Coalition proofness in a class of games with strategic substitutes

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: quartieri.f@alice.it
R. Shinohara (B)
Faculty of Economics, Hosei University, 4342, Aihara-machi, Machida, Tokyo 194-0298, Japan
e-mail: ryusukes@hosei.ac.jp
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F. Quartieri, R. Shinohara
with either strategic substitutes or strategic complements. Despite a sort of duality in
the definition of the two classes of games,many properties that hold true for the games
of one of the two classes need not hold for those of the other. Indeed, apart from the
order-theoretic nature of their definitions, the two classes do not seem to share many
common properties.
Accrediting the importance of monotone externalities (i.e., the monotonicity of
players’ payoff functions in the opponents’ strategies) in many games with strategic
complements of economic interest, the literature has provided an exhaustive investi-gation
of the Pareto and coalition-proofness properties of Nash equilibria in abstract
classes 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 dominated
by other Nash equilibria and that there is a tendency for that equilibrium to be the
unique 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 strategic
substitutes and monotone externalities is still missing in the literature; we are aware
of only some partial results presented in Yi (1999) and in Shinohara (2005), which
will 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 equilibria
in a subclass of games with strategic substitutes and monotone externalities where the
strategic interaction is mediated by interaction functions. The games of this subclass
will be called σ-interactive games with strategic substitutes and increasing/decreasing
externalities. As we shall point out, various models studied in Industrial organization,
Public economics and Network economics are associated to games that belong to such
a subclass.
Somewhat loosely speaking—the reader is referred to Sect. 2.3 for a precise
definition—we say that a game is a σ-interactive game with strategic substitutes and
increasing/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 of
the value attained by a real-valued interaction function σi defined on the joint strategy
set; (iii) each interaction function σi is increasing in all arguments and constant in the
i -th argument; (iv) a change in a joint strategy that increases the value attained by the
interaction function σi entails a “ decrease” of player i ’s best-reply; (v) a change of the
strategies of i ’s opponents that increases the value attained by the interaction function
σi entails an increase/decrease in player i ’s payoff. As we shall observe in Sect. 2.4, if
one additionally assumes that each interaction function σi is also continuous then the
games 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, for
results concerning the coalition-proofness of Nash equilibria—in the sense of Bernheim et al. (1987)—see
Theorem A2 and its subsequent remark in Milgrom and Roberts (1996) and Theorems 1 and 2 and their
respective Corollaries in Quartieri (2013). An appropriate discussion can be found in the last-mentioned
article.
2 For a result on the uniqueness of coalition-proof Nash equilibria in games with strategic complements
that dispenses with the assumption of monotone externalities see also Theorem A1 in Milgrom and Roberts
(1996).
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Coalition-proofness in a class of games
sense of Jensen (2010) (and hence our examination contributes also to that strand of
literature3). However, such a continuity condition—as well as some condition similar
to that implied by Assumption 2 in Jensen (2010)—plays no role in our results and
hence will never be imposed as an assumption in this article.
The formal definition of a coalition-proof Nash equilibrium was firstly provided
in 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 under
strong Pareto dominance (in this Introduction s-CPN equilibria for short) may well
differ from the set of coalition-proof Nash equilibria under weak Pareto dominance
(in this Introduction w-CPN equilibria for short). Indeed, there are various numerical
examples4 of games where the set of s-CPN equilibria and that of w-CPN equilibria
are 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 refinement
of the other). For this reason, in this article we shall inspect both concepts and we shall
also 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 relation
with 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 equilibria
that 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 at
most 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 sets
can well be nonempty and disjoint. Clearly, it is particularly interesting to determine
sufficient 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-CPN
equilibrium.
We shall prove that in every σ-interactive game with strategic substitutes and
increasing externalities the set of Nash equilibria coincides with the set of s-CPN
equilibria (and hence also with the set of s-FN equilibria). Besides we shall prove that
in every σ-interactive game with strategic substitutes and increasing externalities the
set of BR-maximal Nash equilibria (i.e., the Nash equilibria whose components are
the greatest best-replies) is included in both the set of w-FN equilibria and that of
w-CPN equilibria; some examples will show that the previous inclusion relation can
be 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 of
literature. In the eight aformentioned articles one can find economic examples (possibly adding some
conditions) of real σ-interactive games with strategic substitutes and increasing/decreasing externalities
that 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 dual
statements for the case of decreasing externalities.
The previous results are of interest for various reasons. The first—and very
