--- Page 1 ---
CHAPTER 12
Revealed Preference and Systems of
Polynomial Inequalities
It should be apparent by now that systems of linear inequalities emerge
naturally in revealed preference theory. They constitute the essence of Afriat’s
Theorem, for example; and we formulated revealed preference problems using
systems of linear inequalities in Chapters 3, 6, 7, and 8. In this chapter, we
describe how revealed preference problems can generally be understood as
a system of inequalities. From a purely computational perspective, one can
very often solve a revealed preference problem by algorithmically solving the
corresponding system of inequalities.
When the system of inequalities is linear, the problem is easy to solve both
computationally and analytically. Here we develop a GARP-like acyclicity test
(similar to the ones in Chapters 2 and 3). The test will follow from the linearity
of the system of inequalities embodied in the revealed preference question.
We shall discuss an extension of the linear theory to systems of polynomial
inequalities. The theory of polynomial inequalities will be seen to be very
relevant for revealed preference theory (but harder to work with compared to
the theory of linear inequalities).
12.1
LINEAR INEQUALITIES: THE THEOREM OF THE
ALTERNATIVE AND REVEALED PREFERENCE
We start by revisiting the Theorem of the Alternative, or Farkas’ Lemma
from Chapter 1. It is easy to see why it is useful in revealed preference
theory. We then discuss some sources of linear systems for popular models
in economics. The following is a bit weaker than Lemma 1.13. It is written so
as to emphasize that the lemma can be used to “remove existential quantiﬁers;”
it states that an existential statement (one that start with “there is. . . ”) is
equivalent to a universal statement (one that starts with “for all. . . ”). Note that
the discussion of the Tarski–Seidenberg Theorem in Chapter 9 is also about
removing existential quantiﬁers.
Lemma 1.13′
(Integer–Real Farkas) Let {Ai}M
i=1 be a ﬁnite collection of
vectors in QK. The following statements are equivalent:


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176
Revealed Preference and Systems of Polynomial Inequalities
I) There exists y ∈RK such that for all i = 1,...,M, Ai · y > 0.
II) For all z ∈ZM
+ \ {0}, it holds that M
i=1 ziAi ̸= 0.
The vector y represents some unknown quantities. The vectors Ai encode
some properties we want y to have. Typically, y is an unobservable theoretical
object, such as a utility function. The properties in Ai can be theoretical (for
example requiring utility to be monotonic) or come from the data, for example
requiring the utility function to rationalize the data. In the example that we
expand on below, M is the number of observations, and K is the number of
possible alternatives from which an agent chooses. Then every observation
corresponds to a vector Ai. For example, suppose that we observe object h
chosen over object l. Then there is an Ai of the form 1h −1l. See Section 12.1.2
below for more details. The existence of y ∈RK satisfying the inequalities then
translates into the existence of a utility function rationalizing the data.
System I corresponds to the revealed preference formulation of a problem:
A dataset is “rationalizable” if there exists a value for the theoretical object
y that satisﬁes the right properties and explains the data. We call this an
existential formulation of a theory. As stated, it is not falsiﬁable: If one is given
a candidate solution y, it is easy to check whether System I is satisﬁed. If the
candidate is a solution then we are done, but if the candidate y fails to satisfy
System I then we have essentially no clue as to whether System I is solvable
because there are inﬁnitely many other potential solutions to the system. So
when System I has no solution, and the dataset is “not rationalizable,” then
there is no way that checking individual vectors y one by one can tell us that
the system is not solvable.
In contrast II is a universal statement. It says that something has to be true
for all vectors z. Given a single z ∈ZM
+ \ {0} with M
i=1 ziAi = 0, we know by
Lemma 1.13′ that there cannot exist y solving System I. Despite the existential
formulation of the theory, the theory is falsiﬁable. When System I has no
solution, and the dataset is “not rationalizable,” then no single y can certify
that the theory has been falsiﬁed, but thanks to Lemma 1.13′, a single z can do
that.
Lemma 1.13′ says a bit more about z. It says that it is a vector of integers.
This is very often useful in ﬁnding a “combinatorial” revealed preference
axiom, such as GARP or SARP.
Note that veriﬁcation is, in some sense, a dual property to falsiﬁcation. The
existential formulation System I means that an individual y cannot certify that
the system has no solution. But when the system does have a solution, a single
vector y is enough to certify that a solution exists.
12.1.1
Linear systems from ﬁrst-order conditions
First-order conditions are a common source of systems of inequalities in
revealed preference theory. For example, in the revealed preference problem
of rational demand theory, we obtain the system described in Afriat’s Theorem


