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CHAPTER 10
Game Theory
In this chapter we continue the study of the testable implications of models of
collective behavior. We focus here on game-theoretic models. Our ﬁrst results
use a version of the abstract choice environments from Chapter 2, and discuss
testing Nash equilibrium as the prediction of the choices made by a collection
of agents. We then turn to models of bargaining and two-sided matching.
10.1
NASH EQUILIBRIUM
Let N be a ﬁnite set of agents of cardinality n. For each i ∈N, consider some
ﬁnite set ¯Si. The idea is that ¯Si is some “global” set of strategies that may be
available to agent i. In a speciﬁc (observable) instance, player i is restricted to
choosing a strategy from Si ⊆¯Si, a “budget” of possible strategies. Consider
the nonempty product subsets of 
i∈N ¯Si, denoted by S. A typical element of
S has the form S1 × S2 × ··· × Sn; where each Si is a nonempty subset of ¯Si.
These sets are called game forms.
An element of 
i∈N ¯Si is called a strategy proﬁle. As usual in game theory
we use (si,s−i) to denote a strategy proﬁle in which si is i’s strategy and s−i ∈

j∈N\{i} ¯Sj is a strategy proﬁle for players in N \ {i}.
A joint choice function is a mapping c :  ⊆S →2

i∈N ¯Si\{∅} satisfying
c(S) ⊆S. Note that c(S) need not have a product structure, but it is required to
be nonempty. In particular, every joint choice function is a choice function in
the sense of Chapter 2. We will be interested in notions of rationalization that
reﬂect the product structure of S.
Let {⪰i}i∈N be a family of preference relations, each over 
i∈N ¯Si. Note that
these preferences deﬁne a (normal-form) game (N,(Si,⪰i|S)i∈N) for each S =
S1 × S2 × ··· × Sn ∈S. By ⪰i|S we mean the restriction of preferences ⪰i to S.
A strategy proﬁle s = (si)i∈N ∈S is a Nash equilibrium of (N,(Si,⪰i|S)i∈N) if
(si,s−i) ⪰i (s′
i,s−i) for all s′
i ∈Si and all i ∈N.
We say that the family of preference relations {⪰i}i∈N strongly Nash
rationalize a choice function c if for all S = 
i∈N Si ∈S, c(S) coincides with


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Game Theory
the set of Nash equilibria of the game (N,(Si,⪰i|S)i∈N):
c(S) = {(si)i∈N : for all j ∈N and all s′
j ∈Sj,(sj,s−j) ⪰j (s′
j,s−j)}.
We say that c is strongly Nash rationalizable by a class of preference relations
if there exist preference relations ⪰i in that class which strongly Nash
rationalize c.
We say that a family of binary relations ⪰i weakly Nash rationalize c
if for all S ∈S, c(S) ⊆{(si)i∈N : for all j ∈N and all s′
j ∈Sj,(sj,s−j) ⪰j
(s′
j,s−j)}. We will say c is weakly Nash rationalizable by a class of preference
relations if there exist preference relations ⪰i in that class which weakly Nash
rationalize c.
Clearly, weak Nash rationalizability by weak orders has no empirical
content; thus, when we speak of weak Nash rationalization, we shall take
interest in weak Nash rationalization by strict preference relations.
10.1.1
Choice from all game forms
We shall assume that we observe the outcomes of the strategic interaction of
the agents in N for each possible game form. That is,  = S.1 The results in
this subsection are due to Yves Sprumont.
The set S forms a lattice under the set inclusion relation; its meet is given
by S∧S′ = S∩S′. Writing S = 
i∈N Si and S′ = 
i∈N S′
i, the join S∨S′ is given
by S ∨S′ = 
i∈N(Si ∪S′
i).
Say that a joint choice function c satisﬁes persistence under expansion if for
all S,S′ ∈S, c(S) ∩c(S′) ⊆c(S ∨S′). It is obvious from the deﬁnition of Nash
equilibrium that persistence under expansion is a necessary condition for c to
be strongly Nash rationalizable.
We will say that S ∈S is a line if there exists i ∈N such that for all j ̸= i,
Si is a singleton. A line can be understood as a single-agent decision problem.
We say that a joint choice function c satisﬁes persistence under contraction if
the following two properties are satisﬁed:
• for all S,S′ ∈S with S ⊆S′, c(S′) ∩S ⊆c(S).
• If S′ is a line, S ⊆S′, and c(S′) ∩S ̸= ∅, then c(S) ⊆c(S′).
Note that the ﬁrst property of persistence under contraction is condition α of
Chapter 2. The second property, which is only required to hold for lines, is
similar to condition β.
The following theorem characterizes strongly Nash rationalizable joint
choice functions.
1 This assumption is analogous to assuming that the choices from all possible budgets are
observed in Chapter 2.


