Coalition Proofness in a Class of Games With Strategic Substitutes

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Int J Game Theory DOI 10.1007/s00182-014-0452-8 Coalition-proofness in a class of games with strategic substitutes Federico Quartieri · Ryusuke Shinohara Accepted: 11 September 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract We examine the coalition-proofness and Pareto properties of Nash equi- libria in pure strategy σ -interactive games with strategic substitutes and increas- ing/decreasing externalities. For this class of games: (i) we prove the equivalence among the set of Nash equilibria, the set of coalition-proof Nash equilibria under strong Pareto dominance and the set of Nash equilibria that are not strongly Pareto dominated by other Nash equilibria; (ii) we prove that the fixpoints of some “ extremal” selections from the joint best reply correspondence are both coalition-proof Nash equi- libria under weak Pareto dominance and not weakly Pareto dominated by other Nash equilibria. We also provide an order-theoretic characterization of the set of Nash equi- libria and show various applications of our results. Keywords Coalition-proof Nash equilibrium · Pareto dominance · Strategic substitutes · Externalities · Generalized aggregative games 1 Introduction The work of Bulow et al. (1985) provided the seminal notion of a game with strategic substitutes and that of a game with strategic complements. Since then the literature has considerably generalized such notions. Nowadays, indeed, any game that possesses either “ decreasing” or “ increasing” best-replies can be legitimately labelled as a game F. Quartieri Dipartimento di scienze economiche e statistiche, Università degli studi di Napoli Federico II, Naples, Italy e-mail: [email protected] R. Shinohara (B ) Faculty of Economics, Hosei University, 4342, Aihara-machi, Machida, Tokyo 194-0298, Japan e-mail: [email protected] 123

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Coalition.

Transcript of Coalition Proofness in a Class of Games With Strategic Substitutes

Page 1: Coalition Proofness in a Class of Games With Strategic Substitutes

Int J Game TheoryDOI 10.1007/s00182-014-0452-8

Coalition-proofness in a class of games with strategicsubstitutes

Federico Quartieri · Ryusuke Shinohara

Accepted: 11 September 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract We examine the coalition-proofness and Pareto properties of Nash equi-libria in pure strategy σ -interactive games with strategic substitutes and increas-ing/decreasing externalities. For this class of games: (i) we prove the equivalenceamong the set of Nash equilibria, the set of coalition-proof Nash equilibria understrong Pareto dominance and the set of Nash equilibria that are not strongly Paretodominated by other Nash equilibria; (ii) we prove that the fixpoints of some “ extremal”selections from the joint best reply correspondence are both coalition-proof Nash equi-libria under weak Pareto dominance and not weakly Pareto dominated by other Nashequilibria. We also provide an order-theoretic characterization of the set of Nash equi-libria and show various applications of our results.

Keywords Coalition-proof Nash equilibrium · Pareto dominance ·Strategic substitutes · Externalities · Generalized aggregative games

1 Introduction

The work of Bulow et al. (1985) provided the seminal notion of a game with strategicsubstitutes and that of a game with strategic complements. Since then the literature hasconsiderably generalized such notions. Nowadays, indeed, any game that possesseseither “ decreasing” or “ increasing” best-replies can be legitimately labelled as a game

F. QuartieriDipartimento di scienze economiche e statistiche,Università degli studi di Napoli Federico II, Naples, Italye-mail: [email protected]

R. Shinohara (B)Faculty of Economics, Hosei University, 4342, Aihara-machi, Machida, Tokyo 194-0298, Japane-mail: [email protected]

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with either strategic substitutes or strategic complements. Despite a sort of duality inthe definition of the two classes of games, many properties that hold true for the gamesof one of the two classes need not hold for those of the other. Indeed, apart from theorder-theoretic nature of their definitions, the two classes do not seem to share manycommon properties.

Accrediting the importance of monotone externalities (i.e., the monotonicity ofplayers’ payoff functions in the opponents’ strategies) in many games with strategiccomplements of economic interest, the literature has provided an exhaustive investi-gation of the Pareto and coalition-proofness properties of Nash equilibria in abstractclasses of games with strategic complements and monotone externalities.1 In very bru-tal summary, this literature shows that in these games an extremal Nash equilibrium(exists and) is always a coalition-proof Nash equilibrium that is not Pareto dominatedby other Nash equilibria and that there is a tendency for that equilibrium to be theunique coalition-proof Nash equilibrium.2 As a matter of fact, a comparably exhaus-tive examination of these two properties in abstract classes of games with strategicsubstitutes and monotone externalities is still missing in the literature; we are awareof only some partial results presented in Yi (1999) and in Shinohara (2005), whichwill be adequately discussed in Sect. 3.1.1.

The purpose of this article is to provide a better understanding, and a more system-atic investigation, of the Pareto and coalition-proofness properties of Nash equilibriain a subclass of games with strategic substitutes and monotone externalities where thestrategic interaction is mediated by interaction functions. The games of this subclasswill be called σ -interactive games with strategic substitutes and increasing/decreasingexternalities. As we shall point out, various models studied in Industrial organization,Public economics and Network economics are associated to games that belong to sucha subclass.

Somewhat loosely speaking—the reader is referred to Sect. 2.3 for a precisedefinition—we say that a game is a σ -interactive game with strategic substitutes andincreasing/decreasing externalities if: (i) strategy sets are subsets of the real line; (ii)the payoff to each player i can be expressed as a function of the player’s strategy and ofthe value attained by a real-valued interaction function σi defined on the joint strategyset; (iii) each interaction function σi is increasing in all arguments and constant in thei-th argument; (iv) a change in a joint strategy that increases the value attained by theinteraction function σi entails a “ decrease” of player i’s best-reply; (v) a change of thestrategies of i’s opponents that increases the value attained by the interaction functionσi entails an increase/decrease in player i ’s payoff. As we shall observe in Sect. 2.4, ifone additionally assumes that each interaction function σi is also continuous then thegames examined in this article are generalized quasi-aggregative games in the precise

1 See Milgrom and Roberts (1990), Milgrom and Roberts (1996) and Quartieri (2013). In particular, forresults concerning the coalition-proofness of Nash equilibria—in the sense of Bernheim et al. (1987)—seeTheorem A2 and its subsequent remark in Milgrom and Roberts (1996) and Theorems 1 and 2 and theirrespective Corollaries in Quartieri (2013). An appropriate discussion can be found in the last-mentionedarticle.2 For a result on the uniqueness of coalition-proof Nash equilibria in games with strategic complementsthat dispenses with the assumption of monotone externalities see also Theorem A1 in Milgrom and Roberts(1996).

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sense of Jensen (2010) (and hence our examination contributes also to that strand ofliterature3). However, such a continuity condition—as well as some condition similarto that implied by Assumption 2 in Jensen (2010)—plays no role in our results andhence will never be imposed as an assumption in this article.

The formal definition of a coalition-proof Nash equilibrium was firstly providedin Bernheim et al. (1987) and was based on the concept of strong Pareto domi-nance. In the subsequent literature various authors have alternatively based that def-inition on the concept of weak Pareto dominance (just to provide some examples:Milgrom and Roberts (1996), Kukushkin (1997), Furusawa and Konishi (2011)).Konishi et al. (1999) pointed out that the set of coalition-proof Nash equilibria understrong Pareto dominance (in this Introduction s-CPN equilibria for short) may welldiffer from the set of coalition-proof Nash equilibria under weak Pareto dominance(in this Introduction w-CPN equilibria for short). Indeed, there are various numericalexamples4 of games where the set of s-CPN equilibria and that of w-CPN equilibriaare nonempty and disjoint: these examples prove that the concept of an s-CPN equi-librium is distinct from that of a w-CPN equilibrium (i.e., none of them is a refinementof the other). For this reason, in this article we shall inspect both concepts and we shallalso examine how they relate within the class of games under consideration.

A general issue concerning the sets of w- and s-CPN equilibria is their relationwith the Nash equilibria that are not weakly Pareto dominated by other Nash equi-libria (in this Introduction w-FN equilibria for short) and with the Nash equilibriathat are not strongly Pareto dominated by other Nash equilibria (in this Introduc-tion s-FN equilibria for short). While it is not difficult to see that in games with atmost two players the set of w-CPN (resp. s-CPN) equilibria and the set of w-FN(resp. s-FN) equilibria are equivalent, in games with more than two players these setscan well be nonempty and disjoint. Clearly, it is particularly interesting to determinesufficient conditions for a Nash equilibrium to be, at the same time, a w-FN equilib-rium (and a fortiori also an s-FN equilibrium), a w-CPN equilibrium and an s-CPNequilibrium.

We shall prove that in every σ -interactive game with strategic substitutes andincreasing externalities the set of Nash equilibria coincides with the set of s-CPNequilibria (and hence also with the set of s-FN equilibria). Besides we shall prove thatin every σ -interactive game with strategic substitutes and increasing externalities theset of BR-maximal Nash equilibria (i.e., the Nash equilibria whose components arethe greatest best-replies) is included in both the set of w-FN equilibria and that ofw-CPN equilibria; some examples will show that the previous inclusion relation canbe proper and that, in general, the set of w-FN equilibria and that of w-CPN equilibria

3 Just to mention a few other articles in that strand of literature: Corchón (1994); Alós-Ferrer and Ania(2005); Kukushkin (1994); Kukushkin (2005); Jensen (2006); Dubey et al. (2006); Acemoglu and Jensen(2013). Quite interestingly, even Shinohara (2005) and Yi (1999) actually belong also to that strand ofliterature. In the eight aformentioned articles one can find economic examples (possibly adding someconditions) of real σ -interactive games with strategic substitutes and increasing/decreasing externalitiesthat are not discussed in Sect. 4.4 See, e.g., Examples 1 and 2 in Quartieri (2013). See also Example 1 in Konishi et al. (1999); however,note that in Konishi et al. (1999) the terminology seemingly reverses the terminology of the present article.

