the strategic advantage of negatively interdependent preferences

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Journal of Economic Theory 92, 274299 (2000) The Strategic Advantage of Negatively Interdependent Preferences 1 Levent Kockesen Department of Economics, Columbia University, New York, New York 10027 Efe A. Ok Department of Economics, New York University, 269 Mercer Street, New York, New York 10003 okefefasecon.econ.nyu.edu and Rajiv Sethi Department of Economics, Barnard College, Columbia University, New York, New York 10027 Received March 28, 1998; revised August 15, 1999 We study certain classes of supermodular and submodular games which are symmetric with respect to material payoffs but in which not all players seek to maximize their material payoffs. Specifically, a subset of players have negatively interdependent preferences and care not only about their own material payoffs but also about their payoffs relative to others. We identify sufficient conditions under which members of the latter group have a strategic advantage in the following sense: at all intragroup symmetric equilibria of the game, they earn strictly higher material payoffs than do players who seek to maximize their material payoffs. The conditions are satisfied by a number of games of economic importance. We discuss the implications of these findings for the evolutionary theory of preference formation and the theory of strategic delegation. Journal of Economic Literature Classification Numbers: C72, D62. 2000 Academic Press doi:10.1006jeth.1999.2587, available online at http:www.idealibrary.com on 274 0022-053100 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved. 1 We thank an associate editor and an anonymous referee for suggesting a number of improvements. We are also grateful J. P. Benoit, Eddie Dekel, Duncan Foley, Boyan Jovanovic, Roy Radner, Debraj Ray, Ariel Rubinstein, and seminar participants at Alicante, Columbia, Cornell, Duke, Michigan, NYU, Rutgers, Penn State, Princeton, and University College London for helpful comments. Support from the C. V. Starr Center for Applied Economics at New York University is gratefully acknowledged.

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Page 1: The Strategic Advantage of Negatively Interdependent Preferences

Journal of Economic Theory 92, 274�299 (2000)

The Strategic Advantage of Negatively InterdependentPreferences1

Levent Koc� kesen

Department of Economics, Columbia University, New York, New York 10027

Efe A. Ok

Department of Economics, New York University, 269 Mercer Street,New York, New York 10003

okefe�fasecon.econ.nyu.edu

and

Rajiv Sethi

Department of Economics, Barnard College, Columbia University,New York, New York 10027

Received March 28, 1998; revised August 15, 1999

We study certain classes of supermodular and submodular games which aresymmetric with respect to material payoffs but in which not all players seek tomaximize their material payoffs. Specifically, a subset of players have negativelyinterdependent preferences and care not only about their own material payoffs butalso about their payoffs relative to others. We identify sufficient conditions underwhich members of the latter group have a strategic advantage in the followingsense: at all intragroup symmetric equilibria of the game, they earn strictly highermaterial payoffs than do players who seek to maximize their material payoffs. Theconditions are satisfied by a number of games of economic importance. We discussthe implications of these findings for the evolutionary theory of preference formationand the theory of strategic delegation. Journal of Economic Literature ClassificationNumbers: C72, D62. � 2000 Academic Press

doi:10.1006�jeth.1999.2587, available online at http:��www.idealibrary.com on

2740022-0531�00 �35.00Copyright � 2000 by Academic PressAll rights of reproduction in any form reserved.

1 We thank an associate editor and an anonymous referee for suggesting a number ofimprovements. We are also grateful J. P. Benoit, Eddie Dekel, Duncan Foley, BoyanJovanovic, Roy Radner, Debraj Ray, Ariel Rubinstein, and seminar participants at Alicante,Columbia, Cornell, Duke, Michigan, NYU, Rutgers, Penn State, Princeton, and UniversityCollege London for helpful comments. Support from the C. V. Starr Center for AppliedEconomics at New York University is gratefully acknowledged.

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1. INTRODUCTION

A fundamental ingredient of most economic models is the hypothesis ofindependent preferences: agents choose their actions with the sole purposeof maximizing their own material payoffs regardless of how their actionsaffect the payoffs of other individuals. While this postulate is seldom givenexplicit justification, it appears to be based on the intuition that thoseindividuals who are willing to make material sacrifices to affect the payoffsof others will lose wealth relative to those who are unwilling to do so, withthe eventual consequence that the latter will come to dominate the economy.In this case, the maximization of one's own material payoffs would simply bea precondition for survival in an environment where a competitive selectionprocess is at work. While this intuition may be persuasive in the context ofperfectly competitive environments, it can be seriously misleading whenapplied to strategic settings, for it is not generally true in such environ-ments that agents who pursue the maximization of their own materialpayoffs will obtain higher material payoffs in equilibrium than symmetri-cally placed individuals who maximize other objective functions.

This last point has been demonstrated in the literature mostly by meansof particular specifications of Cournot oligopoly models. For instance,Vickers [23] and Fershtman and Judd [7] have shown in linear versionsof such models that a firm whose objective function gives a positive weightto its relative profits or sales will outperform absolute profit maximizers interms of absolute profits. Similar results obtain in some other strategic environ-ments, such as common pool resource and public good games [11], in whichagents with negatively interdependent preferences (that is, those who careabout both absolute and relative payoffs) may obtain greater absolute payoffsin equilibrium than do symmetrically placed absolute payoff maximizers. Insuch environments, interdependent preferences may be said to yield a strategicadvantage to those who possess them.

The broader significance of such findings rests on the extent to whichthey are valid in a large set of economic environments. Accordingly, ouraim in this paper is to provide a general analysis of the conditions underwhich negatively interdependent preferences yield an unambiguous strategicadvantage over independent preferences. We consider classes of supermodularand submodular games in which only a subset of players have independentobjective functions whereas the rest have negatively interdependent preferences.Informally stated, we identify several sets of sufficient conditions under whichthe members of the latter group have a strategic advantage in the followingsense: at all intragroup symmetric equilibria, the interdependent individualsearn higher material payoffs than do players who seek to maximize their ownmaterial payoffs. It turns out that the class of games in which interdependentpreferences have a strategic advantage in this sense is unexpectedly rich.

275NEGATIVELY INTERDEPENDENT PREFERENCES

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To motivate the present inquiry from a purely game-theoretic viewpoint,consider a two-person game in which player 1 wishes to maximize his orher monetary reward while player 2 cares also about ``beating'' player 1,that is, making more money than player 1. The question we ask is: whichof the two players will make more money in equilibrium? It is not difficultto see that the answer is not trivial and depends on the structure of thegame. Using the familiar notions of strategic complementarity and sub-stitutability, we show in this paper that in many (but not all) games ofinterest, it is, in fact, player 2 who will outperform player 1 in equilibrium.Put differently, we identify subclasses of supermodular and submodularnormal form games in which an envious concern with the payoffs of othersleads one to have greater absolute payoffs in equilibrium than those obtainedby (absolute) payoff maximizers. We contend that our results achieve a usefullevel of generality, for our sufficiency conditions are satisfied by a numberof games which play central roles in various branches of economic theory,including strategic market games, search models, input and public goodgames, and arms races.

The analysis of strategic advantage also leads to interesting applications,two of which we study here in some detail. Our first application concernsthe analysis of oligopolistic industries and stems from the fact that executivemanagers may in some circumstances either choose or be given incentivesby owners to incorporate relative profit (or market share) concerns intotheir decision making. For instance, if managers' efforts are unobservableby owners and there is some common uncertainty affecting all firms in theindustry, owners may benefit from making their managers' compensationcontingent upon relative as well as absolute profits (Holmstro� m [9]). Asa corollary of our results, we find here that such compensation schemesmay yield an unplanned strategic advantage to a firm in terms of itsabsolute profits. This occurs in supermodular market games, including thestandard Bertrand model with differentiated products, and in submodulargames, including the standard case of Cournot competition. In the sub-modular case, moreover, our results allow us to show that objective functionswhich value relative profits may arise as perfect equilibrium outcomes intwo-stage delegation games in which owners select the objective functionsof their managers in the first stage. This finding conforms with and extendsto a class of submodular games the results obtained in the theory ofstrategic delegation by Vickers [23] and Fershtman and Judd [7]. It is auseful illustration of the power of the strategic advantage approach that wedevelop here.