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12.1 Linear inequalities: The Theorem of the Alternative
177
from the ﬁrst-order conditions in the consumer’s maximization problem. The
following is a heuristic derivation of the system of linear inequalities in Afriat’s
Theorem.
Recall the setup in Chapter 3. We observe a dataset (xk,pk), k = 1,...,K. We
want to know when there is a function u : Rn
+ →R such that xk is a maximizer
of u in the budget set B(pk,mk) = {x ∈Rn
+ : pk · x ≤mk}, where mk = pk · xk.
If such a u exists, and if it is smooth and concave, then the ﬁrst-order
condition must be satisﬁed at the choices xk:
∇u(xk) −λkpk = 0,
(12.1)
where ∇u(xk) denotes the gradient of u at xk, and λk is the Lagrange multiplier
associated to this maximization problem. The unknowns here are the numbers
∇u(xk) and λk.
We need to ﬁnd values for these unknowns that are compatible with a
concave u. Essentially we can solve for another set of numbers, utility values,
Uk = u(xk), which can be extended to a function deﬁned on all of Rn
+.
Concavity demands the following inequality be satisﬁed for all k and l:
Ul −Uk ≤∇u(xk) · (xl −xk).
(12.2)
Putting the equations of type (12.1), which result from the ﬁrst-order
conditions, together with inequalities of type (12.2) yields the linear system
we solve for in Afriat’s Theorem. By using Lemma 1.13 with this system one
obtains GARP: see 3.1.1.
Another example can be obtained from Cournot oligopoly theory. Suppose
that there are n ﬁrms in a market for a homogeneous good. The theory says that
market price is determined as if each ﬁrm i has a cost function ci : R+ →R,
with ci(0) = 0, and chooses quantity qi to maximize
qiP
⎛
⎝
n

j=1
qj
⎞
⎠−ci(qi),
where P : R+ →R+ is the market inverse demand function. The quantity
qiP
#n
j=1 qj
$
is the revenue of ﬁrm i, so the expression reﬂects that i
maximizes proﬁts.
Suppose that we observe K instances in which these ﬁrms were engaged in
quantity competition. The data we observe are of the form:
(qk
1,...,qk
n,pk) : k = 1,...K;
where qk
i ≥0 is the quantity chosen by ﬁrm i in instance k, and pk is the
prevailing market price. The functions ci and P are not observable.
Now, say that a dataset is Cournot rationalizable if there are convex
functions ci and a smooth and decreasing function P such that qk
i maximizes
the proﬁts of ﬁrm i, given that all other ﬁrms choose the quantities in the
vector (qk
j ).


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Revealed Preference and Systems of Polynomial Inequalities
Now, the ﬁrst-order condition for ﬁrm i is:
qk
i P′
⎛
⎝
n

