Lecture 9: Errors in Decisions
Games against p-intelligent players
In real decision situations, we observe that decisions of
players are not optimal.
Reasons:
 Different levels of information.
 Lack of time (e.g., stock exchange).
One of the modeling approaches: games against a pintelligent players.
Definition:
A player behaving with probability p like a normatively
intelligent player and with probability 1-p like a random
mechanism will be called a p-intelligent player.
For p=0, player behaves as a random mechanism, for
p=1, he is an perfectly intelligent player.
It is not reasonable to apply the same strategies against
the p-intelligent players as against intelligent ones.
Approach P: optimal strategy of normatively intelligent
player is the row maximizing the mean value payoff
(only pure strategies are used).
Approach M: intelligent player applies mixed strategies.
Let us have matrix (constant-sum game):
3 3 3 3
7 1 7 7
3 1 −1 2
8 0 8 8
Player 1 (intelligent) observes strategy of player 2
s(p) = py* + (1 − p) r
y – Nash equilibrium strategy of player 2,
r – rectangular (uniform) distribution.
The equilibrium strategies are:
x* = (1, 0, 0, 0), y* = (0, 1, 0, 0).
If player 2 is p-intelligent, player 1 expects that player 2
is going to use the strategy:
s(p) = p(0, 1, 0, 0) + (1−p)(1/4, 1/4, 1/4, 1/4) =
= (0.25−0.25p, 0.25+0.75p, 0.25−0.25p, 0.25−0.25p).
We can show that:
 the first row is an optimal strategy if p > 5/9;
 the second row is an optimal strategy if 1/3 < p < 5/9;
 the fourth row is an optimal strategy if p < 1/3.