MISTAKES IN COOPERATION: THE STOCHASTIC STABILITY OF EDGEWORTH S RECONTRACTING*

Transcription

1 The Economic Journal, 118 (October), Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. MISTAKES IN COOPERATION: THE STOCHASTIC STABILITY OF EDGEWORTH S RECONTRACTING* Roberto Serrano and Oscar Volij We analyse a dynamic trading process of coalitional recontracting in an exchange economy with indivisible 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 agents who make mistakes and of the seriousness of each mistake. If preferences are always strict, the unique stochastically stable state is the competitive equilibrium allocation. In economies with indifferences, non-core cycles are sometimes stochastically stable, while some core allocations are not. The robustness of these results is confirmed in a weak coalitional recontracting process Edgeworth s Recontracting Process The question of what outcomes will arise in decentralised trade amongst arbitrary groups of agents dates back to Edgeworth (1881). Edgeworth argued that these outcomes consist of the set of Ôfinal settlementsõ, which is referred to as the core, i.e., the set of allocations Ôwhich cannot be varied with advantage to all the contracting partiesõ. Although this definition of Ôfinal settlementsõ applies to a static exchange economy, Edgeworth motivates them using a description of a recontracting process that is inherently dynamic. This process received elegant formalisations in the analysis of Feldman (1974) and Green (1974) where it was shown that under certain assumptions on the economy, Edgeworth s recontracting process converges to a core allocation. 1 In the present article we formalise Edgeworth s recontracting process in a way similar to those of Feldman (1974) and Green (1974), except for two differences: (1) agents make ÔmistakesÕ in their decisions 2 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 economies Feldman s and Green s recontracting systems, as well as ours, fail to converge to core allocations. Indeed, in our specification of the dynamics, along with the core allocations, some long-run predictions recurrent classes arise which consist solely of * We thank Bob Aumann, Geoffroy de Clippel, Leonardo Felli, Philippe Jehiel, Michihiro Kandori, Eric Maskin, Indra Ray and two anonymous referees for useful suggestions. Serrano acknowledges support from Fundacion Banco Herrero, Universidad Carlos III, NSF grant SES and Deutsche Bank, and thanks Universidad Carlos III and CEMFI in Madrid and the Institute for Advanced Study in Princeton for their hospitality. 1 Another aspect of the relationship between dynamics and the core is provided by the literature on the extensive-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. [ 1719 ]

2 1720 THE ECONOMIC JOURNAL [ OCTOBER non-core allocations. 3 One way to deal with this emergence of a large number of recurrent classes is to use refinement techniques, which can be accomplished by adding mistakes and applying the concept of stochastic stability. Moreover, the consideration of mistakes captures a realistic aspect of decision making Stochastic Stability in Economic Models This article applies the concept of stochastic stability to the non-strategic context of a private ownership economy. Stochastic stability was first used in game theory by Kandori et al. (1993) and Young (1993), who applied it to 2 2 games as a selection device. 4 Since then it has been applied successfully to a wide variety of models (Young, 1998). The concept of stochastic stability belongs to the ÔevolutionaryÕ approach to explaining the emergence of equilibria, conventions or social norms. 5 In this approach, agents are not fully rational but ÔprogrammedÕ to behave in a fixed way until they are replaced by other agents who are better prepared to face their environment, or by unlikely mutations. In the usual application of stochastic stability to non-cooperative games one fixes a one-shot game in normal form (strategy sets and payoff functions) played by myopic players and one specifies a stochastic process to describe how players choose their strategies in each period. The process consists of three forces: inertia, selection and randomness. Inertia is the driving force that acts most of the time: typically players automatically repeat what they played in the last period, or copy the strategy played by their immediate ancestors playing the game. At times, however, the system is affected by selection: a player may want to put some thought into his strategy, for example by choosing 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 in playersõ choices. A natural question that then arises is what strategies are most likely to be played in the long run. These strategies are called stochastically stable and have been shown to have interesting properties; see, for example, Kandori et al. (1993) and Young (1993). In this article we similarly consider a one-shot exchange economy (preferences and endowments) with myopic agents, which is repeated over time. We are interested in the evolution of the allocations when they follow a stochastic process that is composed of the three forces discussed above. The main force, within which the other two are embedded, can be seen as a flow of inertia. In each period the agents take their fixed initial endowments and trade them for a given consumption bundle, repeating the pattern of trade of the previous period. These bundles constitute a status quo allocation which is consumed every period, unless something disturbs the system. Inertia is interrupted 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 by Gale 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 and Wentzell (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).