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need not be ordered by an inclusion relation. We shall also provide and prove dualstatements for the case of decreasing externalities.

The previous results are of interest for various reasons. The first—and verygeneral—reason is that it is difficult to say anything about the coalition-proofnessof Nash equilibria in any class of games, and we show that this can be done in a veryprecise and simple way in the class of games considered (see Corollaries 1 and 2). Thesecond is that our results imply new existence results for w- and s-CPN equilibria:we do not prove new Nash equilibrium existence results, but our results easily allowfor the transformation of some known Nash equilibrium existence results into w- ands-CPN equilibrium existence results (see Observation IV and its footnote, Corollaries1 and 2, and Sect. 4). The third is that our results clarify that, quite surprisingly, inthe class of games considered every Nash equilibrium is an s-CPN equilibrium, andhence also an s-FN equilibrium: this implies that the s-CPN and s-FN equilibriumconcepts cannot act as effective refinements of the Nash equilibrium concept in theclass of games considered (see Corollary 1). The fourth is that our results clarify thatin the class of games considered the w-CPN and w-FN equilibrium concepts can stillbe effective refinements, provided some Nash equilibrium is not strict (see Corollary2 and the Examples of Sect. 4). Finally, our results allow a sensible comparison withknown results for games with strategic complements and monotone externalities (seeSect. 3.3).

The rest of this article is organized as follows. Section 2 sets the definitions andsome basic preliminaries. Section 3 contains the main results presented above and alsoan order-theoretic characterization of the sets of equilibria. Section 4 presents variousapplications and examples. An Appendix contains all proofs that are not directly relatedto the main results.

2 Definitions and preliminaries

2.1 Basic standard game-theoretic notions

Henceforth, by � we shall denote a game(M, (Si )i∈M , (ui )i∈M

)where M �= ∅ is the

set of players and, for all i ∈ M , Si �= ∅ is player i’s strategy set and ui : ∏i∈M Si → R

is player i’s payoff function; unless explicitly stated otherwise, M will be assumed tohave finite cardinality m.

Let � be a game, C ⊆ M , l ∈ M and s ∈ ∏i∈M Si . The set

∏i∈C Si is also denoted

by SC . The tuples (si )i∈C , (si )i∈M\C , (si )i∈{l} and (si )i∈M\{l} are also denoted by,respectively, sC , sM\C , sl and s−l . The pair (x, y) ∈ SC × SM\C will alternativelydenote the tuple z ∈ SM such that zC = x and zM\C = y. For all i ∈ M , letbi : SM → 2Si denote player i’s best-reply correspondence, which is defined by

bi : s �→ arg maxz∈Si

ui (z) if M = {i} and bi : s �→ arg maxz∈Si

ui (z, s−i ) otherwise.

Finally, let b : s �→ (bi (s))i∈M denote the joint best-reply correspondence.

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2.2 Basic standard order-theoretic notions

A partial order � on a set X is a reflexive, transitive, and antisymmetric binary relationon X ; the pair

(X,�

)is a partially ordered set, or poset. Henceforth ≤ will exclu-

sively denote the usual partial—in fact total—order on the extended reals R, and <

its asymmetric part; given A ⊆ R, we write inf A, min A, sup A, and max A onlyto respectively denote the infimum, the minimum (if any), the supremum, and themaximum (if any) of A under ≤. Given a poset

(X,�

)and Y ⊆ X we say that Y is

an antichain of(X,�

)if, for any two distinct y′ and y′′ in Y , neither y′ � y′′ nor

y′′ � y′. However, we also say that Y ⊆ Rm is an antichain in R

m if, for any twodistinct y′ and y′′ in Y , y′

i < y′′i for some i and y′′

l < y′l for some l. Henceforth, we

say that a function f : A ⊆ R → B ⊆ R is increasing (resp. strictly increasing,decreasing, strictly decreasing) if, for all x, y ∈ A, x < y implies f (x) ≤ f (y)

(resp. f (x) < f (y), f (y) ≤ f (x), f (y) < f (x)).

2.3 Games with interaction functions

Definition 1 An interaction system σ for a game � is a family {σi }i∈M of functionssuch that, for all i ∈ M , the interaction function σi maps SM into an arbitrary set Ii

and is constant in the i-th argument. A game � is said to have a compatible interactionsystem σ if σ is an interaction system for � and, for all i ∈ M , there exists a function

υi : Si × σi [SM ] → R

such that

ui (s) = υi (si , σi (s)) at all s ∈ SM .

Every game has always at least one compatible interaction system, say σ ∗.5 Ofcourse, a game can have many compatible interaction systems. Clearly, not all possibleinteraction systems for a game are necessarily compatible with it.

Definition 2 A game � is said to be a real σ -interactive game with strategic sub-stitutes if it has a compatible interaction system σ and, for all i ∈ M : (i) Si ⊆ R ⊇ Ii ;(ii) σi is increasing in all arguments and, for all (x, y) ∈ SM × SM ,

v ∈ bi (x) , w ∈ bi (y) and σi (x) < σi (y) implies w ≤ v.

A game is said to be a real σ -interactive game with strategic substitutes andincreasing (resp. decreasing) externalities if it is a real σ -interactive game withstrategic substitutes and, for all i ∈ M , υi is increasing (resp. decreasing) in thesecond argument.

5 E.g.: if |M | > 1 put Ii = SM\{i} and σ∗i : s �→ s−i for all i ∈ M and take the function υi defined by

υi(si , σ

∗i (s)

) = ui (s) at all s ∈ SM for all i ∈ M ; if M = {i} put Ii = {0} and σ∗i : s �→ 0 and take the

function υi defined by υi(si , σ

∗i (s)

) = ui (s) at all s ∈ SM .

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In the previous definition, the strategic substitutability and externalities are mediatedby the interaction functions. Since each player’s interaction function is assumed topreserve the order of the joint strategies of the opponents, both definitions do not departfrom the usual general notions of strategic substitutability and monotone externality.6

On the other hand, our terminology would seem improper without this assumption. Itis worth remarking that the reader might well think of σi as a function of only s−i ;however, defining σi as a function of s—albeit constant in the i th argument—allowsus not to have to distinguish between one-player games and games with two or moreplayers when we deal with the recursive notion of a coalition-proof Nash equilibriumand when we prove our results. Note that up to now the assumption that M is finitehas never been used and we could have dispensed with it.

Notation (�- and �-interactivity) When � is a real σ -interactive game with strategicsubstitutes and σi : s �→ ∑

l∈M\{i} sl (resp. σi : s �→ ∏l∈M\{i} sl ) for all i ∈ M ,

we also say that � is a real �-interactive (resp. �-interactive) game with strategicsubstitutes, agreeing that each player i’s interaction function becomes �i : s �→∑

l∈M\{i} sl (resp. �i : s �→ ∏l∈M\{i} sl ).7

2.4 Relation with quasi-aggregative games

In the literature, one of the most general definitions of an aggregative game is formu-lated in Jensen (2010). Such a definition is sufficiently general to subsume many previ-ous definitions of aggregative games, for more details see Jensen (2010). (Throughoutthis Sect. 2.4 suppose there are many players).

Generalized quasi-aggregative game (Jensen 2010) A game � is said to be a gener-alized quasi-aggregative game with aggregator g : SM → R if, for all i ∈ M , Si is asubset of a Euclidean space and there exist continuous8 functions Fi : Si × R → R

(the shift functions) and ςi : SM\{i} → X−i ⊆ R (the J-interaction-functions) suchthat

ui (s) = ui (si , ςi (s−i )) , where ui : Si × X−i → R,

and g (s) = Vi (s−i ) + Fi (si , ςi (s−i )) for all s ∈ SM and i ∈ M,

where Vi is an arbitrary real-valued function on SM\{i}.

The following Observations I–V (see the Appendix for a proof) provide a clarifi-cation asked by an associate editor on how the games considered in this article relateto the generalized quasi-aggregative games in Jensen (2010).

6 When writing this, we mean, in particular, that games properly characterized by some notion of strategiccomplementarity are ruled out by Definition 2. (Of course, weaker notions of strategic substitutability—andof monotone externality—can be conceived and traced in the literature.)7 We recall that, when {xi }i∈I is an indexed family of reals, by an established convention

∑i∈I xi = 0

and∏

i∈I xi = 1 if I = ∅.8 The reader might even assume that Fi has a continuously differentiable extension to an open superset ofits domain (see Jensen (2012)): the following discussion remains unaltered.

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Observation I Not every real σ -interactive game with strategic substitutes and increas-ing (or alternatively, decreasing) externalities is a generalized quasi-aggregative game.

Observation II Suppose � is a game where each strategy set is a subset of a Euclideanspace. Then � is a generalized quasi-aggregative game if and only if � has a compatibleinteraction system σ where every interaction function σi is real-valued and continuous.(A fortiori, every real σ -interactive game with strategic substitutes and continuousinteraction functions is a generalized quasi-aggregative game; the converse is evidentlyfalse.)

Observation III Suppose � is a game where each strategy set is a subset of a Euclideanspace, and suppose for a moment that the continuity of the J-interaction-functions isdispensed with in the definition of a generalized quasi-aggregative game. Then agame � is a generalized quasi-aggregative game if and only if � has a compatibleinteraction system σ where every interaction function σi is real-valued. (A fortiori,in this case, every real σ -interactive game with strategic substitutes is a generalizedquasi-aggregative game; the converse is evidently false.)

The last two Observations clarify, in particular, the exact relation between thenotion of strategic substitutability employed in Definition 2 above and that employedin Assumption 1’ in Jensen (2010)—which, in some very loose sense, are identical.