j=1
qk
j
⎞
⎠+ P
⎛
⎝
n

j=1
qk
j
⎞
⎠−c′
i(qk
i ) = 0.
Recall that price pk is observed as part of the data, so the unknowns are the
numbers P′(n
j=1 qk
j ) and c′
i(qk
i ). One needs to ﬁnd cost and demand functions
such that pk = P(n
j=1 qk
j ), and derivatives behave properly. As with Afriat’s
Theorem, it is enough to do so by ﬁnding certain real numbers: in our case the
numbers P′(n
j=1 qk
j ) and c′
i(qk
i ), such that certain inequalities are satisﬁed.
We use the notation δk
i for the number c′(qk
i ). The ﬁrst-order condition
implies that
δk
j −pk
qk
j
= δk
i −pk
qk
i
< 0,
(12.3)
for all i, j, and k, as P′(
j qk
j ) does not depend on i and must be negative for
demand to slope down.
It is possible to show that data are Cournot rationalizable iff there are
numbers δk
i , i = 1,...,N, k = 1,...,K such that the linear inequalities (12.3)
are satisﬁed, and such that when qk
i < ql
i then δk
i < δk
j ; the latter requirement
captures that marginal cost must be monotone increasing for cost functions to
be convex.
12.1.2
The existence of a rationalizing utility
We shall illustrate the role of linear inequalities with a very basic example,
dealing with the simplest version of revealed preference for individual decision
making.
Suppose that we are given a ﬁnite set X of alternatives. Suppose we are given
a dataset in the form of an observed revealed preference relation ⪰R. That is,
we observe a set of binary comparisons, where an agent has chosen x over y
iff x ⪰R y. We want to know when R is rationalizable using a utility function
u : X →R. Suppose that the relation R encompasses only strict comparisons,
so x ⪰R y and x ̸= y implies that x ≻R y. Then a utility function u rationalizes
⪰R if u(x) > u(y) whenever x ≻R y.
The problem can be set up as follows. A utility function is a vector in
Euclidean space RX. Deﬁne a matrix B with |X| columns. For every pair
(x,y) ∈≻R, include a row in B which has zeroes in all entries except for a
1 in the column for x and a −1 in the column for y. So the row is the vector
1x −1y. Then there is a utility function that rationalizes the data if and only if
there is a vector u ∈RX that solves the system B·u ≫0 of linear inequalities.1
1 The matrix B is like the upper left submatrix of the matrix constructed in the proof of Afriat’s
Theorem.


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12.1 Linear inequalities: The Theorem of the Alternative
179
By Lemma 1.13′ there is no rationalizing u (that is, no u such that B·u ≫0)
if and only if there is z > 0 such that z · B = 0. So suppose that that is the case,
and let z be the vector promised by Lemma 1.13′. Let (x,y) ∈X ×X correspond
to a row r for which zr > 0; so x ≻R y. There is at least one such pair because
z > 0. There is a 1 in the column for x in row r. Now, no entry of z is negative,
and z·B = 0, so there must be some row r′ with zr′ > 0 in which we have a −1
in the column for x. So there must exist x′ with x′ ≻R x; the row r′ corresponds
to (x′,x). Note that now x′ is in the position that x was in when we started to
make this argument. So there must exist x′′ ∈X with
x′′ ≻R x′ ≻R x ≻R y.
Since X is a ﬁnite set, by repeating the argument we must reach a cycle of ⪰R.
The argument shows that the order pair ⟨⪰R,≻R⟩is not acyclic.
In conclusion, the non-existence of a solution to the system u · B ≫0 is
equivalent to the existence of a cycle of ⪰R. The standard result on acyclicity
therefore emerges as a consequence of Lemma 1.13.
The same reasoning can be applied to many of the results in Chapter 2. We
sketch proofs of Theorems 2.6, 2.8, 2.16, and 2.17 using this device. In fact, the
Theorem of the Alternative is often a useful way of “guessing” the appropriate
concept of rationalization speciﬁed by a particular choice-theoretic axiom.
Consider Theorem 2.6. Suppose that X is a ﬁnite set. Rationalizability in the
sense of Theorem 2.6 is equivalent to the existence of a function u : X →R
for which x ⪰c y implies u(x) ≥u(y) and x ≻c y implies u(x) > u(y). One
can then proceed almost exactly as we did in the previous paragraphs to
obtain the acyclicity result. The only difference is that we now have weak
revealed preference comparisons. Let the order pair ⟨⪰c,≻c⟩be as deﬁned in
Chapter 2. Construct a matrix B with |X| columns, and a row for each ⪰c or
≻c comparison. For comparisons of the type x ⪰c y, we add a row to the matrix
of form 1x −1y; likewise for comparisons of type x ≻c y. We want to ﬁnd a
solution u ∈RX such that for rows Bi of type ⪰c, we have Bi · u ≥0, and for
rows of type ≻c, we have Bi · u > 0. Applying Lemma 1.13 and using similar
techniques as in the preceding proof for removing cycles, we come up with
congruence as the necessary and sufﬁcient condition.
Theorem 2.8 is a bit different. We can no longer work with a utility function,
as it is clear that not every complete binary relation can be so encoded (unless
it is transitive, of course). Instead, we encode the preference via a function
ϕ : X2 →R for which ϕ(x,y) = −ϕ(y,x). The interpretation here is that x ⪰y
iff ϕ(x,y) ≥0. This suggests a matrix B, whose rows are indexed by X2, and
for each pair x,y, there is a row 1(x,y) + 1(y,x). Now, we also add a row for each
comparison of the type x ⪰c y; namely, this row consists of 1(x,y); likewise, we
add a row for each comparison of the type x ≻c y, and again, this row consists
of 1(x,y). We want to ﬁnd a solution ϕ ∈RX2 such that for rows Bi of the ﬁrst
type, we have Bi ·ϕ = 0, for rows Bi of the second type, we have Bi ·ϕ ≥0, and
for rows Bi of the third type, we have Bi ·ϕ > 0. Again, a simple application of
the Theorem of the Alternative leads to the weak axiom of revealed preference.


