Game Development Reference

In-Depth Information

We then focus on the iterated elimination of never best responses, in short

IENBR
, obtained by using the

→
N
relation. The following counterpart of

the IESDS Theorem 1.3 then holds.

Suppose that G
is an outcome of IENBR from

Theorem 1.10
(IENBR)

a strategic game G.

(i) If s is a Nash equilibrium of G, then it is a Nash equilibrium of G
.

(ii) If G is finite and s is a Nash equilibrium of G
, then it is a Nash

equilibrium of G.

(iii) If G is finite and solved by IENBR, then the resulting joint strategy is

a unique Nash equilibrium.

Exercise 1.4

Provide the proof.

Further, as shown by Apt [2005], we have the following analogue of the

Order Independence I Theorem 1.5.

Theorem 1.11
(Order Independence II)
Given a finite strategic game all

iterated eliminations of never best responses yield the same outcome.

In the case of infinite games we encounter the same problems as in the case

of IESDS as Example 1.6 readily applies to IENBR, as well. In particular,

if we solve an infinite game by IENBR we cannot claim that we obtained a

Nash equilibrium. Still, IENBR can be useful in such cases.

Example 1.12
Consider the following infinite variant of the location game

considered in Example 1.4. We assume that the players choose their strategies

from the open interval (0
,
100) and that at each real in (0
,
100) there resides

one customer. We have then the following payoffs that correspond to the

intuition that the customers choose the closest vendor:

⎧

⎨

s
i
+
s
3
−
i

2

if
s
i
<s
3
−i

s
i
+
s
3
−
i

2

p
i
(
s
i
,s
3
−i
):=

100
−

if
s
i
>s
3
−i

⎩

50

if
s
i
=
s
3
−i
.

It is easy to check that in this game no strategy strictly or weakly dominates

another one. On the other hand each strategy 50 is a best response to some

strategy, namely to 50, and no other strategies are best responses. So this

game is solved by IENBR, in one step. We cannot claim automatically

that the resulting joint strategy (50
,
50) is a Nash equilibrium, but it is

straightforward to check that this is the case. Moreover, by the IENBR

Theorem 1.10(
i
) we know that this is a unique Nash equilibrium.

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