Observation IV Suppose � is a game with nonempty-valued best-replies whereevery strategy set is a subset of the real line; besides, suppose � is a generalizedquasi-aggregative game satisfying Assumption 1’ in Jensen (2010) such that every J-interaction-function ςi is increasing in all arguments. Then � is also a real σ -interactivegame with strategic substitutes (and also with increasing/decreasing externalities ifevery ui is also increasing/decreasing in the second argument).9

Observation V Suppose � is a game with nonempty-valued best-replies where everystrategy set is a subset of the real line; besides, suppose � is a real σ -interactive gamewith strategic substitutes (and increasing/decreasing externalities) such that everyinteraction function σi is continuous. Then � is also a generalized quasi-aggregativegame satisfying Assumption 1’ in Jensen (2010) (and every ui is increasing/decreasingin the second argument).

2.5 Equilibrium notions

As usual, s ∈ SM is a Nash equilibrium (resp. strict Nash equilibrium) for a game� if si ∈ bi (s) (resp. {si } = bi (s)) for all i ∈ M .

9 If one additionally assumes that each ui is upper semicontinuous in s and continuous in s−i , that each Siis compact, that each Fi has a continuously differentiable extension and that Assumption 2 in Jensen (2010)holds, then Corollary 1 in Jensen (2010) guarantees that the set of Nash equilibria is nonempty. Clearly,one can alternatively—but not equivalently, see Observation VI—guarantee the nonemptiness of the setof Nash equilibria also assuming other additional conditions (e.g., conditions that allow the application ofKakutani’s fixpoint theorem to b).

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Notation (E�N , E�

ST N , E�N , E

N ) The set of Nash equilibria (resp. strict Nash equilibria)for � is denoted by E�

N (resp. E�ST N ). When Si ⊆ R for all i ∈ M, we put:

E�N = {

s ∈ E�N : s = (inf bi (s))i∈M

}; E

N = {s ∈ E�

N : s = (sup bi (s))i∈M

}.

Each element of E�N (resp. E

N ) is called a BR-minimal (resp. BR-maximal) Nashequilibrium for �. Perhaps it is worth remarking that best-replies might be empty-valued in some real σ -interactive games with strategic substitutes. In this connectionit might be worth recalling that inf ∅ = sup R = +∞ and sup ∅ = inf R = −∞.Thus,10 when strategy sets are subsets of R we have:

• s ∈ E�N ⇐⇒ for all i ∈ M , min bi (s) exists in R and si = min bi (s);

• s ∈ E�

N ⇐⇒ for all i ∈ M , max bi (s) exists in R and si = max bi (s).

Needless to say, E�N ⊇ E�

ST N ⊆ E�

N when strategy sets are subsets of R.Let � be a game. A joint strategy s ∈ SM weakly Pareto dominates in � a joint

strategy z ∈ SM if ui (z) ≤ ui (s) for all i ∈ M and u j (z) < u j (s) for somej ∈ M ; a joint strategy s ∈ SM strongly Pareto dominates in � a joint strategy z ifui (z) < ui (s) for all i ∈ M . Let � be a game, C ∈ 2M\ {∅, M}, s ∈ SM and, for alli ∈ C , ui : SC → R, ui : z �→ ui

(z, sM\C

). The game induced by C at s is the game

�|sM\C := (C, (Si )i∈C , (ui )i∈C

).

Definition 3 Let � be a game. Assume that |M | = 1; then s ∈ SM is a w-coalition-proof (resp. s-coalition-proof ) Nash equilibrium for � if s ∈ E�

N . Assume that|M | ≥ 2 and that a w-coalition-proof (resp. s-coalition-proof) Nash equilibrium hasbeen defined for games with fewer than |M | players; then

• s ∈ SM is a w-self-enforcing (resp. s-self-enforcing) strategy for � if it is a w-coalition-proof (resp. s-coalition-proof) Nash equilibrium for �|sM\C for all non-empty C ⊂ M ;

• s ∈ SM is a w-coalition-proof (resp. s-coalition-proof) Nash equilibrium for � if itis w-self-enforcing (resp. s-self-enforcing) for � and there does not exist anotherw-self-enforcing (resp. s-self-enforcing) strategy for � that weakly (resp. strongly)Pareto dominates s in �.

Notation (wF�N , s F�

N , E�wC P N , E�

sC P N ) For each game �, the set of Nash equilibriathat are not weakly (resp. strongly) Pareto dominated in � by other Nash equilibria isdenoted by wF�

N (resp. s F�N ) and the set of w-coalition-proof (resp. s-coalition-proof)

Nash equilibria is denoted by E�wC P N (resp. E�

sC P N ).

10 Note that Si ⊆ R and bi (x) = ∅ implies that inf bi (x)(= +∞) and sup bi (x)(= −∞) exist in R (butnot in R ⊇ Si ).

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3 Results

3.1 Coalition-proofness and welfare properties

Theorem 1 Suppose � is a real σ -interactive game with strategic substitutes andincreasing externalities. Then,

(i) each BR-maximal Nash equilibrium for � is not weakly Pareto dominated in �

by other Nash equilibria for �, and hence E�

N ⊆ wF�N ;

(ii) each BR-maximal Nash equilibrium for � is a w-coalition-proof Nash equilibrium

for �, and hence E�

N ⊆ E�wC P N ;

(iii) each Nash equilibrium for � is not strongly Pareto dominated in � by other Nashequilibria for �, and hence E�

N = s F�N ;

(iv) each Nash equilibrium for � is an s-coalition-proof Nash equilibrium for �, andhence E�

N = E�sC P N .

Proof (i) By way of contradiction, suppose there exist x ∈ E�

N and y ∈ E�N such that

ui (x) ≤ ui (y) for all i ∈ M (1)

and thatu j (x) < u j (y) for some j ∈ M. (2)

If σ j (y) ≤ σ j (x) then σ j (y) ≤ σ j(y j , x− j

)because σ j is constant in the j-th

argument, and hence u j (y) ≤ u j(y j , x− j

)because of the increasing externality

condition; clearly u j(y j , x− j

) ≤ u j (x) because x ∈ E�

N , and hence

u j (y) ≤ u j (x)

in contradiction with (2). Therefore we must have that

σ j (x) < σ j (y) ,

which implies, by the increasingness of σ j in all arguments, that

xk < yk for some k ∈ M . (3)

Again, if σk (y) ≤ σk (x) then σk (y) ≤ σk (yk, x−k), and hence uk (y) ≤ uk (yk, x−k);

clearly uk (yk, x−k) < uk (x) because x ∈ E�

N and xk = max bk (x) < yk , and hence

uk (y) < uk (x)

in contradiction with (1). Therefore we must have that

σk (x) < σk (y) .

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Since xk ∈ bk (x), yk ∈ bk (y) and σk (x) < σk (y), the strategic substitutabilitycondition implies that yk ≤ xk in contradiction with (3).

(ii) The proof is by induction. Clearly, part (ii) of Theorem 1 is true if |M | = 1.Assume that part (ii) is true when 1 ≤ |M | < n. We shall prove that part (ii) is true when|M | = n. It is not difficult to see that, for every nonempty C ⊂ M and every s ∈ SM ,�|sM\C is a σ -interactive game with strategic substitutes and increasing externalities forthe interaction system σ = {σi }i∈C defined by σi : SC → R, σi : x �→ σ

(x, sM\C

)

for all i ∈ C . Clearly,

if x ∈ E�

N then xC ∈ E�|xM\CN for all nonempty C ⊂ M .

Hence, by the induction hypothesis, E�

N is included in the set of w-self-enforcing

strategies for �. Thus, from part (i) of Theorem 1 it follows easily that E�

N ⊆ E�wC P N .

(iii) By way of contradiction, suppose there exist x ∈ E�N and y ∈ E�

N such that

ui (x) < ui (y) for all i ∈ M . (4)

Take an arbitrary j ∈ M . If σ j (y) ≤ σ j (x) then the increasing externality conditionimplies u j (y) ≤ u j (x) in contradiction with (4). Therefore σ j (x) < σ j (y), whichimplies that

xk < yk for some k ∈ M . (5)

Again, if σk (y) ≤ σk (x) then uk (y) ≤ uk (x) in contradiction with (4). Thereforeσk (x) < σk (y). Since xk ∈ bk (x) and yk ∈ bk (y) and σk (x) < σk (y), the strategicsubstitutability condition implies that yk ≤ xk in contradiction with (5).

(iv) The proof is by induction. Clearly, part (iv) of Theorem 1 is true if |M | = 1.Assume that part (iv) is true when 1 ≤ |M | < n. We shall prove that part (iv) istrue when |M | = n. For every nonempty C ⊂ M and every s ∈ SM , �|sM\C isa σ -interactive game with strategic substitutes and increasing externalities for theinteraction system σ = {σi }i∈C defined by σi : SC → R, σi : x �→ σ

(x, sM\C

)for

all i ∈ C . Clearly,

if x ∈ E�N then xC ∈ E

�|sM\CN for all nonempty C ⊂ M .

Hence, by the induction hypothesis, E�N coincides with the set of s-self-enforcing

strategies for �. Thus, from part (iii) of Theorem 1 it follows easily thatE�

N = E�sC P N . ��

Corollary 1 Suppose � is a real σ -interactive game with strategic substitutes andincreasing externalities, then

E�

N ⊆ E�wC P N ⊆ E�

sC P N = E�N = s F�

N ⊇ wF�N ⊇ E

N .

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Coalition-proofness in a class of games

Suppose � is a real σ -interactive game with strategic substitutes and decreasingexternalities, then

E�N ⊆ E�

wC P N ⊆ E�sC P N = E�

N = s F�N ⊇ wF�

N ⊇ E�N .

Proof In case of increasing externalities Corollary 1 is an immediate consequenceof Theorem 1. Construct �� as in Fact 1 in the Appendix and note that Fact 2guarantees that Corollary 1 is true also in case of decreasing externalities (note alsothat E�

N = −E��N , E�

sC P N = −E��sC P N , E�

wC P N = −E��wC P N , s F�

N = −s F��N ,

wF�N = −wF��

N and in particular E�N = −E

��N ). ��

Corollary 2 Suppose � is a real σ -interactive game with strategic substitutes andeither increasing or decreasing externalities. Besides suppose all Nash equilibria arestrict.11 Then

E�N = E

N = E�wC P N = E�

sC P N = wF�N = s F�

N = E�ST N = E�

N .