general—reason is that it is difficult to say anything about the coalition-proofness
of Nash equilibria in any class of games, and we show that this can be done in a very
precise and simple way in the class of games considered (see Corollaries 1 and 2). The
second 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 allow
for the transformation of some known Nash equilibrium existence results into w- and
s-CPN equilibrium existence results (see Observation IV and its footnote, Corollaries
1 and 2, and Sect. 4). The third is that our results clarify that, quite surprisingly, in
the class of games considered every Nash equilibrium is an s-CPN equilibrium, and
hence also an s-FN equilibrium: this implies that the s-CPN and s-FN equilibrium
concepts cannot act as effective refinements of the Nash equilibrium concept in the
class of games considered (see Corollary 1). The fourth is that our results clarify that
in the class of games considered the w-CPN and w-FN equilibrium concepts can still
be effective refinements, provided some Nash equilibrium is not strict (see Corollary
2 and the Examples of Sect. 4). Finally, our results allow a sensible comparison with
known results for games with strategic complements and monotone externalities (see
Sect. 3.3).
The rest of this article is organized as follows. Section 2 sets the definitions and
some basic preliminaries. Section 3 contains themain results presented above and also
an order-theoretic characterization of the sets of equilibria. Section 4 presents various
applications and examples.AnAppendix contains all proofs that are not directly related
to 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 to
have finite cardinality m.
Let be a game, C ⊆ M, l ∈ M and s ∈ i∈M Si. The seti∈C Si is also denoted
by SC. The tuples (si )i∈C, (si )i∈MC, (si )i∈{l} and (si )i∈M{l} are also denoted by,
respectively, sC, sMC, sl and s−l . The pair (x, y) ∈ SC × SMC will alternatively
denote the tuple z ∈ SM such that zC = x and zMC = y. For all i ∈ M, let
bi : SM → 2Si denote player i ’s best-reply correspondence, which is defined by
bi : s→ argmax
z∈Si
ui (z) if M = {i } and bi : s→ argmax
z∈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 orderon a set X is a reflexive, transitive, and antisymmetric binary relation
on 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 only
to respectively denote the infimum, the minimum (if any), the supremum, and the
maximum (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 Rm if, for any two
distinct 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 functions
such that, for all i ∈ M, the interaction function σi maps SM into an arbitrary set Ii
and is constant in the i -th argument.Agame is said to have a compatible interaction
system 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 Of
course, a game can have many compatible interaction systems. Clearly, not all possible
interaction 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 and
increasing (resp. decreasing) externalities if it is a real σ-interactive game with
strategic substitutes and, for all i ∈ M, υi is increasing (resp. decreasing) in the
second argument.
∗
i
5 E.g.: if |M| 1 put Ii = SM{i } and σ
: s→ s−i for all i ∈ M and take the function υi defined by
∗
i (s) = ui (s) at all s ∈ SM for all i ∈ M; if M = {i } put Ii = {0} and σ
υi si, σ
∗
i
: s→ 0 and take the
∗
i (s) = ui (s) at all s ∈ SM.
function υi defined by υi si, σ
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In the previous definition, the strategic substitutability and externalities are mediated
by the interaction functions. Since each player’s interaction function is assumed to
preserve the order of the joint strategies of the opponents, both definitions do not depart
from the usual general notions of strategic substitutability and monotone externality.6
On the other hand, our terminology would seem improper without this assumption. It
is 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 ith argument—allows
us not to have to distinguish between one-player games and games with two or more
players when we deal with the recursive notion of a coalition-proof Nash equilibrium
and when we prove our results. Note that up to now the assumption that M is finite
has never been used and we could have dispensed with it.
Notation (- and-interactivity) When is a real σ-interactive game with strategic
substitutes 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 strategic
substitutes, 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 subsumemany previ-ous
definitions of aggregative games, for more details see Jensen (2010). (Throughout
this 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 a
subset 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) such
that
ui (s) =
u
i (si, ςi (s−i )), where
i : Si × X−i → R,
u
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 relate
to the generalized quasi-aggregative games in Jensen (2010).
6 When writing this, we mean, in particular, that games properly characterized by some notion of strategic
complementarity are ruled out by Definition 2. (Of course, weaker notions of strategic substitutability—and
of 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 of
its domain (see Jensen (2012)): the following discussion remains unaltered.
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Observation I Not every real σ-interactive gamewith 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 Euclidean
space. Then is a generalized quasi-aggregative game if and only if has a compatible
interaction system σ where every interaction function σi is real-valued and continuous.
(A fortiori, every real σ-interactive game with strategic substitutes and continuous
interaction functions is a generalized quasi-aggregative game; the converse is evidently
false.)
Observation III Suppose is a gamewhere each strategy set is a subset of a Euclidean
