betting, imprecise probabilities and ukasiewicz logickeimel/papers/malcev11a.pdf · ob saja teorija...

Betting, imprecise probabilities and Lukasiewicz logic

Klaus KeimelDepartment of Mathematics

Technische Universitat Darmstadt(joint work with M. Fedel, F. Montagna, W. Roth)

www.mathematik.tu-darmstadt.de/ keimel

October 11, 2011

Klaus Keimel Betting, imprecise probabilities and Lukasiewicz logic

What is the probability thatSergei Goncharov will live for 60 more years?

When preparing this talk, I was surprised to discover that in thepast logicians were intrigued by the foundations of probabilitytheory.

What is the probability thatSergei Goncharov will live for 60 more years?

When preparing this talk, I was surprised to discover that in thepast logicians were intrigued by the foundations of probabilitytheory.

George Boole 1853

An Investigation of theLaws of Thought

on which are foundedThe Mathematical Theories of Logic and Probability

George Boole 1853

An Investigation of theLaws of Thought

on which are foundedThe Mathematical Theories of Logic and Probability

Jan Lukasiewicz 1913

Die logischen Grundlagen der Wahrscheinlichkeitsrechnung,Krakow, 75 pages. English translation in:L. Borkowski (ed.), Selected Works. North-Holland 1970

”This remarkable agreement suggests that probability propositionsare nothing else than indefinite propositions, while probabilityfractions are their truth values. This supposition becomes acertainty when we find that all the difficulties thus far attendingthe laying of the logical foundations of probability theory can beremoved only on the basis of the interpretation of probabilityoffered in this paper.”

Richard von Mises 1928

Wahrscheinlichkeit, Statistik und Wahrheit, Springer-Verlag, Wien

”Mit anderen Worten: fur uns ist die Wahrscheinlichkeitsrechnungeine ganz normale Wissenschaft, gekennzeichnet durch denbesonderen Gegenstand, mit dem sie sich beschaftigt, aber nichtdurch eine besonders geartete Methode des Denkens oder desSchließens.”

Andrei Nikolajewitsch Kolmogoroff 1933

Grundbegriffe der Wahrscheinlichkeitsrechnung.Springer Verlag, Berlin (1933).

Obsaja teorija mery i iscislenie verojatnostei, TrudyKommunist.Akademii. Matematika. – M.: 1929, T. 1, S. 8-21.“The theory of probability, as a mathematical discipline, can andshould be developed from axioms in exactly the same way asGeometry and Algebra.”“There are other postulational systems of the theory of probability,particularly those in which the the concept of a probability is nottreated as one of the basic concepts, but is itself expressed bymeans of other concepts. However, in that case the aim isdifferent, namely, to tie up as closely as possible the mathematicaltheory with the empirical development of the theory of probability.”Sulla forma generale di un processo stocastico omogeneo (Unproblema di Bruno de Finetti) (1932)

Grundbegriffe der Wahrscheinlichkeitsrechnung.Springer Verlag, Berlin (1933).Obsaja teorija mery i iscislenie verojatnostei, TrudyKommunist.Akademii. Matematika. – M.: 1929, T. 1, S. 8-21.

“The theory of probability, as a mathematical discipline, can andshould be developed from axioms in exactly the same way asGeometry and Algebra.”“There are other postulational systems of the theory of probability,particularly those in which the the concept of a probability is nottreated as one of the basic concepts, but is itself expressed bymeans of other concepts. However, in that case the aim isdifferent, namely, to tie up as closely as possible the mathematicaltheory with the empirical development of the theory of probability.”Sulla forma generale di un processo stocastico omogeneo (Unproblema di Bruno de Finetti) (1932)

Grundbegriffe der Wahrscheinlichkeitsrechnung.Springer Verlag, Berlin (1933).Obsaja teorija mery i iscislenie verojatnostei, TrudyKommunist.Akademii. Matematika. – M.: 1929, T. 1, S. 8-21.“The theory of probability, as a mathematical discipline, can andshould be developed from axioms in exactly the same way asGeometry and Algebra.”