Just to avoid misunderstandings, we remark that Theorem 1 and Corollaries 1 and 2do not guarantee that E�

N �= ∅ (and it is well possible that, e.g., a finite real σ -interactivegame with strategic substitutes and either increasing or decreasing externalities doesnot possess Nash equilibria). We shall return to this point in Sect. 4.

3.1.1 Comparison with the relevant literature and tightness of the results

The reader acquainted with the literature on coalition-proof Nash equilibria mightwell want to know how Corollary 1 relates to the Theorem in Yi (1999) and to theProposition in Shinohara (2005). We shall first consider the second-mentioned article.

The Proposition in Shinohara proves that E�wC P N ⊆ E�

sC P N for a proper12 subclassof the class of real �-interactive games with strategic substitutes and either increasingor decreasing externalities. Thus, our Corollary 1 subsumes the Proposition in Shino-hara (2005). In fact, our Corollary 1 shows also that the inclusion relation establishedin the Proposition in Shinohara (2005) is due to the equivalence (under the conditionsof that Proposition) of E�

sC P N and E�N . Thus, one might legitimately wonder whether

it is possible to establish an analogous equivalence between E�wC P N and E�

N under theconditions of Corollary 1. In order to answer this and many other legitimate questionsabout the tightness of the conclusions of Corollary 1 we explicitly claim (and provein the Appendix) the following.

Claim 1 Given the hypotheses of Corollary 1 (and without additional hypotheses), noinclusion relation can be generally established between E�

wC P N and wF�N , and each

of the inclusion relations established in the theses of Corollary 1 cannot be generallyreversed.

11 A sufficient condition for E�N = E�

ST N is that all best-replies are at most single-valued.12 See Example 1 and Remark 2.

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F. Quartieri, R. Shinohara

Let us now turn to a much more difficult comparison with the Theorem in Yi (1999).The difficulty of such a comparison resides in the simple fact that Yi’s Theorem doesnot generally hold true (we refer to Quartieri and Shinohara (2012) for a detailedexamination of this issue). That Theorem states that s F�

N ⊆ E�sC P N for a class of

games with strategic substitutes and monotone externalities (and with only strict Nashequilibria) which is seemingly similar to that considered here (as it is clear from theproofs in Yi (1999), wF�

N and E�wC P N are not examined). Our results do not imply

the Theorem in Yi (1999); also the converse is true, because there certainly existreal σ -interactive games with strategic substitutes and either increasing or decreasingexternalities that do not satisfy the conditions of the Theorem in Yi (1999) (e.g., thegames in Sect. 4.3–5 do not generally satisfy condition (1) of that Theorem and thosein Sect. 4.1–2 and 4.4–5 do not generally satisfy condition (3) of that Theorem).13

Having said this, there is little else to add: a precise comparison between our resultsand the Theorem in Yi (1999) is in fact pointless because of the essential erroneousnessof the statement of that Theorem. Indeed, we claim (and prove in the Appendix) thefollowing.

Claim 2 There exists a game � that satisfies the assumptions of the Theorem inYi (1999) such that s F�

N \E�sC P N �= ∅.

The counterexample illustrated in the proof of Claim 2 shows that the condition of“strategic substitutes in equilibrium” is too general for the validity of Yi’s Theorem.That counterexample is a game with weakly positive externalities in the sense ofYi (1999). Proposition 2 in Quartieri and Shinohara (2012) proves that a statementsimilar to Yi’s Theorem is true in case of weakly negative externalities and convexstrategy sets; anticipating unjustified conjectures based on Proposition 2 in Quartieriand Shinohara (2012) and on the third remark at p. 358 in Yi (1999), we claim (andprove in the Appendix) the following.

Claim 3 There exists a game � (with weakly negative externalities) that satisfiesall conditions of the Theorem in Yi (1999), but not its condition (3), such thats F�

N \E�sC P N �= ∅.

3.1.2 A final remark on mixed-strategies

If a game � satisfies the conditions of Corollary 1 (resp. 2), then Corollary 1 (resp. 2)applies to that game � but does not generally apply to some mixed-strategy extensionof that game, say �′ , which is a distinct game in its own right. It must be remarkedalso that some—of the possibly many—mixed-strategy extensions of some games thatsatisfy the conditions of Corollary 1 (or those of Corollary 2) need not even be well-defined: e.g., in the game in Example 1 there are problems with the integrability ofu1 (for instance, because u1 (·, s−1) is unbounded) relative to all probability measureson 2N0 (i.e., on the sigma algebra generated by the singletons of N0). Needless to

13 Note, however, that the games considered in Sect. 3.1–2 of Yi (1999) satisfy the assumptions of Corollary2; consequently—and this has not been noted in Yi (1999)—in those games the “ Pareto-efficient frontierof the Nash equilibrium set” in the sense of Yi (1999) is equivalent to the entire set of Nash equilibria.

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Coalition-proofness in a class of games

say, if a game � satisfies the conditions of Corollary 2 and the images of all best-reply correspondences of a well-defined mixed-strategy extension, say �′, are alwaysdegenerate mixed strategies then the conclusions of Corollary 2 extend in fact also tothe mixed-strategy extension �′.

3.2 Order-theoretic characterization of E�N

It is well-known—see, e.g., Proposition 1.1 in Dacic (1979)—that the set of fixpoints ofan antitone self-map on a poset

(X,�

)is an antichain of

(X,�

). Many classes of games

that satisfy some notion of strategic substitutability have antitone joint best-replyfunctions and thus their sets of Nash equilibria are antichains. Such an order-theoreticcharacterization of the set of Nash equilibria is emblematic of the situation of strategicconflict inherent in these games. However, when joint best-reply correspondences aremulti-valued, it can well happen that two Nash equilibria of a real σ -interactive gamewith strategic substitutes can be compared under the order of the joint strategy sets:simple examples of such games where E�

N is the Cartesian product of m(> 1) compactproper intervals can be easily constructed by the reader.

Theorem 2 below shows that the set of Nash equilibria of a real σ -interactive game� with strategic substitutes can still be characterized as an antichain when the set ofNash equilibria is endowed with a “natural” order relation on E�

N derived from theinteraction system σ .

Notation (�) Consider a real σ -interactive game � with strategic substitutes. Let �is the binary relation on E�

N such that s∗ � s∗∗ if and only if σi (s∗) < σi (s∗∗) forall i ∈ M, and let � denote the reflexive closure of �. (Therefore � is the binaryrelation on E� such that s∗ � s∗∗ if and only if either s∗ = s∗∗ or σi (s∗) < σi (s∗∗)for all i ∈ M).

Theorem 2 Suppose � is a real σ -interactive game with strategic substitutes. Then,E�

N is an antichain of the poset(E�

N ,�)

(i.e., it is impossible that x and y are Nashequilibria for � and σi (y) < σi (x) for all i ∈ N).

Proof The proof that(E�

N ,�)

is a poset is immediate and is left to the reader. Now,by way of contradiction, suppose x and y are Nash equilibria for � and

σi (y) < σi (x) for all i ∈ M. (6)

As xi ∈ bi (x) and yi ∈ bi (y) for all i ∈ M and � is a real σ -interactive game withstrategic substitutes, (6) implies that xi ≤ yi for all i ∈ M ; hence, by the increasingnessof σi in all arguments, σi (x) ≤ σi (y) for all i ∈ M , in contradiction with (6). ��

Corollary 3 Suppose � is a real σ -interactive game � with strategic substitutes suchthat, for all i ∈ M,

s∗, s∗∗ ∈ SM and s∗l < s∗∗

l for all l ∈ M\ {i} implies σi(s∗) < σi

(s∗∗) ;

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F. Quartieri, R. Shinohara

besides suppose m > 1 (e.g., � is a multiplayer �-interactive game with strategicsubstitutes and nonnegative strategies). Then it is impossible that x, y ∈ E�

N andxi < yi for all i ∈ M.

Proof If xi < yi for all i ∈ M then {i ∈ M : σi (y) < σi (x)} = M , in contradictionwith Theorem 2. ��

Note that an immediate consequence is that, under the assumptions of Corollary 3,there can exist at most one symmetric Nash equilibrium (whether or not the game issymmetric).

Corollary 4 Suppose � is a real σ -interactive game � with strategic substitutes and,for all i ∈ M, σi is strictly increasing in sl for all l ∈ M\ {i} (e.g., � is a real�-interactive game with strategic substitutes). Then, x, y ∈ E�

N implies that

• either xi∗ < yi∗ for some i∗ ∈ M and yi∗∗ < xi∗∗ for some i∗∗ ∈ M,• or x−i = y−i for some i ∈ M.

Proof If x−i �= y−i for all i ∈ M and xl ≤ yl for all l ∈ M\ {i} then σi (x) < σi (y)

for all i ∈ M , in contradiction with Theorem 2. ��Corollary 4 states that—under its assumptions—if x and y are two Nash equilibria

such that xl ≤ yl for all l ∈ M then the two Nash equilibria must be identical exceptfor at most one component. The same thesis is in fact stated also in the Corollary ofTheorem 3 in Jensen (2006) but14 under the hypothesis that the games are strictlysubmodular in the sense of Jensen (2006). Since there exist real �-interactive gameswith strategic substitutes that are not strictly submodular games (see Example 1 andRemark 2), our Corollary 4 is not implied by the Corollary of Theorem 3 in Jensen(2006). Clearly, Corollary 4 does not in the least imply the Corollary of Theorem 3 inJensen (2006).

Theorem 3 Suppose � is a real σ -interactive game � with strategic substitutes.Besides suppose x and y are two distinct strict Nash equilibria for �. Then it isimpossible that xi ≤ yi for all i ∈ M.

