GLOBAL GAMES WITH STRATEGIC SUBSTITUTES
| Date | 01 February 2021 |
| Author | Rodrigo Harrison,Pedro Jara‐Moroni |
| Published date | 01 February 2021 |
| DOI | http://doi.org/10.1111/iere.12481 |
INTERNATIONALECONOMIC REVIEW
Vol. 62, No. 1, February 2021 DOI: 10.1111/iere.12481
GLOBAL GAMES WITH STRATEGIC SUBSTITUTES∗
By Rodrigo Harrison and Pedro Jara-Moroni1
Universidad Adolfo Ibañez, Chile ; Universidad de Santiago de Chile, Chile
We study global games with strategic substitutes. Specifically, for a class of binary-action, N-player games
with strategic substitutes, we prove that under payoff asymmetry, as incomplete information vanishes, the
global games approach selects a unique equilibrium. We characterize this equilibrium profile ; players employ
switching strategies at different cutoff signals, the order of which is directly determined by payoff asymmetry.
We provide examples that illustrate our result and its connection with dominance solvability. We extend the
global game literature, which has thus far been developed for games with strategic complementarities, to new
applications in industrial organization, collective action problems, finance, etc .
1. introduction
Global games are games of incomplete information, originally developed as equilibrium se-
lection devices. Incomplete information comes from a noisy payoff perturbation of a com-
plete information game, such that when the noise vanishes, we recover the original game. In
a global game, each player receives, with a small amount of noise, a private signal of the value
of a payoff’s fundamental. The noise distribution is common knowledge, so each player’s sig-
nal generates beliefs not only about the fundamental but also about the other players’ beliefs
(over the fundamental and beliefs of their rivals and so on). The idea of this approach is to ex-
amine Nash equilibria—of the original complete information game—as a limit of the equilib-
ria of the payoff-perturbed game.2
More precisely, consider a standard game of complete information with multiple equilibria,
in which the payoffs depend on a parameter that represents a fundamental. Instead of players
observing the true value of the parameter, suppose instead that players observe only a private
noisy signal of the fundamental. This is now a game of incomplete information, the equilib-
ria of which are profiles of functions that map signals to profiles of actions.3In particular, by
evaluating each Bayesian equilibrium in the true value of the fundamental, we obtain a set of
∗Manuscript received April 2019; revised June 2020.
1R. Harrison thanks financial support from FONDECYT, Grant No 1151123. P. Jara-Moroni acknowledges fund-
ing from Complex Engineering Systems Institute, ISCI (ICM-FIC: P05-004-F,CONICYT: FB0816). We thank the ref-
erees and the coeditor for comments and suggestions that substantially improved the content and presentation of the
article. We are especially grateful to Roger Lagunoff and Stephen Morris for valuable discussions and helpful com-
ments; we thank Muhamet Yildiz and Andrés Musalem for useful suggestions and seminar participants at Global
Games in Ames. This article incorporates material from the unpublished Ph.D. thesis entitled Essays in Global
Games by Rodrigo Harrison (see as well Harrison, 2003). Please address correspondence to: Rodrigo Harrison, Fac-
ultad de Ingeniería y Ciencias, Universidad Adolfo Ibañez, Diagonal Las Torres 2.700, Peñalolén, Santiago, Región
Metropolitana, Chile. E-mail: rodrigo.harrison@uai.cl.
2Global games were first introduced by Carlsson and van Damme (1993) (CvD hereafter) showing that in binary-
action, two-player games, there is a unique equilibrium selected through the iterative elimination of strictly domi-
nated strategies. This result was generalized by Frankel et al. (2003) to games with many players and many actions
in the context of strategic complementarities. For an excellent description and survey of the early literature on global
games and their applications, see Morris and Shin (2003). The globalgames selection approach has strong experimen-
tal support (see, for instance, Heinemann et al., 2009; Elbittar et al., 2014).
3The existence of Bayesian equilibrium, in the context of global games, has been proven by Hoffmann and Sabar-
wal (2019a).
