A social game is a generalization of a strategic-form game, in which not only the payoff of each player depends upon the strategies chosen by their opponents, but also their set of admissible strategies. Debreu (1952) proves the existence of a Nash equilibrium in social games with continuous strategy spaces. Recently, Polowczuk and Radzik (2004) have proposed a discrete counterpart of Debreu's theorem for two-person social games satisfying some ``convexity properties'. In this note, we define the class of supermodular social games and give an existence theorem for this class of games.Strategic-form games, social games, supermodularity, Nash equilibrium, existence.

Ludovic Renou

English

Research Papers in Economics

 
 
 
 
 
 
School of Economics 
 
 
Working Paper 2004-16 
 
 
Supermodular social games 
 
 
 
Ludovic Renou 
 
 
 
 
 
School of Economics 
University of Adelaide University, 5005 Australia 
 
 
ISSN 1444 8866 Supermodular social games
Ludovic Renou∗
University of Adelaide
28th January 2005
Abstract
A social game is a generalization of a strategic-form game, in which
not only the payoﬀ of each player depends upon the strategies chosen
by their opponents, but also their set of admissible strategies. Debreu
(1952) proves the existence of a Nash equilibrium in social games with
continuous strategy spaces. Recently, Polowczuk and Radzik (2004)
have proposed a discrete counterpart of Debreu’s theorem for two-
person social games satisfying some “convexity properties”. In this
note, we deﬁne the class of supermodular social games and give an
existence theorem for this class of games.
Keywords: Strategic-form games, social games, supermodularity,
Nash equilibrium, existence.
JEL Classiﬁcation Numbers: C7.
∗Department of Economics, Room G52, Napier Building, Adelaide 5005, Australia.
Phone: +61 (0)8 8303 4930, Fax: +61 (0)8 8223 1460. ludovic.renou@adelaide.edu.au
11 Introduction
A social game is a generalization of a strategic-form game, in which the
set of “admissible” strategies of a player is constrained by the strategies of
the other players. Historically, Arrow and Debreu (1954) were the ﬁrst to
introduce the concept of a social game, calling it an abstract economy in
their original paper. To motivate the need for this generalization, Arrow
and Debreu invoked the special position of consumers in an economy. The
strategies of a consumer can be regarded as the choice of diﬀerent bundles
of goods. Theirs is a constrained choice in that the total cost of the goods
chosen at market prices cannot exceed their disposable income. In turn,
market prices and the disposable income are determined by the choices of
other agents in the economy e.g., tax authorities or employers.
Formally, a social game G is a tuple < N,(Xi,ui,Si)i∈N >. N =
{1,...,n} is the set of players, Xi is the set of pure strategies available
to player i. Denote X−i =
Q
j∈N\{i} Xj, and x−i an element of X−i. For
each player i ∈ N, Si is a multi-valued map from the set X−i to sub-
sets of the set Xi, with Si(x−i) the set of pure strategies admissible to
player i when his opponents play x−i. Hence, the map Si represents the
social constraint imposed by player i’s opponents on his behavior. Player
i’s payoﬀ function is ui : Xi × X−i → R. The mixed extension of a ﬁ-
nite social game G is the tuple < N,(∆(Xi),vi,Si)i∈N > where ∆(Xi) is
the set of probabilities on Xi, for all (pi,p−i) ∈
Q
i∈N ∆(Xi), vi(pi,p−i) = P
xi∈Xi
P
x−i∈X−i ui(xi,x−i)pi(xi)p−i(x−i), and
Si(p−i) =
\
x−i∈ supp p−i
Si(x−i).
A proﬁle of strategies x∗ = (x∗
i,x∗
−i) is an Arrow-Debreu-Nash equilibrium,
(hereafter, an equilibrium), of the social game G if for each player i ∈ N,
x∗
i ∈ Si(x∗
−i), and
x
∗
i ∈ BRi(x
∗
−i) := arg max
xi∈Si(x∗
−i)
ui(xi,x
∗
−i).
A proﬁle of strategies is a mixed equilibrium of G if it is a equilibrium of
the mixed extension of G. Formally, (p∗
