The epistemic structure of de Finetti s betting problem

Transcription

1 The epistemic structure of de Finetti s betting problem Tommaso Flaminio 1 Hykel Hosni 2 1 IIIA - CSIC Universitat Autonoma de Barcelona (Spain) tommaso@iiia.csic.es tommaso 2 SNS Scuola Normale Superiore (Italy) hykel.hosni@sns.it homepage.sns.it/hosni CILC 12 Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

2 Motivation De Finetti s foundation of probability Let E 1,..., E n be of events of interest. De Finetti s betting problem is the choice that an idealized agent called bookmaker must make when publishing a book, i.e. when making the assignment B = {(E i, β i ) : i = 1,..., n} such that each E i is given value β i [0, 1]. Once a book has been published, a gambler can place bets on event E i by paying α i β i to the bookmaker. In return for this payment, the gambler will receive α i, if E i occurs and nothing if E i does not occur. The book B = {(E i, β i ) : i = 1,..., n} is coherent (or is not a Dutch Book) iff there is no betting strategy for gambler, ensuring a sure win! iff the assignment B extends to a probability measure on the algebra of formulas. Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

7 Motivation Therefore, in his foundation, de Finetti (tacitly) assumes that each event E satisfies the following conditions: 1 E is unknown at the moment when bookmaker publishes B, 2 Each possible realization of E must be an accessible information to both bookmaker and gambler. The violation of the above conditions leads either to a trivial bet, or to a null bet. In fact: Trivial: How much would you bet on now we are in Rome? and how much on Rome is in France? Null: Would you participate to a betting game on the sentence whose truth (or falsity) would be always unknown? e.g. the electron ɛ has speed s, and position p Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

13 Partial valuations Formalization Let L = {q 1,..., q n } be a classical propositional language. The set of sentences SL = {ϕ, ψ, θ,...} is inductively built up from L through the propositional connectives,,, and, as usual. to denote falsum. [ϕ] is the equivalence class of ϕ (i.e., ψ [ϕ] iff ψ ϕ). A partial valuation on L is a map w : X L {0, 1} which, for a total valuation ω and for every q L is defined by { ω(q) if q X; w(q) = undefined otherwise. The class of all partial valuations over L is denoted by Ω P (L). For w : X {0, 1} and w : Y {0, 1} in Ω P (L) we say that w extends w (and we write w w ) if X Y, and for every x X, w(x) = w (x). Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

17 Formalization Realization Let ϕ SL, and let w Ω P (L). (1) We say that w realizes ϕ, written w ϕ, if w(ϕ) is defined. (2) We say that w realizes [ϕ], written w [ϕ], if w ψ for at least a ψ [ϕ]. In this case we assign w(γ) = w(ψ) for every γ [ϕ]. A partially evaluated frame (pef for short) is a pair (Ω P (L), R), where R is a transitive binary accessibility relation on Ω P (L). A pef K = (Ω P (L), R) is said to be: (M) Monotonic if, for every w, w Ω P (L), if R(w, w ), then w w. (A) Full accessible if, for every w, w Ω P (L), if w w, then R(w, w ). Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

22 Formalization Events, Facts, and Inaccessible propositions Definition Let (Ω P (L), R) be a pef, w Ω P (L) and ϕ SL. Then we say that a proposition [ϕ] is: A w-event iff w [ϕ], and for every w Ω P (L) such that w w and w [ϕ], R(w, w ). A w-fact iff w [ϕ]. A w-inaccessible proposition iff w [ϕ] and for every w such that w [ϕ], R(w, w ) For every w Ω P (L) we denote by E(w) and F(w) the classes of w-events, and w-facts respectively. Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

23 Results Theorem Let B = {[ϕ i ] = β i : i = 1,..., n} be a coherent w-book, let [ψ] be not a w-events, and let B = B {[ψ] = γ} be a propositional book extending B. Then the following hold: (1) If [ψ] is a w-facts, then B is coherent if and only if γ = w(ψ); (2) If [ψ] is w-inaccessible, then B is coherent if and only if γ = 0. Theorem Let (Ω P (L), R) be a fully accessible pef. Then for every w Ω P (L), E(w) is closed under the classical connectives. Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

24 Results Theorem Let B = {[ϕ i ] = β i : i = 1,..., n} be a coherent w-book, let [ψ] be not a w-events, and let B = B {[ψ] = γ} be a propositional book extending B. Then the following hold: (1) If [ψ] is a w-facts, then B is coherent if and only if γ = w(ψ); (2) If [ψ] is w-inaccessible, then B is coherent if and only if γ = 0. Theorem Let (Ω P (L), R) be a fully accessible pef. Then for every w Ω P (L), E(w) is closed under the classical connectives. Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8

25 Results Future Work (Probability!) 1 for each w, and for each w-event [ϕ], the a w-probability P w ([ϕ]) is the probability of [ϕ] conditioned by the w-facts. 2 We can give an answer to the de Finetti-Bayes principle: All probability is conditional on what we already know to be the case. But distinguishing between the probability of a conditional event, and the probability of an event conditioned by a fact! 3 We are formalizing the following intuition by de Finetti: Every [...] probability, is conditional; not only on the mentality or psychology of the individual involved, at the time in question, but also, and especially, on the state of information in which he finds himself at that moment Flaminio, Hosni (IIIA-SNS) CILC - Roma, June 2012 CILC / 8