As a second application, we consider the theory of preference evolution.Evolutionary models of preference formation are typically based on theassumption that the selection dynamics are payoff monotonic: the popula-tion share of those endowed with preferences that are more highly rewarded

276 KOC� KESEN, OK, AND SETHI

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materially increases relative to the population share of those who are lesshighly rewarded. In the presence of such selection dynamics, our results enableus to identify the evolutionary stability properties of absolute-payoff-maximiz-ing behavior in a class of environments in which a finite population interactsstrategically, with each player matched with each other player (the ``playingthe field'' model). Specifically, we identify environments in which the long-runpopulation composition cannot be a monomorphic one composed only ofabsolute payoff maximizers. These findings are then compared with theearlier evolutionary literature on the ``spiteful effect,'' which reachesbroadly similar conclusions. The idea behind the spiteful effect is concep-tually close to the idea of strategic advantage, and it turns out that thepresence of the spiteful effect is necessary but not sufficient for the strategicadvantage of negatively interdependent preferences. Since our sufficientconditions are based on the primitives of a game (in contrast with thespiteful effect), they serve also as a means of identifying games in which thespiteful effect is present.

The paper is organized as follows. In Section 2 we introduce our generalframework and formalize the nature of the present inquiry. Section 3contains our main results, which identify classes of supermodular and sub-modular games in which interdependent players have a strategic advantageover independent players at all intragroup symmetric equilibria. In Section 4,we elaborate on the implications of our main findings for the theories ofoligopolistic competition and preference formation. Section 5 concludes.

2. THE FRAMEWORK

Since our ultimate aim is to compare the performance of different preferencestructures in terms of material outcomes, we shall concentrate on games instrategic form in which no player has an a priori advantage in terms of theprimitives of the game. Consequently, our focus will be on symmetricgames. Given any integer n�2, we let 1 stand for a symmetric n-persongame in normal form. That is,

1#(X, [?r]r=1, ..., n),

where X and ?r : Xn � R are the action space and the absolute payoff func-tion of player r, where ?r(x)=?q(x$) for all r, q=1, ..., n and all x, x$ # X n

such that x$ is obtained from x by exchanging xr and xq . As is usually donein applied and experimental game theory, we interpret ?r as the materialpayoff function of player r. To be able to interpret the notion of ``relative

277NEGATIVELY INTERDEPENDENT PREFERENCES

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payoffs' in the usual manner, we assume that 1 satisfies the following non-negativity conditions:

?r(x)�0 \x # X n and r=1, ..., n. (1)

Moreover, we endow X with a linear order � to obtain a chain.2 The classof all 1 that satisfy these assumptions is denoted G, and we let N(1 ) standfor the set of all Nash equilibria of 1 # G.

For much of this paper we shall assume that the set of players consistsof two different types, namely, independent and (negatively) interdependenttypes. The independent players are those who are absolute payoff maxi-mizers in the usual sense; the objective function of an independent playeri is precisely his or her own material payoff function ?i . On the other hand,(negatively) interdependent players are concerned not only with their absolutepayoffs, but also with how their absolute payoffs compare with the averagepayoff in the game. Let ?� #� ?r�N denote the average (absolute) payofffunction on Xn, and define the relative payoff of player j as follows:3

\j#{?j �?� ,0,

if ?j>0if ?j=0.

(2)

The objective function of an interdependent player j is given by x [ F(?j , \j),where F is an arbitrary strictly increasing real function on R2

+ . This particularway of representing negatively interdependent preferences has recently beenproposed and axiomatically characterized to Ok and Koc� kesen [15]. Inparticular, when 1 is played between individuals (as opposed to, say,firms), the preferences represented in this form can be interpreted as acompromise between the standard case in which individuals are assumed tocare only about their monetary earnings ?j and the other extreme in whichthey are concerned exclusively with their relative payoffs \j .

4 If, on theother hand, 1 is an oligopoly game, then an interdependent player withsuch an objective function can be thought of as a firm (or manager) whichnot only is concerned with the level of its profits, but also cares about itsprofit share in the industry.

278 KOC� KESEN, OK, AND SETHI

2 This is not an excessively demanding structural assumption insofar as applications areconcerned. In many economic contexts one has X�R so that X is linearly ordered in anatural way.

3 We use the convention of setting \ j (x)=0 whenever ?j (x)=0 to avoid the difficulty ofevaluating the indeterminate form 0�0.

4 Special cases of this representation of interdependent preferences are utilized in numerouseconomic contexts ranging from models of optimal income taxation to experimental bargaininggames. We refer the reader to the references cited in Ok and Koc� kesen [15].

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Suppose that precisely k # [1, ..., n&1] players in 1 # G are independent,and denote the set of all strictly increasing F: R2

+ � R by F. For any F # F,we define the n-person normal form game

1F (k)#(X, [ pr]r=1, ..., n)

with

pr #{?r ,F(?r , \r),

if r # Ik

if r # Jk ,(3)

where Ik #[1, ..., k] and Jk #[k+1, ..., n]. Clearly, in 1F (k), the set ofall independent players is Ik , and the set of all interdependent players is Jk .The crucial interpretation is that, while an uninformed outsider may onlyobserve the payoffs associated with the game 1, the players themselves areengaged in playing 1F (k).

In this paper, we wish to analyze the nature of pure strategy Nash equi-libria of an arbitrary 1F (k). We note at the onset that there are twoimmediate difficulties which shall be assumed away in much of the analysisthat follows. First, the existence of a Nash equilibrium of 1F (k) is ratherdifficult to establish in general. Even if we take X�Rl and posit the standardrequirement of quasi concavity of ?r in xr for all r (along with continuityof ?r and compactness and convexity of X), the payoff function pj , j # Jk ,need not inherit this property. Even the deeper existence theorems estab-lished in the literature (such as those of [5, 21]) are not readily helpful insettling this existence problem. It appears that the best strategy at this stageis to ignore the existence problem and search for some qualitative proper-ties of the equilibria of 1F (k), when they exist. In fact, in many examplesof economic interest (such as the Cournot and Bertrand oligopolies, commonpool resource and public good games, and arms races) one can directly verifythat the set of equilibria of 1F (k) is nonempty, and hence our approach isfruitful.

The second difficulty is the analytical intractability of certain asymmetricequilibria of an arbitrary 1F (k). The analysis is greatly simplified when wefocus instead on the intragroup symmetric Nash equilibria of 1F (k), denotedNsym(1F (k)), which are defined as

Nsym(1F (k))#[([a]k , [b]n&k) # N(1F (k)) : a, b # X],

where [t] l denotes the l-replication of the object t. One could, of course,advance a ``focal point'' argument to justify interest in Nsym(1F (k)). Perhapsmore importantly, we shall observe that in most of the economic examplesconsidered below, we actually have N(1F (k))=Nsym(1F (k)), so a focus onintragroup symmetric equilibria is unrestrictive. This is trivially the case inall two-person games.

279NEGATIVELY INTERDEPENDENT PREFERENCES

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Finally, let us clarify what we mean by ``studying the nature of Nsym(1F (k))''given a 1 # G. Put precisely, we are interested in identifying some generalsubclasses of G where the interdependent players have a strategic advantageover the independent players in terms of monetary payoffs, that is, where

?j (x̂)>?i (x̂) \(i, j) # Ik_Jk and x̂ # Nsym(1F (k)). (4)

As noted in the Introduction, there are at least two concrete economicconsiderations motivating this inquiry. Whether interdependent players(who do not directly maximize their absolute payoffs) obtain higher absolutepayoffs than all independent players (who do target the maximization of theirabsolute payoffs) is a question of considerable interest in theories of preferenceevolution and strategic delegation. These applications are discussed inSection 4 below.

