; I; any interior Nash equilibrium of the original market game is also an interior Nash equilibrium of the pseudo-market game. Lemma A.1. Suppose weP in the original exchange economy. Then, there exists E%40 such that for all EpE%; there is some feasible allocation x with jjx wjjoE and uðxÞcuðwÞ such that xANðwÞ in the original market game.


174

Proof. For the purposes of the proof, it is convenient to work with excess demands. Let zil ¼ xil wil : For each individual iAI; consider the set Z% i fzARL : ui ðz þ wi ÞXui ðwi Þg: As ui ð:Þ is strictly monotone and concave, for each i; Z% i is non-empty and convex. It follows, by applying the supporting hyperplane theorem, that there exists a vector p* i ¼ ðp* i1 ; y; p* iL Þa0 such that p* i z%X0 for all z%AZ% i : Next, note that p* il 40 for all T l ¼ 1; y; L:19 Let Z% ¼ iAI Z% i : Z% is convex as it is the intersection of convex upper % 20 Note that for each iAI; contour sets. Remark that 0 is on the boundary of Z: P L i i fzAR : p* z ¼ 0g-Z% ¼ f0g: Consider p* ¼ iAI li p* where li X0 for all iAI and P that fzARL : pz * ¼ 0g-Z% ¼ f0g and therefore, by setting iAI li ¼ 1: It follows P li 40 for all iAI and such that for each i; * iAI li ¼ 1 we can choose pc0 @x ui ðwÞap; * 21 i.e. for each i; fzARL : ui ðz þ wi Þ4ui ðwi Þg-fzARL : pz * ¼ 0g ¼ f: Without loss of generality, by an appropriate normalization, we can set * ðp* 1 ; y; p* L Þ ¼ ð1; p2 ; y; pL Þc0 where pl ¼ pp* l : In the remainder of the proof we 1 work with the vector p ¼ ð1; p2 ; y; pL Þc0: As weP; for each commodity l; l ¼ 2; y; L; there exists two non-empty sets of individuals Il0 and Il00 such that for all 0

i0 AI 0 ;

0

@x2 ui ðwi Þ 0 0 opl @x1 ui ðwi Þ

00

while for each i00 AI 00 ;

00

@x2 ui ðwi Þ 00 00 4pl : @x1 ui ðwi Þ

For each commodity l; it 0

follows that the excess demands vectors in the projection of the set fzARL : ui ðz þ 0 0 0 wi Þ4ui ðwi Þg; i0 AI 0 ; on coordinates 1; l must have opposite sign from the excess 00 00 00 00 demands vectors in the projection of the set fzARL : ui ðz þ wi Þ4ui ðwi Þg; i00 AI 00 ; on coordinates 1; l: Moreover, as for each i;

@x2 ui ðwi Þ apl @x1 ui ðwi Þ

for some commodity l; by

the strict concavity of utility functions, it follows that there exists e%0 40 such that for all epe%0 ; the set ZðeÞ defined as ( ) X 1 I i i i i i i i i ðz ; y; z Þ: jjz jjoe; pz ¼ 0 and u ðz þ w Þ4u ðw Þ 8i; z ¼0 i

zAZðe%0 Þ

is non-empty. Take any and consider a pseudo-exchange economy where P PL i i i i i zˆ1;l ¼ pl zl ; zˆl ¼ zl for all iAI and all l ¼ 2; y; L: Note that Ii¼1 l¼1 zˆ1;l ¼ 0; P i i zˆl ¼ 0; l ¼ 2; y; L and pl ¼

zî1;l zîl

;

l ¼ 2; y; L

whenever zî1;l a0 and zîl a0; l ¼ 2; y; L: For any commodity l; l ¼ 2; y; L; let IðlÞ ¼ Indeed, suppose to the contrary that p* il p0 for some l: Then, there exists z0 AfzARL : p* i z ¼ 0g such that wi þ z0 Xwi for all iAI and therefore, as the utility functions are strongly monotone, ui ðz0 þ wi Þ4ui ðwi Þ for all iAI; a contradiction. 20 Note that even if players have different endowments in the commodity space, in the excess demand space they all have the same status quo at 0: 21 Recall that the utility functions are twice continuously differentiable and smooth at 0 by Assumption 1. 19


