PATHS TO STABILITY IN THE ASSIGNMENT PROBLEM

. We study a labor market with ﬁnitely many heterogeneous workers and ﬁrms to illustrate the decentralized (myopic) blocking dynamics in two- sided one-to-one matching markets with continuous side payments (assignment problems, Shapley and Shubik [24]). Assuming individual rationality, a labor market is unstable if there is at least one blocking pair, that is, a worker and a ﬁrm who would prefer to be matched to each other in order to obtain higher payoﬀs than the payoﬀs they obtain by being matched to their current partners. A blocking path is a sequence of outcomes (specifying matchings and payoﬀs) such that each outcome is obtained from the previous one by satisfying a blocking pair (i.e., by matching the two blocking agents and assigning new payoﬀs to them that are higher than the ones they received before). We are interested in the question if starting from any (unstable) individually rational outcome, there always exists a blocking path that will lead to a stable outcome. In contrast to discrete versions of the model (i.e., for marriage markets, one-to-one matching, or discretized assignment problems), the exis- tence of blocking paths to stability cannot always be guaranteed. We identify a necessary and suﬃcient condition for an assignment problem (the existence of a stable outcome such that all matched agents receive positive payoﬀs) to guarantee the existence of paths to stability and show how to construct such a path whenever this is possible.


1.
Introduction. Many markets involve bilateral relationships where each agent of one side of the market can be matched to any agent of the other side of the market but cannot be matched to any agent from the same side. Examples for such two-sided matching markets include marriage markets (women and men), college admissions markets (colleges and students), auction markets (buyers and sellers), and labor markets (workers and firms). Two-sided matching markets can be partitioned in two main categories: markets without side payments (e.g., marriage and college admissions markets) and markets with side payments (e.g., auction and labor markets). Side payments in the form of prices, fees, or salaries are a natural feature of many economic situations. Here, we study a simple two-sided one-to-one matching market with side payments: a labor market with finitely many heterogeneous workers and firms. To keep the model simple we impose a unit-demand condition such that each worker accepts at most one job and each firm hires at most one worker.
Two-sided matching markets with side payments -assignment problems-have first been analyzed by Shapley and Shubik [24]. In an assignment problem, indivisible objects (e.g., auctioned items or jobs) are exchanged with monetary transfers (e.g., prices or salaries) between two finite sets of agents (e.g., buyers/sellers or workers/firms). Agents are heterogeneous in the sense that each object may have different values to different agents. Each agent either demands or supplies exactly one unit. Thus, agents can form pairs to exchange the corresponding objects and at the same time make monetary transfers of the value they create (alternatively, singletons can execute an outside option).
An outcome for an assignment problem specifies a matching between agents of both sides of the market and, for each agent, a payoff. An outcome is stable if it is individually rational and there is no blocking pair, that is, there are no two agents that are not matched with each other, but in fact would prefer to be. For instance, in a labor market, a worker and a firm form a blocking pair if both could get higher payoffs than the payoffs they obtain by being matched to their current partners (if we matched them with higher payoffs, we would be satisfying a blocking pair ). An outcome is in the core if no coalition of agents can improve their payoffs by rematching among themselves. Shapley and Shubik [24] established many important structural properties of the core. 1 The literature on stability in two-sided matching markets was initiated by Gale and Shapley [11] who proposed a centralized procedure, the famous deferred acceptance algorithm, to find a stable outcome for any marriage or college admissions problem (with responsive preferences). The deferred acceptance algorithm turned out to be the key element for many centralized market clearing houses, e.g., for the National Resident Matching Program (Roth [20]), for school choice programs (Abdulkadiroglu and Sönmez [2] and Abdulkadiroglu et al. [1]), and for auctions and trading networks (Demange et al. [9], Gul and Stacchetti [12], Milgrom [17], Ausubel [3], Sun and Yang [26], and Hatfield et al. [13]). 2 Dynamic changes in real world (two-sided) matching markets are frequently observed. This indicates that outcomes often are not stable. For instance, in a labor market, a worker might switch to a new job if that increases his salary while the 1 Shapley and Shubik [24] showed that (a) the core of an assignment problem and the set of stable outcomes coincide, (b) for any assignment problem, there always exists a stable outcome, (c) the set of stable outcomes is a complete lattice with two extreme points, each of them corresponding to an outcome where all the agents of the same side of the market (e.g., the workers) receive their maximal stable payoffs while the agents of the other side (e.g., the firms) receive their minimal stable payoffs, and (d) at any stable outcome, the matching between the workers and the firms is optimal (i.e., the value created by the pairs in the corresponding matching is maximal). Sotomayor [25] and Wako [27] proved that if there is only one optimal matching, then the core contains infinitely many stable outcomes. Conversely, the core is a singleton only when multiple optimal matchings exist. 2 See also Crawford and Knoer [7], Kelso and Crawford [14], and Crawford [6] for centralized processes in labor markets and Roth and Sotomayor [21] for an excellent survey on two-sided matching theory until 1990. firm who hires him finds his qualification/productivity higher than that of his predecessor. A blocking path for a (two-sided) matching markets is a finite sequence of outcomes where each outcome is obtained from the previous one by satisfying a blocking pair taking into account that agents behave myopically, i.e., agents do not forecast how their decision to block an outcome might influence the future evolution of the market.
Knuth [16] showed that for marriage markets a process of myopic blocking may cycle, i.e., a decentralized process may not converge to a stable outcome. Roth and Vande Vate [22] show that for marriage markets there always exists a blocking path starting from any unstable outcome that leads to a stable outcome in finitely many steps. Assuming that each blocking pair is selected with strict positive probability, this result implies that a decentralized blocking process converges to stability with probability one. 3 Chen et al. [5] and Nax et al. [18] both analyze a similar decentralized blocking process for labor markets with discrete side payments; they construct a blocking path to stability and show that a decentralized blocking process converges to stability with probability one. Biró et al. [4] consider a more general one-sided version of the assignment problem and obtain results that imply those of Chen et al. [5] and Nax et al. [18] with a different proof technique. Apart from looking at a continuous model instead of a discretized one (as Chen et al. [5] and Nax et al. [18] do), a difference between the work of Biró et al. [4], Chen et al. [5], Nax et al. [18] compared to ours is that we consider strict blocking while all these other articles consider weak blocking. 4 This difference induces differentiated results and different proof techniques and we will discuss the exact relation of these articles with ours in Section 4.3. When studying the classical matching models, (discrete) marriage markets (Gale and Shapley [11]) and (continuous) assignment problem (Shapley and Shubik [24]), it is almost always the case that a result in one model has a corresponding result in the other model. We will show that this is not the case for the path to stability result: for continuous assignment problems, even when compared to discrete assignment problems, a path to stability does not always exist.
The paper is organized as follows: In Section 2, we introduce the classical assignment model with continuous side payments (Shapley and Shubik [24]).
In Section 3, we define a generic blocking path and we show with a few examples that a blocking path to stability might not exist for all assignment problems. We then state and prove our main result that, for all assignment problems that satisfy our necessary and sufficient condition (the existence of a stable outcome such that all matched agents receive positive payoffs), a stable outcome can always be obtained through a finite sequence of outcomes, each outcome being obtained from the previous one by satisfying a blocking pair under the strict blocking norm.