Proof By way of contradiction, suppose xi ≤ yi for all i ∈ M . Then, by the increas-ingness of σi in all arguments,

σi (x) ≤ σi (y) for all i ∈ M . (7)

Since x �= y and xi ≤ yi for all i ∈ M , we have that x j < y j for some j ∈ M .Since

{x j

} = b j (x),{

y j} = b j (y) and x j < y j , the assumption that � is a real

σ -interactive game with strategic substitutes implies that

σ j (y) < σ j (x) ,

in contradiction with (7). ��

14 Actually, we are presuming that in the statement of that Corollary in Jensen (2006) the two equilibria(i.e., s∗,1 and s∗,2) are “tacitly” assumed to be ordered.

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Coalition-proofness in a class of games

Corollary 5 Suppose � is a real σ -interactive game � with strategic substitutes.Besides suppose all Nash equilibria are strict. Then E�

N is an antichain in Rm.

It is perhaps worth noting that Corollary 5 is not implied by Theorem 1 of Roy andSabarwal (2008). Indeed—interpreting the joint best-reply correspondence b as oneof their parametrized correspondence g (·, t)—the assumption of Theorem 1 in Royand Sabarwal (2008) that each correspondence g (·, t) is never-increasing excludes thepossibility that—when each strategy set is a subset of the real line—b might assumethe same value at two distinct points of the joint strategy set, say x and y, such thatxi < yi for all i ∈ M (this is clear, in particular, from the end of the second paragraphof Sect. 2.1 in Roy and Sabarwal (2008)). However, there are real �-interactive gameswith strategic substitutes and single-valued best-reply correspondences where the jointbest-reply correspondence b is not never-increasing in the sense of Roy and Sabarwal(2008) (e.g., it can be verified that in Example 1 at p. 182 in Kerschbamer and Puppe(1998)—which is an instance of a real �-interactive game with strategic substitutesdiscussed in Sect. 4—one has b (0.6, 0.6) = b (0.7, 0.7), and hence in that exampleb is not never-increasing). Clearly, Corollary 5 does not in the least imply Theorem 1of Roy and Sabarwal (2008).

3.3 Comparison with previous results in games with strategic complements andmonotone externalities

To provide a sensible comparison between the results of Sect. 3.1–2 and known resultsfor games with strategic complements and monotone externalities, we introduce thefollowing definition.

Definition 4 A game � is said to be a real σ -interactive game with strategic comple-ments if it has a compatible interaction system σ and, for all i ∈ M : (i) Si ⊆ R ⊇ Ii ;(ii) σi is increasing in all arguments and, for all (x, y) ∈ SM × SM ,

v ∈ bi (x) ,w ∈ bi (y) and σi (x) < σi (y) implies v ≤ w;

(iii) Si is compact and bi has nonempty compact values. A game is said to be a realσ -interactive game with strategic complements and increasing (resp. decreasing)externalities if it is a real σ -interactive game with strategic complements and, for alli ∈ M , υi is increasing (resp. decreasing) in the second argument.

The following known result is only a straightforward and very particular conse-quence of Theorem 1 in Quartieri (2013), which is much more general in its original.The similarities and dissimilarities with Corollaries 1 and 2 (and with Theorems 2and 3) are evident.

Result (Quartieri 2013) Suppose � is a real σ -interactive game with strategic com-plements and increasing (resp. decreasing) externalities. Then there exists a greatest

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F. Quartieri, R. Shinohara

Nash equilibrium e ∈ E�

N and a least Nash equilibrium e ∈ E�N ,15 and

{e} ⊆ E�wC P N = wF�

N ⊆ E�sC P N ⊆ s F�

N ⊆ E�N

(resp.{e} ⊆ E�

wC P N = wF�N ⊆ E�

sC P N ⊆ s F�N ⊆ E�

N );

in particular, wF�N coincides with the set of Nash equilibria that are payoff equivalent

to e (resp. e) and every element of wF�N weakly Pareto dominates every element of

E�N \wF�

N . Besides, if all Nash equilibria are strict then

{e} = E�wC P N = wF�

N = E�sC P N ⊆ s F�

N ⊆ E�ST N = E�

N

(resp.{e} = E�

wC P N = wF�N = E�

sC P N ⊆ s F�N ⊆ E�

ST N = E�N ).

What is still not clear to us is whether the additional assumption that “each Si iscompact and each bi has nonempty compact values” might allow one to prove thatE�

wC P N ⊆ wF�N in every real σ -interactive game � with strategic substitutes and

either increasing or decreasing externalities.16 This is still an open issue. We do notexclude that such a possibility can be disproved only by means of a very complexcounterexample with a large number of players, which at the moment we do not have.

4 Applications

We shall present examples of models where our results apply. In particular, we shallconsider economic models of the literature, or extensions thereof, where the structureof the set of coalition-proof Nash equilibria has not been analyzed yet or for whichthere are only some partial results.

It should be clear that the sets E�N and E

N play a special role in our results: Corollary1 shows that the Nash equilibria in these two sets satisfy many desirable properties. Aresult by Kukushkin (2005, Corollary of Theorem 2) will be particularly useful to prove

the nonemptiness of E�N and E

N in almost all our applications. Here below we shallstate only a straightforward and very particular consequence of that more general result.

Existence result I (Kukushkin 2005) Let � be a real σ -interactive game with strategicsubstitutes. Assume that, for all i ∈ M, Si ⊆ R+ is closed and bi is nonempty-valued.Additionally assume that, for all i ∈ M ,

σi : s �→ α∏

l∈M\{i}sl +

l∈M\{i}β

(l)i sl ,

15 I.e., there exist e and e in E�N ( �= ∅) such that, for all e ∈ E�

N ,

min bi(e) = ei ≤ ei ≤ ei = max bi (e) for all i ∈ M.

Clearly σi(e) ≤ σi (e) ≤ σi (e) for all i ∈ M .

16 Certainly, and more importantly, even with these topological conditions we might have that wF�N �

E�wC P N like in Example 3 below and we might have that E�

N = ∅.

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where α ∈ R+ and β(h)k = β

(k)h ∈ R+ for all k, h ∈ M. Then E�

N �= ∅.17 Besides

E�N �= ∅ (resp. E

N �= ∅) if, for all i ∈ M , bi is also closed-valued (resp. compact-valued).

It is perhaps good to remark that in Sect. 4.1–5 we consider games that are gen-eralized quasi-aggregative games in the sense of Jensen (2010). Moreover, in all ourapplications where the above Existence result I is used to show the existence of BR-extremal Nash equilibria (i.e., in Sect. 4.1–2 and 4.4–5) one can show the existence ofa Nash equilibrium also utilizing other Nash equilibrium existence results of the liter-ature (e.g., Corollary 1 in Jensen (2010)), but their—more or less—direct applicationdoes not generally guarantee the existence of BR-extremal Nash equilibria. In Sect.4.3 we shall instead use the following standard result.

(Standard) Existence result II Let � be a real σ -interactive game with strategicsubstitutes. Assume that, for all i ∈ M, Si ⊆ R+ is closed and bi is nonempty-valued.Additionally assume that, for all i ∈ M , Si is convex and bi is convex-valued and

closed (i.e., with a closed graph). Then E�N �= ∅. Besides E�

N = E�

N = E�N =

E�ST N �= ∅ if, for all i ∈ M, bi is also single-valued at each Nash equilibrium.

4.1 Models of Cournot competition

A finite set M �= ∅ of firms produce a homogeneous good. Each firm chooses a levelof production out of its production set Si ⊆ R+ which is assumed to be nonempty andclosed. The price at which an aggregate quantity is entirely demanded is given by acontinuous and decreasing function p : R+ → R+ with nonempty support T . Firm i’scost function is a strictly increasing left-continuous function ci : Si → R+ such thatp (x) x −ci (x) ≤ ci (0) for x large enough if Si is unbounded. Let ui : ∏

i∈M Si → R,ui : s �→ p

(∑l∈M si

)si − ci (si ) be firm i’s profit function, for all i ∈ M . Finally

assume that p is either (i) log-concave and strictly decreasing or (ii) twice differentiableon T \ {0} with T \ {0} �= R++ and D2 p (x) x + Dp (x) < 0 for all x ∈ T \ {0}.

The models of Cournot competition just described are widely studied exten-sions to possibly nonconvex strategy sets of the Cournot models described inNovshek (1985) and Amir (1996). It is well-known that the associated games � =(M, (Si )i∈M , (ui )i∈M

)are real �-interactive games with strategic substitutes and

decreasing externalities, and it is well-known that E�N �= ∅ by the above Existence

result I. What is not well-known is that, by Corollary 1, in the above models one has

E�N ⊆ E�

wC P N ⊆ E�sC P N = E�

N = s F�N ⊇ wF�

N ⊇ E�N .

Remark 1 Consequently—and this is true for all applications presented in Sect. 4,but we shall avoid the inutile repetition of such an immediate consequence—we haveguaranteed also that in the previous models there exists a w-coalition-proof Nashequilibrium which is also an s-coalition-proof Nash equilibrium that is not weakly (and

17 It is interesting to remark that all games in the proof of Proposition 1 satisfy the previous assumptions(and even posses compact strategy sets).

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F. Quartieri, R. Shinohara

hence strongly) Pareto dominated by any Nash equilibrium: also this fact has nevernoted and proved before. In this connection it must be acknowledged that Kukushkin(1997) provides sufficient conditions for a game � under which E�

wC P N �= ∅ andthat one of the applications of the Theorem in Kukushkin (1997) concerns also somemodels of Cournot competition considered above. Example 2 in Kukushkin (1997)shows the importance of the assumption of convexity of strategy sets for the validityof the Theorem in Kukushkin (1997); in fact that result does not generally ensurethe nonemptiness of E�

wC P N (and of E�sC P N ) in the models of Cournot competition

considered above.

4.1.1 Numerical examples

The following example shows that in the games described above a Nash equilibriumneed not be a w-coalition-proof Nash equilibrium.