space, and suppose for a moment that the continuity of the J-interaction-functions is
dispensed with in the definition of a generalized quasi-aggregative game. Then a
game is a generalized quasi-aggregative game if and only if has a compatible
interaction 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 generalized
quasi-aggregative game; the converse is evidently false.)
The last two Observations clarify, in particular, the exact relation between the
notion of strategic substitutability employed in Definition 2 above and that employed
in Assumption 1’ in Jensen (2010)—which, in some very loose sense, are identical.
Observation IV Suppose is a game with nonempty-valued best-replies where
every strategy set is a subset of the real line; besides, suppose is a generalized
quasi-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-interactive
game with strategic substitutes (and also with increasing/decreasing externalities if
every
i is also increasing/decreasing in the second argument).9
u
Observation V Suppose is a game with nonempty-valued best-replies where every
strategy set is a subset of the real line; besides, suppose is a real σ-interactive game
with strategic substitutes (and increasing/decreasing externalities) such that every
interaction function σi is continuous. Then is also a generalized quasi-aggregative
game satisfyingAssumption 1’ in Jensen (2010) (and every
i is increasing/decreasing
u
in 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 Si
is 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 set
of Nash equilibria also assuming other additional conditions (e.g., conditions that allow the application of
Kakutani’s fixpoint theorem to b).
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Notation (E
N , E
N , E
ST N, E
N ) The set ofNash equilibria (resp. strictNash equilibria)
N (resp. E
ST N). When Si ⊆ R for all i ∈ M, we put:
for is denoted by E
E
N
= s ∈ E
N
: s = (inf bi (s))i∈M ; E
N
= s ∈ E
N
: s = (sup bi (s))i∈M .
N (resp. E
Each element of E
N ) is called a BR-minimal (resp. BR-maximal) Nash
equilibrium for . Perhaps it is worth remarking that best-replies might be empty-valued
in some real σ-interactive games with strategic substitutes. In this connection
it might be worth recalling that inf∅ = supR = +∞ and sup∅ = inf R = −∞.
Thus,10 when strategy sets are subsets of R we have:
• s ∈ E
N
⇐⇒ for all i ∈ M, minbi (s) exists in R and si = min bi (s);
• s ∈ E
N
⇐⇒ for all i ∈ M, maxbi (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 some
j ∈ M; a joint strategy s ∈ SM strongly Pareto dominates in a joint strategy z if
ui (z) ui (s) for all i ∈ M. Let be a game, C ∈ 2M {∅, M}, s ∈ SM and, for all
i ∈ C, ˜ ui : SC → R, ˜ ui : z→ ui z, sMC. The game induced by C at s is the game
|sMC
:= 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 has
been 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 |sMC for all non-empty
C ⊂ M;
• s ∈ SM is a w-coalition-proof (resp. s-coalition-proof) Nash equilibrium for if it
is w-self-enforcing (resp. s-self-enforcing) for and there does not exist another
w-self-enforcing (resp. s-self-enforcing) strategy for that weakly (resp. strongly)
Pareto dominates s in .
N , sF
N , E
wCPN, E
sC PN ) For each game , the set of Nash equilibria
Notation (wF
that are not weakly (resp. strongly) Pareto dominated in by other Nash equilibria is
denoted by wF
N (resp. sF
N ) and the set of w-coalition-proof (resp. s-coalition-proof)
Nash equilibria is denoted by E
wCPN (resp. E
sC PN ).
10 Note that Si ⊆ R and bi (x) = ∅ implies that inf bi (x)(= +∞) and supbi (x)(= −∞) exist in R (but
not 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 and
increasing 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-maximalNash equilibrium for is aw-coalition-proof Nash equilibrium
for , and hence E
N
⊆ E
wCPN;
(iii) each Nash equilibrium for is not strongly Pareto dominated in by other Nash
equilibria for , and hence E
N
= sF
N ;
(iv) each Nash equilibrium for is an s-coalition-proof Nash equilibrium for , and
hence E
N
= E
sC PN .
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 that
u j (x) u j (y) for some j ∈ M. (2)
If σj (y) ≤ σj (x) then σj (y) ≤ σj y j , x−j because σj 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
σj (x) σj (y) ,
which implies, by the increasingness of σj 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 substitutability
condition 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,
|sMis a
-interactive gamewith strategic substitutes and increasing externalities for
C σ
the interaction system
σ
= {
i }i∈C defined by
σ
i : SC → R,
σ
i : x→ σ x, sMC
σ
for all i ∈ C. Clearly,
if x ∈ E
N then xC ∈ E
|xMC
N 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
wCPN.
N and y ∈ E
N such that
(iii) By way of contradiction, suppose there exist x ∈ E
ui (x) ui (y) for all i ∈ M. (4)
Take an arbitrary j ∈ M. If σj (y) ≤ σj (x) then the increasing externality condition
implies u j (y) ≤ u j (x) in contradiction with (4). Therefore σj (x) σj (y), which
implies that
xk yk for some k ∈ M. (5)
Again, if σk (y) ≤ σk (x) then uk (y) ≤ uk (x) in contradiction with (4). Therefore
σk (x) σk (y). Since xk ∈ bk (x) and yk ∈ bk (y) and σk (x) σk (y), the strategic
substitutability 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) is
true when |M| = n. For every nonempty C ⊂ M and every s ∈ SM, |sMis
C a
-interactive game with strategic substitutes and increasing externalities for the
σ
interaction system
σ
= {
i }i∈C defined by
σ
i : SC → R,
σ
i : x→ σ x, sMC for
σ
all i ∈ C. Clearly,
if x ∈ E
N then xC ∈ E
|sMC
N for all nonempty C ⊂ M.
N coincides with the set of s-self-enforcing
Hence, by the induction hypothesis, E
strategies for . Thus, from part (iii) of Theorem 1 it follows easily that
E
= N
E
.
sC PN Corollary 1 Suppose is a real σ-interactive game with strategic substitutes and
increasing externalities, then
E
N
⊆ E
wCPN
⊆ E
sC PN
= E
N
= sF
N
⊇ wF
N
⊇ E
N .
123