3. MAIN RESULTS

3.1. Supermodular Games

An n-person normal form game 1 # G is said to be supermodular whenever

?r(x6 y)+?r(x 7 y)�?r(x)+?r( y) \x, y # Xn and r=1, ..., n,

where x 6 y is the lowest upper bound of [x, y] in Xn (with respect to theproduct order induced by � ) and x 7y is the greatest lower bound of[x, y] in Xn.5 We say that 1 is strictly supermodular if the above inequalityholds strictly for all r and x, y # Xn such that [x 6 y, x 7 y]{[x, y].Supermodular games correspond to games in which the actions of twodistinct players are strategic complements in the sense that the best responsecorrespondences of the players are increasing (Bulow et al. [3], Topkis [21],Vives [24]). It is well known that if X�Rl is open and ?r is C2, then 1 issupermodular if and only if �2?r ��xr �xq�0 for all r{q (Topkis [20]).

We next introduce the following subclass of G.

Definition. An n-person normal form game 1 # G is said to bepositively (negatively) action monotonic if, for all x # Xn, xr o ( O ) xq

implies ?r(x)>?q(x).

Action monotonicity is a property that requires a tight connectionbetween payoffs and actions. While it is not a standard condition for

280 KOC� KESEN, OK, AND SETHI

5 This definition is slightly more demanding than the usual one, which requires that ?r haveincreasing differences. As will become clear below, however, the present formulation results inlittle loss of generality and is very convenient for our purposes.

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normal form games, action monotonicity is nevertheless satisfied by a varietyof symmetric games. In general, any 1 # G with ?r(x)=9(xr , �(x)), where9: X_R � R+ is strictly increasing (decreasing) in the first componentand �: Xn � R is symmetric, is positively (negatively) action monotonic.Several widely studied symmetric games, including common pool resourceextraction and public goods games, and Cournot oligopolies with constantaverage costs, are special cases of this general formulation. Thus all of thesegames are action monotonic.

Our first main result provides an answer to the question stated in theprevious section within the class of all action monotonic strictly super-modular games.

Theorem 1. Let k # [1, ..., n&1] and F # F. If 1 # G is strictly super-modular and action monotonic, then for any x̂ # Nsym(1F (k)) with x̂1 {x̂n ,we have ? j (x̂)>?i (x̂) for all (i, j) # Ik_Jk .

Proof. Take any x̂=([a]k , [b]n&k) # Nsym(1F (k)) such that a{b.Since ?1= p1 and x̂ is an equilibrium, we have ?1([a]k , [b]n&k)�?1(b,[a]k&1 , [b]n&k). This, together with symmetry of 1, yields

?n(b, [a]k&1 , [b]n&k&1 , a)�?n(b, [a]k&1 , [b]n&k). (5)

By strict supermodularity, since a{b, we have

?n([a]k , [b]n&k)+?n(b, [a]k&1 , [b]n&k&1 , a)

<?n([a]k , [b]n&k&1 , a)+?n(b, [a]k&1 , [b]n&k),

which, together with (5), implies that

?n(x̂)=?n([a]k , [b]n&k)<?n([a]k , [b]n&k&1 , a)=?n(x̂&n , a). (6)

Now assume that 1 is positively (negatively) action monotonic, andsuppose, by way of contradiction, that \n(x̂)�1. Then we must have either?n(x̂)=0 or ?n(x̂)>0 and, by symmetry,

n?n(x̂)k?1(x̂)+(n&k) ?n(x̂)

�1.

In either case, we obtain ?n(x̂)�?1(x̂). This is possible only if ao ( O ) bby action monotonicity. But then applying action monotonicity again, wefind ?n(x̂&n , a)>?j (x̂&n , a), j=k+1, ..., n&1, while (by symmetry) ?n(x̂&n , a)=?j (x̂&n , a), j=1, ..., k. Hence we have \n(x̂&n , a)�1. Therefore, by (6)and the definition of pr , we have pn(x̂&n , a)>pn(x̂), which contradicts the

281NEGATIVELY INTERDEPENDENT PREFERENCES

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hypothesis that x̂ is an equilibrium. Therefore, we must have \n(x̂)>1 sothat, by symmetry, ?j (x̂)=?n(x̂)>? i (x̂)=?1(x̂) for all (i, j) # Ik_Jk .

Q.E.D

Theorem 1 states that at any intragroup symmetric equilibrium of an action-monotonic strictly supermodular game, the absolute payoffs to interdependentplayers are strictly greater than those to independent players, unless bothgroups take the same equilibrium action.6 The significance of Theorem 1 islimited, however, by the fact that it deals only with intragroup symmetricequilibria and relies on the property of action monotonicity, which severalimportant supermodular games do not satisfy.7 We next show that both ofthese requirements can be relaxed for the case in which only one of theplayers has interdependent preferences.

Theorem 2. If 1 # G is strictly supermodular, F # F, and x̂ # N(1F (n&1))satisfies x̂1 {x̂n , then ?n(x̂)>?i (x̂) for all i # In&1 .

Proof. Take any x̂ # N(1F (n&1)). Suppose first that x̂ is intragroupsymmetric, so we may write x̂=([a]n&1 , b). Since inequality (6) abovewas obtained without requiring action monotonicity, we have ?n(x̂)<?n(x̂&n , a). Since x̂ is an equilibrium, we must therefore have \n(x̂)>\n(x̂&n ,a)=\n([a]n)=1. The theorem then follows from the symmetry of 1.

To complete the proof, we show that x̂ # Nsym(1F (n&1)). This is trivialfor the case n=2, so suppose that n�3. If x̂ is not intragroup symmetric,then there exist i and i $ # In&1 such that x̂i {x̂ i $ . Without loss of generality,let i=1, i $=2, x̂1=a, and x̂2=a$ with a{a$. Since x̂ is an equilibrium and1 is symmetric, we have ?1(a, a$, x̂3 , ..., x̂n)�?1(a$, a$, x̂3 , ..., x̂n) and

?1(a$, a, x̂3 , ..., x̂n)=?2(a, a$, x̂3 , ..., x̂n)�?2(a, a, x̂3 , ..., x̂n)

=?1(a, a, x̂3 , ..., x̂n).

But, given that a{a$, these two inequalities contradict the strict super-modularity of 1. Hence a=a$, and we conclude that x̂ is intragroup symmetric.

Q.E.D

In particular, Theorem 2 shows that the interdependent player unam-biguously holds the upper hand in any strictly supermodular two-persongame. This simple case can be used to provide some intuition for the resultsof this section. Consider a two-person symmetric and strictly supermodular

282 KOC� KESEN, OK, AND SETHI

6 Theorem 1 remains valid if we replace strict supermodularity with the weaker notion ofstrict quasisupermodularity, which only requires that ?r(x)�?r(x 7y) imply ?r(x 6 y)>?r( y),for all r and x, y # Xn such that [x 6y, x 7 y]{[x, y] (Milgrom and Shannon [14]).

7 One can, however, show that the hypothesis of intragroup symmetry can be relaxed inTheorem 1 if we assume that 1 is positively (negatively) action monotonic and has the negative(positive) spillovers property (see Section 3.2), in addition to being strictly supermodular.

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game 1, and let (a, b) # X 2 be any outcome in which the independentplayer 1 has a strictly higher payoff than the interdependent player 2.Clearly, the relative payoff of player 2 is strictly less than 1 at this actionprofile. But, as inequality (6) proves formally, symmetry and (strict) super-modularity ensure the following: if a{b and a is a best response to b, then ais a (strictly) better response to a than b is. Therefore, if (a, b) were an equi-librium, by switching to player 1's action a, player 2 could increase both theabsolute and relative payoffs, which means that (a, b) cannot be an equilibrium.

