Norde, H.W.

Voorneveld, M.

English

Research Papers in Economics

. GAMES AND ECONOMIC BEHAVIOR 19, 235]242 1997
ARTICLE NO. GA970554
NOTE
A Characterization of Ordinal Potential Games
Mark Voorneveld* and Henk Norde
Department of Econometrics, Tilburg Uni¨ersity, P.O. Box 90153, 5000 LE Tilburg,
The Netherlands
Received October 22, 1996
This note characterizes ordinal potential games by the absence of weak improve-
ment cycles and an order condition on the strategy space. This order condition is
automatically satisﬁed if the strategy space is countable. Journal of Economic
Literature Classiﬁcation Number: C72. Q 1997 Academic Press
1. INTRODUCTION
. Monderer and Shapley 1996 introduce several classes of potential
games. A common feature of these classes is the existence of a real-valued
function on the strategy space that incorporates the strategic possibilities
. of all players simultaneously. In their paper, Monderer and Shapley 1996
distinguish between exact and ordinal potential games. As an example of
an exact potential game, consider the two-person game in Fig. 1a, where
the ﬁrst player chooses either T or B, and the second player simultane-
ously and independently chooses either L or R. The numbers in the
corresponding cells are the payoffs to players 1 and 2, respectively. Also,
consider the real-valued function on the strategy space given in Fig. 1b.
Note that the change in the payoff to a unilaterally deviating player exactly
equals the corresponding change in the value of this function. For in-
.. stance, if the second player deviates from T, L to T, R , his payoff
increases by one, just like the function in Fig. 1b. This function is therefore
called an exact potential of the game. Exact potential games are character-
. ized in Monderer and Shapley 1996 by the property that the changes in
payoff to deviating players along a cycle sum to zero, where a cycle in the
*To whom correspondence should be addressed. E-mail: M. Voorneveld@kub.nl. The
authors thank Peter Borm, Peter Wakker, and Stef Tijs for helpful comments.
235
0899-8256r97 $25.00
Copyright Q 1997 by Academic Press
All rights of reproduction in any form reserved.NOTE 236
LR L R
T0 , 2y 1,3 T 0 1
B1 , 0 0 , 1 B 1 2
ab
F IGURE 1
strategy space is a closed sequence of strategy combinations in which
players unilaterally deviate from one point to the next. The game in Fig. 2a
is an example of an ordinal potential game. Consider the function in
Fig. 2b and note that the sign of the change in the payoff to a unilaterally
deviating player exactly matches the sign of the corresponding change in
. this function. For instance, if the second player deviates from T, L to
. T , R , his payoff increases, just like the function in Fig. 2b. Since deviating
. . from T, R to B, R does not change player 1’s payoff, the value of the
function remains the same. For this reason, this function is called an
ordinal potential of the game.
Monderer and Shapley do not give a characterization of ordinal poten-
tial games. The class of ﬁnite ordinal potential games was characterized in
. Voorneveld 1996 through the absence of weak improvement cycles, i.e.,
cycles along which a unilaterally deviating player never incurs a lower
payoff and at least one such player increases his payoff. The necessity of
this condition is immediate, since a potential function would never de-
crease along a weak improvement cycle, but increases at least once. This
gives a contradiction, because a cycle ends up where it started. Proving
sufﬁciency is harder. In this note we characterize the total class of ordinal
potential games. It turns out that countable ordinal potential games are
still characterized by the absence of weak improvement cycles, but that for
uncountable ordinal potential games an additional order condition on the
strategy space is required.
The organization of this note is as follows: Deﬁnitions and some
preliminary results are given in Section 2. In Section 3 we provide a full
characterization of ordinal potential games. In Section 4 we indicate that
LR L R
T0 , 20 , 3 T 0 2
B1 , 00 , 1 B 1 2
ab
F IGURE 2NOTE 237
the absence of weak improvement cycles characterizes ordinal potential
games with a countable strategy space, but not necessarily ordinal poten-
tial games in which the strategy space is uncountable.
