Section IX: Finitely Repeated Games
Daniel Egel
November 9, 2008
1 Individually Rational
I will start by providing a formal deﬁnition and then I will give you a more intuitive deﬁnition.
Finally, as always, we’ll do a few examples.
1.1 Formal Deﬁnition
1
Deﬁne player
i
’s minmax payoﬀ as
v
i
. A payoﬀ proﬁle
w
, which includes the payoﬀs for all
n
players such that
w
= (
w
1
,w
2
,...,w
n
), is
individually rational
if
w
i
≥
v
i
for all
i
.
1.2 Intuitive Deﬁnition and Approach for Finding Individually Rational
Points
Lets consider a game with only two players. The simplest way of writing this that I can think of
is as follows:
•
A point is individually rational if the payoﬀs to both players are at least as good (
≥
) to the
payoﬀ that they would get if they were being minmaxed by the other player
1.3 Some Examples of Individually Rational Payoﬀs
Lets go back to the examples from last week and ﬁnd the payoﬀ proﬁles that are individually
rational.
1.3.1
Example 1
Let’s do an example together. Consider the following game:
Player 1
Player 2
E
F
G
H
A
2,5
10,3
4,5
3, 9
B
3,7
1,4
6, 8
7,7
C
1, 1
0,0
0, 3
4,1
D
4, 6
5,5
8, 8
2, 6
1
This following deﬁnition is based on that provided in “A Course in Game Theory”, Osborne and Rubinstein
(1994) p. 143
1