This kind of an argument underlies the proofs of Theorems 1 and 2. Inparticular, Theorem 1 states that this intuition generalizes for arbitrary nand k provided that 1 is also action monotonic and one restricts attentionto intragroup symmetric equilibria; Theorem 2 states that neither actionmonotonicity nor intragroup symmetry is required when k=n&1. As weshall explain below, the case k=n&1 is important from an evolutionaryperspective, since it helps us identify environments in which the extinctionof players with interdependent preferences cannot occur under any payoff-monotonic evolutionary selection dynamics.

We conclude this section by demonstrating that action monotonicityalone is not sufficient to yield any of the above results. The aim of the followingexample is to illustrate the crucial role played by supermodularity inTheorems 1 and 2.

Example 1. Consider the two-person game 1 # G represented by thebimatrix

1, 1 1, 1�2 3, 2

1�2, 1 1, 1 2, 0

2, 3 0, 2 2, 2

Here the strategy space of each agent is the chain [1, 2, 3]. This game iseasily checked to be (negatively) action monotonic but not supermodular.It has three Nash equilibria, N(1 )=[(1, 3), (3, 1), (2, 2)]. Taking F(z1 , z2)=z1z2 for all z1 , z2�0 and adopting the convention of treating thecolumn player as player 2, the game 1F (1) is represented by the bimatrix

1, 1 1, 1�3 3, 8�5

1�2, 4�3 1, 1 2, 0

2, 18�5 0, 4 2, 2

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Clearly, N(1F (1))=[(1, 3)], and ?1(1, 3)=3>2=?2(1, 3). In this game,therefore, the player with interdependent preferences is subject to a strategicdisadvantage.

Example 1 demonstrates that action monotonicity is consistent with thepossibility of interdependent players having a strategic disadvantage againstindependent players. It should thus be formally clear that our main inquiry,that is, determining a general subclass of G, the members of which satisfy (4),is not a trivial one. The following section deals with submodular games inthis subclass.

3.2. Submodular Games with Spillovers

A variety of economically interesting games exhibit a negative (or positive)spillover effect. In such games, an increase in the level of action taken bya player decreases (or increases) the absolute payoffs of all other players.The strong form of this property is, however, too demanding for our purposessince it is not satisfied by games in which players have at least one potentialaction which would nullify the influence of other players. For instance, in theclassical Cournot model of oligopoly, a firm may completely escape the effectof quantity choices of other firms on its profits simply by choosing to shutdown. For this reason, we shall work here with a slightly weaker notion ofthe spillover effect (which is present in the Cournot game).

Definition. Let A#�n&1k=1 [x # Nsym(1F (k)): F # F]. An n-person

normal form game 1 # G is said to have negative spillovers, if for any x # A,

t1 oxr ot2 implies ?q(x&r , t1)<?q(x)<?q(x&r , t2)

for all r and q{r. Games with positive spillovers are defined dually.

It turns out that in games with negative spillovers, there is a tightconnection between action monotonicity and the possibility of ?j (x� )�?i (x� ) holding for all i # Ik and all j # Jk , as stated in the following result.

Lemma 1. Let k # [1, ..., n&1] and F # F. For any 1 # G with negativespillovers and any x̂ # Nsym(1F (k)),

?j (x̂)�(>) ?i (x̂) \(i, j) # Ik_Jk

holds only if x̂j � ( o ) x̂i for all (i, j) # Ik_Jk . Moreover, if k=n&1, thenfor any i # In&1 , we have ?n(x̂)�(>) ?i (x̂) if and only if x̂n � ( o ) x̂i .

This lemma shows that positive action monotonicity at the equilibriumaction profile is essentially a necessary condition for (4) to hold in the caseof games with negative spillovers. It can be shown similarly that negative

284 KOC� KESEN, OK, AND SETHI

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action monotonicity at the equilibrium action profile is a necessary condi-tion for (4) to hold for games with positive spillovers.

An n-person normal from game 1 # G is said to be submodular whenever

?r(x6 y)+?r(x 7 y)�?r(x)+?r( y) \x, y # Xn and r=1, ..., n.

We say that 1 is strictly submodular if the above inequality holds strictlyfor all r and x, y # Xn such that [x 6 y, x 7 y]{[x, y]. In contrast withsupermodular games, submodular games are those in which actions of anytwo players are strategic substitutes in the sense that the best responsemaps of all players are decreasing (cf. [3]).

Finally, we shall need the following concept for the analysis of thissection.

Definition. An n-person normal form game 1 # G is called symmetricin equilibrium if it does not possess an asymmetric Nash equilibrium.

While symmetry in equilibrium is admittedly a demanding property, it issatisfied by a variety of commonly studied symmetric games such as thestage hunt game, the prisoner's dilemma, the common pool resource game,many symmetric Cournot and Bertrand oligopoly models, and public goodgames. In fact, for a strictly submodular game 1, this property is nothingother than the requirement of uniqueness of equilibrium, for, if ([a]n), ([b]n)# N(1), then

?1((a, [b]n&1) 7 (b, [a]n&1))+?1((a, [b]n&1) 6 (b, [a]n&1))

=?1([a]n)+?1([b]n)

�?1(b, [a]n&1)+?1(a, [b]n&1)

so that unless a=b, 1 cannot be strictly submodular. When combined withsymmetry, this observation leads us to the following.

Lemma 2. Let 1 # G be a strictly submodular game such that N(1 ){<.Then 1 is symmetric in equilibrium if, and only if, it has a unique equilibrium.

Our main result takes as primitives those games in G where the commonstrategy set of the players is a convex and compact subchain of Rl, and thepayoff function of the rth player is continuous and quasi concave in xr , forall r. Denoting the class of all such games by G

*, we are now ready to state

the following strategic advantage result, the interpretation of which issimilar to that of Theorem 1.

Theorem 3. Let 1 # G*

, k # [1, ..., n&1] and F # F. If 1 is a positively(negatively) action monotonic and strictly submodular game with negative

285NEGATIVELY INTERDEPENDENT PREFERENCES

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( positive) spillovers and is symmetric in equilibrium, then, for any x̂ #Nsym(1F (k)) such that x̂1 {x̂n and ?r(x̂)>0 for all r, we have ?j (x̂)>?i (x̂)for all (i, j) # Ik_Jk .

Remark 2. Lemma 1 demonstrates the necessity of action monotonicityfor the conclusion of Theorem 3 to hold. Since we think of submodulargames with spillovers as primitives in the above analysis, the only questionabout the tightness of this result concerns the relaxation of the symmetryin equilibrium condition. To see that this condition too cannot be relaxedin Theorem 3, consider the ``hawk�dove'' game represented by the bimatrix

10, 10 5, 15

15, 5 1, 1

and define 1 as its mixed strategy extension. One can easily verify that 1satisfies all the hypotheses of Theorem 3 except for symmetry in equilibriumand that ((1, 0), (0, 1)) # Nsym(1F (1)), where F(z1 , z2)=(z1+1)(z2+1) forall z1 , z2�0. Hence there exists an equilibrium in which the player with inter-dependent preferences obtains a strictly lower absolute payoff.

To provide intuition for Theorem 3, let us consider a two-person symmetricand strictly submodular game 1 which satisfies positive action monotonicity,negative spillovers, and symmetry in equilibrium, and let (a, b) # X2 be anequilibrium of 1F (1) in which a exceeds b and hence the independent player 1has a strictly higher payoff than the interdependent player 2. By raising hisor her action, player 2 can lower player 1's material payoff (due to negativespillovers) so if (a, b) is to be an equilibrium of 1F (1), any such changemust also lower player 2's material payoff. In fact, since (a, b) cannot be anequilibrium of 1 (due to symmetry in equilibrium), player 2's material-payoff-maximizing response to a must lie strictly below b. If one constructsa sequence in which, starting from (a, b), each player chooses, in turn, amaterial-payoff-maximizing response to the other player's previous choice,it can be shown that (due to the strict submodularity of 1 ) this sequenceleads to increasing choices for player 1 and decreasing choices for player 2,and it converges to an asymmetric equilibrium of 1. Hence, if 1 is sym-metric in equilibrium, the premise that a exceeds b must be false, and hencethe independent player 1 cannot have a higher payoff than the interdependentplayer 2 at an equilibrium of 1F (1). While this particular reasoning appliesonly to the two-person case, it generalizes to yield Theorem 3.