Finally, in Section 4, we discuss some relevant points. First, we consider a specific blocking path where each time a blocking pair is satisfied the blocking agents equally split the surplus they create. We ask whether such a fair blocking path can be always used to construct a path to stability (the answer is no). Second, we discuss the probabilistic interpretation of the blocking path result obtained in Section 3. Third, we discuss in more details the articles by Biró et al. [4], Chen et al. [5], and Nax et al. [18], and show how their results and our results are related. Fourth, we propose a centralized stabilization procedure that uses a so-called median stable outcome as compromise target outcome. Then, we briefly conclude.
2. The assignment problem. We consider a simple labor market model that matches workers and firms. Let W and F be two distinct finite sets containing |W | workers and |F | firms, respectively. Thus, the set of agents equals W ∪F . We denote generic agents by i, j, a generic worker by w, and a generic firm by f . We assume that each worker can work for at most one firm and a firm can employ at most one worker. 5 We denote the set of pairs that agents in W × F can form (including "degenerate" pairs where agents i ∈ W ∪ F form a "pair" (i, i) with themselves) by A characteristic function for W ∪ F is a function π : P (W, F ) → R + such that for each i ∈ W ∪ F , π(i, i) = 0. The characteristic function π describes the value that agents create when forming pairs. In particular, π(i, i) = 0 represents the reservation value of an agent i ∈ W ∪ F . 6 A (two-sided one-to-one) assignment problem is a triple (W, F, π).
A matching µ (for assignment problem (W, F, π)) is a function µ : Two agents i, j ∈ W ∪ F are matched if µ(i) = j (or equivalently µ(j) = i); for convenience, we also use the notation (i, j) ∈ µ. We refer to µ(i) as i's partner at µ. If (w, f ) ∈ µ, then we say that worker w and firm f form a couple at µ. If (i, i) ∈ µ, then we say that agent i remains single at µ. Thus, at any matching µ, the set of agents is partitioned into the set of agents that form couples C(µ) := {i ∈ W ∪ F | µ(i) = i} and the set of agents that remain single S(µ) := {i ∈ W ∪ F | µ(i) = i}; i.e., W ∪ F = C(µ) ∪ S(µ). Let M(W, F ) denote the set of matchings (for W and F ).
A matching µ is optimal for assignment problem (W, F, π) if, for all matchings µ ∈ M(W, F ), If µ is an optimal matching, then we refer to µ(i) as i's optimal partner at µ. We say that a worker w and a firm f are optimal partners if there exists an optimal matching µ such that (w, f ) ∈ µ.
An outcome for assignment problem (W, F, π) is a pair (µ, u) ∈ M(W, F ) × R |W ∪F | where µ is a matching and u is a payoff vector such that 5 This unit-demand assumption has also been made in the following and closely related articles: Shapley and Shubik [24], Crawford and Knoer [7], Chen et al. [5], Biró et al. [4], and Nax et al. [18]. 6 Our assumptions on the characteristic function π are without loss of generality. It is convenient to normalize agents' reservation values to be all equal to zero, i.e., one only measures net gains from the stand alone value each agent can obtain. This normalization, for instance, can be obtained by assuming that for each (w, f ) ∈ W × F , worker w requires a minimal salary s min (w, f ) to work for firm f and firm f is willing to pay a maximal salary smax(w, f ) for worker w. Then, taking the possibility of not forming a pair into account, the joint value created equals π(w, f ) = max{(smax(w, f ) − s min (w, f )), 0} ≥ 0.
The following property is a voluntary participation condition based on the idea that an agent can always enforce his reservation value by staying single. An outcome If, at outcome (µ, u) [at payoff vector u], there is a pair (w, f ) ∈ W × F such that u w + u f < π(w, f ), then w and f have an incentive to form a couple in order to obtain a higher payoff. Then, (w, f ) is a blocking pair for outcome (µ, u) [for payoff vector u] that creates the blocking surplus Let S(W, F, π) denote the set of stable outcomes for assignment problem (W, F, π). 8 The following lemma explains how optimal matchings and stable payoffs are related.
We provide two examples to illustrate the set of stable outcomes and their properties.
Example 1 (An infinite set of stable outcomes). Let (W, F, π) be an assignment problem given by W = {w}, F = {f }, and π(w, f ) = 1; that is, worker w and firm f generate value 1 by forming a pair. In any stable outcome (µ, u), w and f are matched and the set of stable outcomes equals S(W, F, π) := {(µ, u) | (w, f ) ∈ µ and [u w ≥ 0, u f ≥ 0, and u w + u f = 1]}. Note that there exists a unique optimal matching µ, the lowest stable payoff for each agent is 0, and the highest stable payoff for each agent is 1.
Example 2 (A finite set of stable outcomes). Let (W, F, π) be an assignment problem given by W = {w 1 , w 2 }, F = {f }, and π(w 1 , f ) = π(w 2 , f ) = 1. A stable outcome (µ, u) for (W, F, π) matches one of the workers w ∈ W with firm f while the other worker is single and the firm obtains the total value. Formally, 7 Note that in the definition of an outcome we could replace conditions (i) and (ii) by [ i∈W ∪F u i = (i,j)∈µ π(i, j)]. Then, a stable outcome (µ, u) would automatically satisfy (i) and (ii) in the definition of an outcome (Roth and Sotomayor [21], Lemma 8.5). Defining an outcome via conditions (i) and (ii) simplifies our exposition, but it is not essential for our results. 8 The set of stable outcomes coincides with the core (Shapley and Shubik [24]): we could model an assignment problem (W, F, π) as a cooperative game with transferable utility (TU) whose characteristic function υ assigns to each coalition S ⊆ W ∪ F , the number υ(S) ≡ max µ∈M(W ∩S,F ∩S) { (i,j)∈µ π(w, f )} with υ(∅) = 0. The core of assignment problem (W, F, π) is the set C(W, F, π) = {(µ, u) | µ is optimal and for all S ⊆ W ∪ F, i∈S u i ≥ υ(S)}. Thus, for any assignment problem (W, F, π) an outcome (µ, u) is in the core if matching µ is optimal and no coalition of agents S ⊆ W ∪ F can improve their payoffs at u by rematching among themselves. Furthermore, if an agent is single at a stable outcome, then at each stable outcome, he receives his reservation value (Demange and Gale [8]). the set of stable outcomes is S(W, F, π) := {(µ, u) | [either (w 1 , f ) ∈ µ or (w 2 , f ) ∈ µ] and [u w1 = u w2 = 0 and u f = 1]}. Any outcome (µ, u) at which firm f earns u f < 1 is not stable because then there is always a single worker w with u w = 0 such that u w + u f < 1 = π(w, f ). Note that there exist two optimal matchings, the unique stable payoff for each of the workers is 0, and the unique stable payoff for the firm is 1.
In Example 1 the set of stable outcomes is infinite: a unique optimal matching supports an infinite number of stable payoffs. In contrast, the set of stable outcomes in Example 2 is finite: two optimal matchings support the unique stable payoff.
3. Blocking paths to stability. For the following definitions of paths and blocking paths and in the formulation of our results we only focus on non-degenerate blocking pairs of one worker with one firm (a degenerate blocking pair would be a single agent blocking "pair"). Our focus on non-degenerate blocking pairs eases notation without loss of generality since we can always obtain an individually rational outcome whenever needed by "singleton blocking" by those agents who obtain negative payoffs.
