Teoria podejmowania decyzji

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

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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)

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AB

good 1.

good

2.




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A

B

good 1.

good

2.




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B


Good 1.

Good

2.

A

C




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

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• 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

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• 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


good 1.

good

2.



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AB

good 1.

good

2.




18


A

B

good 1.

good

2.




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• 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

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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,b,c} {a} {a} {a,b} {b} {a}


a yes yes no no yes

b yes yes yes yes no

An exercise – check b property for C1-C5

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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

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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}


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,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

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• Rule #1 – meets’em all: WARP, a, b, g(intuitively – the evaluation does not depend on the menu, will be formalized later)

The solution

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• 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

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