Patrone, F.

Pusillo, L.

Tijs, S.H.

English

Research Papers in Economics

Top (2007) 15: 138–145
DOI 10.1007/s11750-007-0008-1
ORIGINAL PAPER
Multicriteria games and potentials
Fioravante Patrone ·Lucia Pusillo ·Stef Tijs
Received: 13 December 2005 / Accepted: 9 January 2007 /
Published online: 12 April 2007
© Sociedad de Estadística e Investigación Operativa 2007
Abstract In this note we study how far the theory of strategic games with potentials,
as reported by Monderer and Shapley (Games Econ Behav 14:124–143, 1996), can
be extended to strategic games with vector payoffs, as reported by Shapley (Nav Res
Logist Q 6:57–61, 1959). The problem of the existence of pure approximate Pareto
equilibria for multicriteria potential games is also studied.
Keywords Noncooperative potential games · Multicriteria · Pareto equilibria ·
Approximate Pareto equilibria · Supermodularity
Mathematics Subject Classiﬁcation (2000) 91A10
1 Introduction
Monderer and Shapley (1996) introduced a subclass of strategic games for which
a potential exists. Such potential games turn out to have interesting properties, among
which the existence of pure (approximate) Nash equilibria in the case of upper
F. Patrone
DIPTEM, Sezione Metodi e Modelli Matematici, Facoltà di Ingegneria, University of Genoa,
P. le Kennedy, pad. D, 16129 Genoa, Italy
e-mail: patrone@diptem.unige.it
L. Pusillo () · S. Tijs
DIMA, Department of Mathematics, University of Genoa, via Dodecaneso 35, 16146 Genoa,
Italy
e-mail: pusillo@dima.unige.it
S. Tijs
CentER and Department of Econometrics and Operations Research, Tilburg University,
P.O. Box 90153, 5000 LE Tilburg, The Netherlands
e-mail: s.h.tijs@uvt.nlMulticriteria games and potentials 139
bounded potentials. Further, they show that there is an interesting relation between
potential games and congestion games introduced by Rosenthal (1973). Our list of
references shows that the Monderer–Shapley paper inspired many game theorists.
The purpose of this paper is to investigate the possibility of extending the potential
theory to multicriteria strategic games introduced by Shapley (1959) and also studied
by Borm et al. (1989). Since the extension of the theory of potential games to multi-
criteria situations is turning out to be obvious in the case we know the extension to the
bicriteria situation, in this paper we restrict ourselves mostly to bicriteria situations.
The outline of the paper is as follows. In Sect. 2, we introduce bicriteria games
with a potential. In Sect. 3 we show that in the case the strategy spaces are ﬁnite,
such a game has at least one pure Pareto equilibrium. Further, for multicriteria games
with an upper bounded (vector) potential, it turns out that  -Pareto equilibria in pure
strategies exist. In Sect. 4, we give a characterization of zero-sum bicriteria games
with a potential and discuss their relations with supermodular bicriteria games. Sec-
tion 5 concludes with some suggestions for further research.
2 Bicriteria games and potentials
A strategic bicriteria game is a tuple Γ =  N,(Ai)i∈N,(ui)i∈N , where N is the
set of players, Ai is the strategy space for player i ∈ N, A is the Cartesian product 
k∈N Ak of the strategy spaces (Ai)i∈N, and ui : A → R2 is the utility function for
player i.
We call Γ =  N,(Ai)i∈N,(ui)i∈N  a potential game if there exists a map
P : A → R2 such that, for all i ∈ N, ai, bi ∈ Ai, and a−i ∈ A−i :=

