teoria podejmowania decyzji
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Teoria podejmowania decyzji. Wykład 2. Decision Theory – the foundation of modern economics. Individual decision making under Certainty C hoice functions Revelead preference and ordinal utility t heory Operations Research , Management Science under Risk - PowerPoint PPT PresentationTRANSCRIPT
Teoria podejmowania decyzji
Wykład 2
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• Decision rules• Uncertainty aversion models
• Interactive decision making– 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)
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• 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)
• …
Exemplary choice functions
5
Choice functions – some intuition (1)
A
B
Out of the gray set, A was chosen (a unique choice)
good 1.
good
2.
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when considered collectively)
6
Choice functions – some intuition (2)
AB
good 1.
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)
7
Choice functions – some intuition (3)
A
B
good 1.
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)
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B
Choice functions – some intuition (4)
Good 1.
Good
2.
A
C
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)
Out of the golden set, C was chosen (a unique choice)
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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?
Homework
10
• Notation:
• (Technical) properties:
• 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)
Choice functions – a formal definition
always a choice
out of a menuBBC )(
set of decision alternatives
available menus (non-empty subsets of X)
choice function, working for every menu
XBB ,2X
BB:C
)(BC
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• 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)
An exercise
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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}
The solution
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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}
The solution
14
• 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
Desirable properties
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Definition (WARP):A pair(B,C()) satisfies WARP, if the following holds:if for some B from B, s.t. x,yB, we have xC(B), than for every B’ from B, s.t. x,yB’, if yC(B’), then xC(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.
WARP – weak axiom of revealed preferences
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WARP – an intuition
A
B
Out of the gray set, A was chosen (a unique choice)
good 1.
good
2.
Out of the blue set, B was chosen (a unique choice)
Do we find these choices confusing? (when considered collectively)
17
WARP – an intuition
AB
good 1.
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)
18
WARP – an intuition
A
B
good 1.
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)
19
• Check which functions C1-C5 do not fulfill WARP, prove by giving exemplary menus
An exercise
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}
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• 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
The solution
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Definition (a property):Assume B=2X. C() meets a, if the following holds:if for some B out of B we have xC(B), then for every B’B, s.t. xB’, we have xC(B’).
Intuitively:if x picked from menu B, then shall be picked from each smaller menu B’ (if present in it).
a property (Chernoff property)
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• If something not picked from menu B’, shan’t be picked from a bigger one:
• 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
a property differently
)(\)'('\' BCBBCBBB
212121 )()(:, BBCBBCBB B
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Prove that the previous definitions are equivalent
Homework
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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
An exercise – check the a property for C1-C5
25
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,yC(B’), than for each B, B’B, we have xC(B) yC(B).
Intuitively:if x and y are picked in a menu B’, then their status is equal in every greater menu B.
b property
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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
An exercise – check b property for C1-C5
27
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 xC(Bi), then for B=Bi we have xC(B).
Intuitively:if x is picked in every menu (in a family of menus), than it is also picked in a joint menu
g property
28
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
An exercise – check g property for C1-C5
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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 noa yes yes no no yesb yes yes yes yes nog yes yes yes no no
The complete solution
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• 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)
Properties and manipulation
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• 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?
An exercise
32
• Rule #1 – meets’em all: WARP, a, b, g(intuitively – the evaluation does not depend on the menu, will be formalized later)
The solution
33
• 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
The solution
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• 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
Summing up
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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