We provide an example of a data set where all the revealed preference relations seem to be consistent with single crossing differences and yet the revealed preference relations cannot be extended to a complete preference obeying that propert

Kukushkin, Nikolai S.

Quah, John K.-H.

Shirai, Koji

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
A counterexample on the completion of
preferences with single crossing
differences
Nikolai S. Kukushkin and John K.-H. Quah and Koji Shirai
Dorodnicyn Computing Centre, FRC CSC RAS, Department of
Economics, Johns Hopkins University, School of Economics, Kwansei
Gakuin University
16 September 2016
Online at https://mpra.ub.uni-muenchen.de/73760/
MPRA Paper No. 73760, posted 18 September 2016 09:08 UTC
A counterexample on the completion of preferences with
single crossing differences
Nikolai S. Kukushkin, John K.-H. Quah, and Koji Shirai
September 15, 2016
Abstract
We provide an example of a data set where all the revealed preference rela-
tions seem to be consistent with single crossing differences and yet the revealed
preference relations cannot be extended to a complete preference obeying that
property.
Keywords: monotone comparative statics, single crossing differences, interval
dominance, supermodular games, lattices
JEL classification numbers: C6, C7, D7
1 Introduction
To motivate the counterexample presented in this note, we first give a quick review of some
concepts and results in monotone comparative statics. Let X  R be the set of all possible actions
of an agent and let Ξ be a partially ordered set which we shall interpret as a set of exogenous
parameters that may affect the agent’s preference over the actions in X. By a preference we
mean a binary relation Á on X  Ξ such that for every ξ P Ξ, it is reflexive, complete, and
 Addresses: Dorodnicyn Computing Centre, FRC CSC RAS, 40, Vavilova, Moscow 119333 Russian Federation
[ququns@inbox.ru]; Department of Economics, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD
21218, USA [john.quah@jhu.edu]; School of Economics, Kwansei Gakuin University, 1-155 Uegahara Ichiban-cho,
Nishinomiya, Hyogo, Japan [kshirai1985@gmail.com]
Financial support from the Russian Science Foundation (project 16-11-10246) is acknowledged.
1
transitive on X. A preference relation Á obeys strict single crossing differences (or SSCD, for
short), if for every x2 ¡ x1 and ξ2 ¡ ξ1,
px2, ξ1q Á px1, ξ1q ùñ px2, ξ2q ¡ px1, ξq. (1)
Suppose the agent has a preference Á; for every A  X and ξ P Ξ, we denote the agent’s set of
best responses by BRpA, ξq, i.e., BRpA, ξq  tx P A : px, ξq Á px1, ξq for all x1 P Au. We say that
the best response set is increasing in ξ if for every ξ2 ¡ ξ1,
x2 P BRpA, ξ2q, x1 P BRpA, ξ1q ùñ x2 ¥ x1. (2)
Milgrom and Shannon (1994) shows that BRpξ, Aq is increasing in ξ for every set A  X if and
only if the agent’s preference Á obeys SSCD. It is possible to weaken the SSCD property if we
consider a smaller class of sets A. For example, we may restrict an agent’s feasible action sets
to intervals of X;1 then BRpA, ξq is increasing in ξ for all intervals A if and only if Á obeys the
strict interval dominance (SID) order (see Quah and Strulovici (2009)). A preference relation Á
obeys SID, if for every x2 ¡ x1 and ξ2 ¡ ξ1,
px2, ξ1q Á px, ξq for all x P rx1, x2s ùñ px2, ξ2q ¡ px1, ξ2q. (3)
Lazzati, Quah, and Shirai (2016) (henceforth LQS) have recently developed a revealed pref-
erence theory of SID. Their starting point is a finite set of observations O  tpxt, At, ξtqutPT