A path for assignment problem (W, F, π) is a (finite!) sequence of outcomes (µ 1 , u 1 ), ..., (µ k , u k ) such that for each l ∈ {1, ..., k − 1}, the outcome (µ l+1 , u l+1 ) is obtained from (µ l , u l ) by matching a pair (w l , f l ). This induces the matching µ l+1 . Thus, at outcome (µ l+1 , u l+1 ), agents w l and f l are matched and generate value π(w l , f l ), their former partners are single and receive zero payoffs, and all the other agents are matched to the same partners and obtain the same payoffs as before.
A blocking path for assignment problem (W, F, π) is a path of individually rational outcomes (µ 1 , u 1 ), ..., (µ k , u k ) such that for each l ∈ {1, ..., k − 1}, the outcome (µ l+1 , u l+1 ) is obtained from (µ l , u l ) by matching a blocking pair (w l , f l ) for (µ l , u l ) such that the corresponding payoffs are u l+1 w l > u l w l and u l+1 f l > u l f l , i.e., the blocking agents w l and f l split their blocking surplus such that each of them is strictly better off at outcome (µ l+1 , u l+1 ). Hence, while Chen et al. [5], Nax et al. [18], and Biró et al. [4] require weak blocking, 9 we require the more demanding strict blocking norm for blocking pairs. We say that a blocking path leads to stability if the last outcome (µ k , u k ) is stable. We give a simple illustration using the assignment problem introduced in Example 1. 9 Under the weak blocking norm, agents payoffs only need to satisfy u l+1 w l ≥ u l w l and u l+1 f l ≥ u l f l with at least one strict inequality.
Example 3 only shows that such a blocking path to stability might exist. In the next example we construct an infinite sequence of outcomes by satisfying blocking pairs. Recall that a blocking path to stability is a finite sequence of outcomes ending in a stable outcome; hence, the following example does not construct a path to stability. In fact, we prove in Theorem 2 that no path to stability exists for the following two examples (Examples 4 and 5).
At each outcome (µ l , u l ), if l is even, then firm f is matched to worker w 1 and if l is odd (except for l = 1), then firm f is matched to worker w 2 . Since for all l ≥ 1,u l f < 1, the outcome (µ l , u l ) is never stable. Therefore, the sequence (µ 1 , u 1 ), (µ 2 , u 2 ), (µ 3 , u 3 ), ... is infinite. Furthermore, lim k→∞ u k f = 1 and, for all w ∈ W , lim k→∞ u k w = 0, i.e., payoffs converge to the unique stable payoffs.
Example 4 shows that a blocking path to stability might not always exist (since the finiteness of such a path for this specific assignment problem is impossible -a statement we will proof formally when proving Theorem 2). In this example, that is the case because it is always possible to make the firm and the respective single worker better off by letting them block the previous outcome. The only way for a path in this example to end in a stable outcome would require the firm to obtain the total value generated by the blocking pair. But then, the worker that blocks with the firm would be indifferent between working for the firm and staying single (he receives zero payoff in both cases) -it is part of the definition of a blocking path that both agents in a blocking pair are better off.
Example 4 also illustrates that an infinite blocking sequence can converge to a stable payoff: the firm's payoff monotonically increases and converges towards 1 (its stable payoff) and the workers' payoffs converge to 0 (their stable payoffs) along the sequence. However, next we show that we cannot always guarantee convergence to stable payoffs: we vary the previous example by constructing an infinite sequence of outcomes that converge to unstable payoffs (in fact, workers' payoffs do not converge at all).
Example 5 (An infinite sequence not converging to stable payoffs). Let a ∈ (0, 1) and consider the assignment problem (W, F, π) in Examples 2 and 4: Unlike in Example 4, we construct a sequence that is not based on equally splitting the blocking surplus. Instead, if at outcome (µ l , u l ) firm f has a payoff u l f < a, then its payoff at the next outcome by blocking with the single worker w s is given by u l+1 i.e., we guarantee the single worker a minimal payoff (1 − a) whenever he blocks with firm f .
Similarly to Example 4, start the sequence with the empty matching (µ 1 , u 1 ) as the initial (unstable) outcome. Select worker w 1 and let (µ 2 , u 2 ) be the outcome obtained from (µ 1 , u 1 ) by satisfying the blocking pair (w 1 , f ), such that µ 2 (f ) = w 1 , u 2 w1 = 1 − 1 2 a, u 2 w2 = 0, and u 2 f = 1 2 a. Continue the construction of the sequence similarly as in the previous example. Table 2 summarizes the payoffs along the sequence. Table 2. Payoffs along the sequence in Example 5.
As in Example 4, f alternates between w 1 and w 2 as its partner. Since for all l ≥ 1, u l f < a < 1, the outcome (µ l , u l ) is never stable. Therefore, the sequence (µ 1 , u 1 ), (µ 2 , u 2 ), (µ 3 , u 3 ), ... is infinite. Furthermore, and in contrast to Example 4, lim k→∞ u k f = a < 1 and for workers w payoffs alternate between 0 and values > (1 − a), i.e., the firm's payoffs converge to an unstable payoff and workers payoffs do not converge.
Note that in Examples 4 and 5, although the payoffs (partially) converge, the matchings do not converge. Along the blocking path firm f alternates between w 1 and w 2 as its partner. Because the path is not finite (and payoffs only converge in the limit, if at all) this oscillation never stops.
Example 2 is an assignment problem with a finite set of stable outcomes because the set of stable payoffs is a singleton. Based on this assignment problem, we have constructed in Examples 4 and 5 infinite sequences of outcomes that either payoff-converge to stability or that diverge in payoffs. In contrast, the set of stable outcomes in Example 1 is infinite because the set of stable payoffs is infinite. Based on the assignment problem described in Example 1 we have shown in Example 3 that a blocking path to stability might exist.
A crucial difference between the assignments problems depicted in Example 1 and Example 2 relates to the characteristics of the stable payoffs. In Example 1, there is an infinite number of interior stable outcomes, i.e., stable outcomes where all the agents obtain strictly positive payoffs, and two extreme outcomes where one of the two matched agents obtains a zero payoff. In Example 2, all the workers obtain zero payoffs at a stable outcome irrespective of which one of the two workers is matched with the firm. Since we require any two blocking agents to be strictly better off when satisfying a blocking pair, stability in Example 4 (based on Example 2) will never be reached because it is impossible to satisfy a blocking pair formed by a single worker and the firm with stable payoffs that make both of them strictly better off. 10 We show that for an assignment problem the existence of a stable outcome that is "away from zero" (zero being the normalized reservation value) for matched agents is a necessary and sufficient condition to guarantee the existence of a blocking path to stability. We formalize this condition as follows.
Property 1 (No reservation value degeneracy). Assignment problem (W, F, π) satisfies no reservation value degeneracy if there exists a stable outcome (µ * , u * ) such that for each agent i ∈ W ∪ F who is not single, i.e., µ * (i) = i, we have u * i > 0. Next, we show that no reservation value degeneracy is sufficient for the existence of blocking paths to stability.
Theorem 1 (Paths to Stability). Let (W, F, π) be an assignment problem satisfying no reservation value degeneracy and (µ, u) an arbitrary individually rational outcome for (W, F, π). Then, there exists a blocking path ( Note that assuming individual rationality for the initial outcome (µ, u) in Theorem 1 is without loss of generality: we can drop individual rationality if we slightly (and naturally) extend the definition of blocking in a blocking path by allowing single agents to block.