2. DEFINITIONS AND PRELIMINARY RESULTS
 i . i .: A strategic game is a tuple G s N, X , u , where N is the ig Ni g N
player set; for each i g N the set of player i’s strategies is X
i, and
u
i:  X
i ª R is player i’s payoff function. ig N
For brevity, we deﬁne X s  X
i and for i g N: X
yi s  X
j. ig Nj g N R  i 4
Let x g X and i g N. With a slight abuse of notation, we sometimes
 i yi. denote x s x , x .
 i . i .: Let G s N, X , u be a strategic game. A path in the ig Ni g N
. strategy space X is a sequence x , x ,... of elements x g X such that 12 k
for all k s 1,2,... the strategy combinations x and x differ in exactly kk q 1
. i  k . . one, say the ikth, coordinate. A path is nondeteriorating if ux F k
i  k . .  . ux for all k s 1,2,... . A ﬁnite path x ,...,x is called a weak kq11 m
i  k . . impro¨ement cycle if it is nondeteriorating, x s x , and ux - 1 mk
i  k . . 4 ux for some k g 1,...,my1. k q 1
Deﬁne a binary relation e on the strategy space X as follows: xe y if
there exists a nondeteriorating path from x to y. The binary relation f
on X is deﬁned by x f y if xe y and ye x.
By checking reﬂexivity, symmetry, and transitivity, one sees that the
binary relation f is an equivalence relation. Denote the equivalence
wx wx  < 4 class of x g X with respect to f by x , i.e., x s y g Xy f x , and
deﬁne a binary relation $ on the set X of equivalence classes as f
wx wx wx wx follows: x $ y if x / y and xe y. To show that this relation is
well-deﬁned, observe that the choice of representatives in the equivalence
classes is of no concern:
;x, x, y, y g X with x f x and y f y: xe y m xe y. ˜˜ ˜ ˜ ˜ ˜
Note, moreover, that the relation $ on X is irreﬂexive and transitive. f
The equivalence relation f plays an important role in the characteriza-
tion of ordinal potential games.
. D EFINITION 1 Monderer and Shapley, 1996 . A strategic game G s
 i . i .: N , X , u is an ordinal potential game if there exists a func- ig Ni g N
tion P: X ª R such that
;i g N, ;x
yi g X
yi, ;x
i, y
i g X
i:
u
i x
i, x
yi ) u
i y
i, x
yi m Px
i , x
y i ) Py
i , x
y i. . . . .
() The function P is called an ordinal potential of the game G.NOTE 238
In other words, if P is an ordinal potential function for G, the sign of
the change in payoff to a unilaterally deviating player matches the sign of
the change in the value of P.
A necessary condition for the existence of an ordinal potential function
is the absence of weak improvement cycles.
 i . i .: L EMMA 2.1. Let G s N, X , u be a strategic game. If G is ig Ni g N
an ordinal potential game, then X contains no weak impro¨ement cycles.
Proof. Assume G is an ordinal potential game and suppose that
. i  k . . x ,...,x is a weak improvement cycle. By deﬁnition, ux F 1 m k
i  k . . 4 ux for all k g 1,...,my1 with strict inequality for at least kq1
.  . one such k. But then Px F Px for all and strict inequality for at kk q 1
4.  . . least one k g 1,...,my1 , implying Px - Px s Px, a contra- 1 m 1
diction. B
In Section 4, we will show that the converse of Lemma 2.1 is true if
. X , $ is properly ordered. f
. D EFINITION 2. Consider a tuple A, $ consisting of a set A and an
. irreﬂexive and transitive binary relation $ . A, $ is properly ordered if
there exists a function F: A ª R that preserves the order $ :
;x, y g A: x $ y « Fx- Fy. . .
Properly ordered sets are a key topic of study in utility theory. Not every
. tuple A, $ with $ irreﬂexive and transitive is properly ordered. A
familiar example is the lexicographic order on R
2. See, e.g., Fishburn
. 1979 for more details. However, if the set A is countable, i.e., if A is
. ﬁnite or if there exists a bijection between A and N, then A, $ is
properly ordered.
LEMMA 2.2. Let A be a countable set and $ be a binary relation on A
. that is irreﬂexi¨e and transiti¨e. Then A, $ is properly ordered.
