Investing before Stable Matching∗

Benjamın Tello†

February 2016

Abstract

We analyze the investment game induced by matching markets where workers invest

and then match to firms in a stable way, and where monetary transfers are not allowed.

We assume that workers have common preferences over firms. We show that a profile

of investments is a strictly strong Nash equilibrium if and only if the matching it

induces is “investment efficient,” and stable in a related market where investments and

partnerships are simultaneously determined. We also characterize its pure strategy

Nash equilibria by stability and a weaker notion of efficiency called unilateral efficiency.

Next, we provide a condition on the domain of preference profiles that generalizes

the notion of lexicographic preferences and ensures the existence of stable and invest-

ment efficient matchings. Finally, we show that the requirement that each firm has

lexicographic preferences cannot be substantially weakened while still guaranteeing the

existence of stable and unilaterally efficient matchings.

Keywords: pre-matching investment, Nash equilibria, stability, investment efficiency,

unilateral efficiency, only-bilateral disagreement, lexicographic preferences.

JEL classification: C78, D47, D60, D82.

∗I am grateful to Flip Klijn for his guidance. I thank David Cantala, Jordi Masso and Isabel Melguizo for

helpful comments. Financial support from the Consejo Nacional de Ciencia y Tecnologıa (CONACyT), Universitat

Autonoma de Barcelona through PIF grant 412-01-9/2010 and the Spanish Ministry of Economy and Competitive-

ness through FPI grant BES-2012-055341 (Project ECO2011-29847-C02) is gratefully acknowledged.†Universitat Autonoma de Barcelona and Barcelona GSE, email: benjamin.tello@uab.cat.

1

1 Introduction

Consider the market for medical residency positions or the assignment of students

to public schools. Participants in these markets often make large human capital

investments well before the matching stage. For example, medical doctors study

for several years before participating in residency matching, and students engage in

extra-curricular activities or prepare for admission tests before applying to schools.

Moreover, in these markets, salaries or prices are fixed by law or they are not the

main factor determining the allocation. Therefore, they are not useful in solving the

matching problem.

The goal of this paper is to study the functioning of matching markets where

participants can make investments prior to matching. Our main contribution is the

characterization of equilibrium investments and equilibrium outcomes. Moreover,

our results describe when equilibrium investments are efficient and when the timing

of investment does not matter.

We assume that, after sinking their investments, workers and firms match in a

stable way. That is, the matching between workers and firms is such that no worker

and firm prefer to be assigned to each other rather than to their current partners.

Stability is a reasonable assumption for decentralized matching markets with no

frictions. The reason is that the only robust predictions that can be made about the

outcome of these markets are stable matchings. It is also reasonable for centralized

matching markets that employ a stable mechanism1 i.e., a mechanism that selects

a stable matching with respect to agents’ reported preferences. The reason is that

under the restriction we impose on workers’ preferences (see next paragraph) all

stable mechanisms coincide (they select the same matching), and for every worker

and firm it is a dominant strategy to report their true preferences to the mechanism.

We represent markets with pre-matching investment by a model of (one-to-one)

matching with contracts where a contract specifies a firm, a worker and the worker’s

investment.2 We impose one restriction on the profile of workers’ preferences called

unanimous separability. This restriction requires that there is a ranking of firms

1Examples of matching markets that employ a stable mechanism are the National Residency Matching Program,

see Roth (1984), and the New York high school assignment system, see Abdulkadiroglu, Pathak, and Roth (2009).2In view of the entry-level labor market interpretation, we call the agents on one side workers, and the agents on

the other side firms.

2

such that for any two contracts that involve the same investment, the contract

that involves the best firm according to the ranking is preferred by all workers to

the other contract. This assumption, while restrictive, arises naturally in various

settings. For example, it holds when firms and workers produce using a technology

that is increasing in firms’ types and the output is split between workers and firms

in fixed proportions. Unanimous separability ensures that once workers have sunk

their investments, there is a single stable matching between workers and firms.

Our main findings are as follows. In general, the investment game may not have a

pure strategy Nash equilibrium (Example 1). We show that an investment profile (a

list of workers’ investments) is a strictly strong Nash equilibrium of the investment

game if and only if the matching it induces is investment efficient,3 and stable at

the complete market4 (Corollary 1).

Corollary 1 gives conditions under which the investment game has a (strictly)

strong (Nash) equilibrium. Moreover, it establishes that the outcome of the strong

equilibrium coincides with the outcome of a single-stage centralized market organized

by means of the worker-proposing deferred acceptance (DA) mechanism of Fleiner

(2003) and Hatfield and Milgrom (2005). Our result suggests that matching markets

whose matching stage is organized by means of a stable mechanism can work well in

the presence of a pre-matching investment phase. In particular, (i) workers do not

need to recur to complex randomizations, (ii) the equilibrium outcomes are efficient

for workers, and (iii) no worker (or even group of workers) has incentives to change

investments. In addition, Corollary 1 and the lattice structure of the set of stable

matchings imply that the strong equilibrium can be efficiently computed, whenever

it exists, via the worker-proposing DA algorithm.

Our second main result characterizes the (pure strategy Nash) equilibria of the in-

vestment game by stability (at the complete market) and a weak notion of efficiency

called unilateral efficiency (Corollary 2).

A natural question is, under which conditions do stable (at the complete market)

and investment efficient matchings exist? To address this question, we provide a re-

striction on the domain of preference profiles called only-bilateral disagreement that

3A matching is investment efficient if there is no investment profile that produces a matching that is weakly

preferred by all workers and strictly preferred by some.4The complete market is the benchmark situation where investments and partnerships are determined simulta-

neously. It is captured by the matching with contracts market where every possible contract is available.

3

ensures the existence of stable and investment efficient matchings. This restriction

requires that for each worker w and each firm f, the best contract for w and the

best contract for f among all contracts that involve w and f are separated in the

preferences of f only by contracts involving w. Thus, the disagreement between f

and w is “bilateral” in the sense that it cannot involve some other worker.