Suppose is a real σ-interactive game with strategic substitutes and decreasing
externalities, then
E
N
⊆ E
wCPN
⊆ E
sC PN
= E
N
= sF
N
⊇ wF
N
⊇ E
N .
Proof In case of increasing externalities Corollary 1 is an immediate consequence
of Theorem 1. Construct
as in Fact 1 in the Appendix and note that Fact 2
guarantees that Corollary 1 is true also in case of decreasing externalities (note also
that E
N

N , E
sC PN
= −E

sC PN , E
wCPN
= −E

wCPN, sF
N
= −E

N ,
= −sF
wF
N

N and in particular E
= −wF
N

N ).
= −E
Corollary 2 Suppose is a real σ-interactive game with strategic substitutes and
either increasing or decreasing externalities. Besides suppose all Nash equilibria are
strict.11 Then
E
N
= E
N
= E
wCPN
= E
sC PN
= wF
N
= sF
N
= E
ST N
= E
N .
Just to avoid misunderstandings, we remark that Theorem 1 and Corollaries 1 and 2
do not guarantee that E
N
= ∅(and it is well possible that, e.g., a finite real σ-interactive
game with strategic substitutes and either increasing or decreasing externalities does
not 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 might
well want to know how Corollary 1 relates to the Theorem in Yi (1999) and to the
Proposition in Shinohara (2005).We shall first consider the second-mentioned article.
The Proposition in Shinohara proves that E
wCPN
sC PN for a proper12 subclass
⊆ E
of the class of real -interactive games with strategic substitutes and either increasing
or decreasing externalities. Thus, our Corollary 1 subsumes the Proposition in Shino-hara
(2005). In fact, our Corollary 1 shows also that the inclusion relation established
in the Proposition in Shinohara (2005) is due to the equivalence (under the conditions
of that Proposition) of E
sC PN and E
N . Thus, one might legitimately wonder whether
wCPN and E
N under the
it is possible to establish an analogous equivalence between E
conditions of Corollary 1. In order to answer this and many other legitimate questions
about the tightness of the conclusions of Corollary 1 we explicitly claim (and prove
in the Appendix) the following.
Claim 1 Given the hypotheses of Corollary 1 (and without additional hypotheses), no
inclusion relation can be generally established between E
wCPN and wF
N , and each
of the inclusion relations established in the theses of Corollary 1 cannot be generally
reversed.
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.
123

Let us nowturn to amuch more difficult comparison with the Theorem in Yi (1999).
The difficulty of such a comparison resides in the simple fact that Yi’s Theorem does
not generally hold true (we refer to Quartieri and Shinohara (2012) for a detailed
examination of this issue). That Theorem states that sF
N
⊆ E
sC PN for a class of
games with strategic substitutes and monotone externalities (and with only strict Nash
equilibria) which is seemingly similar to that considered here (as it is clear from the
proofs in Yi (1999), wF
N and E
wCPN are not examined). Our results do not imply
the Theorem in Yi (1999); also the converse is true, because there certainly exist
real σ-interactive games with strategic substitutes and either increasing or decreasing
externalities that do not satisfy the conditions of the Theorem in Yi (1999) (e.g., the
games in Sect. 4.3–5 do not generally satisfy condition (1) of that Theorem and those
in 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 results
and the Theorem in Yi (1999) is in fact pointless because of the essential erroneousness
of the statement of that Theorem. Indeed, we claim (and prove in the Appendix) the
following.
Claim 2 There exists a game that satisfies the assumptions of the Theorem in
Yi (1999) such that sF
N
E
sC PN
= ∅.
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 of
Yi (1999). Proposition 2 in Quartieri and Shinohara (2012) proves that a statement
similar to Yi’s Theorem is true in case of weakly negative externalities and convex
strategy sets; anticipating unjustified conjectures based on Proposition 2 in Quartieri
and Shinohara (2012) and on the third remark at p. 358 in Yi (1999), we claim (and
prove in the Appendix) the following.
Claim 3 There exists a game (with weakly negative externalities) that satisfies
all conditions of the Theorem in Yi (1999), but not its condition (3), such that
sF
N
E
sC PN
= ∅.
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 extension
of that game, say
, which is a distinct game in its own right. It must be remarked
also that some—of the possiblymany—mixed-strategy extensions of some games that
satisfy 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 of
u1 (for instance, because u1 (·, s−1) is unbounded) relative to all probability measures
on 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 Corollary
2; consequently—and this has not been noted in Yi (1999)—in those games the “ Pareto-efficient frontier
of the Nash equilibrium set” in the sense of Yi (1999) is equivalent to the entire set of Nash equilibria.
123

