## Abstract

This paper extends the roommate problem to include externalities, allowing preferences for a partner to depend on the situation of others. Stability concepts for matchings and partitions of the set of agents are proposed and characterized, conditional on all agents having prudent expectations about other agents’ reactions to deviations. We prove that any roommate problem with externalities has a stable partition and that a stable matching exists if there is a stable partition without odd rings. These results allow us to find restrictions on the space of preferences ensuring the existence of a stable matching. We also show that some classical properties are lost in the presence of externalities: the existence of paths to stability from any unstable matching, the coincidence of the core with the set of stable matchings, and the invariance of the set of agents who are alone in a stable matching.

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In an ordered group \(\{a_1,\ldots , a_k\}\), \(a_{i-1}\) is the predecessor of \(a_i\) for each \(i\in \{1,\ldots , k\}\) (subscripts modulo

*k*). Stability of a partition*P*also requires that \(a_i\) prefers to be matched with \(a_{i+1}\) than with \(a_{i-1}\), for any \(\{a_1,\ldots , a_k\}\in P\) and \(k\ge 3\).Chung (2000) analyses the roommate problem with weak preferences. He determines several types of restrictions on the space of individuals’ preferences that ensure the existence of a stable matching.

The effect of externalities in

*two-sided*matching models has been extensively studied in the last years [see Bando et al. (2016) for a systematic survey].In our framework, the prudence of agents is a necessary and sufficient condition to guarantee solvability in the entire subclass of marriage problems with externalities [see Sasaki and Toda (1996, Proposition 3.1 and Theorem 4.1)].

The case where

*a*blocks \(\mu \) to be alone is captured by taking \(b=a\).*A comment about terminology:*In all roommate problems, with or without externalities,*a*considers*b*acceptable if and only if*a*does not want to deviate from a matching in \({\mathcal {M}}(a,b)\) to be alone. However, only in models without externalities this last property is equivalent to assume that*a*prefers to be matched with*b*than remain alone: \(\mu \succ _a \eta \) for all \((\mu ,\eta )\in {\mathcal {M}}(a,b)\times {\mathcal {M}}(a,a)\). From this perspective, our concept of acceptable partner can be viewed as a weaker notion than the traditional one, as we only require that there exists \(\eta \in {\mathcal {M}}(a,a)\) such that \(\mu \succ _a \eta \) for all \(\mu \in {\mathcal {M}}(a,b)\).All through the text, properties in which subscripts are in a set \(\{1,\ldots ,m\}\) hold

*modulo**m*.For classical roommate problems, Pęski (2017) describes stable partitions through bijective mappings \(f:N\rightarrow N\), that he refers to as

*improper matchings*when \(f(f(a))\ne a\) for some \(a\in N\). Underlying his approach is the following one-to-one correspondence between ordered partitions*P*and bijections \(f:N\rightarrow N\): \(\{a_1,\ldots , a_k\}\in P\) if and only if \(f(a_i)=a_{i-1}\) for all \(i\in \{1,\ldots , k\}\) and \(k\ge 1\) (*f*is referred as the*bijection induced by**P*). Since this one-to-one correspondence does not depend on the presence of externalities, in our context stable partitions can also be described in terms of self-bijections of the set of agents. Indeed, a partition*P*is stable, in the sense of Definition 2 above, if and only if the bijection \(f:N\rightarrow N\) induced by*P*satisfies the following properties:-
(a)
*f*associates to each*a*a potential mate*f*(*a*) who is acceptable to her. -
(b)
If \(f(a)\ne f^{-1}(a)\), then

*a*considers that \(f^{-1}(a)\) is more interesting than*f*(*a*). -
(c)
If

*a*forms a pair with either*f*(*a*) or \(f^{-1}(a)\) in a matching \(\mu \in {\mathcal {M}}(P)\), then*a*cannot block \(\mu \) by deviating with an agent*b*that forms a pair with either*f*(*b*) or \(f^{-1}(b)\) in \(\mu \).

Notice that,

*f*satisfies requirements (a) and (b) if and only if the partition*P*satisfies Definition 2(i).Moreover,

*f*satisfies (c) if and only if any blocking pair of a matching \(\mu \in {\mathcal {M}}(P)\) includes an agent \(a_i\) that is not paired with either \(f(a_i)\) or \(f^{-1}(a_{i})\) in \(\mu \). Since the bijection*f*was induced by*P*, it follows that*f*satisfies (c) if and only if any blocking pair of a matching \(\mu \in {\mathcal {M}}(P)\) includes an agent who is alone under \(\mu \) and belongs to a ring of*P*. Hence, (c) and Definition 2(ii) are equivalent.-
(a)
In the adapted proof, \(\mu _{a,b}\) must be defined as the least preferred matching on \({\mathcal {M}}_a(b)\).

This property is a consequence of Sasaki and Toda (1996, Proposition 3.1), the proof of Corollary 1 (see Sect. 4), and Theorem 1. The first result guarantees that, when individuals have sophisticated beliefs, there are unsolvable marriage markets with externalities. The proof of Corollary 1 ensure that marriage markets with externalities do not allow for odd rings. Therefore, as a consequence of Theorem 1, it is sufficient to consider a roommate problem that is an

*instance*of an unsolvable marriage market (see Sect. 4 for definitions).In the literature of general equilibrium, there are works exploring whether stable matchings and commodity trade can coexist in competitive markets when

*agents’ deviations do not affect endogenous variables*(cf., Gersbach and Haller 2010, 2011; Contreras 2014; Gersbach et al. 2015). These models focus on*marriage markets*and assume that individuals may get utility from private consumption, from discrete actions as job choice, or from externalities associated to partner’s observable characteristics or to the reduction of consumption rivalry.*On stable algorithms.*It is not difficult to verify that a stable matching of the problem without externalities \((N, (\succ ^*)_{a\in N})\) is also stable in \((N,(\succ _a)_{a\in N})\). Hence, when \((N, (\succ ^*)_{a\in N})\) is solvable, the classical algorithm proposed by Irving (1985) can be applied to it in order to find a stable matching for \((N,(\succ _a)_{a\in N})\). Alternatively, if \((N, (\succ ^*)_{a\in N})\) has no stable matching, we can apply any of the algorithms proposed by Tan (1991) and Tan and Hsueh (1995) in order to find a stable partition, which is also stable for \((N, (\succ )_{a\in N})\) (see the proof of Theorem 1). However, the stable partitions of \((N, (\succ ^*)_{a\in N})\) do not necessarily induce stable matchings of \((N, (\succ )_{a\in N})\). For instance, in Example 2 the matchings induced by the stable partitions of \((N, (\succ ^*)_{a\in N})\) are \(\{\mu _1, \mu _2, \mu _3\}\), while \(\mu _0\) is the only stable matching of \((N, (\succ )_{a\in N})\).Indeed, \(\mu _2\) is blocked by (2, 3), while \(\mu _3\) and \(\mu _0\) are blocked by (1, 3).

Notice that, without externalities, if a roommate problem has a non-empty set of stable matchings, then \({\mathcal {A}}\) is an absorbing set if and only if \({\mathcal {A}}=\{\mu \}\) for some stable matching \(\mu \) [see Iñarra et al. (2013, Proposition 2)].

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Contreras, J.L., Torres-Martínez, J.P. The roommate problem with externalities.
*Int J Game Theory* **50**, 149–165 (2021). https://doi.org/10.1007/s00182-020-00743-z

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DOI: https://doi.org/10.1007/s00182-020-00743-z