Example 1 Put M = {1, 2, 3},

p : x �→{

p (x) = 8 − x if x ≤ 7p (x) = e7−x if x > 7,

S1 = S2 = {0, 1, 2, ...}, S3 = {0, 7, 14, ...} , c1 : x �→ x , c2 : x �→ x +max {0, x − 3}and c3 : x �→ e−4x . Clearly p is log-concave and all the assumptions listed above aresatisfied. Note that

E�N = {(2, 2, 0) , (3, 2, 0) , (2, 3, 0) , (0, 0, 7)}

and {(2, 2, 0) , (0, 0, 7)} = E�N = E�

wC P N = wF�N ⊂ E�

N = E�sC P N = s F�

N .

It is well-known that checking the set of w- and s-coalition-proof Nash equilibria ofa game can be very time-consuming (all w- and s-self-enforcing strategies of the gameand of many induced games must be checked). Our results are useful in this regard.For example, to check all sets of equilibria above, one could proceed as follows. Checkthat b1 (x, 0, 0) = {3}, b1 (x, 2, 0) = {2, 3}, b1 (x, 3, 0) = {2}, b1 (x, 0, 7) = {0},b3 (2, 2, x) = {0} and b3 (0, 0, x) = {7}; besides check that the joint strategies (3, 2, 0)

and (2, 3, 0) are weakly Pareto dominated by (2, 2, 0). There is nothing else to bechecked numerically. By symmetry, conclude that b2 (0, x, 0) = {3}, b2 (2, x, 0) ={2, 3}, b2 (3, x, 0) = {2} and b2 (0, x, 7) = {0}. Therefore any number greater than 3 isnever a best-reply for players 1 and 2. Thus {(2, 2, 0) , (3, 2, 0) , (2, 3, 0) , (0, 0, 7)} ⊆E�

N . By Corollary 4, conclude that there cannot exist a fourth Nash equilibrium ssuch that s3 = 0 and that there cannot exist a second Nash equilibrium s such thats3 = 7. Thus E�

N and E�N are exactly the sets defined in Example 1. By Corollary

1, E�N = E�

sC P N = s F�N and E�

wC P N ⊇ E�N ⊆ wF�

N . But then, since (3, 2, 0) and(2, 3, 0) are weakly Pareto dominated by (2, 2, 0), also E�

wC P N and wF�N are exactly

the sets defined in Example 1.

Remark 2 Note that in the game of Cournot competition illustrated in Example 1we have u1 (5, 4, 0) − u1 (4, 4, 0) < u1 (5, 5, 0) − u1 (4, 5, 0). Therefore that game

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is an instance of a real �-interactive game with strategic substitutes and decreasingexternalities that satisfies neither the conditions of the Proposition in Shinohara (2005)nor the condition of “strict submodularity” in Jensen (2006).

Example 2 below shows that, in the games described above, a w-coalition-proofNash equilibrium need not be a BR-minimal Nash equilibrium.

Example 2 Put M = {1, 2, 3}, p : x �→ e−x and ci : x �→ e−x x for all i ∈ M . For alli ∈ M , let Si = [0, 1]. Also in this example p is log-concave. It can be easily verifiedthat

E�N = ([0, 1] × {0} × {0}) ∪ ({0} × [0, 1] × {0}) ∪ ({0} × {0} × [0, 1]) ,

E�N = {(0, 0, 0)} ⊂ E�

N = E�wC P N = wF�

N and (1/2, 0, 0) ∈ E�N \E�

N .

Example 3 below shows that, in the games described above, a Nash equilibriumwhich is not weakly Pareto dominated by other Nash equilibria need not be a w-coalition-proof Nash equilibrium.

Example 3 Consider again Example 1 and modify only the following assumptions:now put S1 = S2 = {0, 1, 2, 3} and S3 = {0, 7} and let c3 : x �→ e−5x . It is left to thereader to verify that

E�N = E�

wC P N = {(0, 0, 7)} ⊂ {(3, 2, 0) , (2, 3, 0) , (0, 0, 7)} = E�N = wF�

N ,

and hence that wF�N \E�

wC P N �= ∅ .

Example 3 is important because it has shown that it is possible that wF�N � E�

wC P Nin some real �-interactive games with strategic substitutes and decreasing externalitieswith a compact set of Nash equilibria and continuous payoff functions.18

4.2 Models of voluntary contribution of a public good

Consider the model of voluntary contribution of a public good analyzed in Proposition1 of Acemoglu and Jensen (2013), and assume that the private good is strictly normal(more precisely, assume that the inequality in (18) of Acemoglu and Jensen (2013) isstrict). Besides assume that the payoff to each individual is increasing in the sum ofthe contributions of the other individuals (more precisely, assume that the functions ui

defined in (16) of Acemoglu and Jensen (2013) are increasing in the second argument).It can be easily verified that under the two previous additional assumptions the games� that can be associated to this model are real �-interactive games with strategic

18 In games with compact sets of Nash equilibria and upper semicontinuous payoff functions the non-emptiness of E�

N implies the nonemptiness of wF�N (thus, in these games, wF�

N ⊆ E�wC P N and E�

N �= ∅together imply E�

wC P N �= ∅).

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F. Quartieri, R. Shinohara

substitutes and increasing externalities (and also in this case E�

N �= ∅ by the aboveExistence result I). Then, our Corollary 1 ensures that

E�

N ⊆ E�wC P N ⊆ E�

sC P N = E�N = s F�

N ⊇ wF�N ⊇ E

N

and our Corollary 4 implies that if a renegotiation of a (w- or s-)coalition-proof Nashequilibrium for � increases the contribution of all agents then it strictly increases thecontribution of exactly one agent.

Similarly, the games associated to the model of voluntary contribution of a publicgood described in Kerschbamer and Puppe (1998) (or of its extension to n players illus-trated in Quartieri and Shinohara (2012)) are real �-interactive games with strategicsubstitutes and increasing externalities with E�

ST N = E�N ( �= ∅ by the above Existence

result II). In this case, our Corollary 2 ensures even that

E�wC P N = E�

sC P N = E�N = wF�

N = s F�N

and our Corollary 5 implies that no renegotiation of a (w- or s-)coalition-proof Nashequilibrium for � can increase the contributions of all agents.

4.3 Games on networks: Convex strategy sets

A finite nonempty set M of agents strategically interact on a network. For each agenti ∈ M , we denote by Ni the set of i’s neighbors, i.e., the agents other than i whostrategically affect i’s payoff; this suffices to describe the (possibly directed) networkin our context. Each agent i chooses an action si from a closed interval Si ⊆ R+ suchthat 0 ∈ Si . The cost of i’s choice is ci (si ), where ci is a continuous, convex andincreasing real-valued function on R+. Put SM = ∏

l∈M Sl and let σ = {σi }i∈M be anarbitrary family of continuous real-valued functions on SM such that, for all i ∈ M : σi

is increasing in every argument; σi is constant in every argument sl with l ∈ M\Ni ;σi vanishes at the origin. The revenue of each agent i at s ∈ SM is ri (si , σi (s)), whereri : R+ × R+ → R is a continuous function such that:

(i) ri is strictly concave in the first argument and increasing in the second argument;(ii) D+

1 ri is decreasing in the second argument;19

(iii) ri (·, 0) − ci is not strictly increasing if Si = R+.

Each agent i ∈ M obtains ui (s), where ui : SM → R, ui : s �→ ri (si , σi (s)) −ci (si ). (Just to provide an example for ci , ri and σi let ci : x �→ x , ri : (si , x) �→2√

si + x , and σi : s �→ max {sl : l ∈ Ni } if Ni �= ∅ while σi : s �→ 0 if Ni = ∅.)It is easily seen that the games � = (

M, (Si )i∈M , (ui )i∈M)

just described arereal σ -interactive games with strategic substitutes and increasing externalities, and itis immediate that E�

N = E�ST N �= ∅ by the above Existence result II. What is not

immediate, but follows directly from our Corollary 2, is that

E�wC P N = E�

sC P N = E�N = wF�

N = s F�N .

19 D+1 ri : R+ → R denotes the (well-defined) right-hand derivative of ri .

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Coalition-proofness in a class of games

Observation VI The games on network we have considered so far need not be best-reply pseudo-potential games with an upper semi-continuous potential; more precisely,there exists a game on network such that for no upper semi-continuous function P :SM → R we have

bi (s) ⊇ arg maxz∈Si

P (z, s−i ) at all s ∈ SM , for all i ∈ M.

(See the Appendix for a proof). Thus, Corollary 1 in Jensen (2010) does not prove thatE�

N �= ∅ in the games on network described above.

Remark 3 The model of provision of a public good on network in Bramoullé andKranton (2007)—which is properly generalized by the model just described—restrictsattention to the special case of undirected networks with σi (s) = ∑

l∈Nisl and Si =

R+ for all i ∈ M and where D+1 ri is strictly decreasing in the second argument, for

all i ∈ M . For this special case Bramoullé and Kranton (2007) exhibit a measure ofsocial welfare for which only some Nash equilibria can possess the highest welfare,while our results point out that E�

N = wF�N (clearly, these two facts are not at odds).

It must be acknowledged that, for the special case previously indicated, also Newton(2010) shows that E�

sC P N = E�N .

4.4 Games on networks: discrete strategy sets

Consider again the model above and remove the assumption that strategy sets areintervals and assumptions (i), (ii) and (iii). Assume instead that the network is undi-rected (i.e., Nk � h ⇐⇒ k ∈ Nh for all h, k ∈ M) and that, for all i ∈ M :Si = {0, 1}; σi (s) = ∑

l∈Nisl at all s ∈ SM ; ri is increasing in the second argument;

[ri (1, ·) − ci (1)]− [ri (0, ·) − ci (0)] vanishes at at most one point and is decreasing.We have sufficient assumptions to conclude that the games on network just describedare real σ -interactive games with strategic substitutes and increasing externalities for

the interaction system σ = {σi }i∈M . In this case, E�

N �= ∅ by the above Existenceresult I and, by our Corollary 1,

E�

N ⊆ E�wC P N ⊆ E�

sC P N = E�N = s F�

N ⊇ wF�N ⊇ E

N .

