Economics Letters 68 (2000) 251–254
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A generalization of supermodularity
*
Elettra Agliardi
Faculty of Economics, University of Bologna, Piazza Scaravilli, 2 I-40126 Bologna, Italy
Received 7 October 1999; accepted 22 February 2000
Abstract
The notion of pseudo-supermodularity is introduced and the existence and order structure of equilibria in
pseudo-supermodular games are analyzed using lattice-theoretical methods. 2000 Elsevier Science S.A. All
rights reserved.
Keywords: Pseudo-supermodularity; Pseudo-supermodular games; Strategic complementarities
JEL classification: C72; C60
1. Introduction
In this paper we propose a generalization of the concept of supermodularity (Topkis, 1998) and introduce the notions of pseudo-supermodularity and pseudo-supermodular games. We analyze the existence and order structure of equilibria for this class of games, in which the set of feasible joint decisions is a lattice, and exploit their monotonicity properties, using lattice-theoretical methods. Such analytical tools have been most powerful in the study of supermodular games, where the payoff function of each player has properties of supermodularity and increasing differences. Supermodular games exhibit ‘strategic complementarities’, which yield monotone increasing best replies; as a result, many of the most important economic applications of non-cooperative game theory are encompassed by this class of game.
In Section 2 we present the main proposition on the existence of equilibria. We show that the condition of pseudo-supermodularity is weaker than supermodularity and is an alternative to the Milgrom and Shannon (1994) notion of quasi-supermodularity. Our analysis allows us to find equilibria with nice comparative statics properties. We provide a monotonicity theorem for pseudo-supermodular objective functions which parallels the result in Milgrom and Shannon (1994).
*Tel.:139-51-209-8668; fax: 139-51-221-968.
E-mail address: agliardi@economia.unibo.it (E. Agliardi)
252 E. Agliardi / Economics Letters 68 (2000) 251 –254
2. Definitions and main results
Let L be a partially ordered set, that is, a set with a partial order $ that is reflexive, antisymmetric and transitive. For x and y elements of L, let xky denote the least upper bound, or ‘join’, of x and y
in L, if it exists, and let xny denote the greatest lower bound, or ‘meet’, of x and y in L, if it exists. A
lattice is a partially ordered set (L, $) any two of whose elements have a meet xny and a join xky
in the set. It is complete if every non-empty subset of L has a meet and a join in L. If a subset S of L has the property that x[S and y[S implies that xky[S and xny[S, then S is a sublattice of L.
n
In many applications, L will be R with the component-wise order. If x and y are real numbers, then
xky5maxhx, yj and xny5minhx, yj. If x and y are n-vectors, then xky5(x1ky , . . . .x1 nky ) andn xny5(x1ny , . . . .x1 nny ). For a lattice L with the given relationn $, with S and S9 subsets of L, we say that S#S9 if, for every x[S and x9[S9, xnx9[S and xkx9[S9. Consider f(x) as a real-valued function on a lattice L. We define pseudo-supermodularity in the following way:
Definition. f(x) is pseudo-supermodular on L if (i) f(x)kf(x9)$f(xnx9) implies f(xkx9)$
f(x)nf(x9) and (ii) f(x)kf(x9).f(xnx9) implies f(xkx9).f(x)nf(x9), for all x[L, x9[L.
Note that when L5Revery function is pseudo-supermodular; when the choice space is of greater dimension, pseudo-supermodularity involves additional multivariate restrictions as well. Pseudo-supermodularity expresses a weak kind of complementarity between the choice variables.
Proposition 1. If f is supermodular then it is pseudo-supermodular.
Proof. Assume that f is not pseudo-supermodular, that is, there exist x,x9[L such that f(x)kf(x9)$
f(xnx9) and f(xkx9),f(x)nf(x9). Since f is supermodular we get f(x)1f(x9)#f(xnx9)1
f(xkx9),f(x)nf(x9)1f(x)kf(x9), which yields f(x)1f(x9),f(x)1f(x9), a contradiction. h
Proposition 2. If f is quasi-supermodular, then it is pseudo-supermodular.
