Abstract
This paper develops a class of equilibriumindependent predictions of competitive equilibrium with indivisibilities. Specifically, we prove an analogue of the “Lone Wolf Theorem” of classical matching theory for the Baldwin and Klemperer (Econometrica 87(3):867–932, 2019) model of exchange economies with transferable utility, showing that any agent who does not participate in trade in some competitive equilibrium must receive her autarky payoff in every competitive equilibrium. Our results extend to approximate equilibria and to settings in which utility is only approximately transferable.
Introduction
In models of exchange and production, results on the uniqueness of equilibria typically require continuity and convexity conditions on agents’ preferences, in addition to further conditions on aggregate demand (see, for example, Section 17.F of MasColell et al. 1995). However, continuity and convexity conditions fail in the presence of indivisibilities, leading to the existence of multiple equilibria. In turn, the presence of multiple equilibria weakens the predictive power of economic models.
To avoid the problems of multiplicity, we often consider predictions that do not depend on which equilibrium is realized.^{Footnote 1} In this paper, we develop a class of equilibriumindependent predictions of competitive equilibrium in the Baldwin and Klemperer (2019) model of exchange economies with indivisible goods and transferable utility.^{Footnote 2} Specifically, we show that any agent who does not participate in trade in some competitive equilibrium must receive her autarky payoff in every competitive equilibrium. Hence, observing that an agent does not participate in trade indicates that there is no competitive equilibrium in which that agent improves on her autarky payoff.
Our result is analogous to the classical “Lone Wolf Theorem” from matching theory, which asserts that in onetoone matching without transfers, any agent who is unmatched in some stable outcome is unmatched in every stable outcome (McVitie and Wilson 1970).^{Footnote 3} Informally, the Lone Wolf Theorem shows that the set of “lone wolves” (unmatched agents) is equilibriumindependent. Since our setting involves continuous transfers, and hence indifferences, we cannot obtain sharp predictions regarding the set of agents that participate in trade in equilibrium. Instead, we show that any agent who does not participate in trade in some equilibrium cannot benefit from trade in any equilibrium.
When utility is perfectly transferable, there is generically a unique efficient allocation of goods among agents, although there may be multiple possible equilibrium price vectors. Hence, the set of agents that participate in trade is generically equilibriumindependent. However, our lone wolf result applies even when there are multiple efficient allocations. Moreover, we show that versions of our Lone Wolf Theorem hold in settings in which multiple allocations can be robustly sustained in equilibrium. Specifically, we consider an approximate equilibrium concept—which may be a reasonable solution when the economy suffers from small frictions—and derive an approximate Lone Wolf Theorem. We also prove an approximate Lone Wolf Theorem for equilibria in economies in which utility is only approximately transferable between agents.
Quasiconversely, we illustrate that some assumption on the transferability of utility is essential to our lone wolf results. Specifically, we show by example that lone wolf results do not generally hold in settings with strong income effects. Hence, while our lone wolf results show that competitive equilibrium has some (approximately) robust predictions when utility is (approximately) transferable, it remains an open question whether we can derive classes of equilibriumindependent predictions more generally.
In a companion paper (Jagadeesan et al. 2018), we used our Lone Wolf Theorem to show that descending salaryadjustment processes are strategyproof in quasilinear Kelso and Crawford (1982) economies. Our argument for strategyproofness rests crucially on applying our Lone Wolf Theorem to (nongeneric) economies with multiple efficient matchings.
The remainder of this paper is organized as follows. Section 2 presents the model. Section 3 presents our lone wolf results. Section 4 discusses the role of the hypothesis that utility is (approximately) transferable. Section 5 connects our results to the matching literature. Proofs omitted from the main text are presented in Appendix A.
Model
We consider a general model of exchange economies with indivisible goods and transferable utility, following Baldwin and Klemperer (2019); this framework embeds the trading network model of Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp (2013), which in turn generalizes the settings of Gul and Stacchetti (1999, 2000) and Sun and Yang (2006, 2009). There is a finite set \(\varGamma \) of goods and a finite set I of agents.