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10.1 Nash equilibrium
145
Theorem 10.1
A joint choice function c is strongly Nash rationalizable by
preference relations iff it satisﬁes persistence under expansion and persistence
under contraction.
Proof. It is easy to verify that if c is strongly Nash rationalizable by preference
relations, then it satisﬁes persistence under expansion and persistence under
contraction. Conversely, suppose c satisﬁes these two properties.
We ﬁrst deﬁne a basic revealed preference, ⪰c
i . For any pair (si,s−i) and
(s′
i,s−i), we deﬁne (si,s−i) ⪰c
i (s′
i,s−i) in the natural way; that is, (si,s−i) ⪰c
i
(s′
i,s−i) iff (si,s−i) ∈c
#
{si,s′
i} × 
j̸=i{sj}
$
. Each ⪰c
i is clearly reﬂexive.
Transitivity is also straightforward and follows similarly to the proof of
Theorem 2.9. Hence, each ⪰i
c has an extension (by Theorem 1.5) to a weak
order ⪰i.
We claim that the orders ⪰i strongly Nash rationalize c. So, suppose we
have given S ∈S and s ∈S such that for all i ∈N and all s′
i ∈Si, (si,s−i) ⪰i
(s′
i,s−i). Hence, for all i ∈N and all s′
i ∈Si, (si,s−i) ⪰c
i (s′
i,s−i), so that
(si,s−i) ∈c
#
{si,s′
i} × 
j̸=i{sj}
$
. Now, S = 3
i∈N
3
s′
i∈Si
#
{si,s′
i} × 
j̸=i{sj}
$
, so
that by persistence under expansion (applied inductively), we have s ∈c(S).
Conversely, suppose that s ∈c(S). Then it follows by persistence under
contraction that s ∈c
#
{si,s′
i} × 
j̸=i{sj}
$
for all i and all s′
i, so that (si,s−i) ⪰c
i
(s′
i,s−i), or (si,s−i) ⪰i (s′
i,s−i).
The following result characterizes weak rationalizability by strict preference
relations. We omit its proof, which is simple.
Theorem 10.2
A joint choice function c is weakly rationalizable by strict
preference relations iff for all S ∈S, all s ∈c(S), all i ∈N, and all s′
i ∈Si,
c
#
{s′
i,si} × 
j̸=i{sj}
$
= {s}.
The notion of Nash rationalizability is based on individual agents’ incentives to
choose strategies. We can instead consider what the group of agents N would
jointly decide to do. One natural property is that the chosen strategy proﬁles
should not be dominated: A joint choice function c is Pareto rationalizable by
preference relations iff there are preference relations ⪰i on 
i∈N ¯Si such that
for all S ∈S, s ∈c(S) iff there is no s′ ∈S for which s′ ⪰i s for all i, with at least
one preference strict. Say it is team rationalizable if there exists a preference
⪰over 
i∈N ¯Si such that for all S ∈S, c(S) is the set of ⪰-maximal elements
of S.
Theorem 10.3
Suppose a joint choice function c satisﬁes |c(S)| = 1 for all
S ∈S. Then c is Pareto rationalizable iff it is team rationalizable.
Proof. If c is team rationalizable, it is clearly Pareto rationalizable. Conversely,
suppose c is Pareto rationalizable, and let {⪰i}i∈N be the rationalizing relations.
Deﬁne s ⪰∗s′ if it is not the case that s ⪰i s′ for all i ∈N with at least one


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Game Theory
preference strict. Clearly then, s ≻∗s′ iff s ⪰i s′ for all i ∈N and there is j ∈N
for which s ≻j s′. Hence, ⪰∗is quasitransitive. By deﬁnition, for every S, c(S)
coincides with the ⪰∗maximal elements of S. The result now follows from
Proposition 2.7.
In fact, Theorem 10.3 does not rely on the product structure of S and could be
proved more generally.
The following theorem establishes that the implications of Nash rationaliz-
ability are stronger than those of Pareto rationalizability. We omit its proof,
which is technical.
Theorem 10.4
Suppose that |N| = 2, and that a joint choice function c
satisﬁes |c(S)| = 1 for all S ∈S. Then if c is Nash rationalizable, it is Pareto
rationalizable.
10.1.2
Choice from a subset of game forms
We now discuss results in the revealed preference approach to game theory
when not all possible game forms in S are observable.
We will say a collection  ⊆S is line-closed if for all S ∈, all s ∈S,
and all i ∈N, Si × 
i̸=j∈N{sj} ∈. Line-closedness is enough to allow us to
generate a meaningful revealed preference relation. A joint choice function can
now be deﬁned on , and it is meaningful to discuss strong rationalizability
by preferences.
The following idea is due to Galambos. We deﬁne the following revealed
preference pair. Deﬁne ⪰c
i by s ⪰c
i s′ if for all j ̸= i, sj = s′
j, and there exists
S ∈ for which s,s′ ∈S and s ∈c(S). We say a joint choice function c deﬁned
on  satisﬁes N-congruence if for all i ∈N and all S, if for s ∈S we have
s

