Decision Making and Desirability in Statistical Foundations

Desirability D. Samet and D. Schmeidler

The two sides of decision making Unobservable probability

Observable preferred choices Bets

The two sides of decision making Unobservable probability qualitative probability E is at least as probable as ?

observable unobservable Pre-Savage Bets probability on consequences qualitative probability relation on events Savage probability-utility desirability Acts relation on events

Qualitative probability - measurable space (?,?) - Qualitative probability relation ?? E ? E is at least as probable as ? 1. is complete and reflexive; If ? ? ? = then E ? E H ? H; 2. 3. E , , ?. Probability ?:? ?represents when E ? ?(?) ?(?) Theorem (Savage 54) Satisfies 1.-3. and P6 if and only if it is represented by a nonatomic probability ?.

Theorem: There exist ?:? ? ? ? ? such that the foundations of the foundations of statistics statistics abridged abridged ? g ?(? ? )??(?) ?(? ? )??(?) ? ? consequences Actions Interim ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? Comprehensive state space ?? Satisfies ??, ,?? Replacing (?,?) by a relation on events?? ?? ?? ?? ?? ?? ? states of the world Ex ante

We say that the constant act ?? is preferred to the constant act ??. How do we say in Savage s model that consequence ?? is preferred to consequence ?? ? ?? ?? ?? ?? ?? ? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? Ex ante Savage s model

We say that the event ?? is preferred to the event ??. Preference on events - desirability Interim ?? ?? ?? ?? ?? ? Comprehensive state space ?? ?? ?? ?? ?? ?? ?? How do we say in a comprehensive ? state space that consequence ?? is preferred to consequence ?? ?

The model - A state space and a -algebra of events (?,?) - Consequences, ? > ? ? = {??, ,??} - A measurable act ?: ? ? Formalities - The event that the consequence is ? ? = ? ?(?) The set of null events is an ideal: Desirability -Desirability relation ?? - closed under union E is at least as desirable as ? - the sub events of null are null E ? E is as desirable as ? E ? E ? and F F ? E ? not F ? E is more desirable than ? ? is null if for each ? and ? such that E ? also Assume that for each ?, ? is non-null ? ? whenever ? ?? ? and ? ?? ?.

Technical axioms Axiom 1. (weak order) is a complete transitive order on the non-null events Axiom 2. (non-triviality) There are ? and ? such that ? ?. Axiom 3. (nonatomiciy) F?? ? ? there exists a partition of ? ? = ??, ,?? such that for each ?: if ??? ?? then ? ? and if ??? ?? then ? ? . Axiom 4. (pairs) If ? and ? are non-null and for each pair ?? and ?? ? (?? ??) and ? ?? ?? are either null or non-null and as desirable, Then ? ?.

Technical axioms Substantial Test your desire: ? Acceptance letter from AER Acceptance letter from AER Life found on Mars Axiom 5. (constant act) For each consequence ?, if ? is non-null and ? ? then ? ?.

Technical axioms Substantial Test your desire: Acceptance letter from TESN Acceptance letter from AER ? Acceptance letter from AER Acceptance letter from AER Acceptance letter from TESN Axiom 6. (intermediacy) If ? ? = then the following conditions are equivalent: ? ? Transactions of the Econometric Society of Novosibirsk ? ? ? ? ? ?

Technical axioms Substantial Test your desire: E Acceptance letter from TESN Acceptance letter from AER ? H Acceptance letter from TESN Life found on Mars Acceptance letter from AER F ? ? ? ? ? ? notation ? ? ? = ? ? ? = ? ?? ? ? ? ? ? ? ? ?

Technical axioms Substantial Axiom 7. (persistency) If ? ? ? and ? ? ? and ? ? ? = and ? ? ? = . Then ? ?? implies ? ??. ? ? ? ? ? ? notation ? ? ? = ? ? ? = ? ?? ? ? ? ? ? ? ? ?

Representation theorem The expectation of ? given ? is a desirability relation there exist a probability ?:? ? and utility ?:? ? such that: the ?-null events are the null events, and ? ? if and only if ??? ? ? ? | ? ? ? | ? ? ? ??

The pairs (?,?) that represent Assume that ?? ?? ? ?? For each ?, the conditionals ?( |??) are uniquely determined. For each ? the utility ? is uniquely determined up to transformations Let ? = ??, ,?? = (?(??), ,?(??)) ? ?? + ?(?, ,?) for ? > ? The vectors (?? ??,?? ??, ) are ordered by LRD in a reverse order of the corresponding ? s. ?likelihood ratio dominates (LRD) ? (? ?) if for each ?, There are ??+? ?? > ? is more optimistic than ?. ??= , ,? ??+???. ??= ?, , and all ? are in the open interval such that ?? ?? (?? ??) if ? ? Then ? ?

Consistent family of desirability relations ? a family of acts ? = ? ? ? } Axiom 2. (richness) If (??) is a partition into non-null events, then there is ? ? such that for each ?, ??= ? ?(?) . Axiom 1. (joint null events) All desirability relations in ? have the same null events. Axiom 4. (joint likelihood) ?,? ?, Axiom 3. (joint desirability) ?,? ?, ?? for some ?, ? ? ?|?= ?|? and ?|?= ?|? ?? for some ? , ? ? ? ?? ? ??

Representation theorem A family of desirability relations { ?} is consistent there exists a pair ?,? that represents each relation ?. Moreover, the probability ? is uniquely determined and the utility ? is uniquely determined up to a positive affine transformation.

Neighbor exchange rate Keeping probability fixed ? =? = 1/2 2 ? ? Keeping utility fixed ? ? ?? ? ? ?? ? ? ?? -1 3 ? ? ? ? = ?(? ? + ? ?) 1 ? ? ? ? 1 = ?(? 0 + ? ?)1/2 ? utility 0 1/2 1 Consequence events ?? ?? ?? ? with Lebesgue measure Prob of consequence 1 1/3 2/3 0 1/3 1/3 1/3

. The same probability and expected utility Neighbor exchange rate . (?,? ?,? ? =? = 1/2 ?) 2 The same conditional expected utility ? ? ? ? ??+ ??= ??? . . 3 (? ?,? ?? (?,? ?,? ?? ?,?) ?) . . ? = (? ?,? ?,? ?) 1 (? ?,?,? ?) ? = (?,? ?,?) utility 0 1/2 1 .(? ?, ?,? ?? ?) Consequence events ?? ?? ? with Lebesgue measure Prob of consequence 1 1/3 2/3 0 1/3 1/3 1/3