175

fiAI: zîl a0g: Now consider the equations @xl ui ðzi þ wi Þ Qˆ l; i ¼ ðpl Þ2 ; iAIðlÞ; l ¼ 2; y; L; i i i @x1 u ðz þ w Þ Bˆ l; i ! bîl i i ; iAIðlÞ; l ¼ 2; y; L: zˆl ¼ qˆ l þ pl P As i zîl ¼ 0; l ¼ 2; y; L; Bˆ l pl ¼ ; l ¼ 2; y; L: Qˆ l Solving these equations we obtain that P ðpl Þ2 ð jai zˆlj Þ ˆ Bl; i ¼ ; iAIðlÞ; l ¼ 2; y; L; @ ui ðzi þwi Þ ð@ xl ui ðzi þwi ÞÞ pl x1 X j Bˆ l; i Qˆ l; i ¼ zˆl þ ; iAIðlÞ; l ¼ 2; y; L: pl jai After some manipulation, we obtain that iAIðlÞ; l ¼ 2; y; L @xl ui ðzi þ wi Þ Qˆ l; i pl pl ¼ ðpl Þ2 i i i @x1 u ðz þ w Þ Bˆ l; i ! pl Qˆ l; i ¼ pl 1 : Bˆ l; i Now,

P

jai

zˆlj ¼

Bˆ l; i pl ð1

pl Qˆ l; i Þ Bˆ l; i

and therefore,

P

@ ui ðzi þwi Þ

jai

zˆlj and pl ð@ xl ui ðzi þwi ÞÞ have x1

the same sign. Therefore, Bˆ l; i 40; l ¼ 2; y; L: Consider now the excess demand vectors on the supporting hyperplane in Zðe%0 Þ and consider the associated set of ˆ e%0 Þ defined as the set excess demand vectors in the pseudo-exchange economy Zð fðˆz1 ; y; zÎ Þ: zî1;l ¼ pl zil ; zîl ¼ zil ; iAIðlÞ; l ¼ 2; y; L; zAZðe%0 Þg: As w is not Pareto optimal and uðxÞcuðwÞ in the original exchange economy, we @ ui ðwi Þ

must have that pl @ xl ui ðwi Þa0: Therefore, x1

lim

zˆ-0;ˆzAZˆ

Bˆ l; i ¼

lim

zˆ-0;ˆzAZˆ

Qˆ l; i ¼ 0;

iAIðlÞ; l ¼ 2; y; L:

j g40 and wˆ l; i ¼ minjai fwˆ lj g40; l ¼ 2; y; L: It follows Let wˆ 1;l; i ¼ minjai fwˆ 1;l zî

that for each iAIðlÞ; l ¼ 2; y; L; there exists zî1;l a0; zîl a0; l ¼ 2; y; L and pl ¼ zˆ1;li l

such that 0oBˆ l; i oðI 1Þwˆ 1;l; i ; 0oQˆ l; i oðI 1Þwˆ l; i ; and therefore 0obîl owˆ i1;l and 0oqˆ i owˆ i : For each l ¼ 2; y; L and ieIðlÞ; set bî ¼ pl qˆ i which implies zî ¼ 0: It l

l

l

l

l

follows that xˆ i1;l ¼ wˆ i1;l þ zî1;l ; xˆ il ¼ wˆ il þ zîl for all iAI; l ¼ 2; y; L as an interior Nash



