Games with strategic complements and substitutes

TLDR: In this article, the authors studied games with both strategic substitutes and strategic complements, and more generally, games with strategic heterogeneity (GSH), showing that the equilibrium set in a GSH is totally unordered (no two equilibria are comparable in the standard product order), and under mild assumptions (on one player only), parameterized GSH do not allow decreasing equilibrium selections.

Abstract: This paper studies games with both strategic substitutes and strategic complements, and more generally, games with strategic heterogeneity (GSH). Such games may behave differently from either games with strategic complements or games with strategic substitutes. Under mild assumptions (on one or two players only), the equilibrium set in a GSH is totally unordered (no two equilibria are comparable in the standard product order). Moreover, under mild assumptions (on one player only), parameterized GSH do not allow decreasing equilibrium selections. In general, this cannot be strengthened to conclude increasing selections. Monotone comparative statics results are presented for games in which some players exhibit strategic substitutes and others exhibit strategic complements. For two-player games with linearly ordered strategy spaces, there is a characterization. More generally, there are sufficient conditions. The conditions apply only to players exhibiting strategic substitutes; no additional conditions are needed for players with strategic complements. Several examples highlight the results.

Figures

Figure 3: Cournot Duopoly with Spillovers

Figure 7: Increasing Equilibria: Differentiated Duopoly

Figure 8: Non-increasing Equilibria: Differentiated Duopoly

Figure 4: Matching Pennies: Double-or-Nothing, Part 2

Figure 12: Equilibrium Outcome of Strategic Complement Player Goes Down

Figure 6: Non-Decreasing Equilibria For Cournot Duopoly With Spillovers

Full text

Games with Strategic Complements and Substitutes
By
Andrew J. Monaco Tarun Sabarwal
Department of Economics Department of Economics
University of Puget Sound University of Kansas
Tacoma, WA, 98416, USA Lawrence KS, 66045, USA
amonaco@pugetsound.edu sabarwal@ku.edu
Abstract
This paper studies games with both strategic substitutes and strategic complements,
and more generally, games with strategic heterogeneity (GSH). Such games may behave
differently from either games with strategic complements or games with strategic sub-
stitutes. Under mild assumptions (on one or two players only), the equilibrium set in
a GSH is totally unordered (no two equilibria are comparable in the standard product
order). Moreover, under mild assumptions (on one player only), parameterized GSH do
not allow decreasing equilibrium selections. In general, this cannot be strengthened to
conclude increasing selections. Monotone comparative statics results ar e presented fo r
games in which some players exhibit strategic substitutes and others exhibit strategic
complements. For two-player games with linearly ordered strategy spaces, there is a char-
acterization. More generally, there are sufficient conditions. The conditions apply only
to players exhibiting strategic substitutes; no additional conditions are needed for players
with strategic complements. Several examples highlight the results.
JEL Numbers: C70, C72
Keywords: Lattice games, strat egic complements, strategic substitutes, strategic hetero-
geneity, equilibrium set, monotone comparative statics
First Draft: April 2 010
This Version: January 26, 2015

1 Introduction
Games with strategic substitutes (GSS) and games with strategic complements (GSC)
formalize two basic strategic interactions and have widespread applications. In GSC, best
response of each player is weakly increasing in actions of the other players, whereas GSS
have the characteristic that the best response of each player is weakly decreasing in the
actions of the other players.
1
This paper focuses on games with bot h strategic substitutes and strategic comple-
ments. Relatively little is known about such games even though several classes of inter-
actions fall in this category. For example, a classic application in Singh and Vives (1984)
considers a duopoly in which one firm behaves as a Cournot-firm (exhibiting strategic
substitutes) and the other as a Bertrand-firm (with stra t egic complements). Variations of
the classic matching pennies game provide other examples. A Becker (1968) type game of
crime and law enforcement is another example: the criminal exhibits strategic substitutes
(the greater is law enforcement, the lower is crime) and the police exhibit strategic com-
plements (the gr eater is crime, the greater is law enforcement). Such games also arise in
studies of pre-commitment in industries with learning effects, see Tombak (2006). More-
over, Fudenberg and Tirole (1984) and Dixit (1987) present examples of pre-commitment
where the strategic property of one player’s action is opposite to that of the other player.
More recent examples are found in Shadmehr and Bernhardt ( 2011), analyzing collective
1
There is a long literature developing the theor y of GSC. Some of this work can be seen in Topkis
(1978), Topkis (1979), Bulow, Geanakoplos, and Klemperer (1985), Lippman, Mamer, and McCardle
(1987), Sobel (1988), Milgrom and Robe rts (1990), Vives (1990), Milgro m and Shannon (1994), Mil-
grom and Roberts (1994), Zhou (1994), Shannon (1995), Villas-Boas (1997), Edlin and Shannon (19 98),
Echenique (2002), Echenique (2004), Quah (2007), and Quah and Strulovici (2009), among others. Ex-
tensive bibliographies are available in Topkis (1998), in Vives (1999), and in Vives (2005). There is a
growing literature on GSS: confer Amir (1996), Villas-Boas (1997), Amir and Lambson (2000), Schipper
(2003), Zimper (2007), Roy and Sabarwal (2008), Acemoglu and Jensen (2009), Amir, Garcia, and Knauff
(2010), Acemoglu and Jensen (2010), Roy and Sabarwal (2010), Jensen (2010), and Roy and Sabarwal
(2012), among others.
1