3 2008] EDGEWORTH S RECONTRACTING PROCESS 1721 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 new status quo allocation, which will be repeated each period until the next disturbance takes place. Finally, randomness is represented by a process of mistakes. When a coalition is selected to look for a profitable recontract, there is a small probability that some of its members will agree to a reallocation that makes them worse off. This persistent 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 stability then identifies those states (allocations, in this case) that are visited by the system a positive proportion of time in the very long run. Fixing the preferences and endowment of the economy allows us to compare the stochastically stable allocations to the ones prescribed by classical solution concepts, such as the core or competitive equilibrium. Therefore, our goal is to identify those allocations that will be visited by Edgeworth s recontracting process a significant proportion of time in the long run if mistakes are small probability events that all agents may make all the time Summary of Results We present two kinds of results. First, under certain conditions, stochastic stability will serve as a powerful refinement tool and provide a new foundation of competitive outcomes. And second, more generally, stochastic stability will highlight that some solution concepts, such as the core, might be lacking an aspect of coalitional interactions that stochastic dynamics may help to capture. To be more precise, when we analyse the recontracting process free of mistakes, each core allocation constitutes an absorbing state and, in addition to them, subsets of noncore allocations constitute non-singleton recurrent classes. However, when mistakes are introduced into the model our results will depend on the class of economies under consideration. In economies where all preferences are strict, we are able to obtain a remarkable refinement of the set of recurrent classes by perturbing the system with mistakes. Theorem 1 states that the unique stochastically stable state of the recontracting process with mistakes is the competitive allocation, provided that serious mistakes (those by which agents end up worse off) are sufficiently more costly than minor ones (those by which agents end up neither worse nor better off). This result relies on the property of Ôglobal dominanceõ of the competitive allocation in economies with no indifferences, as identified by Roth and Postlewaite (1977). Global dominance states that for every feasible allocation, there exists a coalition whose members can (weakly) improve upon the status quo by using their competitive bundles. This property is of course confined to the housing market setting and cannot be extended to general economies. Nonetheless, the housing market is of interest and within this setting our result provides a new foundation for competitive allocations. 6 In contexts closer to a social choice model, where only preferences and not the endowments are part of the primitives, Ben-Shoham et al. (2004) and Kandori et al. (2008) explore related models that use mistakes in decision making or random utility maximisation.

4 1722 THE ECONOMIC JOURNAL [ OCTOBER The conclusions of our analysis are quite different for the more general case of economies with indifferences, where we obtain non-competitive results. We provide a series of examples to illustrate that the predictions of stochastic stability will not coincide with any of the classical solution concepts. In particular, we regard Example 5 as the other main result of the article in that it shows that non-core (and hence, noncompetitive) cycles are sometimes stochastically stable, whereas some core allocations are not. This is worth stressing for two reasons. First, when mistakes are introduced into the recontracting process the economy will frequently visit coalitionally unstable allocations. Second, entire regions of the core will be reached a zero proportion of time in the very long run. This suggests that there is a different level of ÔdecentralisationÕ behind each core allocation. Whereas some allocations may arise as a result of a simple sequence of trade and are therefore easy to reach, others will require complicated transactions and, hence, will not be reached as easily. As a robustness check of our results, we also study a second process based on weak coalitional recontracting. We find that the stochastic stability criterion always selects a non-empty subset of competitive allocations when the unperturbed process contains only singleton recurrent classes (in particular, in economies with only strict preferences, the competitive allocation is the unique recurrent class of this unperturbed process). If the mistake-free process exhibits also non-singleton recurrent classes, however, stochastic stability is compatible with the appearance of cycles containing non-competitive allocations Recontracting with Mistakes and Edgeworth s Tradition In this subsection we argue that our process of recontracting with mistakes is a faithful formalisation of the recontracting process envisioned by Edgeworth. In particular, we devote 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 clear that he has a static exchange economy in mind:...the disposition and circumstances of the parties are assumed to remain throughout 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 periodically over time. This is also suggested by the fact that Edgeworth chose the labour market to exemplify his concepts. Workers and entrepreneurs show up in the market every period with the same endowment and preferences. In particular, entrepreneurs