⪰c
i
T (s′
i,s−i) for all i and all s′
i ∈Si, then s ∈c(S).2
Theorem 10.5
Suppose that  is a line-closed domain. Then a joint
choice function is strongly rationalizable by preference relations iff it satisﬁes
N-congruence.
Proof. Suppose a joint choice function is strongly Nash rationalizable by weak
orders ⪰i. Clearly,

⪰c
i
T ⊆⪰i. Consequently, if s ∈S satisﬁes s

⪰c
i
T (s′
i,s−i)
for all s′
i ∈Si, it follows that s ⪰i (s′
i,s−i) for all s′
i ∈Si, so that s ∈c(S), since
the {⪰i}i∈N strongly Nash rationalize c.
Conversely, suppose that choice function c satisﬁes N-congruence. If we
deﬁne ≻c
i by (si,s−i) ≻c
i (s′
i,s−i) if there exists S for which s,(s′
i,s−i) ∈S and s ∈
c(S) but (s′
i,s−i) ̸∈c(S), then by N-congruence, order pair ⟨⪰c
i ,≻c
i ⟩is acyclic, so
that by Theorem 1.5, there is a weak order ⪰i for which ⪰c
i ⊆⪰i and ≻c
i ⊆≻i.
We claim that the {⪰i}i∈N strongly Nash rationalize c. Suppose that s ∈c(S).
We claim that for all i and all s′
i ∈Si, (si,s−i) ⪰i (s′
i,s−i). This follows as
2 Recall that for a binary relation ⪰, ⪰T denotes the transitive closure.


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10.2 Bayesian Nash equilibrium
147
(si,s−i) ⪰c
i (s′
i,s−i) and ⪰c
i ⊆⪰i. On the other hand, suppose that s ∈S satisﬁes
(si,s−i) ⪰i (s′
i,s−i) for all i and all s′
i ∈Si. We claim that s ∈c(S). Since 
is line-closed, it follows that Si × 
j̸=i{sj} ∈. Then s ∈c
#
Si × 
j̸=i{sj}
$
.
Otherwise, since c is nonempty-valued, there exists s′
i ∈Si for which (s′
i,s−i) ∈
#
Si × 
j̸=i{sj}
$
, from which it follows that (s′
i,s−i) ≻c
i s, contradicting the
deﬁnition of ⪰i. Consequently, we know that s ⪰c
i (s′
i,s−i) for all i ∈N, so
that by N-congruence, s ∈c(S).
10.1.3
Zero-sum games
The preceding results impose no structure on the rationalizing games, but one
may want to investigate the joint implications of Nash behavior and some
properties of the strategic environment. One of the most natural restrictions
is the property that the game be a zero-sum game. Zero-sum games are the
subject of intense study in game theory; they also arise naturally if one believes
that players care about relative, not absolute, payoffs.3
Fix N = {1,2}. Suppose that  = S again, so that we are given choice from
all possible game forms. A choice function c is strongly Nash rationalizable by
a zero-sum game if there is a preference relation ⪰on ¯S1 × ¯S2 such that, for all
S ∈S, c(S) is the set of Nash equilibria of ({1,2},(S1,⪰|S),(S2,⪯|S)); where
we denote by ⪯the dual binary relation deﬁned from ⪰, i.e. x ⪯y iff y ⪰x.
It turns out that to characterize strong zero-sum rationalizability, all we need
is one additional restriction on c. Say that c is interchangeable if, for all S ∈S,
and all s,s′ ∈c(S), {s} ∨{s′} ⊆c(S). Our next result is due to SangMok Lee.
Theorem 10.6
A choice function is strongly Nash rationalizable by a
zero-sum game iff it satisﬁes persistence under expansion, persistence under
contraction, and it is interchangeable.
We omit the proof of Theorem 10.6, which is somewhat technical.
10.2
BAYESIAN NASH EQUILIBRIUM
The next class of problems we tackle are motivated by mechanism design.
Given a function from agents’ types to outcomes, we want to know if the
function could be an incentive-compatible direct revelation mechanism, for
some preferences of the agents.
Formally, let N be a set of agents, and X a ﬁnite set of outcomes. For each
agent i ∈N, suppose we are given a ﬁnite type space Ti. A direct revelation
mechanism is a function g : T = 
i∈N Ti →X. The question we ask here is
simple. Suppose we observe N, T = 
i∈N Ti, and g. Can we rationalize g
as a strongly incentive-compatible direct revelation mechanism for some list
3 For example, suppose that “material” payoffs are pi and there are two players; i = 1,2. If each
player i cares about the difference pi −p3−i, then the sum of payoffs will be zero.


