Strategy-Proofness of Worker-Optimal Matching with Continuously Transferable Utility

We give a direct proof of one-sided strategy-proofness for worker-ﬁrm matching under continuously transferable utility. A new “Lone Wolf” theorem (Jagadeesan et al. (2017)) for settings with transferable utility allows us to adapt the method of proving one-sided strategy-proofness that is typically used in settings with discrete transfers.

4 While many people believed or expected strategy-proofness results to hold for more general markets with continuous transfers, we are not aware that any such results were known formally prior to the work of Hatfield, Kojima, and Kominers (2017) (see also the discussion in Footnote 9). 5 On the other hand, the Vickrey-Clarke-Groves (VCG) mechanism is available in settings with continuously Recently, Hatfield, Kojima, and Kominers (2017) proved one-sided strategy-proofness of deferred acceptance for unit-supply sellers in the trading network framework of Hatfield et al. (2013); 6,7 however, their proof strategy is quite indirect. 8,9 Here, we give a direct proof of one-sided strategy-proofness in a context with continuously transferable utility. We consider worker-firm matching with quasilinear utility, and adapt the Hatfield and Milgrom (2005) method of proving one-sided strategy-proofness in settings with discrete contracts. Our approach makes use of a version of the Lone Wolf Theorem that we recently developed for settings with transferable utility (Jagadeesan et al. (2017)).
In recent independent work, Schlegel (2016) has shown that worker-optimal core-selecting mechanisms exist and are strategy-proof for workers in non-quasilinear settings that satisfy a preference substitutability assumption (see Theorem 5 of Schlegel (2016)). Schlegel's (2016) proof combines results on the strategy-proofness of worker-optimal core-selecting mechanisms in settings with discrete transfers (Hatfield and Milgrom (2005)) with a limiting argument.
Our approach extends to Schlegel's (2016) setting by replacing our Lone Wolf Theorem (Jagadeesan et al. (2017)) with Schlegel's (2016) version of the lone wolf result.
Our results show that the matching-theoretic approach to proving strategy-proofness generalizes to settings with transferable utility. In contrast, Ausubel (2004Ausubel ( , 2006 and Sun and Yang (2014) work with multi-unit demand settings, in which the matching-theoretic approach does not yield strategy-proofness results (Roth (1984a)). 6 The Hatfield et al. (2013) model that Hatfield, Kojima, and Kominers (2017) work with is quite general-it embeds the worker-firm matching setting of Crawford and Knoer (1981) and the object allocation settings of Koopmans and Beckmann (1957), Shapley and Shubik (1971), Stacchetti (1999, 2000) and Yang (2006, 2009). 7 Hatfield et al. (2013) assumed quasilinear utility (as we do), so their model does not fully generalize the model of Kelso and Crawford (1982); they (and we) only address the quasilinear case of Kelso and Crawford (1982).
8 Effectively, Hatfield, Kojima, and Kominers (2017) showed that the seller-optimal stable matching mechanism coincides with VCG when all sellers have unit supply. We instead apply matching-theoretic arguments to prove strategy-proofness directly; by the Green-Laffont-Holmström Theorem, it then follows that the seller-optimal stable matching mechanism coincides with VCG. 9 The results of Ausubel (2004Ausubel ( , 2006 for the case of unit-demand bidders and no bid information imply the strategy-proofness of deferred acceptance when firms have linear values for workers. Indeed, in this case, the firms' valuations can be interpreted as reserve prices; then deferred acceptance corresponds to an ascending clock auction (Hatfield and Milgrom (2005)), and so the strategy-proofness results of Ausubel (2004Ausubel ( , 2006 apply. The remainder of this paper is organized as follows: Section 2 presents an example that illustrates the idea and proof of our main result. Section 3 introduces a transferableutility version of the Kelso and Crawford (1982) model of worker-firm matching. Section 4 introduces the Lone Wolf Theorem and proves our strategy-proofness result. Section 5 discusses antecedents and implications of our results.

An Illustrative Example
We begin with an example economy that illustrates our main result and proof strategy. We suppose that there are three workers (or as we might say, illustrators), Al, Bob, and Charles, and two firms, Gale Gallery and Shapley Studios. Firms' valuations are as given in Table 1; workers' costs of working are as presented in Table 2.  We say that an employment outcome is in the core if no set of workers and firms can profitably recontract among themselves; such an outcome is worker-optimal if it is mostpreferred by all workers among all core outcomes. Two worker-optimal core outcomes exist in our economy: 10 outcome S, in which Al and Bob work for Shapley Studios at salaries of $700 and $500, respectively, and outcome T , in which Bob works for Gale Gallery at a salary of $1000 and Al and Charles work for Shapley Studios at salaries of $700 and $300, respectively.
Note that, given their cost functions, all workers are indifferent between S and T .
The Jagadeesan et al. (2017) Lone Wolf Theorem shows that if any agent is unmatched in one core outcome, then he receives 0 net utility in every core outcome. 11 For example, because Charles is unmatched in S, he must receive 0 net utility in any core outcome. 12 One mechanism for finding a worker-optimal core outcome is the worker-proposing deferred acceptance mechanism (with salaries), also called the descending salary adjustment process (see Crawford and Knoer (1981); Kelso and Crawford (1982)); under this mechanism, salaries start at their highest possible level and descend until all firms are willing to pay the salaries of the workers that prefer to work for them at those salaries, and the market clears. In our setting, if all workers reveal their true costs of employment (in their choices between firms at proposed salaries), then worker-proposing deferred acceptance selects either S or T .
Our main result shows that worker-proposing deferred acceptance-indeed, any workeroptimal core-selecting mechanism-is strategy-proof for workers, that is, it is a dominant strategy for all workers to reveal their true costs/preferences. We explain the underlying logic in the case of Charles: 13 We can show by construction that if there is a strictly profitable deviation for Charles that assigns him to firm f at salary s, then there is a strictly profitable deviation in which Charles 10 Note that both firms in our example have substitutable valuations (in the sense of Kelso and Crawford (1982)), so worker-optimal core outcomes are guaranteed to exist (see Kelso and Crawford (1982) and Hatfield et al. (2013)).
11 The Jagadeesan et al. (2017) Lone Wolf Theorem applies to competitive equilibria, but competitive equilibrium outcomes correspond to core outcomes (and to stable outcomes) in many-to-one matching with transfers (Kelso and Crawford (1982); Roth (1984b); Hatfield et al. (2013)). 12 In particular, observe that Charles receives 0 net utility under T . 13 The interested reader might consider possible deviation strategies to observe that none make Charles better-off.
1. represents that only firm f is acceptable and 2. correctly represents his cost of working at firm f.
As Charles is unemployed under S, which is in the core when he reports his true preferences, S is clearly in the core when he misrepresents his preferences by shading his willingness to work. By the Lone Wolf Theorem, Charles must receive 0 net utility under any outcome in the core under his misrepresented preferences-but this contradicts our hypothesis that Charles has a strictly profitable deviation. 14

Model
We work with a model of many-to-one matching with continuously transferable utility, following Crawford and Knoer (1981) and Kelso and Crawford (1982). There is a finite set F of firms and a finite set W of workers. We let I ≡ F ∪ W denote the full set of agents.
A firm f and a worker w can sign a contract (f, w, s) indicating that w will work for f at salary s ∈ R. The full set of contracts is X ≡ F × W × R. Given a set of contracts A ⊆ X, we denote the sets of contracts in A associated to firm f ∈ F and worker w ∈ W by respectively. An outcome is a set of contracts A ⊆ X under which each worker is employed by at most one firm, i.e., a set of contracts A for which |A w | ≤ 1 for all workers w ∈ W .
14 The fact that Charles receives 0 net utility under truthful reporting in our example simplifies the logic here. In the general argument, we first (again by construction) inflate costs of work to recover the 0-net-utility case.

Preferences
Each worker w ∈ W has a valuation over firms, as in our example, this valuation may encode the costs of producing products at each employer.
The valuation v w induces a quasilinear utility function u w : The utility function u w naturally extends to outcomes A ⊆ X. Throughout, we use the convention that u w (∅) = 0.
Each firm f has a valuation function over sets of workers where w(Y ) denotes the set of workers associated to contracts in Y . As with worker utility functions, the utility function u f naturally extends to outcomes A. By construction, we have u f (∅) = 0. 15 Here, the notation ℘ denotes the power set, ℘(R) ≡ {R : R ⊆ R} for any set R.

The Core
An outcome A is in the core (under the valuation profile v) if there does not exist a core block (for A), i.e., a set of agents J ⊆ I and an outcome B ⊆ X such that

Mechanisms
The set of possible valuations for worker w is V w ≡ (R ∪ {−∞}) F . We let V ≡ × w∈W V w be the set of possible valuation profiles. A mechanism is a function M : V → ℘(X) that maps valuation profiles to outcomes. A mechanism is core-selecting if it always returns core outcomes.

Strategy-Proofness of Worker-Optimal Mechanisms
A core-selecting mechanism M is optimal for worker w if for any core allocation A and any v ∈ V , we have A mechanism M is strategy-proof for worker w if reporting truthfully is (weakly) dominant for worker w under M-that is, if Our main result is as follows.
Theorem 1. Any core-selecting mechanism that is optimal for worker w is strategy-proof for w.