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Revealed Preference and Systems of Polynomial Inequalities
Theorems 2.16 and 2.17 can be proved using similar techniques to the
preceding – the distinction is that in the case of Theorem 2.16, we search for
a u ∈RX for which x ≻c y implies u(x) > u(y) (no requirement is made for
x ⪰c y), and in the case of Theorem 2.17, we search for a ϕ ∈RX2 for which
x ≻c y implies ϕ(x,y) > 0 (again no requirement is made for x ⪰c y).
The general form of the requirement in these theorems is something along
the following lines: “There exists some function such that for all pairs,
the function is related to revealed preference in some way.” The existential
statement quantiﬁes an unobservable object. The universal (for all) statement
quantiﬁes observable objects. The Theorem of the Alternative is used to show
that these statements are logically equivalent to a statement which is universal
and stated only in terms of observable objects; for example, Theorem 2.6 states
that existence of such a utility is equivalent to the statement: “For all x1,...,xn,
it is not the case that for i = 1,...,n −1, xi ⪰c xi+1 and xn ≻c x1.” Though
existential statements are in principle problematic from the viewpoint of
falsiﬁability, we see that in general this might not be true when the existential
operator quantiﬁes unobservable objects. Existential operators operating on
unobservables can sometimes be “removed,” and turned into an equivalent
universal statement involving only observables.
12.2
POLYNOMIAL INEQUALITIES: THE
POSITIVSTELLENSATZ
Some revealed preference problems give rise to nonlinear, speciﬁcally
polynomial, systems of inequalities. Below we give a detailed example taken
from Nash bargaining theory (see Chapter 10). For polynomial systems of
inequalities, there is a result like the Theorem of the Alternative; it is called
the Positivstellensatz.
In order to state the Positivstellensatz, we need a bit of notation. Given is a
collection of variables, say {x1,...,xn}. We assume that the notion of a polyno-
mial is understood. We will describe one version of the Positivstellensatz.
Given a collection of polynomials {f1,...,fm} over the variables {x1,...,xn},
we deﬁne the ideal of {f1,...,fm} to be the collection of all polynomials which
can be written in the form:
m

i=1
gifi,
where gi is a polynomial. We deﬁne the cone generated by f1,...,fm to be
the smallest set of polynomials that (1) includes all sums of squares of
polynomials, (2) includes all polynomials f1,...,fm, and (3) is closed under
addition and multiplication. It is easy to see that any such element can be
written as