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Game Theory
of preferences? In particular, is it possible that these mappings could be the
unique Bayesian Nash equilibrium for some preferences in the direct revelation
game? This answer and its question are due to John Ledyard.
Formally, let us deﬁne a Bayesian environment to be a pair of functions, one
for each agent, denoted by ui : X × T →R and pi : Ti →(T−i). The function
ui is meant to be i’s utility function over outcomes. The function pi gives i’s
beliefs about other agents’ types.
Two points are worth mentioning here. First, we allow agent i’s utility
to depend on the entire proﬁle of types. Second, pi carries each type to a
probability measure over the others’ types. We denote the probability of type
proﬁle t−i by pi(t−i|ti). Note, however, that we are not making any common
prior assumption.
A Bayesian environment strongly rationalizes direct revelation mechanism
g if for all ti,t′
i ∈Ti, if g(t) ̸= g(t′
i,t−i) for some t−i, then

t−i∈T−i
ui(g(ti,t−i),t)p(t−i|ti) >

t−i∈T−i
ui(g(t′
i,t−i),t)p(t−i|ti).
(10.1)
Theorem 10.7
For any direct revelation mechanism, there exists a Bayesian
environment strongly rationalizing it.
Proof. Rewrite Equation (10.1), so that we have:

t−i∈T−i

x∈X
ui(x,t)p(t−i|ti)1g(ti,t−i)=x >

t−i∈T−i

x∈X
ui(x,t)p(t−i|ti)1g(t′
i,t−i)=x;
equivalently:

t−i∈T−i

x∈X
ui(x,t)p(t−i|ti)[1g(ti,t−i)=x −1g(t′
i,t−i)=x] > 0.
(10.2)
First, observe that given any ti, we can ﬁnd ui and pi (depending on ti) which
satisfy equation (10.2) iff there is a function wti : X ×T−i →R such that for all
t′
i for which g(ti,t−i) ̸= g(t′
i,t−i) for some t−i, then

t−i∈T−i

x∈X
wti(x,t−i)[1g(ti,t−i)=x −1g(t′
i,t−i)=x] > 0.
(10.3)
This follows because it is simple to then ﬁnd ui(x,t) and pi(t−i|ti) for which
ui(x,t)pi(t−i|ti) = wti(x,t−i) (for example, deﬁne pi(t−i|ti) > 0 arbitrarily so
that 
t−i pi(t−i|ti) = 1, and then deﬁne ui(x,t) =
wti(x,t−i)
pi(t−i|ti) ). Hence, we turn to
equation (10.3). For every ti, there is a list of such equations.
If we can solve the corresponding list of equations for each ti, then we have
proved that the direct revelation mechanism is strongly rationalizable. So, ﬁx
ti. For each t′
i for which there exists t−i such that g(ti,t−i) ̸= g(t′
i,t−i), there is
an equation of type (10.3). Denote the set of such t′
i by T′
i(ti).
In equation (10.3), view w as a vector in RX×T−i. The quantities [1g(ti,t−i)=x −
1g(t′
i,t−i)=x] play the role of coefﬁcients. Applying Lemma 1.12, if for some ti,


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10.3 Bargaining theory
149
a solution to the system described by equation (10.3) does not exist, then there
exists, for each t′
i ∈T′
i(ti), ηt′
i ≥0, not all of which are zero, such that for all
t−i,x,

t′
i∈T′
i(ti)
ηt′
i[1g(ti,t−i)=x −1g(t′
i,t−i)=x] = 0.
(10.4)
By assumption, for any t′′
i ∈T′
i(ti), there are t−i and x for which g(ti,t−i) =
x ̸= g(t′′
i ,t−i). Pick such t−i and x. Because 1g(ti,t−i)=x = 1, we then have
that [1g(ti,t−i)=x −1g(t′′
i ,t−i)=x] = 1, and further, for any t′
i ∈T′
i(ti), we have
[1g(ti,t−i)=x −1g(t′
i,t−i)=x] ≥0. Hence, equation 10.4 implies ηt′
i = 0. We conclude
that η = 0, a contradiction.
10.3
BARGAINING THEORY
We now turn to a formulation of the revealed preference problem for
bargaining theory. We can work out the empirical consequences of the most
commonly used cooperative theories of bargaining.
Suppose n agents bargain over a ﬁxed quantity of a single-dimensional
resource: think of bargaining over a ﬁxed monetary amount, which needs to
be allocated among n agents. There is a given disagreement point, a point
that speciﬁes monetary outcomes for all the agents in the event that there
is no agreement. We imagine that we observe similar agents bargaining in
different circumstances, for example workers and ﬁrms in wage bargaining.
The question is when observed outcomes can be rationalized as consistent with
standard bargaining theory.
We shall ﬁrst describe the theories under consideration, then deﬁne the
kinds of data we might use to test them. Then, we present a result that
characterizes the observable implications of these theories in the case where
the disagreement points are ﬁxed and the same for all agents. The surprising
implication is that all of these theories are observationally equivalent. Each
theory tries to capture a distinct economic phenomenon or criterion, but they
turn out to have rather weak empirical consequences, to the point that they are
all equivalent.
In fact, we uncover a particularly striking form of observational equivalence.
We ﬁnd preferences (utility functions) that serve to rationalize the data as
coming from any of the theories. We might expect that two theories are
observationally equivalent because a given dataset can be rationalized by
one theory or by the other, but normally each rationalization will involve
different values for the unobservable variables in the theories (preferences in
our case). In the case of bargaining theory, it turns out that we can choose the
unobservables so that they work for all rationalizations.


