We demonstrate that iterated elimination of strictly dominated strategies is an order dependent procedure. We also prove that order does not matter if strategy spaces are compact and payoff functions continous. Examples show that this result is tight.game theory; iterated strict dominance; order independence

Dufwenberg, Martin

Stegeman, Mark

English

Research Papers in Economics

WHEN ORDER MATTERS FOR ITERATED STRICT DOMINANCE *
Martin Dufwenberg** & Mark Stegeman***
February 1999
Abstract: We demonstrate that iterated elimination of strictly dominated strategies is an order
dependent procedure. We also prove that order does not matter if strategy spaces are compact
and payoff functions continuous. Examples show that this result is tight.
Keywords: game theory, iterated strict dominance, order independence
JEL code: C72
                                                
*   We thank Geir Asheim and the members of the VPI theory seminar for helpful comments.
**  Stockholm University; md@ne.su.se
*** Virginia Polytechnic Institute; stegeman@vt.edu1
1. INTRODUCTION
There seems to be widespread belief that the outcome of iterated elimination of strictly
dominated strategies (IESDS) does not depend on the order of elimination. Nevertheless, this
assertion has not been formally proved. We show that, in fact, order may matter. One of the
examples is taken from Stegeman (1990). We also prove that for games with compact strategy
spaces and continuous payoff functions order does not matter. This result covers the cases of
finite games and their mixed extensions. Examples show that the result is tight.
The work most closely related to ours is that of Gilboa, Kalai & Zemel (1990) (GKZ), and in a
separate section we connect to their contribution. GKZ consider a variety of elimination
procedures and provide sufficient conditions for order independence. Among the procedures
considered by GKZ is a form of IESDS, and they prove that for finite games this procedure is
order invariant. GKZ, however, impose a bound on the rate of elimination, that is, they
establish invariance for only a subset of possible elimination sequences. It follows from our
aforementioned result that this bound is irrelevant for finite games. We generalize this finding.
GKZ consider IESDS only for a finite number of eliminations rounds, but in games with
infinite strategy spaces it is natural to allow an infinite sequence of elimination rounds, and
GKZ's definition is easily generalized to allow this. Given this modification, we prove that
GKZ's bound on the rate of elimination is irrelevant for all games with compact strategy
spaces and continuous payoff functions.
We show that order may matter for IESDS in Section 2, prove that order does not matter in
games with compact strategy spaces and continuous payoff functions in Section 3, discuss the
contribution of GKZ in Section 4, and offer concluding remarks in Section 5.2
2. WHEN ORDER MATTERS
Our first example is the simplest game we can think of for which order matters for IESDS.
The example shows that order can matter if strategy sets are not closed.
Example 1. Consider a one-player game with strategy set  G1=(0,1) and payoff function u1:G1
ﬁR defined by ui(x)=x for all x˛G1. In this game every strategy is strictly dominated. For any
x˛G1, eliminate in round one all strategies in the set G1\{x}, and only x survives IESDS.
Our next example shows that closing all strategy sets is not enough to ensure order
independence. This example shows that using IESDS "to simplify" a two-player game may
not be innocuous even if the game possesses a Nash equilibrium. IESDS generates not only
ambiguous residual games but also ambiguous sets of Nash equilibria.