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 always
degenerate mixed strategies then the conclusions of Corollary 2 extend in fact also to
the mixed-strategy extension
.
3.2 Order-theoretic characterization of E
N
It is well-known—see, e.g., Proposition 1.1 in Daci´c (1979)—that the set of fixpoints of
an 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-reply
functions and thus their sets of Nash equilibria are antichains. Such an order-theoretic
characterization of the set of Nash equilibria is emblematic of the situation of strategic
conflict inherent in these games. However, when joint best-reply correspondences are
multi-valued, it can well happen that two Nash equilibria of a real σ-interactive game
with 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) compact
proper 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 of
Nash equilibria is endowed with a “natural” order relation on E
N derived from the
interaction 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∗∗
) for
all i ∈ M, and let denote the reflexive closure of . (Therefore is the binary
relation 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 Nash
equilibria for and σi (y) σi (x) for all i ∈ N).
N , is a poset is immediate and is left to the reader. Now,
Proof The proof that E
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 with
strategic substitutes, (6) implies that xi ≤ yi for all i ∈ M; hence, by the increasingness
of 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 such
that, for all i ∈ M,
s∗
, s∗∗ ∈ SM and s∗
l s∗∗
l for all l ∈ M {i } implies σi s∗ σi s∗∗ ;
123

besides suppose m 1 (e.g., is a multiplayer -interactive game with strategic
substitutes and nonnegative strategies). Then it is impossible that x, y ∈ E
N and
xi yi for all i ∈ M.
Proof If xi yi for all i ∈ M then {i ∈ M : σi (y) σi (x)} = M, in contradiction
with 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 is
symmetric).
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 except
for at most one component. The same thesis is in fact stated also in the Corollary of
Theorem 3 in Jensen (2006) but14 under the hypothesis that the games are strictly
submodular in the sense of Jensen (2006). Since there exist real -interactive games
with strategic substitutes that are not strictly submodular games (see Example 1 and
Remark 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 in
Jensen (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 is
impossible 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 = bj (x), y j = bj (y) and x j y j , the assumption that is a real
σ-interactive game with strategic substitutes implies that
σj (y) σj (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.
123

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 and
Sabarwal (2008). Indeed—interpreting the joint best-reply correspondence b as one
of their parametrized correspondence g (·, t)—the assumption of Theorem 1 in Roy
and Sabarwal (2008) that each correspondence g (·, t) is never-increasing excludes the
possibility that—when each strategy set is a subset of the real line—b might assume
the same value at two distinct points of the joint strategy set, say x and y, such that
xi yi for all i ∈ M (this is clear, in particular, from the end of the second paragraph
of Sect. 2.1 in Roy and Sabarwal (2008)). However, there are real -interactive games
with strategic substitutes and single-valued best-reply correspondences where the joint
best-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 substitutes
discussed in Sect. 4—one has b (0.6, 0.6) = b (0.7, 0.7), and hence in that example
b is not never-increasing). Clearly, Corollary 5 does not in the least imply Theorem 1
of Roy and Sabarwal (2008).
3.3 Comparison with previous results in games with strategic complements and
monotone externalities
To provide a sensible comparison between the results of Sect. 3.1–2 and known results
for games with strategic complements and monotone externalities, we introduce the
following definition.
Definition 4 Agame is said to be a real σ-interactive gamewith 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 real
σ-interactive game with strategic complements and increasing (resp. decreasing)
externalities if it is a real σ-interactive game with strategic complements and, for all
i ∈ 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 2
and 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
123

Nash equilibrium e ∈ E
N ,15 and
N and a least Nash equilibrium e ∈ E
{e} ⊆ E
wCPN
= wF
N
⊆ E
sC PN
⊆ sF
N
⊆ E
N
(resp. e ⊆ E
wCPN
= wF
N
⊆ E
sC PN
⊆ sF
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
wCPN
= wF
N
= E
sC PN
⊆ sF
N
⊆ E
ST N
= E
N
(resp. e = E
wCPN
= wF
N
= E
sC PN
⊆ sF
N
⊆ E
ST N
= E
N ).
What is still not clear to us is whether the additional assumption that “each Si is
compact and each bi has nonempty compact values” might allow one to prove that
E
⊆
wCPN
wFN in every real σ-interactive game with strategic substitutes and
either increasing or decreasing externalities.16 This is still an open issue. We do not
exclude that such a possibility can be disproved only by means of a very complex
counterexample 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 shall
consider economic models of the literature, or extensions thereof, where the structure
of the set of coalition-proof Nash equilibria has not been analyzed yet or for which
there are only some partial results.
N and E
It should be clear that the sets E
N play a special role in our results:Corollary
1 shows that the Nash equilibria in these two sets satisfy many desirable properties. A
result byKukushkin (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 shall
state only a straightforward and very particular consequence of that more general result.
Existence result I (Kukushkin 2005) Let be a real σ-interactive gamewith strategic
substitutes. 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
EwCPN like in Example 3 below and we might have that E
N
= ∅.
123