In closing, we note that one can again relax the requirement of actionmonotonicity when there is only one interdependent player in the game.The following is then a counterpart to Theorem 2.

286 KOC� KESEN, OK, AND SETHI

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Theorem 4. Let F # F. If 1 # G*

is a strictly submodular game with negativeor positive spillovers and is symmetric in equilibrium, then, for any x̂ #Nsym(1F (n&1)) such that x̂1 {x̂n and ?r(x̂)>0 for all r, we have ?n(x̂)>?i (x̂) for all i # In&1 .

Theorem 4 provides sufficient conditions for the single player with inter-dependent preferences to have a strategic advantage with respect to theremaining players in submodular games. In particular, it implies that if1 # G

*is a two-person strictly submodular game with spillovers and a

unique equilibrium, then we have ?2(x̂)>?1(x̂) for any Nash equilibriumx̂ of 1F (1) with x̂1 {x̂2 and min[?1(x̂), ?2(x̂)]>0. This is analogous toour earlier observation about two-person supermodular games.

4. APPLICATIONS

The usefulness of the theorems established above hinges on the degree towhich they may be applied to games of economic interest. There are avariety of environments to which Theorems 1�4 apply, among which areinput and public good games, search models, and arms races. In each ofthese examples, it is possible to verify that, under standard conditions,interdependent preferences yield a strategic advantage over independentpreferences.8 To illustrate, we shall discuss here the implications of thestrategic advantage results of the previous section for certain oligopolygames, and in particular, provide an application to the theory of strategicdelegation. We shall then briefly discuss the implications of our results forthe theory of preference evolution.

4.1. Strategic Advantage and Delegation in Market Games

4.1.1. Strategic Advantage

Objective functions which incorporate relative payoff concerns areparticularly easy to justify in the case of firms that separate managementfrom ownership. For instance, owners may benefit from writing contractswith managers in which the compensation of the latter is based, in part, onthe performance of their firm relative to that of other firms or relative tosome industry average (Holmstro� m [9]). This, in turn, would provide anincentive for managers to pursue the maximization of interdependentobjective functions. The results of Section 3 can thus be used to show thatsuch contracts may have the unplanned effect of yielding a strategic advantage

287NEGATIVELY INTERDEPENDENT PREFERENCES

8 For a detailed analysis of these examples we refer the reader to Koc� kesen et al. [11].

Page 15: The Strategic Advantage of Negatively Interdependent Preferences

to a firm, enabling it to achieve a higher level of profitability than its profit-maximizing competitors.

To illustrate, consider the case of Cournot competition. For expositionalsimplicity, we consider a duopolistic industry composed of two firms withidentical cost structures producing a homogeneous product.9 Firm r choosesan output level xr # X#[0, Q� ], 0<Q� <�, where Q� is interpreted as thecapacity limit on a firm's output level. The profit function of firm r is givenby ?r(x)#xrP(x1+x2)&C(xr), x # X2, where the inverse demand functionP is a strictly positive and twice differentiable function on [0, 2Q� ], and thecost function C is a twice differentiable function on [0, Q� ]. We make thestandard assumptions that demand is downward sloping and average costis non-decreasing: P$<0, C(0)=0, C$>0, C"�0. We also assume that thegame is strictly submodular.10 These assumptions imply that the game hasa unique equilibrium (cf. [4], Proposition 1.3), which must therefore besymmetric. Let us denote the resulting Cournot game by 1 C # G and makethe additional assumptions that P(2Q� )>C(Q� )�Q� to ensure positive profitsfor each firm at any output profile, and P(2Q� )+Q� P$(2Q� )<C$(Q� ) andP(0)>C$(0) to ensure that the equilibrium occurs in the interior. The lastcondition also guarantees that 1 C has the negative spillovers property.It is a straightforward matter to show that under the stated conditions,any equilibrium x̂ # N(1 C

F(1)) satisfies x̂1 {x̂2 .11 Since 1 C is strictly sub-modular with negative spillovers and is symmetric in equilibrium, thefollowing then follows from Theorem 4.

Proposition 1. Take any Cournot duopoly 1 C and F # F. Then, at anyx̂ # N(1 C(1)), we have ?1(x̂)<?2(x̂), i.e., the firm with interdependentpreferences obtains a strictly higher profit than does the independent firm atany equilibrium.

While the analysis above pertains to the classical Cournot duopoly,similar results may be obtained for other market games. For instance, inthe case of a Bertrand duopoly with differentiated products and constantmarginal costs, we may apply Theorem 2 (under the standard conditions

288 KOC� KESEN, OK, AND SETHI

9 All of our results extend with minor modifications to n-firm industries; we focus on theduopoly case only to avoid uninteresting technical details.

10 That is, P$(x1+x2)+xrP"(x1+x2)<0 for all x # [0, Q� ]2.11 Suppose x̂=(a, a). The boundary conditions posited above imply a # (0, Q� ) so that

�?1(a, a)��x1=0 and �p2(a, a)��x2=0. Since, by symmetry, the former equation impliesthat �?2(a, a)��x2=0, the latter equation holds only if (?2F2 �?� 2) �?1 ��x2=0 at the actionprofile (a, a). Yet this is a contradiction, for we have ?i (a, a)>0, i=1, 2, F2>0, and�?1(a, a)��x2<0 in this model.

Page 16: The Strategic Advantage of Negatively Interdependent Preferences

that ensure the supermodularity of the game) to conclude that the inter-dependent firms will be more profitable than profit-maximizing firms. Insuch environments, therefore, managers who include relative profit considera-tions in their decision making process (say, due to incentive contracts) willobtain higher profits in equilibrium than those who do not. This finding hasan immediate implication for industries in which the entry and exit of firmsoccurs on the basis of profitability: behavior that corresponds to an inter-dependent objective function will thrive at the expense of profit-maximizingbehavior in the long run.

4.1.2. Strategic Delegation

An interesting application of Proposition 1 concerns the theory of strategicdelegation as developed by Vickers [23] and Fershtman and Judd [7].Consider an extension of the Cournot duopoly game 1 C in which one ofthe firms has the option of hiring a manager and delegating the outputchoice to him or her. In the first stage of the game, the owner of firm 2(henceforth, owner 2) gives a compensation contract to his or her manager,the nature of which is common knowledge. In the second stage, the managerand firm 1 engage in a standard Cournot competition in which the manager'sobjective is to maximize compensation, which is determined by the objectivefunction

6(x, %)#(1&%) ?2(x)+%\2(x) \(x, %) # [0, Q� ]2_[0, 1], (7)

where \2 is the relative profit of firm 2 as defined in (2). While owner 2chooses % to maximize ?2 , we assume for simplicity that the owner offirm 1 is a standard profit maximizer.

We denote the resulting 3-person extensive from game by 1 D. The sub-game of 1 D that is reached when owner 2 chooses % # [0, 1] is denoted by1 D

% . Clearly, 1 D% belongs to the family of 2-person normal-form games

considered in Proposition 1. More formally, we have 1 D% =1 C

F%(1), where

F% is defined on R2+ by F% (z1 , z2)=(1&%) z1+%z2 . In particular, we have

1 D0 =1 C. In what follows, we refer to 1 D as a delegation game and assume

that 1 D0 carries the structure of the Cournot duopoly model described in

the beginning of this section. The main issue in the theory of strategicdelegation is the question of whether owner 2 will choose to delegate inequilibria of 1 D, that is, whether he or she will set %>0. Given earlierfindings in the literature, our answer will not come as a surprise: delegationoccurs in any perfect equilibrium of 1 D.

Proposition 2. At any subgame perfect equilibrium of 1 D, the owner offirm 2 chooses to delegate.