Example 2. Consider a two-player game with strategy sets G1=G2=[0,1], and payoff functions
ui:Gi·GjﬁR with i,j=1,2 and i„j, defined by
ui(x,y) = x if x<1
ui(1,y) = 0 if y<1
ui(1,1) = 1
The strategy profile (1,1) is the game's unique Nash equilibrium, and every strategy except 1 is
strictly dominated. Eliminating Gi\{1,x} for some x<1, for i=1,2, leaves the following 2·2
game, which cannot be further reduced:
  1   x
1, 1 0, x
x, x x, 0
  1
  x3
Suppose one applies an equilibrium selection theory which favours ”risk dominance” in the 2·2
game. Then the profile (x,x) is selected iff x is large enough.
The game in Example 2 has discontinuous payoffs. Our final example, taken from Stegeman
(1990), shows that even with continuous payoff functions and closed strategy sets order
matters. Again, what set of Nash equilibria obtains in the reduced game created by IESDS
depends on the order of elimination.
Example 3. Consider a two-player game in which player 1 chooses x˛R+, 2 chooses y˛R+,
and each player receives the common payoff u(x,y)=(max{x,1-x-y})/(1+x). The payoff
function is continuous and has range [0,1]. If y>0 then player 1's optimal action is undefined,
and it follows directly that the unique Nash equilibrium of the game is (x,y)=(0,0). One way to
perform IESDS is as follows: eliminate every x>0 as it is strictly dominated by some x¢>x.
Given that x=0, every y>0 is then strictly dominated by y=0. IESDS thus eliminates all except
Nash play. Another way to perform IESDS is: eliminate every x>0 except x=1, leaving the
strategy sets {0,1} for player 1 and R+ for player 2. No more eliminations are possible and the
residual game now has many Nash equilibria: (x,y)=(0,0) and (x,y)=(1,y) for all y‡½.
3. WHEN ORDER DOES NOT MATTER
In this section we prove that order does not matter for IESDS in games with compact strategy
spaces and continuous payoff functions. Preliminary definitions follow.
Games, subgames, and strict dominance.  A game is a triple G=(I,(Gi)i˛I,(ui)i˛I), where
I={1,2,...n} is the set of players, Gi˝R
m (Gi„˘) is player i’s strategy set for some integer m‡1,
and ui:PiGiﬁR is the payoff to player i.  We call the game G compact and continuous if Gi is
compact and ui is continuous "i˛I. For convenience, assume that the players' strategy sets are
disjoint.  A subgame of G is a game H=(I,(Hi)i˛I,(ui¢)i˛I), where Hi˝Gi and ui¢ is the restriction4
of ui to Pi Hi, "i˛I.  For any subgame H, let H-i”Pj„iHj.  Let S(G) denote the set of all
subgames of G.  Given a subgame H of G, and x,y˛Gi: yfHx if ui(y,s-i)>ui(x,s-i) "s-i˛H-i.
(The reordering of the arguments of ui simplifies notation, where no confusion is possible.)
The relation (fH) embodies the notion of strict dominance given rivals’ options in game H.
Reduction.  Consider subgames H,H¢˛S(G), such that Hi¢˝Hi "i˛I.  HﬁH¢ if, for each
x˛Hi\Hi¢, $y˛Hi such that yfHx. We use the symbol ﬁ* as follows: Hﬁ*H¢ if there exists a
(finite or infinite) sequence of subgames, A
t˛S(G), t=0,1,2..., such that A
0=H, A
tﬁA
t+1 for all
t, and Hi¢=˙tHi
t "i.  H is a maximal (ﬁ)-reduction of G if Gﬁ*H and HﬁH¢ only for H¢=H.
The following Lemma is the key result behind both of our theorems.
Lemma.  If Gﬁ*H for some compact and continuous game G, and yfHx for some x,y˛Gi and
i˛I, then $z*˛Hi such that z / fHz*fHx "z˛Hi.
Proof.  Given H as described, let A
t˛S(G), t=0,1,2..., be the implied sequence of subgames.
Let Z”{z˛Gi‰ui(z,s-i)‡ui(y,s-i) "s-i˛H-i}.  Clearly y˛Z, and the continuity of ui and
compactness of Gi imply that Z is compact.  Define f:ZﬁR by f(z)=ui(z,s*) for some fixed
and arbitrary s*˛H-i.  The continuity of ui, implies that f is continuous, which with Z compact
implies that f reaches a maximum f* at some z*˛Z.  z*˛Z and yfHx imply z*fHx.  If zfHz*