S⊆{1,...,m}
1
gS
"
i∈S
fi
2
,


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12.2 Polynomial inequalities: The Positivstellensatz
181
where gS is a sum of squares of polynomials. Finally, we deﬁne the
multiplicative monoid generated by f1,...,fm to be the collection of polynomials
of the form m
i=1 f ai
i , where each ai is a non-negative integer.
Let fi, gj, and hl be polynomials over {x1,...,xn}, for i = 1,...,m, j = 1,...,k,
and l = 1,...,q.
Theorem 12.1 (Positivstellensatz).
A collection of inequalities fi(x) = 0, i =
1,...,m, gi(x) ≥0, i = 1,...,k, hl(x) ̸= 0, i = 1,...,q has no solution iff there exist
polynomials f in the ideal of {f1,...,fm}, g in the cone generated by {g1,...,gk},
and h in the multiplicative monoid generated by {h1,...,hq} for which f + g +
h = 0.
The Positivstellensatz thus provides an alternative system of polynomial
inequalities, such that the ﬁrst system has no solution iff the second system has
a solution. The statement is similar in spirit to the Theorem of the Alternative,
but the unknowns in the second system are polynomials, not real numbers.
From a computational perspective, the problem is clearly harder to deal with.
Conceptually, the Positivstellensatz has implications that are similar to those
we highlighted for the Theorem of the Alternative in Section 12.1: it turns
an existential, veriﬁable theory into a universal, falsiﬁable statement. Thus,
if a system of polynomial inequalities cannot be satisﬁed, it is possible to
demonstrate, or certify, its infeasibility.
To actually ﬁnd a falsiﬁcation is challenging computationally. One practical
approach is to search for a solution to the second system of inequalities among
polynomials of bounded degree. This approach is common in the engineering
literature (see the references in Section 12.3). So one would look for f in
the ideal of {f1,...,fm}, g in the cone generated by {g1,...,gk}, and h in the
multiplicative monoid generated by {h1,...,hq} for which f + g + h = 0; but
restricting such a search to polynomials of degree smaller than some given
bound. The search for a solution among polynomials of bounded degree can
be formulated as a semideﬁnite program, and thus there are efﬁcient algorithms
for solving the problem.
Of course, the approach only works when one ﬁnds a solution: when one
ﬁnds polynomials certifying a solution f + g + h = 0, thus certifying that the
ﬁrst system of inequalities (the one we are really interested in) has no solution.
If one does not ﬁnd a solution among polynomials of bounded degree, then it
is possible that there is a solution of a higher degree, or that the original system
in fact has a solution.
12.2.1
Application: Nash bargaining
To get a sense for how these ideas might be applied in economics, let us
consider the problem of testing Nash bargaining theory, as in Section 10.3
of Chapter 10. In Chapter 10, we assumed that the disagreement point was
ﬁxed across observations, and that it was the same for all agents. If we relax
that assumption then things get more complicated. We can still formulate the


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182
Revealed Preference and Systems of Polynomial Inequalities
problem as a polynomial system of inequalities and use the Positivstellensatz
to obtain an answer. The alternative, or dual, system from the Positivstellensatz
gives a test for Nash bargaining theory.
Assume a ﬁnite set N of agents is given. Generalizing the setup from
Section 10.3, suppose that a dataset is a set D = {(xk,dk) : k = 1,...,K}; where
for each k, xk and dk are vectors in RN
+ such that xk ≫dk ≥0. These are
observations of K bargaining instances; in the kth instance, agent i received
xk
i dollars, while his disagreement outcome was dk
i .
A dataset is Nash bargaining rationalizable if there are concave and
monotonic functions ui : R+ →R, i ∈N, such that
n
"
i=1

ui(xk
i ) −ui(dk
i )

≥
n
"
i=1

ui(xi) −ui(dk
i )

,
for all x ∈RN
+ such that xi ≥dk
i and 
i∈N xi ≤
i∈N xk
i .
Proposition 12.2
A dataset D is Nash rationalizable if and only if for all
i ∈N, there are numbers Ui(dk
i ),Ui(xk
i ),Mi(dk
i ), and Mi(xk
i ) for k = 1,...,K
which solve the following equations: for all z ∈{xk
i }, z′ ∈{xk
i ,dk
i }, and all i, j,
and k,
Mi(xk
i )
Ui(xk
i ) −Ui(dk
i ) =
Mj(xk
j )
Uj(xk
j ) −Uj(dk
j )
and for all z,z′ ∈K
i=1{dk
i ,xk
i }

