Jansen, M.J.M.

Vermeulen, A.J.

English

Research Papers in Economics

Journal of Mathematical Economics 36 (2001) 161–167
An ordinal selection of stable sets in
the sense of Hillas
A.J. Vermeulen∗, M.J.M. Jansen
Department of Quantitative Economics, Faculty of Economics and Business Administration,
Maastricht University, P.O. Box 616, 6200 Maastricht, MD, The Netherlands
Received 24 March 2000; accepted 19 July 2001
Abstract
In this paper, it is shown in an example that the original deﬁnition of stable sets in Hillas [Econo-
metrica 58 (1990) 1365–1391] does not satisfy (an even slightly weakenened version of) the invari-
ance condition proposed in Mertens [Ordinality in noncooperative games, Core Discussion Paper
8728,CORELouvaindelaNeuve,Belgium,1987].However,itisalsoshownthatthebasicstability
condition of Hillas underlying his deﬁnition of stable sets does admit a selection that is invariant in
the strong sense, and even ordinal. © 2001 Elsevier Science B.V. All rights reserved.
Keywords: Game theory; Stable sets; Invariance
1. Introduction
The development of the theory of stable sets was initiated by the paper of (Kohlberg and
Mertens,1986).Theiraimwastoconstructasolution 1 thatsatisﬁesanumberofconditions
they considered to be absolutely essential for any reasonable solution.
One of these conditions is invariance. Originally, at least in the paper of Kohlberg and
Mertens, a solution was said to be (KM-)invariant if it only depends on the reduced normal
form of the game. They even argued that it should only depend on the reduced strategic
form of the game. To be a bit more formal, the identiﬁcation of payoff-equivalent strategy
proﬁles of a game induces a natural projection from the space of strategy proﬁles of the
game onto the space of strategy proﬁles of its reduced strategic form. A solution is now said
∗ Corresponding author. Fax: +31-4338-84874.
E-mail address: D.Vermeulen@ke.unimaas.nl (A.J. Vermeulen).
1 In this context, a solution is a rule that assigns to each normal form game a collection of sets of strategy proﬁles
of that game, the solution sets.
0304-4068/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.
PII: S0304-4068(01)00073-8162 A.J. Vermeulen, M.J.M. Jansen/Journal of Mathematical Economics 36 (2001) 161–167
to be KM-invariant if for any two games having the same reduced strategic form the images
of the solution sets of each of these games coincide under their associated projections onto
their mutual reduced strategic form.
In an attempt to construct a solution that satisﬁes all conditions required in the paper of
(Kohlberg and Mertens, 1986) (Hillas, 1990) deﬁned a new type of stable sets. Basically he
required sets of equilibria to be robust against any sufﬁciently small perturbation of the best
reply correspondence. We will refer to sets that are robust in this sense as BR-sets. Now the
usualwaytoextractareasonablesolutionfromsuchabasicdeﬁnitionwastoconsiderthose
setsthatareminimalwithinthecollectionofallsetsofequilibriathatarerobustinthesense
under consideration. However, it is not clear whether the resulting solution is KM-invariant
or not.
Therefore,HillasalsoappliedatechniquethatwasalreadyusedbyKohlbergandMertens
in their deﬁnition of hyperstable sets. Although the formulation of this technique is some-
what ambiguous in both papers, the idea can best be described as follows. Given a (normal
form) game, consider the equivalence class of all (normal form) games that have the same
reduced strategic form as the given game. Then, within the strategy space of the reduced
strategicformgame,thereisatleastonesetthat,foranygameintheequivalenceclass,isthe
projection of some BR-set of that game (namely, the whole strategy space itself). Consider
the collection of sets within this strategy space that also have this property and select the
minimal ones w.r.t. set inclusion in this collection. Now, for the given (normal form) game
the solution sets are deﬁned to be those BR-sets that project onto one of these minimal sets
in the strategy space of the reduced strategic form game. It is fairly straightforward to show
that the resulting solution is KM-invariant.
In (Mertens, 1987), the author gave a further strengthening of the invariance condition.
He argued that a reasonable solution should be ordinal and showed that a solution is ordinal
if it is both invariant in a strong sense and admissible-best-reply invariant (abr-invariant).
See also (Vermeulen and Jansen, 2000).
Inthispaper,wewillﬁrstgiveanexamplethatshowsthatthesolutionproposedbyHillas
is not invariant in the strong sense. Although we use a slightly different way to construct a
KM-invariant solution given the basic deﬁnition, the same example also works for Hillas’
original deﬁnition. We deviate from the original deﬁnition simply because it would take
too much time to work out the details of the original deﬁnition and because the resulting
differences between Hillas’ deﬁnition and the one we use are not relevant for the arguments
given in the example. As already observed by (Mertens, 1987), the real culprit that causes
lack of invariance is the minimality condition.
Secondly, we will show that there is a quite natural way to select BR-sets such that the
resulting solution is both invariant in the strong sense and abr-invariant, and therefore also
ordinal.
2. Preliminaries
A normal form game with player set N is a pair Γ =  A,u  such that A :=
 