“There are other postulational systems of the theory of probability,particularly those in which the the concept of a probability is nottreated as one of the basic concepts, but is itself expressed bymeans of other concepts. However, in that case the aim isdifferent, namely, to tie up as closely as possible the mathematicaltheory with the empirical development of the theory of probability.”Sulla forma generale di un processo stocastico omogeneo (Unproblema di Bruno de Finetti) (1932)

Grundbegriffe der Wahrscheinlichkeitsrechnung.Springer Verlag, Berlin (1933).Obsaja teorija mery i iscislenie verojatnostei, TrudyKommunist.Akademii. Matematika. – M.: 1929, T. 1, S. 8-21.“The theory of probability, as a mathematical discipline, can andshould be developed from axioms in exactly the same way asGeometry and Algebra.”“There are other postulational systems of the theory of probability,particularly those in which the the concept of a probability is nottreated as one of the basic concepts, but is itself expressed bymeans of other concepts. However, in that case the aim isdifferent, namely, to tie up as closely as possible the mathematicaltheory with the empirical development of the theory of probability.”

Sulla forma generale di un processo stocastico omogeneo (Unproblema di Bruno de Finetti) (1932)

Grundbegriffe der Wahrscheinlichkeitsrechnung.Springer Verlag, Berlin (1933).Obsaja teorija mery i iscislenie verojatnostei, TrudyKommunist.Akademii. Matematika. – M.: 1929, T. 1, S. 8-21.“The theory of probability, as a mathematical discipline, can andshould be developed from axioms in exactly the same way asGeometry and Algebra.”“There are other postulational systems of the theory of probability,particularly those in which the the concept of a probability is nottreated as one of the basic concepts, but is itself expressed bymeans of other concepts. However, in that case the aim isdifferent, namely, to tie up as closely as possible the mathematicaltheory with the empirical development of the theory of probability.”Sulla forma generale di un processo stocastico omogeneo (Unproblema di Bruno de Finetti) (1932)

Bruno De Finetti 1937

La prevision: ses lois logiques, ses sources subjectives.Annales de l’Institut Henri Poincare, tome 7, pages 1–68

De Finetti’s reversible betting scheme

The bookmaker K(laus) offers bets on a set A of events A thatmay or may not occur in the future. Put

v(A) =

{1 if A occurs

0 if A does not occur

Events are expressed by propositions. Events may be composedfrom other events by means of ’and’, ’or’,’not’. Thus, A can beseen as a subset of a Boolean algebra B of propositions (modlogical equivalence). v will be a valuation, a homomorphism of Bto the two-element Boolean algebra.

De Finetti’s reversible betting scheme

The bookmaker K(laus) offers bets on a set A of events A thatmay or may not occur in the future. Put

v(A) =

{1 if A occurs

0 if A does not occur

Events are expressed by propositions. Events may be composedfrom other events by means of ’and’, ’or’,’not’. Thus, A can beseen as a subset of a Boolean algebra B of propositions (modlogical equivalence). v will be a valuation, a homomorphism of Bto the two-element Boolean algebra.

De Finetti betting scheme (ctd.)

The bookmaker K chooses a set A of events and, for every A ∈ A,a real number p(A) between 0 and 1, his subjective prevision forthe probability that the event will occur. (A, p) is called a book.

The bettor S(ergei) chooses events A1, . . . ,An ∈ Aand he bets amounts (of rubles) he wants to win r1, . . . rn ∈ RHe has to pay to K the sum p(A1)r1 + · · ·+ p(An)rn.