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 our
applications 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 of
a 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 application
does 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 strategic
substitutes. 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 level
of production out of its production set Si ⊆ R+ which is assumed to be nonempty and
closed. The price at which an aggregate quantity is entirely demanded is given by a
continuous and decreasing function p : R+ → R+ with nonempty support T . Firmi ’s
cost function is a strictly increasing left-continuous function ci : Si → R+ such that
p (x) x−ci (x) ≤ ci (0) for x large enough if Si is unbounded. Let ui : i∈M Si → R,
ui : s→ p l∈Msi 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 differentiable
on 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 in
Novshek (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
wCPN
⊆ E
sC PN
= E
N
= sF
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 have
guaranteed also that in the previous models there exists a w-coalition-proof Nash
equilibrium 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).
123

hence strongly) Pareto dominated by any Nash equilibrium: also this fact has never
noted and proved before. In this connection it must be acknowledged that Kukushkin
(1997) provides sufficient conditions for a game under which E
wCPN
= ∅ and
that one of the applications of the Theorem in Kukushkin (1997) concerns also some
models of Cournot competition considered above. Example 2 in Kukushkin (1997)
shows the importance of the assumption of convexity of strategy sets for the validity
of the Theorem in Kukushkin (1997); in fact that result does not generally ensure
the nonemptiness of E
wCPN (and of E
sC PN ) 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 equilibrium
need not be a w-coalition-proof Nash equilibrium.
Example 1 Put M = {1, 2, 3},
p : x→

p (x) = 8 − x if x ≤ 7
p (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 are
satisfied. 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
wCPN
= wF
N
⊂ E
N
= E
sC PN
= sF
N .
It is well-known that checking the set of w- and s-coalition-proof Nash equilibria of
a game can be very time-consuming (all w- and s-self-enforcing strategies of the game
and 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. Check
that 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 be
checked 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 is
never 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 s
such that s3 = 0 and that there cannot exist a second Nash equilibrium s such that
s3 = 7. Thus E
N and E
N are exactly the sets defined in Example 1. By Corollary
1, E
N
= E
sC PN
= sF
N and E
wCPN
⊇ E
N
⊆ wF
N . But then, since (3, 2, 0) and
wCPN and wF
N are exactly
(2, 3, 0) are weakly Pareto dominated by (2, 2, 0), also E
the sets defined in Example 1.
Remark 2 Note that in the game of Cournot competition illustrated in Example 1
we have u1 (5, 4, 0) − u1 (4, 4, 0) u1 (5, 5, 0) − u1 (4, 5, 0). Therefore that game
123

is an instance of a real -interactive game with strategic substitutes and decreasing
externalities 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-proof
Nash 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 all
i ∈ M, let Si = [0, 1]. Also in this example p is log-concave. It can be easily verified
that
E
N
= ([0, 1] × {0} × {0}) ∪ ({0} × [0, 1] × {0}) ∪ ({0} × {0} × [0, 1]) ,
E
N
= {(0, 0, 0)} ⊂ E
N
= E
wCPN
= wF
N and (1/2, 0, 0) ∈ E
N
E
N .
Example 3 below shows that, in the games described above, a Nash equilibrium
which 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 the
reader to verify that
E
N
= E
wCPN
= {(0, 0, 7)} ⊂ {(3, 2, 0) , (2, 3, 0) , (0, 0, 7)} = E
N
= wF
N ,
and hence that wF
N
E
wCPN
= ∅ .
Example 3 is important because it has shown that it is possible that wF
N E
wCPN
in some real-interactive games with strategic substitutes and decreasing externalities
with a compact set of Nash equilibria and continuous payoff functions.18
4.2 Models of voluntary contribution of a public good
Consider themodel of voluntary contribution of a public good analyzed in Proposition
1 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) is
strict). Besides assume that the payoff to each individual is increasing in the sum of
the 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
⊆ Ew
CPN and E
N
= ∅
together imply Ew
CPN
= ∅).
123

substitutes and increasing externalities (and also in this case E
N
= ∅ by the above
Existence result I). Then, our Corollary 1 ensures that
E
N
⊆ E
wCPN
⊆ E
sC PN
= E
N
= sF
N
⊇ wF
N
⊇ E
N
and our Corollary 4 implies that if a renegotiation of a (w- or s-)coalition-proof Nash
equilibrium for increases the contribution of all agents then it strictly increases the
contribution of exactly one agent.
Similarly, the games associated to the model of voluntary contribution of a public
good described inKerschbamer and Puppe (1998) (or of its extension to n players illus-trated
in Quartieri and Shinohara (2012)) are real -interactive games with strategic
substitutes 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
wCPN
= E
sC PN
= E
N
= wF
N
= sF
N
and our Corollary 5 implies that no renegotiation of a (w- or s-)coalition-proof Nash
equilibrium 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 agent
i ∈ M, we denote by Ni the set of i ’s neighbors, i.e., the agents other than i who
strategically affect i ’s payoff; this suffices to describe the (possibly directed) network
in our context. Each agent i chooses an action si from a closed interval Si ⊆ R+ such
that 0 ∈ Si . The cost of i ’s choice is ci (si ), where ci is a continuous, convex and
increasing real-valued function on R+. Put SM = l∈M Sl and let σ = {σi }i∈M be an
arbitrary 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 ∈ MNi ;
σi vanishes at the origin. The revenue of each agent i at s ∈ SM is ri (si, σi (s)), where
ri : 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 are
real σ-interactive games with strategic substitutes and increasing externalities, and it
is 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
wCPN
= E
sC PN
= E
N
= wF
N
= sF
N .
19 D+
1 ri : R+ →R denotes the (well-defined) right-hand derivative of ri .
123