Each agent has an endowment \(e^{i} \in \mathbb {Z}^\varGamma \) and a valuation
We assume that valuations are bounded above and that \(v^{i}(e^{i}) > \infty \); however, we do not impose any further conditions on them. The valuation \(v^i\) induces a quasilinear utility function \(u^{i}\) by
where \(t^{i}\) is the transfer received by agent i. We allow \(t^{i}\) to be arbitrarily negative, so agents have unlimited budgets.
A competitive equilibrium consists of (1) an allocation of goods to agents and (2) prices for each good such that the allocation maximizes each agent’s utility given prices.
Definition 1
A competitive equilibrium is a pair \(\left[ q^{};p_{}\right] ,\) where \(q = (q^i)_{i \in I},\) at which

each agent i demands her bundle \(q^{i}\) at prevailing prices \(p_{}\), i.e.,
$$\begin{aligned} q^{i} \in \mathop {\mathrm{arg\,max}}\limits _{\bar{q}^{i} \in \mathbb {Z}^\varGamma } \left\{ u^{i}(\bar{q}^{i},(e^{i}\bar{q}^{i}) \cdot p_{})\right\} \end{aligned}$$(1)for all \(i \in I\), and

the market clears, i.e.,
$$\begin{aligned} \sum _{i \in I} q^{i} = \sum _{i \in I} e^{i}. \end{aligned}$$(2)
Baldwin and Klemperer (2019) have provided sufficient conditions for the existence of competitive equilibria in the model we consider here. For example, competitive equilibria exist if units of goods are substitutable and only finitely many bundles have value greater than \(\infty \) (see also Kelso and Crawford 1982; Gul and Stacchetti 1999; Milgrom and Strulovici 2009).
Results
The classical Lone Wolf Theorem from the theory of twosided matching without transfers asserts that any agent who is matched in some equilibrium outcome is matched in every equilibrium (McVitie and Wilson 1970; see also Roth 1984, 1986; Klaus and Klijn 2010). In this section, we prove analogues of the Lone Wolf Theorem for exchange economies.
An exact Lone Wolf Theorem
For a pair \(\left[ q^{};p_{}\right] \), we say that an agent \(i\in I\)does not participate in trade in\(\left[ q^{};p_{}\right] \) if \(q^{i} = e^{i}\). Such an agent is a “lone wolf” in \(\left[ q^{};p_{}\right] .\)
Our first result asserts that any agent who does not participate in trade in some competitive equilibrium receives her autarky payoff in every competitive equilibrium. Thus, we show that any agent who is a “lone wolf” in some equilibrium cannot benefit from trade in any equilibrium.
Theorem 1
Let \(\left[ q^{};p_{}\right] \) and \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) be competitive equilibria, and let \(j \in I\) be an agent. If j does not participate in trade in \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) (i.e., if \(\hat{q^{}}^{j} = e^{j}\)), then \(u^{j}(q^{j},t^{j}) = u^{j}(e^{j},0),\) where \(t^{j} = \left( e^{j}q^{j}\right) \cdot p_{}\) is the net transfer to agent j at equilibrium \(\left[ q^{};p_{}\right] .\)
We prove Theorem 1 as a corollary of a more general result (Theorem 2) that allows for optimization error in equilibrium.
An approximate Lone Wolf Theorem
Note that Theorem 1 has bite only in economies in which there are multiple competitive equilibrium allocations. As competitive equilibrium allocations are generically unique in transferable utility economies, Theorem 1 has nontrivial consequences only for nongeneric economies.
However, as we show in this section, our lone wolf result holds more generally. In particular, we prove a version of Theorem 1 for approximate equilibria; as approximate equilibria are only approximately efficient, this result has nontrivial consequences in settings in which equilibrium allocations are robustly nonunique.
We relax the definition of competitive equilibrium by allowing agents’ total maximization error to be positive but bounded above by \(\varepsilon \).
Definition 2
An \(\varepsilon \)equilibrium consists of a pair \(\left[ q^{};p_{}\right] \) for which
and (2) is satisfied (i.e., the market clears).