5 2008] EDGEWORTH S RECONTRACTING PROCESS 1723 do not sell the labour units that they bought in the previous period. The renewal of contracts is also consistent with the landlord-tenant relationship, another example mentioned by Edgeworth (see below). Further, in his Mathematical Psychics (see page 17, for instance), Edgeworth uses the expression Ôdealers in commodity xõ and Ôdealers in commodity yõ to refer to the economic agents, emphasising the persistence of the property rights based on the initial endowment allocation. Edgeworth is interested in the Ôfinal settlementsõ. According to Edgeworth a Ôfinal settlementõ consists of a set of agreements which cannot be varied with advantage to all the recontracting parties. (Edgeworth, 1925, p. 314.) The market presumably reaches a final settlement after going through various ÔsettlementsÕ or Ôtemporary equilibriaõ. Edgeworth, however, is not clear as to whether consumption takes place at each temporary equilibria or only at a final settlement. Indeed, when he explains the meaning of recontracting (or blocking, in the modern economist s jargon) he gives two examples which point to different directions concerning 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 a price if no higher bid) recontracts with a higher bidder. So a landlord on expiry of 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) between the auctioneer and the last bidder, due to the appearance of a higher bidder. In the second example, in contrast, there is a transaction every period between a landlord and a tenant that leads to consumption. Here, although each period s contract is just a settlement, it is not necessarily a final one. Edgeworth proposed the core as an alternative to the competitive equilibrium allocations, identified several years earlier by Walras (1874). He found that most of the times the core was indeterminate as a solution concept, since it prescribes a large set of final settlements. This contrasted with the more determinate concept due to Walras (which was believed to be unique). For this reason, Edgeworth was especially interested in this question of uniqueness of the core and was the first to notice the connection between the core and competitive allocations in large economies. 7 For Edgeworth s main purpose, the timing of consumption is irrelevant. One can interpret his temporary equilibria, very much in the competitive tattonement tradition, as agreements that are not carried out but which are intermediate steps leading to a final settlement; and alternatively, one can interpret his settlements as agreements that are carried out each period without affecting the Ôdisposition and circumstancesõ of the parties for the next period. Edgeworth s comparison of the core to Walras s equilibrium 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. See Anderson (1992) and Aumann (1987) for surveys. Although the robustness of the equivalence result is remarkable, several violations thereof have been identified, from which one can learn the role of certain frictions in markets. These references include Anderson and Zame (1997) for infinite dimensional commodity spaces, Manelli (1991) for instances of non-monotonicity in preferences, and Serrano et al. (2001) for asymmetric information.

6 1724 THE ECONOMIC JOURNAL [ OCTOBER However, when one focuses on the dynamic process of recontracting, the interpretation of the states of the system gains importance. The dynamic systems proposed by 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. Therefore they are consistent with an interpretation in which consumption only takes place after convergence is achieved. In our model, when agents make random mistakes, the system has no absorbing states. Therefore an interpretation more in line with the landlord tenant example, in which there are transactions and consumption every period, is more appropriate. There is yet another conceptual advantage to our approach. Essential to Edgeworth s recontracting process is the idea that any individual is free to contract and recontract without the consent of third parties. However, note that within the first interpretation there is no consumption until all agents reach an agreement. That is, no one can consume until it is certain that every agent has reached a final settlement. In fact, it may well be the case that along the path toward the final settlement, some agents hold bundles that are strictly preferred to the bundle that is consumed by them at any final settlement. Postponing consumption until the very end seems to contradict the spirit of recontracting without 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 model and basic definitions, the unperturbed recontracting process, and its perturbed version with mistakes. Section 2 contains our main results. To check their robustness, Section 3 analyses 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 EhN,H,( i,e i ) i2n i, where N is a finite set of individuals, H is a finite set of houses with the number of houses (jh j) being equal to the number of individuals (jn j). For each individual i 2 N, i is a complete and transitive preference relation over H, with C i denoting its associated strict preference relation and i its indifference relation. Finally, (e i ) i2n is the profile of individual endowments: 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. Afeasible allocation for coalition S is a redistribution of the coalitional endowment (e i ) i2s among the members of S. Denote the set of feasible allocations for coalition S by A S. We simply write A for A N. An allocation x 2 A is individually rational if there is no individual i 2 N for whom e i C i x i.