The proof of the theorem proceeds in three steps. In Step 1, if the starting outcome is not stable, we first unmatch as many couples as possible via blocking, i.e., we maximize the number of single agents. In some cases, using no reservation value degeneracy, we might already be able to rematch single agents via blocking to obtain a stable outcome. If this is not immediately possible, in Step 2 we then apply a stabilization process that deals with non optimal matchings, unstable payoffs, and single agents who cannot immediately be matched via blocking because they would have to receive a stable zero payoff to reach stability directly, which is not possible according to our strict blocking norm (and hence, some extra steps are needed to move the process towards a positive stable payoff for such agents). We prove that the stabilization process ends in finitely many steps by induction. In the final Step 3, using no reservation value degeneracy, we complete the construction of the blocking path by matching remaining single agents with optimal partners. The result is a stable outcome. Throughout the proof, a stable outcome (µ * , u * ) as specified by no reservation value degeneracy serves as a target and outcomes along the blocking path are getting closer to the target outcome along the way. We prove Theorem 1 in Appendix A and provide an example that illustrates the paths to stability construction in Appendix B.
One distinguishing difference between our results (Theorems 1 and 2) and the results by Biró et al. [4] (Theorem 1), Chen et al. [5] (Theorem 1), and Nax et al. [18] Theorem 1 is that with the strict blocking norm and continuity, paths to stability do not always exist (they only do exist if and only if no reservation value degeneracy is satisfied).
Theorem 2 (No Path to Stability). Let (W, F, π) be an assignment problem violating no reservation value degeneracy. Then, there exists an individually rational outcome (µ, u) for (W, F, π) such that no blocking path starting from (µ, u) leads to stability.
We prove Theorem 2 in Appendix A. We give a survey of the existence of paths to stability results for different assignment model specifications in Table 3. 4.1. Fair blocking paths. We have targeted throughout the proof of Theorem 1 an outcome (µ * , u * ) "away from zero" for matched agents (according to no reservation value degeneracy). In some steps of our blocking path we had to align payoffs according to the stable payoff vector u * ; in other words, at times we have used very specific payoff splits for certain blocking pairs. It is a natural question to ask if our result could also be obtained via an equal division blocking dynamics (as used in Examples 3 and 4).
We call a fair blocking path a sequence of outcomes, such that each outcome is obtained from the previous one by satisfying a blocking pair with the additional condition that the blocking agents equally split the blocking surplus. The following example shows that a fair blocking path might not lead to stability.
Example 6 (A fair blocking path with an infinite sequence of outcomes). Let (W, F, π) be an assignment problem given by matches each worker with any of the two firms and both workers (respectively both firms) obtain the same payoffs, i.e., stable payoffs must be aligned. Formally, the set of stable outcomes is S(W, F, π) Any outcome (µ, u) at which the two workers obtain different payoffs is not stable because then there is always a worker . Thus, the worker who gets the smallest payoff always forms a blocking pair with the firm that is matched with the other worker.
Note that (i, j) is the unique blocking pair for (µ 2 , u 2 ). Satisfy this blocking pair with equal surplus splitting to obtain the next outcome ( 1 2 , 1}, and the two remaining agents are single and obtain zero payoffs. Hence, outcome (µ 3 , u 3 ) has the same structure as outcome (µ 1 , u 1 ).
Hence, in this example, any blocking path that starts from the unstable outcome (µ 1 , u 1 ) at some point always reaches an unstable outcome that has the same structure as outcome (µ 1 , u 1 ). Hence, the sequence (µ 1 , u 1 ), (µ 2 , u 2 ), (µ 3 , u 3 ), ... is infinite and a fair blocking path in this example cannot lead to stability.
We have assumed in Example 6 that u 1 w1 / ∈ {0, 1 2 , 1} and we have constructed an infinite sequence of outcomes. Of course, a fair blocking path might lead to stability in finitely many steps in some instances. Consider the following sequence: let (µ 1 , u 1 ) be an outcome such that worker w 1 is matched with firm f 1 and both of them have payoffs u 1 w1 = u 1 f1 = 1 2 , and worker w 2 and firm f 2 are single and obtain zero payoffs. Suppose the next outcome (µ 2 , u 2 ) is obtained from (µ 1 , u 1 ) by satisfying the blocking pair (w 2 , f 2 ) with equal surplus splitting. At outcome (µ 2 , u 2 ), worker w 1 is matched with firm f 1 , worker w 2 is matched with firm f 2 and all the agents obtain stable payoffs u 2 i = 1 2 , i ∈ W ∪ F . Hence, the fair blocking path (µ 1 , u 1 ), (µ 2 , u 2 ) leads to stability in one step.

Probabilistic interpretation. A central question addressed by Roth and
Vande Vate [22], Diamantoudi et al. [10], Chen et al. [5] (Theorem 1), and Nax et al. [18] (Theorem 1) is whether a decentralized process where each blocking pair (and each possible blocking surplus split in the latter two models) is randomly selected with a strictly positive probability converges to a stable outcome. All four papers answer this question in the affirmative. In each of these papers the authors construct a blocking path that leads to stability in finitely many steps. Since each blocking pair is selected with strictly positive probability in a decentralized process, the blocking path they construct converges to stability with probability one. However, the fact that each blocking pair is selected with positive probability relies precisely on the underlying assumptions of the models. In the marriage problem of Roth and Vande Vate [22] and the roommate problem of Diamantoudi et al. [10], agents have ordinal preferences over the (finite) set of agents with whom they can form a blocking pair. In the assignment problem of Chen et al. [5] and Nax et al. [18] side payments are discrete and the number of possible divisions of a blocking surplus is finite. Those assumptions imply that for each outcome there is always a finite number of blocking pairs (including no blocking pairs if the outcome is stable).
In our assignment problem with continuous side payments two blocking agents can split the blocking surplus in infinitely many ways. We replace the assumption of the above discrete models that any blocking pair and surplus split is chosen with a positive probability with the assumption that blocking pairs and surplus splits are based on a probability distribution with full support over all blocking pairs and surplus splits. Now, the probabilistic interpretation based on the existence of a path to stability turns out to be problematic in our model if in some step of our blocking path constructed stable payoffs have to be aligned (see our next example for such a situation). More precisely, if for some blocking pair there is a unique division of a blocking surplus that leads to stability, then, given the continuity of payoffs, the point probability that such a blocking pair is selected is zero. Hence, in our model, we cannot deduct a probabilistic convergence to stability result from the existence of a blocking path to stability. The following example illustrates the situation.