P equilibrium allocation in the pseudo-market game and ð Ll¼2 xˆ 1;l ¼ x1 ; x2 ; y; xL Þ; xl ¼ xˆ l ; l ¼ 2; y; L; is an interior Nash equilibrium allocation in the original market game such that uðxÞcuðwÞ: In other words, there exists e%pe%0 such that 8eAð0; e% (xANðwÞ; jjx wjjoe; such that uðxÞcuðwÞ: & By the preceding lemma, there exists x0 ANðw0 Þ such that u%i ðw%i Þou%i ðw0%i Þou%i ðx0%i Þ for the same individual %i: As the utility of each individual is continuous, there exists E#40 such that for all feasible allocations x with jjx wjjo#E; u%i ðx%i Þou%i ðx0%i Þ: But, then, by the preceding lemma, there exists E% such that for all EpE% ; there exists xANðwÞ such that jjx wjjoE and uðxÞcuðwÞ: Let E0 ¼ minf#E; E%g: It immediately follows that there exists xANðwÞ such that jjx wjjoE0 ; uðxÞcuðwÞ and u%i ðx%i Þou%i ðx0%i Þ: & Before going concluding this part of the appendix, it is worth pointing the following consequence of Lemma A.1. Suppose, we can find a feasible allocation xˆ such that ui ðxˆ i Þ4ui ðwi Þ for some iAIˆ (where Iˆ is a non-empty subset of individuals) with ui ðxˆ i ÞXui ðwi Þ for all i and further, there exists a strictly positive vector pc0 (with price of commodity one normalized to one) such that pxˆ i ¼ pwi for all iAI: Applying Lemma A.1 to individuals in Iˆ and setting the ratio of (strictly positive) bids and offers of individuals in I\Iˆ to be the equal to pl ; l ¼ 2; y; L; it follows that there exists E% 40 such that for any feasible allocation x with jjx wjjoE% ; ˆ (c) pxi ¼ pwi ; then (b) ui ðxi ÞXui ðwi Þ for all iAI (with strict inequality for iAI) xANðwÞ: A.2. Convergence with myopic retrading: an alternative proof Let U˜ be the set of strictly monotone, strictly concave, C r ; rXLI; utility functions endowed with the topology of uniform convergence on compacta (see [15] for a definition). Let U 0 be the subset of utility functions in U˜ which have the property that all uAU i are finite in the corresponding norm. Then, as Dubey and Rogawski [7] note, by Theorem 10.2 in [1], U 0 is an open subset of a Banach set. Let U ¼ U 1 ? U I where for each i; U i ¼ U 0 : Proposition A.1. There is a countable intersection of a collection of open and dense subsets of U such that for all in this set and wARLI þþ ; if NðwÞ satisfies (P1) and (P2) whenever weP; there exists a sequence of allocations fy˜ t g; t ¼ 0; 1; y; y˜ t ANðy˜ t 1 Þ for all tX1; with w ¼ y0 such that, for any E40; there is a T40 with dðuðy˜ t Þ; uðP-IRÞÞoE for all t4T: Proof. By Propositions 3 and 4, Remark 5 and Section 5.1 in [6], for every wARLI þþ ; there is an open and dense subset of U so that for each u% in this subset, each interior Nash equilibrium profile of strategies can be represented as the transverse intersection of an appropriately chosen map, Zw;u% ; with an appropriately chosen


