Decision Theory
Lecture 4
Decision Theory – the foundation of
modern economics
•
Individual decision making
– under Certainty
•
•
•
Choice functions
Revelead preference and ordinal utility theory
Operations Research, Management Science
– under Risk
•
•
•
•
Expected Utility Theory (objective probabilities)
Bayesian decision theory
Prospect Theory and other behavioral theories
Subjective Expected Utility (subjective probabilities)
– under Uncertainty
•
•
•
Interactive decision making
–
–
–
–
•
Decision rules
Uncertainty aversion models
Non-cooperative game theory
Cooperative game theory
Matching
Bargaining
Group decision making (Social choice theory)
– Group decisions (Arrow, Maskin, etc.)
– Voting theory
– Welfare functions
• Individual decision making
– under Certainty
• Choice functions
Choice
Choice function
Weak axiom of revealed preference (WARP)
Exemplary choice functions
• Pick the cheapest (e.g. public tenders)
• Pick the second cheapest (wine for a party)
• Maximize the IRR (investment projects)
• Pick whoever gets majority of votes (Talent
shows on TV)
• …
4
good 2.
Choice functions – some intuition (1)
Out of the gray set, A was chosen (a unique choice)
B
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when
considered collectively)
A
good 1.
5
Choice functions – some intuition (2)
good 2.
Out of the gray set, A was chosen (a unique choice)
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when
considered collectively)
B
A
good 1.
6
Choice functions – some intuition (3)
good 2.
Out of the gray set, A was chosen (a unique choice)
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when
considered collectively)
A
B
good 1.
7
Good 2.
Choice functions – some intuition (4)
Out of the gray set, A was chosen (a unique choice)
A
Out of the blue set, B was chosen (a unique choice)
Out of the golden set, C was chosen (a unique choice)
B
Do we find these choices confusing? (when
considered collectively)
C
Good 1.
8
Homework
1. Can we, using only linear budget constraints,
construct such an example for two goods,
that there is a „consistency problem” when
considering more than two alternatives, and
no problem when considering only each two
alternatives separately?
2. And when considering three goods?
9
Choice functions – a formal definition
• Notation:
X
B  2X ,  B
C :B B
set of decision alternatives
available menus (non-empty subsets of X)
choice function, working for every menu
• (Technical) properties:
C (B )  
always a choice
C ( B)  B
out of a menu
• If C(B) contains a single element  this is the choice
• If more elements  these are possible choices (not simultaneously, the
decision maker picks one in the way which is not described here)
10
An exercise
• Let X={a,b,c}, B=2X
• Write down the following choice functions:
– C1: always a (if possible), if not – it doesn’t matter
– C2: always the first one in the alphabetical order
– C3: whatever but not the last one in the alphabetical order (unless there is
just one alternative available)
– C4: second first alphabetically
(unless there is just one alternative)
– C5: disregard c (if technically it is possible), and if you do disregard c, also
disregard b (if technically possible)
11
The solution
B
C1(B)
C2(B)
C3(B)
C4(B)
C5(B)
{a}
{a}
{a}
{a}
{a}
{a}
{b}
{b}
{b}
{b}
{b}
{b}
{c}
{c}
{c}
{c}
{c}
{c}
{a,b}
{a}
{a}
{a}
{b}
{a,b}
{a,c}
{a}
{a}
{a}
{c}
{a}
{b,c}
{b,c}
{b}
{b}
{c}
{b}
{a,b,c}
{a}
{a}
{a,b}
{b}
{a}
12
The solution
B
C1(B)
C2(B)
C3(B)
C4(B)
C5(B)
{a}
{a}
{a}
{a}
{a}
{a}
{b}
{b}
{b}
{b}
{b}
{b}
{c}
{c}
{c}
{c}
{c}
{c}
{a,b}
{a}
{a}
{a}
{b}
{a,b}
{a,c}
{a}
{a}
{a}
{c}
{a}
{b,c}
{b,c}
{b}
{b}
{c}
{b}
{a,b,c}
{a}
{a}
{a,b}
{b}
{a}
13
Desirable properties
• Sometimes an internal consistency is postulated
• Why so?