A firm’s preferences are lexicographic if she ranks all contracts involving the

same worker consecutively in her preferences. If the preferences of all firms are lexi-

cographic, then only-bilateral disagreement is satisfied regardless of the preferences

of workers. Thus, only-bilateral disagreement is a generalization of lexicographic

preferences. Pakzad-Hurson (2014) shows that in a model of many-to-one matching

with contracts of which our framework is a special case, suitable generalizations of

lexicographic preferences and Ergin’s (2002) acyclicity are sufficient for the existence

of stable and Pareto efficient matchings. In addition, he shows that these two con-

ditions are necessary in the following sense: if the profile of firms’ preferences is not

lexicographic or has an Ergin cycle, then there are preferences for workers such that

no stable matching is efficient.

Lexicographic preferences and Ergin acyclicity are not necessary for the existence

of stable and investment efficient matchings.5 We show that if one firm’s preferences

are not lexicographic and satisfy a mild condition, then there are lexicographic

preferences for all other firms and unanimously separable preferences for all workers

such that no stable matching is unilaterally efficient (Proposition 1). This result

shows that it is not possible to weaken the only-bilateral disagreement condition

substantially while still guaranteeing the existence of an equilibrium.

1.1 Related literature

Investment in matching markets has been the subject of several studies. Partic-

ular attention has been paid to the case where firms can pay continuous salaries

and equilibrium is competitive (agents take salaries as given) (Cole, Mailath, and

Postlewaite, 2001a,b; Noldeke and Samuelson, 2015). In this setting efficient invest-

ment, in the sense of maximizing social surplus, is always an equilibrium. However,

because of coordination failures, inefficient equilibria may also arise.

5Example 4 in the Appendix exhibits a profile of firms’ preferences that has Ergin cycles for which a stable and

investment efficient matching exists regardless of workers’ preferences.

4

If hospitals cannot pay continuous salaries, then efficient investments are not an

equilibrium in gerenal. However, Hatfield, Kominers, and Kojima (2015) show that

if hospitals can pay discrete salaries, then approximately efficient investments are

an equilibrium. A remarkable implication of this result is that under the worker

proposing DA mechanism, workers have incentives to make approximately efficient

investments before the matching stage.

The case where transfers are not possible in the matching stage is analyzed by

Peters and Siow (2002); Peters (2007) and Noldeke and Samuelson (2015). Peters

(2007) studies a model similar to ours where transfers are not possible and where

agents on both sides engage in costly investment and are then matched assortatively.

The author considers the mixed strategy Nash equilibria of the induced investment

game and shows that it converges to a degenerate pure strategy Nash equilibrium

in which the two sides of the market invest too much. This differs from and comple-

ments our study as it focuses on the mixed strategy Nash equilibria of an investment

game, whereas we focus on pure strategy Nash equilibria.

The model of Noldeke and Samuelson (2015) with non-transferabilities subsumes

ours.6 However, the equilibrium concept they consider is competitive: agents take

other agents’ utilities as given. Since our equilibrium concept is non-competitive,

our analysis and results are different from theirs.

2 Model

2.1 A matching with contracts market

Let W and F be two disjoint sets of workers and firms such that |W | = |F | = m.

Let N = W ∪F be the set of agents. Let T be the set of investments or investment

types. For each w ∈ W, let tw ∈ T be an investment for worker w. Denote by

t = (tw)w∈W an investment profile and let TW be the set of all investment

profiles.

A feasible contract is a triplet (w, f, tw) = x ∈ X where

X ⊆ X ≡ W × F × T

is a feasible set of contracts. We only consider feasible sets of contracts that contain

at least one contract between each worker and each firm. Moreover, we assume that6Noldeke and Samuelson (2015) consider both the transferable and the non-transferable utility case.

5

no agent (worker or firm) is assigned more than one contract and that there are no

outside options.

We write w(x), f(x) and t(x) to denote the worker, firm and investment involved

in contract x, respectively. For each X ⊆ X ,

Xw ≡ {x ∈ X : w(x) = w} and Xf ≡ {x ∈ X : f(x) = f}

denote the sets of contracts within X involving worker w and firm f , respectively.

Each agent i ∈ N has a complete, transitive and strict preference relation Pi

over the set Xi. For x, x′ ∈ Xi we write xPi x′ if agent i prefers x to x′ (x 6= x′),

and xRi x′ if i finds x at least as good as x′, i.e., xPi x

′ or x = x′. We denote

profiles of workers’ and profiles of firms’ preferences by PW = (Pw)w∈W and PF =

(Pf )f∈F , respectively. Let P = (PW , PF ) be a preference profile and P be the

set of all preference profiles. We represent agents’ preferences by ordered lists of

feasible contracts; for example, Pf : (w, f, tw), (w′, f, t′w′), (w, f, t′w), . . . indicates

that (w, f, tw)Pf (w′, f, t′w′) Pf (w, f, t′w) . . .

We impose one restriction on workers’ preference profiles called unanimous sep-

arability. It requires that there is a common ranking of firms such that for each

worker and for any two contracts that involve the same investment, one contract

is preferred over the other if and only if the former contract involves a better firm

according to the common ranking. Formally, a profile of workers’ preferences PW

satisfies unanimous separability if there is a linear order � over F such that for all

w ∈ W, all tw ∈ T and all f, f ′ ∈ F,

f � f ′ if and only if (w, f, tw)Pw (w, f ′, tw).

Let � denote the weak relation associated with �.

We fix W , F and T . Therefore, a market is completely described by a feasible

set of contracts X ⊆ X and a preference profile P ∈ P . We denote a market by a

pair (X,P ).

A matching µ for (X,P ) is a mapping from N to X such that

m1. for each i ∈ N, µ(i) ∈ Xi

m2. for each w ∈ W and each f ∈ F, if x ∈ Xw ∩Xf , then µ(w) = x if and only

if µ(f) = x.

6

LetM be the set of (all) matchings for (X , P ). Note that any matching µ for (X,P )

with X ⊆ X is an element of M.

A matching µ is blocked by w ∈ W, f ∈ F and x ∈ Xw ∩Xf at (X, P ) if

b1. xPw µ(w) and

b2. xPf µ(f).

If a matching µ is blocked by w, f and x, we may write “µ is blocked by w and

f via t(x)”. A matching is stable at market (X,P ) if it is not blocked at (X,P ).

Let S(X, P ) be the set of (all) stable matchings at (X,P ). By Theorem 1 in Kelso

and Crawford (1982) or Theorem 3 in Hatfield and Milgrom (2005), S(X,P ) is

non-empty for any (X,P ).

For each matching µ, t(µ) ≡(t(µ(w))

)w∈W ∈ T

W denotes the investment profile

associated with µ.