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Game Theory
The model is as follows. We assume some quantity m ∈R+ of money, and a
vector d = (d1,...,dn) that represents the disagreement point. The set
B(m,d) = {(x1,...,xn) ∈Rn
+ :
n

i=1
xi ≤m and, for all i,xi ≥di}
is the set of all allocations of m amongst n agents, in which everyone gets at
least their disagreement outcomes. The set B is therefore a set of feasible and
“individually rational” allocations.
A bargaining theory uses information on agents’ preferences to predict
an outcome in B(m,d). Suppose that each agent i is described by a strictly
increasing and concave utility function ui : R+ →R. We shall focus on three
theories: utilitarianism, Nash bargaining, and egalitarianism.
The utilitarian theory calls for maximizing the sum of agents’ utilities. It
predicts that m is allocated so as to maximize the sum n
i=1 ui(xi) over B(m,d).
In fact, for reasons that will become clear later, we consider a generalization
of the utilitarian theory, where for some function g : A ⊆R →R, the sum
n
i=1 g(ui(xi) −ui(di)) is maximized over B(m,d).
The Nash bargaining theory predicts a choice in B(m,d) that maximizes
n
"
i=1
[ui(xi) −ui(di)].
The expression being maximized is termed the Nash product. Note that the
Nash bargaining theory is a special case of our generalization of the utilitarian
theory, letting g = log.
Finally, the egalitarian (or maxmin) theory says that x ∈B(m,d) should be
chosen to maximize
min
i∈N [ui(xi) −ui(di)].
We assume a set of K observations of bargaining outcomes. Each outcome
represents a split of some monetary quantity. We assume that the disagreement
points are ﬁxed and the same for all agents: we can, for all intents and purposes,
take the disagreement outcome to be zero for all agents. A dataset is then a
set D = {xk : k = 1,...,K}. Each observation k speciﬁes an allocation xk =
(xk
1,...,xk
n) ∈Rn
+ of the total amount of money n
i=1 xk
i . Let N = {1,...,n}.
Let g : R+ →R ∪{−∞} be a strictly increasing, smooth, and concave
function. We say that data {xk}K
k=1 are g-rationalizable if there exist strictly
increasing, smooth, and strictly concave functions ui for which ui(0) = 0 and
u′
i(0) = ∞(Inada conditions), and for which 
i∈N g(ui(xk
i )) ≥
i∈N g(ui(yi))
for all allocations (y1,...,yn) ∈B(
i xk
i ,0) and k = 1,...,K. The utilitarian and
Nash models are special cases of g-rationalizability. Note that the assumption
of rationalizability already reﬂects our assumption that the disagreement point
is ﬁxed and the same for all agents.
On the other hand, data {xk}K
k=1 are maxmin rationalizable if there exist
strictly increasing and strictly concave ui, normalized so that ui(0) = 0, for


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151
which mini∈N ui(xk
i ) ≥mini∈N ui(yi) for all (y1,...,yn) ∈B(
i xk
i ,0) and k =
1,...,K.
We say that data {xk}K
k=1 are comonotonic if for all i,j ∈N and all k,l,
xk
i < xl
i implies xk
j < xl
j, and for all i,j ∈N, xk
i = 0 iff xk
j = 0. Comonotonicity
requires that outcomes are perfectly strictly ordinally correlated (when 0 is also
considered an outcome).
The following result characterizes the data that are g-rationalizable or
maxmin rationalizable.
Theorem 10.8
Given data {xk}K
k=1 and a strictly increasing concave g, the
following are equivalent:
I) The data are comonotonic.
II) The data are g-rationalizable.
III) The data are maxmin rationalizable.
The proof shows more than is stated here. In the proof we construct
rationalizing utilities that work for any function g, as well as for the maxmin
model. The resulting observational equivalence is therefore unusually strong.
We can ﬁnd unobservable preferences that rationalize the data using any of the
models under consideration.
Proof. It follows from the ﬁrst-order conditions that if the data are either
g-rationalizable or maxmin rationalizable, then they are comonotonic.
For the other direction, we show something slightly stronger: If the data
are comonotonic, then there exist strictly concave, continuous, and increasing
functions ui such that, if ϕ : [0,∞) →R ∪{−∞} is an increasing, symmetric,
and quasiconcave function, then ϕ(u1(xk
1),...,un(xk
n)) ≥ϕ(u1(y1),...,un(yn))
for all allocations (y1,...,yn) satisfying 
i∈N xk
i = 
i∈N yi.4 As a special case,
we have ϕ(z1,...,zn) = n
i=1 g(zi). Note the order of the quantiﬁers used
above: the same proﬁle of utility functions u1,...,un works across all ϕ.
To this end, we suppose the data are comonotonic, and ignore replications as
well as points where every agent consumes 0. Without loss of generality, let us
suppose that x1
i < x2
i < ... < xK
i for all i ∈N (that this is possible follows from
comonotonicity). Below we construct a proﬁle of utility functions u1,...,un
with the property that for all k = 1,...,K, 
i∈N ui(xk
i ) is maximal across all
allocations y1,...,yn for which 
i∈N xk
i = 
i∈N yi, and mini∈N ui(xk
i ) is also
maximal across all such allocations; it follows that, since each ui is strictly
increasing, ui(xk
i ) = uj(xk
j ) for all i,j ∈N.
We ﬁrst argue that such a construction sufﬁces to establish the result: Let ϕ
be as above, and suppose by way of contradiction that there is a k and a feasible
allocation (y1,...,yn) for which ϕ(u1(y1),...,un(yn)) > ϕ(u1(xk
1),...,un(xk
n)).
Note then, by symmetry of ϕ, that for any permutation of the agents σ : N →N,
4 Symmetry means that if σ is a permutation on {1,...,n} then ϕ(xσ(1),...,xσ(n)) =
ϕ(x1,...,xn). Increasing here means that if xi > yi for all i, then ϕ(x1,...,xn) > ϕ(y1,...,yn).