141
© (2020) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of So-
cial and Economic Research Association
142 harrison and jara-moroni
action profiles. Then, equilibrium selection through this “perturbed game” is obtained when
the limit—as noise vanishes—of this set of action profiles is a strict subset of the set of equilib-
ria of the original complete information game.4In time, global games have become as well a
useful methodology for simplifying the analysis of high-order beliefs in strategic settings. Our
interest relates to their equilibrium selection application.
This approach has been proven to be very useful for modeling situations such as bank runs
(Goldstein and Pauzner, 2004, 2005), currency crises (Morris and Shin, 1998), herding behav-
ior (Chamley, 1999), platform pricing (Jullien and Pavan, 2019), and regime change (Szkup
and Trevino, 2015). Nevertheless, the literature on global games has been developed primar-
ily under the assumption of strategic complementarities (Kim, 1996; Frankel et al., 2003), this
is, in situations where the incentive to choose a higher action is increasing in the actions of the
opponents.5
In this article, we study the application of global games to equilibrium selection in games
with strategic substitutes. These are games in which each player’s marginal payoff from in-
creasing her own action is decreasing in the other players’ actions. It is well known that rel-
ative to the case with complementarities, the study of equilibria in environments with strate-
gic substitutes (even beyond the global games literature) is much more intricate. This article’s
contribution to the literature on global games is to prove and characterize the unique selec-
tion of an equilibrium in this context. Our framework accommodates several important eco-
nomic examples, such as the provision of a public good, entry games, and commons problems.
Our setting is based on Frankel et al. (2003) (FMP hereafter) where we pass from strate-
gic complements to strategic substitutes. For tractability, we include two variations: (i) instead
of any countable union of subsets of the [0,1] interval, the action set is {0,1}, and (ii) the
payoffs in our setting depend on the number of players that take action 1, whereas in FMP,
there is no specific dependence on strategy profiles, apart from requiring strategic comple-
ments. The key insight of this article is to show that if players’ payoffs display a certain com-
monly known asymmetry, then a unique equilibrium selection may be obtained in games with
strategic substitutes applying global games ideas. Specifically, we show that as the noise goes
to zero, any equilibrium profile of the Bayesian global game converges to a single profile of
switching strategies. In such a profile, each player has a threshold signal (cutoff point), above
which she takes the “higher” action and below which she takes the “lower” action.6A very
important characteristic of this profile is that each player has a different cutoff point. Inter-
estingly, the order of these cutoff points is directly determined by payoff asymmetry. A simple
setting, commonly used in economic models, that satisfies our requirement of payoff asymme-
try is when the player’s marginal payoff from increasing her own action takes the form of a
common marginal net payoff minus an idiosyncratic cost (i.e., πi=π −ci).
To better motivate this work, think of an entry game in which firms first decide whether
to enter or not a market and then if entering they engage in oligopolistic competition. The
multiplicity of equilibria is a natural feature of these games under complete information,
which produces an identification problem in the structural empirical analysis. This problem
has been addressed in the empirical industrial organization literature (Bresnahan and Reiss,
1990; Berry, 1992; Scott Morton, 1999; Berry and Reiss, 2007), and some solutions have been
proposed in the theoretical literature. Espin-Sanchez and Parra (2018) provide equilibrium se-
lection by adding private information (see footnote ), and Quint and Einav (2005) propose a
war of attrition in the entry game that has a unique equilibrium outcome. Generally, payoffs
in an entry game are the market equilibrium payoff minus the entry cost if entering and zero
otherwise; thus, if firms differ only in the entry cost, then such a game fits the setting described
4Our analysis includes the private values case, in which the payoffs might depend directly on the signal instead of
on the fundamental.
5Recent results include those of Angeletos and Pavan (2013), Basteck et al. (2013), Oury (2013), Frankel (2012),
Honda (2011), Oyama and Takahashi(2011), and Basteck and Daniëls (2011).