289NEGATIVELY INTERDEPENDENT PREFERENCES

Page 17: The Strategic Advantage of Negatively Interdependent Preferences

Provided that the contract space is limited to that assumed above, thereis reason to expect managerial compensation schemes to embody relativeprofitability concerns at least to some extent.12 Since we exogenously limitourselves to a particular contract space, Proposition 2 cannot be used topredict the form of the equilibrium contract in general. It tells us, however,that no-delegation cannot be an equilibrium so long as the contract spaceis rich enough to include those contracts given by (7). In contrast to Vickers(1984) and Fershtman and Judd (1987), Proposition 2 shows that this conclu-sion obtains without assuming the linearity of the demand and cost func-tions.13

Remark 3. The findings of [23] have also been generalized by SalasFumas [18], whose Proposition 1(a) is similar to our Proposition 2.Neither result, however, implies the other. In particular, our hypotheses areposited only on the primitive Cournot game 1 D

0 (as opposed to all the sub-games 1 D

% ). Perhaps more importantly, our proof (given in the appendix)is quite elementary and hence readily generalizes to the usual n-firm scenario.In fact, it is easy to see that the same proof (with only minor modifications)enables one to replace the Cournot game considered in Proposition 2 withessentially any strictly submodular game that satisfies the hypotheses ofTheorem 4.

4.2. Preference Evolution and the Spiteful Effect

While it is conventional in economic models to posit that individuals arematerial payoff maximizers, there is now mounting experimental evidencethat contradicts this strong ``independence'' hypothesis. The theoreticalplausibility of this assumption has accordingly been questioned recently byseveral economists on evolutionary grounds.14 The theory of preference evolu-tion is based on the premise that individual preferences come into being as aresult of an unplanned process in which children inherit the preferences oftheir parents either by genetic transmission or by imitation. The populationcomposition is typically assumed to evolve according to a payoff-monotonicselection dynamic: those behaviors which yield the highest material rewards

290 KOC� KESEN, OK, AND SETHI

12 We are, of course, neglecting objections that could be raised on the grounds of contractunobservability (see Katz [10] and Koc� kesen and Ok [13] for more on this issue).

13 Formally speaking, however, Proposition 2 does not directly generalize the correspondingresult of, say, Vickers [23] since Vickers identifies a compensation contract with the mappingx [ ?2(x)&%?� (x), %�0, as opposed to (7). However, the proof of Proposition 2 may beeasily adapted (by setting F% (z1 , z2)#z1(1&%�z2) so that Proposition 1 may be applied) toaccount for this case.

14 See, for instance, Bester and Gu� th [2], Fershtman and Weiss [8], and Koc� kesen et al.[12].

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are replicated with greatest frequency from one generation to the next. Theevolutionary scenario we consider here is one in which each individual inter-acts with each other member of the population in each period, the so-called``playing the field'' model. This is an environment in which strategic advan-tage has particularly transparent evolutionary implications. Dispensingwith formalities for brevity, we briefly discuss these implications next.

Consider a finite population of size n, such that the constituent individualsare engaged in playing an n-person game 1 # G. Not all players need bematerial-payoff maximizers; we allow for the possibility that some playersare negatively interdependent. With the initial preference distribution in thepopulation given exogenously, suppose that this distribution evolves overtime in accordance with the hypothesis of payoff monotonicity. It is apparentfrom Theorem 2 that if 1 is strictly supermodular, then the long-run popula-tion composition cannot consist exclusively of material-payoff maximizersunless the initial state consists exclusively of such preferences. If, due to a``mutation,'' a single individual in the population happens to be negativelyinterdependent (denote the resulting game by 1F), then this individual willobtain at least as great a material payoff as any independent individual, andconsequently, independent preferences could not expel such a mutant.15 On thebasis of Theorem 4, the same conclusion holds if 1 is strictly submodular, issymmetric in equilibrium, and has negative or positive spillovers, provided thatall independent agents take the same action at any equilibrium of 1F . We there-fore conclude that there are evolutionary reasons to expect that the populationwill not be composed only of absolute payoff maximizers in the long run.16

Broadly similar conclusions have been reached earlier in the evolutionaryliterature on the ``spiteful effect.'' The spiteful effect occurs when it is possiblefor a player to deviate from a Nash equilibrium action profile in such amanner as to reduce the payoffs of other players more severely than thepayoffs of the deviating player (Rhode and Stegeman [16], Vega-Redondo[22]). In this case payoff-monotonic selection dynamics will lead to thespread of a ``spiteful'' mutant who adopts such a deviation. It turns out that

291NEGATIVELY INTERDEPENDENT PREFERENCES

15 While the basic idea here is quite transparent, it is nevertheless informal. To examine theissue at hand formally, we need a set-valued extension of the concept of an evolutionarilystable state for a finite population (cf. [17, 19] ) since, if the preference space is sufficientlyrich, there will typically exist preferences that are behaviorally indistinguishable in particulargame forms. (For instance, in the multi-person prisoners' dilemma, defection will be a domi-nant strategy for a very wide range of objective functions including all negatively interdepen-dent and independent preferences.) Such a stability concept is introduced in Kc� kesen et al.[11], where the evolutionary ideas discussed above are formalized.

16 Other interesting environments in which this issue can be investigated include randommatching models as well as models of local interaction. Results obtained by Fershtman andWeiss [8] and Bester and Gu� th [2] for pairwise random matching and by Eshel et al. [6] forlocal interaction on a circle suggest that even altruistic preferences can survive in such cases.

Page 19: The Strategic Advantage of Negatively Interdependent Preferences

the presence of the spiteful effect is necessary but not sufficient for thestrategic advantage of negatively interdependent preferences. To see that itis necessary, consider for simplicity a two-person game in which the spitefuleffect is not present at any equilibrium of 1. In particular, it is not presentat any equilibrium x̂ at which ?1(x̂)�?2(x̂). Since 1 is symmetric, theremust exist at least one such equilibrium. In the absence of the spitefuleffect, any deviation from this equilibrium by player 2 will lower his or herpayoffs at least as much as it lowers the payoff of player 1. Such a deviationcan raise neither the absolute payoff nor the relative payoff of the deviatingplayer. Hence x̂ is also an equilibrium of 1F when player 1 is independentand player 2 negatively interdependent. As a result, interdependent playerscannot have a strategic advantage in the absence of the spiteful effect.

To see that the spiteful effect is not sufficient for strategic advantage, oneneed only examine Example 1 above. The game 1 has three Nash equilibriawhen both players have independent preferences and the spiteful effect ispresent at each one of them. However, if exactly one of the players is negativelyinterdependent, this player may end up with strictly lower material payoffsthan his or her opponent at the unique equilibrium of the resulting game 1F .Hence the presence of the spiteful effect does not guarantee the existence ofstrategic advantage for negatively interdependent preferences. The reasonthe two criteria yield different predictions is that the spiteful effect is basedon the hypothesis that the incumbent population does not respond optimallyto the appearance of the mutant (and continues, instead, to adopt the samestrategies that were previously optimal). In contrast, we postulate rationalbehavior by all players and study the properties of the equilibrium set condi-tional on the underlying preference distribution. The introduction of anindividual with interdependent preferences into an incumbent population ofabsolute payoff maximizers causes the latter to adjust their actions in sucha manner as to locate a new equilibrium. Since the notion of strategicadvantage is global (having been defined conditional on the equilibriaof 1F), it cannot be deduced by examining the consequence of local deviationsfrom equilibria of 1, which is what defines the spiteful effect. Furthermore,verification of the presence of the spiteful effect requires one to examine theproperties of equilibria, which are not primitives of a game. In contrast, thestrategic advantage results of Section 3 are obtained by positing conditionson the payoff (fitness) functions and are therefore easily verified in arbitrarygames.17 Given that the spiteful effect is necessary for the strategic advantage

292 KOC� KESEN, OK, AND SETHI

17 The only non-primitive property that we used above is symmetry of equilibrium. But thisproperty too can be replaced by any set of primitive assumptions that ensure uniqueness ofequilibrium in our context (recall Lemma 2). For instance, in Theorems 3 and 4, the symmetryof equilibrium property can be replaced with the requirements that (i) payoff functions are strictlyquasi concave in own actions and (ii) the associated best response functions are contractionmaps.