– positive approach – non-consistent will go bankrupt
– normative – in order not to go bankrupt
• We’ll discuss the following:
–
–
–
–
weak axiom of revealed preferences
a property
b property
g property
14
WARP – weak axiom of revealed preferences
Definition (WARP):
A pair(B,C()) satisfies WARP, if the following holds:
if for some B from B, s.t. x,yB, we have xC(B),
than for every B’ from B, s.t. x,yB’, if yC(B’), then
xC(B’).
Intuitively:
if x was shown to be at least as willingly picked as y
(for a menu B), then for every menu B’ containing
x,y, if y is picked, so does x have to be.
15
good 2.
WARP – an intuition
Out of the gray set, A was chosen (a unique choice)
B
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when
considered collectively)
A
good 1.
16
WARP – an intuition
good 2.
Out of the gray set, A was chosen (a unique choice)
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when
considered collectively)
B
A
good 1.
17
WARP – an intuition
good 2.
Out of the gray set, A was chosen (a unique choice)
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when
considered collectively)
A
B
good 1.
18
An exercise
• Check which functions C1-C5 do not fulfill
WARP, prove by giving exemplary menus
B
C1(B)
C2(B)
C3(B)
C4(B)
C5(B)
{a}
{a}
{a}
{a}
{a}
{a}
{b}
{b}
{b}
{b}
{b}
{b}
{c}
{c}
{c}
{c}
{c}
{c}
{a,b}
{a}
{a}
{a}
{b}
{a,b}
{a,c}
{a}
{a}
{a}
{c}
{a}
{b,c}
{b,c}
{b}
{b}
{c}
{b}
{a,b,c}
{a}
{a}
{a,b}
{b}
{a}
19
The solution
• C1 – fulfils
• C2 – fulfils
• C3 – doesn’t! b picked from {a,b,c} and not from {a,b}
• C4 – doesn’t! b picked from {a,b,c} and not from {b,c}
• C5 – doesn’t! b picked from {a,b} and not from {a,b,c},
while a picked
20
a property (Chernoff property)
Definition (a property):
Assume B=2X. C() meets a, if the following holds:
if for some B out of B we have xC(B),
then for every B’B, s.t. xB’, we have xC(B’).
Intuitively:
if x picked from menu B, then shall be picked from
each smaller menu B’ (if present in it).
21
a property differently
• If something not picked from menu B’, shan’t
be picked from a bigger one:
B'  B  B'\C ( B' )  B \ C ( B)
• If we add to B1 some new alternatives B2,
then the choice will either not change, or
something out of new alternatives should be
picked
B1 , B2 B : C( B1  B2 )  C( B1 )  B2
22
Homework
Prove that the previous definitions are
equivalent
23
An exercise – check the a property for C1-C5
B
C1(B)
C2(B)
C3(B)
C4(B)
C5(B)
{a}
{a}
{a}
{a}
{a}
{a}
{b}
{b}
{b}
{b}
{b}
{b}
{c}
{c}
{c}
{c}
{c}
{c}
{a,b}
{a}
{a}
{a}
{b}
{a,b}
{a,c}
{a}
{a}
{a}
{c}
{a}
{b,c}
{b,c}
{b}
{b}
{c}
{b}
{a,b,c}
{a}
{a}
{a,b}
{b}
{a}
WARP
yes
yes
no
no
no
a
yes
yes
no
no
yes
24
b property
Conclusion for the previous exercise – a and WARP differ
(let’s look for other properties)
Definition (b property):
Take B=2X. C() meets b property, if the following holds:
if form some B’ in B we have x,yC(B’),
than for each B, B’B, we have xC(B)  yC(B).
Intuitively:
if x and y are picked in a menu B’,
then their status is equal in every greater menu B.