--- Page 10 ---
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Game Theory
ϕ(uσ(1)(yσ(1)),...,uσ(n)(yσ(n))) = ϕ(u1(y1),...,un(yn)). Quasiconcavity of ϕ
then implies that
ϕ
1
i∈N
ui(yi)
n
,...,

i∈N
ui(yi)
n
2
> ϕ(u1(xk
1),...,un(xk
n)).
By the strictly increasing property of ϕ, and using the fact that ui(xk
i ) = uj(xk
j )
for all i,j ∈N, this implies that

i∈N
ui(yi)
n
>

i∈N
ui(xk
i )
n
,
contradicting

i∈N
ui(xk
i ) ≥

i∈N
ui(yi)
for all feasible allocations y1,...,yn.
We ﬁnish the proof by constructing, for each i, a strictly decreasing,
continuous, and positive function fi, with the property that if we set ui to be the
integral of fi, then the proﬁle of utility functions (u1,...,un) works as required
by the ﬁrst part of the proof.
We proceed by induction. We ensure that, for each i ∈N and each k, the
following are true:
I)
0 xk
i
0 fi(x)dx =
0 xk
j
0 fj(x)dx
II) fi(xk
i ) = fj(xk
j ).
In the ﬁrst place, for k = 1, we deﬁne for each agent j, fj(0) = +∞. The
construction is done in a series of steps, labeled (I) to (VI).
I) For K, deﬁne fi(xK
i ) = 1 for all i ∈N;
II) for x > xK
i , we deﬁne fi(x) to be any strictly decreasing function,
taking values everywhere less than 1 and making fi continuous.
III) We proceed by induction. Let k > 1 be arbitrary, and suppose that
fi(x) has been deﬁned for all x ≥xk
i . We assume that for all k′ ≥k,
fi(xk′
i ) = fj(xk′
j ) and
, xK
i
xk
i
fi(x)dx =
, xK
j
xk
j
fj(x)dx for all i,j ∈N.
Recall that we have x1
i < x2
i < ... < xK
i . We choose a ﬁnite fj(xk−1
j
)
but we must choose it to be sufﬁciently large. Speciﬁcally, let z be
large enough so that there is ε > 0 for which z(xk
j −xk−1
j
) −ε >
maxi∈N fi(xk
i )(xk
i −xk−1
i
) + ε for all j. We can then set fj(xk−1
j
) = z for
all j.
IV) Observe that, given fj(xk−1
j
) and fj(xk
j ), for any ε > 0 and any
y ∈
#
fj(xk
j )(xk
j −xk−1
j
) + ε,fj(xk−1
j
)(xk
j −xk−1
j
) −ε
$
,


--- Page 11 ---
10.4 Stable matching theory
153
we may deﬁne fj continuous and decreasing on x ∈(xk−1
j
,xk
j ) so that
, xk
j
xk−1
j
fj(x)dx = y.
This follows as we may choose the integral as close as possible
to fj(xk−1
j
)(xk
j −xk−1
j
) by taking a sequence of decreasing continu-
ous functions approaching the constant value fj(xk−1
j
) pointwise in
(xk−1
j
,xk
j ); likewise we may choose the integral as close as possible
to fj(xk
j )(xk
j −xk−1
j
).
V) Complete fj(x) on x ∈(xk−1
j
,xk
j ) so that
, xk
j
xk−1
j
fj(x)dx
is equalized across all agents, by picking
y ∈

i∈N

fi(xk
i )(xk
i −xk−1
i
) + ε,fi(xk−1
i
)(xk
i −xk−1
i
) −ε

and choosing fj(x) on x ∈(xk−1
j
,xk
j ) so that
, xk
j
xk−1
j
fj(x)dx = y.
VI) In the case of k = 1, we must also maintain that
, x1
j
0
fj(x)dx < +∞.
The functions fj so constructed satisfy the conditions we ask for: that for all k,
fj(xk
j ) is equalized across j, and
, xk
j
0
fj(x)dx
is equalized across j. By setting
uj(x) =
, x
0
fj(x)dx,
we have the required uj.
10.4
STABLE MATCHING THEORY
We now turn to stable matching theory. Stable matchings ﬁnd very important
normative applications in economics, but the theory provides a basic predictive
framework as well. Many markets, such as labor markets and the marriage