177

manifold Nw; u% (see [7, p. 295]). As the domain of Zw;u% is compact, by the openness of transversal intersections [15, p. 43], it follows that there is an E40 such that for each w0 with jjw0 wjjoE; for each u% in the open and dense subset of U associated with w; each interior Nash equilibrium profile of strategies with endowments w0 can be represented as the transverse intersection of Zw0 ;u% with the manifold Nw 0 ;u% : Note that the set of rational numbers in F ; Fk ; is a dense subset of F : For each allocation in wAFk ; (a) there exists an open and dense subset of U such that each interior Nash equilibrium profile of strategies can be represented as the transverse intersection of an appropriately chosen map, Zw;u% ; with an appropriately chosen manifold Nw; u% and 0 0 (b) there is an E40 such that for all w with jjw wjjoE; for each u% in the open and dense subset of U associated with w; each interior Nash equilibrium profile of strategies can be represented as the transverse intersection of the Zw0 ;u% with the manifold Nw 0 ;u% identified in step (a). It follows that by taking the countable intersection of the open and dense subsets of U corresponding to each wAFk ; we obtain, by the Baire property [15, p. 10], a non-empty dense subset of U such that each u in this set and every yAF -RLI þþ ; each interior Nash equilibrium profile of strategies can be represented as the transverse intersection of an appropriately and further, Zy;u is also chosen map, Zy;u ; with an appropriately chosen manifold Ny;u transverse to every submanifold of Ny;u : therefore, Zy;u satisfies the definition of transverse stability. Fix u in this dense subset of U: Let weP: Consider the sequence of sets N1 ; y; Nt ; y; with y0 ¼ w; and Nt ¼ fx: xANðyÞ; for some yANt 1 g; t ¼ 1; y; with the associated sequence of sets uðN1 Þ; y; uðNt Þ; y in utility space RI : By (P2), we can extract a sequence u˜ t ; t ¼ 0; 1; y such that u˜ t AuðNt Þ and u˜ tþ1 4u˜ t ; at each t; with y0 ; y; yt ; y the associated sequence of allocations. Note that for each iAI; the sequence u˜ it ; t ¼ 0; 1; y is bounded above, as the utility of each individual is continuous and the set of feasible allocations is compact. Let u% i denote the supremum of the sequence u˜ it ; t ¼ 0; 1; y . As every increasing sequence converges to the supremum, it follows that the sequence u˜ t ; t ¼ 0; 1; y; converges to u% ¼ ðu% 1 ; y; u% I Þ; the component-wise supremum of u˜ t ¼ ðu˜ 1t ; y; u˜ It Þ; t ¼ 0; 1; y . Moreover, we may assume that the associated sequence of allocations yt ; t ¼ 0; 1; y has a limit point allocation y% such that uðyÞ % ¼ u: % By considering every sequence of utilities and the corresponding sequence of allocations generated by myopic retrading which satisfy (P2), we obtain a set of allocations Y% which consists of the limit allocations of each sequence of allocations yt ; t ¼ 0; 1; y . We have to show that for every E40; there is some yA % Y% such that dðuðyÞ; % uðP-IRÞÞoE: Suppose to the contrary, that inf yAY% dðuðyÞ; uðP-IRÞÞ40: Consider y% 0 Aarg inf yAY% dðuðyÞ; uðP-IRÞÞ: Then, by i 0i ˆ ˆ (P2), there exists an allocation yANð y% 0 Þ and iAI such that ui ðyÞ4u ðy% Þ: Moreover at 0 y% ; each interior Nash equilibrium profile of strategies can be represented as the transverse intersection of an appropriately chosen map, Zy%0 ;u ; with an appropriately chosen manifold Ny%0 ;u and further, Zy%0 ;u is also transverse to every submanifold of Ny%0 ;u : therefore, Zy%0 ;u satisfies the definition of transverse stability. It follows that there exists E* 40 such that for all jjy y% 0 jjo*E; there exists yANðyÞ ˜ with ui ðy˜ i Þ4ui ðy% 0i Þ:


178

˜ y00 ¼ y0 ; for t ¼ T˜ þ 1; Consider the sequence y000 ; y; y00t ; y where for tpT; t t i 0i y00t ANðy0t 1 Þ such that ui ðy00i % Þ and for t4T˜ þ 1; y00t ANðy00t 1 Þ and t Þ4u ðy i 00i ui ðy00i ˜ 00t ; t ¼ 0; 1; y be the associated sequence of allocations. t Þ4u ðyt 1 Þ: Let u 0 Remark that u% is no longer the component-wise supremum of u˜ 00t ; t ¼ 0; 1; y . Therefore, the sequence u˜ 00t ; t ¼ 0; 1; y must converge to u% 00 ¼ ðu% 001 ; y; u% 00I Þ; the component-wise supremum of u˜ 00t ¼ ðu˜ 00t ; y; u˜ 00I t Þ; t ¼ 0; 1; y and the associated sequence of allocations y000 ; y; y00t ; y has a limit point y% 00 such that uðy% 00 Þ ¼ u% 00 such that dðu% 00 ; uðP-IRÞÞodðuðy% 0 Þ; uðP-IRÞÞ; a contradiction. It follows that for every E40; there is some yA % Y% such that dðuðyÞ; % uðP-IRÞÞoE and therefore, there will be some T40 and some sequence of allocations generated by myopic retrading yt ; t ¼ 0; 1; y with a limit point y% such that (a) dðuðyÞ; % uðP-IRÞÞo2E and (b) for all t4T; dðuðyt Þ; uðP-IRÞÞoE: &

