ECON6036
IESDAs in General Game
Thus far we have restricted ourselves to finite games (games with finite number of
players and each with a finite action set). However, in many applications games may
have infinite number of players and each player’s action set may not be finite. Thus, it
is important to see if the order independence property of IESDAs we just obtained
under finite games still holds. Here we state without giving the proof (see
Dufwenberg and Stegeman 2002):
Theorem: Suppose the action space is compact (closed and bounded), and the payoff
function is continuous or uppersemicontinuous in own player’s action. Then the set of
action profile that survives IESDAs is order independent.
There are still important classes of games that are not covered by theorem. For
instance, in first price sealed bid auction, a bidder’s payoff function is not continuous.
Thus it is worthwhile considering more general games. It turns out for these general
games, IESDAs may not give order independent surviving set. To illustrate our basic
points, consider a simple one-person game where the action space is (0,1) and the
payoff function is u ( x)  x for every action x . Clearly, every action is never-best
reply and is dominated by some action. Eliminate in round one all strategies except a
particular action x  (0,1) . After this first round, there is only one action left and it is
impossible to have further elimination. Hence, the set survived IESDA is {x} .
However, the first round of elimination could have been proceeded such that all
strategies other than x '  x are eliminated. In this order of elimination, the survival
set, { x '} , is different. Hence, IESDA is not order independent.
One solution one may think of to get out of this difficulty is to restrict the order in
which strategies are eliminated. For instance, we may require all dominated strategies
to be eliminated at the same time. This is sometimes referred to as “fast elimination”
in the literature. Another restriction one may think of is to require that a action that is
used to eliminate a action in this round cannot be eliminated in the same round.
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Here we report a third way due to Chen, Long, and Luo. In a recent paper (2005),
they argue that the problem of order dependence is a result of the way IESDA is
normally defined in the literature. They argue that the problem is solved if in each
round of elimination we allow also strategies that have been eliminated in previous
rounds to eliminate strategies. Thus, according to CLL, the effect of successive
rounds of elimination is only to narrow the set of beliefs about one’s opponents, not to
narrow the range of choices available to the player.
Go back to the one-person (0,1) game. Since x which survives the first round of
elimination is dominated by ( x  1) / 2 , according to CLL, it will be eliminated in the
second round. And we are left with the empty set after the IESDA is completed, and
the difficulty of order dependence no longer exists.
There is another problem associated with conventional definition of IESDA procedure.
In the one person game, x survives IESDA and is thus a “spurious Nash equilibrium”
— i.e. a Nash equilibrium of the reduced game after iterated elimination of strictly
dominated strategies is not a Nash equilibrium of the original game. Under the IESDA
in the fashion of CLL, in round two, x is further eliminated, and thus our “maximal
reduction” yields an empty set of strategies, indicating (correctly) that the game has
no Nash equilibrium. This makes sense since x cannot be justified as a best reply
(and hence cannot be justified by any higher order knowledge of “rationality”).
Consequently, this example shows that the conventional procedure that eliminates
only those strategies that are dominated by some uneliminated action is rather
restrictive.
We find the altered definition of IESDA proposed by CLL makes sense. In any case,
the process of IESDA is just a thought experiment. What matters is the final outcome
of the thought process, and we should allow as much flexibility as possible before the
process ends. Since there is no reason why strategies eliminated in previous rounds
should not be resurrected in a new round, they can be used as dominators too.
For the example of one person (0,1) game, the “fast elimination” just performs as well
as the CLL altered definition. In general they are not equivalent. The following
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example shows that so long as previously eliminated strategies are not allowed to be
dominators, “fast elimination” still gives problematic predictions.
Example 2. Consider a two-person symmetric game: G  ( N ,{ X i }iN ,{ui }iN ) , where
N  {1, 2}, X1  X 2  [0,1] , and for all xi , x j  [0,1], i, j  1, 2 , and
i  j (cf. Fig. 1)
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1

xi ,
if xi  2 or x j  2

ui (xi , x j )   12 min{xi , x j }, if xi  12 and x j  12 .

if x  1 or x  1
0,

i
j
2
2
xj
1
xi 2
1
xj 2
xi
xi
2
0
1
2
1
xi
Fig. 1. Payoff function ui ( xi , x j ) .
In this game it is easy to see that any action xi in [0,1 2) is dominated by
yi  14 
xi
2
 xi for xi  12 . After eliminating all these dominated strategies, 1 2 is
dominated by 1 since (i) ui (1,1 2)  1  1 2  ui (1 2,1 2) if x j  1 2 , and (ii)
ui (1, x j )  x j 2  1 4  ui (1 2, x j ) if x j  1 2 . After eliminating the action 1 2 , no
xi  (1 2,1] is strictly dominated by some action xi  (1/ 2, 1] , because in the joint
action set (1 2, 1]  (1 2,1] , setting x j  xi , we have ui ( xi , x j )  xi / 2  ui ( xi, x j ) for
all xi  (1/ 2, 1] . Hence, (1/ 2,1]  (1/ 2,1] is the unique “maximal reduction” under
the “fast elimination” procedure in the conventional way.
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However, any xi  (1/ 2,1) is dominated by the previously eliminated action
yi  (1  xi ) / 4  [0,1/ 2) since, for all x j  (1/ 2,1], ui ( xi , x j )  xi / 2
 (1  xi ) / 4  ui ( yi , x j ) . Therefore, the unique maximal reduction is {(1,1)} . In fact,
(1,1) is the unique Nash equilibrium. (In this game, the payoff function ui (., x j ) is not
upper-semicontinuous since limsup x1/ 2 ui ( xi , x j )  1/ 2  0  ui (1/ 2, x j ) for all
x j  1/ 2 .
CLL also show that every Nash equilibrium survives their notion of IESDA, denoted
by IESDA*, and hence remains a Nash equilibrium in the reduced game after the
iterated elimination procedure.
As explained earlier, in addition to the order dependence issue, the conventional
definition of IESDA has the problem of having spurious Nash equilibrium in the
reduced game. IESDA* eliminates this problem for many games but not for all. The
following example shows that IESDA∗ can generate spurious Nash equilibria.
Example 4. Consider a two-person symmetric game: G  (N,{Xi }iN ,{ui }iN ) , where
N  {1,2} , X1  X2  [0,1] , and for all xi , x j [0,1],i, j  1,2 , and i  j (cf. Fig. 2)
if xi [1 / 2,1] and x j [1 / 2,1]
1,

ui (xi , x j )  1  xi , if xi [0,1 / 2) and x j (2 / 3, 5 / 6) .
x ,
otherwise
 i
xj
1
5/6
2/3
xi
1  xi
1
1/ 2
xi
1/ 2
1
xi
Fig. 2. Payoff function ui (xi , x j ) .
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It is easily verified that the unique maximal reduction D  [1 / 2,1]  [1 / 2,1] since any
yi [0,1 / 2) is dominated. That is, IESDA* leaves the reduced game
G
D
 (N,{Di }iN ,{ui
D
}iN ) that cannot be further reduced, where ui
D
is the payoff
function ui restricted on D . Clearly, D is the set of Nash equilibria in the reduced
game G
D
since ui
D
is a constant function. However, it is easy to see that the set of
Nash equilibria in game G is {x D x1 , x2 (2 / 3, 5 / 6)} . Thus, IESDA* still
generates spurious Nash equilibria x D where some xi (2 / 3, 5 / 6) .
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