After the results are known, S will receive from K for the first betthe amount r1 if A1 occurs, 0 if A1 does not occur, etc.Total amount paid out by K to S: v(A1)r1 + · · ·+ v(An)rn

Thus, at the end, the balance (gain or loss) of S will be

Payoff =n∑

(v(Ai )− p(Ai )) · ri

The bettor S(ergei) chooses events A1, . . . ,An ∈ Aand he bets amounts (of rubles) he wants to win r1, . . . rn ∈ RHe has to pay to K the sum p(A1)r1

+ · · ·+ p(An)rn.

Payoff =n∑

(v(Ai )− p(Ai )) · ri

Payoff =n∑

(v(Ai )− p(Ai )) · ri

After the results are known, S will receive from K for the first betthe amount r1 if A1 occurs, 0 if A1 does not occur, etc.Total amount paid out by K to S: v(A1)r1

+ · · ·+ v(An)rn

Payoff =n∑

(v(Ai )− p(Ai )) · ri

Payoff =n∑

(v(Ai )− p(Ai )) · ri

Payoff =n∑

(v(Ai )− p(Ai )) · ri

Example

In the first Italian football league AC Milano will end up

events

firstfirst or secondfirst or third

Payoff

previsions

1/25/65/6

1−1−1

all possible outcomes

0 1 0 00 1 1 00 1 0 1

7/6 1/6 1/6 1/6

The Payoff for S is strictly positive no matter what will happen,that is, S has a sure win strategy.

Example

events

Payoff

previsions

1/25/65/6

1−1−1

0 1 0 00 1 1 00 1 0 1

7/6 1/6 1/6 1/6

Example

events

Payoff

previsions

1/25/65/6

1−1−1

0 1 0 00 1 1 00 1 0 1

7/6 1/6 1/6 1/6

Example

events

Payoff

previsions

1/25/65/6

1−1−1

0 1 0 00 1 1 00 1 0 1

7/6 1/6 1/6 1/6

Example

events

Payoff

previsions

1/25/65/6

1−1−1

0 1 0 00 1 1 00 1 0 1

7/6 1/6 1/6 1/6

Coherence

The book (A, p) is incoherentif S has a sure win strategy,

that is,if there are finitely many events A1, . . . ,An in A and real numbersr1, . . . , rn such that, for every valuation v ,

Payoff =n∑

(v(Ai )− p(Ai )) · ri > 0

De Finetti’s Theorem

The book (A, p) is coherent⇐⇒ there is a finitely additive probability measure p on the

Boolean algebra B such that p(A) = p(A) for all a ∈ A.

Coherence

The book (A, p) is incoherentif S has a sure win strategy, that is,if there are finitely many events A1, . . . ,An in A and real numbersr1, . . . , rn such that, for every valuation v ,

Payoff =n∑

(v(Ai )− p(Ai )) · ri > 0

Coherence

The book (A, p) is incoherentif S has a sure win strategy, that is,if there are finitely many events A1, . . . ,An in A and real numbersr1, . . . , rn such that, for every valuation v ,

Payoff =n∑

(v(Ai )− p(Ai )) · ri > 0

Irreversible betting games

A game in which positive and negative betting odds are allowed iscalled reversible. We now consider the more realistic situation ofan irreversible betting game: We have the same game as beforeexcept for restricting betting odds to positive numbers ri > 0.

Now, sure win strategies do not exist any more. Nevertheless, notevery book should be rationnal or coherent.

New coherence condition?

Bad bets

Betting (say r > 0) on an event A ∈ A is a bad bet for S,if there is another bet that is better whatever will happen,

that is,if there are A1, . . . ,An ∈ A and betting odds r1, . . . , rn > 0 suchthat, for every valuation v ,

n∑i=1

(v(Ai )− p(Ai )) · ri > (v(A)− p(A)) · r

In a reversible game, the following are equivalent:— There is no sure win strategy.— There is no sure loss strategy.— There is no bad bet.— There is no good bet.