Proof. Suppose that f is not pseudo-supermodular, that is, there exist x,x9[L such that f(x)kf(x9)$
f(xnx9) and f(xkx9),f(x)nf(x9). Since f is quasi-supermodular, then, if f(x)$f(xnx9), we have
f(x)kf(x9)$f(x)$f(xnx9) implies f(xkx9)$f(x9); therefore, we get f(x)nf(x9)#f(x9)#f(xkx9) and f(x)nf(x9).f(xkx9), a contradiction. If f(x),f(xnx9), then f(x9)$f(xnx9) implies f(xkx9)$
f(x)$f(x)nf(x9), contradicting f(xkx9),f(x)nf(x9). Thus, it is proved that if f is quasi-supermodu-lar, then it is pseudo-supermodular as for definition (i); similarly, it can be proved for (ii). h
Remark 1. Quasi-supermodularity is not a necessary condition for pseudo-supermodularity, which
implies that our results cannot be captured by the analysis in Milgrom and Shannon (1994) and Shannon (1995). We give here an example of a pseudo-supermodular function that is not quasi-supermodular. Consider the following lattice L5h0, a, b, cj, where 05(0,0), a5(1,0), b5(0,1),
c5(1,1), with the usual partial ordering. Suppose that f :L→R is such that f(0),f(a),f(c),f(b). It can be easily verified that f is pseudo-supermodular. However, it is not quasi-supermodular. Indeed, taking x5a, x9 5b we have f(x).f(xnx9) but f(xkx9),f(x9).
E. Agliardi / Economics Letters 68 (2000) 251 –254 253
property in (x,t) if for x9 .x0 and t9 .t0, (i) f(x9,t0)$f(x0,t0) implies f(x9,t9)$f(x0,t9) and (ii)
f(x9,t0).f(x0,t0) implies f(x9,t9).f(x0,t9). Furthermore, we say that f satisfies the additional (A) property in (x,t) if for t9 .t and for x,x9[L such that f(x9,t).f(x,t): (i) f(xkx9,t)$f(x,t) implies f(xkx9,t9)$f(x9,t9); and (ii) f(xkx9,t).f(x,t) implies f(xkx9,t9).f(x9,t9). Condition (A) implies that increasing a parameter (t) more than raises the marginal returns in activities.
Now we can state our first main result:
Proposition 3. Let f :LxT→R, where L is a lattice, T a partially ordered set and S,L, and letw(t,S )
be the set of optimal solutions to the problem maxx[S f(x,t). If f is pseudo-supermodular in x and satisfies the (SC ) and (A) properties in (x,t), thenw(t,S ) is monotone non-decreasing in (t,S ), that is, if
S#S9, t,t9, then w(t,S )#w(t9,S9).
Proof. Let S9 $S, t9 .t, x[w(t,S ) and x9[w(t9,S9). Consider xkx9; since x[w(t,S ) and S#S9, then f(x,t)$f(xnx9,t). By pseudo-supermodularity and the definitions of join and meet, we obtain the following string of inequalities: f(x,t)kf(x9,t)$f(x,t)$f(xnx9,t)5 .f(xkx9,t)$f(x,t)nf(x9,t). By the (SC) and (A) properties we get: f(xkx9,t)$f(x,t)nf(x9,t)5 .f(xkx9,t9)$f(x9,t9), hence
xkx9[w(t9,S9).
Similarly, we can show that xnx9[w(t,S ). Indeed, consider xnx9; since x9[w(t9,S9) and S#S9, then f(x9,t9)$f(xkx9,t9). By the (SC) and (A) properties, we get f(xkx9,t)#f(x9,t)nf(x,t), and by
pseudo-supermodularity: f(xkx9,t)#f(x,t)nf(x9,t)5 .f(x,t)#f(x,t)kf(x9,t)#f(xnx9,t), hence,
xnx9[w(t,S ). h
Remark 2. Let us give an example of a function that satisfies pseudo-supermodularity and the (SC ) and (A) properties, but fails quasi-supermodularity, and whose optimal solutions have the mono-tonicity properties as stated in Proposition 3.
