Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic.


 Melvin Montgomery
 2 years ago
 Views:
Transcription
1 Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Martina Fedel joint work with K.Keimel,F.Montagna,W.Roth Martina Fedel (UNISI) 1 / 32
2 Goal The goal of this talk is to extend de Finetti s theorem about coherent probability assignments to events of manyvalued logic characterized by a condition of uncertainty. We axiomatically define imprecise probabilities (upper and lower probabilities) and we characterize them in terms of states. Is it possible to give an interpretation in terms of bets of imprecise probabilities? Can we prove a de Finettistyle coherence criterion for imprecise probabilities? Martina Fedel (UNISI) 2 / 32
3 Goal The goal of this talk is to extend de Finetti s theorem about coherent probability assignments to events of manyvalued logic characterized by a condition of uncertainty. We axiomatically define imprecise probabilities (upper and lower probabilities) and we characterize them in terms of states. Is it possible to give an interpretation in terms of bets of imprecise probabilities? Can we prove a de Finettistyle coherence criterion for imprecise probabilities? Martina Fedel (UNISI) 2 / 32
4 Outline 1 Preliminaries 2 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Tools from functional analysis Imprecise probabilities and sets of states 3 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets An adequate coherence criterion Martina Fedel (UNISI) 3 / 32
5 Preliminaries Definition An MValgebra is an algebra A = (A,,, 0) where: (A,, 0) is a commutative monoid, is an involutive unary operation 1 = 0 is an element such that a 1 = 1, a b = ( a) b, is such that (a b) b = (b a) a. In any MValgebra A, we further define a b = ( a b), a b = ( a b) Martina Fedel (UNISI) 4 / 32
6 Preliminaries Definition A valuation on an MValgebra A is an homomorphism v : A [0, 1]. Proposition Let X A the set of all valuations on the MValgebra A. X A is a close subset of [0, 1] A, thus an Hausdorff compact space with respect to the topology of pointwise convergence. Martina Fedel (UNISI) 5 / 32
7 Preliminaries Remark In a rational bettinggame one will always identify events that have a priori the same value under every possible valuation, hence events will be the terms of the LindenbaumTarski algebra of the Łukasiewicz logic, denoted by Fm L / L. Proposition Fm L / L is an MValgebra. Thus events may be regarded as elements of an MValgebra. Martina Fedel (UNISI) 6 / 32
8 Preliminaries Remark For technical reasons, it is convenient to consider a wider class of events and to work in 2divisible MValgebras, i.e. MValgebras in which we can multiply every element by 1 2. Problem: why can we do this? Every MValgebra A is embeddable in a 2divisible one. Fuzzy events may be regarded as continuous functions from the Hausdorff compact space X A in [0,1]. Thus all dyadic numbers q = m 2 in the 2divisible MValgebra A n may be regarded as consant events, i.e. events that have a priori the same value under every possible valuation. Martina Fedel (UNISI) 7 / 32
9 Preliminaries Remark For technical reasons, it is convenient to consider a wider class of events and to work in 2divisible MValgebras, i.e. MValgebras in which we can multiply every element by 1 2. Problem: why can we do this? Every MValgebra A is embeddable in a 2divisible one. Fuzzy events may be regarded as continuous functions from the Hausdorff compact space X A in [0,1]. Thus all dyadic numbers q = m 2 in the 2divisible MValgebra A n may be regarded as consant events, i.e. events that have a priori the same value under every possible valuation. Martina Fedel (UNISI) 7 / 32
10 Outline A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras 1 Preliminaries 2 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Tools from functional analysis Imprecise probabilities and sets of states 3 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets An adequate coherence criterion Martina Fedel (UNISI) 8 / 32
11 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Definition A state on the MValgebra A is a map P : A [0, 1] such that for all x, y A: (i) P(1) = 1 (ii) If x y = 0 then P(x y) = P(x) + P(y) Theorem (PantiKroupa) Let A an MValgebra. There is a canonical bijective correspondence between the set of states on A and the set of regular probability measures on the Hausdorff compact space X A. States play the same role for MValgebras as probabilities for Boolean algebras. Martina Fedel (UNISI) 9 / 32
12 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Definition A state on the MValgebra A is a map P : A [0, 1] such that for all x, y A: (i) P(1) = 1 (ii) If x y = 0 then P(x y) = P(x) + P(y) Theorem (PantiKroupa) Let A an MValgebra. There is a canonical bijective correspondence between the set of states on A and the set of regular probability measures on the Hausdorff compact space X A. States play the same role for MValgebras as probabilities for Boolean algebras. Martina Fedel (UNISI) 9 / 32