Our Approximate Lone Wolf Theorem asserts that if there exists an \(\varepsilon \)equilibrium in which no agents in \(J\subseteq I\) participate in trade, then the difference between the total utility of agents in J and the total autarky payoff of agents in J is bounded above by \((\delta +\varepsilon )\) in every \(\delta \)equilibrium.
Theorem 2
Let \(\left[ q^{};p_{}\right] \) be a \(\delta \)equilibrium, let \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) be an \(\varepsilon \)equilibrium, and let \(J \subseteq I\) be a set of agents. If no agent in J participates in trade in \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) (i.e., if \(\hat{q^{}}^{j} = e^{j}\) for all \(j\in J\)), then
where \(t^{j} = \left( e^{j}q^{j}\right) \cdot p_{}\) is the net transfer to agent j at approximate equilibrium \(\left[ q^{};p_{}\right] .\)
The key to the proof of Theorem 2 is a lemma that allows us to produce a new approximate equilibrium from any two approximate equilibria. Our lemma, which we prove in Appendix A, is an approximate version of the wellknown fact that \(\left[ \hat{q^{}}^{};p_{}\right] \) is a competitive equilibrium whenever \(\left[ q^{};p_{}\right] \) and \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) are competitive equilibria [see page 3 of Shapley (1964), as well as Gul and Stacchetti (1999), Sun and Yang (2006), and Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp (2013)].
Lemma 1
If \(\left[ q^{};p_{}\right] \) is a \(\delta \)equilibrium and \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) is an \(\varepsilon \)equilibrium, then \(\left[ \hat{q^{}}^{};p_{}\right] \) is a \((\delta +\varepsilon )\)equilibrium.
To prove Theorem 2, we exploit the fact that \(\left[ \hat{q^{}}^{};p_{}\right] \) is a \((\delta +\varepsilon )\)equilibrium (by Lemma 1), so the allocation \(\hat{q^{}}^{}\) must approximately maximize j’s utility given the price vector \(p_{}\).
Proof of Theorem 2
By Lemma 1, \(\left[ \hat{q^{}}^{};p_{}\right] \) is a \((\delta +\varepsilon )\)equilibrium. As \(\hat{q^{}}^{j} = e^{j}\) for all \(j \in J\) by assumption, we have that
Meanwhile, with \(t^{j} = \left( e^{j}q^{j}\right) \cdot p_{},\) we have that
Combining (3) and (4), we see that
as desired.
Taking \(J = \{j\}\) in Theorem 2 yields the following corollary.
Corollary 1
Let \(\left[ q^{};p_{}\right] \) be a \(\delta \)equilibrium, let \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) be an \(\varepsilon \)equilibrium, and let \(j \in I\) be an agent. If j does not participate in trade in \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) (i.e., if \(\hat{q^{}}^{j} = e^{j}\)), then
where \(t^{j} = \left( e^{j}q^{j}\right) \cdot p_{}.\)
Note that the \(\delta = \varepsilon = 0\) case of Corollary 1 is Theorem 1.
A Lone Wolf Theorem for economies with approximately transferable utility
Theorem 2 also implies a Lone Wolf Theorem for economies in which utility is “close to transferable” in a formal sense. Specifically, we consider economies in which agents’ utility functions can be wellapproximated by quasilinear utility functions.
Definition 3
A utility function \(u^{i}\) is quasilinear within\(\eta \) if there exists a quasilinear utility function \(\check{u}^{i}\) for which
When utility functions are approximately quasilinear, we obtain another approximate Lone Wolf Theorem.^{Footnote 4}
Theorem 3
Let \(\left[ q^{};p_{}\right] \) and \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) be competitive equilibria, and let \(J \subseteq I\) be a set of agents. If no agent in J participates in trade in \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) (i.e., if \(\hat{q^{}}^{j} = e^{j}\) for all \(j\in J\)) and \(u^{i}\) is quasilinear within \(\eta \) for all \(i \in I\), then
where \(t^{j} = \left( e^{j}q^{j}\right) \cdot p_{}\) is the net transfer to agent j at equilibrium \(\left[ q^{};p_{}\right] .\)
To prove Theorem 3, we construct an approximating economy in which utility functions are quasilinear. Competitive equilibria in the original economy give rise to approximate equilibria in the approximating economy; Theorem 3 then follows from Theorem 2.