Note that if one additionally assumes that the network is also connected then the setof Nash equilibria is characterized as in Corollary 3; therefore, when the game justdescribed is used as the abstract structure of a model of provision of a public goodon a connected network with many agents, one has that no renegotiation of a (w- ors-)coalition-proof Nash equilibrium for � can increase strictly the contributions of allagents.

Just to provide a specific example—possibly for unconnected networks—let, forall i ∈ M : Si = {0, 1}; ci : si �→ γ si for some fixed γ > 0; ri : (si , σi (s)) �→min {ti , si + σi (s)} with ti > 0. (Note that when ti = 1 for all i ∈ M and γ < 1 onehas exactly the “ Best shot” public good game on network illustrated in Example 2 in

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F. Quartieri, R. Shinohara

Jackson and Zenou (2014); note also that in that particular case all Nash equilibria arestrict and hence that even Corollaries 2 and 5 apply).

4.5 Team projects

Consider the teamwork project as it is exactly described in the first nine lines of Sect. 5.1in Jensen (2010), and with the topological assumptions of that article. Besides assumethat there are at least two players and that: (i) each player has exactly one task; (ii)each πi —in the notation of Sect. 2 in Jensen (2010)—is increasing (resp. decreasing)in the second argument; (iii) Assumption 1’ of Jensen (2010) holds. We already havesufficient assumptions to conclude that the games described are real �-interactivegames with strategic substitutes and increasing (resp. decreasing) externalities. Since

players have exactly one task, it is well-known that E�

N �= ∅ �= E�N by the above

Existence result I. By our Corollary 1, in the model just described one has

E�

N ⊆ E�wC P N ⊆ E�

sC P N = E�N = s F�

N ⊇ wF�N ⊇ E

N

(resp.E�N ⊆ E�

wC P N ⊆ E�sC P N = E�

N = s F�N ⊇ wF�

N ⊇ E�N ).

Theorem 2 and Corollary 3 characterizes E�N . In particular Theorem 2 implies that:

if there exists a Nash equilibrium for � where the project fails with certainty (i.e., atleast one player is inactive) because at least two players are inactive, then the projectmust fail with certainty at all Nash equilibria for � (i.e., then at each Nash equilibriumfor � at least one player must be inactive).

4.5.1 Numerical example

The set of Nash equilibria in the games just described need not be characterized as inCorollary 4 as long as Nash equilibria exist at which the project fails with certaintybecause of the inactivity of two players. The following example well illustrates thepoint.

Example 4 Let � = (M, (Si )i∈M , (ui )i∈M

)be a game where M = {1, 2, 3} and for

all i ∈ M , Si = [0, 1] and ui : s �→ −si · �i (s). Each of the previous assumptions issatisfied and one has20

E�N = {s ∈ [0, 1]3 : min {s1, s2, s3} = 0}.

Note, in particular, that (0, 0, 0) ∈ E�N � (1, 1, 0) and hence

(min S1, min S2, min S3) ∈ E�N � (max S1, max S2, min S3)

(with min Si < max Si for all i ∈ M).

20 The reader might enjoy a comparison with E�N in Example 2.

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Coalition-proofness in a class of games

Corollary 4 implies that for no order-preserving transformation of the payoff func-tions, or of each strategy set, the game in Example 4 can be represented as a �-interactive game with strategic substitutes. Note also that, in the game in Example4, each ui is constantly zero on E�

N = {s ∈ [0, 1]3 : min {s1, s2, s3} = 0}; hencethe condition of “(strong) strategic substitutes in equilibrium” of the Theorem in Yi(1999) is not satisfied.

Finally, if we consider a variant of the game in Example 4 where ui : s �→ si −si · �i (s) for all i ∈ {1, 2, 3}, we obtain a game which still satisfies the previousassumptions but which has at least one Nash equilibrium where the project succeedswith certainty (e.g., the joint strategy (1, 1, 1)) and at least one Nash equilibrium wherethe project fails with certainty (e.g., the joint strategy (1, 1, 0)). Variants of Example4 where all players are active at each Nash equilibrium can be easily constructed bythe reader.

Acknowledgments The present version of this paper considerably benefited from discerning commentsand remarks of two anonymous reviewers. The second author gratefully acknowledges financial supportfrom Grant-in-Aid for Young Scientists (21730156, 24730165) from the Japan Society for Promotion ofScience.

Appendix

Fact 1 Let � = (M, (Si )i∈M , (ui )i∈M ) be a real σ -interactive game with strategicsubstitutes. We can define a game

�� = (M, (S�i )i∈M , (u�

i )i∈M )

such that S�i = −Si and u�

i : S�M → R , u�

i : s �→ ui (−s) , for all i ∈ M.21

Besides we can define the family

σ� = {σ�i }i∈M

such that σ�i : S�

M → R, σ�i : s �→ −σi (−s) for all i ∈ M. Indeed, also �� is a

real σ�-interactive game with strategic substitutes.

Proof Since � is a real σ -interactive game with strategic substitutes, there existsυi : Si × σi [SM ] → R such that ui (s) = υi (si , σi (s)) at all s ∈ SM , for all

i ∈ M . Letting υ�i : S�

i × σ�i

[S�

M

]→ R, υ

�i : (x, y) �→ υi (−x,−y) for all

i ∈ M , it can be easily verified that �� is a real σ�-interactive game with strategicsubstitutes. ��Fact 2 Let � = (M, (Si )i∈M , (ui )i∈M ) be a real σ -interactive game with strategicsubstitutes and increasing (resp. decreasing) externalities, and define �� and σ�as in Fact 1. Then �� is a real σ�-interactive game with strategic substitutes anddecreasing (resp. increasing) externalities.

21 Clearly, S�M denotes

∏i∈M S�

i .

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F. Quartieri, R. Shinohara

Proof Define υ�i for all i ∈ M as in the proof of Fact 1. Then Fact 2 is an immediate

consequence of Fact 1 and of the decreasingness (resp. increasingness) of each υ�i in

the second argument. ��Proof of Observation I Consider the game � with M = {1, 2, 3}, S1 = S2 = [0, 1],S3 = {0}, ui : s �→ 0, u2 : s �→ 0 and u3 : s �→ 1A (s) (i.e., u3 is the indicatorfunction 1A of A ⊂ SM ) where

B = {s ∈ SM : max {s1, s2} �= 1} ∪ {(1, 0, 0)} and A = SM\B.

It is immediate that the game is a real σ -interactive game with strategic substitutesand increasing externalities for the interaction system σ such that σi : s �→ ui (s) forall i ∈ M (just put υi : (si , t) �→ t for all i ∈ M). On the other hand, it is also quitesimple to notice that there cannot exist a continuous function ς3 : SM\{3} → R suchthat u3 (s) = u3 (s3, ς3 (s−3)) for some u3: by way of contradiction suppose on thecontrary that such ς3 exists; notice that ς3 (1, 0) �= ς3 (0, 1) (as u3 (0, ς3 (1, 0)) �=u3 (0, ς3 (0, 1))); infer that, by the continuity of ς3, there must exist a point z∗ ∈((0, 1) × {1}) ∪ {(1, 1)} ∪ ({1} × (0, 1)) such that

min {ς3 (1, 0) , ς3 (0, 1)} < ς3(z∗) < max {ς3 (1, 0) , ς3 (0, 1)}

and a point z∗∗ ∈ {z ∈ (0, 1) × (0, 1) : z1 + z2 = 1} such that

ς3(z∗) = ς3

(z∗∗) ;

finally, conclude that we obtained the following impossible equalities

1 = 1A(z∗

1, z∗2, 0

) = u3(0, ς3

(z∗)) = u3

(0, ς3

(z∗∗)) = 1A

(z∗∗

1 , z∗∗2 , 0

) = 0.

This completes the proof (for the case of increasing externalities, clearly Fact 2 guaran-tees that we can construct an analogous example with decreasingexternalities). ��Proof of Observation II Proof of the if part. Suppose � has a compatible interactionsystem σ where interaction functions are real-valued and continuous. Let g : SM → R,g : s �→ 0. For all i ∈ M , take an arbitrary si ∈ Si and let:

• ςi : SM\{i} → X−i := σi [SM ], ςi : s−i �→ σi (si , s−i );• Vi : SM\{i} → R, Vi : s−i �→ 0;• Fi : Si × R → R, Fi : (si , x) �→ 0.

Finally, for all i ∈ M , let ui : Si × X−i → R be the function defined byui (si , ςi (s−i )) = υi (si , σi (s)) at all s ∈ SM and conclude that � is a generalizedquasi-aggregative game with aggregator g. Clearly, since each interaction function σi

is continuous, also each J-interaction function ςi is continuous.Proof of the only if part. Suppose � is a generalized quasi-aggregative game and let

σi : SM → Ii := R, σi : s �→ ςi (s−i ) for all i ∈ M . Let υi : Si ×σi [SM ] → R be the

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Coalition-proofness in a class of games

function defined by υi (si , σi (s)) = ui (si , ςi (s−i )) at all s ∈ SM , for all i ∈ M . Letσ := {σi }i∈M and conclude that σ is an interaction system which is compatible with� and that each interaction function σi is real-valued and continuous. The continuityof each σi can be easily verified by the reader considering that σi is, by construction,constant in si (and not just merely continuous in si ) and continuous in s−i . ��Proof of Observation III In fact, the same proof of Observation II (without involvingcontinuity arguments). ��Proof of Observation IV It is left to the reader to notice that, constructing again eachσi as in the proof of the only if part of Observation II, the proof is immediate. ��Proof of Observation V It is left to the reader to notice that, constructing again eachςi as in the proof of the if part of Observation II, the proof is immediate. ��Proof of Observation VI For example, construct the following game on network. PutM = {1, 2, 3}, N1 = {2}, N2 = {3}, N3 = {1}, σ1 : s �→ s2, σ2 : s �→ s3, σ3 : s �→ s1and, for all i ∈ M , Si = [0, 1], ri : (si , x) �→ 2

√si + x and ci : x �→ x . By way of

contradiction, suppose there exists an upper semicontinuous function P : SM → R

such that

bi (s) ⊇ arg maxz∈Si

P (z, s−i ) at all s ∈ SM , for all i ∈ M .

