Complementarity with incomplete preferences
Sayantan Ghosal
University of Warwick
April 23, 2010
Abstract
This paper extends the formulation of complementarity in Milgrom
and Shannon (1994) to the case of incomplete but acyclic preferences.
It is shown that the problem can be reformulated as one with complete,
irre‡exive, acyclic but not necessarily transitive preferences. However,
now, quasi-supermodularity and single-crossing on their own do not
guarantee either monotone comparative statics or equilibrium existence in pure strategies: an additional condition, monotone closure, is
required. The results obtained here relax the requirement of convexity
in Shafer and Sonnenschein (1975)’s existence result with incomplete
preferences. In an application, it is shown that pure strategy equilibria
exist in incomplete information games with Knightian uncertainty.
JEL classi…cation numbers: C62, C70.
Keywords: complementarity, preferences, intransitive, acyclic, incomplete, comparative statics, existence.
I would like to thank Chuck Blackorby, Anirban Kar and Mike Waterson for many
helpful comments. All remaining errors are mine. E-mail: S.Ghosal@warwick.ac.uk
1
1
Introduction
This paper extends the formulation of complementarity in Milgrom and
Shannon (1994) to the case of incomplete1 , acyclic preferences. It is shown
that the problem can be reformulated as one with complete but intransitive preferences. However, now, quasi-supermodularity and single-crossing
on their own no longer su¢ ce to guarantee either monotone comparative
statics or existence: an additional condition, monotone closure, is required.
Taken together, quasi-supermodularity, single-crossing and monotone closure guarantee both monotone comparative statics and equilibrium existence
(via Tarski’s (1955) theorem) thus extending the existence results contained
in Vives (1990), Milgrom and Shannon (1994), Topkis (1998).
The results obtained here relax the requirement of convexity2 in Shafer
and Sonnenschein (1975)’s existence result.
As an application, it is shown that pure strategy equilibria exist in incomplete information games with Knightian uncertainty where the formulation
of Knightian uncertainty follows Bewley (2002)3 .
In what follows, section 2 studies monotone comparative statics and existence while section 3 applies the results obtained in section 2 to incomplete
information games with Knightian uncertainty.
2
Comparative statics and existence
In this section, I study comparative statics and existence of pure strategy
equilibria with intransitive preferences.
2.1
A decision problem with incomplete preferences
To begin with, consider a single decision maker who has to choose an action
from a set A
<k . The preferences of the decision-maker is described
by a map, : P ! A A, where P (indexed by p) is a set of preference
parameters, P <n , and for each p 2 P , p describes a preference relation
1
There is a large and growing literature on games and markets where agents have
incomplete preferences. A selective list of references (not referred to elsewhere in this
paper) includes Shapley (1959), Aumann (1962), Gale and MasCollel (1975) and Shafer
and Sonnenschein (1975). Incomplete preferences arise in an essential way in models of
coalition (and party) formation as in Ray and Vohra (1997), Roemer (1999) and Levy
(2004).
2
The convexity assumption used by Shafer and Sonnenschein (1975) rules out scenarios with indivisibilities, with increasing returns, with the fundamental non-convexity of
feasible sets in the presence of externalities identi…ed by Starret (1972) and the possible
non-convexity of reference-dependent preferences in mutli-dimensional choice with loss
aversion (concave over gains but convex over losses) studied by Khaneman and Tversky
(1979, 1991).
3
Applications of Knightian uncertainty (Knight (1921)) that use Bewley’s approach
include Rigotti and Shannon (2005) and Lopomo, Rigotti and Shannon (2006).
2
over A. The expression (a; a0 ) 2 p is written as a p a0 and is to be read
as "a is preferred to a0 by the decision-maker when the utility parameter is
p". De…ne the sets p (a) = fa0 2 A : a0 p ag (the upper section of p ),
1
0
0
p ). It is assumed that
p (a) = fa 2 A : a p a g (the lower section of
for each p 2 P , (i) p is acyclic i.e. there is no …nite set a1 ; :::; aT such
that at 1 p at , t = 2; :::; T , and aT p a1 , and (ii) p 1 (a) is open relative
to A i.e. p has an open lower section4 . Write a0 2
= p (a) as a p a0 .