Ui(z) −Ui(z′) > 0
if z < z′,
Mi(z′)(z −z′) ≥Ui(z) −Ui(z′).
Proposition 12.2 is straightforward. We simply ask for numbers Ui(z) to
signify levels of utility, and Mi(z) for supergradients, or marginal utilities.
The ﬁrst system of equalities ensures that the ﬁrst-order conditions for the
maximization of the Nash product hold. The second set of inequalities make
sure that utility is monotonic and that marginal utilities are a supergradient of
the utility.
Proof. We ﬁrst show that if we are given increasing and concave utility
functions ui, then xk
1,...,xk
n is a solution to max
i∈N xi=M

i∈N[ui(xi) −ui(dk
i )]
if and only if for each i, there is a supergradient μi of ui at xk
i for which
μi
Ui(xk
i ) −Ui(dk
i ) =
μj
Uj(xk
j ) −Uj(dk
j ).
To this end, deﬁne U = {

u1(x1) −u1(dk
1),...,un(xn) −un(dk
n)

: 
i∈N xi =
M,xi ≥dk
i }, and consider maximizing the function f(y) = 
i∈N yi subject to
y ∈U. A point u ∈U maximizes f if and only if
#
i̸=1 ui,...,
i̸=n ui
$
supports
U at u (by deﬁnition). Because f is strictly convex, and since U is convex and


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12.2 Polynomial inequalities: The Positivstellensatz
183
compact, there is a unique such maximizer u∗. It is clear that u∗
i > 0 for all
i ∈N.
This states that there is a unique solution xk
1,...,xk
n to the Nash problem
for which ui(xk
i ) −ui(dk
i ) = u∗
i . We deﬁne λj = 
i̸=j[ui(xk
i ) −ui(dk
i )]. We
know that 
i∈N λiui(xi) is maximized at xk
1,...,xk
n across all xi for which

i∈N xi = M. Our next step is to show that this can happen if and only if the
vector (1/λ1,...,1/λn) is proportional to a vector of supergradients.
Since the constraints xi ≥dk
i are not binding, we can set up the Lagrangian
for the problem, say L(x,μ) = 
i∈N λiui(xi) + μ(M −
i∈N xi), and note that
it is equal to L(x,μ) = 
i∈N[λiui(xi) −μxi] + μM. We know the constraint

i∈N xi = M is binding, so that the solution to the maxmin problem features
μ∗> 0. For μ∗, we know that maxx L(x,μ∗) is equal to the maximum Nash
product subject to the constraint, and has the same solution. This is equivalent
to saying that (λi/μ∗)ui(xk
i ) −xk
i ≥(λi/μ∗)ui(xi) −xi for all xi, or, rewriting:
ui(xi) + (μ∗/λi)(xk
i −xi) ≤ui(xk
i ).
This is equivalent to saying that μ∗/λi is a supergradient, or that the vector
(1/λ1,...,1/λn) is proportional to a supergradient.
Another way of saying that (1/λ1,...,1/λn) is proportional to a vector of
supergradients is saying that for all i ∈N, there is a supergradient Mi(xk
i ) of
ui at xk
i such that for all i,j, λi
λj =
Mj(xk
j )
Mi(xk
i ). Writing out the explicit form of λ and
eliminating terms, this is equivalent to saying that
Mi(xk
i )
ui(xk
i )−ui(dk
i ) =
Mj(xk
j )
uj(xk
j )−uj(dk
j ),
which is precisely the condition in the theorem. The other conditions simply
say that Mi is a supergradient, and that ui is strictly increasing.
Conversely, the details of how to construct a utility function from these
numbers essentially follow from Afriat, deﬁning ui(x) = infz∈K
k=1{xk
i ,dk
i } Ui(z)+
Mi(z)(x −z), where the inﬁmum is taken over all data points. It is then simple
to verify by construction that for all z ∈K
k=1{xk
i ,dk
i }, Mi(z) is a supergradient
of ui at z. From this, the fact that the equality in the statement of the theorem is
solved implies that the Nash product is maximized for this collection of utility
functions (by the previous argument).
From Proposition 12.2 and the Positivstellensatz we can obtain a test for Nash
bargaining. The conditions in the proposition involve polynomials in 4|N|K
variables, namely Ui(dk
i ),Ui(xk
i ),Mi(dk
i ), and Mi(xk
i ). We have the system of
polynomial equalities
Mi(xk
i )
Ui(xk
i ) −Ui(dk
i ) =
Mj(xk
j )
Uj(xk
j ) −Uj(dk
j )
and the inequalities for all z,z′ ∈K
i=1{dk
i ,xk
i },