i,p∗
−i) is a mixed equilibrium of G if
2for each player i ∈ N, supp p∗
i ⊂ Si(p∗
−i), and p∗
i ∈ BRi(p∗
−i). This note is
concerned with the equilibrium existence in social games.
In “Social equilibrium existence theorem” (1952), Debreu provides suf-
ﬁcient conditions for the existence of an equilibrium in a social game. The
suﬃcient conditions for existence are as follows. For each player i ∈ N,
Xi is a contractible polyhedron, Si is a semi-continuous multi-valued map
(i.e., its graph is closed), ui is continuous, and for each x−i, the set Si(x−i)
is contractible. (See Theorem, p. 888 in Debreu (1952).) In an historical
note, Debreu also mentions the existence of an equilibrium if for each player
i ∈ N, Xi is a non-empty, compact, convex subset of a ﬁnite Euclidean
space, ui is continuous and quasi-concave in xi, and the multi-valued map Si
is semi-continuous, non-empty and convex-valued. This last statement is now
familiar to game theorists as it relies on the celebrated Kakutani ﬁxed-point
theorem.
Recently, Polowczuk and Radzik (2004) have provided a counterpart of
Debreu’s theorem for two-player non-zero sum games with ﬁnite strategy
spaces. Their main assumptions are: Z1) symmetry i.e., for all x2 ∈ S2(x1),
x1 ∈ S1(x2) and for all x1 ∈ S1(x2), x2 ∈ S2(x1), Z2) sections convexity i.e., a
discrete counterpart of the convex-valuedness of Si, and Z3) game convexity
i.e., a discrete counterpart of the quasi-concavity of ui. Assuming Z1-Z3, they
prove the existence of an equilibrium in (mixed) strategies consisting of two
two-adjoining pure strategies i.e., mixed strategies assigning strictly positive
probability to only two consecutive pure strategies x and x + 1. However,
their theorem (Theorem 4) does not hold true for three players or more. For
instance, consider the following three-player game (Figure 1).
0 1
0 1,1,0 ?,?,?
1 0,1,1 1,0,1
0
0 1
0 1,0,1 0,1,1
1 0,0,0 1,1,0
1
Figure 1: A three-player social game with no equilibrium.
Each player has two strategies 0 and 1. Player 1 chooses a row, player 2
a column and player 3 a matrix. Moreover, the proﬁle of strategies (0,1,0)
3is not admissible i.e., we have S1(1,0) = {1}, S2(0,0) = {0}, S3(0,1) = {1}
and for each player i, Si(x−i) = {0,1}, otherwise. We can easily check that
this game has no Nash equilibrium.
In the next section, we deﬁne the class of supermodular social games and
prove the existence of a Nash equilibrium. In particular, our result holds
for n-player games with ﬁnite unidimensional strategy spaces if payoﬀs have
increasing diﬀerences.
2 Supermodular social games and equilibrium
existence
The following deﬁnition of a supermodular social game generalizes the deﬁ-
nition of a supermodular game introduced in Milgrom and Roberts (1990).
Both deﬁnitions coincide when there is no social constraints i.e., if for all
players, for all x−i ∈ X−i, Si(x−i) = Xi. We refer the reader to Topkis
(1998) for the deﬁnitions of concepts introduced below.
Deﬁnition 1 A social game G =< N,(ui,Xi,Si)i∈N > is supermodular,
if for each player i ∈ N,
(A1) Xi together with the order ≥i is a non-empty complete lattice;
(A2) ui : X → R ∪ {−∞} is order upper semi-continuous in xi (for a ﬁxed
x−i), order-continuous in x−i (for ﬁxed xi), and has a ﬁnite upper-
bound;
(A3) ui is supermodular in xi (for ﬁxed x−i);
(A4) ui has increasing diﬀerences in xi and x−i;
(A5) Si is ascending in x−i, and, for each x−i, Si(xi) is a non-empty complete
sublattice of Xi.
Conditions (A1)-(A4) are equivalent to conditions (A1)-(A4) of Milgrom
and Roberts (1990). However, condition (A5) has no equivalence in Milgrom
and Roberts as they do not consider social games. The ﬁrst part of (A5)
4together with (A3)-(A4) insures that player i’s best-reply map BRi is non-
decreasing on the set {x−i ∈ X−i : BRi(x−i) 6= ∅}. The second part of