A.3. Proof of Proposition 6 Fix some T41: We construct a strategy profile s that has the following phases: Phase (1): Players choose s˜t at each t that supports the sequence of allocations generated by myopic (interior) Nash retrading which converges to the Pareto frontier along the equilibrium path of play namely, fy˜ t g; t ¼ 0; 1; y . Such a sequence of allocations exists by Proposition 2 and is, by construction, retrade-proof be the corresponding sequence of allocations. At each time t; if the history of play is as in phase (1), continue choosing s˜t : Phase (2): In phase (2), some player ið1Þ has deviated from the sequence of bids and offers in phase (1). Consider a deviation from phase (1) at some tð1ÞoT by 0ið1Þ ið1Þ player ið1Þ to some stð1Þ a˜stð1Þ : As s˜tð1Þ is a myopic (interior) Nash equilibrium at tð1Þ; 0ið1Þ

ð2Þ;j

0ið1Þ

uið1Þ ðxið1Þ ðstð1Þ ; s˜ ið1Þ;tð1Þ ÞÞouið1Þ ðxið1Þ ð˜stð1Þ ÞÞ: Let y˜ tð1Þ ¼ x j ðstð1Þ ; s˜ ið1Þ;tð1Þ Þ; j ¼ 1; y; I: ð2Þ

ð2Þ

Let y˜ tð1Þ ¼ ðy˜ ð2Þ;j : jAIÞ: Remark that y˜ tð1Þ c0 as at tð1Þ all traders iaið1Þ are choosing strictly positive bids and offers. Therefore, by Proposition 1, there exists a ð2Þ ð2Þ sequence of allocations fy˜ t g starting from y˜ tð1Þ ; t ¼ tð1Þ; tð1Þ þ 1; y; T; such ð2Þ

ð2Þ

ð2Þ;ið1Þ

ið1Þ

Þouið1Þ ðy˜ t Þ for all tXtð1Þ: that (i) y˜ t ANðy˜ t 1 Þ for all t4tð1Þ; (ii) uið1Þ ðy˜ t Fix the sequence of allocations generated by s in a phase (2) subgame to ð2Þ be fy˜ t g: y Phase ðKÞ: Consider a deviation from phase ðK 1Þ at some tðK 1Þ4tðK 2Þ 0iðK 1Þ iðK 1Þ and tðK 1ÞoT by player iðK 1Þ to some stðK 1Þ a˜stðK 1Þ : As s˜tðKÞ is a myopic 0iðK 1Þ

(interior) Nash equilibrium at tðK 1Þ; uiðK 1Þ ðxiðK 1Þ ðstðK 1Þ ; s˜ iðK 1Þ;tðK 1Þ ÞÞ ðKÞ;j

0iðK 1Þ

ouiðK 1Þ ðxiðK 1Þ ð˜stðK 1Þ ÞÞ: Let y˜ tðK 1Þ ¼ x j ðstðK 1Þ ; s˜ iðKÞ;tðK 1Þ Þ; j ¼ 1; y; I: Let ðKÞ

ðKÞ;j

ðKÞ

y˜ tðK 1Þ ¼ ðy˜ tðK 1Þ : jAIÞ: Remark that y˜ tðK 1Þ c0 as at tðK 1Þ all traders


179

iaiðK 1Þ are choosing strictly positive bids and offers. Therefore, by Proposition ðKÞ ðKÞ 1, there exists a sequence of allocations fy˜ t g starting from y˜ tðK 1Þ ; t ¼ tðK ðKÞ