for some z˛Gi, then ui(z,s-i)>ui(z*,s-i) "s-i˛H-i, implying z˛Z and f(z)>f(z*)=f*, a
contradiction.  Therefore, z / fHz* "z˛Gi (and hence "z˛Hi), implying z / f
Atz* "z˛Gi, "t
(because H-i˝A-i
t), implying z*˛Ai
t, "t, implying z*˛Hi. ž
Theorem 1.  If G is compact and continuous, then any maximal (ﬁ)-reduction of G is unique.
Proof.  Let H and H¢ be maximal (ﬁ)-reductions of G.  Given Gﬁ*H¢, let A
t˛S(G),
t=0,1,2..., be the implied finite or infinite sequence of subgames.  Suppose that Hi / ˝Hi¢ for
some i.  Then Hi / ˝Ai
t "t>T, for some T such that Ai
T+1 is well-defined.  Let T take the largest5
value such that Hi˝A
T "i.  Choose i˛I and x˛Hi\Ai
T+1.  Then x˛Ai
T\Ai
T+1, implying from
A
TﬁA
T+1 that $y˛Ai
T such that yf
A
Tx, which with Hi˝Ai
T "i implies yfHx.  The Lemma
implies $z*˛Hi such that z*fHx, contradicting that H is a maximal reduction.  Therefore,
Hi˝Hi¢ "i.  Similarly, Hi¢˝Hi "i, implying H=H¢. ž
Theorem 1 says that IESDS is an order independent procedure for compact and continuous
games. Note that this result covers finite games and their mixed extensions. The three
examples of Section 2 show that Theorem 1 is tight with respect to closedness and
boundedness of the players' strategy sets, as well as with respect to continuity of the payoff
functions.
4. GKZ REDUCTIONS
In this section we connect to the work of GKZ. They define a notion of reduction which
bounds the rate of elimination, unlike the textbook (ﬁ)-reduction we have considered so far.
We shall use the symbol ￿ for GKZ's reduction. Intuitively, the difference between a (￿)-
reduction and a (ﬁ)-reduction is that the former, but not the latter, requires that for any
strictly dominated strategy x which is eliminated there exists a strategy y which strictly
dominates x and which is not eliminated.
GKZ Reduction.  Consider subgames H,H¢˛S(G), such that Hi¢˝Hi "i˛I.  H￿H¢ if, for each
x˛Hi\Hi¢, $y˛Hi¢ such that yfHx.  We use the symbol ￿* as follows: H￿*H¢ if there exists a
(finite or infinite) sequence of subgames, A
t˛S(G), t=0,1,2..., such that A
0=H, A
t￿A
t+1 for all
t, and Hi¢=˙tAi
t "i.  H is a maximal (￿)-reduction of G if G￿*H and H￿H¢ only for H¢=H.
H￿H¢ and H￿*H¢ imply, respectively, HﬁH¢ and Hﬁ*H¢. The present definition of a
maximal (￿)-reduction is more general than that used by GKZ in that infinite sequences of
subgames are allowed. GKZ consider only finite sequences. We now prove that, although6
(￿)-reductions are more restrictive than (ﬁ)-reductions, the two produce identical maximal
reductions, and hence identical results for IESDS, in compact and continuous games.
Theorem 2.  If G is compact and continuous, then Gﬁ*H if and only if G￿*H.
Proof.  G￿*H immediately implies Gﬁ*H.  Going the other way, suppose Gﬁ*H, and let
A
t˛S(G), t=0,1,2..., be the implied sequence of subgames.  It is sufficient to show that A¢￿A†
for any two consecutive elements of this sequence.  Consider such A¢ and A†.  If A¢=A†, then
A¢￿A† trivially.  If not, then choose i˛I and x˛Ai¢\Ai†.  A¢ﬁA† implies $y˛Ai¢ such that
yfA¢x.  The Lemma implies that $z*˛Ai¢ such that z / ¢ fA z*f ¢ A x "z˛Ai¢, and A¢ﬁA† then
implies z*˛Ai†.  Hence, x˛Ai¢\Ai†, any i˛I, implies $z*˛Ai† such that z*f ¢ A x.  Therefore,
A¢￿A†.   ž
Hence, if there is an advantage to GKZ reductions, it must be based on games outside the
compact and continuous class. We close this section with a few comments about such games
and about GKZ reductions. In Example 1, the problematic (ﬁ)-reduction would not be
permitted as a (￿)-reduction, but IESDS based on (￿)-reductions does not escape the
problem of order dependence. To see this, consider the following infinite sequence of  (￿)-
reduced strategy sets: (0, 1), [x, 1), {x}¨[1-(1-x)/2, 1), {x}¨[1-(1-x)/3, 1), {x}¨[1-(1-x)/4,
