Game Development Reference

In-Depth Information

(i) Then m is a Nash equilibrium of G iff it is a Nash equilibrium of G
RAT
.

(ii) If each player has in G
RAT
exactly one strategy, then the resulting joint

strategy is a unique Nash equilibrium of G.

Exercise 1.8

Provide the proof.

In the context of rationalizability a joint mixed strategy of the opponents

is referred to as a
belief
. The definition of rationalizability is generic in

the class of beliefs w.r.t. which best responses are collected. For example,

we could use here joint pure strategies of the opponents, or probability

distributions over the Cartesian product of the opponents' strategy sets,

so the elements of the set Δ
S
−i
(extending in an expected way the payoff

functions). In the first case we talk about
point beliefs
and in the second

case about
correlated beliefs
.

In the case of point beliefs we can apply the elimination procedure entailed

by

to arbitrary games. To avoid discussion of the outcomes reached in

the case of infinite iterations we focus on a result for a limited case. We refer

here to Nash equilibria in pure strategies.

RAT

Theorem 1.22
Assume a strategic game G. Consider the definition of the

RAT operator for the case of point beliefs and suppose that the outcome

G
RAT

is reached in finitely many steps.

(i) Then s is a Nash equilibrium of G iff it is a Nash equilibrium of G
RAT
.

(ii) If each player is left in G
RAT
with exactly one strategy, then the

resulting joint strategy is a unique Nash equilibrium of G.

Exercise 1.9

Provide the proof.

A subtle point is that when
G
is infinite, the restriction
G
RAT

may have

empty strategy sets (and hence no joint strategy).

Example 1.23
Bertrand competition
, originally proposed by Bertrand

[1883], is a game concerned with a simultaneous selection of prices for the

same product by two firms. The product is then sold by the firm that chose

a lower price. In the case of a tie the product is sold by both firms and the

profits are split.

Consider a version in which the range of possible prices is the left-open

real interval (0
,
100] and the demand equals 100

p
, where
p
is the lower

price. So in this game
G
there are two players, each with the set (0
,
100] of

strategies and the payoff functions are defined by:

−

Search Nedrilad ::

Custom Search