Sketch of Proof
The key new ingredient in our proof of Theorem 1 is the following "Lone Wolf Theorem," which we have derived in other work.
Theorem 2 (Jagadeesan et al. (2017)). If A and A are both core outcomes, w is a worker, and we have u w (A w ) > 0, then we must have A w = ∅. That is, if a worker receives strictly positive utility in some core outcome, then that worker is matched in every core outcome. 16 Theorem 2 is an analogue of the classical Lone Wolf Theorem (see McVitie and Wilson (1970) and Gale and Sotomayor (1985), as well as Roth (1984aRoth ( , 1986, Alkan (2002) (2016)). 17 As Hatfield and Milgrom (2005) showed in the context of many-to-one matching with (discrete) contracts, the Lone Wolf Theorem is useful in proving one-sided strategy-proofness.
We adapt the Hatfield-Milgrom (2005) proof of (one-sided) strategy-proofness to deduce Theorem 1 from Theorem 2. Indeed, we suppose for the sake of deriving a contradiction that there exists a profitable deviation for worker w under core-selecting mechanism M, namely reporting valuationv w instead of v w . By inflating w's costs of work sufficiently, we can assume that w is unmatched under truthful revelation but can obtain an individually rational match by deviating; from that individually rational match, we construct a core outcome under which w is matched, contradicting the Lone Wolf Theorem (Theorem 2).
16 While Jagadeesan et al. (2017) formally state their Lone Wolf Theorem for competitive equilibria, Kelso and Crawford (1982) have shown that competitive equilibrium outcomes coincide with core outcomes in many-to-one matching markets with continuous transfers. Hence, the main result of Jagadeesan et al. (2017) directly implies Theorem 2.
17 Recently, Schlegel (2016) has shown that Theorem 2 holds when workers' utility functions are continuous but not necessarily quasilinear, provided that all firms' valuations are (grossly) substitutable (in the sense of Kelso and Crawford (1982); see Footnote 20). Schlegel (2016) proved his result (Theorem 4 of Schlegel (2016)) by combining the Rural Hospitals Theorem for matching with contracts (Theorem 8 of Hatfield and Milgrom (2005)) with a limiting argument. As we have shown (Jagadeesan et al. (2017)), transferable utility substitutes for substitutability in Theorem 2, yielding a simpler proof.
We have shown that any worker-optimal core-selecting mechanism is strategy-proof for workers in the context of matching with continuous transfers. We used a proof strategy based on the Lone Wolf Theorem. Due to the absence of a suitable Lone Wolf Theorem, it has not previously been clear whether such a strategy could be used in matching settings with continuous transfers, even though such a strategy has been applied in many other matching contexts.

Antecedents
Worker-optimal matching mechanisms have long been believed to be strategy-proof in manyto-one environments with transferable utility, but to our knowledge the only prior proof of this fact is a recent indirect argument due to Hatfield, Kojima, and Kominers (2017)-we give a direct proof. 18 Theorem 1 generalizes results of Demange (1982) and Leonard (1983) for one-to-one matching contexts (see also Demange et al. (1986) and Ausubel and Milgrom (2002)) and results of Gul and Stacchetti (2000) for object allocation settings. Our result is an analogue of Theorem 11 of Hatfield and Milgrom (2005), which gives the corresponding result for worker-firm matching with discrete (rather than continuous) transfers. 19 Our approach to Theorem 1 adapts the proof strategy of Hatfield and Milgrom (2005) to the context of matching with continuous transfers. 18 Hatfield, Kojima, and Kominers (2017) proved Theorem 1 by showing that worker-optimal mechanisms are efficient and guarantee each worker the full marginal surplus from a change in valuation; these observations together imply strategy-proofness by the main result of Hatfield, Kojima, and Kominers (2017). As we discussed in Footnote 9, the results of Ausubel (2004Ausubel ( , 2006 imply the same result when firms have linear valuations over workers. 19 As we discussed in Section 1, analogues of Theorem 1 have been found in a range of matching contexts without transferable utility, including marriage and college admissions matching (Dubins and Freedman (1981); Roth (1982); Martínez et al. (2004)), many-to-one matching with contracts (Hatfield and Milgrom (2005); Kojima (2009, 2010); Hatfield and Kominers (2015); Hatfield, Kominers, and Westkamp (2016)), and supply chain matching (Hatfield and Kominers (2012); see also Ostrovsky (2008)).