Example 7 (Probabilistic Interpretation). Consider the assignment problem (W, F, π) in Example 6: W = {w 1 , w 2 }, F = {f 1 , f 2 }, and, for all (w, f ) ∈ W × F , π(w, f ) = 1. Recall that at any stable outcome the workers (the firms) must obtain the same stable payoffs, i.e., stable payoffs are aligned. Consider the following unstable outcome (µ, u): worker w 1 and firm f 1 are matched, w 1 obtains a payoff u w1 = a ∈ [0, 1], f 1 obtains a payoff u f1 = 1 − a ∈ [0, 1], and the remaining agents w 2 and f 2 are single and obtain zero payoffs. Graphically, (µ, u) is represented as follows: - The set of agents that form couples is C(µ) = {w 1 , f 1 }, w 1 and f 1 are optimal partners and receive stable payoffs, such that w 1 and f 1 are matched according to the stable outcome (µ * , u * ) where is an optimal matching, and (ii) u * is a stable payoff vector, i.e., a ∈ (0, 1), [for all i ∈ W , u * i = a], and [for all j ∈ F , u * j = 1 − a]. Hence, following our blocking path (Step 3, Matching completion process) it suffices to match worker w 2 with firm f 2 with stable payoffs u w2 = a and u f2 = 1 − a in order to reach the stable outcome (µ * , u * ) in one step. However, since payoffs are continuous, the probability that the blocking pair (w 2 , f 2 ) is satisfied with exactly those stable payoffs is zero. Similarly as in Example 6 one can show that any path to stability would require such a "zero probability" alignment step. Hence, probabilistically, convergence to stability in a decentralized process cannot be obtained.

A discussion of three closely related papers.
Discretized two-sided assignment with weak blocking: Chen et al. [5] and Nax et al. [18]. Our main result (Theorem 1) is related to Chen et al. [5] (Theorem 1) and Nax et al. [18] (Theorem 1) in that it implies the paths to stability results that they also obtain for their assignment model specifications. 11 Since both these papers obtain the same paths to stability result with its associated probabilistic interpretation (see our discussion in Section 4.2) using essentially the same proof technique, we explain the difference between their results and ours by referring to Chen et al. [5].
Chen et al. [5] study a labor market with finitely many heterogeneous workers and firms to illustrate the blocking dynamics in assignment problems. They prove the existence of blocking paths to stability for assignment problems, as we do. They make two main assumptions that make their model different from our model. First, Chen et al. [5] consider an assignment problem with discrete side payments. Second, they use a weak blocking norm: two agents form a blocking pair if satisfying this blocking pair makes at least one of them strictly better off. As Chen et al. [5] we also study an assignment problem with side payments, but side payments are continuous and our strict blocking norm requires that two agents form a blocking pair if and only if satisfying this blocking pair makes both agents strictly better off. In contrast to Chen et al. [5] we have shown that with our assumptions a blocking path to stability does not always exists in the assignment problem. We identified a necessary and sufficient condition to guarantee the existence of blocking paths to stability. As discussed in Section 4.2, the probabilistic interpretation of the path to stability result that Chen et al. [5] (Theorem 1) obtain does not apply to our continuous assignment model.
Although our results and the results of Chen et al. [5] are closely related, we use a different proof technique. Chen et al. [5] essentially construct an algorithm that targets a side optimal outcome. Each time an agent, say agent i, is selected to block an outcome, he will choose to block with his most preferred partner, i.e., the blocking partner with whom he can generate the largest surplus, and offers this best blocking partner the smallest payoff consistent with the incentives to block, such that i obtains in the next outcome the largest payoff while forming a blocking pair (this element of the proof is similar to the approach in Roth and Vande Vate [22] but the authors additionally need to take care of tie-breaking to avoid cycling). Our proof is more akin to the one used by Diamantoudi et al. [10] in the sense that our algorithm (as well as theirs) uses a target stable outcome to avoid cycling.
Despite the fact that our proof technique differs from the proof in Chen et al. [5] , our proof works well for their environment. To see how our blocking path construction works for assignment problems with discrete side payments is straight forward. If payoffs were discretized in our model, then any blocking surplus can only be divided in finitely many ways; switching from continuous to discrete payoffs in our model has the simplifying effect to reduce the number of blocking possibilities to a finite number. When applying our proof construction to a discretized assignment problem, our blocking path will still lead to stability but in possibly fewer steps (the reason why we then also can drop our necessary and sufficient assumption of no reservation value degeneracy is that with discrete payoffs the convergence problems we indicate in Section 3 cannot occur). Hence, our result extends that of Chen et al. [5] to assignment problems with continuous side payments.
Chen et al. [5] allow weak blocking pairs to be formed. In contrast, we focused on strict payoff improvement for a blocking pair to be satisfied. Therefore, we need to target stable payoffs away from zero (and assume their existence by our necessary and sufficient assumption of no reservation value degeneracy) to ensure that agents that obtain zero payoffs at some point and are matched at a stable outcome have a clear incentive to form a blocking pair by obtaining a strictly higher payoff. This crucial situation of having to match zero payoff stable partners may appear multiple times along our blocking path (see our proof of Theorem 1 in Appendix A). 12 In our model, allowing for weak blocking pairs would simply make no reservation value degeneracy unnecessary because then it would always be possible to satisfy a blocking pair of optimal partners with one of them being single even though the single agent obtains zero payoff at a stable outcome. For instance, recall that in Example 4 we constructed an infinite sequence of outcomes that converges to stable payoffs. The infiniteness of the sequence is precisely due to the fact that at a stable outcome the worker who is matched obtains a zero payoff. Thus, in this example a blocking path would exist under the weak blocking norm.
Continuous one-sided assignment with weak blocking: Biró et al. [4]. Biró et al. [4] establish the existence of blocking paths to stability for one-sided assignment problems using the weak blocking norm. Hence, part of their model is more general than ours because they fully work out the one-sided assignment setting (we only remark that our proof technique can easily be used in a corresponding onesided setting) and part of our model is more general since we use the more stringent strict blocking norm. Hence, our results are very close, but somewhat incomparable (in addition to being independently obtained).
Note that using a proof technique with a target stable outcome is absolutely necessary for one-sided assignment problems (for one-sided assignment problems the set of stable outcomes may be empty and if it is nonempty, then it does not need to form a complete lattice with two extreme points that reflect the polarization between both sides of the assignment problem). However, in spite of being a more general proof technique (since it also applies to one-sided assignment problems), the target outcome proof technique comes at the price of a less intuitive myopic blocking behavior along the constructed blocking path because at times blocking payoffs have to be aligned according to the stable target payoffs. In contrast, the classical twosided proof technique used by Chen et al. [5] and Nax et al. [18] requires an actively blocking agent to find a blocking partner with the highest blocking surplus and then extract the highest possible blocking payoff from this blocking partner (with the weak blocking norm this corresponds to the active blocking partner taking the 12 First, in Cases 2.C and 3.C within Step 2 (Stabilization Process) we have defined a short blocking sequence (2 steps) in order to rematch two optimal partners according to our stable target payoff vector. In the second phase of this short blocking sequence, we rematch one agent who forms a couple (but not with an optimal partner) with his optimal partner who is single and obtains zero payoff. Thus, without the weak blocking assumption, this rematching step is feasible only if a stable payoff away from zero exists (no reservation value degeneracy). Second, in Step 3 (Matching Completion Process) we complete the blocking path by matching single agents who are optimal partners. Given that all single agents obtain zero payoffs, no reservation value degeneracy is needed in order to ensure a strict payoff improvement for each blocking agent.
whole blocking surplus -this extreme blocking surplus extraction is not possible under the strict blocking norm that we use in contrast to the other articles we have been discussing in this section).

4.4.
