What are the implications of rationality and players’ knowledge of payoffs? What
can we infer more if we also assume that players know that the other players are rational?
What is the limit of predictive power we obtain as we make more and more assumptions
about players’ knowledge about the fellow players’ rationality? These notes try to explore
these questions.
1 Rationality and Dominance
We say that a player is rational if and only if he maximizes the expected value of his
payoffs (given his beliefs about the other players’ play.) For example, consider the
following game.
1\2 L R
T 2,0 -1,1
M 0,10 0,0
B -1,-6 2,0
(1)
Consider Player 1. He is contemplating about whether to play T, or M, or B. A quick
inspection of his payoffs reveals that his best play depends on what he thinks the other
player does. Let’s then write p for the probability he assigns to L (as Player 2’s play.)
Then, his expected payoffs from playing T, M, and B are
U
T
= 2p − (1 − p) = 3p − 1,
U
M
= 0,
U
B
= −p + 2(1 − p) = 2 − 3p,
1
respectively. These values as a function of p are plotted in the following graph:
U
0 p 1
U
M
0
-1
U 2 U
B T
As it is clear from the graph, U
T
is the largest when p > 1/2, and U
B
is the largest
when p < 1/2. At p = 1/2, U
T
= U
B
> 0. Hence, if player 1 is rational, then he will
play B when p < 1/2, D when p > 1/2, and B or D if p = 1/2. Notice that, if Player
1 is rational, then he will never play M–no matter what he believes about the Player
2’s play. Therefore, if we assume that Player 1 is rational (and that the game is as it is
described above), then we can conclude that Player 1 will not play M. This is because
M is a strictly dominated strategy, a concept that we define now.
Let us use the notation s
−i
to mean the list of strategies s
j
played by all the players
j other than i, i.e.,
s
−i
= (s
1
, .
..s
i−1
, s
i+1
, .
..s
n
).
Definition 1 A strategy s