Game Development Reference
In-Depth Information
we have p 1 (3 , 8) = 5 and p 2 (3 , 8) = 6. When the vendors 'share' a customer,
they end up with a fractional payoff.
In general, we have the following game:
each set of strategies consists of the set
{
1 ,...,n
}
,
each payoff function p i is defined by:
s i + s 3 −i
1
if s i <s 3 −i
2
s i + s 3 −i
1
p i ( s i ,s 3 −i ):=
n −
if s i >s 3 −i
2
n
2
if s i = s 3 −i .
It is easy to see that for n =2 k + 1 this game is solved by k rounds of
IESDS, and that each player is left with the 'middle' strategy k . In each
round both 'outer' strategies are eliminated, so first 1 and n , and so on.
There is one more natural question that we left so far unanswered. Is the
outcome of an iterated elimination of strictly dominated strategies unique,
or in game theory parlance: is strict dominance order independent ? The
answer is positive. The following result was established independently by
Gilboa et al. [1990] and Stegeman [1990].
Theorem 1.5 (Order Independence I) Given a finite strategic game all
iterated eliminations of strictly dominated strategies yield the same outcome.
As noted by Dufwenberg and Stegeman [2002] the above result does not
hold for infinite strategic games.
Example 1.6 Consider a game in which the set of strategies for each
player is the set of natural numbers. The payoff to each player is the number
(strategy) he selected.
Note that in this game every strategy is strictly dominated. Consider now
three ways of using IESDS:
by removing in one step all strategies that are strictly dominated,
by removing in one step all strategies different from 0 that are strictly
dominated,
by removing in each step exactly one strategy.
In the first case we obtain the restriction with the empty strategy sets,
in the second one we end up with the restriction in which each player has
just one strategy, 0, and in the third case we obtain an infinite sequence of
reductions.