ðKÞ

1Þ; tðK 1Þ þ 1; y; T; such that (i) y˜ t ANðy˜ t 1 Þ for all t4tðK 1Þ; (ii) ðKÞ;iðK 1Þ

iðK 1Þ

uiðK 1Þ ðy˜ t ÞouiðK 1Þ ðy˜ t Þ for all tXtðK 1Þ: Fix the sequence of allocaðKÞ tions generated by s in a phase ðKÞ subgame to be fy˜ t g: y Phase ðTÞ: In phase ðTÞ; some player iðT 1Þ has deviated from the sequence of bids and offers in phase ðT 1Þ: Consider a deviation from phase ðT 1Þ at T 1 0iðT 1Þ iðT 1Þ by player iðT 1Þ to some sT 1 a˜sT 1 : As s˜T 1 is a myopic (interior) Nash 0iðT 1Þ

equilibrium at T 1; uiðT 1Þ ðxiðT 1Þ ðsT 1 ; s˜ iðT 1Þ;T 1 ÞÞouiðT 1Þ ðxiðT 1Þ ð˜sT 1 ÞÞ: Let

ðTÞ;j

0iðT 1Þ

y˜ T 1 ¼ x j ðsT 1 ; s˜ iðT 1Þ;T 1 Þ;

j ¼ 1; y; I:

Let

ðTÞ

ðTÞ;j

y˜ T 1 ¼ ðy˜ T 1 : jAIÞ:

ðTÞ

Remark that y˜ T 1 c0 as at T 1 all traders iaiðT 1Þ are choosing strictly ðTÞ

positive bids and offers. Therefore, by Proposition 1, there exists an allocation y˜ t ðTÞ ðTÞ ðTÞ T;iðT 1Þ starting from y˜ T 1 ; t ¼ T 1; T; such that (i) y˜ T ANðy˜ T 1 Þ; (ii) uiðK 1Þ ðy˜ T Þo ðT 1Þ;iðT 1Þ

uiðT 1Þ ðy˜ T Þ: Remark that the sequence of bids and offers chosen at each phase of the strategy profile s will depend on the identity of the deviator from the path of play specified in the preceding phase. The existence of such a strategy profile s is guaranteed by Proposition 1. At each phase ðKÞ; KpT; by construction, the current profile of bids and offers is a one-shot Nash equilibrium. It follows that no player will have an incentive to deviate at phase ðTÞ: But, then, for any value of d; no player will have an incentive to deviate at any phase ðKÞ; Ko1: following a deviation at phase ðKÞ; in the continuation subgame, the other traders punish the deviating trader by coordinating at each subsequent round of trade on a interior myopic Nash equilibrium at which the deviating trader is worseoff. It only remains to check that no player will stop trading within a phase before time T: Consider the sequence of allocations in phase (1) fy˜ t g; t ¼ 0; 1; y; T: In Proposition 5, we have already shown that for every E40; there is a T and dð1Þ such % i will % have a that if T ¼ T; (i) dðuðy˜ T Þ; uðPÞÞoE; (ii) for all dAðdð1Þ; 1Þ; no player % % unilateral incentive to stop trading at some toT (iii) fy˜ t g; t ¼ 0; 1; y; T is retrade % for each phase ðKÞ; K41; % proof. By a similar argument, it also follows that KoT; ð2Þ ð2Þ ðK 1Þ ðKÞ % the sequence of allocations y˜ 0 ; y; y˜ tð1Þ 1 ; y˜ tð1Þ ; y; y˜ tð2Þ 1 ; y; y˜ tðK 1Þ 1 ; y˜ tðK 1Þ ; ðKÞ

y; y˜ T there is a dðKÞ such that for all dAðdðKÞ; 1Þ; (i) no player i will % % % have a unilateral incentive to stop trading at some toT (ii) the sequence of ð2Þ ð2Þ ðK 1Þ ðKÞ % ðKÞ allocations y˜ 0 ; y; y˜ tð1Þ 1 ; y˜ tð1Þ ; y; y˜ tð2Þ 1 ; y; y˜ tðK 1Þ 1 ; y˜ tðK 1Þ ; y; y˜ T is retrade % proof. Fix d ¼ minKpT fdð1Þ; y; dðKÞ; y; dðTÞg: Remark that do1: It % % % % % follows that for all d; dodo1; s satisfies the unimprovability % by one-shot deviations and is, therefore, subgame perfect. But, then, for every E40; there is a T and d and yAX˜ R ðd; w; TÞ such that dðuðyÞ; uðPÞÞoE for all dAðd; 1Þ; % % % TXT: & %