actions in citizen protests and r evolutions, and Baliga and Sjostrom ( 2012), analyzing
third-party incentives to manipulate conflict between two players.
Games with both strategic substitutes and strategic complements are the basis for
our more general notion of a game with strat egic heterogeneity (GSH), which, in prin-
ciple, allows for arbitrary strategic heterogeneity amo ng players. Moreover, the unified
framework of GSH helps clarify the scope of results found separately for GSC or GSS.
We present three main results.
First, we show that under mild conditions, the equilibrium set in a GSH is totally
unordered (no two equilibria ar e comparable in the standard product order). These con-
ditions can ta ke one of two forms: either just o ne player has strictly decreasing and
singleton-valued best response, o r one player has strictly decreasing best response and
one player has strictly increasing best response; in either case there are no restrictions on
strategic interactions among other players. Three implications of this result are no table.
Firstly, the nice order and structure properties of the equilibrium set in GSC
2
do not sur-
vive a minimal introduction of strategic substitutes, in the sense that if we modify a GSC
so that just one player has strict strategic substitutes
3
and ha s a singleton-valued best
response, then the order structure of the equilibrium set is destroyed completely; no two
equilibria are comparable. Similarly, if we modify a GSC to require that one player has
strict strategic complements,
4
and another has strict strategic substitutes, then again the
order structure of the equilibrium set is destroyed completely. Secondly, the non-ordered
nature of equilibria implies that starting from one equilibrium, algorithms to compute
another equilibrium may be made more efficient by discarding two areas of the strategy
2
The equilibrium set in a GSC always has a smallest and a largest equilibrium, and more generally,
the equilibrium set is a non-empty, complete lattice. These properties are useful to provide simple and
intuitive algor ithms to compute equilibria and to show monoto ne comparative statics of equilibria in
GSC. In contrast, in GSS, the equilibrium set is totally unordered: no two equilibria are comparable in
the sta ndard product order.
3
Intuitively, best response is strictly decreasing in o ther player strategies.
4
Intuitively, best response is strictly increasing in other player strategies.
2

space. Thirdly, if player strat egy spaces are linearly ordered,
5
then the set of symmetric
equilibria is no n-empty, if, and only if, there is a unique symmetric equilibrium.
6
There-
fore, in such cases, there is at most one symmetric equilibrium. In t his regard, a game
with both stra t egic substitutes and strategic complements is different from a GSC and
resembles more the results for a GSS.
Second, we show that under mild conditions, parameterized GSH do not allow decreas-
ing equilibrium selections (as the parameter increases, equilibria do not decrease). These
conditions can take one of two forms: either just one player has strict strategic substitutes
and singleton-valued best response, or just one player has strict strategic substitutes and
strict single-crossing property in (own variable; parameter); in either case, there a r e no
restrictions on strategic interaction among other players. Recall that in a GSC, (leaving
aside stability issues,) it is possible to find a higher equilibrium at a lower parameter and
a lower equilibrium at a higher parameter. In a GSS, however, there are no decreasing
equilibrium selections. Therefore, our second result implies that decreasing selections in
a GSC ar e eliminated with a “minimal” intr oduction of strat egic substitutes. Moreover,
an example shows that our second result cannot be strengthened to yield increasing equi-
libria more generally. In this regard, too, a GSH is different from a GSC and more closely
resembles known results for a GSS.
Third, we present monotone comparative statics results (at a hig her parameter value,
there are equilibria in which all players take a higher action) for games in which some
players exhibit strategic substitutes and others exhibit strategic complements. For two-
player games in which one player exhibits strategic substitutes, the other player exhibits
strategic complements, and each player ha s a linearly ordered strategy space, we char-
acterize mo notone comparative statics via a condition on the best response of only the
player with strategic substitutes. (No additional condition is imposed on the player with
5
As usual, a partially ordered set is linearly ordered, if the partial or de r is complete; that is, every
two elements are comparable.
6
As usual, in a symmetric equilibrium, each player play s the same strategy.
3