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) ⊇ argmax
z∈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 that
E
= ∅ N
in the games on network described above.
Remark 3 The model of provision of a public good on network in Bramoullé and
Kranton (2007)—which is properly generalized by themodel just described—restricts
attention to the special case of undirected networks with σi (s) = l∈Ni sl 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 of
social 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 PN
= E
N .
4.4 Games on networks: discrete strategy sets
Consider again the model above and remove the assumption that strategy sets are
intervals 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∈Ni sl 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 described
are 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 Existence
result I and, by our Corollary 1,
E
N
⊆ E
wCPN
⊆ E
sC PN
= E
N
= sF
N
⊇ wF
N
⊇ E
N .
Note that if one additionally assumes that the network is also connected then the set
of Nash equilibria is characterized as in Corollary 3; therefore, when the game just
described is used as the abstract structure of a model of provision of a public good
on a connected network with many agents, one has that no renegotiation of a (w- or
s-)coalition-proof Nash equilibrium for can increase strictly the contributions of all
agents.
Just to provide a specific example—possibly for unconnected networks—let, for
all 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 andγ 1 one
has exactly the “ Best shot” public good game on network illustrated in Example 2 in
123

Jackson and Zenou (2014); note also that in that particular case all Nash equilibria are
strict 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. 5.1
in Jensen (2010), and with the topological assumptions of that article. Besides assume
that 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 have
sufficient assumptions to conclude that the games described are real -interactive
games 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
wCPN
⊆ E
sC PN
= E
N
= sF
N
⊇ wF
N
⊇ E
N
(resp.E
N
⊆ E
wCPN
⊆ E
sC PN
= E
N
= sF
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., at
least one player is inactive) because at least two players are inactive, then the project
must fail with certainty at all Nash equilibria for (i.e., then at each Nash equilibrium
for 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 in
Corollary 4 as long as Nash equilibria exist at which the project fails with certainty
because of the inactivity of two players. The following example well illustrates the
point.
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 is
satisfied 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.
123

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 Example
4, each ui is constantly zero on E
N
= {s ∈ [0, 1]3 : min {s1, s2, s3} = 0}; hence
the 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 previous
assumptions but which has at least one Nash equilibrium where the project succeeds
with certainty (e.g., the joint strategy (1, 1, 1)) and at least one Nash equilibrium where
the project fails with certainty (e.g., the joint strategy (1, 1, 0)). Variants of Example
4 where all players are active at each Nash equilibrium can be easily constructed by
the reader.
Acknowledgments The present version of this paper considerably benefited from discerning comments
and remarks of two anonymous reviewers. The second author gratefully acknowledges financial support
from Grant-in-Aid for Young Scientists (21730156, 24730165) from the Japan Society for Promotion of
Science.
Appendix
Fact 1 Let = (M, (Si )i∈M, (ui )i∈M) be a real σ-interactive game with strategic
substitutes. We can define a game

i )i∈M, (u
= (M, (S

i )i∈M)

i
such that S
= −Si and u

i
: S

M
→ R , u

i
: s→ ui (−s) , for all i ∈ M.21
Besides we can define the family
σ

i
= {σ
}i∈M

i
such that σ
: S

M

i
→ R, σ
: 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 exists
υi : Si × σi [SM] → R such that ui (s) = υi (si, σi (s)) at all s ∈ SM, for all
i ∈ M. Letting υ

i

i
: S

i
× σ

S

M

i
→ R, υ
: (x, y)→ υi (−x,−y) for all
i ∈ M, it can be easily verified that
is a real σ
-interactive game with strategic
substitutes.
Fact 2 Let = (M, (Si )i∈M, (ui )i∈M) be a real σ-interactive game with strategic
substitutes and increasing (resp. decreasing) externalities, and define
and σ

as in Fact 1. Then
is a real σ
-interactive game with strategic substitutes and
decreasing (resp. increasing) externalities.
21 Clearly, S