collected from an agent, where xt is the action chosen by the agent from a compact interval At
(contained in a set X  R of all possible actions) when ξt is the prevailing parameter value. LQS
identify a condition on O that is necessary and sufficient for O to be rationalizable by an SID
preference; this means that there exists an SID preference relation Á on X  Ξ such that, with
this preference, xt is a best response from the interval At when the parameter is ξt.
The condition obtained by LQS involves a restriction on the revealed preference relations. We
define the direct revealed preference relation ÁR on X  Ξ in the following manner: px2, ξq ÁR
px1, ξq if there exists some t P T with xt  x2, ξt  ξ, and x1 P At. The indirect revealed
1A is an interval of X if for every x2, x1 P A with x2 ¡ x1, if z P X satisfies x1   z   x2 then z P A.
2
preference relation ÁRT is the transitive closure of ÁR, i.e. px2, ξq ÁRT px1, ξq, if there exists a
finite sequence of actions z1, z2, ..., zK such that x2  z1, x1  zk, and pzk, ξq ÁR pzk 1, ξq for
every k P t1, 2, ..., K  1u. It is obvious that so long as an agent is maximizing some preference
(whether or not it is SID) then ÁRTÁ. LQS show that O can be rationalized by an SID
preference if and only if it satisfies the following property, which they call the axiom of revealed
complementarity (ARC):
ξt ¡ ξs, xt   xs, and pxt, ξtq ÁRT pxs, ξtq ùñ pxs, ξsq ÃRT pxt, ξsq. (4)
In fact, LQS reach a stronger conclusion: whenever O obeys ARC, it can be rationalized by a
preference Á obeying SSCD.
Suppose that, instead of being intervals, the observed feasible action sets At in the data set
O are arbitrary nonempty sets. In this case, ARC is no longer necessary for rationalizability by
an SID preference but it is obvious that the property is still necessary for O to be rationalized
by an SSCD preference. Given this, it would be natural to conjecture that, even allowing for
arbitrary feasible sets, ARC is both necessary and sufficient for O to be rationalized by an SSCD
preference. However, Example 2 in LQS shows that this conjecture is false.
We may think that this conjecture fails because ARC does not capture all the implications
of rationalizability by an SSCD preference and that there is a more stringent but still intuitive
and easy-to-check condition on the data set that is both necessary and sufficient for this ratio-
nalizability property. An ideal scenario would be for the characterization to take the form of
a no-cycling condition, similar to the ones used in Afriat’s (1967) Theorem or Richter’s (1966)
Theorem. Indeed, there is a natural way of constructing a condition of this type, which we shall
now explain. Given ÁRT , we define its single crossing extension as the binary relation ¡RTS such
that for x2 ¡ p qx1, px2, ξq ¡RTS px1, ξq, if there exists some ξ1   p¡qξ with px2, ξ1q ÁRT px1, ξ1q.
We also define ÁRTSÁRT Y ¡RTS. (Note that ¡RTS is not the asymmetric part of ÁRTS.) It
is straightforward to check that if O is rationalizable by an SSCD preference Á, then ÁRTSÁ
and ¡RTS¡ (where ¡ is the asymmetric part of Á). It follows that ÁRTS must be cyclically
3
consistent in the sense that
pz1, ξq ÁRTS pz2, ξq ÁRTS    ÁRTS pzK , ξq ùñ pzK , ξq £RTS pz1, ξq. (5)
A data set that violates ARC will not have ÁRTS obeying cyclical consistency but it is possible
for a data set (with non-interval feasible sets) to obey ARC and yet have ÁRTS be cyclically
inconsistent (see Example 2 in LQS). So perhaps data sets that are rationalizable by SSCD
preferences are characterized by the cyclical consistency of ÁRTS, rather than ARC? But we can