i∈NAi is
the product of non-empty and ﬁnite pure strategy sets Ai and u = (ui)i∈N is an N-tuple of
payoff functions ui: A → R.A.J. Vermeulen, M.J.M. Jansen/Journal of Mathematical Economics 36 (2001) 161–167 163
For a strategy proﬁle x = (xi)i∈N ∈   :=
 
j∈N j, the expected payoff of player i is
denoted by ui(x). Here,  i is the set of probability distributions on Ai.
A solution is a map σ that assigns to each game Γ a collection σ(Γ)of (non-empty and)
closed subsets (called solution sets) of the strategy space of Γ. In (Mertens, 1987), it is
shown that a solution is ordinal if it is both invariant and admissible-best-reply invariant
(abr-invariant). Since these last two conditions are in fact the ones that will be used in this
paper, we will ﬁrst recall their deﬁnitions.
A solution is called abr-invariant if the solution sets coincide for those (so-called abr-
equivalent) games for which the best replies against a completely mixed strategy proﬁle
coincide.
In order to deﬁne invariance, let Γ and Γ   be two games. A map f = (fi)i∈N from   
to   is called a reduction map from Γ   to Γ if each map fi is linear and onto from   
i to
 i and the payoff function u 
i equals the composition ui ◦f of the payoff function ui with
f. Now a solution σ is called invariant if for any reduction map f from a game Γ   to a
game Γ, the set f(T)is a solution set of the game Γ for every solution set T of the game
Γ   and, conversely, for every solution set S of the game Γ the set f −1(S) is equal to the
union over all solution sets T of the game Γ   for which f(T)= S.
Following (Hillas, 1990), we deﬁne best-reply sets (BR-sets) as follows. A closed set
of strategy proﬁles is called a BR-set if any neighborhood of that set contains at least
one ﬁxed point of any compact and convex-valued upper hemicontinuous correspondence
whose pointwise Hausdorff distance to the best reply correspondence is sufﬁciently small.
A BR-set S is called extensible (cf. (Vermeulen et al., 1995)) if f −1(S) is also a BR-set for
any reduction map f.
3. An example
In this section, we will show that the (slightly changed 2 ) deﬁnition of stability in the
sense of (Hillas, 1990) does not satisfy invariance. In fact we will consider two ways to
deﬁne solution sets using BR-sets. On the one side we consider the minimal extensible
BR-sets as solution sets and on the other hand the minimal BR-sets.
We will show now that the ﬁrst solution—although it does satisfy the (weaker) type of
invariance mentioned in the paper of (Kohlberg and Mertens, 1986)—is not invariant in
the sense of the previous section. The same example can also be used to show that neither
the second solution is invariant.
Consider the 3 × 4-bimatrix game
Γ   :




(0,2)( 4,0)( 2,1)( 2,1)
(4,0)( 0,2)( 2,1)( 2,1)
(3,2)( 3,2)( 3,2)( 3,2)