2.2 Markets induced by investment profiles

An investment profile t induces a feasible set of contracts

X(t) ≡ {(w, f, tw) ∈ X : w ∈ W and f ∈ F}.

We say that (X(t), P ) is the market induced by investment profile t. In this market

all available contracts for a worker involve the same investment. Therefore, this

market represents a situation where workers’ investments are fixed in the matching

stage. By contrast, the market (X , P ) represents a situation where investments are

flexible, or determined together with worker-firm matches in a single stage. We

(sometimes) refer to this market as the complete market.

Given our assumption on workers’ preferences (unanimous separability) the pref-

erences of each worker w over contracts in Xw(t) are straightforwardly induced by

�. Therefore, we have the following Lemma.

Lemma 1. Let t ∈ TW . For each market (X(t), P ), S(X(t), P ) is a singleton.7

Proof: A market (X(t), P ) corresponds to a one-to-one matching without con-

tracts market where all workers have the same preferences over firms. Therefore, by

Eeckhout (2000), S(X(t), P ) is a singleton.

7From now on, in view of Lemma 1, we slightly abuse notation by treating S(X(t), P ) as a matching instead of

a singleton.

7

Remark 1. If a matching µ is stable at (X , P ), then µ = S(X(t(µ)), P ). In

words, any matching that is stable at the complete market is also stable at the

market induced by its associated investment profile. This observation together with

Lemma 1 imply that for any two different matchings µ, µ′ that are stable at (X , P ),

t(µ) 6= t(µ′).

2.3 The investment game

The investment decisions of workers induce a market with a unique stable outcome

that determines how workers and firms match (Lemma 1). Workers, in anticipation,

choose investments strategically. Formally, they play a complete information normal

form game Γ(P ) = (W,T, P ) where W is the set of players and T is the set of

strategies for each player. Given an investment (strategy) profile t the outcome

of this game is determined by S(X(t), P ). Each worker w evaluates the outcome

according to his true preferences Pw.

A coalition is a nonempty subset of workers I ⊆ W . Given an investment profile

t, a coalition I has a profitable deviation at t if there exists t′I ∈ TI such that

1. for each w ∈ I, µ′(w)Rw µ(w) and

2. for some w ∈ I, µ′(w)Pw µ(w),

where µ = S(X(t), P ) and µ′ = S(X(t′I , tW\I), P ).

An investment profile t is a (pure strategy Nash) equilibrium of Γ(P ) if no

coalition I with |I| = 1 has a profitable deviation at t. It is a (strictly) strong

(Nash) equilibrium of Γ(P ) if no coalition I has a profitable deviation at t.

2.4 Efficiency

We consider three different notions of efficiency. Let µ, µ′ ∈ M. Then, µ Pareto

dominates µ′ if

p.1. for each w ∈ W, µ(w)Rw µ′(w) and

p.2. for some w ∈ W, µ(w)Pw µ′(w).

A matching µ ∈M is

e.1. efficient if no other matching µ′ ∈M Pareto dominates µ.

8

e.2. investment efficient if there is no investment profile t ∈ T such that S(X(t), P )

Pareto dominates µ.

e.3. unilaterally efficient if there is no tw ∈ TW such that S(X(tw, t(µ)−w), P )

Pareto dominates µ.

Clearly, efficiency implies investment efficiency and investment efficiency implies

unilateral efficiency. However, no converse to either of these implications holds. Ex-

ample 3 exhibits a matching that is unilaterally efficient but not investment efficient

and Example 2 exhibits a matching that is investment efficient but not efficient.

Remark 2. By Theorems 1 and 4 in Kelso and Crawford (1982) or Theorems 3 and

4 in Hatfield and Milgrom (2005), S(X , P ) forms a non-empty lattice with respect

to the Pareto domination relation. By Remark 1, each matching in S(X , P ) is asso-

ciated with a different investment profile. Thus, the investment profiles associated

with matchings in S(X , P ) also form a non-empty lattice with respect to the Pareto

domination relation.

2.5 Deferred acceptance

We describe a worker-proposing deferred acceptance algorithm which is a general-

ization of Gale and Shapley’s (1962) deferred acceptance algorithm to markets with

contracts. Fleiner (2003) and Hatfield and Milgrom (2005) show that this algorithm

produces a stable matching that Pareto dominates any other stable matching. The

description of the algorithm (below) is based on Pakzad-Hurson (2014).

Let X ⊆ X be a feasible set of contracts. For each i ∈ N , let Chi(X,Pi) be i’s

most preferred contract in Xi, i.e.,

Chi(X,Pi) = argmaxPi

{Xi}.

When it is clear from the context we suppress the dependence of Chi from Pi.

The worker proposing deferred acceptance (DA) algorithm

Input: A market (X,P ).

Step 1: An arbitrary worker w1 ∈ W proposes his most preferred contract in Xw1 .

This contract involves some firm say f1 ∈ F . Let firm f1 hold contract x1. Set

y2(f1) = x1 and set y2(f) = ∅ for each f 6= f1.

9

Step k: Let Ik be the set of workers involved in a contract which is held by any

firm after Step k− 1. An arbitrary worker wk ∈ W \ Ik proposes his most preferred

contract xk ∈ Xwk which he has not proposed in a previous step. This contract

involves some firm fk ∈ F . Firm fk holds the contract x ∈ Chf({yk(fk)} ∪ {xk}

),

and rejects the other (if any). All other f 6= fk continue to hold the contract

they held at the end of Step k − 1. Set yk+1(fk) = Chf ({yk(fk)} ∪ {xk}) and set

yk+1(f) = yk(f) for each f 6= fk.

The algorithm terminates at some step K when no worker proposes any new

contract. Given that there is an equal number of firms and workers and that there

are no outside options, each worker is matched to some firm at the end of the

algorithm. The function µ(f) = yK(f) gives the final matching and this matching

is called the worker optimal stable matching at (X,P ).

The firm proposing DA algorithm is defined symmetrically by exchanging the

roles of workers and firms in the worker proposing DA algorithm. Hatfield and Mil-

grom (2005) show that the firm proposing DA algorithm produces a stable matching

at (X,P ) that is Pareto dominated by any other stable matching at (X,P ).

The next example illustrates the investment game and the possibility that no

equilibrium exists.

Example 1 (The investment game and the non-existence of equilibrium). Consider

a market with W = {w1, w2}, F = {f1, f2}, T = {t1, t2}, and preferences P given

by the columns in Table 1. Vertical dots mean that preferences can be arbitrary.

Both workers have the same preferences compatible with unanimous separability

and f1 � f2 .

Table 1: Preferences P in Example 1

Pf1 Pf2 Pw1Pw2

(w1, f1, t1) (w2, f2, t2) (w1, f1, t2) (w2, f1, t2)

(w2, f1, t1)... (w1, f1, t1) (w2, f1, t1)

(w1, f1, t2) (w1, f2, t2) (w2, f2, t2)

(w2, f1, t2) (w1, f2, t1) (w2, f2, t1)

There is a unique stable matching at (X , P ) given by

10

w1 w2

| |µ : t1 t2

| |f1 f2

which is the boxed matching in Table 1. This can be verified by running the worker

and the firm proposing DA algorithms with input (X , P ) and observing that the

outcome of both algorithms is the same. By Remark 1, µ = S(X(t1, t2), P ).

Matching µ is not investment efficient. For example, it is Pareto dominated by

the matching µ′ = S(X(t2, t2), P ) given by

w1 w2

| |µ′ : t2 t2

| |f1 f2

which is the bold face matching in Table 1.

Table 2 depicts the firm matched with each worker at S(X(t), P ) for each in-

vestment profile t.

Table 2: Stable matchings at (X(t), P )

w1 \ w2 t1 t2

t1 (f1, f2) (f1, f2)

t2 (f2, f1) (f1, f2)

Using Tables 1 and 2 one can verify that no investment profile is an equilibrium

of the game Γ(P ). Therefore, this game has no equilibria. �

3 Results

First we show that if a matching is investment efficient and stable at the complete

market, then its associated investment profile is a strong equilibrium of the invest-

ment game. In particular, this result establishes conditions under which a strong

equilibrium exists and under which the equilibrium outcome of the two stage market

coincides with the outcome of a single stage market that is organized by means of

the worker proposing DA mechanism and where the agents are truthful.

11

Theorem 1. If µ is investment efficient and stable at (X , P ), then t(µ) is a strictly

strong Nash equilibrium of Γ(P ).

The proof of Theorem 1 is in the Appendix.

The next result, Theorem 2, is a partial converse to Theorem 1. It establishes

that every equilibrium (and in particular every strong equilibrium) induces a market

such that its unique stable matching is stable at the complete market.

Theorem 2. If t ∈ TW is a pure strategy Nash equilibrium of Γ(P ), then S(X(t), P )

is stable at (X , P ).

The proof of Theorem 2 is in the Appendix.

Theorems 1 and 2 and the fact that every strong equilibrium is investment effi-

cient (otherwise the set of all workers would have a profitable deviation) deliver the

following characterization of the strong equilibria of the investment game in terms

of stability and investment efficiency.

Corollary 1. Let t ∈ TW . Then, t is a strictly strong Nash equilibrium of Γ(P ) if

and only if S(X(t), P ) is stable at (X , P ) and investment efficient.

Proof. The if statement follows from Theorem 1. Let µ = S(X(t), P ). For the only

if statement observe that Theorem 2 implies that µ is stable at (X , P ). Assume

by contradiction that µ is not investment efficient. Then, the coalition W has a

profitable deviation at t, contradicting that t is a strong equilibrium.

Corollary 1 and the lattice structure of the set of stable matchings (Remark

2) imply that whenever a strong equilibrium exists it is unique. In fact, it is the

investment profile associated with the worker optimal stable matching at (X , P ).

Hence, Corollary 1 gives an easy way to check whether a strong equilibrium exists.

There may be equilibria that fail to be investment efficient (see Example 3).

Theorem 2 allows us to restrict our search of equilibria to stable matchings of the

complete market. Next, we characterize equilibria by stability and a weaker notion

of efficiency, unilateral efficiency.

Corollary 2. Let t ∈ TW . Then, t is a pure strategy Nash equilibrium of Γ(P ) if

and only if S(X(t), P ) is stable at (X , P ) and unilaterally efficient.

Proof. The proof of the if statement is a slight modification of the proof of Theorem

1 and therefore we omit it. Let µ = S(X(t), P ). For the only if statement observe

12

again that Theorem 2 implies that µ is stable at (X , P ). Assume by contradiction

that µ is not unilaterally efficient. Then, there is a worker w with a profitable

deviation, which contradicts that t is an equilibrium.

We give an example of a market that has a matching that is stable at the com-

plete market, investment efficient, but not efficient. This example also serves as an

illustration of Theorem 1.

Example 2 (A stable and investment efficient strong equilibrium outcome). Con-

sider a market with preferences P given by the columns in Table 3. Both workers’

preferences are compatible with unanimous separability and f1 � f2.

Table 3: Preferences P in Example 2

Pf1 Pf2 Pw1 Pw2

(w1, f1, t2) (w2, f2, t2) (w1, f1, t1) (w2, f1, t2)

(w2, f1, t1)... (w1, f1, t2) (w2, f1, t1)

(w2, f1, t2) (w1, f2, t1) (w2, f2, t2)

(w1, f1, t1) (w1, f2, t2) (w2, f2, t1)

Consider the matching given by:

w1 w2

| |µ : t2 t2

| |f1 f2

which is the boxed matching in Table 3. Since f1 and f2 obtain their most preferred

contracts, µ is stable at (X , P ). Matching µ is not efficient, as it is Pareto dominated

by the matching

w1 w2

| |µ′ : t1 t2

| |f1 f2

which is the bold face matching in Table 3. However, µ is investment efficient. To

see this observe that no investment profile induces a matching that Pareto dominates

µ. In particular any profile where w1 makes investment t1 induces a matching under

which w1 matches f2 and therefore is worse off. For example, (t1, t2) induces the

matching indicated by thick boxes in Table 3.

13

Table 4 depicts the firm matched with each worker at S(X(t), P ) for each in-

vestment profile t.

Table 4: Stable matchings at (X(t), P )

w1 \ w2 t1 t2

t1 (f2, f1) (f2, f1)

t2 (f1, f2) (f1, f2)

Using Tables 3 and 4 one can verify that the investment profile t(µ) = (t2, t2)

is the unique equilibrium of the game Γ(P ). Therefore, no coalition formed by one

agent has profitable deviations. Since µ is investment efficient, the coalition formed

by workers w1 and w2 has no profitable deviations either. Thus, t(µ) is also a strong

equilibrium. �

In the next example we exhibit a market where no stable matching is investment

efficient, but where the investment profile associated with the worker optimal stable

matching is an equilibrium.

Example 3 (A stable and investment inefficient equilibrium outcome). Consider

a market with W = {w1, w2, w3}, F = {f1, f2, f3}, T = {t1, t2}, and prefer-

ences P given by the columns of Table 5. Workers’ preferences are compatible with

unanimous separability and f1 � f2 � f3 .

Table 5: Preferences P in Example 3

Pf1 Pf2 Pf3 Pw1Pw2

Pw3

(w1, f1, t1) (w2, f2, t1) (w1, f1, t2) (w2, f1, t1) (w3, f1, t1)

(w2, f1, t2) (w1, f2, t2)... (w1, f2, t2) (w2, f1, t2) (w3, f2, t1)

(w3, f1, t1) (w3, f2, t1)... (w1, f1, t1) (w2, f2, t1) (w3, f3, t1)

......

......

...

The worker optimal stable matching at (X , P ) is given by:

w1 w2 w3

| | |µ : t1 t1 t1

| | |f1 f2 f3

14

which is the boxed matching in Table 5. By Remark 1, µ = S(X(t1, t1, t1), P ).

The investment profile t(µ) = (t1, t1, t1) is a NE. To see this observe that w3 can

never profit by deviating to t2. Thus, we analyze the incentives of w1 and w2 to

deviate to t2 given that w3 chooses t1. Table 6 gives the firm matched with w1 and

w2 for each pair (t1, t1), (t1, t2), (t2, t1), (t2, t2).

Table 6: Stable matchings for w1 and w2 at (X(t), P ) given tw3= t1

w1 \ w2 t1 t2

t1 (f1, f2) (f3, f2)

t2 (f1, f3) (f2, f2)

If w1 deviates to t2, he obtains the contract (w1, f3, t2), but µ(w1)Pw1 (w1, f3, t2).

If w2 deviates to t2 he obtains contract (w2, f3, t2), but µ(w2)Pw2 (w2, f3, t2). Since

no worker has incentives to deviate, t(µ) is a NE. By these same arguments one can

verify that t(µ) is unilaterally efficient. However, t(µ) is not investment efficient as

the matching µ′ = S(X(t2, t2, t1), P ) given by

w1 w2 w3

| | |µ′ : t2 t2 t1

| | |f2 f1 f3

Pareto dominates µ (bold face matching in Table 5). �

3.1 Restricted domains of preference profiles

Under which conditions on preference profiles do stable and investment efficient

matchings exist? We provide a restriction on preference profiles, called only-bilateral

disagreement, that ensures the existence of stable and investment efficient match-

ings. Thus, by Theorem 1 the investment games associated with preference profiles

satisfying this restriction have a strong equilibrium.

Our restriction requires that each firm and each worker partially agree on what

would be the best investment if they were to match. More precisely, for any worker

w and any firm f , let x be the best contract for f and let y be the best contract for

w, among all contracts that involve f and w. Then, any contract in between x and

15

y in the preferences of f must involve w. In this sense, the disagreement between f

and w is bilateral.

Formally, a pair (w, f) ∈ W ×F only-bilaterally disagrees at (Pw, Pf ) if there

is no z ∈ Xf \ Xw such that

Chf (Xw ∩ Xf, Pf ) Pf z Pf Chw(Xw ∩ Xf, Pw)

A preference profile P ∈ P satisfies only-bilateral disagreement if each pair

(w, f) ∈ W × F only-bilaterally disagrees at (Pw, Pf ).

Theorem 3. Let P ∈ P . If P satisfies only-bilateral disagreement, then there is

an investment efficient and stable matching at (X , P ). Thus, the game Γ(P ) has a

strictly strong Nash equilibrium.

The proof of Theorem 3 is in the Appendix.

Only-bilateral disagreement generalizes the notion of lexicographic preferences.

A firm’s preferences are lexicographic if she ranks consecutively all contracts that

involve the same worker in her preference.8 Formally, Pf is lexicographic if for

any two contracts x, y ∈ Xw ∩ Xf with xPf y there is no z ∈ Xf \ Xw such that

xPf z Pf y. A profile of firms’ preferences PF is lexicographic if for each f ∈ F,

Pf is lexicographic.

Remark 3. If PF is lexicographic, then P satisfies only-bilateral disagreement

regardless of the preferences of workers.

The preferences of a firm f have a brick if in between any two contracts involving

worker w (in the preference of f) there are all contracts involving another worker

w′. The presence of bricks is a form of non lexicographic preferences that does not

preclude the existence of stable and unilaterally efficient matchings. In fact, the

preferences of firm f1 in Example 2 violate lexicographic preferences by having a

brick, but a stable and unilaterally efficient matching exist in that market.

Formally, a firm’s preferences Pf have a brick if there are two different workers

w,w′ ∈ W and two contracts x, y ∈ Xw ∩ Xf with xPf y such that for all z ∈

Xw′ ∩Xf , x Pf z Pf y. Next, we show that if firms’ preferences have no bricks, then

it is not possible to weaken lexicographicity and still guarantee the existence of

stable and unilaterally efficient matchings.8See Pakzad-Hurson (2014) for a general definition of lexicographic preferences in a many-to-one matching with

contracts framework.

16

Proposition 1. Suppose that there are at least two workers, two firms and two

investments. Moreover, assume that one firm’s preferences are not lexicographic

and have no bricks. Then, there are lexicographic preferences for all other firms

and unanimously separable preferences for all workers such that no stable matching

at (X , P ) is unilaterally efficient. Thus, there is no equilibrium of the induced

investment game.

The proof of Proposition 1 is in the Appendix.

To have an idea of the power of Proposition 1, consider a market with two workers

w and w′ and T = {t1, t2, . . . , tk}. Consider the following preference for a firm f :

Pf : (w, f, t1), (w′, f, t1), . . . , (w

′, f, tk), (w, f, t2), . . . (w, f, tk).

The preference Pf has a brick. Moreover, it is possible to obtain 2k − 1 non lexico-

graphic preferences from Pf by placing some (possibly empty) subset of contracts

involving w′ below the contract (w, f, t2). This gives a sense in which Proposition 1

holds for most violations of lexicographic preferences. In particular, if the number

of investments is large, then bricks represent a small fraction of non lexicographic

preferences.