Proof of Theorem 3
We define an auxiliary economy in which utility functions are given by quasilinear \((\check{u}^{i})_{i \in I}\), with each \(\check{u}^{i}\) as in Definition 3. By construction, \(\left[ q^{};p_{}\right] \) and \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) are \(2\eta I\)equilibria in the auxiliary economy. Theorem 2 then guarantees that
As \(\sum _{j \in J} u^{j}(q^{j},t^{j})  \check{u}^{j}(q^{j},t^{j}) \le \eta I\) and \(\sum _{j \in J} u^{j}(e^{j},0)  \check{u}^{j}(e^{j},0) \le \eta I\) by our choice of \((\check{u}^{i})_{i \in I}\), we see from (5) that
as desired.
The role of utility transferability
The proofs of our lone wolf results rely on the assumption that utility is (at least approximately) transferable. In fact, the Lone Wolf Theorem does not always hold when utility is not transferable. For example, the result can fail in the presence of income effects.
Consider an economy with four agents, a house builder \(\textsf {B},\) two real estate agents \(\textsf {HighEnd}\) and \(\textsf {LowEnd},\) and a consumer \(\textsf {C}\). Real estate agent \(\textsf {HighEnd}\) specializes in highend properties, while \(\textsf {LowEnd}\) specializes in lowend properties. The builder \(\textsf {B}\) can construct a highend property or a lowend property and sell it to the consumer \(\textsf {C}\) via \(\textsf {HighEnd}\) or \(\textsf {LowEnd},\) respectively. These possible interactions can be summarized in the following network:^{Footnote 5}
It costs \(\textsf {B}\) 110 to construct a highend property and 50 to construct a lowend property. It costs each real estate agent 0 to intermediate between \(\textsf {B}\) and \(\textsf {C}\). We suppose that \(\textsf {C}\) values the highend property more than the lowend property but experiences income effects—when prices are high, \(\textsf {C}\) prefers to buy the lowend property. Intuitively, the highend property might require additional fees, such as maintenance costs and property taxes, which could cause \(\textsf {C}\) to prefer the lowend property when prices are high, but not when prices are low. Moreover, \(\textsf {C}\) cannot costlessly convert a highend property into a lowend property.^{Footnote 6} Formally, \(\textsf {C}\) has utility function
As there is only one real estate agent of each type (or, more generally, as we do not assume that there is free entry in the markets for real estate agents), it may be possible for the real estate agents to extract rents. Thus, a competitive equilibrium must specify prices that the consumer faces for each type of property separately from the prices that real estate agents face.
There are two possible equilibrium allocations: either \(\textsf {B}\) can construct a highend property and sell it to \(\textsf {C}\) via \(\textsf {HighEnd},\) or \(\textsf {B}\) can construct a lowend property and sell it to \(\textsf {C}\) via \(\textsf {LowEnd}.\) For example, two competitive equilibria are:
(Here, we denote competitive equilibria by writing a price for each interaction pictured in (6) and boxing the prices of interactions that occur.) In equilibrium (I), \(\textsf {HighEnd}\) extracts a rent of 5 from intermediating between \(\textsf {B}\) and \(\textsf {C}\); in equilibrium (II), \(\textsf {LowEnd}\) extracts a rent of 5 from intermediating. However, \(\textsf {HighEnd}\) does not trade in equilibrium (II), while \(\textsf {LowEnd}\) does not trade in equilibrium (I). Thus, there are agents who do not trade in one equilibrium but receive utility strictly greater than their autarky payoffs in the other equilibrium—that is, the lone wolf result fails.
In our example, the builder is able to extract more surplus when the lowend property is traded because the consumer is willing to pay more for the lowend property due to the failure of free disposal. Similarly, the consumer is able to extract more surplus from the builder when the highend property is traded due to having higher marginal utility of wealth after buying the highend property. Therefore, the highend real estate agent is able to improve the consumer’s utility, while the lowend real estate agent improves the builder’s utility, making it possible for either real estate agent to extract rents.^{Footnote 7}
In contrast, our Lone Wolf Theorem shows that if utility were transferable, then at most one of the real estate agents would be able to extract rents. Intuitively, if utility were transferable, then the builder and consumer would agree on which possible trade is better.^{Footnote 8} In the presence of income effects, on the other hand, agents can contribute to the social surplus in one allocation but not in another. For example, \(\textsf {HighEnd}\) must contribute to the economy when the highend property is traded, as she is able to extract rents in equilibrium (I). However, because \(\textsf {HighEnd}\) does not participate in trade in equilibrium (II), she cannot possibly contribute to the economy when the lowend property is traded.
