MISTAKES IN COOPERATION: THE STOCHASTIC ... IN COOPERATION: THE STOCHASTIC STABILITY OF EDGEWORTH’S RECONTRACTING* Roberto Serrano and Oscar Volij We analyse a dynamic trading process

MISTAKES IN COOPERATION: THE STOCHASTICSTABILITY OF EDGEWORTH’S RECONTRACTING*

Roberto Serrano and Oscar Volij

We analyse a dynamic trading process of coalitional recontracting in an exchange economy withindivisible goods, where agents may make mistakes with small probability. According to this process,the resistance of a transition from one allocation to another is a function of the number of agentswho make mistakes and of the seriousness of each mistake. If preferences are always strict, the uniquestochastically stable state is the competitive equilibrium allocation. In economies with indifferences,non-core cycles are sometimes stochastically stable, while some core allocations are not. Therobustness of these results is confirmed in a weak coalitional recontracting process.

0.1. Edgeworth’s Recontracting Process

The question of what outcomes will arise in decentralised trade amongst arbitrarygroups of agents dates back to Edgeworth (1881). Edgeworth argued that theseoutcomes consist of the set of �final settlements�, which is referred to as the core, i.e.,the set of allocations �which cannot be varied with advantage to all the contractingparties�. Although this definition of �final settlements� applies to a static exchangeeconomy, Edgeworth motivates them using a description of a recontracting pro-cess that is inherently dynamic. This process received elegant formalisations in theanalysis of Feldman (1974) and Green (1974) where it was shown that under certainassumptions on the economy, Edgeworth’s recontracting process converges to a coreallocation.1

In the present article we formalise Edgeworth’s recontracting process in a way similarto those of Feldman (1974) and Green (1974), except for two differences:

(1) agents make �mistakes� in their decisions2 and(2) the process is applied to the class of housing economies introduced by Shapley

and Scarf (1974).

One feature of these economies is that, since their set of feasible allocations is finite,they do not satisfy some of the Feldman-Green conditions. As a result, in these eco-nomies Feldman’s and Green’s recontracting systems, as well as ours, fail to converge tocore allocations. Indeed, in our specification of the dynamics, along with the coreallocations, some long-run predictions – recurrent classes – arise which consist solely of

* We thank Bob Aumann, Geoffroy de Clippel, Leonardo Felli, Philippe Jehiel, Michihiro Kandori, EricMaskin, Indra Ray and two anonymous referees for useful suggestions. Serrano acknowledges support fromFundaci�on Banco Herrero, Universidad Carlos III, NSF grant SES-0133113 and Deutsche Bank, and thanksUniversidad Carlos III and CEMFI in Madrid and the Institute for Advanced Study in Princeton for theirhospitality.

1 Another aspect of the relationship between dynamics and the core is provided by the literature on theextensive-form non-cooperative implementation of the core, e.g., Perry and Reny (1994) and Dagan et al.(2000).

2 Indeed, papers in cooperative game theory have no mistakes. We shall depart from this noble tradition.

The Economic Journal, 118 (October), 1719–1741. � The Author(s). Journal compilation � Royal Economic Society 2008. Published

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non-core allocations.3 One way to deal with this emergence of a large number ofrecurrent classes is to use refinement techniques, which can be accomplishedby adding mistakes and applying the concept of stochastic stability. Moreover, theconsideration of mistakes captures a realistic aspect of decision making.

0.2. Stochastic Stability in Economic Models

This article applies the concept of stochastic stability to the non-strategic context of aprivate ownership economy. Stochastic stability was first used in game theory by Kan-dori et al. (1993) and Young (1993), who applied it to 2 � 2 games as a selectiondevice.4 Since then it has been applied successfully to a wide variety of models (Young,1998). The concept of stochastic stability belongs to the �evolutionary� approach toexplaining the emergence of equilibria, conventions or social norms.5 In this approach,agents are not fully rational but �programmed� to behave in a fixed way until they arereplaced by other agents who are better prepared to face their environment, or byunlikely mutations.

In the usual application of stochastic stability to non-cooperative games one fixes aone-shot game in normal form (strategy sets and payoff functions) played by myopicplayers and one specifies a stochastic process to describe how players choose theirstrategies in each period. The process consists of three forces: inertia, selection andrandomness. Inertia is the driving force that acts most of the time: typically playersautomatically repeat what they played in the last period, or copy the strategy played bytheir immediate ancestors playing the game. At times, however, the system is affected byselection: a player may want to put some thought into his strategy, for example bychoosing it as a best response to a representative sample of past history. Finally,randomness further perturbs the system in the form of mutations or experimentation inplayers� choices. A natural question that then arises is what strategies are most likely tobe played in the long run. These strategies are called stochastically stable and havebeen shown to have interesting properties; see, for example, Kandori et al. (1993) andYoung (1993).

In this article we similarly consider a one-shot exchange economy (preferences andendowments) with myopic agents, which is repeated over time. We are interested in theevolution of the allocations when they follow a stochastic process that is composed ofthe three forces discussed above. The main force, within which the other two areembedded, can be seen as a flow of inertia. In each period the agents take their fixedinitial endowments and trade them for a given consumption bundle, repeating thepattern of trade of the previous period. These bundles constitute a status quo alloca-tion which is consumed every period, unless something disturbs the system. Inertia isinterrupted by a process of selection, based on coalitional recontracting. That is,in each period a coalition is chosen with some small probability, and it has the

3 This should be compared to the results obtained within the two-sided matching model introduced byGale and Shapley (1962). Roth and Vande Vate (1990), for instance, show that from any initial matching,there is always a path of blocking pairs leading to a matching in the core.

4 This methodology, based on the techniques developed for stochastic dynamical systems by Freidlin andWentzell (1984), was first applied to evolutionary biology by Foster and Young (1990).

5 Other references for evolutionary game theory in general are Weibull (1995), Vega-Redondo (1996),Samuelson (1997) and Young (1998).

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opportunity to reallocate its resources in a way that makes its members better off.If they find one such reallocation, they carry it out, thus sending the economy to a newstatus quo allocation, which will be repeated each period until the next disturbancetakes place. Finally, randomness is represented by a process of mistakes. When acoalition is selected to look for a profitable recontract, there is a small probability thatsome of its members will agree to a reallocation that makes them worse off. Thispersistent randomness ensures that the system does not get stuck at any given state.Instead, it keeps transiting all the time from one state to the next. Stochastic stabilitythen identifies those states (allocations, in this case) that are visited by the system apositive proportion of time in the very long run.

Fixing the preferences and endowment of the economy allows us to compare thestochastically stable allocations to the ones prescribed by classical solution concepts,such as the core or competitive equilibrium. Therefore, our goal is to identify thoseallocations that will be visited by Edgeworth’s recontracting process a significant pro-portion of time in the long run if mistakes are small probability events that all agentsmay make all the time.6

0.3. Summary of Results

We present two kinds of results. First, under certain conditions, stochastic stability willserve as a powerful refinement tool and provide a new foundation of competitiveoutcomes. And second, more generally, stochastic stability will highlight that somesolution concepts, such as the core, might be lacking an aspect of coalitional inter-actions that stochastic dynamics may help to capture.

To be more precise, when we analyse the recontracting process free of mistakes, eachcore allocation constitutes an absorbing state and, in addition to them, subsets of non-core allocations constitute non-singleton recurrent classes. However, when mistakes areintroduced into the model our results will depend on the class of economies underconsideration.