7 2008] EDGEWORTH S RECONTRACTING PROCESS 1725 An allocation x 2 A is a core allocation if there is no coalition S and no feasible allocation for S, y 2 A S, such that y i C i x i for all i 2 S. An allocation x 2 A is a strong core allocation if there is no coalition S and no feasible allocation for S, y 2 A S, such that y i C i x i for all i 2 S and y j C j x j for some j 2 S. An allocation x 2 A is a competitive allocation if there exist positive competitive equilibrium prices p such that for all i 2 Np xi ¼ p ei, and for all i 2 N and for all h 2 H, h C i x i implies p h > p ei. 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 of trading cycles. That is, if and only if there exists a partition of the set of agents fs 1,S 2,...,S m g such that for every k ¼ 1,2,...,m, x Sk 2 A Sk ; and for every j 2 S k ; x j j x i for every i 2 S k [...S m : In words, the agents in S 1 redistribute their endowments and get their most preferred houses; the agents in S 2 redistribute their endowments and get their most preferred houses out of the endowments of S 2 [...S m, and so forth. In the following subsections, we shall define two Markov processes for any given housing economy, one being a perturbation of the other. The states of the process are the allocations of the housing economy. In each period a coalition of agents is selected at random and the system moves from one state to another when the matched agents recontract. In the unperturbed Markov process M 0 of Subsection 1.2, agents do not make mistakes in their coalitional meetings: they sign a contract if and only if there is a strictly beneficial coalitional recontracting opportunity. In the perturbed process M of Subsection 1.3, agents will make mistakes with a small probability and sign a contract with a coalition even when they do not improve as a result. In Section 3 we study Markov 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 the case for M 0 ) has many long-run predictions recurrent classes and thus many stationary or invariant distributions. As such, the long-run behaviour of the system crucially depends on the initial state. On the other hand, for each small >0, the perturbed process M is ergodic, which implies that it has a unique stationary distribution, denoted by l. This stationary distribution, which is independent of the initial state, represents the proportion of time that the system will spend on each of its states in the long run. It also represents the long-run probability that the process will be at each allocation. In order to define the stochastically stable states, which are our objects of interest, we check the behaviour of the stationary distribution l as goes to 0. It is known that lim!0 l exists and further that it is one of the stationary distributions of the unperturbed process M 0. The stochastically stable states of the system M are defined to be those states that are assigned positive probability by this limit distribution. We are interested in identifying these allocations because they are the ones that are expected to be observed in the long run Ômost of the timeõ An Unperturbed Recontracting Process Consider the following unperturbed Markov process M 0, adapted from Feldman (1974) and Green (1974). In each period t, if the system is at the allocation x(t), one

8 1726 THE ECONOMIC JOURNAL [ OCTOBER coalition is chosen according to an arbitrary distribution which assigns positive probability to each coalition. Suppose coalition S is chosen. (i) If there exists an S-allocation y S 2 A S such that y i C i x i (t) for all i 2 S, the coalition moves with positive probability to one of such y S in that period. Then, the new state is either xðt þ 1Þ ¼ðy S ; x S ðtþþ if x S ðtþ 2A S ; or (ii) Otherwise, x(t þ 1) ¼ x(t). xðt þ 1Þ ¼ðy S ; e S Þ if x S ðtþ j2 A S : The interpretation of the process is one of coalitional recontracting. Following a status quo x(t), a coalition can form and modify the status quo if all members of the coalition improve as a result. When this happens, upon coalition S forming, the complement coalition N ns continues to have the same houses as before if this is feasible for them. Otherwise, N ns breaks apart and each of the agents in it receives his individual endowment. 8 If after coalition S forms all its agents cannot find any strict improvement, the original status quo persists A Perturbed Recontracting Process Next we introduce the perturbed Markov process M for an arbitrary small 2 (0,1), a perturbation of M 0. 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 ÔmistakeÕ when he signs a contract that either leaves him indifferent to the same house he already had or he becomes worse off upon signing. Each of the members of S may make one of these ÔmistakesÕ with a small probability, as a function of >0, independently of the others. Specifically, for a small fixed 2 (0,1), we shall postulate that an agent s probability of agreeing to a new allocation that leaves him indifferent is, 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 less likely than the former, while both are rare events in the agent s decisionmaking process. 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 coalitions such that, if chosen, can induce the perturbed system to transit from z to z 0 in one step. Note that it is always the case that T(z, z 0 ) is non-empty, since N 2 T(z, z 0 ) for any z and z 0. 8 Our treatment of the complement coalition constitutes a small difference with respect to the processes in Feldman (1974) and Green (1974): in these papers, when recontracting happens, the complement coalition is sent back always to their individual endowments in Feldman (1974), or to a Pareto efficient allocation of their resources in Green (1974). Our results are robust to different specifications; for instance, if only those agents in N ns who are directly or indirectly affected by the reallocation proposed by S are sent to their initial endowments, 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 than one in which the only frictions are indifferences.