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of negatively interdependent preferences, the main results of this paper providesufficient conditions for the presence of the spiteful effect in a given normalform game.

5. CONCLUSION

We have aimed in this paper to uncover the generality of the statementthat negatively interdependent preferences provide individuals with a strategicadvantage over others who are motivated exclusively by a concern with theirown material payoffs. While the plausibility of this phenomenon has beennoted earlier in a variety of contexts, the conditions under which it ariseshave not previously been systematically explored. It turns out that there isa broad class of strategic environments in which such an advantage exists,although this class is not exhaustive. Our results identify the properties ofstrategic complementarity and substitutability as particularly relevant tothe possibility that interdependent preferences earn greater absolute payoffsthan do (absolute) payoff maximizers. While we find this observation interest-ing from a purely game-theoretic viewpoint, it also has important implicationsfor the theories of strategic delegation and preference evolution. We note,however, that out results fall short of characterizing the class of games inwhich interdependent players have a strategic advantage over independentplayers (cf. [12]). Determining precisely the class of all normal-form gamesfor which this phenomenon occurs remains an open problem.

APPENDIX: PROOFS

Proof of Lemma 1. Assume that x̂=([a]k , [b]n&k) for some a, b # Xwith aob. Then, by hypothesis, negative spillovers effect, and symmetryof 1,

?1([a]k , [b]n&k)�?n([a]k , [b]n&k)<?n(b, [a]k&1 , [b]n&k)

=?1(b, [a]k&1 , [b]n&k)

and this contradicts that playing a is a best response for player 1 against([a]k&1 , [b]n&k). The first assertion follows by the completeness andantisymmetry of � .

To prove the second assertion, let k=n&1 and notice that all we haveto show is that ?n(x̂)>?1(x̂) whenever boa. Assume then for contradic-tion that

boa and ?n([a]n&1 , b)�?1([a]n&1 , b). (8)

293NEGATIVELY INTERDEPENDENT PREFERENCES

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Since : [ :�({+:) is a strictly increasing mapping in :�0 for any{>0, (8) implies that

\n(x̂)n

=?n(x̂)

(n&1) ?1(x̂)+?n(x̂)�

?1(x̂)(n&1) ?1(x̂)+?1(x̂)

=1n

.

Hence \n(x̂)�1 and since b is a best response of player n against [a]n&1

in 1F (n&1), we must have ?n([a]n&1 , b)�?n([a]n). Therefore, by thenegative spillovers effect and symmetry of 1, we have ?1([a]n&1 , b)<?1([a]n)=?n([a]n)�?n([a]n&1 , b), which contradicts (8). Q.E.D

Proof of Theorem 3. This theorem is an immediate consequence of thefollowing

Lemma A. Let 1 # G*

, k # [1, ..., n&1] and take any strictly increasingF: R2

+ � R. If 1 is a strictly submodular game with negative spillovers andis symmetric in equilibrium, then, for any x̂ # Nsym(1F (k)) with ?r(x̂)>0 forall r, we have

x̂j � x̂i \(i, j) # Ik_Jk .

Proof. Let x̂=([a]k , [b]n&k) # Nsym(1F (k)) for some a, b # X. Clearly,since a is a best response of player 1 against ([a]k&1 , [b]n&k) in 1F (k), wehave

?1([a]k , [b]n&k)�?1(t, [a]k&1 , [b]n&k) \t # X. (9)

We claim that

?n([a]k , [b]n&k)>?n([a]k , [b]n&k&1 , t) \t # [t$ # X : t$ob]. (10)

To see this, let us assume for contradiction that

?n(x̂&n , t)=?n([a]k , [b]n&k&1 , t)�?n([a]k , [b]n&k)=?n(x̂)>0 (11)

holds for some t # X with tob. Since : [ :�({+:) is a strictly increasingmapping in :�0 for any {>0, we must then have

?n(x̂)�n&1

r=1 ?r(x̂)+?n(x̂)�

?n(x̂&n , t)�n&1

r=1 ?r(x̂)+?n(x̂&n , t). (12)

But since tob, the negative spillovers effect yields ?r(x̂)>?r(x̂&n , t) for allr{n, so that

?n(x̂&n , t)�n&1

r=1 ?r(x̂)+?n(x̂&n , t)<

?n(x̂&n , t)�n&1

r=1 ?r(x̂&n , t)+?n(x̂&n , t).

294 KOC� KESEN, OK, AND SETHI

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By this inequality, (12) and (11) yield pn(x̂&n , t)>pn(x̂), which contradictsthat x̂ is a Nash equilibrium of 1F (k). Therefore, we must conclude that(10) holds.

Now, for any :, ; # X, define the correspondences K; : X��X andL: : X��X as

K;(:)#arg maxt # X

?1(t, [:]k&1 , [;]n&k)

and

L:(;)#arg maxt # X

?n([:]k , [;]n&k&1 , t).

We define next the double sequence (am , bm) # X2 recursively as follows:

a0=a, b0=b, am # Kbm(am), and bm # Lam&1

(bm), m=1, 2, ...

Claim 1. (am , bm) is well defined.

Proof of Claim 1. Fix any :, ; # X. Since X is a convex compact set,and ?1 and ?n are continuous, K; and L: must be nonempty (byWeierstrass' theorem) and must have closed graphs (by Berge's maximumtheorem). Moreover, quasi concavity of ?1 and ?n entail that K; and L: areconvex-valued. Therefore, by Kakutani's fixed point theorem, there existfixed points of K; and L: . Since : and ; were arbitrary in this reasoning,we may conclude that (am) and (bm) are well defined sequences.

Let Br: Xn&1��X be the best response correspondence of player r in 1.We note that, for any :, ; # X,

Bi ([:]k&1 , [;]n&k)=K;(:) \i # Ik (13)

and

B j ([:]k , [;]n&k&1)=L:(;) \j # Jk (14)

hold by symmetry of 1.

Claim 2. If aob, then a0 Oa1 Oa2 O } } } and } } } Ob2 Ob1 Ob0 .

Proof of Claim 2. Let aob. We shall first establish that b1 {b0 . Ifb1=b, then b1 # La(b1) implies by (14) that b # B j ([a]k , [b]n&k&1) for allj # Jk . But then since a # Bi ([a]k&1 , [b]n&k) for all i # Ik , it follows that([a]k , [b]n&k) # N(1 ), contradicting that 1 is symmetric in equilibrium. If,on the other hand, b1 ob, then (10) yields that ?n(x̂)>?n([a]k , [b]n&k&1 , b1).But then by submodularity of ?n , ?n([a]k , [b1]n&k)<?n([a]k , [b1]n&k&1 , b)which, in turn, contradicts that b1 # La(b1). We thus conclude that bob1 .

295NEGATIVELY INTERDEPENDENT PREFERENCES

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Next, we claim that a1 � a0 . By (9) and the fact that a1 # Kb1(a1), we

have ?1([a]k , [b]n&k)�?1(a1 , [a]k&1 , [b]n&k) and ?1([a1]k , [b1]n&k)�?1(a, [a1]k&1 , [b1]n&k). Clearly, given that bob1 , if aoa1 , the lasttwo inequalities would contradict the strict submodularity of ?1 . Therefore,a1 � a must hold. In fact, a1 {a, for otherwise, a1 # Kb1

(a1) and b1 #La(b1) would yield that ([a]k , [b1]n&k) # N(1), and this would contradict1 's symmetry in equilibrium since then aobob1 would have to hold. Bylinearity of � , therefore, we have a1 oa0 .