--- Page 12 ---
154
Game Theory
“market,” have two sets of agents who pair up and who may have preferences
over who they form a pair with. We describe the basic notion of a stable
matching, and carry out a simple revealed preference exercise.
Our version of the model assumes a set M of types of men and a set W of
types of women. The sets M and W are ﬁnite and disjoint. We assume a number
Km of men of type m, and Kw of women of type w. The primitives of the model
are then given by a tuple ⟨M,W,P,K⟩, in which M and W denote sets as before,
K = (Ki)i∈M∪W is a list of non-negative integers, and P is a preference proﬁle:
a list of preferences >m for every m ∈M and >w for every w ∈W. Each >m is
a linear order over W, and each >w is a linear order over M.5
The standard prediction concept, or theory of which matchings to expect, is
the notion of stable matching. A matching is an |M| × |W| matrix X = (xm,w)
such that xm,w ∈Z+, 
w xm,w = Km for all m, and 
m xm,w = Kw for all
w. The number xm,w is the number of men of type m matched to women of
type w.
Stability requires the deﬁnition of blocking. A pair (m,w) ∈M × W is a
blocking pair for X if there are m′ and w′ such that m >w m′, w >m w′, xm,w′ >
0, and xm′,w > 0. The matching X is stable if there are no blocking pairs for
X. To keep the presentation simple, we ignore individual rationality and single
(non-matched) agents.
A matching X is stable-rationalizable if there exists a preference proﬁle
P = ((>m)m∈M,(>w)w∈W) such that X is a stable matching in ⟨M,W,P,K⟩.
A (undirected) graph is a pair G = (V,L), where V is a set and L ⊆V ×V is a
non-reﬂexive and symmetric binary relation on V. Elements of V are referred
to as vertices and elements of L as edges. A path in G is a sequence p =
⟨v0,...,vN⟩such that (vn,vn+1) ∈L for all n ∈{0,...,N −1}. We denote by
v ∈p that v is a vertex in p. A path ⟨v0,...,vN⟩connects the vertices v0 and vN.
A path ⟨v0,...,vN⟩is minimal if there is no proper subsequence of ⟨v0,...,vN⟩
which also connects v0 and vN.
A cycle in G is a path c = ⟨v0,...,vN⟩with v0 = vN. A cycle is minimal if
for any two vertices vn and vn′ in c, the paths in c from vn to vn′ and from vn′ to
vn are distinct and minimal. If c and c′ are two cycles, and there is a path from
a vertex in c to a vertex in c′, then we say that c and c′ are connected.
For a matching X, we consider the graph deﬁned by letting the vertices be
all the nonzero elements of X; and by letting there be an edge between two
vertices when they lie on the same row or column of X. Formally, to each
matching X we associate a graph (V,L) deﬁned as follows. The set of vertices V
is {(m,w) : m ∈M,w ∈W such that xm,w > 0}, and an edge ((m,w),(m′,w′)) ∈L
is formed for every pair of vertices (m,w) and (m′,w′) with m = m′ or w = w′
(but not both).
5 The most basic model assumes that there is only one agent of each type, but the revealed
preference question is more interesting with many agents of each type.


--- Page 13 ---
10.4 Stable matching theory
155
Theorem 10.9
A matching is stable-rationalizable iff its associated graph
does not contain two connected distinct minimal cycles.
We omit the proof of Theorem 10.9, but the intuition behind the necessity
direction is simple, and worth conveying here. Consider for example the
matching on the left below:
11
9
10
0
22
41
13
91
0
11
9
10
0
22
41
13
91
0
The matching has three types of men and three types of women. There are 11
men of type 1 matched to women of type 1, 9 men of type 1 are matched to
women of type 2, and so on. The graph associated to this matching contains
a cycle, shown on the right. In fact, it has more than one cycle, and they are
connected, but focus now on the cycle depicted.
If we are to ﬁnd rationalizing preferences, we need to decide how men of
type 1 rank women of type 1 and 2. Say that women of type 2 are preferred
to 1; we can denote this preference by orienting the horizontal edge on the
graph as pointing from 11 to 9. Now consider the preferences of women of
type 2. Is it possible for them to rank men of type 1 above men of type 3?
The answer is negative, as there are type 2 women matched to type 3 men, and,
simultaneously, type 1 men matched to type 2 women. If women of type 3 were
to prefer type 1 men, then some of the latter would form blocking pairs with
type 1 men who are matched to type 1 women. Thus the vertical edge from 9
to 91 must point down, away from the direction of the ﬁrst edge, which points
from 11 to 9. In fact, the important implication of stability is that for any two
consecutive edges which form a right angle in the graph, one must be oriented
to point away from the other.
The reasoning above extends to the horizontal edge between 91 and 13, and
then to the vertical edge between 13 and 11. The former must point to the left,
and the latter must then point up. As a result, a cycle must be oriented as a
“ﬂow.” If we start, as we did, by having type 1 men prefer type 2 to type 1
women, then we obtain a clockwise ﬂow. If we instead had started with the
opposite preference, then we would have obtained a counterclockwise ﬂow.
Now, we can consider a path leaving the cycle, for example the horizontal
edge between 11 and 10 in the graph. An orientation of this edge amounts to
specifying type 1 men’s preferences between women of type 1 and 3. Recall
that we have oriented the cycle in a clockwise fashion, so that the edge between
13 and 11 points up. Then the edge leaving the cycle, and going from 11 to 10,
cannot be oriented to the left. The reason is that we would then have a blocking
pair using some of the 10 men of type 1 who are matched with women of type