25
An exercise – check b property for C1-C5
B
C1(B)
C2(B)
C3(B)
C4(B)
C5(B)
{a}
{b}
{a}
{b}
{a}
{b}
{a}
{b}
{a}
{b}
{a}
{b}
{c}
{a,b}
{a,c}
{c}
{a}
{a}
{c}
{a}
{a}
{c}
{a}
{a}
{c}
{b}
{c}
{c}
{a,b}
{a}
{b,c}
{a,b,c}
{b,c}
{a}
{b}
{a}
{b}
{a,b}
{c}
{b}
{b}
{a}
WARP
yes
yes
no
no
no
a
yes
yes
no
no
yes
b
yes
yes
yes
yes
no
26
g property
Definition (g property):
Assume B=2X. C() meets g, if the following holds:
if for every menu Bi out of a family of menus we
have xC(Bi),
then for B=Bi we have xC(B).
Intuitively:
if x is picked in every menu (in a family of menus),
than it is also picked in a joint menu
27
An exercise – check g property for C1-C5
B
C1(B)
C2(B)
C3(B)
C4(B)
C5(B)
{a}
{a}
{a}
{a}
{a}
{a}
{b}
{b}
{b}
{b}
{b}
{b}
{c}
{c}
{c}
{c}
{c}
{c}
{a,b}
{a}
{a}
{a}
{b}
{a,b}
{a,c}
{a}
{a}
{a}
{c}
{a}
{b,c}
{b,c}
{b}
{b}
{c}
{b}
{a,b,c}
{a}
{a}
{a,b}
{b}
{a}
WARP
yes
yes
no
no
no
a
yes
yes
no
no
yes
b
yes
yes
yes
yes
no
g
yes
yes
yes
no
no
28
The complete solution
B
C1(B)
C2(B)
C3(B)
C4(B)
C5(B)
{a}
{a}
{a}
{a}
{a}
{a}
{b}
{b}
{b}
{b}
{b}
{b}
{c}
{c}
{c}
{c}
{c}
{c}
{a,b}
{a}
{a}
{a}
{b}
{a,b}
{a,c}
{a}
{a}
{a}
{c}
{a}
{b,c}
{b,c}
{b}
{b}
{c}
{b}
{a,b,c}
{a}
{a}
{a,b}
{b}
{a}
WARP
yes
yes
no
no
no
a
yes
yes
no
no
yes
b
yes
yes
yes
yes
no
g
yes
yes
yes
no
29
no
Properties and manipulation
• Assume C1-C5 can be used in a public tender
(a,b,c denote offers)
• Take C3({a,b})={a}, C3({b,c})={b}, C3({a,b,c})={a,b}
– different choice for a complete problem (b may be
selected),
– different when short listing
– … pairise comparisons also change the outcome – b
„better than” c, a „better than” b, hence a
– putting c on the table impacts the chocie (favours b –
possible alliance)
30
An exercise
• Public tender
• Alternatives – offers described by: price and time to deliver
(quality is constant)
• Rule #1:
– minimize the expression a  pricei + b  timei (for some weights a>0,
b>0 determined irrespectively of set of offers)
• Rule #2:
– calculated the minimal price (MP) and minimal time (MT) for all offers
(assume MP>0 and MT>0)
– minimize the expression pricei/MP + timei/MT
• Which rule do you like?
31
The solution
• Rule #1 – meets’em all: WARP, a, b, g
(intuitively – the evaluation does not depend
on the menu, will be formalized later)
32
The solution
• Rule #2 – doesn’t meet a single one
• Take B={x,y,z}, x=(4,4), y=(1,9), z=(16,1)
– what will be selected?
• Try to find some modifications in order to
show how a, b, g are broken
33
Summing up
• Different views on decision making
– choice and choice functions
– preferences
– utility function
• We can judge not only alternatives, but also
choice rules
– not meeting some properties yields a risk of being
manipulated
– different properties, not all of them equivalent
34
Materials
• Compulsory:
– A. MasColell, M. Whinston, J. Green
Microeconomic Theory, Oxford University Press,
1995, rozdz. 1
• Supplementary:
– A. Sen, Choice Functions and Revealed
Preference, The Review of Economic Studies,
1971, 38(3), s. 307-317
35