Finally, we claim that b1 ob2 . (Since we used (10) in establishing thatb0 ob1 , this step is necessary to be able to complete the proof by induc-tion.) This claim follows from the fact that b1 # La(b1) and b2 # La1

(b2)imply that ?n([a]k , [b1]n&k)�?n([a]k , [b1]n&k&1 , b2) and ?n([a1]k ,[b2]n&k)�?n([a1]k , [b2]n&k&1 , b1), respectively. If b2 ob1 held, giventhat a1 oa, these inequalities would contradict the strict submodularity of?n . Moreover, if b1=b2 , then b1 # La1

(b1) holds, and since a1 # Kb1(a1),

we obtain ([a1]k , [b1]n&k) # N(1), contradicting that 1 is symmetric inequilibrium (because a1 oaobob1). We conclude that b1 ob2 .

Proof is completed by a straightforward induction argument.

Since X is compact, there exist convergent subsequences (avm) and (bvm

)such that (avm

, bvm) � (a*, b*) # X2 as m � �. We now claim that (a*, b*)

# N(1). To see this, notice that avm# Kbvm

(avm) implies that

avm# B1([avm

]k&1 , [bvm]n&k) m=1, 2, ...

But since X is compact and ?1 is continuous, B1 must have a closed graph,and therefore,

a*= limm � �

avm# B1( lim

m � �([avm

]k&1 , [bvm]n&k))=B1([a*]k&1 , [b*]n&k).

Moreover, by symmetry of 1, a* # Bi ([a*]k&1 , [b*]n&k) for all i # Ik .Similarly, we can show that b* # Bi ([a*]k , [b*]n&k&1) for all i # Jk . Wethus conclude that (a*, b*) # N(1 ) as is sought. Therefore, if aob held, byClaim 2 there would exist an (a*, b*) # N(1 ) with a*ob*, contradictingthat 1 is symmetric in equilibrium. Q.E.D

Remark A. (a) If n=2, we may drop the hypotheses of convexity ofX and quasiconcavity of ?r 's from the statement of Lemma A, for then onedoes not need Kakutani's fixed point theorem in proving that (am , bm) iswell defined.

(b) Lemma A remains valid if the chain X is any compact convexsubset of a metric space, and ?r 's are continuous with respect to the sub-space topology. The proof of this claim is essentially identical to that of

296 KOC� KESEN, OK, AND SETHI

Page 24: The Strategic Advantage of Negatively Interdependent Preferences

Lemma A, the only major modification being the use of the Tychonoff�Fanfixed point theorem [1, p. 251] instead of Kakutani's theorem in provingClaim 1.

(c) Continuity of ?r can be weakened in Lemma A to upper semicon-tinuity of ?r and lower semicontinuity of Vr(x&r)#maxa # X ?r(x&r , a) forall x&r # X n&1, which is well defined when ?r is upper semicontinuous. Weonly need to check that these conditions guarantee the nonemptiness of K;

(and L:) and the closed graph property of Br. The former is immediatefrom upper semicontinuity of ?r . To see the latter, take any sequence xm

in Xn such that xm � x* as m � �, and let xmr # Br(xm

&r) for all m. Ifxr* � Br(x*&r), then it must be the case that Vr(x*&r)>?r(x*&r , xr*). But thenusing the upper semicontinuity of ?r , the fact that xm

r # Br(xm&r) for all m,

and the lower semicontinuity of Vr , we reach the following contradiction:18

?r(x*&r , xr*)�lim supm � �

?r(xm&r , xm

r )=lim supm � �

Vr(xm&r)

�lim infm � �

?r(xm&r , xm

r )�Vr(x*&r)>?r(x*&r , xr*).

Proof of Theorem 4. Immediate from Lemma 1 and Lemma A. Q.E.D

Proof of Proposition 2. Let (a, a) # N(1 C) and notice that we must havea # (0, Q� ). Let b1 denote the restriction of the best response correspondenceof player 1 to (0, Q� ) in the game 1 C. By strict concavity of ?2 , b1 can beconsidered as a function. Define the function .: (0, Q� ) � R by .(q)#?2(b1(q), q). By the implicit function theorem, b1 and . are C1 on (0, Q� ),and since a=b1(a) and �?2(a, a)��x2=0, we have .$(a)=aP$(2a) b$1(a)>0,where the inequality follows from P$<0 and strict submodularity of P. Inview of continuity of ., we then have

Claim 1. There exists an =� >0 such that ?2(b1(a+=), a+=)>?2(b1(a), a)for all = # (0, =� ].

Claim 2. There exists a %� # (0, 1] such that N(1 D% ){< for all % # [0, %� ].

Proof of Claim 2. Take any = # (0, Q� ) and % # [0, 1], and consider thetwo-person game 1 D

%, = #(X 2= , [?1 , 6( } , %)]) where X= #[=, Q� ]. First, note

that 6( } , %) is continuous on X 2= .19 Second, since

�26(x, %)�x2

2

=(1&%)�2?2(x)

�x22

+%�2\2(x)

�x22

and�2?2(x)

�x22

<0,

297NEGATIVELY INTERDEPENDENT PREFERENCES

18 Dasgupta and Maskin [5] use analogous reasoning in proving their Theorem 2.19 Notice, however, that 6( } , %) is not continuous at (0, 0).

Page 25: The Strategic Advantage of Negatively Interdependent Preferences

there exists a %(=) # (0, 1] such that 6(x1 , } , %) is concave on X= for all% # [0, %(=)]. We can therefore use Nash's existence theorem to concludethat N(1 D

%, =){< for all % # [0, %(=)]. Now, choose any x(=) # N(1 D%, =) with

% # [0, %(=)]. Since

lim= a 0

supx2 # [0, =)

6((x1(=), x2), %)=0<6(x(=), %)

there exists an =>0 such that 6((x1(=), x2), %)�6(x(=), %) for all x2[0, =).Choosing such an =>0, therefore, x2(=) is a best response of player 2 tox1(=) in the game 1 D

% for all % # [0, %(=)]. That x1(=) # arg maxx1 # X ?1(x1 ,x2(=)) is, on the other hand, easily verified. Hence, by choosing %� =%(=), wehave N(1 D

% ){< for all % # [0, %� ].

Claim 3. Let x̂(%) # N(1 D% ) for any % # (0, %� ]. There exists a %0 # (0, %� ]

such that ?2(x̂(%))>?2(a, a) for all % # (0, %0].

Proof of Claim 3. Take any sequence %n # [0, %� ] that converges to 0,and note that, by Claim 2, N(1 D

%n){< for all n. Let x̂(%n) # N(1 D

%n), n�1.

Since b1 is strictly decreasing (due to strict submodularity of 1 C), Proposition1 implies that x̂2(%n)>a for all n�1. There must then exist a subsequencex̂2(%nl

) such that lim x̂2(%nl)�a. Let y2 #lim x̂2(%nl

), and notice that x̂(%nl)

� (b1( y2), y2) (as l � �) by continuity of b1 . But then we must have(b1( y2), y2) # N(1 C) since Nash equilibrium correspondence has a closedgraph. Given the uniqueness of the equilibrium of 1 C, it follows that(b1( y2), y2)=(a, a). Thus, since x̂2(%n)>a for all n�1, there exists apositive integer l* such that x̂2(%nl

) # (a, a+=� ] for all l�l*, where =� isdefined as in Claim 1. Applying Claim 1, we find ?2(x̂2(%nl

))>?2(a, a) forall l�l*. Choose %0=%nl*

and the claim follows.

The proof of Proposition 2 is now easily completed. The Nash equi-librium of the subgame induced by the choice of %=0, i.e., N(1 C), is givenby (a, a), where the payoff of owner 2 is ?2(a, a). However, Claim 3 showsthat there exists a %0 # (0, 1] such that ?2(x̂(%))>?2(a, a) for all % # (0, %0]which implies that %=0 cannot obtain in any subgame perfect equilibriumof 1 D. Q.E.D

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