Bad bets

Betting (say r > 0) on an event A ∈ A is a bad bet for S,if there is another bet that is better whatever will happen,that is,if there are A1, . . . ,An ∈ A and betting odds r1, . . . , rn > 0 suchthat, for every valuation v ,

n∑i=1

(v(Ai )− p(Ai )) · ri > (v(A)− p(A)) · r

Bad bets

Betting (say r > 0) on an event A ∈ A is a bad bet for S,if there is another bet that is better whatever will happen,that is,if there are A1, . . . ,An ∈ A and betting odds r1, . . . , rn > 0 suchthat, for every valuation v ,

n∑i=1

(v(Ai )− p(Ai )) · ri > (v(A)− p(A)) · r

Imprecise probabilities

A set Q of (finitely additive) probability measures on B may beconsidered to be an imprecise probability. With such a set weassociate an upper probability p(B) = supp∈Qp(B)and a lower probability p(B) = infp∈Qp(B).The interval [p(B), p(B)] represent the imprecise probability of B.

Theorem

In an irreversible game with a book (A, p):There is no bad bet for S ⇐⇒ there is an upper probability p onB such that p(A) = p(A) for all A ∈ A.

There is no good bet for S ⇐⇒ there is a lower probability p onB such that p(A) = p(A) for all A ∈ A.

There neither a good nor a bad bet for S ⇐⇒ there is aprobability measure p on B such that p(A) = p(A) for all A ∈ A.

Theorem

Questions

1. In the previous results, probabilities were finitely additive. Butin probability theory they are usually supposed to be countablyadditive?

2. Different sets Q of probabilities may represent the same’imprecise probability’?

3. Is there a characterization of upper and lower probabilitieswithout referrring to sets of probabilities?

We will address these question in a more general setting.

Daniele Mundici 2006

generalized De Finettis result in another direction: Admit (truth)values between 0 and 1 and use Lukasiewicz propositional logic.

On the unit interval [0, 1] interpret negation and disjunction by

¬r = 1− r , r ⊕ s = min(r + s, 1)

Boolean algebras are generalized to MV-algebras:Let (G , u) be a unital `-group, that is, a lattice-ordered abeliangroup with strong order unit u, that is an element such that forevery element g ∈ G there is a natural number n such thatn · u ≥ g .

Then the ’unit interval’ [0, u] = {g ∈ G | 0 ≤ g ≤ u} with theoperations

¬g = u − g , g ⊕ h = (g + h) ∧ u

is an MV-algebra and G is called the enveloping unital `-group.Klaus Keimel Betting, imprecise probabilities and Lukasiewicz logic

Valuations on MV-algebras

A valuation on a unital `-group (G , u) is an `-grouphomomorphism v : G → R such that v(u) = 1.

Restricting a valuation on G to its unit interval yields anMV-algebra homomorphim [0, u]→ [0, 1], an MV-algebravaluation. Every MV-algebra homomorphim [0, u]→ [0, 1] is therestriction of a unique valuation on G (Mundici).

The space of valuations

The set XG of all valuations on G (∼= the set of all MV-algebravaluations on [0, u]) is a compact Hausdorff space for the topologyof pointwise convergence.

The Yoshida Representation

For a compact Hausdorff spaxe X , the set C (X ) of all continuousreal-valued functions on X is an `-group, the constant function 1being a strong unit. The interval [0, 1] of functons f with0 ≤ f ≤ 1 is an MV-algebra.

To every element g ∈ G we assign its Gelfand transform

g : XG → R defined by g(v) = v(g) for all v ∈ XG

yielding an `-homomorphism g 7→ g : G → C (XG ), mapping theunit u of G to the constant function 1.

Restricting this `-homomorphism to the MV-algebra [0, u] yields anMV-algebra homomorphism from [0, u] into [0, 1] ⊆ C (XG ).

Boolean algebras as MV-algebras

Every Boolean algebra B is an MV-algebra.

The valuations are the homomorphisms v from B onto the twoelement Boolean algebra {0, 1} ⊆ [0, 1]. The space of valuations isthe Stone space XB.

The enveloping lattice-ordered group of B is C (XB,Z), thelattice-ordered group of all continuous functions f : XB → Z.