0
De…ne a map : P ! A, where (p) = fa 2 A : p (a0 ) = ;g: for each
p 2 P , (p) is the set of maximal elements of the preference relation p .
Consider the following extension of : for each p, de…ne p on A by
a
p
a0 , a0
p
a:
As for each p, p is acyclic and therefore irre‡exive, it follows that p
is complete and acyclic but not necessarily transitive.
Let ^ p denote the strict preference relation corresponding to p i.e.
a ^ p a0 if and only if a p a0 but a0 p a. For each p 2 P and a; a0 2 A,
a p a0 if and only if a ^ p a0 .
De…ne a map ^ : P ! A, where ^ (p) = fa0 2 A : ^ p (a0 ) = ;g: for each
p 2 P , ^ (p) is the set of maximal elements of the preference relation ^ p .
Note that for each p, p and ^ p are equivalent and have the same set
of maximal elements i.e. for each p 2 P and a; a0 2 A, a p a0 if and only if
a ^ p a0 and therefore, (p) = ^ (p).
It follows that the decision problem with incomplete but acyclic preferences is rephrased as a decision problem with complete and acyclic but not
necessarily transitive preferences.
Consider the following assumptions:
(A1) A is a complete, compact lattice with the vector ordering5 ;
(A2) For each p, and a; a0 , (i) if a p inf(a; a0 ), then sup(a; a0 ) p a0 (ii)
if a p sup (a; a0 ) then inf (a; a0 ) p a0 (quasi-supermodularity);
(A3) For each a a0 and p p0 , (i) if a p0 a0 , then a p a0 and (ii) if
0
a p a then a0 p0 a (single-crossing property);
(A4) For each p and a
a0 , (i) if p (a0 ) = ; and a p a0 , then
0
0
p (a) = ;, and (ii)
p (a) = ; and a
p a, then
p (a ) = ;,(monotone
closure).
4
The continuity assumption, that p has an open lower section, is weaker than the
continuity assumption made by Debreu (1959) (who requires that preferences have both
open upper and lower sections), which in turn is weaker than the assumption by Shafer
and Sonnenschein (1975) (who assume that preferences have open graphs). Note that assuming p has an open lower section is consistent with p being a lexicographic preference
ordering over A.
5
A lattice is a partially ordered subset of <k with the vector ordering (the usual component wise ordering: x y if and only if xi yi for each i = 1; ::; K, and x > y if and
only if both x y and x 6= y, and x
y if and only if xi > yi for each i = 1; ::; K) which
contains the supremum and in…mum of any two of its elements. A lattice that is compact
(in the usual topology) is a compact lattice.
3
Assumptions (A2)-(A3) are quasi-supermodularity and single-crossing
property de…ned by Milgrom and Shannon (1994).
Assumption (A4) is new. Consider a pair of actions such that the …rst
action is greater (in the ussual vector ordering) than the second action. For
a …xed p, suppose the two actions are unranked by p . Then, assumption
(A4) requires that either both actions are maximal elements for p or neither
is.
The role played by assumption (A4) in obtaining the monotone comparative statics with incomplete preferences is clari…ed by the following examples. There preferences and action sets in each example satisfy assumptions
(A1)-(A3). However, assumption (A4) fails to hold in either example.
Example 1: ( (p) needn’t be a lattice.)
P is single valued and A is the four point lattice in <2
f(e; e) ; (f; e) ; (e; f ) ; (f; f )g
where f > e. Suppose that (f; f ) (e; e) but no other pair is ranked. Then,
consists of f(f; e) ; (e; f ) ; (f; f )g clearly not a lattice. Note that in this
case, preferences satisfy acyclicity and quasi-supermodularity (and trivially,
single-crossing property). However, preferences do not satisfy monotone closure: (f; e) (e; e), with ((f; e)) = ; and (e; e) (f; e), but ((e; e)) 6= ;.
The preceding example demonstrates that without the additional assumption of monotone closure, quasi-supermodularity on its own cannot
ensure that the set of maximal elements of is a sublattice of A even when
is acyclic. The example also demonstrates that can be acyclic without
necessarily satisfying monotone closure and therefore, the two are distinct
conditions on preferences.