In economies where all preferences are strict, we are able to obtain a remarkablerefinement of the set of recurrent classes by perturbing the system with mistakes.Theorem 1 states that the unique stochastically stable state of the recontracting processwith mistakes is the competitive allocation, provided that serious mistakes (those bywhich agents end up worse off) are sufficiently more costly than minor ones (those bywhich agents end up neither worse nor better off). This result relies on the property of�global dominance� of the competitive allocation in economies with no indifferences, asidentified by Roth and Postlewaite (1977). Global dominance states that for everyfeasible allocation, there exists a coalition whose members can (weakly) improve uponthe status quo by using their competitive bundles. This property is of course confinedto the housing market setting and cannot be extended to general economies.Nonetheless, the housing market is of interest and within this setting our resultprovides a new foundation for competitive allocations.

6 In contexts closer to a social choice model, where only preferences and not the endowments are part ofthe primitives, Ben-Shoham et al. (2004) and Kandori et al. (2008) explore related models that use mistakes indecision making or random utility maximisation.

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The conclusions of our analysis are quite different for the more general case ofeconomies with indifferences, where we obtain non-competitive results. We provide aseries of examples to illustrate that the predictions of stochastic stability will notcoincide with any of the classical solution concepts. In particular, we regard Example 5as the other main result of the article in that it shows that non-core (and hence, non-competitive) cycles are sometimes stochastically stable, whereas some core allocationsare not. This is worth stressing for two reasons. First, when mistakes are introduced intothe recontracting process the economy will frequently visit coalitionally unstableallocations. Second, entire regions of the core will be reached a zero proportion of timein the very long run. This suggests that there is a different level of �decentralisation�behind each core allocation. Whereas some allocations may arise as a result of a simplesequence of trade and are therefore easy to reach, others will require complicatedtransactions and, hence, will not be reached as easily.

As a robustness check of our results, we also study a second process based on weakcoalitional recontracting. We find that the stochastic stability criterion always selects anon-empty subset of competitive allocations when the unperturbed process containsonly singleton recurrent classes (in particular, in economies with only strict prefer-ences, the competitive allocation is the unique recurrent class of this unperturbedprocess). If the mistake-free process exhibits also non-singleton recurrent classes,however, stochastic stability is compatible with the appearance of cycles containingnon-competitive allocations.

0.4. Recontracting with Mistakes and Edgeworth’s Tradition

In this subsection we argue that our process of recontracting with mistakes is a faithfulformalisation of the recontracting process envisioned by Edgeworth. In particular, wedevote special attention to the issue of the timing of consumption.

Our model has several features that deserve emphasis:

1 There is a static exchange economy.2 This economy is repeated over time.3 In each period trade results in a potentially different allocation and the

resulting bundles are, presumably, consumed.

When Edgeworth describes the process of contracting and recontracting, it is clearthat he has a static exchange economy in mind:

. . .the disposition and circumstances of the parties are assumed to remainthroughout constant. (Edgeworth, 1925, p. 313.)

In this context, the recontracting process consists of a series of

. . .agreements [that] are renewed or varied many times. (Edgeworth, 1925,p. 314.)

Note that the renewal of contracts is consistent with a market that takes place peri-odically over time. This is also suggested by the fact that Edgeworth chose the labourmarket to exemplify his concepts. Workers and entrepreneurs show up in the marketevery period with the same endowment and preferences. In particular, entrepreneurs



do not sell the labour units that they bought in the previous period. The renewal ofcontracts is also consistent with the landlord-tenant relationship, another examplementioned by Edgeworth (see below). Further, in his Mathematical Psychics (see page 17,for instance), Edgeworth uses the expression �dealers in commodity x� and �dealers incommodity y� to refer to the economic agents, emphasising the persistence of theproperty rights based on the initial endowment allocation.

Edgeworth is interested in the �final settlements�. According to Edgeworth a �finalsettlement� consists of

a set of agreements which cannot be varied with advantage to all the recon-tracting parties. (Edgeworth, 1925, p. 314.)

The market presumably reaches a final settlement after going through various �set-tlements� or �temporary equilibria�. Edgeworth, however, is not clear as to whetherconsumption takes place at each temporary equilibria or only at a final settlement.Indeed, when he explains the meaning of recontracting (or blocking, in the moderneconomist’s jargon) he gives two examples which point to different directionsconcerning the issue of the timing of consumption (emphasis in the original):

Thus an auctioneer having been in contact with the last bidder (to sell at such aprice if no higher bid) recontracts with a higher bidder. So a landlord on expiryof lease recontracts, it may be, with a new tenant. (Edgeworth, 1881, p. 17.)

In the first example there is no transaction (and a fortiori no consumption) betweenthe auctioneer and the last bidder, due to the appearance of a higher bidder. In thesecond example, in contrast, there is a transaction every period between a landlord anda tenant that leads to consumption. Here, although each period’s contract is just asettlement, it is not necessarily a final one.

Edgeworth proposed the core as an alternative to the competitive equilibriumallocations, identified several years earlier by Walras (1874). He found that most of thetimes the core was indeterminate as a solution concept, since it prescribes a large set offinal settlements. This contrasted with the more determinate concept due to Walras(which was believed to be unique). For this reason, Edgeworth was especially interestedin this question of uniqueness of the core and was the first to notice the connectionbetween the core and competitive allocations in large economies.7

For Edgeworth’s main purpose, the timing of consumption is irrelevant. One caninterpret his temporary equilibria, very much in the competitive tattonement tradition,as agreements that are not carried out but which are intermediate steps leading to afinal settlement; and alternatively, one can interpret his settlements as agreements thatare carried out each period without affecting the �disposition and circumstances� of theparties for the next period. Edgeworth’s comparison of the core to Walras’s equilib-rium can be understood adopting either interpretation.

7 This observation gave rise to the important core convergence/equivalence literature (Debreu and Scarf,1963; Aumann, 1964) as one of the leading game theoretic justifications of competitive equilibrium. SeeAnderson (1992) and Aumann (1987) for surveys. Although the robustness of the equivalence result isremarkable, several violations thereof have been identified, from which one can learn the role of certainfrictions in markets. These references include Anderson and Zame (1997) for infinite dimensional com-modity spaces, Manelli (1991) for instances of non-monotonicity in preferences, and Serrano et al. (2001) forasymmetric information.



However, when one focuses on the dynamic process of recontracting, the inter-pretation of the states of the system gains importance. The dynamic systems proposedby Feldman (1974) and Green (1974), since they do not have random perturbations,may potentially (and in fact, they actually do) converge to an absorbing state. There-fore they are consistent with an interpretation in which consumption only takes placeafter convergence is achieved. In our model, when agents make random mistakes, thesystem has no absorbing states. Therefore an interpretation more in line with thelandlord–tenant example, in which there are transactions and consumption everyperiod, is more appropriate.

There is yet another conceptual advantage to our approach. Essential to Edgeworth’srecontracting process is the idea that any individual is free to contract and recontractwithout the consent of third parties. However, note that within the first interpretation there isno consumption until all agents reach an agreement. That is, no one can consume untilit is certain that every agent has reached a final settlement. In fact, it may well be the casethat along the path toward the final settlement, some agents hold bundles that arestrictly preferred to the bundle that is consumed by them at any final settlement. Post-poning consumption until the very end seems to contradict the spirit of recontractingwithout the consent of others. If one can recontract without the permission of others,then it would appear that one should be able also to consume without their consent.