push this logic further.
Assuming that ÁRTS is cyclically consistent, we can construct its transitive closure ÁRTST ,
which is defined as follows: px2i , ξiq Á
RTST
i px
1
i, ξiq if there exists a sequence z
1
i , z
2
i , ..., z
k
i such that
px2i , ξiq Á
RTS
i pz
1
i , ξiq Á
RTS
i pz
2
i , ξiq Á
RTS
i ... Á
RTS
i pz
k
i , ξiq Á
RTS
i px
1
i, ξiq. (6)
If we can find at least one strict relation ¡RTSi in the sequence (6), then, we let px
2
i , ξiq ¡
RTST
i
px1i, ξiq. (Once again, note that ¡
RTST
i is not the asymmetric part of Á
RTST
i .) Given Á
RTST , we
can construct its single crossing extension ÁRTSTS; this relation must also be cyclically consistent
if O is rationalizable by an SSCD preference. If so, we can take its transitive closure, the single
crossing extension of that transitive closure, check for cyclical consistency and, if it passes, repeat
the process. Since the data set has only finitely many observations, this process will terminate
after a finite number of steps. Furthermore, if O is rationalizable by an SSCD preference then
the process will terminate without a violation of cyclical consistency. It is natural to speculate
that this property (i.e., the absence of violations of cyclical consistency) is also sufficient for
O to be rationalized by an SSCD preference. We provide a counter example to show that this
conjecture is also false.
2 A counter-example
Consider the following direct revealed preference relations:
(a) ξ  0: p5, 0q ÁR p1, 0q, p1, 0q ÁR p5, 0q,p3, 0q ÁR p1, 0q, p1, 0q ÁR p3, 0q. p1, 0q ÁR p2, 0q,
p1, 0q ÁR p4, 0q, p9, 0q ÁR p6, 0q, p8, 0q ÁR p7, 0q, p6, 0q ÁR p9, 0q, p2, 0q ÁR
4
p1, 0q, p4, 0q ÁR p1, 0q, and p7, 0q ÁR p8, 0q.
(b) ξ  1: p6, 1q ÁR p3, 1q, p2, 1q ÁR p7, 1q, p8, 1q ÁR p5, 1q, p4, 1q ÁR p9, 1q, p7, 1q ÁR p2, 1q,
p3, 1q ÁR p6, 1q, p9, 1q ÁR p4, 1q, and p5, 1q ÁR p8, 1q.
These relations can be formed from observing an agent choosing from pairs of alternatives; for
example, p5, 0q ÁR p1, 0q because 5 was chosen from the feasible action set t1, 5u when the
prevailing parameter value is 0. Figures 1 and 2 show the graph of ÁR at each value of ξi. We
claim that this data set has the following features:
(i) it is not rationalizable by an SSCD preference,
(ii) the binary relation ÁRTS obeys cyclical consistency, and
(iii) the relation ÁRTST does not have a nontrivial single crossing extension.
To establish claim (i), suppose that there exists an SSCD preference relation Á that ratio-
nalizes the data. We show that is not possible for either p1, 0q Á p1, 0q or p1, 0q Á p1, 0q
to hold. First, we show that p1, 0q Á p1, 0q is impossible. If p1, 0q Á p1, 0q, then it im-
plies that p3, 0q Á p2, 0q and p5, 0q Á p4, 0q since p3, 0q ÁR p1, 0q, p1, 0q ÁR p2, 0q, and Á is
transitive. Since Á obeys SSCD, we obtain p3, 1q ¡ p2, 1q and p5, 1q ¡ p4, 1q. The relations
p6, 1q ÁR p3, 1q and p2, 1q ÁR p7, 1q and the transitivity of Á in turn imply that p6, 1q ¡ p7, 1q
and p8, 1q ¡ p9, 1q. Again by SSCD, it holds that p6, 0q ¡ p7, 0q and p8, 0q ¡ p9, 0q. These
relations, together with p9, 0q ÁR p6, 0q and p7, 0q ÁR p8, 0q together with transitivity imply that
p6, 0q ¡ p6, 0q, which is impossible.
Suppose instead that p1, 0q Á p1, 0q. By p2, 0q ÁR p1, 0q, p4, 0q ÁR p1, 0q, and the
transitivity of Á, we obtain p2, 0q Á p3, 0q and p4, 0q Á p5, 0q. SSCD then guarantees
that p2, 1q ¡ p3, 1q and p4, 1q ¡ p5, 1q. The transitivity of Á, p7, 1q ÁR p2, 1q and
p3, 1q ÁR p6, 1q together imply that p7, 1q ¡ p6, 1q. Similarly, p9, 1q ÁR p4, 1q and