2 Thischangeisaresultofbothachangeinthedeﬁnitionofinvarianceandanambiguityintheoriginaldeﬁnition
of stable sets. A justiﬁcation of these adjustments is given in the introduction.164 A.J. Vermeulen, M.J.M. Jansen/Journal of Mathematical Economics 36 (2001) 161–167
and let T be the Cartesian product of the one-point set {(0,0,1)} (for player 1) and the set
inthestrategyspaceofplayer2consistingofthefollowingthreelinesegments:theﬁrstone
from(3/4,1/4,0,0)to(1/16,5/16,1/2,1/8),thesecondonefrom(1/16,5/16,1/2,1/8)
to (3/8,1/8,1/2,0) and the third one from (3/8,1/8,1/2,0) to (1/4,3/4,0,0). The ﬁrst
two line segments are depicted below by the coarsely dotted lines and the third one is the
line segment in the front face of the tetrahedron.
The following is the strategy space of player 2:
It can be shown that T is a stable set in the sense of (Mertens, 1989) (and therefore a
BR-set).BytheproofofProposition11of(Mertens,1987),itfollowsthat,foranyreduction
map f from a game Γ    to Γ  , f −1(T) is a BR-set. So T is an extensible BR-set. We will
show that T is also minimal. To this end, note that the two-point set
S :={ (0,0,1)}×{ (3
4, 1
4,0,0),(1
4, 3
4,0,0)}
is the only stable set in the sense of Kohlberg and Mertens of the game Γ   that is contained
in T. However, by Proposition 3 of (Hillas, 1990), we know that any stable set in the sense
of Hillas is connected and T is the only connected subset of T that contains S. Hence, T
must be minimal.
Now consider the 3 × 3-bimatrix game
Γ :




(0,2)( 4,0)( 2,1)
(4,0)( 0,2)( 2,1)
(3,2)( 3,2)( 3,2)