0.5 Plan of the Article

The article is organised as follows. Section 1 is concerned with preliminaries: the modeland basic definitions, the unperturbed recontracting process, and its perturbed versionwith mistakes. Section 2 contains our main results. To check their robustness, Section 3analyses an alternative model of weak recontracting and mistakes. Section 4 concludes.The proof of the main theorem appears in the Appendix.

1. Preliminaries

1.1. A Housing Economy

We shall pose our questions in the context of the housing model of Shapley and Scarf(1974), namely a simple exchange economy with only indivisible goods; see Roth et al.(2004) for a recent successful application of the model.

Formally, a housing economy is a 4-tuple E � hN,H,(�i,ei)i2Ni, where N is a finite set ofindividuals, H is a finite set of houses with the number of houses (jH j) being equal tothe number of individuals (jN j). For each individual i 2 N, �i is a complete andtransitive preference relation over H, with Ci denoting its associated strict preferencerelation and �i its indifference relation. Finally, (ei)i2N is the profile of individualendowments: each individual is endowed with one house.

A coalition S of agents is a non-empty subset of N. The complement coalition of S,N nS, will sometimes be denoted by �S. A feasible allocation for coalition S is a redis-tribution of the coalitional endowment (ei)i2S among the members of S. Denote the setof feasible allocations for coalition S by AS. We simply write A for AN. An allocationx 2 A is individually rational if there is no individual i 2 N for whom ei Ci xi.



An allocation x 2 A is a core allocation if there is no coalition S and no feasible allocationfor S, y 2 AS, such that yi Ci xi for all i 2 S. An allocation x 2 A is a strong core allocation ifthere is no coalition S and no feasible allocation for S, y 2 AS, such that yi Ci xi for alli 2 S and yj Cj xj for some j 2 S. An allocation x 2 A is a competitive allocation if thereexist positive competitive equilibrium prices p such that for all i 2 N pxi

¼ pei, and for all

i 2 N and for all h 2 H, h Ci xi implies ph > pei. It has been shown (Shapley and Scarf,

1974) that an allocation x is competitive if and only if it can be obtained as a result oftrading cycles. That is, if and only if there exists a partition of the set of agentsfS1,S2,. . .,Smg such that for every k ¼ 1,2,. . .,m,

xSk 2 ASk ; and for every j 2 Sk ; xj �j xi for every i 2 Sk [ . . . Sm :

In words, the agents in S1 redistribute their endowments and get their most preferredhouses; the agents in S2 redistribute their endowments and get their most preferredhouses out of the endowments of S2 [. . .Sm, and so forth.

In the following subsections, we shall define two Markov processes for any givenhousing economy, one being a perturbation of the other. The states of the process arethe allocations of the housing economy. In each period a coalition of agents is selectedat random and the system moves from one state to another when the matched agentsrecontract. In the unperturbed Markov process M0 of Subsection 1.2, agents do notmake mistakes in their coalitional meetings: they sign a contract if and only if there is astrictly beneficial coalitional recontracting opportunity. In the perturbed processM� ofSubsection 1.3, agents will make mistakes with a small probability and sign a contractwith a coalition even when they do not improve as a result. In Section 3 we studyMarkov trading processes in which coalitions can find a weak (rather than strict)coalitional recontracting move.

It is often the case that an unperturbed Markov process (and it will certainly be thecase for M0) has many long-run predictions – recurrent classes – and thus manystationary or invariant distributions. As such, the long-run behaviour of the systemcrucially depends on the initial state. On the other hand, for each small � > 0, theperturbed process M� is ergodic, which implies that it has a unique stationary dis-tribution, denoted by l�. This stationary distribution, which is independent of theinitial state, represents the proportion of time that the system will spend on each of itsstates in the long run. It also represents the long-run probability that the process willbe at each allocation. In order to define the stochastically stable states, which are ourobjects of interest, we check the behaviour of the stationary distribution l� as � goes to0. It is known that lim�!0 l� exists and further that it is one of the stationarydistributions of the unperturbed processM0. The stochastically stable states of the systemM� are defined to be those states that are assigned positive probability by this limitdistribution. We are interested in identifying these allocations because they are theones that are expected to be observed in the long run �most of the time�.

1.2. An Unperturbed Recontracting Process

Consider the following unperturbed Markov process M0, adapted from Feldman(1974) and Green (1974). In each period t, if the system is at the allocation x(t), one



coalition is chosen according to an arbitrary distribution which assigns positive prob-ability to each coalition. Suppose coalition S is chosen.

(i) If there exists an S-allocation yS 2 AS such that yi Ci xi(t) for all i 2 S, the coa-lition moves with positive probability to one of such yS in that period. Then, thenew state is either

xðt þ 1Þ ¼ ðyS ; x�SðtÞÞ if x�SðtÞ 2 A�S ; or

xðt þ 1Þ ¼ ðyS ; e�SÞ if x�SðtÞ j2A�S :

(ii) Otherwise, x(t þ 1) ¼ x(t).

The interpretation of the process is one of coalitional recontracting. Following astatus quo x(t), a coalition can form and modify the status quo if all members of thecoalition improve as a result. When this happens, upon coalition S forming, thecomplement coalition N nS continues to have the same houses as before if this isfeasible for them. Otherwise, N nS breaks apart and each of the agents in it receives hisindividual endowment.8 If after coalition S forms all its agents cannot find any strictimprovement, the original status quo persists.

1.3. A Perturbed Recontracting Process

Next we introduce the perturbed Markov process M� for an arbitrary small� 2 (0,1), a perturbation of M0. Suppose the state of the system is the allocation xand that coalition S meets. We shall say that a member of S makes a �mistake� whenhe signs a contract that either leaves him indifferent to the same house he alreadyhad or he becomes worse off upon signing. Each of the members of S may makeone of these �mistakes� with a small probability, as a function of � > 0, indepen-dently of the others. Specifically, for a small fixed � 2 (0,1), we shall postulate thatan agent’s probability of agreeing to a new allocation that leaves him indifferentis �, while the probability of agreeing to an allocation that makes him worse off is �k

for a sufficiently large positive integer k. That is, the latter mistakes are much lesslikely than the former, while both are rare events in the agent’s decisionmakingprocess.9

Before we define the perturbed process, we need some notation and definitions.Consider an arbitrary pair of allocations z and z 0. Let T(z, z 0) be the set of coalitionssuch that, if chosen, can induce the perturbed system to transit from z to z0 in onestep. Note that it is always the case that T(z, z0) is non-empty, since N 2 T(z, z0) forany z and z 0.

8 Our treatment of the complement coalition constitutes a small difference with respect to the processes inFeldman (1974) and Green (1974): in these papers, when recontracting happens, the complement coalitionis sent back always to their individual endowments in Feldman (1974), or to a Pareto efficient allocation oftheir resources in Green (1974). Our results are robust to different specifications; for instance, if only thoseagents in N nS who are directly or indirectly affected by the reallocation proposed by S are sent to their initialendowments, while the rest of the economy stays put.

9 Following Bergin and Lipman (1996), the results will depend on the specification of these probabilities.Our results are robust to any specification where a transition that makes an agent worse off is less likely thanone in which the only frictions are indifferences.