Example 2: (No increasing selection from (:):)
P = fp; p0 g, p < p0 , and A is the …ve point lattice in <2
f(e; e) ; (f; e) ; (e; f ) ; (f; f ) ; (g; g)g
where g > f > e. Preferences are such that: (i) (g; g) p (f; f ) p (e; e), ,
(f; e) p (e; f ), (f; e) p (g; g), (e; f ) p (g; g) but the pairs f(f; e) ; (e; e)g,
f(e; f ) ; (e; e)g, f(f; e) ; (f; f )g and f(e; f ) ; (f; f )g aren’t ranked by p ; (ii)
(g; g) p0 (f; f ) p0 (e; e), (e; f ) p0 (f; e), (f; e) p0 (g; g), (e; f ) p0 (g; g)
but the pairs f(f; e) ; (e; e)g, f(e; f ) ; (e; e)g, f(f; e) ; (f; f )g and f(e; f ) ; (f; f )g
aren’t ranked by p0 . Note that in this case, both p and p0 satisfy acyclicity (but not transitivity), quasi-supermodularity and the single-crossing
property. It follows that (p) = f(f; e)g and (p0 ) = f(e; f )g (i.e. for
both p and p0 the set of maximal elements is a singleton and hence, trivially
a lattice). Therefore, (:) does not admit an increasing selection. Observe that neither p nor p0 satisfy monotone closure: (f; f ) (f; e), with
(e; f ),
p ((f; e)) = ; and (f; f ) p (f; e), but
p ((f; f )) 6= ; and (f; f )
0
0
0
with p ((e; f )) = ; and (f; f ) p (e; f ), but p ((f; f )) 6= ;.
4
The preceding example demonstrates that with incomplete but acyclic
preferences, quasi-supermodularity and single crossing on their own cannot
ensure an increasing selection from the set of maximal elements.
The following result shows that assumptions (A1)-(A4), taken together,
are su¢ cient to ensure monotone comparative statics with incomplete preferences:
Proposition 1: Under assumptions (A1)-(A4), each p 2 P , (p) is nonempty and a compact sublattice of A where both the maximal and minimal
elements, denoted by a(p) and a(p) respectively, are increasing functions on
P.
Proof. By assumption, for each p, p is acyclic, p 1 (a) are open
relative to A and A is compact. By Bergstrom (1975), it follows that (p)
is non-empty. As Bergstrom (1975) doesn’t contain an explicit proof that
(p) is compact, a proof of this claim follows next. To this end, note that the
complement of the set (p) in A is the set c (p) = fa0 2 A : p (a0 ) 6= ;g.
If c (p) = ;, then (p) = A is necessarily compact. So suppose c (p) 6= ;.
For each a0 2 c (p), there is a00 2 A such that a00 p a0 . By assumption,
1
00
c (p).
(p), p 1 (a00 )
p (a ) is open relative to A. By de…nition of
1
00
0
c
Therefore, p (a ) is a non-empty neighborhood of a 2
(p) and it is
clear that c (p) is open and therefore, (p) is closed. As A is compact,
(p) is also compact. Next, it is shown that for p
p0 if a 2 (p) and
0
0
0
0
a 2 (p ), then sup (a; a ) 2 (p) and inf (a; a ) 2 (p0 ). Note that as
a0 2 (p0 ), a0 p0 inf (a; a0 ). By quasi-supermodularity, sup (a; a0 ) p0 a.
By single-crossing, sup (a; a0 ) p a. As a 2 (p), p (a) = ;, and by
monotone closure sup (a; a0 ) p a and p (sup (a; a0 )) = ;, it follows that
sup (a; a0 ) 2 (p). Next, note that as a 2 (p), a p sup (a; a0 ). By singlecrossing, a p0 sup (a; a0 ) and by quasi-supermodularity, inf (a; a0 ) p0 a0 .
As a0 2 (p0 ), p0 (a0 ) = ;, and by monotone closure inf (a; a0 ) p0 a0 and
0
0
(p0 ). Therefore, (i) (p)
p0 (inf (a; a )) = ;, it follows that inf (a; a ) 2
is ordered, (ii) (p) is a compact sublattice of A and has a maximal and
minimal element (in the usual component wise vector ordering) denoted by
a(p) and a(p), and (iii) both a(p) and a(p) are increasing functions from P
to A.