Then, since best-replies are single-valued and since P is an upper semicontinuousfunction on a compact set, we must have that

bi (s) = arg maxz∈Si

P (z, s−i ) at all s ∈ SM , for all i ∈ M .

Note that, for all i ∈ M , bi (s) = {0} if σi (s) = 1 and bi (s) = {1} if σi (s) = 0.Therefore:

• P (1, 0, 0) < P (1, 1, 0) as b2 (1, x, 0) = {1};• P (1, 1, 0) < P (0, 1, 0) as b1 (x, 1, 0) = {0};• P (0, 1, 0) < P (0, 1, 1) as b3 (0, 1, x) = {1};• P (0, 1, 1) < P (0, 0, 1) as b2 (0, x, 1) = {0};• P (0, 0, 1) < P (1, 0, 1) as b1 (x, 0, 1) = {1};• P (1, 0, 1) < P (1, 0, 0) as b3 (1, 0, x) = {0}.

But this is impossible because we obtain P (1, 0, 0) < P (1, 0, 0). ��Proof of Claim 1 A consequence of Proposition 1 below and of Fact 2. ��Proposition 1 The following statements are true:

(i) there exists a real �-interactive game with strategic substitutes and decreasingexternalities where E�

wC P N = wF�N ⊂ E�

N ;(ii) there exists a real �-interactive game with strategic substitutes and decreasing

externalities where E�N ⊂ E�

wC P N = wF�N ;

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F. Quartieri, R. Shinohara

(iii) there exists a real �-interactive game with strategic substitutes and decreasingexternalities where wF�

N \E�wC P N �= ∅;

(iv) there exists a real �-interactive game with strategic substitutes and decreasingexternalities where E�

wC P N \wF�N �= ∅.

Proof (i) See Example 1 in Sect. 4.(ii) See Example 2 in Sect. 4.

(iii) See Example 3 in Sect. 4.(iv) Consider the game � with M = {1, 2, 3}, S1 = S2 = [0, 1], S3 = [0, 1] ∪ [7, 8],

ui (s) ={

si (1 − �i (s)) − 1 if �i (s) ≥ 32

si (1 − �i (s)) if �i (s) < 32

for i ∈ {1, 2} and

u3 (s) =

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

−1 if s3 = 0 and �3 (s) > 1− 1

2 if s3 = 0 and �3 (s) = 10 if s3 = 0 and �3 (s) < 1s3 (1 − �3 (s)) − 1 if s3 ∈ (0, 1) ∪ {8} and �3 (s) > 1s3 (1 − �3 (s)) if s3 ∈ (0, 1) ∪ {8} and �3 (s) ≤ 1−9 if s3 ∈ {1} ∪ [7, 8) .

Define Q1 := {( 12 , 1

2 , 12

)}, Q2 := {(0, 0, 8)}, Q3 := {(1, 0, x) : 0 < x < 1}, Q4 :=

{(0, 1, x) : 0 < x < 1}, Q5 := {(x, 1, 0) : 0 < x < 1} and Q6 := {(1, x, 0) : 0 < x≤ 1}. Noting that

bi (s) =⎧⎨

{0} if �i (s) > 1[0, 1] if �i (s) = 1{1} if �i (s) < 1

for i ∈ {1, 2} and

b3 (s) =⎧⎨

{0} if �3 (s) > 1(0, 1) ∪ {8} if �3 (s) = 1{8} if �3 (s) < 1,

conclude that � is a real �-interactive game with strategic substitutes and decreasingexternalities. Note that E�

ST N = Q2 ⊂ ⋃6i=1 Qi = E�

N ; thus E�wC P N ⊇ Q2 ⊆ wF�

Nby Theorem 1. Note that

• (ui (s))i∈M = (0, 0, 0) if s ∈ Q1,• (ui (s))i∈M = (−1,−1, 8) if s ∈ Q2,• (ui (s))i∈M = (1 − s3, 0, 0) if s ∈ Q3 and s3 ∈ (

0, 12

)

• (ui (s))i∈M = (1 − s3,−1, 0) if s ∈ Q3 and s3 ∈ [ 12 , 1

),

• (ui (s))i∈M = (0, 1 − s3, 0) if s ∈ Q4 and s3 ∈ (0, 1

2

),

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Coalition-proofness in a class of games

• (ui (s))i∈M = (−1, 1 − s3, 0) if s ∈ Q4 and s3 ∈ [ 12 , 1

),

• (ui (s))i∈M = (0, 1 − s1,−1) if s ∈ Q5 and s1 ∈ (0, 1),• (ui (s))i∈M = (1 − s2, 0,−1) if s ∈ Q6 and s2 ∈ (0, 1],

and conclude that wF�N = Q2. Note that

• s∗ ∈ Q2 ∪ Q5 ∪ Q6 and s∗∗ ∈ Q1 implies that s∗ does not weakly Pareto dominates∗∗ in �,

• if s ∈ Q1 then s is w-self-enforcing for � as, for all i ∈ M , s−i is not weakly Paretodominated in �|si by any other Nash equilibrium for �|si ,

• every strategy s ∈ Q3 is not w-self-enforcing for � as s−2 is weakly Pareto domi-

nated in �|s2 by(s1,

12 s3

) ∈ E�|s2N ,

• and every strategy s ∈ Q4 is not w-self-enforcing for � as s−1 is weakly Pareto

dominated in �|s1 by(s2,

12 s3

) ∈ E�|s1N ,

and conclude that Q1 ⊆ E�wC P N . Note that (Q5 ∪ Q6) ∩ E�

wC P N = ∅ (consider thedeviating coalition {1, 2}). Thus Q2 = wF�

N ⊂ E�wC P N = Q1 ∪ Q2 ⊂ E�

N and inparticular E�

wC P N \wF�N �= ∅. ��

Proof of Claim 2 Consider the game � with M = {1, 2, 3}, S1 = S2 = [0, 1], S3 =[−1, 0], and

u1 (s) = 99 (s2 + s3)2 + 400 (s2 + s3) + 2s1 (s2 + s3) − s2

1 ,

u2 (s) = 99 (s1 + s3)2 + 400 (s1 + s3) + 2s2 (s1 + s3) − s2

2 ,

u3 (s) = 99 (s1 + s2)2 + 400 (s1 + s2) − s3 (s1 + s2) − s2

3 .

In this game b1 (s) = {max {0, s2 + s3}}, b2 (s) = {max {0, s1 + s3}} and b3 (s) ={− 12 (s1 + s2)

}. Let e := (0, 0, 0). It can be easily verified that E�

N = {e} = s F�N .

All conditions of the Theorem in Yi (1999) hold (in particular the condition of “(strong) strategic substitutes in equilibrium” holds vacuously); however the thesis of

the Theorem in Yi (1999) does not hold: e /∈ E�sC P N = ∅ since (1, 1) ∈ E

�|e−3N and

(1, 1) strongly Pareto dominates e−3 in �|e−3 . Therefore the statement of Yi’s theoremis false.

We can provide also a second counterexample with four players (in fact it sufficesto add a player): consider the game � with M = {1, 2, 3, 4}, S1 = S2 = S4 = [0, 1],S3 = [−1, 0], and

u1 (s) = 99 (s2 + s3 + s4)2 + 400 (s2 + s3 + s4) + 2s1 (s2 + s3 + s4) − s2

1 ,

u2 (s) = 99 (s1 + s3 + s4)2 + 400 (s1 + s3 + s4) + 2s2 (s1 + s3 + s4) − s2

2 ,

u3 (s) = 99 (s1 + s2 + s4)2 + 400 (s1 + s2 + s4) − s3 (s1 + s2 + s4) − s2

3 ,

u4 (s) = s1 + s2 + s3 − s24 .

All hypotheses of Yi’s Theorem hold but E�N = {(0, 0, 0, 0)} = s F�

N �= ∅ and∅ = E�

sC P N (consider again the deviating coalition {1, 2}). ��

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F. Quartieri, R. Shinohara

Table 1 Counterexample to a remark in Yi (1999)

s∗∗4

s∗3 s∗∗

3

s∗2 s∗∗

2 s∗2 s∗∗

2

s∗∗1 0, 1, 1, 0 0, 0, 1, 0 0, 1, 0, 0 0, 0, 0, 0

s∗1 1, 1, 1, 0 1, 0, 1, 0 1, 1, 0, 0 1, 0, 0, 0

s∗4

s∗3 s∗∗

3

s∗2 s∗∗

2 s∗2 s∗∗

2

s∗∗1 7, 5, 5, 6 5, 5, 5, 6 3, 2, 2, 6 3, 3, 2, 6

s∗1 6, 6, 6, 6 5, 7, 5, 6 6, 6, 6, 6 2, 3, 2, 6

Proof of Claim 3 Consider the game � with M = {1, 2, 3, 4} and, for all i ∈ M ,Si = {

s∗i , s∗∗

i

}and ui is specified by Table 1 (the l-th number in each entry is the

l-th player’s payoff). Put, for all i ∈ M , s∗i = 0 and s∗∗

i = ∑il=110l−1. (Note that

every four-player game with the just defined strategy sets satisfies condition (1) inthe statement of Yi’s theorem). Apart from condition (3), all the conditions of Yi’sTheorem hold and we have

(s∗∗

1 , s∗∗2 , s∗

3 , s∗4

) ∈ s F�N \E�

sC P N �= ∅ (this time considerthe deviating coalition {1, 2, 3}). ��

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