and the reduction map f from Γ   to Γ with f1 equal to identity and f2 is the linear map
f2(q1,q 2,q 3,q 4) = (q1 + 1
2q4,q 2 + 1
2q4,q 3).A.J. Vermeulen, M.J.M. Jansen/Journal of Mathematical Economics 36 (2001) 161–167 165
It is easy to check that the set M := M1 ∪ M2 with
M1 :={ (0,0,1)}×{ (1 − s)(3
4, 1
4,0) + s(1
3, 1
3, 1
3)|0 ≤ s ≤ 1}
M2 :={ (0,0,1)}×{ (1 − s)(1
3, 1
3, 1
3) + s(1
4, 3
4,0)|0 ≤ s ≤ 1}
is a proper subset of f(T). Moreover, M is a minimal extensible BR-set by the same line
of reasoning used above. Hence, f(T)is not a minimal extensible BR-set which shows that
the solution that assigns to each game its minimal extensible BR-sets is not invariant.
4. The ordinal selection
SincethesolutionthatassignstoagameitscollectionofextensibleBR-setsis,aswewill
alsoseeinthissection,invariantandabr-invarianttheaboveexamplesuggeststhatthelackof
invariance of stable sets in the minimal extensible BR-sets is mainly due to the minimality
condition. Such an incompatibility of minimality with invariance was already noted by
(Mertens,1989).Leavingouttheentireminimalityconditionisnotverysatisfactorythough,
sincethesolutionwe,thus,getviolatesseveralotherrequirementsoftheKohlberg–Mertens
program.
In this section, we will apply a relaxation of the minimality condition similar to the one
usedbyMertens.Infactwewillconsiderthesolutionτ thatassignstoagamethecollection
of extensible BR-sets that are a connected subset of the set of perfect equilibria of the game
inquestion.Fromnowonwewillsimplyrefertotheelementsofτ(Γ)assolutionsetsofΓ.
WewillshowthatthiswayofselectingBR-setsyieldsaninvariantandabr-invariantsolution
that also satisﬁes the other relevant properties of Hillas’ solution. Then, by Theorem 2 of
(Mertens, 1987), it follows that τ is ordinal.
Using the fact that minimal BR-sets are connected sets of perfect equilibria it can be
showed (as in Theorems 6 and 7 of (Vermeulen et al., 1995)) that minimal extensible
BR-sets are also connected sets of perfect equilibria. Hence, every game has at least one
solution set and the solution τ is well-deﬁned.
Next, we will show the following theorem.
Theorem 1. The solution τ is invariant.
Proof. Since a reduction map f from a game Γ   to a game Γ preserves closedness, con-
nectedness, perfectness, BR-sets and extendibility (cf. Proposition 1 of (Vermeulen et al.,
1995)),f(T)isasolutionsetifT is.Sincef −1 hasthesamepreservationpropertiesf −1(S)
is a solution set if S is. 
Theorem 2. The solution τ is abr-invariant.
Proof. Let Γ and Γ ∗ be two abr-equivalent games and let S be a solution set of the game
Γ ∗.Then,byTheorem3of(VermeulenandJansen,1997),itmustbeaclosedandconnected
set of perfect equilibria of the game Γ. In order to show that S is also an extensible set
of the game Γ, take a reduction map f from a game Γ   to Γ. Now let (Γ ∗)  be the game166 A.J. Vermeulen, M.J.M. Jansen/Journal of Mathematical Economics 36 (2001) 161–167
with the same strategy space as Γ   for which the payoff function of a player equals the
composition of the payoff function of that player in the game Γ   and f. Since each fi
preserves best replies (by Lemma 1 of (Vermeulen and Jansen, 1997)), the games (Γ ∗)  and
Γ   are abr-equivalent. Since, moreover, f is a reduction map from (Γ ∗)  to Γ ∗, f −1(S) is a
BR-setofthegame(Γ ∗) .Thisimpliesthatf −1(S)isalsoaBR-setofthegameΓ  .Hence,
S is an extensible set for Γ and, therefore, a solution set of the game Γ. By symmetry the
proof is complete. 
Finally, we will show that the solution τ satisﬁes all the relevant properties of Hillas’
solution. First of all solutions sets are connected sets of perfect equilibria by deﬁnition and
a solution set—being a BR-set—contains a proper equilibrium.
In order to show that the solution τ is independent of inadmissible strategies, we need
someterminology.Thedeletionofapurestrategy,sayd,ofthesetofpurestrategiesofsome
player i in a game Γ induces a game Γ ∗ for which the payoff function of each player is the
obvious restriction of the original payoff function. Furthermore, a subset S of   induces
the subset S∗ of the strategy space  ∗ of Γ ∗ obtained by skipping the dth coordinate of xi
in every strategy proﬁle x ∈ S for which this coordinate equals zero.
A solution is independent of inadmissible strategies if for any solution set S of a game
Γ for which there is a pure strategy d that is not an admissible best reply against S the set
S∗ (corresponding with the deletion of d) contains a solution set of the game Γ ∗. Here, a
pure strategy b of a player is called an admissible pure best reply against S if there exists a
strategy proﬁle x in S against which b is an admissible best reply. Note that for a BR-set S,
S∗ is a non-empty BR-set. Now we have the following theorem.
Theorem 3. The solution τ is independent of inadmissible strategies.
Proof. Let S be a solution set of the game Γ, and suppose that the pure strategy d of player
j is not an admissible best reply against S.
1. We will show that S∗ is an extensible BR-set of the game Γ ∗.
So, let f be a reduction map from a game (Γ ∗)  to Γ ∗. Now let Γ   be the game
obtained from (Γ ∗)  by adding the strategy d as a pure strategy for player j and where
thepayofffunctionofaplayerequalsthecompositionofthepayofffunctionofthatplayer
in the game (Γ ∗)  and the reduction map g from Γ   to Γ that coincides with f and with
gj(d) = d. Then g−1(S) is a BR-set of the game Γ  . So, by Lemma 6 of (Vermeulen
et al., 1995), d is not an admissible best reply against g−1(S). Since, by Proposition 4 of
(Hillas, 1990), [g−1(S)]∗ contains a BR-set of the game (Γ ∗) ,[ g−1(S)]∗ = f −1(S∗)
is a BR-set of the game (Γ ∗) . Hence, S∗ is an extensible BR-set of Γ ∗.
2. Since—as we observed at the beginning of this section—a BR-set contains a minimal
BR-set and such a set is a solution set, the proof is complete. 
References
Hillas, J., 1990. On the deﬁnition of the strategic stability of equilibria. Econometrica 58, 1365–1391.
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Belgium.A.J. Vermeulen, M.J.M. Jansen/Journal of Mathematical Economics 36 (2001) 161–167 167
Kohlberg, E., Mertens, J.F., 1986. On strategic stability of equilibria. Econometrica 54, 1003–1037.
Vermeulen, A.J., Jansen, M.J.M., Potters, J.A.M., 1995. Making solutions invariant, Report 9519, Department of
Mathematics, University of Nijmegen, Nijmegen.
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Sciences 33, 251–267.
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An ordinal selection of stable sets in the sense of Hillas.

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