In the direct transition from z to z0 and for each S 2 T(z,z0), define the followingnumbers:

nI ðS ; z; z0Þ ¼ jfi 2 S : zi �i z0igj;nW ðS ; z; z0Þ ¼ jfi 2 S : zi �i z0igj;

nðS ; z; z0Þ ¼ knW ðS ; z; z0Þ þ nI ðS ; z; z0Þ:

Here nI(S, z, z0) is the number of individuals in S that are indifferent between alloca-tions z and z0. Similarly, nW(S, z, z0) is the number of members of S that make a mistakeby agreeing to move from z to z0, worse for them. Finally, n(S, z, z0) is the weightednumber of mistakes that are made in the transition from z to z0, where k > 1 is theweight given to �serious� mistakes relative to those that lead to just indifference.

In the perturbed Markov process M� the transition probabilities are calculated asfollows. Suppose that the system is in allocation z. All coalitions are chosen with a fixedpositive probability. Assume coalition S is chosen. If S j2 T(z, z0), then S moves to z0 withprobability 0. If S 2 T(z, z0) and n(S, z, z0) > 0, then coalition S agrees to move to z0

with probability �n(S,z,z0). If S 2 T(z, z0) and n(S, z, z0) ¼ 0, coalition S moves to those z0

with some (possibly state-dependent) probability d, where 0 < d < 1/jAj.For all � 2 (0,1) small enough, the system M� is a well-defined irreducible Markov

process, i.e., one can go from any allocation to any other one with positive probabilityin a finite number of periods. As such, it has a unique invariant distribution. Thisdistribution is a good estimate of the probability that the system is in each of theallocations in the long run. We are interested in the limit of the correspondinginvariant distributions as � tends to 0. More precisely, our focus will be on theallocations that are assigned positive probability by this limiting distribution. Theseallocations are called stochastically stable allocations.

In order to obtain our results, we will use the techniques developed by Kandori et al.(1993) and Young (1993). But before that, we need some additional definitions.

Note that by the definition of the perturbed Markov process M�, for every twoallocations z and z0, the direct transition probability lz,z0(�) converges to the limittransition probability lz,z0(0) of the unperturbed processM0 at an exponential rate. Inparticular, for all allocations z, z0 such that lz,z0(0) ¼ 0, the convergence is at a rater(z, z0) ¼ minS2T(z,z0)knW (S, z, z0) þ nI(S, z, z0), that is, the weighted number of mis-takes associated with the most likely direct transition. We call the value r(z, z0) theresistance of the direct transition from allocation z to allocation z0. For any two allocationsz, z0, a (z, z0)-path is a sequence of allocations n ¼ (i0,i1,. . .,im) such that i0 ¼ z, im ¼ z0.The resistance of the path n is the sum of the resistances of its transitions. We seek to findthe path of least resistance between any two allocations.

Let Z 0 ¼ fE 0, E 1, . . ., E Qg be the set of recurrent classes of the unperturbed processM0 – each recurrent class E j is a set of states with the property that if the process M0

reaches a state in the class, it will never get out of that class. Consider the completedirected graph with vertex set Z 0. We want to define the resistance of each one of theedges in this graph. For this, let E i and E j be two elements of Z 0. The resistance of the edge(E i,E j) in the graph, r(E i, E j), is the minimum resistance over all the resistances of the(zi, z j)-paths, where zi 2 E i and z j 2 E j. A spanning tree rooted at E j, or an E j-tree, is aset of directed edges such that from every recurrent class different from E j, there is a



unique directed path in the tree to E j. The resistance of an E j-tree is the sum of theresistances of its edges. The stochastic potential of the recurrent class E j is the minimumresistance attained by an E j-tree: it captures the smallest friction to get to class E j fromany of the other recurrent classes. As shown in Young (1993), the set of stochasticallystable states of the perturbed process M� consists of those states belonging to therecurrent classes with minimum stochastic potential.

2. Main Results

Let us begin by analysing the long-run predictions of the unperturbed processM0. It isclear that the absorbing states of this unperturbed process are precisely the coreallocations of the economy. However, the absorbing states are not the only recurrentclasses of M0, as shown by the following example.

Example 1. Let N ¼ f1, 2, 3g and denote by the individual endowment allocation(e1, e2, e3). Let the agents� preferences be as follows:

e3 �1 e2 �1 e1;e1 �2 e3 �2 e2;e2 �3 e1 �3 e3:

Consider the following three allocations: x ¼ (e1, e3, e2), y ¼ (e2, e1, e3) and z ¼(e3, e2, e1). These three allocations constitute a recurrent class: if the system is at x, thestate changes only when coalition f1, 2g meets, yielding y. At y, the system can moveonly to z, when coalition f1, 3g meets. Finally, the system will move out of z only bygoing back to x, when coalition f2, 3g meets.

Note that the unique competitive allocation w ¼ (e3, e1, e2) also constitutes a sin-gleton recurrent class.

We can prove the following result, which characterises the recurrent classes of theunperturbed process M0:

Proposition 1. The recurrent classes of the unperturbed recontracting process M0 take thefollowing two forms:

(i ) Singleton recurrent classes, each of which contains each core allocation.(ii ) Non-singleton recurrent classes: in each of which, the allocations are individually rational

but are not core allocations.

Proof. It is clear that each core allocation constitutes an absorbing state of M0 andthat every absorbing state must be a core allocation. For the second form of recurrentclass, note that, by construction of the system, no state in a recurrent class can ever benon-individually rational: if at some state x in which ei Ci xi the coalition fig is chosen,then, the system moves to the individual endowment e, never to return to a non-individually rational allocation. It is also clear that each of the states in the recurrentclass cannot be absorbing, i.e., cannot be a core allocation.

Thus, each core allocation is an absorbing state of the unperturbed Markov processM0 and, in principle, there may be additional non-singleton recurrent classes, such as



that shown in Example 1. Note also that as soon as the economy has more than onecore allocation, the system M0 has many stationary distributions. This multiplicity oflong-run predictions justifies the consideration of the perturbed process based onmistakes, whose results we shall report next.

2.1. Economies with Strict Preferences

In this subsection we shall assume that for every agent i 2 N the preference relation �i

is antisymmetric, which implies that all indifference sets are singletons: for all i 2 N,h �i h0 if and only if h ¼ h0. Making this assumption, Roth and Postlewaite (1977)proved the following result:

Lemma 1. Let E be a housing economy where all preferences are strict. Then,

(i ) There is a unique competitive allocation w.(ii ) The allocation w is the only strong core allocation.

(iii ) Global dominance� of w: For every allocation x 2 A, x 6¼ w, there exists a coalition Ssuch that wS is feasible for S and satisfies wi �i xi for all i 2 S and wj Cj xj for somej 2 S.

Lemma 1 will be useful in proving our first main result, which we state as follows:

Theorem 1. Let E be a housing economy where all preferences are strict. Suppose thatk > jN j � 2. Then, the unique stochastically stable allocation of the perturbed recontractingprocess with mistakes M� is the competitive allocation w.

The proof of Theorem 1 is in the Appendix. It combines the different parts ofLemma 1. Specifically, its parts (ii) and (iii) imply that the competitive allocation w is astrong dynamic attractor in the trading process with mistakes. Theorem 1 provides aremarkable refinement of the set of long-run predictions of the unperturbed process,as identified in Proposition 1.

