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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.

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Choice under uncertainty – complete ignorance

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  1. Choice underuncertainty – completeignorance

  2. 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

  3. Savage (1954) „Yourwifehasjustbroken 5 goodeggsinto a bowlwhenyoucome in and volunteer to finish the omelet. A sixthegg, which for somereasonmust be eitherused for the omeletorwastedaltogether, liesunbrokenbeside the bowl. Youmustdecidewhat to do with thisunbrokenegg…”

  4. 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

  5. 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

  6. 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

  7. Choice function– intuition • Choice functionis a functionwhichassigns a decision to each set of feasibleoptions • Requirements: • Nonempty set of decisions • The set of decisionshas to be contained in the set of feasibleoptions • The choice functionshouldgeneratetransitivechoices • The choice functionshould be immune to manipulation (addinganirrelevantalternativeshould not change the choice) – Independence of IrrelevantAlternatives

  8. ??? A small trattoria,whichyoudon’tknow. Menu: • bistecca • pollo The cookarrives and announcesthat he canadditionallyprepare • trippa alla fiorentina

  9. Consistency and attractiveness of the choice rules 1) Maximin – itsdisadvantageispessimism 2) Hurwiczrule • How to chooseα? • Critique 1 A1 isoptimal, A2 also, but the combination of the twois not.

  10. 2) Hurwiczrule • Critique 2 • Both actionsoptimal • A1 seems to be much better • But we haveassumedcompleteignorance, so the aboveisequivalent to:

  11. 3) Minimaxregret (Savage) • Proposed to improveoverMaximin • Critique 1: Are the differences in utilities/payoffgoodmeasures of regret? • Critique 2: A small advantage in one stateexceeds the big one in anotherstate Payofftables: Regrettables:

  12. 3) Minimaxregret (Savage) • Proposed to improveoverMaximin • Critique 1: Are the differences in utilities/payoffgoodmeasures of regret? • Critique 2: A small advantage in one stateexceeds the big one in anotherstate Payofftables: Regrettables:

  13. 3) Minimaxregret (Savage) • Three kinds of rescuetransports • Aircraft • Short range • Longrange • Trucks • Headquarters: Wales • Three areas of earthquakerisk: • Wales • Iberianpeninsula • Azerbaijan

  14. 3) Minimaxregret (Savage) • Critique 3: The presence of anunwantedalternativemayhave influence on the chosenaction Payofftable: Regrettable: Payofftable: Regrettable:

  15. 3) Minimaxregret (Savage) • Critique 3: The presence of anunwantedalternativemayhave influence on the chosenaction Payofftable: Regrettable: Payofftable: Regrettable

  16. Solution to the problem?: Instead of comparingthemalltogether, comparethem in pairs Planebetterthanhelicopter Helicopterbetterthan trucks Trucks betterthanplane

  17. 4) Laplace rule (principle of insufficientreason) 5) The decisionmakercannotmakeuphismindwhichrule to use: Maximin, Hurwicz (α=0.75) or Laplace • He decides to choosethisactionwhichwins in pairwisecontest States of natureshould be chosencarefully Instransitive

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