4 Conclusion

We consider a stylized model of a labor (matching) market where workers have to

invest in their human capital and then match to firms. Is there a straightforward

choice or advise for workers about what investment to make? Our results tell when

the answer is affirmative and in that case what the choice or advice should be.

Our main result establishes that a profile of investments is a strictly strong Nash

equilibrium of the investment game if and only if the matching it produces is invest-

ment efficient and stable in the complete market (Corollary 1). From the prescriptive

point of view, this means that when there is no tension between stability and invest-

ment efficiency, we can advise an investment for each worker, such that no worker

and even no group of workers can do better than following our advice given that

the other workers do. Moreover, such advice can be easily found by means of the

worker-proposing DA algorithm.

Unfortunately, when a market does not admit stable matchings that satisfy a

17

weak notion of efficiency (unilateral efficiency), straightforward advice is not possible

(Corollary 2). In this case, coordination failures may lead to inefficient outcomes.

Appendix

Proof of Theorem 1

We prove that if µ is investment efficient and stable at (X , P ), then t(µ) is a strong

equilibrium of Γ(P ).

Suppose by contradiction that some coalition I ⊆ W has a profitable deviation

t′I ∈ TI at t(µ). Let t′ = (t′I , tW\I(µ)) and µ′ = S(X(t′), P ).

We first show the following claim.

Claim 1. Suppose that for some worker w /∈ I, µ(w)Pw µ′(w). Let f = f(µ(w))

and w′ = w(µ′(f)). Then,

(i) w′ /∈ I and

(ii) µ(w′)Pw′ µ′(w′).

Proof of Claim 1. Since w /∈ I, w makes the same investment under t and t′.

Therefore, µ(w)Pw µ′(w) and unanimous separability imply that

f(µ(w)) � f(µ′(w)). (1)

Moreover, by (1) and the definition of w′

w 6= w′. (2)

Suppose that µ(f)Pf µ′(f). Then, f and w block µ′ at (X(t′), P ) contradicting that

µ′ is stable at (X(t′), P ). Hence,

µ′(f)Pf µ(f). (3)

Suppose by contradiction to (i) that w′ ∈ I. Then,

µ′(w′)Pw′ µ(w′). (4)

By (3) and (4), f and w′ block µ at (X , P ) via investment t′w′ , which contradicts

that µ is stable at (X , P ). Hence, w′ /∈ I. This shows (i).

Suppose by contradiction to (ii) that µ′(w′)Pw′ µ(w′). Then, by (3), f and w′

block µ at (X , P ), contradicting that µ is stable at (X , P ). This shows (ii). N

18

We continue with the proof of Theorem 1. Since µ is investment efficient, there

is some w0 /∈ I such that

µ(w0)Pw0 µ′(w0). (5)

Consider the sequence (wk)∞k=1 defined by wk = w(µ′(f(µ(wk−1)))) for k ≥ 1. By

repeatedly applying Claim 1 to w0, w1, . . . one can see that the following hold for

any wk in the sequence (wk)∞k=1

(a) wk /∈ I and

(b) µ(wk)Pwk µ′(wk).

Conditions (a), (b) and unanimous separability imply that for each k ≥ 1,

f(µ(wk)) � f(µ′(wk)). By definition, f(µ′(wk)) = f(µ(wk−1)).9 So, we conclude

that for each k ≥ 1

f(µ(wk)) � f(µ(wk−1)).

That is, the induced sequence(f(µ(wk))

)∞k=1

is strictly increasing in �, but this

is impossible as there is only a finite number of firms. Hence, there are no profitable

deviations for coalition I.

Proof of Theorem 2

We prove that if t ∈ TW is an equilibrium of Γ(P ), then µ = S(X(t), P ) is stable

at (X , P ).

Suppose by contradiction that µ is not stable at (X , P ). Then, there are w′ ∈ W

and f ′ ∈ F that block µ via some t′w′ ∈ T . Let

t′ = (t′w′ , t−w′) and µ′ = S(X(t′), P ). (6)

We show that f(µ′(w′)) � f ′.

Suppose by contradiction that

f ′ � f(µ′(w′)). (7)

Step 1. We show that

µ′(f ′)Pf ′ (w′, f ′, t′w′)Pf ′ µ(f ′). (8)

9By definition wk = w(µ′(f(µ(wk−1)))), applying µ′ to both sides we obtain µ′(wk) = µ′(w(µ′(f(µ(wk−1))))),

the right hand side of this relation is equal to µ′(f(µ(wk−1))). Applying f to both sides we obtain f(µ′(wk)) =

f(µ′(f(µ(wk−1)))). The right hand side of this last relation is equal to f(µ(wk−1)) as desired.

19

By 6, (7) and unanimous separability we have

(w′, f ′, t′w′)Pw′ µ′(w′) . (9)

Moreover, (7) implies

f ′ 6= f(µ′(w′)). (10)

Suppose that the first part of (8) does not hold, i.e., (w′, f ′, t′w′)Pf ′ µ′(f ′) [strictly

by (10)]. Therefore, by (9), w′ and f ′ would block µ′ at (X(t′), P ), contradicting

that µ′ is stable at (X(t′), P ). The second part of (8) follows from the fact that w′

and f ′ block µ at (X , P ) via t′w′ .

Step 2. We state and prove a Claim.

Let f0 = f ′ and w0 = w(µ′(f0)). Consider the sequences of firms (fk)∞k=0 and

workers (wk)∞k=0 defined by

fk+1 = f(µ(wk)), for k ≥ 0 and wk = w(µ′(fk)), for k ≥ 1. (11)

Note that

fk = f(µ′(wk)). (12)

In words, fk is the firm “matched” with wk under µ′ and fk+1 is the firm “matched”

with wk under µ.

Claim 2. For each k ≥ 0.

(i) wk 6= w′,

(ii) fk+1 � fk.

We prove Claim 2 using induction on k.

Basis. We show that (i) and (ii) hold for k = 0.

By (10) and the definition of w0, (i) holds for w0. Suppose (ii) does not hold

for k = 0. That is, f0 � f1.10 By (i), w0 does not change investment from t to t′.