9 2008] EDGEWORTH S RECONTRACTING PROCESS 1727 In the direct transition from z to z 0 and for each S 2 T(z,z 0 ), define the following numbers: n I ðs; z; z 0 Þ¼jfi 2 S : z i i zi 0 gj; n W ðs; z; z 0 Þ¼jfi 2 S : z i i zi 0 gj; nðs; z; z 0 Þ¼kn W ðs; z; z 0 Þþn I ðs; z; z 0 Þ: Here n I (S, z, z 0 ) is the number of individuals in S that are indifferent between allocations z and z 0. Similarly, n W (S, z, z 0 ) is the number of members of S that make a mistake by agreeing to move from z to z 0, worse for them. Finally, n(s, z, z 0 ) is the weighted number of mistakes that are made in the transition from z to z 0, where k > 1 is the weight given to ÔseriousÕ mistakes relative to those that lead to just indifference. In the perturbed Markov process M the transition probabilities are calculated as follows. Suppose that the system is in allocation z. All coalitions are chosen with a fixed positive probability. Assume coalition S is chosen. If S j2 T(z, z 0 ), then S moves to z 0 with probability 0. If S 2 T(z, z 0 ) and n(s, z, z 0 ) > 0, then coalition S agrees to move to z 0 with probability n(s,z,z0).ifs 2 T(z, z 0 ) and n(s, z, z 0 ) ¼ 0, coalition S moves to those z 0 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 probability in a finite number of periods. As such, it has a unique invariant distribution. This distribution is a good estimate of the probability that the system is in each of the allocations in the long run. We are interested in the limit of the corresponding invariant distributions as tends to 0. More precisely, our focus will be on the allocations that are assigned positive probability by this limiting distribution. These allocations 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 two allocations z and z 0, the direct transition probability l z,z 0() converges to the limit transition probability l z,z 0(0) of the unperturbed process M 0 at an exponential rate. In particular, for all allocations z, z 0 such that l z,z 0(0) ¼ 0, the convergence is at a rate r(z, z 0 ) ¼ min S2T(z,z 0 )kn W (S, z, z 0 ) þ n I (S, z, z 0 ), that is, the weighted number of mistakes associated with the most likely direct transition. We call the value r(z, z 0 )the resistance of the direct transition from allocation z to allocation z 0. For any two allocations z, z 0,a(z, z 0 )-path is a sequence of allocations n ¼ (i 0,i 1,...,i m ) such that i 0 ¼ z, i m ¼ z 0. The resistance of the path n is the sum of the resistances of its transitions. We seek to find the path of least resistance between any two allocations. Let Z 0 ¼fE 0, E 1,..., E Q g be the set of recurrent classes of the unperturbed process M 0 each recurrent class E j is a set of states with the property that if the process M 0 reaches a state in the class, it will never get out of that class. Consider the complete directed graph with vertex set Z 0. We want to define the resistance of each one of the edges 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 (z i, z j )-paths, where z i 2 E i and z j 2 E j. A spanning tree rooted at E j,orane j -tree, is a set of directed edges such that from every recurrent class different from E j, there is a

10 1728 THE ECONOMIC JOURNAL [ OCTOBER unique directed path in the tree to E j. The resistance of an E j -tree is the sum of the resistances of its edges. The stochastic potential of the recurrent class E j is the minimum resistance attained by an E j -tree: it captures the smallest friction to get to class E j from any of the other recurrent classes. As shown in Young (1993), the set of stochastically stable states of the perturbed process M consists of those states belonging to the recurrent classes with minimum stochastic potential. 2. Main Results Let us begin by analysing the long-run predictions of the unperturbed process M 0.Itis clear that the absorbing states of this unperturbed process are precisely the core allocations of the economy. However, the absorbing states are not the only recurrent classes of M 0, as shown by the following example. Example 1. Let N ¼ f1, 2, 3g and denote by the individual endowment allocation (e 1, e 2, e 3 ). Let the agentsõ preferences be as follows: e 3 1 e 2 1 e 1 ; e 1 2 e 3 2 e 2 ; e 2 3 e 1 3 e 3 : Consider the following three allocations: x ¼ (e 1, e 3, e 2 ), y ¼ (e 2, e 1, e 3 ) and z ¼ (e 3, e 2, e 1 ). These three allocations constitute a recurrent class: if the 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, 3g meets. Finally, the system will move out of z only by going back to x, when coalition f2, 3g meets. Note that the unique competitive allocation w ¼ (e 3, e 1, e 2 ) also constitutes a singleton recurrent class. We can prove the following result, which characterises the recurrent classes of the unperturbed process M 0 : Proposition 1. The recurrent classes of the unperturbed recontracting process M 0 take the following 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 M 0 and that every absorbing state must be a core allocation. For the second form of recurrent class, note that, by construction of the system, no state in a recurrent class can ever be non-individually rational: if at some state x in which e i C i x i the coalition fig is chosen, then, the system moves to the individual endowment e, never to return to a nonindividually rational allocation. It is also clear that each of the states in the recurrent class cannot be absorbing, i.e., cannot be a core allocation. Thus, each core allocation is an absorbing state of the unperturbed Markov process M 0 and, in principle, there may be additional non-singleton recurrent classes, such as

