=Paper=
{{Paper
|id=None
|storemode=property
|title=The
Epistemic Structure of de Finetti's Betting Problem
|pdfUrl=https://ceur-ws.org/Vol-857/paper_s02.pdf
|volume=Vol-857
|dblpUrl=https://dblp.org/rec/conf/cilc/FlaminioH12
}}
==The
Epistemic Structure of de Finetti's Betting Problem==
https://ceur-ws.org/Vol-857/paper_s02.pdf
The epistemic structure of de Finetti’s betting
problem
Tommaso Flaminio1 and Hykel Hosni2
1
IIIA - CSIC
Campus de la Univ. Autònoma de Barcelona s/n 08193 Bellaterra, Spain.
Email: tommaso@iiia.csic.es
2
Scuola Normale Superiore,
Piazza dei Cavalieri 7 Pisa, Italy.
Email hykel.hosni@sns.it
Abstract. De Finetti’s conception of events is one of the most distinc-
tive aspects of his theory of probability, yet it appears to be somewhat
elusive. The purpose of this note is to set up a formal framework in
which a rigorous characterisation of this notion, and its cognate mod-
elling assumptions, gives rise to a detailed formalisation of the betting
problem which underlies the celebrated Dutch Book Argument. As our
main result shows, this refinement captures an intuitive condition which
de Finetti imposed on the betting problem, namely that it is irrational
to bet on an event which may be true, but whose truth will never be
ascertained by the players.
Keywords: Uncertain reasoning, Events; De Finetti’s betting problem;
Partial valuations; Epistemic structures.
1 Introduction and motivation
Let E1 , . . . , En be of events of interest. De Finetti’s betting problem is the choice
that an idealised agent called bookmaker must make when publishing a book,
i.e. when making the assignment B = {(Ei , βi ) : i = 1, . . . , n} such that each
Ei is given value βi ∈ [0, 1]. Once a book has been published, a gambler can
place bets on event Ei by paying αi βi to the bookmaker. In return for this
payment, the gambler will receive αi , if Ei occurs and nothing otherwise. De
Finetti constructs the betting problem in such a way as to force the bookmaker
to publish fair betting odds for book B. To this end, two modelling assumptions
are built into the problem, namely that (i) the bookmaker is forced to accept
any number of bets on B and (ii) when betting on Ei , gamblers can choose the
sign, as well as the magnitude of (monetary) stakes αi . Conditions (i-ii) force
the bookmaker to publish books with zero-expectation, for doing otherwise may
offer the gambler the possibility of making a sure profit, possibily by choosing
negative stakes thereby unilaterally imposing a payoff swap to the bookmaker. As
the game is zero-sum, this is equivalent to forcing the bookmaker into sure loss.
In this context, de Finetti proves that the axioms of probability are necessary
and sufficient to secure the bookmaker against this possibility.
The crux of the Dutch book argument is the identification of the agent’s
degrees of belief with the price they are willing to pay for an uncertain reward
which depends on the future truth value of some presently unknown proposi-
tions – the events on which the agents are betting. This clearly suggests that
the semantics of events, which bears directly on the definition of probability,
is implicitly endowed with an epistemic structure. The purpose of this paper is
to give this structure an explicit formal characterisation and to show how the
resulting framework helps us making de Finetti’s elusive notion of event much
clearer. In particular we shall be able to give a formal setting to the following
remark:
[T]he characteristic feature of what I refer to as an “event” is that the
circumstances under which the event will turn out to be “verified” or
“disproved” have been fixed in advance. [1]
1.1 Formal preliminaries
Let L = {q1 , . . . , qn } 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. We use ⊥ to denote falsum. Valuations
are mappings ω form L into {0, 1} that naturally extend to SL by the truth-
functionality of the propositional connectives (with the usual stipulation that
ω(⊥) = 0, for all valuations ω). We denote by Ω(L) the class of all valuations
over L.
A partial valuation on L is a map ν : X ⊆ L → {0, 1} which, for ω ∈ Ω and
for every q ∈ L is defined by
ω(q) if q ∈ X;
ν(q) =
undefined otherwise.
The class of all partial valuations over L is denoted by Ω P (L). For ν : X → {0, 1}
and µ : Y → {0, 1} in Ω P (L) we say that µ extends ν (and we write ν ⊆ µ) if
X ⊆ Y , and for every x ∈ X, ν(x) = µ(x).
Finally, for every formula ϕ we set [ϕ] = {ψ ∈ SL : `ϕ ↔ ψ}, where as
usual, ` denotes the classical provability relation. We conform to the custom of
referring to the equivalence classes [ϕ] as to the proposition ϕ.