p5, 1q ÁR p8, 1q imply that p9, 1q ¡ p8, 1q. Again by SSCD, it holds that p7, 0q ¡
p6, 0q and p9, 0q ¡ p8, 0q. Finally, by p6, 0q ÁR p9, 0q and p7, 0q ÁR p8, 0q, we have
p7, 0q ¡ p7, 0q, which is impossible.
To show that claims (ii) and (iii) are also true, we first take the transitive closure of ÁR.
This gives the following relations besides the ones already related by ÁR at ξ  0: p5, 0q ÁRT
p5, 0q, p5, 0q ÁRT p3, 0q, p3, 0q ÁRT p3, 0q, p3, 0q ÁRT p5, 0q, p2, 0q ÁRT p4, 0q, p2, 0q ÁRT
5
p2, 0q, p4, 0q ÁRT p4, 0q, p4, 0q ÁRT p2, 0q, p9, 0q ÁRT p6, 0q, and p6, 0q ÁRT p9, 0q. Note
that, at ξ  1, no new relation is added by taking the transitive closure.
By taking the single crossing extension, we obtain the binary relation ÁRTS. In the list below,
for each value of ξ, Type 1 relations consist of pairs that are also related by ÁRT and where the
relation could be “transimitted” to the other value of ξ via single crossing extension; Type 2
consists of pairs related by ÁRT which cannot be transimitted to the other value of ξ via single
crossing extension, and Type 3 consists of pairs related by ¡RTS, which are formed via extension
from the other value of ξ.
(a) ξ  0:
• Type 1: p5, 0q ÁRTS p1, 0q, p1, 0q ÁRTS p5, 0q, p3, 0q ÁRTS p1, 0q, p1, 0q ÁRTS p3, 0q,
p5, 0q ÁRTS p5, 0q, p5, 0q ÁRTS p3, 0q, p3, 0q ÁRTS p3, 0q, p3, 0q ÁRTS p5, 0q,
p9, 0q ÁRTS p6, 0q, and p6, 0q ÁRTS p9, 0q.
• Type 2: p1, 0q ÁRTS p2, 0q, p8, 0q ÁRTS p7, 0q, p2, 0q ÁRTS p1, 0q, p4, 0q ÁRTS
p1, 0q, p7, 0q ÁRTS p8, 0q, p2, 0q ÁRTS p4, 0q, p2, 0q ÁRTS p2, 0q, p4, 0q ÁRTS p4, 0q,
and p4, 0q ÁRTS p2, 0q.
• Type 3: p2, 0q ¡RTS p7, 0q, p7, 0q ¡RTS p2, 0q, p4, 0q ¡RTS p9, 0q, and p9, 0q ¡RTS
p4, 0q.
(b) ξ  1
• Type 1: p2, 1q ÁRTS p7, 1q, p7, 1q ÁRTS p2, 1q, p4, 1q ÁRTS p9, 1q, and p9, 1q ÁRTS
p4, 1q,
• Type 2: p6, 1q ÁRTS p3, 1q, p8, 1q ÁRTS p5, 1q, p3, 1q ÁRTS p6, 1q, and p5, 1q ÁRTS
p8, 1q,
• Type 3: p5, 1q ¡RTS p1, 1q, p1, 1q ¡RTS p5, 1q, p3, 1q ¡RTS p1, 1q, p1, 1q ¡RTS p3, 1q,
p5, 1q ¡RTS p5, 1q, p5, 1q ¡RTS p3, 1q, p3, 1q ¡RTS p3, 1q, p3, 1q ¡RTS p5, 1q,
p9, 1q ¡RTS p6, 1q, and p6, 1q ¡RTS p9, 1q
These relations are depicted in Figures 3 and 4, where thick arrows represent ¡RTS, while dotted
arrows represent ÁRTS z ¡RTS (which equals ÁRT ). We can check from these graphs that ÁRTS
is cyclically consistent and that ÁRTST obeys single crossing differences in the following sense:
if x2 ¡ x1 and ξ2 ¡ ξ1, then (a) px2, ξ1q ÁRTST px1, ξ1q implies that px2, ξ2q ¡RTST px1, ξ2q and
6
(b) px1, ξ2q ÁRTST px2, ξ2q implies that px1, ξ1q ¡RTST px2, ξ1q. It follows that ÁRTST admits no
further single crossing extension.
References
[1] Afriat, S.N. (1967): The construction of a utility from expenditure data, International of
Economic Review, 8, 67-77.
[2] Lazzati, N., J. K-H. Quah, and K. Shirai, (2016): A revealed preference theory of monotone
choice and strategic complementarity. Mimeo.
[3] Milgrom, P. and C. Shannon, (1994): Monotone comparative statics. Econometrica, 62, 157-
180.
[4] Quah, J. K.-H. and B. Strulovici, (2009): Comparative statics, informativeness, and the
interval dominance order. Econometrica, 77, 1949-1999.
[5] Richter, M.K. (1966): Revealed preference theory. Econometrica, 34, 635645.
7
Figure 1: ÁR at ξ  0
Figure 2: ÁR at ξ  1
Figure 3: ÁRTS at ξ  0
Figure 4: ÁRTS at ξ  1
8


A counterexample on the completion of preferences with single crossing differences

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A counterexample on the completion of preferences with single crossing differences

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