j∈N\{i}Aj,
we have
ui(ai,a−i)−ui(bi,a−i) = P(ai,a−i)−P(bi,a−i). (2.1)
In the classical theory, potential games can be also characterized with the aid of con-
ditions on paths of strategy proﬁles (cf. Monderer and Shapley 1996 and Voorneveld
1999). Since the extension to bicriteria games is straightforward, we do not pay at-
tention to this.
Special classes of potential games are bicriteria coordination games and bicriteria
dummy games. A game  N,(Ai)i∈N,(ui)i∈N  is a coordination game if the utility
functions of all players are the same. Clearly, for such a game, there is a potential
P : A → R2. Take, e.g., P = u1.
The game  N,(Ai)i∈N,(ui)i∈N  is a dummy game if ui(ai,a−i) = ui(bi,a−i) for
all i ∈ N, ai, bi ∈ Ai, and a−i ∈ A−i, that is, the strategy choice of player i does
not inﬂuence his payoff. A potential for such a game is the null function P with
P(a)= (0,0) for each a ∈ A.
Note that  N,(Ai)i∈N,(ui + u 
i)i∈N  is a bicriteria potential game with potential
P + P  if  N,(Ai)i∈N,(ui)i∈N  is a bicriteria potential game with potential P and
 N,(Ai)i∈N,(u 
i)i∈N  a bicriteria potential game with potential P .
In particular, the sum of a bicriteria coordination game and a bicriteria dummy
game is a potential bicriteria game.
The converse also holds as we see in the next theorem. This theorem extends the
result of Facchini et al. (1997) and Slade (1994).140 F. Patrone et al.
Theorem 2.1 (Decomposition theorem). Let  N,(Ai)i∈N,(ui)i∈N  be a bicrite-
ria potential game. Then there is a coordination game  N,(Ai)i∈N,(P)i∈N  and
a dummy game  N,(Ai)i∈N,(Di)i∈N  such that ui = P +Di for each i ∈ N.
Proof Take for P a potential of the original game and deﬁne Di = ui −P. Then from
(2.1) it follows that  N,(Ai)i∈N,(Di)i∈N  is a dummy game. 
Example 2.2 Consider the game where N consists of two players (the row player and
column player) and each player has two strategies (player I: row 1 (A) and row 2 (B);
player II: column 1 (C) and column 2 (D)), and where the utility functions are de-
scribed in the following payoff matrix:
CD
A (0,3)( 0,5) (4,5)( 1,3)
B (0,0)( 1,3) (5,5)( 3,4)
A decomposition of the game into a coordination game and a dummy game is
given by
(0,3)( 0,5) (4,5)( 1,3)
(0,0)( 1,3) (5,5)( 3,4) =
(0,3)( 0,3) (1,1)( 1,1)
(0,0)( 0,0) (2,1)( 2,1)
Bicriteria coordination game
+
(0,0)( 0,2) (3,4)( 0,2)
(0,0)( 1,3) (3,4)( 1,3)
Bicriteria dummy game
3 Existence of (approximate) pure Pareto equilibria
We recall the deﬁnition of Pareto equilibrium of a bicriteria game.
Deﬁnition 3.1 Let Γ =  N,(Ai)i∈N,(ui)i∈N  be a bicriteria strategic game. Then
ˆ a ∈

i∈N Ai is called a Pareto equilibrium (PE for short) of the game if, for each
i ∈ N,w eh a v eˆ ai ∈ PB(ˆ a−i). Here the set PB(ˆ a−i) of Pareto best answers to ˆ a−i is
the set of ai ∈ Ai such that ui(bi, ˆ a−i)/ ∈ ui(ai, ˆ a−i)+(R2
+ \{0}) for all bi ∈ Ai.
We denote by PE(Γ ) the set of Pareto equilibria of Γ .
The following theorem, which extends a result of Monderer and Shapley (1996),
says that ﬁnite bicriteria potential games have at least one Pareto equilibrium.
Theorem 3.2 Let Γ =  N,(Ai)i∈N,(ui)i∈N  be a ﬁnite bicriteria potential game.
Then there exists an ˆ a ∈

i∈N Ai such that ˆ a is a pure Pareto equilibrium of the
game Γ .
Proof Let P : A → R2 be a potential for Γ .Multicriteria games and potentials 141
Take ˆ a ∈ argmaxa∈A(P1(a) + P2(a)). Next, we prove that ˆ a ∈ PE(Γ ). Suppose
that ˆ a/ ∈ PE(Γ ). Then there are i ∈ N and ai ∈ Ai such that
ui(ai, ˆ a−i)−ui(ˆ a)∈ R2
+ \{0}.
But then
P(ai, ˆ a−i)−P(ˆ a)∈ R2
+ \{0},
so
2 
k=1

Pk(ai, ˆ a−i)−Pk(ˆ a)

> 0,
which is in contradiction with the deﬁnition of ˆ a. 
The proof of the following theorem is left to the reader.
Theorem 3.3 Let Γ =  N,(Ai)i∈N,(ui)i∈N  be a bicriteria strategic potentialgame
with potential function P. Then
PE(Γ ) = PE

Γ P
,
where Γ P =  N,(Ai)i∈N,(P)i∈N .
Now we concentrate ourselves on the existence of approximate Pareto equilibria for
bicriteria potential games. We introduce the following deﬁnition for approximate
equilibria.
Deﬁnition 3.4 Let Γ =  N,(Ai)i∈N,(ui)i∈N  be a bicriteria strategic game and
 >0. Then ˆ a ∈

i∈N Ai is called an  -Pareto equilibrium of the game Γ if, for
each i ∈ N,w eh a v eˆ ai ∈ PB (ˆ a−i). Here the set PB (ˆ a−i) of  -Pareto best answers
to ˆ a−i is the set of ai ∈ Ai such that ui(bi, ˆ a−i)/ ∈ ui(ai, ˆ a−i)+R2
+,  for all bi ∈ Ai.
Here R2
+, = R2
+ \([0, ]×[0, ]).
We further denote by PE (Γ ) the set of  -Pareto equilibria of a game Γ .
Deﬁnition 3.5 For f : X → R, we deﬁne
argsup 
x∈Xf(x)=