Let S denote the strong set order introduced by Veinott (1989). For
A a lattice, with C and D elements of the power set (A), C S D if for
every c 2 C and d 2 D, inf(c; d) 2 C and sup(c; d) 2 D. The following
corollary follows from the Proposition 1 above and Theorem 4 in Milgrom
and Shannon (1994). Given a partially ordered set P , a function M : P !
(A) is monotone increasing if p p0 implies M (p) S M (p0 ).
Corollary 1. Let p be de…ned over a complete, compact lattice A
where p 2 P a partially ordered set. Let C
A. Then (p; C) =
0
0
fa 2 C : p (a ) = ;g is monotone increasing in (p; S) if p satis…es assumptions (A2)-(A4).
5
2.2
Normal-form games with incomplete preferences
The set up is as follows. There is a set of I players (indexed by i) and for
each player i, a pure action set Ai (indexed by ai ) where Ai <K , a …nite
dimensional Euclidian space. Let A = i2I Ai and A i = j6=i Aj .
Each player i is endowed with a preference relation i : A i ! Ai Ai .
The expression (ai ; a0i ) 2 i;a i is written as ai i;a i a0i and is to be read as
"ai is preferred to a0i by the decision-maker when the action pro…le chosen
by other players is a i ". Let i;a i (respectively, i;a i ) denote the weak
preference relation (respectively, indi¤erence preference relation) associated
with i;a i . A game is = I; (Ai ; i )i2I .
De…nition 1: A pure strategy equilibrium is a pro…le of actions a such
that for each i 2 I, given a i , i;a i (ai ) = ;.
Proposition 2: For each i 2 I, assume that Ai is a compact, complete
lattice and both i;a i and i;a i satisfy assumptions (A2)-(A4). Then, a
pure strategy equilibrium exists.
Proof. De…ne a map
: A ! A, (a) = ( i (a i ) : i 2 I) as follows: for each a, i (a i ) = a0i 2 Ai : i;a i (a0i ) = ; . By proposition 1,
for each i 2 I and a i 2 A i , i (a i ) is non-empty and compact and for
a i a0 i if ai 2 i (a i ) and a0i 2 1 (a0 i ), then sup (ai ; a0i ) 2 i (a i ) and
inf (ai ; a0i ) 2 1 (a0 i ). Therefore, the map a (a) = (ai (a i ) : i 2 I) is an increasing function from A to itself and as A is a compact, complete lattice, by
applying Tarski’s …x-point theorem, it follows that a = a(a) is a …x-point of
. By a symmetric argument, a(a) is an increasing function from A to itself
and a = a(a) is also a …x-point of . Moreover, a and a are respectively the
largest and smallest …x-points of .
The following corollary follows from the Proposition 2 above and Theorem 13 in Milgrom and Shannon (1994).
Corollary 2: Let t = I; (Ai ; i;t )i2I be a family of games with
strategic complementarities such that for each t assumptions (A1)-(A4) are
satis…ed and i;t satis…es the single crossing property in (ai ; a i ; t) for all
i 2 I. Then a(t) and a(t), respectively the largest and smallest pure strategy
equilibria, are increasing functions of the parameter t.
Remark
Scho…eld (1984) shows that if action sets are convex or are smooth manifolds with a special topological property, the (global) convexity assumption
made by Shafer and Sonnenschein (1975) can be replaced by a "local" convexity restriction, which, in turn, is equivalent to a local version of acyclicity
(and which guarantees the existence of a maximal element). However, here,
as action sets are not necessarily convex and are allowed to be a collection
of discrete points, Scho…eld’s equivalence result does not apply.
A di¤erent approach to the existence of equilibrium would be to deduce
the existence result for games with incomplete preferences from the standard existence result for games with complete preferences as in Bade (2005)
6
who requires that the underlying incomplete preferences be transitive and
admit a concave vector representation. In the set up studied in this paper
(incomplete but acyclic preferences, discrete action sets), both assumptions
are violated.
