Game Development Reference
In-Depth Information
The above example shows that in the limit of an infinite sequence of
reductions different outcomes can be reached. So for infinite games the
definition of the order independence has to be modified. An interested reader
is referred to Dufwenberg and Stegeman [2002] and Apt [2007] where two
different options are proposed and some limited order independence results
are established.
The above example also shows that in the IESDS Theorem 1.3( ii ) and ( iii )
we cannot drop the assumption that the game is finite. Indeed, the above
infinite game has no Nash equilibria, while the game in which each player
has exactly one strategy has a Nash equilibrium.
1.3.2 Elimination of weakly dominated strategies
Analogous considerations can be carried out for the elimination of weakly
dominated strategies, by considering the appropriate reduction relation
W
defined in the expected way. Below we abbreviate iterated elimination of
weakly dominated strategies to IEWDS .
However, in the case of IEWDS some complications arise. To illustrate
them consider the following game that results from equipping each player in
the Matching Pennies game with a third strategy E (for Edge):
H
T
E
H
1 ,
1
1 ,
1
1 ,
1
T
1 ,
1
1 ,
1
1 ,
1
E
1 ,− 1
1 ,− 1
1 ,− 1
Note that
( E,E ) is its only Nash equilibrium,
for each player, E is the only strategy that is weakly dominated.
Any form of elimination of these two E strategies, simultaneous or iterated,
yields the same outcome, namely the Matching Pennies game, that, as we
have already noticed, has no Nash equilibrium. So during this eliminating
process we 'lost' the only Nash equilibrium. In other words, part ( i )ofthe
IESDS Theorem 1.3 does not hold when reformulated for weak dominance.
On the other hand, some partial results are still valid here.
Theorem 1.7 (IEWDS)
Suppose that G is a finite strategic game.
(i) If G is an outcome of IEWDS from G and s is a Nash equilibrium of
G , then s is a Nash equilibrium of G.
 
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