11 2008] EDGEWORTH S RECONTRACTING PROCESS 1729 that shown in Example 1. Note also that as soon as the economy has more than one core allocation, the system M 0 has many stationary distributions. This multiplicity of long-run predictions justifies the consideration of the perturbed process based on mistakes, whose results we shall report next 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 h 0 if and only if h ¼ h 0. 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 dominanceõ of w: For every allocation x 2 A, x 6¼ w, there exists a coalition S such that w S is feasible for S and satisfies w i i x i for all i 2 S and w j C j x j for some j 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 that k > jn j 2. Then, the unique stochastically stable allocation of the perturbed recontracting process with mistakes M is the competitive allocation w. The proof of Theorem 1 is in the Appendix. It combines the different parts of Lemma 1. Specifically, its parts (ii) and (iii) imply that the competitive allocation w is a strong dynamic attractor in the trading process with mistakes. Theorem 1 provides a remarkable refinement of the set of long-run predictions of the unperturbed process, as identified in Proposition Economies with Indifferences In this subsection we explore how the stochastic process of recontracting with mistakes, M, performs over the class of economies that allow non-singleton indifference sets 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 is guaranteed, there may be multiple competitive allocations. Second, the strong core may also contain multiple allocations, while it may sometimes be empty. Third, the Ôglobal dominanceõ property of competitive allocations as specified in Lemma 1, part (iii), is also lost. In economies with only strict preferences, the competitive allocation correspondence and the strong core coincide and, under our assumption on k, the same allocation is the only one that passes the test of stochastic stability. It is convenient, therefore, to examine the larger class of economies to disentangle the different forces at work in the selection of the stochastically stable allocations. We find that the recontracting system with mistakes gives rise to complicated dynamics and no general result of equivalence

12 1730 THE ECONOMIC JOURNAL [ OCTOBER can be established. Therefore, we learn that the paths of least resistance followed in our stochastic dynamic analysis are not intrinsically associated with the strong core property or 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 these examples, one agent has a completely flat indifference map but this is only for simplicity of exposition. Also, in the examples we shall use the notation z! r S z0 to express that the transition of least resistance from z to z 0 takes place through coalition S with a resistance r. We begin by showing that the set of stochastically stable allocations is not the strong core. As just mentioned, the strong core may be empty in these economies, while stochastic stability always selects at least one allocation; but even when the strong core is non-empty, one can generate examples where it does not coincide with the set of stochastically stable states of M. Example 2. In this example, a non-empty strong core is strictly contained in the set of stochastically stable allocations. Let N ¼f1,2g and agentsõ preferences be described as follows: e 1 1 e 2 ; e 1 2 e 2 : In this economy there are two allocations, both of which are competitive, but only the one resulting from trade is in the strong core. Note that both allocations are stochastically stable: ðe 1 ; e 2 Þ! 1 N ðe 2; e 1 Þ and ðe 2 ; e 1 Þ! 1 f1g ðe 1; e 2 Þ. The next example shows that the set of stochastically stable allocations may be a strict subset of the strong core and of the set of competitive allocations. Example 3. Let N ¼f1, 2, 3g and the agentsõ preferences be given by: e 1 1 e 2 1 e 3 ; e 1 2 e 3 2 e 2 ; e 1 3 e 2 3 e 3 : In this economy, all allocations except the initial endowment allocation are competitive and belong to the core. The strong core consists of the following three allocations: (e 2, e 3, e 1 ), (e 3, e 1, e 2 ) and (e 1, e 3, e 2 ). The unique stochastically stable allocation is x ¼ (e 1, e 3, e 2 ). To see that x is the only allocation with minimum stochastic potential, one can construct an x-tree as follows. First, we note that the only recurrent classes of M 0 are the five absorbing states corresponding to each competitive allocation. Next, note that to go from (e 2, e 3, e 1 ) to x can be done with a resistance of 1 (only one indifference): ðe 2 ; e 3 ; e 1 Þ! 1 f1g ðe 1; e 2 ; e 3 Þ! 0 f2;3g x. The same goes for the transition (e 3, e 1, e 2 )tox: ðe 3 ; e 1 ; e 2 Þ! 1 f1g ðe 1; e 2 ; e 3 Þ! 0 f2;3g x. As for the other transitions, we have ðe 3 ; e 2 ; e 1 Þ! 1 f2;3g x and ðe 2; e 1 ; e 3 Þ! 1 f2;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, the resistance will always be at least 2, i.e., at least two indifferences, which implies that in constructing a tree for any of the other recurrent classes its resistance must be at least 5.