1), .... For any choice of x˛(0,1), the intersection {x} is the strategy set corresponding to a
maximal (￿)-reduction. Order matters. In similar fashion one may readily show that, for
IESDS based on (￿)-reductions, order matters also in the games of Examples 2 and 3.
If one returns to GKZ's original definition, which requires maximal (￿)-reductions to end in a
finite number of steps, then it is not possible to get ambiguous maximal (￿)-reductions of the
games in the Examples 1-3, simply because these games have no maximal (￿)-reduction in
finite steps. The restriction to finite steps seems unnatural, however, because in some games
infinite reduction sequences lead to maximal reductions that could not be obtained via a finite
number of eliminations. The following well-known example illustrates the point. It requires7
an infinite sequence of reductions to find the unique maximal (ﬁ)-reduction (which by
Theorem 2 is also the unique maximal (￿)-reduction).
Example 4. (Cournot competition) I={1,2}, G1=G2=[0,1], ui:Gi·GjﬁR with i,j=1,2 and i„j
defined by ui(x,y) = x(1-x-y). The following is a infinite sequence of (ﬁ)-reduced strategy
sets starting with this game: [0, 1], [0, 1/2], [1/4, 1/2], [1/4, 3/8], [5/16, 3/8], .... Taking the
intersection we get for each player {1/3} as the strategy set associated with the maximal (ﬁ)-
reduction. The strategy profile (1/3, 1/3) is the game's unique Nash equilibrium. It is easy to
show that while there are many alternative sequences of (ﬁ)-reductions, they all require an
infinite number of elimination rounds.
Summarizing, complete reduction of some games (including compact and continuous games)
requires an infinite sequence of deletions, but the GKZ bound on the rate of deletions does not
solve the problem of order dependence if infinite sequences of deletions are permitted. Hence,
we cannot find a compelling case for abandoning the standard IESDS definition in favor of
one based on GKZ reductions.
5. CONCLUDING REMARKS
Many textbooks do not recommend iterated elimination of weakly dominated strategies
(IEWDS) as a solution concept, and one important reason is that there are games where order
matters for that procedure. Our examples show that the same criticism applies to IESDS. For
IEWDS, the finding that order matters has prompted researchers to investigate for what class
of games order independence holds, partly on the presumption that it is relatively innocuous to
apply IEWDS in those games (see, for example, Marx & Swinkels, 1997). Adopting this view,
our result of Section 3 provides consolation: Order does not matter for IESDS in compact and
continuous games, so IESDS is a sensible procedure for this large class of games.8
It is unclear what is the proper definition and role of iterated strict dominance in games that
are not compact and continuous. Our Example 1 shows that there are games for which the
concept is intrinsically unsound. The identification of general classes of non-
compact/continuous games for which IESDS is an attractive procedure remains an open
problem. For compact and continuous games, while we have answered the question of
uniqueness, the existence of a maximal reduction remains an open question.
REFERENCES
Gilboa, I., E. Kalai, & E. Zemel, 1990, "On the Order of Eliminating Dominated Strategies",
Operations Research Letters 9, 85-89.
Marx, L. & J. Swinkels, 1997, ”Order Dependence of Iterated Weak Dominance”, Games and
Economic Behavior 18, 219-45.
Stegeman, M., 1990, "Deleting Strictly Dominated Strategies", Working Paper 1990/6,
Department of Economics, University of North Carolina.

When Order matters for Iterated Strict Dominance

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