3
Existence of pure strategy equilibria in a game
with Knightian uncertainty
In this section, the pure strategy equilibria are shown to exist in a incomplete
information game with Knightian uncertainty.
As before, there is a set of I players (indexed by i) and for each player
the …nite pure action set Ai (indexed by ai ). Let A = i2I Ai and A i =
j6=i Aj . There is a …nite set of types for each player Ti (indexed by ti ) with
T = i2I Ti (indexed by t) denoting the set of type pro…les and T i = j6=i Tj
(indexed by t i ) the set of types of all other players excluding player i.
The type of a player describes her private information and thus, associated with each i and ti 2 Ti is a closed and convex set of probability
distributions i;ti
(T i ) (indexed by i;ti ) over the type pro…le of other
players6 . A strategy is a map i : Ti ! Ai (with i the corresponding
set). Let = ( i : i 2 I) (with the corresponding set) denote a pro…le of
strategies for all players and
i = ( j : j 6= i) (with
i the corresponding
set) denote the strategy pro…le for all players excluding player i.
0 if
Following Vives (1990), de…ne the natural ordering on i as
0
and only if (ti )
(ti ) for all ti 2 T .
For each
,i,t
,
it
is possible to directly specify an incomplete preferi
i
ence i;ti ; i ranking pairs of actions in Ai . Instead, in what follows, I apply
Bewley (2002)’s approach to Knightian uncertainty (Knight (1921)). Each
player has a utility function ui : A T ! <. For each
i , i, ti , ai 2 Ai and
i;ti 2
i;ti (T i ), let
X
vi (ai ; i ; t i ; i;ti jti ) =
i;ti (t i ) ui (ai ;
i (t i ) ; ti ; t i )
t
For each
i ,i,ti
i 2T i
and two actions ai ; a0i 2 Ai ,
ai
vi (ai ;
i ; t;
i;ti jti )
i;ti ;
i
> vi (ai ;
a0i ,
i ; t;
i;ti jti )
for all i;ti 2 i;ti (T i ). In the standard Bayesian set-up, i;ti (T i ) is a
singleton for each i; ti and corresponds to the case when there is no Knightian
uncertainty. In general, the preference relation i;ti ; i , de…ned over Ai Ai
6
This formulation is consistent with Ahn (2007) who shows that under common knowledge of coherence each type of each player encodes a compact set of beliefs over other
player’s types.
7
is incomplete. Following Bewley, the preference relation i;ti ; i is complete
if
cl: a; a0 2 Ai : either a i;ti ; i a0 or a0 i;ti ; i a = Ai :
When preferences are complete, a game with Knightian uncertainty reduces
to a Bayesian game.
Suppose the following assumptions hold:
(B2) (Supermodularity) For each t 2 T , i 2 I, a i 2 A i and ai ; a0i 2 Ai ,
ui inf(ai ; a0i ); a i ; t + ui sup(ai ; a0i ); a i ; t
ui (ai ; a i ; t) + ui a0i ; a i ; t ;
(B3) (Increasing di¤erences) For each t 2 T , i 2 I, a
ui ai ; a0 i ; t
ui (a0i ; a0 i ; t)
ui (ai ; a i ; t)
i
a0 i ,
ui a0i ; a i ; t :
Assumptions (B2) and (B3) ensure that for each type pro…le, the utility
function for each individual satis…es supermodularity and increasing di¤erences as in Vives (1990).
De…nition 2: A pure-strategy equilibrium with Knightian uncertainty
is a
such that i;ti ; i ( i (ti )) = ; for all ti 2 Ti and i 2 I.
The following proposition provides an existence theorem in pure strategies for games with Knightian uncertainty.
Proposition 3: Assume that for each i 2 I, Ai is a compact lattice and
for each t 2 T , i 2 I, ui (ai ; a i ; t) satis…es (B2) and (B3). Then, a pure
strategy equilibrium with Knightian uncertainty exists.