y ∈ X : f(y)≥ sup
x∈X
f(x)− 

.
The following theorem extends a result of Norde and Tijs (1998).
Theorem 3.6 Let Γ =  N,(Ai)i∈N,(ui)i∈N  be a bicriteria strategic potential
game. Suppose that the potential functions are upper bounded. Then PE (Γ )  =∅
for all  >0.142 F. Patrone et al.
Proof Let P be a potential for Γ and  >0.
Take ˆ a ∈ argsup 
a∈A(P1(a)+P2(a)). We prove that ˆ a ∈ PE (Γ ).
Take i ∈ N and ai ∈ Ai. Suppose that
ui(ai, ˆ a−i) ∈ ui(ˆ a)+R2
+, .
Then
P(ai, ˆ a−i)−P(ˆ a)= ui(ai, ˆ a−i)−ui(ˆ a)∈ R2
+, .
But then
2 
k=1

Pk(ai, ˆ a−i)−Pk(ˆ a)

> ,
a contradiction to the deﬁnition of ˆ a. So, for each i ∈ N, ˆ ai ∈ PB (ˆ a−i), i.e.,
ˆ a ∈ PE (Γ ). 
Example 3.7 Let  {1,2},(0,1),(0,1),u1,u2  be the 2-person potential game on the
open unit square with u1(x1,x2) = (x1,x2
1) and u2(x1,x2) = (x2,x3
2).
This game has a potential P : (0,1)×(0,1) → R2 given by P(x1,x2) = (x1 +x2,
x2
1 +x3
2) for all (x1,x2) ∈ (0,1)×(0,1). This game has no Pareto equilibria but there
exist  -Pareto equilibria for every  >0.
We shall say that a game is weakly determined if the set of its  -Pareto equilibria is
nonempty for each  >0. The next theorem implies that Γ is (weakly) determined if
and only if Γ P is (weakly) determined.
Theorem 3.8 Let Γ =  N,(Ai)i∈N,(ui)i∈N  be a bicriteria strategic potentialgame
with potential P and Γ P =  N,(Ai)i∈N,(P)i∈N . Then PE (Γ )=P E  (Γ P) for all
  ≥ 0.
Proof Let Di = ui −P for each i ∈ N.
The following ﬁve assertions are equivalent:
(i) ˆ a ∈ PE (Γ );
(ii) ui(ai, ˆ a−i)/ ∈ ui(ˆ a)+R2
+,  for all i ∈ N and ai ∈ Ai;
(iii) P(ai, ˆ a−i)+Di(ai, ˆ a−i)/ ∈ P(ˆ a)+Di(ˆ a)+R2
+,  for all i ∈ N and ai ∈ Ai;
(iv) P(ai, ˆ a−i)/ ∈ P(ˆ a)+R2
+,  for all i ∈ N and ai ∈ Ai;
(v) ˆ a ∈ PE (Γ P).

Monderer and Shapley (1996) relate potential games with congestion models in-
troduced by Rosenthal (1973). We can generalize them to the multicriteria case in
a straightforward way.
For the classical ﬁnite potential games and congestion games, it was shown (Mon-
derer and Shapley 1996; Voorneveld 1999) that every ﬁnite potential game is iso-
morphic to a congestion game. One can extend this result to bicriteria situations by
adapting the beautiful proof of Voorneveld and using Theorem 2.1.Multicriteria games and potentials 143
Monderer and Shapley (1996) present a class of Cournot games which consists of
potential games. It is also possible to connect multicriteria Cournot games of special
type with multicriteria potential games in a straightforward way.
4 Zero-sum bicriteria games and supermodularity
A bicriteria game Γ =  { 1,2},A1,A2,u1,u2  with u1(a) + u2(a) = (0,0) for each
a ∈ A1 ×A2 is called a zero-sum bicriteria game.
For a zero-sum bicriteria game with an exact potential, the payoff functions have
aspecialformasweseeinthenexttheorem.ItextendsaresultofPottersetal.(1999).
Theorem 4.1 Let Γ =  { 1,2},A1,A2,u1,u2  be a two-person zero-sum bicriteria
game. Then the following assertions are equivalent:
(i) Γ is a potential game;
(ii) there exist maps f : A1 → R2 and g : A2 → R2 such that u1(a1,a2) = f(a 1) −
g(a2) =− u2(a1,a2) for all (a1,a2) ∈ A1 ×A2 (Separation property).
Proof To see that (ii) implies (i), it sufﬁces to note that P : A → R2 given by
P(a1,a2) = f(a 1)+g(a2) for each (a1,a2) ∈ A1×A2 is a potential for the game Γ .
Conversely, suppose that (i) holds and let P : A1 × A2 → R2 be a potential. Take
(a∗
1,a∗
2) ∈ A1 ×A2 and deﬁne f : A1 → R2 and g : A2 → R2 by
f(a 1) = 1/2