Decentralized versus centralized matching and median stable target outcomes. In the previous section we have discussed that the classical "greedy" two-sided proof employed in Chen at al. [5] and Nax et al. [18] needs to be replaced by a target outcome proof in our context because we use the strict blocking norm (another reason would be the consideration of the one-sided version of our model). Hence, to find such a path in a decentralized market, agents would have to know about the target stable outcome. Unfortunately, this isn't a realistic informational requirement in a decentralized market. Furthermore, as pointed out in Section 4.2, the probability to find a path to stability by a random blocking dynamics might be zero. Therefore, our results could be interpreted as "impossibility of finding a path to stability" results for some decentralized markets. For these markets, centralization would seem like a good alternative.
Here we argue that then the target method can be used in a centralized assignment market to stabilize an unstable outcome using a "compromise" target outcome. For two-sided assignment problems it is well-known that two extreme stable payoff vectors (and associated outcomes) exist (Shapley and Shubik [24]): the worker-optimal stable payoff vector and the firm-optimal stable payoff vector. These are the stable payoff vectors that are most unequal within the set of stable payoff vectors. Schwarz and Yenmez [23] define the median stable payoff vector (and associated outcomes) as a compromise solution and prove that they are well defined and exist.
Assume that in a centralized labor market we detect that current payoffs are not stable. A centralized adjustment process could then use the median stable payoff vector in our paths to stability algorithm to move to a stable outcome (and by doing so, the originally unstable payoffs, in the stabilization process, would be moved closer to the median stable payoffs). The rationale behind such a centralized stabilization procedure would then be that the resulting outcome could be obtained via decentralized blocking that targets a compromise stable outcome for the current situation.

4.5.
Concluding remarks. We have studied two-sided one-to-one matching problems with continuous side payments. We have considered the existence of blocking paths to stability for such assignment problems under the strict blocking norm. In contrast to weak blocking paths results by Biró et al. [4], Chen et al. [5], and Nax et al. [18], the existence of a blocking path to stability cannot always be guaranteed. We identified a necessary and sufficient condition (no reservation value degeneracy) for the existence of a blocking path to stability.
With no reservation value degeneracy, we distinguish between two types of stable outcomes for any given assignment problem: stable outcomes that involve matched agents with zero payoffs versus those stable outcomes where all matched agents receive strictly positive payoffs (recall that the role of zero here is that of an agent's reservation value in our normalized setup). We find that if stable outcomes are exclusively of the first type, then no path to stability exists (Theorem 2), while the existence of a stable outcome of the second type guarantees the existence of a path to stability (Theorem 1). Even when a path to stability is guaranteed to exist, our results show that finding or constructing such a path might not be trivial (the Proof of Theorem 1 demonstrates that the path construction could be rather involved and requires the use of a target stable outcome satisfying no reservation value degeneracy). Moreover, with examples such as Example 6 we show that an intuitively fair blocking dynamics might never converge to a stable outcome. While these results seem to be bad news for stability as the result of our (myopic) decentralized process, our proof technique has the potential to be used in a centralized market in which a central planner may deliberately choose a specific stable target outcome, e.g., a median stable outcome, for the stabilization process described in the Proof of Theorem 1 (Appendix A).
With suitable modifications of our model (i.e., allowing for weak blocking or specifying a discrete payoff structure), our results imply the results of Chen et al. [5] and Nax et al. [18]. However, the converse is not true: modifying their model to coincide with ours (i.e., imposing strict blocking and allowing continuous transfers) would not allow them to easily adapt their very different proofs to obtain our results. Our proof technique is somewhat similar to that of Biró et al. [4]. The main difference with Biró et al. [4] is that we have to deal with the more stringent requirement of strict blocking, which is the reason why in our model a necessary and sufficient condition is added to obtain the existence of paths to stability. Even though we formulate our model as a two-sided model, our proof technique does not depend on the two-sidedness of the market and hence we could easily obtain corresponding results for a one-sided modelà la Biró et al. [4].
Note that whenever we use the generic notation (i, j) for a pair, then (i, j) ∈ W × F or i = j ∈ W ∪ F are both possible. On a few occasions in the sequel we will also use the specific notation i, j when it is clear that either (i, j) ∈ W × F or (j, i) ∈ W × F , but it is not important which is the case. With some abuse of notation we will not adjust the notation for the corresponding value π(i, j) agents i and j create.
Proof of Theorem 1. Let (W, F, π) be an assignment problem satisfying no reservation value degeneracy and (µ, u) an arbitrary individually rational outcome for (W, F, π). By no reservation value degeneracy, there exists a stable outcome (µ * , u * ) such that for each agent i ∈ W ∪ F who is not single, i.e., µ * (i) = i, we have If is (µ, u) is stable we have nothing to prove. Hence, assume that (µ, u) is individually rational but not stable.
Since at each Step 1.l (l ≥ 1) the number of agents that form couples is reduced by 2, the unmatch process reaches an outcome (µ a , u a ) (a ≥ 1) such that (µ a , u a ) is stable within the set C(µ a ) in finitely many steps. The unmatch process (if starting from a non-empty matching) generates an outcome (µ a , u a ) with at least one couple and C(µ a ) = ∅.

After
Step 1, we distinguish three cases for outcome (µ a , u a ). The first one, Case (A), allows us to easily complete our blocking path using the stable target outcome (µ * , u * ).
If outcome (µ a , u a ) is not in Case A, then we will apply Step 2 in order to appropriately stabilize the outcome through blocking. We distinguish two remaining cases for outcome (µ a , u a ). First, Case (B) deals with instability caused by nonoptimal matching. Second, Case C deals with "problematic" payoffs, i.e., either payoffs are not stable or some agents receive zero stable payoffs.
Case B. There does not exist an optimal matchingμ such thatμ| C(µ a ) = µ a | C(µ a ) , i.e., the agents in C(µ a ) are not matched according to an optimal matching. Case C. There exists an optimal matchingμ such thatμ| C(µ a ) = µ a | C(µ a ) but there exist no stable payoffsũ such thatũ| C(µ a ) = u a | C(µ a ) and for all i ∈ C(μ) \ C(µ a ), u i > 0. 15 The next step will use an induction argument to construct an outcome (µ c , u c ) that belongs to Case A. 13 If we would allow for an individually irrational initial outcome (µ, u), then we would use singleton blockings in this step to unmatch any agent who receives a negative payoff.
14 By (i) in the definition of an outcome, µ l (w l ) = f l . 15 The reason why with zero stable payoffs we might not be able to directly proceed with the completion of our blocking path by going to Step 3 is that it might require to match two singles such that one of them receives a zero payoff; an example of such a situation is: W = {w 1 , w 2 }, F = {f 1 , f 2 }, π(w i , f j ) = 1 for all i, j ∈ {1, 2}, (µ a , u a ) such that C(µ a ) = {w 1 , f 1 }, and u a such that u a w 1 = 1 and u a w 2 = u a f 1 = u a f 2 = 0. Proceeding as in the later Step 3 would require that w 2 and f 2 match with a zero payoff for f 2 ; this would not be a strict blocking pair.
Step 2. Stabilization process. We continue our blocking path (µ a , u a ), (µ a+1 , u a+1 ), ... with the aim to stabilize the set of agents that form couples. Throughout this step, we use a stable outcome (µ * , u * ) satisfying inequality (*) for agents matched at µ * . Note that whenever we refer to (µ * , u * ) (and inequality (*)) we are applying no reservation value degeneracy.