Note that de Finetti’s notion of event is captured by propositions, rather than
sentences. In fact, in the present logical framework, the “circumstances” under
which the propositions turn out to be true or false are nothing but the valuations
in Ω(L). As a consequence, de Finetti’s notion of book is formally defined by
(probability) assignments on a (finite) set of propositions. We will reserve the
expression propositional books to refer to these particular assignments.
Definition 1. Let ϕ ∈ SL, and let ν ∈ Ω P (L). (1) We say that ν realizes ϕ,
written ν ϕ, if ν(ϕ) is defined.
(2) We say that ν realizes [ϕ], written ν [ϕ], if there exists at least a ψ ∈ [ϕ]
such that ν ψ. In this case we assign ν(γ) = ν(ψ) for every γ ∈ [ϕ].
Let W be a finite set of nodes interpreted, as usual, as possible worlds. Let
e : W → Ω P (L) such that for every w ∈ W , e(w) = νw : Xw ⊆ L → {0, 1} is
a partial valuation. Note that to avoid cumbersome notation we will write νw
instead of e(w) to denote the partial valuation associated to w, and similarly, we
denote by Xw the subset of L for which νw is defined. Finally, let R ⊆ W × W
be an accessibility relation. We call a triplet (W, e, R) a partially evaluated frame
(pef for short).
Let w ∈ W and let [ϕ] be a proposition. We say that that w decides [ϕ], if
νw [ϕ].
Definition 2 (Events and Facts). Let (W, e, R) be a pef, w ∈ W and ϕ ∈ SL.
Then we say that a proposition [ϕ] is:
– A w-event iff νw 6 [ϕ], and for every ν ∈ Ω P (L) such that ν ⊇ νw and
ν [ϕ], there exists w0 such that R(w, w0 ), and νw0 = ν.
– A w-fact iff νw [ϕ].
For every w ∈ W we denote by E(w) and F(w) the classes of w-events, and
w-facts respectively.
An important consequence of Definition 2 is that in pef events (and facts)
are relativised to a specific state of the world.
A pef K = (W, e, R) is said to be monotonic if satisfies:
(M) for every w, w0 ∈ W , if R(w, w0 ), then νw ⊆ νw0 .
Recall that in de Finetti’s intuitive characterisation, at the time (i.e. world
in W ) at which the contract is signed, bookmaker and gambler agree on which
conditions will realise the events in the book. This clearly presupposes some
form of monotonic persistence of the underlying structure, which Property (M)
guarantees. In particular, in every monotonic pef, w-facts are w0 -facts in each
w0 which is accessible from w. In addition their truth value once determined, is
fixed throughout the frame.
Definition 3. Let (W, e, R) be any monotonic pef, and let w ∈ W . A w-book
is a propositional book B = {[ϕi ] = βi : i = 1, . . . , n} where the propositions [ϕi ]
are w-events.
Finally, we say that a pef (W, e, R) is complete if the following property is
satisfied:
(C) for every ϕ ∈ SL and for every ν ∈ Ω P such that ν [ϕ], there exists a
w[ϕ] ∈ W such that ν = νw[ϕ] [ϕ], i.e. [ϕ] is a w[ϕ] -fact.
Definition 4 (Inaccessible propositions). Let (W, e, R) be a pef and w ∈
W . A proposition [ϕ] is said to be w-inaccessible if νw 6 [ϕ] and for every w0
such that νw0 [ϕ], ¬R(w, w0 ).
Inaccessible propositions relativise the betting problem to the specific informa-
tion available to the gambler and the bookmaker. This is clearly in consonance
with de Finetti’s rather strict subjectivism according to which all that matters
for the determination of the final payoff is that the agents agree on which events
are realised.
2 No bets on inaccessible propositions
Our first result singles out the conditions under which a coherent w-book B can
be extended, either by w-facts or by w-inaccessible propositions to a coherent
propositional book B 0 .
Theorem 1. Let B = {[ϕi ] = βi : i = 1, . . . , n} be a coherent w-book, let
[ψ1 ], . . . , [ψr ] be propositions that are not w-events, and let B 0 = B ∪ {[ψj ] = γj :
j = 1, . . . , r} be a propositional book extending B. Then the following hold:
(1) If all propositions [ψj ] are w-facts, then B 0 is coherent if and only if for
every j = 1, . . . , r, γj = νw (ψj );
(2) If all propositions [ψj ] are w-inaccessible, then B 0 is coherent if and only if
for every j = 1 . . . , r, γj = 0.
Proof. We only prove the direction from left-to-right, the converse being is im-
mediate in both cases.