Discussion and conclusion
We developed a class of Lone Wolf Theorems that provide equilibriumindependent predictions of competitive equilibrium analysis in contexts with indivisibilities. Our results show that when utility is perfectly transferable, any agent who does not participate in trade in some competitive equilibrium must receive her autarky payoff in every competitive equilibrium; moreover, this result holds approximately under approximate solution concepts and in settings in which utility is only approximately transferable. Approximate transferability of utility is essential for our results—the lone wolf conclusion fails when utility is not transferable (e.g., in the presence of income effects).
Relationship to the lone wolf and rural hospitals results of matching theory
Our lone wolf results extend a classical matchingtheoretic insight of McVitie and Wilson (1970) to exchange economies. Other analogues of the Lone Wolf Theorem have been developed in matching, but those results have different hypotheses and conclusions from ours.
The mostgeneral matchingtheoretic generalizations—developed by Hatfield and Kominers (2012) and Fleiner et al. (2018)—state that for every agent in a trading network (without transfers), the difference between the numbers of the goods bought and sold is invariant across stable outcomes.^{Footnote 9} Our results extend the Lone Wolf Theorem to exchange economies (with transfers) by assessing when agents participate in (profitable) trade instead of analyzing the amounts that agents trade. Furthermore, the matchingtheoretic results rely on two regularity conditions—some form of (gross) substitutability (Kelso and Crawford 1982), and a regularity condition called the “law of aggregate demand”^{Footnote 10}—which we do not require. We instead require that utility is at least approximately transferable.
Recently, Schlegel (2016) has proven a closely related lone wolf result for manytoone matching with continuous transfers. While Schlegel (2016) allowed workers (i.e., agents on the unitdemand side) to experience income effects, he required that firms (i.e., agents on the multiunitdemand side) have utility functions that are not only quasilinear but also gross substitutable. Thus, the Schlegel (2016) lone wolf result is logically independent of ours.
Application to strategyproofness
It has been well known since the work of Dubins and Freedman (1981) and Roth (1982) that the Gale–Shapley (1962) deferred acceptance mechanism is dominantstrategy incentive compatible for all unitdemand agents on the “proposing” side of the market; this onesided strategyproofness result has been important in practice (see, e.g., Roth and Peranson 1999; Abdulkadiroğlu, Pathak, and Roth 2005; Abdulkadiroğlu, Pathak, Roth, and Sönmez 2005; Pathak and Sönmez 2008) and has been extended to settings with discrete transfers (or other discrete contracts; see Hatfield and Milgrom 2005; Hatfield and Kojima 2010; Hatfield and Kominers 2012, 2019).
Onesided strategyproofness results have heretofore been difficult to derive in matching settings with continuous transfers—in part because the nowstandard proof of onesided strategyproofness (due to Hatfield and Milgrom 2005) relies on the matchingtheoretic Lone Wolf Theorem, and prior to our work there was no lone wolf result for settings with transfers.^{Footnote 11} In a companion paper (Jagadeesan et al. 2018), we used the results developed here to give a direct proof of onesided strategyproofness for worker–firm matching with continuously transferable utility.^{Footnote 12}
Divisible goods
Finally, we note that while we assumed that the domains of agents’ valuations consist of integral quantity vectors, identical arguments would apply if agents were instead to maximize over real quantity vectors. Thus, although our model formally requires that goods be indivisible, our results apply in settings with divisible goods as well.