--- Page 14 ---
156
Game Theory
3, and some women of type 1 who are matched to men of type 3. The general
principle is that any path leaving a cycle must point away from the cycle.
It should now be clear that we cannot have two cycles connected by a
minimal path. Such a path would have to point away from both cycles. Then
there would be two consecutive edges on the path, such that each edge point
to their common vertex. This situation would imply the existence of some
blocking pair, just as in the examples we discussed above.
10.5
CHAPTER REFERENCES
Persistence under expansion was introduced by Yanovskaya (1980), while
Theorem 10.1 is a generalization of her result due to Sprumont (2000).
Persistence under expansion and persistence under contraction are closely
related to the consistency concepts discussed by Peleg and Tijs (1996).
Persistence under expansion is clearly related to condition γ from social
choice theory (see, e.g., Sen, 1971). The ﬁrst part of persistence under
contraction is similar to condition α for single-agent choice functions, while
the second part is related to condition β. Theorems 10.2, 10.3, and 10.4 are
also from Sprumont (2000). Ray and Zhou (2001) establishes related results
for extensive-form games and subgame perfection. Xu and Zhou (2007),
Bossert and Sprumont (2013), Rehbeck (2014), and Xiong (2013) study the
extensive-form question when the game itself is not observable. Instead, the
primitive is a classical choice function.
Theorem 10.5 and the notion of N-congruence are due to Galambos (2010).
Galambos (2010) describes a more sophisticated version of Theorem 10.5
which appears in his dissertation. It need not rely on the assumption of a
line-closed domain.
Theorem 10.6 is due to Lee (2011). Interchangeability is a well-known
property of the Nash equilibria of zero-sum games. The contribution in Lee’s
work is to show that it is all that the property of zero-sum adds, from the
revealed preference perspective.
Theorem 10.7 is due to Ledyard (1986). More complicated theorems
appear there, dealing with certain restrictions on the forms of the ui and pi
functions. For example, he obtains the necessary and sufﬁcient conditions
required for a direct revelation mechanism to be strongly rationalized by
a Bayesian environment when the ordinal, but not cardinal, structure of
each ui is known conditional on each type proﬁle t (as would be the case
in standard private-valued single-dimensional consumption environments).
Such characterizations are based on Lemma 1.12. However, a few important
questions seem to remain unresolved. More importantly, the results described
here consist of a single observation. It is not terribly surprising that anything is
rationalizable in this case. For an analogy with choice theory, a choice function
deﬁned only on one budget is always rationalizable, unless there is sufﬁcient
structure on the class of rationalizing relations. An interesting idea would be


--- Page 15 ---
10.5 Chapter references
157
to understand what happens when multiple observations are possible, possibly
when some parameters of the environment change, but others remain ﬁxed.
In fact, a simple method of doing this would be to consider strategy spaces
Si, one for each agent, and consider Bayesian strategies σi : Ti →Si. It is
then meaningful to discuss notions of strict Bayesian Nash equilibrium, and
by varying the spaces of strategies, one may come up with interesting testable
implications.
Section 10.3 is based on Chambers and Echenique (2014b). The paper
includes results for the cases where disagreement points may vary in an
observable way (when disagreement points are unobservable the theories
become non-testable). The proof of Theorem 10.8 and the observation that
the same list of utility functions works to rationalize each environment, taken
here from Chambers and Echenique (2014b), were suggested to us by an
anonymous referee. For a continuous version of the problem, see Chiappori,
Donni, and Komunjer (2012). We discuss other approaches in Chapter 12.
Theorem 10.9 and the discussion in Section 10.4 is taken from Echenique,
Lee, Shum, and Yenmez (2013). That paper also includes results on which
matchings are rationalizable as optimal for one side of the market, and
rationalizable with transfers. These notions turn out to be observationally
equivalent: a matching is rationalizable using monetary transfers iff it is
rationalizable as stable and optimal for one side of the market (men or women).
We have assumed that observed matches consist of matrices of non-negative
integers. We can instead suppose that multiple matching among the same
individuals (or types of individuals) are observed. Then we can insist on
rationalizing preferences that make all of them stable. This exercise is carried
out in Echenique (2008). The same problem for a model with transfers is
worked out in Chambers and Echenique (2014a).