6In any binary-action setting, we can label one action as the “higher” action and the other as the “lower” action ac-
cordingly.
global games, strategic substitutes 143
at the end of the previous paragraph. Under the standard assumption that market equilibrium
payoffs are decreasing in the number of firms that enter the market, the first-stage entry game
becomes one of strategic substitutes. Thus, by adding “a small amount” of incomplete infor-
mation on market demand, the global games approach developed in this article may then pro-
vide a method to select a unique equilibrium in the entry game, which might be useful for em-
pirical analysis (see Example 2 in Section 4).7
Finally, it is important to note that the equilibrium selection result of FMP is obtained
through the iterative elimination of strictly dominated strategies. We show below that in our
context, as is common in the strategic substitutes environment, the iterative elimination of
strictly dominated strategies may not provide a unique outcome, so this technique cannot
be used to prove unique equilibrium selection as in the strategic complements literature. In
games with strategic complements, dominance solvability is equivalent to the uniqueness of
equilibrium (Milgrom and Roberts, 1990), whereas games with strategic substitutes may have
a unique equilibrium that is not the unique rationalizable strategy profile (Zimper, 2007;
Guesnerie and Jara-Moroni, 2011).8Consequently, our method relies on a closely related con-
cept of elimination of nonequilibrium strategies, the result of which is still guaranteed to be
the unique (Bayesian) Nash equilibrium but possibly not the unique rationalizable strategy
profile. In this work, we provide a proof of the unique equilibrium selection using the global
games approach applied to a class of N-player games with strategic substitutes.
1.1. Relation with the Literature. Global games with strategic substitutes have not been as
thoroughly studied as have global games with strategic complements. From this literature, we
may point to Karp et al. (2007), which studies a game that presents both strategic comple-
ments and strategic substitutes, with an emphasis on the existence of equilibrium. The dif-
ference from our work is that our interest is on unique equilibrium selection under strategic
substitutes. Recently, Hoffmann and Sabarwal (2019b) studied equilibrium selection in global
games with strategic substitutes and/or complements. In a different approach, they use a p-
dominant condition as a selection criterion, which does not necessarily deliver a unique so-
lution over the entire uncertainty set as our result does. Finally, Morris and Shin (2009) and
Harrison and Jara-Moroni (2015) explore the concept of strong rationality (Guesnerie, 1992)
in the context of global games with strategic substitutes instead of equilibrium selection.9
Our main result allows us to study strategic situations that are not addressed in the previ-
ous literature. Below, we provide two examples: the voluntary contribution to the provision
of a public good and entry games in oligopolistic markets. Other situations in which strate-
gic substitutes may naturally arise are games with negative externalities (polluting and con-
gestion games, commons problems, etc.), free-riding problems (team work games and public
good provision), and imperfect competition (entry games and Cournot competition).
The remainder of the article is organized as follows: In Section 2, we motivate through
an example. In Section 3, we present the general framework. In Section 4, we establish our
main result and provide illustrative examples. In Section 5, we discuss the relationship be-
tween our result and the question of dominance solvability, a concept that is directly re-
lated to equilibrium uniqueness under strategic complements (Milgrom and Roberts, 1990;
7A detailed study of this type of games is developed in Harrison and Jara-Moroni (2019).
8In games with either strategic complements or substitutes, the processes of elimination of nonrationalizable strat-
egy profiles may be carried out using the best reply function of the game. Starting from the “greatest” and “small-
est” strategy profiles, at each step, one iterates this operator, generating two sequences of strategy profiles such that
the rationalizable set is bounded by the elements of these sequences.In games of strategic complements, the best re-
ply function is increasing, so these processes stop at fixed points of the best reply function, which are equilibria. Thus,
if the equilibrium is unique, then the game is dominance-solvable. In contrast, in games of strategic substitutes, the
best reply function is decreasing, so these processes stop at fixed points of the second iterate of the best reply func-
tion, which are rationalizable but not necessarily equilibria. Therefore, in games of strategic substitutes, we may have
a unique equilibrium that is not the unique profile that survives the iterative elimination of strictly dominated strate-
gies.
9This is global game equilibrium selection through the iterative elimination of strictly dominated strategies.
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