Game Development Reference

In-Depth Information

This brings us to the following notion, where given a binary relation

→

→
∗
its transitive reflexive closure. Consider a strategic game

G
. Suppose that
G

we denote by

→
S
R
, i.e.,
R
is obtained by an iterated elimination of

strictly dominated strategies, in short
IESDS
, starting with
G
.

If for no restriction
R
of
G
,
R

→
S
R
holds, we say that
R
is
an outcome

•

of IESDS from
G
.

•

If each player is left in
R
with exactly one strategy, we say that
G
is

solved by IESDS
.

The following simple result clarifies the relation between the IESDS and

Nash equilibrium.

Suppose that G
is an outcome of IESDS from a

Theorem 1.3
(IESDS)

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 IESDS, then the resulting joint strategy is a

unique Nash equilibrium.

Exercise 1.1

Provide the proof.

Example 1.4
A nice example of a game that is solved by IESDS is the

location game
due to Hotelling [1929]. Assume that that the players are two

vendors who simultaneously choose a location. Then the customers choose

the closest vendor. The profit for each vendor equals the number of customers

it attracted.

To be more specific we assume that the vendors choose a location from

the set

of natural numbers, viewed as points on a real line, and

that at each location there is exactly one customer. For example, for
n
=11

we have 11 locations:

{

1
,...,n

}

and when the players choose respectively the locations 3 and 8:

Search Nedrilad ::

Custom Search