strategic complements.) The condition is intuitive and is based on a tr ade-off between
the direct parameter effect and the indirect strategic substitute effect. Notably, the same
condition works for GSS in a similar setting when best responses a re singleton-valued.
We present examples to show that this characterization does not hold when there are
more than two players or when strategy spaces are not linearly ordered. For the more
general case, when some players exhibit strategic substitutes and others exhibit strategic
complements, we present sufficient conditions that guarantee monotone comparative stat-
ics. As in the two-player case, these conditions are needed only for players with strategic
substitutes. The conditions ar e stronger than in the two-player case, but still involve a
trade-off between the direct parameter effect and the indirect strategic substitute effect.
In this regard, games with both strategic substitutes and complements behave differently
from either GSC or GSS.
Recall that if an analyst can choose a new order, then Echenique (20 04) shows that
there may exist partial orders in which a strategic game can be viewed as a GSC. This
approach is useful when a partial order is not intrinsic to the game, and its choice does not
materially aff ect the interpretation of “more” and “less”. The framework in this paper
is more a ppropriate when there is a natural order on a player’s strat egy space and an
interest in equilibrium predictions and comparative statics in this or der. For example,
when considering the impact of ta xes or subsidies on firm output, a natural order on
output space is the standard order on the real numbers, (and not some ot her order in
which the game may be viewed as a GSC). In our fr amework, the order on a player’s
strategy space is considered a fixed primitive of the game.
The paper proceeds a s follows. Section 2 defines games with strategic heterogene-
ity and presents the first main result o n the structure of the equilibrium set in such
games. Section 3 defines parameterized games with strategic heterogeneity, sub-section
3.1 presents the second main result on non-decreasing equilibrium selections, and sub-
section 3.2 presents the third main result on monotone comparative stat ics. Section 4
concludes.
4

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Games with strategic complements and substitutes" ?

This paper studies games with both strategic substitutes and strategic complements, and more generally, games with strategic heterogeneity ( GSH ). 

Q2. What is the response correspondence of player i?

Recall that if the payoff function of player i is quasi-supermodular in xi, and satisfies the single-crossing property in (xi; x−i), then the best response correspondence of player i is nondecreasing in the induced set order. 

Q3. What is the order structure of the equilibrium set in a game with strategic complements?

If the authors modify this game to require that just one player has strict strategic substitutes, and that player’s best response is singleton-valued (perhaps because that payoff function is strictly quasi-concave), then the order structure of the equilibrium set is destroyed completely. 

Q4. What is the response of player i?

Let us formalize this by saying that player i has strategic substitutes, if player i’s best response correspondence βi is non-increasing in x−i in the induced set order. 

Q5. What are the standard assumptions used to guarantee existence of equilibrium?

Toward the end of the paper, in theorems 5 and 6, the authors make standard assumptions to guarantee existence of equilibrium; these are used to guarantee existence of a “higher” equilibrium. 

Q6. What is the case in case 2?

without loss of generality, that player 1 has strict strategic substitutes, player 2 has strict strategic complements, and suppose the distinct equilibria are comparable, with x̂ ≺ x∗. 

Q7. How can the authors construct a similar example with three players?

For every t∗ t̂ and every x∗ ∈ E(t∗), let ŷ = (ŷi) The authori=1 be defined as follows: ŷi = β i t̂ (x∗−i),19It is possible to formulate a similar example with three players, each with linearly ordered strategy space. 

Q8. What is the first result of the theorem?

Their first result, theorem 1 shows how a single player with (strict) strategic substitutescan destroy the order structure of the equilibrium set. 

Q9. What is the response of the player 2 to the game of double-or-no?

If both players go for double-or-nothing and the pennies match (that is, the outcome is (H,H) and (H,H), or (T, T ) and (T, T )), player 2 wins $2 from player 1, and if both pennies do not match (the outcome is (H,H); (T, T ), or (T, T ); (H,H)), player 1 wins$2 from player 2. 

Q10. What is the first result of the GSH?

without loss of generality, that player 1 has strict strategic substitutes with singleton-valued best response, and suppose the distinct equilibria x̂ and x∗ are comparable, with x̂ ≺ x∗. 

Q11. What is the response of the two players?

Let t = a1 = a2, and rewrite best responses as follows: for firm 1, β 1 t (q2) = t−cq2 2 , and for firm 2, β2t (p1) = tb1−tc+cp1 2(b1b2−c2) , and notice that best response of both players is increasing in t. Suppose t = 2, b1 = b2 = 2, and c = 1. 

Q12. What is the response in the duopoly with spillovers example?

Recall the best responses in the Cournot duopoly with spillovers example above, β1(x2) = a−bx2−c12b , andβ2(x1) = a−bx1−c2s(x1)2b , and the spillover function, s(x1) = 2 3 x31−x 2 1− x1 2 +3, and consider aas the parameter.