Proof. Since A is countable, we can label its elements and write
4 4 A s x , x ,... .ForkgNdeﬁne A s x ,...,x . We deﬁne F: A ª R 12 k 1 k
. by an inductive argument. Deﬁne Fx s 0. Let k g N and assume F has 1
already been deﬁned on A such that k
;x, y g A : x $ y « Fx- Fy.1 . . . k
We extend F to A . Deﬁne kq1
< L s x g Ax $ x 4 kk k q 1
< U s x g Ax $ x . 4 kk k q 1NOTE 239
. . If L / B, take l s max Fz and x g argmax Fz.I fU/B , kz g Lz g Lk kk
. . take u s min Fz and y g argmin Fz. If both L and U are zgUz g Uk k kk
nonempty, then x $ x and x $ y imply x $ y by transitivity; so kq1 kq1
. . . given that F satisﬁes 1 , l s Fx- Fys u .
v . If L s B and U s B, take Fx s 0. kk k q 1
v .  . If L s B and U / B, take Fx gy ` , u. kk k q 1
v .  . If L / B and U s B, take Fx gl , ` . kk k q 1
v .  . If L / B and U / B, take Fx gl , u . kk k q 1
Note that F is now correctly deﬁned on A : kq1
;x, y g A : x $ y « Fx- Fy. . . k q 1
It follows that by proceeding in this way we ﬁnd a function F on A as in
the theorem. B
. In Example 4.1, we will give an example of a game in which X , $ is f
. not properly ordered. A sufﬁcient condition for an uncountable set A, $
to be properly ordered is the existence of a countable subset B of A such
that if x $ z, x f B, z f B, there exists a y g B such that x $ y, y $ z.
Such a set B is $ -order dense in A.
LEMMA 2.3. Let A be a set and $ a binary relation on A that is
irreﬂexi¨e and transiti¨e. If there exists a countable subset of A that is
. $ - order dense in A, then A, $ is properly ordered.
. Proof. This is a corollary of Theorem 3.2 in Fishburn 1979 . B
3. CHARACTERIZATION OF ORDINAL POTENTIAL
GAMES
This section contains a characterization of ordinal potential games. The
. absence of weak improvement cycles was a necessary condition. If X , $ f
is properly ordered, this is also a sufﬁcient condition.
 i . i .: T HEOREM 3.1. A strategic game G s N, X , u is an ordi- ig Ni g N
nal potential game if and only if the following two conditions are satisﬁed:
1. X contains no weak impro¨ement cycles;
. 2. X , $ is properly ordered. f
. Proof. « Assume P is an ordinal potential for G. X contains no
weak improvement cycles by Lemma 2.1. Deﬁne F: X ª R by taking for fNOTE 240
wx  wx . . wx all x g X : Fxs Px. To see that F is well-deﬁned, let y, z g x . f
Since y f z there is a nondeteriorating path from y to z and vice versa.
But since the game has no weak improvement cycles, all changes in the
. payoff to the deviating players along these paths must be zero: Pys
. Pz.
wx wx wx wx Now take x , y g X with x $ y . Since xe y, there is a non- f
. . deteriorating path from x to y,s oPxF Py . Moreover, since x and y
are in different equivalence classes, some player must have gained from
. .  wx . wx . deviating along this path: Px- Py . Hence Fx- Fy .
.  . ¥Assume that the two conditions hold. Since X , $ is prop- f
erly ordered, there exists a function F: X ª R that preserves the order f
.  wx . y i $ . Deﬁne P: X ª R by Pxs Fx for all x g X. Let i g N, x g
X
yi, and x
i, y
i g X
i.
v iiy ii i y ii y ii y i . . . . If ux , x y uy , x ) 0, then y , x e x , x , and by
 i yi. i y i . the absence of weak improvement cycles, not x , x e y , x . Hence
w i yi.xw  iy i . x  iy i . i y i . y , x $ x , x , which implies Px ,x y Py ,x s
 w i y i . x . w  i y i . x . Fx , x y Fy , x ) 0.
v i yii y ii y ii y i . . w . xw . x Assume Px , x y Py , x ) 0. Then x , x / y , x ,
i i yi. i i yi. i i yi. i i yi. so ux , x / uy , x .I fux , x - uy , x ,t h e n
 i y i . i y i . w  i y i . xw  iy i . x  iy i . x , xe y , x, and hence x , x $ y , x . But then Px , x
i y i . w  i y i . x . w  i y i . x . y Py , x s Fx , x y Fy , x - 0, a contradiction. Hence
i i yi. i i yi. ux , x y uy , x ) 0.
Conclude that P is an ordinal potential for the game G. B
The ﬁrst condition in Theorem 3.1 involving cycles closely resembles a
. characterization of exact potential games in Monderer and Shapley 1996 .
A strategic game is an exact potential game if and only if the payoff
changes to deviating players along a cycle sum to zero. In fact, it sufﬁces to
look at cycles involving only four deviations. The next example indicates
that the absence of weak improvement cycles involving four deviations
only is not sufﬁcient to characterize ordinal potential games.
