### Skaidrė 1

Rational Choice Sociology
Lecture 4
Rational Choice under Uncertainty
Is the Problem of Rational Choice under Uncertainty
Important?
An actor chooses under uncertainty if she is not able
to ascribe probabilities to the possible outcomes
of her alternative actions
How often the situations of choice under uncertainty happen?
According to proponents of the theory subjective expected utility (seu),
almost never, because actors almost always have subjective probabilities
(it is even difficult to find an undisputable example of such situation).
There are no situations where special rules for choice under uncertainty
could be applied. The Bayesian rule (maximization of seu) is universal
According to proponents of objectivistic version, Bayesian rule is applicable
only if the actor has statistical data about the relative frequencies. If her
subjective probabilities are not grounded in such data, she chooses under
uncertainty. Situations of the choice of uncertainty include those where
probabilities of the unique or rare events are involved.
For such situations, special rule of choice is needed.
Unsufficient reason rule
Under uncertainty, consider all outcomes as equally probable. Multiply u by
equal p and choose the action with greatest expected utility index;
Or: choose the action with prospect that has greatest average utility
C1
C2
C3
C4
Action
1
12
0
9
2
EU1=5,75
Action
2
5
6
1
9
EU2=5,25
Maximin/Minimax rule
Under uncertainty, choose the action with best worst
outcome
(or: choose the best among worst)
Alternatively: maximize the greatest minimal payoff
(maximin)
or
Minimize the greatest maximal loss (minimax)
Maximin/Minimax rule: example 1
C1
C2
C3
C4
C5
Action1 12
0 X
3
2
6
Action2 8
2
1 X
3
4
Action3 12
22
33
4X!
6
Maximin/Minimax rule: example 2
C1
C2
C3
C4
C5
Action1 -5
-9
-6
-8
-10X
Action2 -9X!
-1
0
-7
-8
Action3 -5
-8
-9
-12X
-1
Maximax rule
C1
C2
C3
C4
C5
Action1 22
-9
9
11
81 X !
Action2 3
-7
1
7X
4
Action3 1
4
5
-6
8X
What if there are two prospects with equally good worst or best
outcomes?
Apply leximin and leximax rules: choose the action which has better
second worst outcome (leximin) or better second best outcome
C1
C2
C3
C4
C5
Action 1
8
9+
3
0 XX
-1 X
Action 2
12
78+
44 ++!
-10 X
8
9
2 XX!
9
22+
-1 X
78+
33 ++
-2 X
4
12
Best according
leximax
Action 3
Best according
leximin
Action 4
Criticism of maximin/minimax and maximax rules
C1
C2
Action1
0
1 000 000
Action2
0,01 X
1
C1
C1
Action1
2
1
Action2
-1000 000
Or: to
break the
backbone
3X
Minimax regret rule:
choose the action that brings minimal maximal regret if
choice will be unsuccesfull:
table left: utility indexes; table right: regret indexes
C1
C2
Action
1
0
100
Action
2X
1
1
C1
C2
Action
1
-1 X
0
Action
2
0
-99
Calculation of regret indexes (transformation of utility matrix into
regret matrix)
and choice according minimax regret rule
NB: utility indexes should be interval scale




Regret index Rij= uij-umaxj
In each column, find the outcome with greatest utility
index; subtract this index from the utility indexes of each
outcome in the same column.
Find in each prospect the greatest (worst) regret index
(=maximal regret)
Choose the action whose greatest regret index is least
(=least maximal regret)
Minimax regret rule: example
C1
C2
C3
C4
A1
-2
5
-8
-7
A2
-5
90
11
A3
4
20
7
C1
C2
C3
C4
A1
-6
-85 X
-19
-21
-12
A2
-9
0
0
-26X!
14
A3
0
-70 X
-4
0
Optimism-pessimism rule
There are more candidates to play the role “middle between
pessimism and optimism” rules, but they are increasingly complex
and have demanding conditions of application (utility indexes
measured at interval level and sometimes some complementary
data).
E.g. The optimism/pessimism rule:
Choose the action with the greatest sum of the utilities of best and
worst oucome weighted by optimism and pessimism indexes
Action Ai > Aj, if a × u max (Ai) + 1-a ×umin (Ai) > a × u max (Aj) + 1-a
×umin (Aj)
Where a is optimism index, 1-a is pessimism index
If a=0, then optimism-pessimism rule collapses into
maximin/minimax rule; if 1-a=0, then optimism-pessimism rule
collapses into maximax rule
Optimism-pessimism rule
(example)
C1
C2
C3
C4
-2
5
-8
-7
Uop(A1)= 0,3×5 + 0,7×8=1,5+ (-5,6)= -4,1
A2 ! -5
90
11
-12
A3
20
7
14
Uop(A2)= 0,3×90 + 0,7×-12
=27+ (-8,4)= 18,6
Uop(A3)= 0,3×20 + 0,7×-4
=6+ (-2,8)= 3,2
A1
-4