13 2008] EDGEWORTH S RECONTRACTING PROCESS 1731 Our next example illustrates the only substantive difference in results between the recontracting process M and its counterpart based on weak recontracting that will be studied in the following Section. Specifically, we now construct an economy for which all recurrent classes of M 0 are singletons and where the set of stochastically stable allocations of M contains an allocation that is not competitive. Example 4. Let N ¼f1, 2, 3, 4g and let the agentsõ preferences be described as follows: e 1 1 e 3 1 e 4 1 e 2 ; e 1 2 e 4 2 e 3 2 e 2 ; e 2 3 e 1 3 e 3 3 e 4 ; e 2 4 e 3 4 e 4 4 e 1 : One can check that there are four competitive allocations: w 1 ¼ (e 3, e 4, e 1, e 2 ), w 2 ¼ (e 1, e 4, e 3, e 2 ), w 3 ¼ (e 4, e 1, e 3, e 2 ), and w 4 ¼ (e 3, e 1, e 2, e 4 ). In addition, there are three other core allocations that are not competitive: y 1 ¼ (e 1, e 4, e 2, e 3 ), y 2 ¼ (e 4, e 1, e 2, e 3 ), and y 3 ¼ (e 4, e 3, e 1, e 2 ). Since Proposition 1 also applies to the economies considered in this subsection, it follows that each of these seven allocations constitutes an absorbing state of M 0. Further, it can be checked that there are no additional recurrent classes. Apart from the seven core allocations, the only absorbing states of M 0, there are four more individually rational allocations. One is e ¼ (e 1, e 2, e 3, e 4 ) - the initial endowment allocation - and the other three are x 1 ¼ (e 1, e 3, e 2, e 4 ), x 2 ¼ (e 3, e 2, e 1, e 4 ) and x 3 ¼ (e 4, e 2, e 1, e 3 ). It can be checked that from each of these four allocations one can go with zero resistance to one of the core allocations. Consider now the following fy 1 g-tree comprising the following edges: w 1! y 1, w 2! w 1, w 3! w 1, w 4! w 2, y 2! w 2, y 3! y 1. We check next that each of these edges has resistance 1 (corresponding to a transition with only one indifference), thereby implying that the tree has minimum stochastic potential. These transitions can be detailed as follows: w 1! 1 f1g e!0 f2;3;4g y 1; w 2! 1 f1;3g x 2! 0 f2;4g w 1; w 3! 1 f1;3g x 2! 0 f2;4g w 1; w 4! 1 f2;4g w 2; y 2! 1 f2;4g w 2; y 3! 1 f1g e!0 f2;3;4g y 1: Thus, y 1 is stochastically stable, although it is not a competitive allocation. It is not in the strong core either; indeed, the strong core is empty in this economy.

14 1732 THE ECONOMIC JOURNAL [ OCTOBER The next example shows that in a dynamic model where agents may make mistakes in decision making, the core may not agree with the set of states that are visited by the process a positive proportion of time. In contrast, some non-core allocations may fare better in this sense than some core allocations. Example This example, an outgrowth of Example 1, shows that a cycle of noncore allocations may be stochastically stable, at the same time as some core allocations having higher stochastic potential. Let N ¼f1, 2, 3, 4g, and let the agentsõ preferences be as follows: e 4 1 e 3 1 e 2 1 e 1 ; e 1 2 e 3 2 e 2 2 e 4 ; e 2 3 e 1 3 e 3 3 e 4 ; e 1 4 e 2 4 e 3 4 e 4 : Consider the following three allocations: x ¼ (e 1, e 3, e 2, e 4 ), y ¼ (e 2, e 1, e 3, e 4 ) and z ¼ (e 3, e 2, e 1, e 4 ). Since agent 4 cannot strictly improve, he cannot be part of any blocking coalition. In fact, as in Example 1, these three allocations constitute a recurrent class: if the system is at x, the state changes only when coalition f1, 2g meets, yielding y. Aty, the system can move only to z, when coalition f1, 3g meets. Finally, the system will move out of z only by going back to x, when coalition f2, 3g meets. That is, x! 0 f1;2g y!0 f1;3g z!0 f2;3g x. The core consists of the following five allocations: c 1 ¼ (e 3, e 1, e 2, e 4 ), c 2 ¼ (e 4, e 1, e 2, e 3 ), c 3 ¼ (e 4, e 1, e 3, e 2 ), c 4 ¼ (e 4, e 3, e 1, e 2 ) and c 5 ¼ (e 4, e 3, e 2, e 1 ). It is easy to see that these five absorbing states i.e., the core allocations and the cycle are the only recurrent classes of the unperturbed system: the twelve allocations where e 4 is allocated to either agent 2 or agent 3 are not even individually rational. And from each of the remaining four allocations, one gets to one of the already identified recurrent classes with 0 resistance: a 1 ¼ðe 1 ; e 2 ; e 3 ; e 4 Þ! 0 f1;2;3g c 1, a 2 ¼ðe 2 ; e 3 ; e 1 ; e 4 Þ! 0 f1;2;3g c 1, a 3 ¼ðe 4 ; e 2 ; e 1 ; e 3 Þ! 0 f2;3g x and a 4 ¼ðe 4 ; e 2 ; e 3 ; e 1 Þ! 0 f2;3g c 5. Let E ¼ fx, y, zg be the non-singleton recurrent class consisting of non-core allocations. 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. We detail the transitions below: c 1! 1 f1;4g a 4! 0 f2;3g c 5; c j! 1 f4g a 1! 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: c 1, c 2, c 3 and c 5. 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 three two-person coalitions improving the status quo. The three players involved in the cycle are the two wives of a man and a third party who buys the man s estate. The cycle occurs after the man died and left his estate. The deceased agent naturally corresponds to our agent 4, who has a flat indifference map. See also Binmore (1985) for a more recent related problem.