13 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Definition An upper probability over a 2divisible MValgebra A is a functional P + : A [0, 1] such that for all x, y A and for all dyadic rational numbers q = m 2 in the unit interval: n Definition (Ax1) is order preserving (Ax2) P + (1) = 1 (Ax3) P + (qx) = qp + (x) (Ax4) If x y = 0 then P + (x y) P + (x) + P + (y) (Ax5) If q x = 0 allora P + (q x) = q + P + (x) A lower probability over a 2divisible MValgebra A is a functional P : A [0, 1] with the same properties of P + except (Ax4) that becomes: (Ax4) If x y = 0 then P (x y) P (x) + P (y) Martina Fedel (UNISI) 10 / 32
14 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Definition An upper probability over a 2divisible MValgebra A is a functional P + : A [0, 1] such that for all x, y A and for all dyadic rational numbers q = m 2 in the unit interval: n Definition (Ax1) is order preserving (Ax2) P + (1) = 1 (Ax3) P + (qx) = qp + (x) (Ax4) If x y = 0 then P + (x y) P + (x) + P + (y) (Ax5) If q x = 0 allora P + (q x) = q + P + (x) A lower probability over a 2divisible MValgebra A is a functional P : A [0, 1] with the same properties of P + except (Ax4) that becomes: (Ax4) If x y = 0 then P (x y) P (x) + P (y) Martina Fedel (UNISI) 10 / 32
15 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Definition Let X an Hausdorff compact space. A functional u : C(X) R is: order preserving if f g implies u(f ) u(g) normalized if u(1) = 1 subadditive (resp.superadditive) if u(f + g) u(f ) + u(g) (resp. ) homogeneous if u(rf ) = ru(f ) for all f C(X) and all r 0, sublinear (resp. superlinear) if u is homogeneous and subadditive (resp. superadditive) linear if u is sublinear and superlinear. weakly linear if u(f + r 1) = u(f ) + r for all r R and for all f C(X). Martina Fedel (UNISI) 11 / 32
16 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Definition Let X an Hausdorff compact space. A functional u : C(X) R is: order preserving if f g implies u(f ) u(g) normalized if u(1) = 1 subadditive (resp.superadditive) if u(f + g) u(f ) + u(g) (resp. ) homogeneous if u(rf ) = ru(f ) for all f C(X) and all r 0, sublinear (resp. superlinear) if u is homogeneous and subadditive (resp. superadditive) linear if u is sublinear and superlinear. weakly linear if u(f + r 1) = u(f ) + r for all r R and for all f C(X). Martina Fedel (UNISI) 11 / 32
17 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Theorem Let A an MValgebra There is a canonical bijection between states on A and order preversing, normalized, homogeneous and linear functionals on C(X A ). If A is a 2divisible MValgebra There is a canonical bijection between upper (resp. lower) probabilities on A and order preversing, normalized, homogeneous, weakly linear and sublinear (resp. superlinear) functionals on C(X A ). Remark In the 2divisible case, we are in the position to apply classical tools of functional analysis to deal with states and imprecise probabilities on MValgebras thanks to their counterparts in the dual of C(X), where X is an Hausdorff compact space. Martina Fedel (UNISI) 12 / 32
18 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Theorem Let A an MValgebra There is a canonical bijection between states on A and order preversing, normalized, homogeneous and linear functionals on C(X A ). If A is a 2divisible MValgebra There is a canonical bijection between upper (resp. lower) probabilities on A and order preversing, normalized, homogeneous, weakly linear and sublinear (resp. superlinear) functionals on C(X A ). Remark In the 2divisible case, we are in the position to apply classical tools of functional analysis to deal with states and imprecise probabilities on MValgebras thanks to their counterparts in the dual of C(X), where X is an Hausdorff compact space. Martina Fedel (UNISI) 12 / 32
19 Outline A Representation for Imprecise Probabilities Tools from functional analysis 1 Preliminaries 2 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Tools from functional analysis Imprecise probabilities and sets of states 3 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets An adequate coherence criterion Martina Fedel (UNISI) 13 / 32
20 A Representation for Imprecise Probabilities Tools from functional analysis Let X an Hausdorff compact space and C(X) the dual vector space of C(X). We endow C(X) with the weak topology. Definition The weak topology is the weakest topology such that the evaluation maps µ µ(f ): C(X) R are continuous for all f C(X). Proposition Let P(X) the set of all order preserving, normalized and linear functionals on C(X). P(X) is a convex subset of M(X) which is compact in the weak topology. Martina Fedel (UNISI) 14 / 32