The Yoshida representation reduces to the Stone representation ofB by the set of all continuous maps from XB to the discrete twoelement space {0, 1}.

Finitely additive probability measures on B ∼= states

A state on a unital `-group is a positive homomorphism s : G → Rwith s(u) = 1. A state on the MV-algebra [0, u] is an additive maps : [0, u]→ [0, 1] such that s(u) = 1. (Restricting states s on G to[0, u] yields a bijection between states on G and states on [0, u].)

Riesz representation theorem

For a compact Hausdorf space X , there is a canonical bijectionbetween regular Borel probability measures and states on C (X )(= positive linear functionals s on C (X ) with s(1) = 1).

The Yoshida representation and Stone-Weierstraß yield:

Generalized Riesz representation theorem

There is a canonical bijection betweenregular Borel probability measures on the valuation space XG andstates of a unital `-group G (states of the MV-algebra[0, u]).

The state space SG

The states on a unital `-group (on an MV-algebra [0, u]) form acompact convex subset SG of RG for the topology of pointwiseconvergence (affinely homeormorphic to the compact convex set ofregular Borel probability measures on XG with the vague topology).

Imprecise Probabilities

Imprecise probability ∼= set Q of prob. measures ∼= set Q of states

Imprecise probabilty ∼= upper state

For every nonempty set Q of states, the function s : G → R definedby s(g) = sups∈Q s(g) for all g ∈ G is an upper state on G .

An upper state on the MV-algebra [0, u] is by definition therestriction of an upper state to the MV-algebra [0, u].

Lower states s are defined analogously replacing sup by inf.Lower and upper states are conjugate in the sense thats(g) = 1− s(u − g).

Imprecise probability ∼= compact convex set of states.

Two sets of states represent the same imprecise probability in thesense that they define the same upper (lower) state if and only ifthe have the same closed convex hull in the state space.

Functional characterization of upper states

There is a greatest upper state SUP, the supremum of all states:SUP(g) = sups∈SG s(g) = supv∈XG

v(g) = maxv∈XGg(v).

Theorem

For a real-valued function s on a unital `-group, the following areequivalent:

— s is an upper state.

— s is subadditive: s(g + h) ≤ s(g) + s(h),N-homogeneous: s(ng) = ns(g) for n ∈ Nupper bounded: s(g) ≤ SUP(g).

— s is subadditive, N-homogeneous, order preserving andstrongly normalized: s(u) = 1 , s(−u) = −1,

For MV-algebras in general I do not know a similar characterizationof functionals that are upper states. It works under a divisibilityhypothesis: An MV-algebra [0, u] is 2-divisible if for everyg ∈ [0, u] there is an h such that g = h + h.

Theorem

For a function s : [0, u]→ [0, 1] on a 2-divisible MV-algebra, thefollowing are equivalent:

— s is an upper state.

— s is subadditive, N-homogeneous, order preserving andstrongly normalized: s(g + 1

2n ) = s(g) + 12n ,

Final result

Theorem

For an irreversible betting game with book (A, p) with events in anMV-algebra,

There is no bad bet for S ⇐⇒there is an upper state s such that p(A) = s(A) for all A ∈ A.

There is no good bet for S ⇐⇒there is an lower state s such that p(A) = s(A) for all A ∈ A.

There is neither a good nor a bad bet for S ⇐⇒there is a state s such that p(A) = s(A) for all A ∈ A.

The proof uses similar results from Peter Walley Reasoning withImprecise Probabilities, Monographs on Statistics and AppliedProbability, Chapman and Hall, London, 1991.

Conclusion

What is the probablity that Sergei will live for 60 more years?

As a bookmaker, I propose my subjective upper probability p = 12

in this irreversible game. What are you ready to bet?

In this irreversible game I take any positive betting add.

But I will win almost surely, as there is a very very high probabilitythat I will not live any more to pay you out.

Conclusion

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