Letμ be the matching that is obtained from the optimal target matching µ * by rematching agents in {w, µ * (f ), f, µ * (w)} according to matching µ a , i.e., Since there does not exist an optimal matchingμ such thatμ| {w,f } = µ a | {w,f } , matchingμ is not optimal. Hence, First, assume that |{w, µ * (f ), f, µ * (w)}| = 4, i.e., µ * (w) = w and µ * (f ) = f . Hence, (w, f ), (µ * (w), µ * (w)), (µ * (f ), µ * (f )) ∈μ. By construction (2.1) ofμ, matchingsμ and µ * coincide for all agents i / ∈ {w, µ * (f ), f, µ * (w)}). Thus, Hence, u a w + u a µ * (w) + u a µ * (f ) + u a f < π(w, µ * (w)) + π(µ * (f ), f ). (2.4) stable payoff vector, by construction (2.6), any blocking pair for outcome (μ,ū) involves either worker w or firm f . Assume that agent i ∈ {w, f } with u b i > u * i is part of a blocking pair i, j for outcome (μ,ū). Then, π(i, j) >ū i +ū j = u b i + u * j > u * i + u * j , contradicting the stability of payoff vector u * . Hence, u b i < u * i for the agent i ∈ {w, f } who participates in blocking pair i, j for outcome (μ,ū). Note that j ∈ S(µ b ) and u b j = 0. Thus, π(i, j) >ū i +ū j ≥ u b i + u b j and i, j is also a blocking pair for outcome (µ b , u b ). Now, starting from outcome (µ b , u b ), satisfy this blocking pair to obtain the next outcome (µ b+1 , u b+1 ) with the condition that u b+1 , and note that at outcome (µ b+1 , u b+1 ) agent i's previous partner µ b (i) is single and receives u b+1 µ b (i) = 0. By construction, u b+1 i < u * i and u b+1 µ b (i) = 0 < u * µ b (i) . Thus, i, µ b (i) = (w, f ) is a blocking pair for outcome (µ b+1 , u b+1 ) that we can satisfy to obtain outcome (µ b+2 , u b+2 ) with the condition that u b+2 Thus, outcome (µ b+2 , u b+2 ) is a Case A outcome. Set (µ c , u c ) := (µ b+2 , u b+2 ) and go to Step 3, where we complete our blocking path to stability by matching single agents in S(µ c ) according to (μ, u * ). Case 3 (t > 1). At the internally stable outcome (µ a , u a ), there are t > 1 couples and agents in C(µ a ) are not matched according to a stable outcome as in Case A. We will use an induction argument to continue our blocking path (µ a , u a ), (µ a+1 , u a+1 ), ... in order to construct an outcome (µ c , u c ) that belongs to Case A. Induction Basis (t = 1). For t = 1, we can construct a blocking sequence (µ a , u a ), ..., (µ c , u c ) (Case 2) such that the set of agents that form couples are stabilized as in Case A, i.e., there exists some stable outcome (μ,ũ) such that (μ,ũ)| C(µ c ) = (µ c , u c )| C(µ c ) and for all i ∈ C(μ) \ C(µ c ),ũ i > 0. Induction Hypothesis (t ≥ 1). Assume that for t ≥ 1 and starting with an internally stable outcome (µ a , u a ) we can construct a blocking sequence (µ a , u a ), ..., (µ c , u c ) such that the set of agents that form couples are stabilized as in Case A, i.e., there exists some stable outcome (μ,ũ) such that (μ,ũ)| C(µ c ) = (µ c , u c )| C(µ c ) and for all i ∈ C(μ) \ C(µ c ),ũ i > 0. Induction Step (t → t + 1). We now assume that at outcome (µ a , u a ), there are t + 1 > 1 couples. Since there does not exist any stable outcome (μ,ũ) such that (μ,ũ)| C(µ a ) = (µ a , u a )| C(µ a ) and for all i ∈ C(μ) \ C(µ a ),ũ i > 0 one of the following holds: (3.B) there does not exist an optimal matchingμ such thatμ| C(µ a ) = µ a | C(µ a ) or (3.C) there exists an optimal matchingμ such thatμ| C(µ a ) = µ a | C(µ a ) but there exist no stable payoffsũ such thatũ| C(µ a ) = u a | C(µ a ) and for all i ∈ C(μ) \ C(µ a ),ũ i > 0. We now start to stabilize the set of agents that form couples as we did in Case 2. During this process, we might create blocking pairs that reduce the number of couples. Whenever this happens, we apply the induction hypothesis to obtain a blocking sequence that results in an outcome (µ c , u c ) such that the set of agents that form couples are stabilized as in Case A. We can then directly proceed to Step 3. Case 3.B. Consider the optimal target matching µ * and denote the set of single agents at µ a that at µ * are matched to agents in C(µ a ) by Note that each agent i ∈ C(µ a ) ∪ S * (µ a ) has his optimal µ * -partner in C(µ a ) ∪ S * (µ a ).
Letμ be the matching that is obtained from the optimal target matching µ * by rematching agents in C(µ a ) ∪ S * (µ a ) according to matching µ a , i.e., Since there does not exist an optimal matchingμ such thatμ| C(µ a ) = µ a | C(µ a ) , matchingμ is not optimal. Hence, By construction (3.1) ofμ, matchingsμ and µ * coincide for all agents i / ∈ C(µ a ) ∪ S * (µ a ). Thus, Note that This implies that we can rewrite L1 as We can now rewrite (3.3) as Next, as shown in Table 4, we map terms on the left side to terms on the right side of inequality (3.4) as follows (to be precise, we define a bijection between terms on the left and the right side of the inequality): Table 4. A bijection between terms on both sides of inequality (3.4).
Since at each Step 2.3.B.l (l ≥ 1) the number of agents that form couples and are matched according to µ * strictly increases by 2, we reach an outcome (µ b , u b ) (b > a) such thatμ| C(µ b ) = µ b | C(µ b ) for some optimal matchingμ in at most t steps (unless we apply the induction hypothesis and go to Step 3).
If (as in Case A) there exists some stable outcome (μ,ũ) such that (μ,ũ)| C(µ b ) = (µ b , u b )| C(µ b ) and for all i ∈ C(μ) \ C(µ b ),ũ i > 0, then set (µ c , u c ) := (µ b , u b ) and go to Step 3, where we complete our blocking path to stability by matching single agents in S(µ c ) according to (μ,ũ). Otherwise, continue the blocking path as described next in Case 3.C. Case 3.C. If at outcome (µ a , u a ) there exists an optimal matchingμ such that µ| C(µ a ) = µ a | C(µ a ) , then first set (µ b , u b ) := (µ a , u a ). We now continue the blocking path with (µ b , u b ) as the initial (internally stable) outcome (note that (µ b , u b ) can come either directly from Step 1 or from Case 3.B within Step 2).