2.2. Economies with Indifferences

In this subsection we explore how the stochastic process of recontracting with mis-takes, M�, performs over the class of economies that allow non-singleton indifferencesets for some agents. Over this larger class of economies, Proposition 1 still holds.However, the conclusions of Lemma 1 do not extend. First, although existence isguaranteed, there may be multiple competitive allocations. Second, the strong coremay also contain multiple allocations, while it may sometimes be empty. Third, the�global dominance� property of competitive allocations as specified in Lemma 1, part(iii), is also lost.

In economies with only strict preferences, the competitive allocation correspondenceand the strong core coincide and, under our assumption on k, the same allocation isthe only one that passes the test of stochastic stability. It is convenient, therefore, toexamine the larger class of economies to disentangle the different forces at work in theselection of the stochastically stable allocations. We find that the recontracting systemwith mistakes gives rise to complicated dynamics and no general result of equivalence



can be established. Therefore, we learn that the paths of least resistance followed in ourstochastic dynamic analysis are not intrinsically associated with the strong core propertyor with the competitive property of allocations.

We present four examples. We arrange them by increasing difficulty and relevance.Indeed, we regard Example 5 as the other main result of the article. In some of theseexamples, one agent has a completely flat indifference map but this is only for sim-plicity of exposition. Also, in the examples we shall use the notation z !r

S z0 to expressthat the transition of least resistance from z to z0 takes place through coalition S with aresistance r.

We begin by showing that the set of stochastically stable allocations is not the strongcore. As just mentioned, the strong core may be empty in these economies, whilestochastic stability always selects at least one allocation; but even when the strong core isnon-empty, one can generate examples where it does not coincide with the set ofstochastically stable states of M�.

Example 2. In this example, a non-empty strong core is strictly contained in the setof stochastically stable allocations. Let N ¼ f1,2g and agents� preferences be describedas follows:

e1 �1 e2;

e1 �2 e2:

In this economy there are two allocations, both of which are competitive, but onlythe one resulting from trade is in the strong core. Note that both allocations arestochastically stable: ðe1; e2Þ !1

N ðe2; e1Þ and ðe2; e1Þ !1f1g ðe1; e2Þ.

The next example shows that the set of stochastically stable allocations may be a strictsubset of the strong core and of the set of competitive allocations.

Example 3. Let N ¼ f1, 2, 3g and the agents� preferences be given by:

e1 �1 e2 �1 e3;

e1 �2 e3 �2 e2;

e1 �3 e2 �3 e3:

In this economy, all allocations except the initial endowment allocation are competitiveand belong to the core. The strong core consists of the following three allocations:(e2, e3, e1), (e3, e1, e2) and (e1, e3, e2). The unique stochastically stable allocation is x ¼(e1, e3, e2). To see that x is the only allocation with minimum stochastic potential, onecan construct an x-tree as follows. First, we note that the only recurrent classes of M0

are the five absorbing states corresponding to each competitive allocation. Next, notethat to go from (e2, e3, e1) to x can be done with a resistance of 1 (only one in-difference): ðe2; e3; e1Þ !1

f1g ðe1; e2; e3Þ !0f2;3g x. The same goes for the transition

(e3, e1, e2) to x: ðe3; e1; e2Þ !1f1g ðe1; e2; e3Þ !0

f2;3g x. As for the other transitions, we haveðe3; e2; e1Þ !1

f2;3g x and ðe2; e1; e3Þ !1f2;3g x. Therefore, the resistance of this x-tree is 4 and

one cannot build a cheaper tree than that. On the other hand, to get out of x, theresistance will always be at least 2, i.e., at least two indifferences, which implies that inconstructing a tree for any of the other recurrent classes its resistance must be at least 5.



Our next example illustrates the only substantive difference in results between therecontracting processM� and its counterpart based on weak recontracting that will bestudied in the following Section. Specifically, we now construct an economy for whichall recurrent classes of M0 are singletons and where the set of stochastically stableallocations of M� contains an allocation that is not competitive.

Example 4. Let N ¼ f1, 2, 3, 4g and let the agents� preferences be described asfollows:

e1 �1 e3 �1 e4 �1 e2;

e1 �2 e4 �2 e3 �2 e2;

e2 �3 e1 �3 e3 �3 e4;

e2 �4 e3 �4 e4 �4 e1:

One can check that there are four competitive allocations: w1 ¼ (e3, e4, e1, e2), w2 ¼(e1, e4, e3, e2), w3 ¼ (e4, e1, e3, e2), and w4 ¼ (e3, e1, e2, e4). In addition, there are threeother core allocations that are not competitive: y1 ¼ (e1, e4, e2, e3), y2 ¼ (e4, e1, e2, e3),and y3 ¼ (e4, e3, e1, e2). Since Proposition 1 also applies to the economies considered inthis subsection, it follows that each of these seven allocations constitutes an absorbingstate of M0. Further, it can be checked that there are no additional recurrent classes.Apart from the seven core allocations, the only absorbing states of M0, there are fourmore individually rational allocations. One is e ¼ (e1, e2, e3, e4) - the initial endowmentallocation - and the other three are x1 ¼ (e1, e3, e2, e4), x2 ¼ (e3, e2, e1, e4) and x3 ¼(e4, e2, e1, e3). It can be checked that from each of these four allocations one can gowith zero resistance to one of the core allocations.

Consider now the following fy1g-tree comprising the following edges: w1 ! y1,w2 ! w1, w3 ! w1, w4 ! w2, y2 ! w2, y3 ! y1. We check next that each of these edgeshas resistance 1 (corresponding to a transition with only one indifference), therebyimplying that the tree has minimum stochastic potential. These transitions can bedetailed as follows:

w1 !1f1g e !0

f2;3;4g y1;

w2 !1f1;3g x2 !0

f2;4g w1;

w3 !1f1;3g x2 !0

f2;4g w1;

w4 !1f2;4g w2;

y2 !1f2;4g w2;

y3 !1f1g e !0

f2;3;4g y1:

Thus, y1 is stochastically stable, although it is not a competitive allocation. It is not inthe strong core either; indeed, the strong core is empty in this economy.



The next example shows that in a dynamic model where agents may make mistakesin decision making, the core may not agree with the set of states that are visited by theprocess a positive proportion of time. In contrast, some non-core allocations may farebetter in this sense than some core allocations.