15 2008] EDGEWORTH S RECONTRACTING PROCESS 1733 However, apart from E, the only stochastically stable allocations are the four competitive allocations c 1, c 2, c 3 and c 5 : there are allocations in the core that are not visited in the long run but a zero proportion of time. In particular, it can be checked that it takes at least two indifferences to get out of other recurrent classes to go to c 4. This implies that c 4 cannot be stochastically stable. To illustrate, we construct a fc 4 g-tree as follows: c 1! 1 f1;4g a 4! 0 f2;3g c 5; c j! 1 f4g a 1! 0 f1;2g ðy 2 EÞ; for j ¼ 2; 3; 5; ðz 2 EÞ! 2 N c 4: Thus, this dynamic analysis, which sheds light on how each allocation emerges, makes a distinction among different core allocations. Some core allocations (like c 4 in the example) are actually very hard to get to, when compared to other allocations in the core or even outside of the core. The point is that the usual emphasis in the definition 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. Dynamics should have something to say about the matter and Example 5 is a confirmation that this is indeed the case. 3. Recontracting Based on Weak Blocking As a robustness check of our results, this Section analyses a Markov process in which transitions take place whenever there is an instance of weak coalitional blocking, instead of the strict version thereof, as was the case in the process M The Unperturbed Process Consider the following unperturbed Markov process M 0 W, which represents the Ôweak recontractingõ version of the process presented in Subsection 1.2. In each period t, the system is at the allocation x(t), and one coalition is chosen according to an arbitrary probability distribution that assigns positive probability to each coalition. Suppose coalition S is chosen. (i) If there exists an S-allocation y S 2 A S such that y i i x i (t) for all i 2 S and y j C j x j (t) for some j 2 S, the coalition moves with positive probability to some of such y S in that period. Then, the new state is either xðt þ 1Þ ¼ðy S ; x S ðtþþ if x S ðtþ 2A S ; or (ii) Otherwise, x(t þ 1) ¼ x(t). xðt þ 1Þ ¼ðy S ; e S Þ if x S ðtþ j2 A S :

16 1734 THE ECONOMIC JOURNAL [ OCTOBER 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 one member of the coalition improves as a result, and if none of the other members is made worse off. Apart from this change, the process is identical to the one in Subsection 2.2. That is, when the weak coalitional recontracting move occurs, upon coalition S forming, the complement coalition N ns continues to have the same houses as before, if this is feasible for them. Otherwise, N ns breaks apart and each of the agents in it receives his individual endowment (the same comment as in footnote 8 applies here in terms of alternative specifications). If after coalition S forms, it cannot find any such weak coalitional improvement, the original status quo persists. The absorbing states of the unperturbed process M 0 W are precisely the strong core allocations of the economy. However, the absorbing states are not the only recurrent classes of M 0 W. The easiest way to see this is to note that there are economies in which the strong core is empty. We can prove the following result, similar to Proposition 1, which characterises the recurrent classes of the unperturbed process M 0 W : Proposition 2. The recurrent classes of the unperturbed weak recontracting process M 0 W 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 Markov process M 0 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, the system M 0 W has many stationary distributions. The conclusions of Proposition 2 extend to all economies where preferences are complete and transitive, with or without indifferences. 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, the unperturbed weak recontracting process M 0 W is ergodic and the only recurrent class is the singleton consisting of the unique competitive allocation. Proof. The proof follows from the definition of the transitions in the process and from Lemma 1. Notice how, as in Theorem 1, the Ôglobal dominanceõ property of the competitive allocation in these economies makes it the only prediction of the dynamic process, this time as the strong attractor of the unperturbed system (see Example 1 for an illustration).