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Choice under uncertainty – complete ignorance

This analysis explores decision-making under uncertainty and complete ignorance, using a case study of a husband faced with an egg dilemma. It examines various criteria for evaluating choices, including Savage's minimax regret and other decision rules. We discuss the implications of different choice functions, the role of probabilities, and the critique of each approach. By analyzing payoff tables and using examples like selecting a restaurant, we highlight the challenges of making optimal decisions when information is incomplete.

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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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