Example 5.10 This example, an outgrowth of Example 1, shows that a cycle of non-

core allocations may be stochastically stable, at the same time as some core allocationshaving higher stochastic potential. Let N ¼ f1, 2, 3, 4g, and let the agents� preferencesbe as follows:

e4 �1 e3 �1 e2 �1 e1;

e1 �2 e3 �2 e2 �2 e4;

e2 �3 e1 �3 e3 �3 e4;

e1 �4 e2 �4 e3 �4 e4:

Consider the following three allocations: x ¼ (e1, e3, e2, e4), y ¼ (e2, e1, e3, e4) and z ¼(e3, e2, e1, e4). Since agent 4 cannot strictly improve, he cannot be part of any blockingcoalition. In fact, as in Example 1, these three allocations constitute a recurrent class: ifthe system is at x, the state changes only when coalition f1, 2g meets, yielding y. At y,the system can move only to z, when coalition f1, 3gmeets. Finally, the system will moveout of z only by going back to x, when coalition f2, 3g meets. That is,x !0

f1;2g y !0f1;3g z !0

f2;3g x.The core consists of the following five allocations: c1 ¼ (e3, e1, e2, e4), c2 ¼

(e4, e1, e2, e3), c3 ¼ (e4, e1, e3, e2), c4 ¼ (e4, e3, e1, e2) and c5 ¼ (e4, e3, e2, e1). It is easy tosee that these five absorbing states i.e., the core allocations and the cycle are the onlyrecurrent classes of the unperturbed system: the twelve allocations where e4 is allocatedto either agent 2 or agent 3 are not even individually rational. And from each ofthe remaining four allocations, one gets to one of the already identified recurrentclasses with 0 resistance: a1 ¼ ðe1; e2; e3; e4Þ !0

f1;2;3g c1, a2 ¼ ðe2; e3; e1; e4Þ !0f1;2;3g c1,

a3 ¼ ðe4; e2; e1; e3Þ !0f2;3g x and a4 ¼ ðe4; e2; e3; e1Þ !0

f2;3g c5.Let E ¼ fx, y, zg be the non-singleton recurrent class consisting of non-core alloca-

tions. Next, we construct an E-tree and show that it has minimum stochastic potential.This tree must have five edges, coming out of each of the five core allocations. Wedetail the transitions below:

c1 !1f1;4g a4 !0

f2;3g c5;

cj !1f4g a1 !0

f1;2g y; for j ¼ 2; 3; 4; 5:

Therefore, the class E has minimum stochastic potential.Note also that there are four competitive allocations: c1, c2, c3 and c5. Thus, the

example also shows that there are non-competitive stochastically stable allocations.

10 As communicated to us by Bob Aumann, a similar story is told in the Talmud: there is a cycle of threetwo-person coalitions improving the status quo. The three players involved in the cycle are the two wives of aman and a third party who buys the man’s estate. The cycle occurs after the man died and left his estate. Thedeceased agent naturally corresponds to our agent 4, who has a flat indifference map. See also Binmore(1985) for a more recent related problem.



However, apart from E, the only stochastically stable allocations are the four com-petitive allocations c1, c2, c3 and c5: there are allocations in the core that are not visitedin the long run but a zero proportion of time. In particular, it can be checked that ittakes at least two indifferences to get out of other recurrent classes to go to c4. Thisimplies that c4 cannot be stochastically stable. To illustrate, we construct a fc4g-tree asfollows:

c1 !1f1;4g a4 !0

f2;3g c5;

cj !1f4g a1 !0

f1;2g ðy 2 EÞ; for j ¼ 2; 3; 5;

ðz 2 EÞ !2N c4:

Thus, this dynamic analysis, which sheds light on how each allocation emerges,makes a distinction among different core allocations. Some core allocations (like c4 inthe example) are actually very hard to get to, when compared to other allocations inthe core or even outside of the core. The point is that the usual emphasis in thedefinition of the core is given to the non-existence of a coalitional blocking move,whereas the issue of how the allocation in question emerges is disregarded. Dynamicsshould have something to say about the matter and Example 5 is a confirmation thatthis is indeed the case.

3. Recontracting Based on Weak Blocking

As a robustness check of our results, this Section analyses a Markov process in whichtransitions take place whenever there is an instance of weak coalitional blocking,instead of the strict version thereof, as was the case in the process M0.

3.1. The Unperturbed Process

Consider the following unperturbed Markov process M0W , which represents the �weak

recontracting� version of the process presented in Subsection 1.2. In each period t, thesystem is at the allocation x(t), and one coalition is chosen according to an arbitraryprobability distribution that assigns positive probability to each coalition. Supposecoalition S is chosen.

(i) If there exists an S-allocation yS 2 AS such that yi �i xi(t) for all i 2 S andyj Cj xj(t) for some j 2 S, the coalition moves with positive probability to someof such yS in that period. Then, the new state is either

xðt þ 1Þ ¼ ðyS ; x�SðtÞÞ if x�SðtÞ 2 A�S ; or

xðt þ 1Þ ¼ ðyS ; e�SÞ if x�SðtÞ j2A�S :

(ii) Otherwise, x(t þ 1) ¼ x(t).



The interpretation of this new process is one of weak coalitional recontracting.Following a status quo, a coalition can form and modify the status quo if at least onemember of the coalition improves as a result, and if none of the other members ismade worse off. Apart from this change, the process is identical to the one inSubsection 2.2. That is, when the weak coalitional recontracting move occurs, uponcoalition S forming, the complement coalition N nS continues to have the samehouses as before, if this is feasible for them. Otherwise, N nS breaks apart and eachof the agents in it receives his individual endowment (the same comment as infootnote 8 applies here in terms of alternative specifications). If after coalition Sforms, it cannot find any such weak coalitional improvement, the original status quopersists.

The absorbing states of the unperturbed process M0W are precisely the strong core

allocations of the economy. However, the absorbing states are not the only recurrentclasses ofM0

W . The easiest way to see this is to note that there are economies in whichthe strong core is empty.

We can prove the following result, similar to Proposition 1, which characterises therecurrent classes of the unperturbed process M0

W :

Proposition 2. The recurrent classes of the unperturbed weak recontracting processM0W take

the following two forms:

(i) Singleton recurrent classes, each of which contains each strong core allocation.(ii) Non-singleton recurrent classes, in each of which, the allocations are individually rational

but are not strong core allocations.

Proof. The proof is similar to that of Proposition 1.

Thus, each strong core allocation is an absorbing state of the unperturbed MarkovprocessM0

W , and in principle there may be additional non-singleton recurrent classes.Note also that as soon as the economy has more than one strong core allocation, thesystemM0

W has many stationary distributions. The conclusions of Proposition 2 extendto all economies where preferences are complete and transitive, with or withoutindifferences.

One can now obtain a simple generalisation of the conclusions of Theorem 1. In fact,the competitive result is obtained even before mistakes are added to the process.

Proposition 3. Let E be a housing economy where all preferences are strict. Then, theunperturbed weak recontracting processM0

W is ergodic and the only recurrent class is the singletonconsisting of the unique competitive allocation.

Proof. The proof follows from the definition of the transitions in the process andfrom Lemma 1.

Notice how, as in Theorem 1, the �global dominance� property of the competitiveallocation in these economies makes it the only prediction of the dynamic process, thistime as the strong attractor of the unperturbed system (see Example 1 for an illus-tration).



3.2. A Perturbed Process

We next attempt to generalise the non-competitive conclusions obtained by ourdynamic analysis in economies with indifferences. For this purpose, we introduce theperturbed Markov process M�

W for an arbitrary small � 2 (0,1), a perturbation ofM0

W . Suppose the state of the system is the allocation x and that coalition S meets.We shall say that a member of S makes a �mistake� when he signs a contract thatmakes him worse off upon signing. Each of the members of S may make a �mistake�with a small probability, as a function of � > 0, independently of the others. Speci-fically, for a small fixed � 2 (0,1), we shall postulate that an agent’s probability ofagreeing to an allocation that makes him worse off is �. In addition, we shall say thatthe coalition S makes an �indifference-based coalitional mistake� if it agrees to a tradewhich leaves each and every one of its members exactly indifferent. In consonancewith our assumption on k made in Subsection 1.3 (see footnote 9), indifference-basedcoalitional mistakes are somewhat less serious than individual mistakes. On the otherhand, they will be more unlikely the larger the size of the coalition involved. Detailsare provided below.