Letū be the payoff vector that is obtained from the stable target payoff vector u * by replacing the payoffs of the agents in C(µ b ) at u * with those at u b , i.e., If payoff vectorū is stable, then outcome (μ,ū) is stable and such that (μ,ū) would be a Case A outcome and we would have continued to Step 3 directly from Step 1. Hence, there is at least one blocking pair i, j for outcome (μ,ū). Since u * is a stable payoff vector, by construction (3.6), any blocking pair for outcome (μ,ū) involves a matched agent i 1 ∈ C(µ b ) such that u b i1 = u * i1 . Assume that agent i 1 ∈ C(µ b ) with u b i1 > u * i1 is part of a blocking pair i 1 , j 1 for outcome (μ,ū). Then, π(i 1 , j 1 ) >ū i1 +ū j1 = u b i1 + u * j1 > u * i1 + u * j1 , contradicting the stability of payoff vector u * . Hence, u b i1 < u * i1 for the agent i 1 ∈ C(µ b ) who participates in blocking pair i 1 , j 1 for outcome (μ,ū). Note that j 1 ∈ S(µ b ) and u b j1 = 0. Thus, π(i 1 , j 1 ) >ū i1 +ū j1 ≥ u b i1 + u b j1 and i 1 , j 1 is also a blocking pair for outcome (µ b , u b ). Now, starting from outcome (µ b , u b ), satisfy this blocking pair to obtain the next outcome (µ b+1 , u b+1 ) with the condition that u b+1 and note that at outcome (µ b+1 , u b+1 ) agent i 1 's previous partner µ b (i 1 ) is single and receives u b+1 is a blocking pair for outcome (µ b+1 , u b+1 ) that we can satisfy to obtain outcome (µ b+2 , u b+2 ) with the condition that u b+2 . To summarize, we can always identify two consecutive blocking pairs i 1 , j 1 (such that i 1 ∈ C(µ b ), u b i1 < u * i1 , and j 1 ∈ S(µ b )) and i 1 , µ b (i 1 ) such that the resulting outcome rematches the original couple i 1 , µ b (i 1 ) with payoffs u * i1 and u * µ b (i1) . If (µ b+2 , u b+2 ) is not internally stable, we can use the unmatch process in Step 1 to obtain an internally stable outcome with fewer couples. We can then use the induction hypothesis to obtain a blocking sequence that results in an outcome (µ c , u c ) such that the set of agents that form couples are stabilized as in Case A. We can then directly proceed to Step 3. Note that |C(µ b+2 )| ∈ {|C(µ b )| − 2, |C(µ b )|}.
If |C(µ b+2 )| = |C(µ b )| − 2 we can again apply the induction hypothesis and proceed to Step 3. Thus, assume that |C(µ b+2 )| = |C(µ b )| and that outcome (µ b+2 , u b+2 ) is internally stable. If outcome (µ b+2 , u b+2 ) is a Case A outcome, we again proceed to Step 3. Finally, if outcome (µ b+2 , u b+2 ) is again a Case C outcome, then we can repeat the same arguments to find another short stabilizing blocking sequence to obtain an outcome (µ b+4 , u b+4 ), etc., as follows: Step 2.3. C.l. For all l ≥ 1, if outcome (µ b+2(l−1) , u b+2(l−1) ) is a Case C outcome, then let (µ b+2l , u b+2l ) be the outcome obtained by satisfying a short stabilizing blocking sequence with blocking pairs i l , j l (such that i l ∈ C(µ b+2l−2 ), u b+2l−2 i l < u * i l , and j l ∈ S(µ b+2l−2 )) and i l , µ b (i l ) such that the resulting outcome rematches the original couple i l , µ b (i l ) with payoffs u * i l and u * µ b (i l ) . Assume that |C(µ b+2l )| = |C(µ b+2l−2 )| and that (µ b+2l , u b+2l ) is stable within the set C(µ b+2l ) (otherwise we apply the induction hypothesis and go to Step 3). Note that we have strictly increased the number of agents that form couples and receive payoffs according to Since at each Step 2.3.C.l (l ≥ 1) the number of agents that form couples and receive payoffs according to u * strictly increases by 2, we reach an outcome (µ c , u c ) (c > b) such thatμ| C(µ c ) = µ c | C(µ c ) for some optimal matchingμ and u * | C(µ c ) = u c | C(µ c ) in at most t steps, where in each step two blocking pairs are satisfied (unless we apply the induction hypothesis and go to Step 3).
Step 3.l. For all l ≥ 1, if there exists (w l , f l ) such that w l , f l ∈ C(μ) \ C(µ c+l−1 ) andμ(w l ) = f l , then (w l , f l ) is a blocking pair for (µ c+l−1 , u c+l−1 ) such that w l and f l are single at µ c+l−1 . Satisfy this blocking pair to obtain (µ c+l , u c+l ) with the property that for i ∈ {w l , f l }, u c+l i =ũ i . Note that outcome (µ c+l , u c+l ) is internally stable and the new set of agents that form couples C(µ c+l ) = C(µ c+l−1 ) ∪ {w l , f l } contains more agents: |C(µ c+l )| = |C(µ c+l−1 )| + 2.
The matching completion process increases the number of couples without perturbing the stability within the set of matched agents. The process terminates when all agents have been (re)matched according to the stable outcome (μ,ũ). Hence, the matching completion process terminates in finitely many steps resulting in a stable outcome (μ,ũ).
Proof of Theorem 2. Let (W, F, π) be an assignment problem violating no reservation value degeneracy. Hence, there exists no stable outcome (µ * , u * ) such that for each agent i ∈ C(µ * ) we have u * i > 0. Equivalently, for all stable outcomes (µ * , u * ) there exists an agent i ∈ C(µ * ) such that u * i = 0.
Then, from outcome (μ,ũ) no blocking path leads to stability. The reason for this is that in order to end in a stable outcome (µ * , u * ), one of the agents in X(μ,û) needs to be matched at a zero payoff along the blocking path. This, however, violates the strict blocking norm.
Step 1. Unmatch process. First we unmatch as many couples as possible.
Step 2. Stabilization process. Second, we stabilize the set of agents that form couples. For instance, two blocking pairs of optimal partners at µ * for (µ a , u a ) are: (w 2 , µ * (w 2 )) = (w 2 , f 2 ) and (w 3 , µ * (w 3 )) = (w 3 , f 3 ). If we satisfy one of those blocking pairs, we increase the number of agents who are matched to an optimal partner at µ * . As we will see, depending on which blocking pair we satisfy, that is either (w 2 , f 2 ) or (w 3 , f 3 ), the blocking path might take different routes. We investigate the two cases.
Case 2. Let (w 3 , f 3 ) block (µ a , u a ). Satisfy the blocking pair (w 3 , f 3 ) to obtain outcome (µ a+1 , u a+1 ), such that w 3 and f 3 obtain payoffs u a+1 w3 = 0.1 and u a+1 f3 = 1.9. Hence, at outcome (µ a+1 , u a+1 ), w 3 and f 3 form a couple, and f 3 's previous partner w 2 is single and obtains zero payoffs. All the other agents are matched to the same partners and earn the same payoffs as before. Outcome (µ a+1 , u a+1 ) is such that (w 3 , f 4 ) is a blocking pair for (µ a+1 , u a+1 ). Then, satisfy this blocking pair to reduce the set of agents that form couples by one couple and apply the induction hypothesis to obtain a blocking sequence that stabilizes the set of agents that form couples (the application of the induction hypothesis is not further worked out in this example).
Step 3. Matching completion process. We continue the construction of the blocking path with (µ c , u c ) := (µ a+1 , u a+1 ), where (µ a+1 , u a+1 ) was obtained from Case 1. In this step, we complete the blocking path to stability by matching the single agents in S(µ c ) according to the stable outcome (µ * ,ũ).