Implications
Theorem 4 of Hatfield et al. (2013) implies the existence of worker-optimal competitive equilibria if all firms have substitutable valuations. 20 Moreover, Kelso and Crawford (1982) showed that there exist competitive equilibrium prices that support any core allocation. Thus, a worker-optimal core-selecting mechanism exists-and hence, by our results, strategy-proof worker-firm matching is possible-if all firms have substitutable valuations. In particular, our results show that the worker-proposing deferred acceptance mechanism (or equivalently, the worker-optimal core-selecting matching mechanism) is strategy-proof for workers in the Kelso-Crawford (1982) setting with continuous transfers and quasilinear utility.

Matching with Contracts
Our proof of Theorem 1 extends verbatim to settings in which workers and firms also negotiate over non-pecuniary contract terms (Roth, 1984b;Hatfield and Milgrom, 2005;Hatfield et al., 2013). However, we need to maintain the assumption that all agents' preferences over transfers are quasilinear. Demange and Gale (1985) and Demange (1987) proved group strategy-proofness for one-to-one matching with continuous transfers. 21 However, no group strategy-proofness result is known in our setting.

Group Strategy-Proofness
Group strategy-proofness results have been found in nearly all the matching contexts 20 The valuation of firm f is (grossly) substitutable if the corresponding indirect utility function is submodular, or equivalently if an increase in the salary f must pay some worker cannot reduce f 's demand for workers whose salaries are unchanged (see, e.g., Kelso and Crawford (1982); Ausubel and Milgrom (2002); ).
for some w ∈ S-that is, there is no deviation v ∈ × w∈S V w that makes all workers in S strictly better off than under truthful reporting.
with discrete contracts that have individual strategy-proofness results (see Section 5.1 and Footnote 19 for references), via an argument first introduced by Hatfield and Kojima (2009 show that worker-optimal core-selecting mechanisms are group strategy-proof in settings with continuous transfers-at least when all firms' valuations are substitutable and quasilinear.

A Proof of Theorem 1
Throughout this Appendix, we fix a worker w ∈ W . We use the convention that utility functions u w ,ū w andû w are associated to valuations v w ,v w , andv w , respectively, via the utility function construction established in Section 3.

A.1 Preliminaries
The proof of Theorem 1 makes use of two lemmata. Lemma A.1 asserts that any core outcome remains in the core if firms become weakly less desirable to the workers that are not their employees. Lemma A.2 asserts that increasing the utility of the outside option only shrinks the core by making certain outcomes fail to be individually rational.
Lemma A.1. Letv w ∈ V w be a valuation withv w ≤ v w and let A be an outcome with Proof. We prove the contrapositive. Assume that A is an outcome with u w (A w ) =ū w (A w ) and that coalition J and outcome B together constitute a core block for A under the valuation If w / ∈ J, then J and B clearly constitute a core block of A for the valuation profile v, so that A is not a core outcome for the valuation profile v. Thus, we suppose that w ∈ J. As v w ≤ v w by assumption, we have It follows from (1) Supposing that w ∈ J, we havē where the the second inequality in (2)

A.2 Main Argument
We let M be a core-selecting mechanism that is optimal for w. We let v ∈ V andv w ∈ V w be arbitrary, and take A = M(v) and A = M (v w , v W {w} ) . We suppose for the sake of deriving a contradiction that u w (A w ) < u w (A w ).
As u w (A w ) > u w (A w ) ≥ 0, worker w receives strictly positive utility under outcome A .
Thus, w is matched to a firm under outcome A ; we denote this firm by f w . We let η > 0 be such that We let A = M (v w − η, v W {w} ) .

Claim. We have
Proof. Suppose for the sake of deriving a contradiction that A w = ∅. Because M is coreselecting, Lemma A.2 then implies that A is a core outcome with respect to the valuation profile v. The individual rationality of A with respect to v w − η, v W {w} implies that u w (A w ) > η, which contradicts the worker-optimality of M, as η > u w (A w ) (recall (3)).

Now, we define valuationv
effectively, underv w , w values only f w -and values f w at an amount strictly bounded above by v w (f w ). By construction, we havev We letη =v w (f w ) − v w (f w ) + η. With (4) and our choice ofη, we havê v w ≤ v w − η =v w −η; moreover, we haveû Now, as A is a core outcome under valuation profile (v w , v W {w} ) and w receives strictly positive utility at A under the valuationv w −η, we know from our choice of η that A must also be a core outcome under the valuation profile v w −η, v W {w} . By Lemma A.1, it then follows from (5) and (6) that A is a core outcome under the valuation profile (v w , v W {w} ).
As A w = ∅, combining Lemma A.1 and (5) implies that A is also a core outcome under the valuation profile (v w , v W {w} ). 24 But then, we have both A and A in the core under the valuation profile (v w , v W {w} ).
As u w (A w ) > 0 by (7) and A w = ∅, this contradicts the Lone Wolf Theorem (Theorem 2).