Before we define the perturbed process, we need some notation and definitions.Consider an arbitrary pair of allocations z and z0. Let T(z, z0) be the set of coalitionssuch that, if chosen, can induce the system to transit from z to z0 in one step. Again,note that it is always the case that T(z, z0) is non-empty since N 2 T(z, z0) for any zand z0.

In the direct transition from z to z0 and for each S 2 T(z, z0), define the followingnumbers:

nI ðS ; z; z0Þ ¼ jfi 2 S : zi �i z0igj;nW ðS ; z; z0Þ ¼ jfi 2 S : zi �i z0igj

In the perturbed Markov process M�W the transition probabilities are calculated as

follows. Suppose that the system is in allocation z. All coalitions are chosen with a fixedpositive probability. Assume coalition S is chosen. If S j2 T(z, z0), then the system movesto z0 with probability 0. If S 2 T(z, z0) and nW(S, z, z0) > 0, then coalition S agrees tomove to z0 with probability �nW(S,z,z0). If S 2 T(z, z0), nW(S, z, z0) ¼ 0, and nI(S, z, z0) ¼ jSjcoalition S moves to z0 with probability �jSj/jNj. Finally, if S 2 T(z, z0), nW(S, z, z0) ¼ 0,and nI(S, z, z0) < jSj, coalition S moves to z0 with some (possibly state-dependent)probability d, where 0 < d < 1/jAj.

For all � 2 (0,1) small enough, the system M�W is a well-defined irreducible Markov

process. Therefore, it has a unique invariant distribution l�W . We are now interested inidentifying the stochastically stable states of M�

W , i.e., those allocations in the supportof lim�!0 l�W . Of course, we already know the answer for economies with no in-differences: from Proposition 3, stochastic stability, which always selects a non-empty setof states, yields the unique competitive allocation. The rest of the Section considerseconomies with indifferences.

Note that by the definition of the perturbed Markov process M�W , for every two

allocations z and z0, the direct transition probability lz,z0(�) converges to the limittransition probability lz,z0(0) of the unperturbed process M0

W at an exponential rate.In particular, if we let



rðS ; z; z0Þ ¼nW ðS; z; z0Þ if nW ðS ; z; z0Þ > 0jSj=jN j if nW ðS ; z; z0Þ ¼ 0 and nI ðS ; z; z0Þ ¼ jS j1 otherwise

(

for all allocations z, z0 such that lz,z0(0) ¼ 0, the convergence is at a rate r(z, z0) ¼minS2T(z,z0)r(S, z, z0). As in previous Sections, we call the value r(z, z0) the resistance of thedirect transition from allocation z to allocation z0. The resistance of a path and of anE j-tree for each recurrent class E j ofM0

W , and the stochastically stable states ofM�W are

defined similarly, as in Subsection 1.3.Now we shall show that suitable generalisations of our previous results are obtainable.

It will be instructive to separate the analysis into two: economies in which the recurrentclasses ofM0

W are all singletons and those in which non-singleton recurrent classes alsoexist. As it turns out, in the former we find the only substantive difference in resultsbetween the strict and weak processes of recontracting.

3.2.1. Economies where all recurrent classes are singletonsFor this case, the criterion of stochastic stability will select a subset of the strong core,which is a subset of the competitive allocations. This is different from the resultsobtained with the process M� based on strict recontracting – recall Example 4. Spe-cifically, one can show the following result.

Proposition 4. Let E be a housing economy in which all recurrent classes of M0W are

singletons. The stochastically stable states of the weak recontracting process with mistakesM�W are

competitive allocations.

Proof. By Proposition 2, if all recurrent classes of the unperturbed process M0W are

singletons, they correspond to each of the strong core allocations. To finish the proof,we show now that all the strong core allocations are competitive. To see this, take astrong core allocation x of an economy E ¼ hN,H,(�i,ei)i2Ni with indifferences, andconsider an economy E0 ¼ hN ;H ; ð�0i ; eiÞi2N i that is obtained from E by undoing allindifferences in any way compatible with the following conditions: if for an agent i 2 N,xi �i h, then xi �0i h; and, for all i 2 N, h0 �0i xi if and only if h Ci xi. In the resultingeconomy E0, x is a strong core allocation. To see this, note that if coalition S can weaklyblock x in the economy E0, it could also block in the economy E, contradicting that x isin the strong core of E. Since E0 is an economy with no indifferences, by Lemma 1, parts(i) and (ii), x is the only strong core allocation and the only competitive allocation of E0.But then x must be competitive in E, because the same prices that support it in E0 alsosupport it in E.

Thus, in economies where all recurrent classes of M0W are singletons, stochastic

stability will select a subset, always non-empty, of the strong core and, therefore, of theset of competitive allocations. In particular, note how Proposition 3 follows from thisresult.

The next example shows that the set of stochastically stable allocations of M�W may

be a strict subset of the strong core (and a fortiori, of the set of competitive alloca-tions).



Example 6. This example studies the process M �W on the same economy as in

Example 3. That is, let N ¼ f1,2,3g and the agents� preferences be given by:

e1 �1 e2 �1 e3;

e1 �2 e3 �2 e2;

e1 �3 e2 �3 e3:

Recall that in this economy, all allocations except the initial endowment allocation arecompetitive and the strong core consists of the following three allocations: (e2, e3, e1),(e3, e1, e2) and (e1, e3, e2). In each of these allocations, each agent receives one of histop-ranked houses. Further, each of these allocations Pareto dominates the other threeallocations. However, while the three allocations constitute the only recurring classes ofthe system, only (e1, e3, e2) is stochastically stable in M �

W . The system transits from eachof the other two strong core allocations to the endowment with a resistance of only oneindifference (that of agent 1) and from there to (e1, e3, e2) with a resistance of 0. On theother hand, in order to leave (e1, e3, e2), the system encounters a resistance of at least 2:two indifferences in a two-agent coalition containing agent 1, after which the grandcoalition recontracts at no cost leading to one of the other recurrent classes.

3.2.2. Economies with non-singleton recurrent classesFor this class of economies, we shall show that some non-singleton recurrent classes areselected by the criterion of stochastic stability. When this happens, the weak recon-tracting process with mistakes identifies extra allocations that the economy will visit apositive proportion of time in the long run, even though none of them are coalitionallystable in the sense of the strong core. The next example shows that the set ofstochastically stable states may contain non-competitive allocations, as well as alloca-tions that are not Pareto indifferent to competitive ones.

Example 7. Let N ¼ f1, 2, 3g and suppose the agents� preferences are as follows:

e2 �1 e3 �1 e1;

e1 �2 e3 �2 e2;

e1 �3 e2 �3 e3:

Here, the strong core is empty because either coalition f1, 2g or f1, 3g always improvesweakly upon any allocation. There are two competitive allocations: (e2, e1, e3) and(e3, e2, e1). The endowment allocation and the allocation (e1, e3, e2) are not in anyrecurrent class of the unperturbed process based on weak blocking: one cannot get tothe endowment from anywhere else by means of weak improvements of any coalition;and one can get to (e1, e3, e2) in that way only from the endowment allocation. How-ever, the other four allocations constitute a recurrent class. We detail the coalitionsinvolved in each transition:

ðe2; e1; e3Þ !N ðe3; e1; e2Þ !f1;3g ðe3; e2; e1Þ !N ðe2; e3; e1Þ !f1;2g ðe2; e1; e3Þ:

Therefore, the four allocations in the unique recurrent class are stochastically stable inthe process M �

W .



