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: