choice under uncertainty – complete ignorance

Choice under uncertainty – complete ignorance

Problem

• The set of possible actions/acts• The set of states of nature• For each action and state of nature a consequence • For each consequence a payoff/utility• A criterion according to which a decision maker

evaluates alternative actions

State 1 State 2 State 3 State 4Action 1 U11 U12 U13 U14Action 2 U21 U22 U23 U24Action 3 U31 U32 U33 U34

Savage (1954)

„Your wife has just broken 5 good eggs into a bowl when you come in and volunteer to finish the omelet. A sixth egg, which for some reason must be either used for the omelet or wasted altogether, lies unbroken beside the bowl. You must decide what to do with this unbroken egg…”

Good egg Bad eggBreak the egg directly into the

bowl 6 eggs omelet No omelet, 5 eggs thrown away

Break the egg to a separate bowl

6 eggs omelet and an additional bowl to clean

5 eggs omelet and an additional bowl to clean

Throw the egg away 5 eggs omelet, 1 good egg thrown away 5 eggs omelet

• Husband – agricultural scientist knows that in a randomly chosen sample of 6 eggs the probability of the sixth egg being bad conditional on first 5 eggs being good is 0.008.– Situation of risk

• Husband – city guy, does not know anything about eggs; additionally the 5 eggs already broken into a bowl are white whereas the sixth egg has brownish dots on its shell and (according to the husband) seems to be of an extraordinary size – Situation of complete ignorance

Good egg Bad egg Maximin Maximax Hurwicz Savage LaplaceBreak the egg

directly 100 -20 -20 100 -20α+100(1-α) 100 ½*(-20)+½*100

Break the egg to a separate bowl 90 70 70 90 70α+90(1-α) 10 ½*70+½*90

Throw the egg away 60 80 60 80 80α+60(1-α) 40 ½*60+½*80

pessimism optimismpessimism -

optimism index

minimax regret

principle of insufficient

reasonRegret table Good egg Bad eggBreak the egg

directly 0 100 0.00≤α≤0.10

Break the egg to a separate bowl 10 10 0.10≤α≤0.75

Throw the egg away 40 0 0.75≤α≤1.00

Exercise• Omelet with X eggs: 20*X• Cleaning Y bowls:

– -10-10Y, if X>0– 0, if X=0

• Throwing away a good egg: -20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-40

-20

0

20

40

60

80

100

120

Choice function– intuition

• Choice function is a function which assigns a decision to each set of feasible options

• Requirements:– Nonempty set of decisions– The set of decisions has to be contained in the set of

feasible options – The choice function should generate transitive choices – The choice function should be immune to manipulation

(adding an irrelevant alternative should not change the choice) – Independence of Irrelevant Alternatives

A small trattoria,which you don’t know. Menu:• bistecca• pollo

???

The cook arrives and announces that he can additionally prepare• trippa alla fiorentina

Consistency and attractiveness of the choice rules

1) Maximin – its disadvantage is pessimism

2) Hurwicz rulea) How to choose α?b) Critique 1

s1 s2A1 0 1000A2 0.1 0.1

s1 s2A1 0 1A2 x x

s1 s2 s3A1 0 1 0A2 1 0 0

0.5A1+0.5A2 0.5 0.5 0A1 is optimal, A2 also, but the combination of the two is not.

2) Hurwicz rule b) Critique 2

s1 s2 s3 … s100A1 0 1 1 … 1A2 1 0 0 … 0

• Both actions optimal• A1 seems to be much better• But we have assumed complete ignorance, so the above is equivalent to:

s1' s2'A1 0 1A2 1 0

3) Minimax regret (Savage)a) Proposed to improve over Maximinb) Critique 1: Are the differences in utilities/payoff

good measures of regret?c) Critique 2: A small advantage in one state

exceeds the big one in another state

s1 s2 s1 s2A1 0 1000 A1 0.01 1000A2 0.1 0.1 A2 0.1 999.9

s1 s2 s1 s2A1 0.1 0 A1 0.09 0A2 0 999.9 A2 0 0.1

Payoff tables:

Regret tables:

3) Minimax regret (Savage)• Three kinds of rescue transports– Aircraft

• Short range• Long range

– Trucks

• Headquarters: Wales

• Three areas of earthquake risk:– Wales– Iberian peninsula– Azerbaijan

Regret table:

3) Minimax regret (Savage)d) Critique 3: The presence of an unwanted alternative

may have influence on the chosen actionIberian Peninsula Azerbaijan Wales

Helicopter 100 40 30Plane 70 80 20Trucks 0 0 110

Payoff table:

Regret table:

Iberian Peninsula Azerbaijan Wales Max regretHelicopter 0 40 80 80Plane 30 0 90 90Trucks 100 80 0 100

Iberian Peninsula Azerbaijan WalesHelicopter 100 40 30Plane 70 80 20

Iberian Peninsula Azerbaijan Wales Max regretHelicopter 0 40 0 40Plane 30 0 10 30

Payoff table:

Regret table

3) Minimax regret (Savage)d) Critique 3: The presence of an unwanted alternative

may have influence on the chosen actionIberian Peninsula Azerbaijan Wales

Helicopter 100 40 30Plane 70 80 20Trucks 0 0 110

Regret table:

Iberian Peninsula Azerbaijan Wales Max regretHelicopter 0 40 80 80Plane 30 0 90 90Trucks 100 80 0 100



Payoff table:

Payoff table:

Iberian Peninsula Azerbaijan WalesHelicopter 100 40 30Trucks 0 0 110

Iberian Peninsula Azerbaijan Wales Max regretHelicopter 0 0 80 80Trucks 100 40 0 100

Iberian Peninsula Azerbaijan WalesPlane 70 80 20Trucks 0 0 110

Iberian Peninsula Azerbaijan Wales Max regretPlane 0 0 90 90Trucks 70 80 0 80



Solution to the problem?: Instead of comparing them all together, compare them in pairs

Plane better than helicopter

Helicopter better than trucks

Trucks better than plane

4) Laplace rule (principle of insufficient reason)

5) The decision maker cannot make up his mind which rule to use: Maximin, Hurwicz (α=0.75) or Laplace– He decides to choose this action which wins in pairwise contest

Iberian Peninsula Azerbaijan Laplacehelicopter 100 40 0.5*100+0.5*40=70plane 70 80 0.5*70+0.5*80=75

Spain Portugal Azerbaijan Laplace

helicopter 100 100 40 80plane 70 70 80 73.33

States of nature should be chosen carefully

s1 s2 s3 Maximin Hurwicz LaplaceA1 2 12 -3 -3 0.75*(-3)+0.25*12=0.75 3.667A2 5 5 -1 -1 0.75*(-1)+0.25*5=0.5 3.000A3 0 10 -2 -2 0.75*(-2)+0.25*10=1 2.667

Instransitive

s1 s2 s3 Maximin Hurwicz LaplaceA1 2 12 -3 3 2 1A2 5 5 -1 1 3 2A3 0 10 -2 2 1 3

choice under uncertainty – complete ignorance

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