Proof. Both supermodularity and increasing di¤erences are preserved
under summation. It follows that for each i;ti 2 (T i ), vi (ai ; i ; t i ; i;ti jti )
satis…es the assumptions of supermodularity and increasing di¤erences. Moreover, any supermodular function is quasi-supermodular and any function
that satis…es increasing di¤erences also satis…es single crossing. Therefore,
~ i;ti ; i satis…es quasi-supermodularity and single-crossing as well. That
i;ti ; i satis…es monotone closure follows from the fact that for each i,
i,
0
t, if ^ i 2 arg maxai 2Ai vi (ai ; i ; t i ; i;ti jti ) and vi (ai ; i ; t i ; i;ti jti )
vi (ai ; i ; t i ; i;ti jti ), then necessarily a0i 2 arg maxai 2Ai vi (ai ; i ; t i ; i;ti jti ).
By proposition 1, the set of maximizing actions given ti ,
i is a non-empty,
compact and complete lattice and its supremum and in…mum are themselves
maximizers. De…ne a map : ! , ( ) = ( i ( i ) : i 2 I) as follows:
0 2
0
for each , i ( i ) =
i : i;ti ; i ( i (ti )) = ;; ti 2 Ti . It follows
i
that for each i 2 I and
i 2
i,
i(
i ) is non-empty and compact
and for i ; 0i 2 i ( i ), sup ( i ; 0i ) 2 i ( i ) and inf ( i ; 0i ) 2 i ( 0 i ).
Therefore, the map ( ) = ( i ( i ) : i 2 I) is an increasing function from
to itself and as is a compact, complete lattice, by applying Tarski’s …xpoint theorem, it follows that = ( ) is a …x-point of . By a symmetric
argument, ( ) is an increasing function from A to itself and = ( ) is
also a …x-point of . Moreover, and are respectively the largest and
smallest …x-points of .
8
Proposition 3 only establishes that a pure strategy cannot be improved
upon by another pure strategy. This without loss of generality in a normal
form game. However, such a restriction could be problematic in a game with
Knightian uncertainty.
Observe that mixed strategies aren’t explicitly considered in de…nition 2.
Given that the objective of the work reported here was to extend Milgrom
and Shannon to the case incomplete and acyclic preferences, there is a good
(technical) reason for this: the set of mixed strategies cannot be a lattice.
However, the restriction to pure strategies may matter in Bewley’s framework. Fix the actions of other players and type ti for player i such that i;ti
consists of exactly two probability distributions. Consider an action ai (a
1+"
candidate best response in pure strategies) with expected payo¤s 1+"
2 ; 2
where " is a small but positive number less than 14 . Suppose there are only
two pure actions a0i ; a00i (with payo¤s 21 ; 32 and 23 ; 12 ) which are not comparable to ai . Then, the convex combination (i.e. a mixed action) 21 a0i + 12 a00i
with payo¤s (1; 1) is preferred to ai .
Therefore, a best response in pure strategies may be dominated by a
mixed strategy.
However, a similar conceptual problem would arise even if players are
allowed to choose best-responses in mixed strategies. Consider a situation
where for a …xed con…guration of actions of other players and type ti for
player i (such that i;ti consists of exactly two probability distributions)
the expected payo¤s to player from his set of pure actions fb; c; d; eg is
f("; 1) ; (1; ") ; (0; 2) ; (2; 0)g where where " is a small but positive number
less than 41 . Consider the mixed strategy 12 b + 12 c with expected payo¤s
1
3
1 3
1+" 1+"
2 ; 2 , the mixed strategy 4 d + 4 e with expected payo¤s 2 ; 2 and
3
1
3 1
mixed strategy 4 d + 4 e with expected payo¤s 2 ; 2 . Clearly, 12 b + 12 c,
3
3
1
1
4 d + 4 e and 4 d + 4 e are all best-responses and not comparable but the
convex combination 21 14 d + 43 e + 21 34 d + 14 e with expected payo¤s (1; 1)
dominates 21 b + 12 c.
Therefore, a best response in mixed strategies may be dominated by a
further round of randomization over the set of mixed strategies.
Although the issue beyond the scope of this paper, the above discussion
raises the general and di¢ cult issue of what is the appropriate de…nition of
equilibrium in a game with Knightian uncertainty, a topic for future research.
Proposition 3 only establishes that a pure strategy cannot be improved upon
by another pure strategy.
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9
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