Notes
In structural estimation, for example, it is possible to avoid the problem of multiplicity of equilibria by performing inference based on equilibriumindependent predictions of the model (see, for example, Bresnahan and Reiss 1991). A related alternative approach, which is valid even when there are few equilibriumindependent predictions, is to obtain setidentification of the parameters of interest by assuming that the observation is one of many possible equilibria (see, for example, Tamer 2003; Ciliberto and Tamer 2009; Pakes 2010; Galichon and Henry 2011; Pakes et al. 2015; Bontemps and Magnac 2017).
To the best of our knowledge, the term “Lone Wolf Theorem” was coined by Klaus and Klijn (2010).
We rule out intentionalflooding and dynamitebased solutions by assumption.
Formally, the extremal equilibria are
Equilibrium (III) is the equilibrium with lowest prices (and is in particular buyeroptimal), while equilibrium (IV) is the equilibrium with highest prices (and is in particular selleroptimal). Note that the highend property is traded in equilibrium (III) while the lowend property is traded in equilibrium (IV).
It might be the case that both trades generate the same social surplus—but when utility were transferable, the builder and the consumer would agree on this fact.
These results are typically called Rural Hospitals Theorems because they imply that clearinghouses cannot improve the recruitment total of rural hospitals—which are often less attractive to doctors—by switching to different (stable) marketclearing mechanisms. The first Rural Hospitals Theorems were proven by Roth (1984, 1986) for manytoone matching with responsive preferences. Rural Hospitals Theorems were also developed for manytomany matching (Alkan 2002; Klijn and Yazıcı 2014), manytoone matching with contracts (Hatfield and Milgrom 2005; Hatfield and Kojima 2010), and manytomany matching with contracts (Hatfield and Kominers 2017).
Indeed, it appears that until Hatfield, Kojima, and Kominers (2019) found an indirect argument by way of a version of Holmström’s (1979) lemma, onesided strategyproofness of deferred acceptance in the presence of continuously transferable utility was known only for onetoone markets (Demange 1982; Leonard 1983; Demange and Gale 1985; Demange 1987).
Intuitively, this argument shows that the allocation \((\hat{q^{}}^{i})_{i \in I}\) must be within \(\varepsilon \) utils of maximizing the sum of agents’ values over all allocations—i.e., that the allocation is approximately efficient.
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Acknowledgements
The authors thank Mohammad Akbarpour, John William Hatfield, Bettina Klaus, Shengwu Li, Paul Milgrom, Robbie Minton, Casey Mulligan, Alvin E. Roth, Alexander Teytelboym, the editor, Marc Fleurbaey, and the anonymous referee for helpful comments.
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This paper was previously circulated under the more amusing title “If You’re Happy and You Know It, Then You’re Matched.” This research was conducted while Jagadeesan and RheingansYoo were Economic Design Fellows at the Harvard Center of Mathematical Sciences and Applications (CMSA). Jagadeesan gratefully acknowledges the support of a National Science Foundation Graduate Research Fellowship under grant DGE1745303, as well as travel support from Harvard Business School and the Harvard Mathematics Department. Kominers gratefully acknowledges the support of National Science Foundation grant SES1459912 and the Ng Fund and the Mathematics in Economics Research Fund of the CMSA. Jagadeesan and Kominers both gratefully acknowledge the support of the Washington Center for Equitable Growth.
Proof of Lemma 1
Proof of Lemma 1
As \(\left[ \hat{q^{}}^{};\hat{p}_{}\right] \) is an \(\varepsilon \)equilibrium, we have that
where the last equality holds as \(\sum _{i \in I} q^{i} = \sum _{i \in I} e^{i} = \sum _{i \in I} \hat{q^{}}^{i}.\)^{Footnote 13} Hence, we have that
It follows that
Grouping (7) by agents, we have that
As \(\left[ q^{};p_{}\right] \) is a \(\delta \)equilibrium, we have that
Summing (8) and (9), we have that
Hence, \(\left[ \hat{q^{}}^{};p_{}\right] \) is a \((\delta +\varepsilon )\)equilibrium, as claimed.
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Jagadeesan, R., Kominers, S.D. & RheingansYoo, R. Lone wolves in competitive equilibria. Soc Choice Welf 55, 215–228 (2020). https://doi.org/10.1007/s0035501901228z
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DOI: https://doi.org/10.1007/s0035501901228z