P

a1,a∗
2

−u2

a1,a∗
2

for each a1 ∈ A1;
g(a2) = 1/2

P

a∗
1,a2

−u1

a∗
1,a2

for each a2 ∈ A2.
Since P is a potential, we have that, for each (a1,a2) ∈ A1 ×A2,
u1(a1,a2)−u1

a∗
1,a2

= P(a1,a2)−P

a∗
1,a2

and
u2(a1,a2)−u2

a1,a∗
2

= P(a1,a2)−P

a1,a∗
2

,
from which follows that
u1(a1,a2) = P(a1,a2)−2g(a2), u2(a1,a2) = P(a1,a2)−2f(a 1).
From these last two equalities and from the fact that u2 =− u1 we obtain
P(a1,a2) = f(a 1)+g(a2) for all (a1,a2) ∈ A1 ×A2
and then
u1(a1,a2) = f(a 1)−g(a2) for all (a1,a2) ∈ A1 ×A2.
Hence, (ii) holds. 
Note that Theorem 4.1 extends the results for uni-criteria games in Theorem 1 of
Branzei et al. (2003). In that paper, this separation theorem is used to show that each144 F. Patrone et al.
classical two-person zero-sum game with an exact potential is strategically equivalent
to a supermodular game.
For more details about supermodularity, see Topkis (1998). This result can be
extended for special bicriteria zero-sum potential games, namely, for bicriteria games
with parallel criteria. We say that Γ =  { 1,2},A1,A2,u1,−u1  has parallel criteria
if, for all a1,a 
1 ∈ A1 and a2,a 
2 ∈ A2,w eh a v e :
(i) u11(a1,a2) ≥ u11(a 
1,a2) ⇔ u12(a1,a2) ≥ u12(a 
1,a2);
(ii) u11(a1,a2) ≥ u11(a1,a 
2) ⇔ u12(a1,a2) ≥ u12(a1,a 
2).
Here u1 = (u11,u12).
By adapting the proof of Branzei et al., we straightforwardly obtain the following:
Theorem 4.2 Each zero-sum bicriteria potential game with parallel criteria is
strategically equivalent to a supermodular game.
5 Concluding remarks
In this paper, we deﬁne and study bicriteria strategic games with a potential. It turns
out that, under certain conditions, these games have pure approximate Pareto equilib-
ria. Moreover, there is a close relation between these games and bicriteria congestion
games.
For zero-sum bicriteria games, the separation property holds. A subclass of these
games can be related to supermodular games.
Monderer and Shapley (1996) also consider ﬁnite ordinal potential games which
turn out to have pure Nash equilibria. Extensions to inﬁnite games are studied in
Voorneveld and Norde (1997) and in Norde and Patrone (2002). What about an
extension to multicriteria ordinal potential games and to other classes of potential
games?
Let us call a bicriteria game G = (X1,X2,u1,u2) an ordinal potential bicriteria
game if there exists a function P : X1 ×X2 → R2 such that
u1(x1,x2)−u1(y1,x2) ≥ 0 ⇔ P(x1,x2)−P(y1,x2) ≥ 0
and
u2(x1,x2)−u2(x1,y2) ≥ 0 ⇔ P(x1,x2)−P(x1,y2) ≥ 0
for all
x1,y1 ∈ X1,x2,y2 ∈ X2.
Thenonecanprove(similarlytoTheorem3.2)thateveryﬁniteordinalpotentialgame
possesses a Pareto equilibrium.
There are still manyquestionsleft that canbe interesting topicsfor possiblefurther
research:
(i) In Peleg et al. (1996), an axiomatic approach is taken for the set of Nash equi-
libria corresponding to the points where the potential is maximal.
An axiomatization of the Pareto optimum set corresponding to a potential for
a multicriteria game could be a very interesting question.Multicriteria games and potentials 145
(ii) Extensions of well-posedness results in Margiocco and Pusillo (2007) to multi-
criteria potential games may be possible and interesting.
(iii) The work of Facchini et al. (1997) can be extended to Bayesian multicriteria
potential games.
(iv) The work of Voorneveld et al. (1998) on sequential production situations and
potentials is possibly extendable to bicriteria situations.
Acknowledgements This paper is partially supported by MIUR (Ministero dell’Istruzione, dell’Univer-
sità e della Ricerca) and by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e Ap-
plicazioni).
The authors gratefully thank three anonymous referees and Rodica Branzei for their helpful comments
and suggestions to improve the paper.
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