M denotes i∈M S

i .
123


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

i in
consequence of Fact 1 and of the decreasingness (resp. increasingness) of each υ
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 indicator
function 1A of A ⊂ SM) where
B = {s ∈ SM : max {s1, s2}= 1} ∪ {(1, 0, 0)} and A = SMB.
It is immediate that the game is a real σ-interactive game with strategic substitutes
and increasing externalities for the interaction system σ such that σi : s→ ui (s) for
all i ∈ M (just put υi : (si , t)→ t for all i ∈ M). On the other hand, it is also quite
simple to notice that there cannot exist a continuous function ς3 : SM{3} → R such
that u3 (s) =
3 (s3, ς3 (s−3)) for some
u
3: by way of contradiction suppose on the
u
contrary that such ς3 exists; notice that ς3 (1, 0)= ς3 (0, 1) (as
3 (0, ς3 (1, 0))=
u
3 (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
u
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 =
3 0, ς3 z∗ =
u
3 0, ς3 z∗∗ = 1A z∗∗
u
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 decreasing
externalities).
Proof of Observation II Proof of the if part. Suppose has a compatible interaction
system 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
i : Si × X−i → R be the function defined by
u
i (si, ςi (s−i )) = υi (si, σi (s)) at all s ∈ SM and conclude that is a generalized
quasi-aggregative game with aggregator g. Clearly, since each interaction function σi
is continuous, also each J-interaction function ςi is continuous.
u
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
123

function defined by υi (si, σi (s)) =
i (si, ςi (s−i )) at all s ∈ SM, for all i ∈ M. Let
u
σ := {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 continuity
of 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 involving
continuity arguments).
Proof of Observation IV It is left to the reader to notice that, constructing again each
σi 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 each
ςi 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. Put
M = {1, 2, 3}, N1 = {2}, N2 = {3}, N3 = {1}, σ1 √
: s→ s2, σ2 : s→ s3, σ3 : s→ s1
and, 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) ⊇ argmax
z∈Si
Then, since best-replies are single-valued and since P is an upper semicontinuous
function on a compact set, we must have that
bi (s) = argmax
z∈Si
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 decreasing
externalities where E
wCPN
= wF
N
⊂ E
N ;
(ii) there exists a real -interactive game with strategic substitutes and decreasing
N
⊂ E
wCPN
= wF
N ;
123

(iii) there exists a real -interactive game with strategic substitutes and decreasing
externalities where wF
N
E
wCPN
= ∅;
(iv) there exists a real -interactive game with strategic substitutes and decreasing
wCPN
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 ifi (s) ≥ 3
2
si (1 − i (s)) if i (s) 3
2
for i ∈ {1, 2} and
u3 (s) =
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
−1 if s3 = 0 and 3 (s) 1
−1
2 if s3 = 0 and 3 (s) = 1
0 if s3 = 0 and 3 (s) 1
s3 (1 − 3 (s)) −1 ifs3 ∈ (0, 1) ∪ {8} and 3 (s) 1
s3 (1 − 3 (s)) if s3 ∈ (0, 1) ∪ {8} and 3 (s) ≤ 1
−9 if s3 ∈ {1} ∪ [7, 8) .
Define Q1 := 1
2 , 1
2 , 1
2 , 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 decreasing
externalities. Note that E
ST N
= Q2 ⊂ 6
i=1Qi = E
N ; thus E
wCPN
⊇ Q2 ⊆ wF
N
by 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, 1
2
• (ui (s))i∈M = (1 − s3,−1, 0) if s ∈ Q3 and s3 ∈ 1
2 , 1,
• (ui (s))i∈M = (0, 1 − s3, 0) if s ∈ Q4 and s3 ∈ 0, 1
2 ,
123

• (ui (s))i∈M = (−1, 1 − s3, 0) if s ∈ Q4 and s3 ∈ 1
2 , 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 dominate
s∗∗ in ,
• if s ∈ Q1 then s is w-self-enforcing for as, for all i ∈ M, s−i is not weakly Pareto
dominated 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
2 s3 ∈ E|s2
in |s2 by s1, 1
N ,
• and every strategy s ∈ Q4 is not w-self-enforcing for as s−1 is weakly Pareto
dominated in |s1 by s2, 1
2 s3 ∈ E|s1
N ,
and conclude that Q1 ⊆ E
wCPN. Note that (Q5 ∪ Q6) ∩ E
wCPN
= ∅ (consider the
deviating coalition {1, 2}). Thus Q2 = wF
N
⊂ E
wCPN
= Q1 ∪ Q2 ⊂ E
N and in
particular E
wCPN
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) =
−1
2 (s1 + s2). Let e := (0, 0, 0). It can be easily verified that E
N
= {e} = sF
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 PN
= ∅ since (1, 1) ∈ E
|e−3
N and
(1, 1) strongly Pareto dominates e−3 in |e−3 . Therefore the statement ofYi’s theorem
is false.
We can provide also a second counterexample with four players (in fact it suffices
to 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 − s2
4 .
All hypotheses of Yi’s Theorem hold but E
N
= {(0, 0, 0, 0)} = sF
N
= ∅ and
sC PN (consider again the deviating coalition {1, 2}).
∅ = E
123

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
= i
l=110l−1. (Note that
every four-player game with the just defined strategy sets satisfies condition (1) in
the statement of Yi’s theorem). Apart from condition (3), all the conditions of Yi’s
Theorem hold and we have s∗∗
1 , s∗∗
2 , s∗
3 , s∗
4 ∈ sF
N
E
sC PN
= ∅ (this time consider
the deviating coalition {1, 2, 3}).
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