EXAMPLE 3.1. Suppose P is an ordinal potential of the game in Fig. 3.
.. . . Then P has to satisfy PT ,L)PT ,RsPM , Rs PM , Ms
.  .  . PB , Ms PB , Ls PT ,L, which is clearly impossible: this is not an
ordinal potential game. It is easy to check, however, that the order
condition is satisﬁed and that there are no weak improvement cycles
involving exactly four deviations.NOTE 241
LMR
T0 , 11 , 20 , 0
M1 , 10 , 00 , 0
B0 , 00 , 01 , 1
F IGURE 3
4. COUNTABLE AND UNCOUNTABLE GAMES
. Lemmas 2.2 and 2.3 give sufﬁcient conditions for X , $ to be f
properly ordered. A consequence of Lemma 2.2 is that a game G with a
countable strategy space X is an ordinal potential game if and only if it
contains no weak improvement cycles. The strategy space X is countable if
the set N of players is ﬁnite and every player i g N has a countable set
X
i of strategies.
 i . i .: T HEOREM 4.1. Let G s N, X , u be a strategic game. If X ig Ni g N
is countable, then G is an ordinal potential game if and only if X contains no
weak impro¨ement cycles.
Proof. If X is countable, X is countable. According to Lemma 2.2, f
. X , $ is properly ordered, so the result now follows from Theo- f
rem 3.1. B
. Theorem 4.1 generalizes the analogous result from Voorneveld 1996
for ﬁnite games. The mixed extension of a ﬁnite ordinal potential game
. may not be an ordinal potential game, as shown in Sela 1992 .
. A consequence of Lemma 2.3 is that if X , $ contains a countable f
$ -order dense subset, then the absence of weak improvement cycles is
once again enough to characterize ordinal potential games.
 i . i .: T HEOREM 4.2. Let G s N, X , u be a strategic game. If ig Ni g N
. X , $ contains a countable $ -order dense subset, then G is an ordinal f
potential game if and only if X contains no weak impro¨ement cycles.
. Proof. By Lemma 2.3, X , $ is properly ordered. The result follows f
from Theorem 3.1. B
This section is concluded with an example of a game with an uncount-
able strategy space in which no weak improvement cycles exist, but which
. is not an ordinal potential game since X , $ is not properly ordered. fNOTE 242
1 42 E XAMPLE 4.1. Consider the two-player game G with X s 0,1 , X s
R, and payoff functions deﬁned by
x if y g Q 1 ux , y s and . y x if y f Q
2 4 ux , y s y for all x, y g 0,1 =R. . .
This game has no weak improvement cycles, since every weak improve-
ment cycle trivially has to include deviations by at least two players. But if
the second player deviates once and improves his payoff, he has to return
to his initial strategy eventually, thereby reducing his payoff.
We show that this game nevertheless is not an ordinal potential game.
Suppose, to the contrary, that P is an ordinal potential for G. We show
that this implies the existence of an injective function f from the uncount-
able set R R Q to the countable set Q, a contradiction.
1. 1 . . For each y g R R Q, u 0, y s 0 ) y1 s u 1, y ,s oP 0, y )
.  .w . . x P 1, y . Fix fygP 1, y , P 0, y l Q. In order to show that f: R R Q
ª Q is injective, let x, z g R R Q, x - z. Then there exists a number
. y g x , z l Q . However,
¡ 22 u 0, x -u 0, y . .¡ P 0, x -P 0, y . .
11 ~ ~ - P 1, y . « u 0, y -u 1, y . .¢ ¢ 22 - P 1, z . . u 1, y -u 1, z . .
. w. . x . w . . x Since fxgP 1, x , P 0, x and fzgP1, z , P 0, z , it follows that
. . fx- fz.S ofis injective, a contradiction.
REFERENCES
. Fishburn, P. 1979 . Utility Theory for Decision Making. New York: Krieger.
. Monderer, D., and Shapley, L. S. 1996 . ‘‘Potential Games,’’ Games Econ. Beha¨. 14,
124]143.
. w Sela, A. 1992 . Learning Processes in Games, Master’s thesis, Haifa; The Technion in
x Hebrew .
. Voorneveld, M. 1996 . Strategic Games and Potentials, Master’s thesis, Tilburg: Tilburg
University.

A characterisation of ordinal potential games.

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