Consider the lattice L introduced in Remark 1, let T5h1,2j, and take the function F :L3T→R,
tf (x) if x±b
F(x,t)5h , where f(0),f(a),f(c),f(b),2f(c), f(b).2f(0), for every t[T. It can be f (b ) if x5b
easily verified that F is pseudo-supermodular in x, satisfies the (SC ) and (A) properties in (x,t) and that the set of optimal solutions to the problem max F(x,t), s.t. x[S,L is non-decreasing in (t,S ), as stated in Proposition 3. However, F is not quasi-supermodular in x. Indeed, taking x5a, x9 5b, we have F(x,1).F(xnx9,1) but F(xkx9,1),F(x9,1).
Remark 3. Observe that the (SC ) property is also a necessary condition for w(t,S ) to be monotone non-decreasing in (t,S ). Indeed, let S5hx,x0j with x0 $x. Then f(x0,t)2f(x,t)$(or .)0 implies that
x0[w(t,S )#w(t0,S ) for t0 .t, so f(x0,t0)2f(x,t0)$(or .)0. Notice that pseudo-supermodularity is
also a necessary condition for non-decreasing monotonicity of w(t,S ), as defined in Milgrom and
Shannon (1994). Indeed, suppose that pseudo-supermodularity does not hold, that is, there exist x,x9,
with f(x,t)$f(x9,t), such that f(xnx9,t)#(or ,)f(x,t)kf(x9,t)5f(x,t) and f(xkx9,t),(or
#)f(x,t)nf(x9,t)5f(x9,t). Let S5hx, xnx9j and S9 5hx9, xkx9j; then S#S9. Thus argmaxS
f(x,t)#argmax f(x,t). However, x#x9 cannot hold, because it yields xnx9 5x, xkx9 5x9 and S9
therefore f(xkx9,t)5f(x9,t),f(x9,t), (or f(xnx9,t)5f(x,t),f(x,t)), a contradiction.
254 E. Agliardi / Economics Letters 68 (2000) 251 –254
m -vector x . The joint decision is xi i 5(x , . . . x ) and the set of feasible joint decisions is a subset L of1 n n
m
R where m5o i51m . Elements of L are called strategy profiles, Li 5L13 ? ? ? 3L . Each strategyn
set L is partially ordered byi $ and the strategy profiles are endowed with the product order, that is,
9
x$x9 means xi$x for all i. We assume that (L ,i i $) is a complete lattice for all i. The feasible decisions for a given player may depend on the decisions chosen by the other players. Let f (x) be thei
payoff player i gets as a result of a joint decision x[L, where f (x) is a real-valued function on L fori i51,..n. Let ( y , xi 2i)5(x ,... x1 i21, y , xi i11, . . . x ). A feasible joint decision xn [L is an
equilibrium point if f (x)i $f ( y , xi i 2i) for all yi[L (x), where L (x)i i 5hy :( y , xi i 2i)[Ljand i51, . . . n.
Definition. A game is a pseudo-supermodular game if for each i, L is a complete lattice, f (x) isi i pseudo-supermodular in x , and f (x) satisfies the (SC) and (A) properties in (x ,xi i i 2i).
Now, it is easy to prove the following result:
Proposition 4. Suppose a game is pseudo-supermodular and, for each i, L is compact, f is upperi i semi-continuous in x for xi 2i fixed, and continuous in x2i for fixed x . Then for each i there existi
¯
strategies x and x which are the smallest and largest strategies which survive strict pure iterated
i i
to z; then, it proceeds as in Milgrom and Roberts (1990), using the result of the previous step.] h
Proposition 4 states that all strategies which survive strict pure iterated admissibility lie in an
¯
interval (x,x ) whose maximum and minimum points are the largest and smallest Nash equilibria.]
Remark 4. Notice that the set of Nash equilibria of a pseudo-supermodular game need not be a
sublattice of L. Here is an example adapted from Topkis (1998). Let n53, m15m25m351,
Li5[0,1] for each i, and f (x)1 5f (x)2 5f (x)3 5x x x . Then (1,0,0) and (0,1,0) are Nash equilibria1 2 3 but (1,1,0)5(1,0,0)k(0,1,0) is not a Nash equilibrium, so the set of equilibrium points is not a
sublattice of L.
References
Milgrom, P., Roberts, J., 1990. Rationalizability, learning and equilibrium in games with strategic complementarities. Econometrica 58, 1255–1277.