Ui(z) −Ui(z′) > 0
if z < z′,
Mi(z′)(z −z′) ≥Ui(z) −Ui(z′).


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Revealed Preference and Systems of Polynomial Inequalities
12.3
CHAPTER REFERENCES
In general, Lemma 1.13 allows us to ﬁnd the exact empirical content of many
linear models. Scott (1964) is a classic reference. The issue with Lemma 1.13
is that the universal quantiﬁer does not typically operate on observables (here, z
is simply a vector – but in our example, observed data were revealed preference
comparisons). It turns out though, that since z is integer-valued, this universal
quantiﬁer can be translated directly into observables. For example, in the
revealed preference example in Section 12.1.2, the fact that for all for all z ∈ZK
+
with K
i=L+1 zi > 0, we have K
i=1 ziAi ̸= 0 is the same as saying there are no
preference cycles. In fact, it is often the case that one can require the universal
quantiﬁer on z to operate over a ﬁnite number of z.
The discussion of the Cournot model in Section 12.1.1 borrows from the
paper of Carvajal, Deb, Fenske, and Quah (2013). These authors prove that
the condition we have stated (the existence of a solution to the system of
inequalities) is equivalent to Cournot rationalizability.
The discussion of existential and universal axioms is inspired by the work
of Popper (1959). Popper claimed famously that existential theories are not
falsiﬁable: consider the theory that claims that there is a black swan. No matter
how many non-black swans one observes, they do not disprove the theory.
Universal theories are falsiﬁable, for example the theory that claims that all
swans are white. The observation of a single non-white swan would disprove
the theory. The papers by Van Benthem (1976) and Chambers, Echenique,
and Shmaya (2012) present a general result on how existential quantiﬁers over
theoretical objects can be removed, and a theory may be shown to be falsiﬁable.
See also Craig and Vaught (1958), Lemma 3.4 for a related result in a more
restrictive environment.
The Positivstellensatz as formulated in Theorem 12.1 can be found, for
example, in Bochnak, Coste, and Roy (1998), Theorem 4.4.2. It is originally
due to Krivine (1964) and rediscovered by Stengle (1974). It is interesting that,
while many economists know and apply the Theorem of the Alternative, there
are almost no applications of the Positivstellensatz (Richter, 1975 is a notable
exception, which itself builds on the work of Tversky, 1967). The theorem
is potentially useful to applied economists, who would use the algorithms in
Parrilo (2004) to carry out tests on actual datasets.
The Positivstellensatz is closely related to, but distinct from, the
Tarski–Seidenberg Theorem; see our discussion in Chapter 9. Brown
and Matzkin (1996) (see also Brown and Kubler (2008)) exploit this technique
to ﬁnd testable implications of equilibrium behavior. They also explain how
the well-known equivalence of the strong axiom of revealed preference and
rationality is a special case of Tarski–Seidenberg.
The approach of searching among polynomials of bounded degree is
described in Parrilo (2003). Indeed, it can be shown to revert to a classical
semideﬁnite programming problem. This approach is outlined in Parrilo
(2003), a shorter introduction is provided in Parrilo (2004); see especially


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12.3 Chapter references
185
Example 1 there. Marshall (2008), Chapter 10 provides a detailed explanation.
Also see Sturmfels (2002).
Proposition 12.2 is from Chambers and Echenique (2014b). There are
similar results in Carvajal and Gonz´alez (2014) and Cherchye, Demuynck,
and De Rock (2013). Cherchye, Demuynck, and De Rock (2013) propose
a different computational approach than the one we have emphasized here.
They propose instead using integer programming to obtain a test for Nash
bargaining.