(1) Suppose, to the contrary, that exists j such that, γj 6= νw (ψj ), and in partic-
ular suppose that νw (ψj ) = 1, so that γj < 1. Then, the gambler can secure sure
win by betting a positive α on ψj . In this case in fact, since the pef is monotonic
by the definition of w-book, νw0 (ϕi ) = 1 holds in every world w0 which is acces-
sible from w. Thus the gambler pays α · γj in order to surely receive α in any
w0 accessible from w. Conversely, if νw (ψj ) = 0, then γj > 0 and in that case
it is easy to see that a sure-winning choice for the gambler consists in swapping
payoffs with the bookmaker, i.e. to bet a negative amount of money on [ψj ].
(2) As above suppose to the contrary that γj > 0 for some j, and that the
gambler bets −α on [ψj ]. By contract, this means that the bookmaker must pay
α · γj to the gambler, thus incurring sure loss, since [ψj ] will not be decided in
any world w0 such that R(w, w0 ).
Theorem 1 captures the key property identified by de Finetti in his informal
characterisation of events, namely that no monetary betting is rational unless
the conditions under which the relevant events will be decided are known to the
bookmaker and the gambler.
3 The language of w-events
A gambler and a bookmaker interpreted on a complete and monotonic pef are
guaranteed that: (1) as soon as a proposition is realized in w, it will stay so across
the accessible worlds from w (monotonicity), and (2) for every sentence ϕ, there
exists a world w that realizes [ϕ] (completeness). In accordance with the above
informal discussion of the betting problem, not only gamblers and bookmaker
must agree that a world w in which the events of interest are realized exists. They
also must agree on the conditions under which this will happen, as captured by
Theorem 1.
Example 1 ([2]). Consider an electron , and a world w. We are interested the
position and the energy of at w. Let [ϕ] and [ψ] be the propositions expressing
those measurements, respectively. Moreover let us assume that both [ϕ] and [ψ]
are w-events. Indeed if at w we are uncertain about the position and the energy
of , we can certainly perform experiments to determine them. But, what about
[ϕ] ∧ [ψ]? Position and energy are represented by non-commuting operators in
quantum theory, and we can assign an electron a definite position and a definite
energy, but not both. This fact can be modelled in complete and monotonic pef
K, by forcing [ϕ] ∧ [ψ] = [ϕ ∧ ψ] to be w-inaccessible.
Definition 5. A pef (W, e, R) is fully accessible if R satisfies:
(A) for all w, w0 ∈ W , if νw ⊆ νw0 , then R(w, w0 ).
It is customary [3] to characterise the coherence of a propositional book B in
terms of its extension to a (finitely additive) probability measure on the Boolean
algebra spanned by the events in B. As our second result shows, completeness
and full accessibility are sufficient conditions in order for the algebra generated
by the w-events in a w-book B to contain only w-events, and therefore, situations
like that of Example 1 cannot be modelled in this context.
Theorem 2. Let K = (W, e, R) be a complete pef. If K is fully accessible, then
for every w ∈ W , E(w) is closed under the classical connectives.
Proof. Let w be any world such that E(w) 6= ∅ and let [ϕ1 ] and [ϕ2 ] be w-
events. Without loss of generality we can assume that ϕ1 and ϕ2 are written
in conjunctive normal form. We want to prove that [ϕ1 ] ∧ [ϕ2 ] = [ϕ1 ∧ ϕ2 ] is a
w-event. Clearly νw 6 [ϕ1 ∧ ϕ2 ], and hence we only need to show that for every
partial valuation ν 0 ∈ Ω P such that ν 0 ⊇ νw , and ν 0 [ϕ1 ∧ ϕ2 ], there exists a
world w∗ such that R(w, w∗ ), and ν 0 = νw∗ .
For every ν 0 ∈ Ω P such that ν 0 [ϕ1 ∧ ϕ2 ], (C) ensures the existence of a
∗
w (that we would have denoted w[ϕ1 ∧ϕ2 ] using the terminonlogy of (C)) such
that ν ∗ [ϕ1 ∧ ϕ2 ], and νw ⊆ νw∗ = ν 0 . Since (W, e, R) satisfies (A), νw ⊆ νw∗
ensures R(w, w∗ ) and hence our claim is settled.
Acknowledgment
Flaminio acknowledges partial support from the Spanish projects TASSAT (TIN2010-
20967-C04-01), ARINF (TIN2009-14704-C03-03), as well as from the Juan de la Cierva
Program of the Spanish MICINN.
References
1. B. de Finetti. Philosophical lectures on probability. Springer Verlag, 2008.
2. F. Hellmann, M. Mandragon, A. Perez, C. Rovelli. Multiple-event probability in
general-relativistic quantum mechanics. Physical Review D 75, 084033, 2007. gr-
qc/0610140
3. J.B. Paris. The uncertain reasoner’s companion: A mathematical perspective. Cam-
bridge University Press, 1994.