Then, by (11), observation (12) and unanimous separability, µ′(w0)Pw0 µ(w0). This

together with (8) implies that w0 and f ′ block µ at (X(t), P ), contradicting that µ

is stable at (X(t), P ). Hence (ii) holds for k = 0.

Induction step. Assume (i) and (ii) hold for all 0, 1, . . . , k− 1 for some k ≥ 1. We

show that (i) and (ii) hold for k.

10Strict because f0 = f ′ = f(µ′(w0)) 6= f(µ(w0)) = f1.

20

To see that (i) holds for k observe that wk and w′ obtain different contracts

under µ′ and hence they are different workers. Formally, by induction assumption

(ii), fk � f0 = f ′. Hence, by (7) and (12), wk 6= w′. So, (i) holds for k.

Induction assumptions (i), (ii) and unanimous separability imply that

µ(wk−1)Pwk−1 µ′(wk−1).

If µ(fk)Pfk µ′(fk), then fk and wk−1 block µ′ at (X(t′), P ). Hence,

µ′(fk)Pfk µ(fk).

Since (i) holds for k, wk does not change investment from µ to µ′. If

µ′(wk)Pwk µ(wk)

then, wk and fk would block µ at (X(t), P ), contradicting that µ is stable at

(X(t), P ). Hence (ii) holds for k. N

The sequence of firms (fk)∞k=0 is increasing in �, but this is impossible as the

number of firms is finite. Therefore, we conclude that

f(µ′(w′)) � f ′. (13)

By (13) and unanimous separability, µ′(w′)Rw′ (w′, f ′, t′w′).Moreover, since (w′, f ′, t′w′)

blocks µ at (X , P ), we have (w′, f ′, t′w′)Pw′ µ(w′). Putting these two relations to-

gether

µ′(w′)Rw′ (w′, f ′, t′w′)Pw′ µ(w′).

Therefore, t′w′ is a profitable deviation for w′ at t. This contradicts that t is a NE

of Γ(P ). We conclude that µ is stable at (X , P ).

Proof of Theorem 3

We show that if P satisfies only-bilateral disagreement, then there is a matching

that is stable at (X , P ) and investment efficient.

We order firms by � (workers’ common ranking of firms) as f1, f2, . . . , fm such

that

f1 � f2 � . . . � fm.

Consider the matching µ∗ generated by the following modified serial dictatorship

algorithm:

21

Input. A market (X , P )

Step 1. Let X1 = X . Let w1 be the worker involved in contract Chf1(X1). Let f1

hold the contract that is most preferred by w1 among contracts involving w1 and

f1. Set

µ∗(f1) ≡ Chw1(Xw1 ∩ Xf1).

Step k. Let Ik ⊆ W be the set of workers with a contract in {µ∗(f1), . . . , µ∗(fk−1)}.

Let Xk be the set of contracts that does not involve workers in Ik,

Xk ≡(W \ Ik

)× F × T.

Let wk be the worker involved in contract Chfk(Xk). Let fk hold the contract that

is most preferred by wk among contracts involving wk and fk. Set

µ∗(fk) ≡ Chwk(Xwk ∩ Xfk).

The algorithm terminates after m steps, and produces the matching µ∗.

For each k, let tk = t(µ∗(fk)), . Thus,

µ∗(fk) = (wk, fk, tk), for each k = 1, . . . ,m.

We shall show that µ∗ is investment efficient and stable at (X , P ). First, we show

µ∗ is stable at (X , P ). Suppose by contradiction that some wi and some fj block

µ∗ at (X , P ) via some t ∈ T . Then,

(wi, fj , t)Pwi (wi, fi, ti) and (wi, fj , t)Pfj (wj , fj , tj). (14)

Case 1. j = i. Then we have

(wi, fi, t)Pwi (wi, fi, ti).

which violates the definition of ti.

Case 2. i < j. Then we have

(wi, fi, ti)Rwi (wi, fi, t)Pwi (wi, fj , t). (15)

The first part of (15) follows from the fact that under the modified serial dictator-

ship fi is assigned the contract that is most preferred by wi among all contracts that

involve wi and fi. The second part follows from unanimous separability. However,

(15) contradicts the first relation in (14).

22

Case 3. j < i. At step j of the modified serial dictatorship algorithm, fj is assigned

(wj , fj , tj). Then, by definition it holds that

Chfj(Xj)Rfj (wj , fj , tj).

Since Chfj(Xj) involves a contract with wj , only-bilateral disagreement implies

(wj , fj , tj)Pfj (wi, fj , t). (16)

However, (16) contradicts the second relation in (14). Therefore, µ∗ is stable at

(X , P ). N

Now we show that µ∗ is investment efficient. In particular, we show the following

claim from which investment efficiency follows.

Claim 3. Let t′ ∈ TW and µ′ = S(X(t′), P ). Suppose that for wk ∈ W,

µ′(wk)Pwk (wk, fk, tk). Then, there is wl ∈ W such that (wl, fl, tl)Pl µ′(wl).

We show Claim 3 by induction on k.

Basis. Claim 3 holds for k = 1.

By unanimous separability and only-bilateral disagreement (w1, f1, t1) is the best

contract for w1. Hence, w1 cannot be made better off.

Induction step. Assume that Claim 3 holds for all 1, . . . , k − 1 for some k ≥ 2.

We show that Claim 3 holds for k.

Suppose

µ′(wk)Pwk (wk, fk, tk). (17)

Let fj = f(µ′(wk)). Thus, µ′(wk) = (wk, fj , t′wk

).

Case 1. k = j. Then we have

(wk, fk, t′wk

)Pwk (wk, fk, tk)

which violates the definition of tk.

Case 2. k < j. Then we have

(wk, fk, tk)Rwk (wk, fk, t′wk

)Pwk (wk, fj , t′wk

). (18)

The first part of (18) follows from the fact that under the modified serial dictator-

ship fk is assigned the contract that is most preferred by wk among all contracts that

involve wk and fk. The second part follows from unanimous separability. However,

(18) contradicts (17).

23

Case 3. j < k. If wj is worse off under µ′ relative to µ∗, then Claim 3 holds for

k. So, suppose wj is better off under µ′ relative to µ∗ (strictly because wj changes

contract from µ∗ to µ′). Then, by the induction assumption some other worker is

worse off under µ′ relative to µ∗. Therefore, Claim 3 holds for k. N

We have shown that µ∗ is stable at (X , P ) and investment efficient. Therefore,

Theorem 3 holds.