4. Concluding Remarks

Theorem 1 uses the �global dominance� property of the competitive allocation, asspecified in Lemma 1, part (iii). Although the models are very different, the globaldominance of the competitive allocation resembles the main driving force of the resultin Vega-Redondo (1997). His paper proposes an evolutionary process based on imi-tation, and its competitive result relies on the fact that if a firm produces the com-petitive output in a symmetric oligopoly, its profit is always higher than that of thosefirms that produce any other output level.

Note that the sufficient condition on the cost of a serious mistake, k > jNj � 2, usedto obtain Theorem 1, is jeopardised when jNj grows. Thus, for a given specification of k,the system may get stuck at other allocations because the number of indifferencesrequired for the economy to abandon a non-competitive allocation and go to thecompetitive allocation grows. If one fixes k, making the economy large is an obstacle tothe competitive result since other allocations could also be visited by the process apositive fraction of time in the long run. This contrasts with the core convergenceliterature, where the results are based on the fact that more agents imply morepotential blocking coalitions.

Along the same lines, if indifferences are present in the agents� preferences, theexamples in Subsection 2.2 demonstrate that stochastic stability may yield a variety ofpatterns, and that the long-run prediction may be compatible with the emergence ofnon-competitive allocations. In particular, Example 5 suggests that the core may not beexactly capturing coalitional stability in a model where mistakes are allowed. Thus,dynamic analysis may be a useful complement to the static recommendations of cooper-ative game theory like the core. It may happen that some regions of the core are hard toaccess, while some non-core allocations may have strong dynamic attractor properties.11

Our results are robust if the transition rule in the unperturbed process is that ofcoalitional weak blocking, instead of the strict blocking specified inM0. As the analysisof the processM0

W in Section 3 shows, one can find conditions under which only somecompetitive allocations are stochastically stable, while in general one must also con-template the possibility of stochastically stable cycles that include non-competitiveallocations.

Appendix

Proof of Theorem 1. Since when jNj ¼ 2, the core consists of the singleton w and there are onlytwo feasible allocations, it is clear that the statement holds, by Proposition 1. Thus, assume thatjNj � 3. Recall that we denote the set of recurrent classes of the unperturbed process M0 byfE 0,E 1,. . .,E Qg, and let E 0 be the singleton recurrent class containing the competitive allocationw. Denote the stochastic potential of E 0 by sp(E 0). We show that the stochastic potential of any

11 In a different setting that includes externalities, Gomes and Jehiel (2005) analyse a dynamic game ofcoalition formation. In their model, the agents are forward-looking and the Markov perfect equilibria do notprovide support to the core. They interpret their result as an instance of how forward-looking agents maydisturb the core predictions. In contrast, our agents are myopic but the stochastic dimension – the presenceof mistakes – also leads to questioning the core. See also Ray and Vohra (2001) and Konishi and Ray (2003).There is also a literature that seeks to identify the allocations that are immune only to �farsighted� coalitionaldeviations. See, for example, Dutta et al. (1989), Chwe (1994) and Xue (1998). Since our agents are myopic,we do not expect our stochastically stable allocations to offer many insights into farsighted coalitional stability.



other recurrent class E k, k ¼ 1,. . .,Q, is greater than sp(E 0). Let E k 6¼ E 0 be an arbitrary recurrentclass. Consider an E k -tree of stochastic potential sp(E k). Introduce in it the following twomodifications:

(i) Delete the edge that connects the class E0 to its successor E j on the path to E k.(ii) Add a directed edge going from E k to E 0.

Note that the resulting graph is an E0-tree. Moreover, the resistance of this new E 0-tree r(T )equals

rðT Þ ¼ spðEkÞ � r ðE0;EjÞ þ r ðEk ;E0Þ:

To finish the proof, we show in the following two lemmas that r(E0,Ej) > r(Ek,E0). This meansthat we would have constructed an E0-tree whose resistance is less than sp(Ek), thereby showingthat sp(E0) < sp(Ek).

Lemma 2. Consider the edge E 0 ! E j that is deleted from the E k-tree. Then, r (E 0,E j ) � k.

Proof. Consider a path that attains the resistance r(E 0,E j), of the edge that connects thecompetitive allocation, w, to the recurrent class E j, and let x1 be the first allocation in thispath. We claim that, in the direct transition from w to x1, at least one agent becomes worseoff. Let the coalition involved in this transition be S1. If it were the case that x1

i �i wi for everyi 2 S1, we would be saying that w is not a strong core allocation, contradicting Lemma 1, part(ii). Therefore, at least one agent becomes worse off in this direct transition, from which itfollows that r(E 0,E j) � k.

Lemma 3. Consider the edge Ek ! E0 that is added to the Ek-tree. Then, r(Ek,E0) jNj � 2.

Proof. We calculate an upper bound for r(Ek,E0) as follows. Let x 2 Ek. By Lemma 1, part (iii),there exists a coalition S such that (wi)i2S 2 AS, wi ¤i xi for all i 2 S and wj Cj xj for some j 2 S.In the next paragraphs, we refer to S as one of the maximal (in the sense of set inclusion) suchcoalitions. We have two cases.

Case 1: S ¼ N. In this case, the maximum possible resistance associated with the direct tran-sition from x to w is (jNj � 2), i.e., the one given by the highest number of indifferences that canoccur in N.

Case 2: S 6¼ N. This case admits two subcases:Subcase 2.1: Suppose that x�S j2 A�S. Then, when coalition S meets, the system moves to y ¼

(wS, e�S) with positive probability. The resistance of this transition cannot be greater than(jN j � 2) because, within S, one can have at most (jSj � 2) indifferences. But note that from y,the system can move to w with a resistance no bigger than jN nS j � 2: if necessary, the coalitionT NnS of agents who are not receiving their competitive house at y will be partitioned in subsets(according to the trading cycles), each of which to perform the necessary trade so that the finalresult is w. Therefore, since jS j jN j � 2, the number of indifferences found in this transitionis at most (jN j � 4).

Subcase 2.2: x�S 2 A�S. In this case, coalition S meets and the system moves to y ¼ (wS, x�S) withpositive probability. But then, by our choice of S and Lemma 1, part (iii) applied to the subec-onomy consisting of agents NnS, it must necessarily be the case that x�S ¼ w�S. Therefore, NnS ¼; because otherwise S would not be a maximal blocking coalition. So in this case S ¼ N and weare back in case 1.

Therefore, the resistance of the transition Ek ! E0 is bounded above by the maximum of thetwo expressions involved in the two cases analysed, which is (jNj � 2). n



In consequence, it follows from our assumption on the size of k that r(E k,E 0) < r(E 0,E j),which concludes the proof.

Brown University and IMDEA-Ciencias SocialesIowa State University and Ben-Gurion University of the Negev

Submitted: 6 March 2006Accepted: 9 August 2007

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MISTAKES IN COOPERATION: THE STOCHASTIC ... IN COOPERATION: THE STOCHASTIC STABILITY OF EDGEWORTH’S RECONTRACTING* Roberto Serrano and Oscar Volij We analyse a dynamic trading process