180

A.4. Proof of Proposition 8 ˜ 0 ; w; TÞ; then y0 AXðd ˜ 00 ; w; TÞ: In order to show this, the We show that if y0 AXðd following lemma comes handy. Consider the strategy profile sðd # 0 ; y0 Þ which is identical to a SPE sðd0 ; y0 Þ on the equilibrium path but differs off the equilibrium path in that, after any deviation from the equilibrium path of play at some time # denote the corresponding set of toTðsðd0 ; y0 ÞÞ; bit0 ¼ qit0 ¼ 0; for all t0 4t: Let S strategies. ˜ 0 ; w; TÞ if and only if there is a Lemma A.2. For any ToN; for all d0 A½0; 1 ; y0 AXðd 0 0 0 # sðd # ; y ÞAS that supports y : Proof. When d ¼ 0; all traders stop trading at t ¼ 0; implying that x0 ANðwÞ: It # Suppose d0 Að0; 1 : If follows that any SPE strategy profile must be an element of S: # that supports y0 ; by definition y0 AXðd ˜ 0 ; w; TÞ: Next, suppose there is a sðd # 0 ; y0 ÞAS 0 0 0 0 ˜ that sðd ; y Þ is a SPE strategy profile that yields y AXðd ; w; TÞ: Then, sðd # 0 ; y0 Þ is also 0 0 ˜ ; w; TÞ: By construction, observe that no player has a SPE strategy that yields y AXðd an incentive to deviate after Tðsðd0 ; y0 ÞÞ þ 1 or after observing a deviation from the equilibrium path of play. Therefore, suppose player i deviates at t choosing some action s0it : As bit% ¼ qit% ¼ 0; iAI; for all t%4t; denote i’s maximum payoff from such a 0 0 tþ1 i i 0i u ðx ðst ; s i;t ðsðd0 ; y0 ÞÞÞ where xi ðs0it ; s i;t ðsðd0 ; y0 ÞÞÞ be deviation by ud;i t ðd Þ ¼ ðd Þ the resulting allocation for i when he chooses s0it while all other players choose according to sðd # 0 ; y0 Þ: Observe that as sðd0 ; y0 Þ is itself a SPE, it must be the case that i’s maximum payoff from deviating from the equilibrium path of play under the 0 strategy profile sðd0 ; y0 Þ cannot be less than ud;i t ðd Þ: Therefore, if player i has no incentive to deviate from the equilibrium path of play under sðd0 ; y0 Þ; she cannot have an incentive to deviate from the equilibrium path of play under sðd # 0 ; y0 Þ: &

˜ 0 ; w; TÞ is supported by Given Lemma A.2, we can assume w.l.o.g. that any y0 AXðd 0 0 0 0 a SPE profile sðd # ; y Þ: We need to show that sðd * ; y Þ remains a SPE strategy profile 00 ˜ when d ¼ d : For each i; let Ti denote the final period when sit ðsðd * 0 ; y0 ÞÞa0: Then we ˜ must have, at each tpT˜ i ; ui ðxi ðst ðsðd * 0 ; y0 ÞÞÞpðd0 ÞTi t ui ðxi ðs ˜ ðsðd * 0 ; y0 ÞÞÞ and for all 0

Ti

t 4t; t pT˜ i ; ui ðxi ðs0i;t ; s i;t ðsðd * 0 ; y0 ÞÞÞpðd0 Þt t ui ðxi ðst0 ðsðd * 0 ; y0 ÞÞÞ: Finally, note that as 00 0 d 4d ; the above inequalities continue to hold when d0 is replaced by d00 : & 0

0

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