Proof of Proposition 1

Label the elements of W and F as {w1, w2, . . . , wm} and {f1, f2, . . . , fm} respec-

tively. Assume that the preferences of firm f1 are not lexicographic. Without loss

of generality

(w1, f1, t1)Pf1 (w2, f1, t3)Pf1(w1, f1, t2), (19)

where t1, t2, t3 ∈ T and t1 6= t2.

Moreover, assume that Pf1 has no bricks. Then, there is t4 ∈ T \ {t3} such that

(w1, f1, t1)Pf1 (w2, f1, t3)Pf1(w1, f1, t2)Pf1 (w2, f1, t4). (20)

We fix the preferences of all other firms.

• Let firm f2 have any lexicographic preferences such that (w2, f2, t4), is her most

preferred contract.

• Let each firm f3, f4, . . . , fm have lexicographic preferences such that each firm

fi with i ≥ 3 ranks first all contracts that involve wm consecutively, then all

contracts that involve wm−1 consecutively and so on.

We fix the preferences of all workers. Let the common ranking of firms be given

by �: fm, fm−1, . . . , f3, f1, f2. That is, given investments all workers prefer fm, then

fm−1 and so on until f3, then they prefer f3 to f1 and f1 to f2.

• Let worker w1 have unanimously separable preferences with common ranking

� such that

(w1, f1, t2)Pw1 (w1, f1, t1)Pw1 (w1, f′, t′),

where f ′ ∈ {f1, f2} , t′ ∈ T and (w1, f′, t′) 6= (w1, f1, t1), (w1, f1, t2).

24

• Let worker w2 have unanimously separable preferences with common ranking

� such that

(w2, f1, t3)Pw2 (w2, f1, t4)Pw2 (w2, f2, t4)Pw2 (w2, f′′, t′′),

where f ′′ ∈ {f1, f2}, t′′ ∈ T and (w2, f′′, t′′) 6= (w2, f1, t3), (w2, f1, t4), (w2, f2, t4).

• Let each worker w3, w4, . . . , wm have lexicographic (in particular unanimously

separable ) preferences with respect to the common ranking �. Also let the

preferences of each worker wi, i ≥ 3 be such that wi agrees with fi on what is

the best investment if wi and fi were to match.

The preferences of w1, w2, f1 and f2 are sketched in Table 8.

Table 7: Preferences of w1, w2, f1, f2

Pf1 Pf2 Pw1Pw2

... (w2, f2, t4)...

...

(w1, f1, t1)... (w1, f1, t2) (w2, f1, t3)

... (w1, f1, t1) (w2, f1, t4)

(w2, f1, t3)... (w2, f2, t4)

... (w1, f′, t′)

...

(w1, f1, t2)... (w2, f

′′, t′′)...

...

(w2, f1, t4)...

Let P denote the preference profile we have constructed. We claim that there is

a unique stable matching at (X , P ) given by:

(i) µ(w1) = (w1, f1, t1),

(ii) µ(w2) = (w2, f2, t4),

(iii) µ(wi) = (wi, fi, ti), for all i ≥ 3 where ti ∈ T denotes the most preferred

investment for wi when matched to fi.

To show this claim suppose by contradiction to (iii) that wi and fj with i 6=

j block µ via some t ∈ T . The case i = j is not possible because ti is al-

ready the best investment for both. If i > j, then because Pwi is lexicographic,

25

(wi, fi, ti)Pwi (wi, fj , t). Thus, wi and fj cannot block µ via any t ∈ T . If i < j,

then because Pfj is lexicographic, (wj , fj , tj)Pfj (wi, fj , t). Thus, wi and fj cannot

block µ via any t ∈ T . Therefore, (iii) holds. Moreover, by the same arguments any

matching under which wi and fj with i 6= j are matched is not stable.

Since (iii) holds, we can consider w1, w2, f1 and f2 in isolation. The unique

stable matching in the isolated market is given by (i) and (ii) (the boxed matching

in Table 7). This completes the proof of the claim.

Consider the investment profile t∗ = (t2, t(µ)−w1) = (t2, t4, (ti)mi=3). We claim

that the unique stable matching in the market induced by t∗, µ∗ = S(X(t∗), P ) is

given by

(iv) µ∗(w1) = (w1, f1, t2),

(v) µ∗(w2) = µ(w2),

(vi) µ∗(wi) = µ(wi), for all i ≥ 3.

By the same arguments as in (iii), (vi) holds and any matching under which wi

and fj with i 6= j are matched is not stable. So again, we can consider w1, w2, f1

and f2 in isolation. The unique stable matching in the isolated market induced by

investments t2 for w1 and t4 for w2 is given by (iv) and (v) (the bold face matching

in Table 7). This completes the proof of the claim.

The matching µ∗ Pareto dominates µ. Moreover, t(µ∗) differs from t(µ) only

in the investment of worker w1. Thus, µ is not unilaterally efficient and hence no

stable matching at (X , P ) is unilaterally efficient. Therefore, by Corollary 2, there

is no equilibrium of the investment game Γ(P ).

4.1 Example 4

The next example exhibits a profile of firms’ preferences that has Ergin cycles for

which regardless of workers’ preferences a stable and investment efficient matching

exists.

Example 4. Consider a market with W = {i, j, k}, F = {f1, f2, f3}, and

T = {t1, . . . tl}, l ≥ 2. Let the preferences of f1, f2 and f3 be lexicographic.

In particular, let f1 rank all contracts with i consecutively, then all contracts with

26

j consecutively and then all contracts with k consecutively. Let f2 rank all con-

tracts with k consecutively, then all contracts with i consecutively, and then all

contracts with j consecutively. We illustrate such preference profile in Table 8 with

t1, t2, t3, t4, t5 ∈ T .

The firms’ preference profile contains Ergin cycles.11 Since the firms’ preference

profile is lexicographic, only-bilateral disagreement is satisfied regardless of workers’

preferences. Thus, by Theorem 3 a stable and investment efficient matching exists

regardless of workers’ preferences. However, it is possible to find preferences for

workers such that no stable matching is efficient.

Table 8: Preferences P in Example 4

Pf1 Pf2 Pf3

(i, f1, t1) (k, f2, t4)...

......

(j, f1, t2) (i, f2, t5)...

......

(k, f1, t3)

�

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