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.

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