(A5) together with (A1)-(A2) insures that best-reply maps are everywhere
non-empty valued. The following example illustrates the importance of this
condition. Let R be the extended real line i.e., R := {−∞} ∪ R ∪ {+∞}.
Together with the usual order ≥, R is a complete lattice. Consider the subset
S = {−2} ∪ (−1,+1) ∪ {+2} of R. S is a complete lattice, a sublattice of
the extended real line, but not a complete sublattice as supS(0,1) = 2 6= 1 =
supR(0,1). It is then easy to see that the function x 7→ f(x) = −x2 + 2x
is order continuous on R, while it is not order upper semi-continuous on S;
and f has no maximum in S, while it has a maximum in R.
Lemma If f is an order upper semi-continuous, supermodular function on
a complete lattice X, then any restriction of f to a complete sublattice S of
X is order upper semi-continuous and supermodular on S.
Proof Let C ⊆ S be a chain, i.e., a totally ordered subset of S. Note that
C is also a chain of X. Since S is a complete sublattice of X, we have that
supS(C) = supX(C). It follows that
lim sup
x∈C, x↑supS(C)
f(x) = lim sup
x∈C, x↑supX(C)
f(x) ≤ f(supX(C)) = f(supS(C)),
since f is order upper semi-continuous on X. A similar reasoning holds for
the convergence to infS(C), hence f is order upper semi-continuous on S.
Finally, it is trivial to prove that f is supermodular on S. 
We can now state our main theorem, which proves the existence of an
equilibrium in supermodular social games.
Theorem A supermodular social game has an equilibrium.
Proof Fix a x−i ∈ X−i. Since Si(x−i) is a complete sublattice of Xi, hence
a complete lattice in its own right, and ui is supermodular and order upper
semi-continuous on Si(x−i) by Lemma, a direct application of Theorem 1
(p1262) of Milgrom and Roberts (1990) proves the existence of a maximum
of ui(·,x−i) in Si(x−i). It follows that BRi(x−i) 6= ∅ for all x−i ∈ X−i. From
Theorem 6.1 in Topkis (1978), we have that BRi is ascending in x−i on the
set {x−i : BRi(x−i) 6= ∅}. Existence of a Nash equilibrium then follows by
Tarski ﬁxed-point theorem as in Topkis (1979). 
5As a ﬁnal remark, it is worth noting that a similar result can already be
found in Topkis (1979, Theorem 3.1 (p. 781)), although it seems that Topkis
did not realize the relation of his result with social games. Moreover, our
theorem slightly improves upon Topkis’ theorem as Topkis considers com-
pact intervals of ﬁnite Euclidean spaces for strategy spaces and continuous
payoﬀ functions, while we consider more general strategy spaces and payoﬀ
functions.
References
[1] K. Arrow and G. Debreu, Existence of an equilibrium for a competitive
economy, Econometrica, 22, 1954, 265-290.
[2] G. Debreu, A social equilibrium existence theorem, Proceedings of the
National Academy of Science, 38, 1952, 886-893.
[3] P. Milgrom and J., Roberts, “Rationalizability, Learning, and Equilib-
rium in Games with Strategic Complementarities”, Econometrica, 1990,
58, 1255-1277.
[4] W. Polowczuk and T. Radzik, Equilibria in constrained bimatrix games,
Mimeo, Institute of Mathematics, Wroclaw University of Technology,
2004.
[5] D.M. Topkis, Minimizing a Submodular Function on a Lattice, Operations
Research, 26, 1978, 305-321.
[6] D.M. Topkis, Equilibrium Points in Nonzero-Sum n-Person Submodular
Games, SIAM Journal on Control and Optimization, 1979, 17(6), p773-
787.
[7] D.M. Topkis, Supermodularity and Complementarity, 1998, MIT Press,
Cambridge, Massachussets.
6

A social equilibrium existence theorem,

Equilibria in constrained bimatrix games,

Equilibrium Points in Nonzero-Sum n-Person Submodular Games,

Existence of an equilibrium for a competitive economy,

Minimizing a Submodular Function on a Lattice,

Supermodularity and Complementarity,

Supermodular social games

http://128.118.178.162/eps/game/papers/0502/0502002.pdf

Supermodular social games

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