21 A Representation for Imprecise Probabilities Tools from functional analysis Let X an Hausdorff compact space and C(X) the dual vector space of C(X). We endow C(X) with the weak topology. Definition The weak topology is the weakest topology such that the evaluation maps µ µ(f ): C(X) R are continuous for all f C(X). Proposition Let P(X) the set of all order preserving, normalized and linear functionals on C(X). P(X) is a convex subset of M(X) which is compact in the weak topology. Martina Fedel (UNISI) 14 / 32
22 A Representation for Imprecise Probabilities Tools from functional analysis Proposition For every weak closed subset K of P(X), the functional K + : C(X) R defined by K + (f ) = max µ K µ(f ) is well defined. Theorem K + (f ) = max µ K µ(f ) is an order preserving, normalized, sublinear and weakly linear functional. Conversely, for every order preserving, normalized, sublinear and weakly linear functional u : C(X) R, u = {µ P(X) : µ u} is a weak closed convex subset of P(X). Martina Fedel (UNISI) 15 / 32
23 A Representation for Imprecise Probabilities Tools from functional analysis Proposition For every weak closed subset K of P(X), the functional K + : C(X) R defined by K + (f ) = max µ K µ(f ) is well defined. Theorem K + (f ) = max µ K µ(f ) is an order preserving, normalized, sublinear and weakly linear functional. Conversely, for every order preserving, normalized, sublinear and weakly linear functional u : C(X) R, u = {µ P(X) : µ u} is a weak closed convex subset of P(X). Martina Fedel (UNISI) 15 / 32
24 A Representation for Imprecise Probabilities Tools from functional analysis Theorem For every weak closed subset K of P(X) the functional defined by K (f ) = min µ K µ(f ) is an order preserving, normalized, superlinear and weakly linear functional. Conversly, for every order preserving, normalized, superlinear and weakly linear functional u : C(X) R, u = {µ P(X) : µ(f ) u(f ) for all f C(X)} is a nonempty weak closed convex subset of P(X). Theorem The maps K (K +, K ) and u (u, u ) are mutually inverse bijections. Martina Fedel (UNISI) 16 / 32
25 Outline A Representation for Imprecise Probabilities Imprecise probabilities and sets of states 1 Preliminaries 2 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Tools from functional analysis Imprecise probabilities and sets of states 3 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets An adequate coherence criterion Martina Fedel (UNISI) 17 / 32
26 A Representation for Imprecise Probabilities Imprecise probabilities and sets of states Theorem Let A a 2divisible MValgebra There is a canonical bijection between upper (resp. lower) probabilities on A and order preversing, normalized, homogeneous, sublinear (resp. superlinear) and weakly linear functionals on C(X A ). There is a canonical bijection between states on A and order preversing, normalized, homogeneous and linear functionals on C(X A ). Martina Fedel (UNISI) 18 / 32
27 A Representation for Imprecise Probabilities Imprecise probabilities and sets of states Theorem (Representation Theorem for Upper and Lower Probabilities) An upper probability is the maximum of a nonempty convex set of states, close with respect to the weak topology and viceversa. A lower probability is the minimum of a nonempty convex set of states, close with respect to the weak topology and viceversa. Remark Two sets A and B of states on an MValgebra A represent the same imprecise probability if and only if they have the same closed convex hull. Martina Fedel (UNISI) 19 / 32
28 A Representation for Imprecise Probabilities Imprecise probabilities and sets of states Theorem (Representation Theorem for Upper and Lower Probabilities) An upper probability is the maximum of a nonempty convex set of states, close with respect to the weak topology and viceversa. A lower probability is the minimum of a nonempty convex set of states, close with respect to the weak topology and viceversa. Remark Two sets A and B of states on an MValgebra A represent the same imprecise probability if and only if they have the same closed convex hull. Martina Fedel (UNISI) 19 / 32
29 Outline Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets 1 Preliminaries 2 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Tools from functional analysis Imprecise probabilities and sets of states 3 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets An adequate coherence criterion Martina Fedel (UNISI) 20 / 32
30 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets Example Let us suppose to know that the extraction of the bingo are biased. If we don t know how it is biased, the event A great number will be drawn out is a fuzzy event under particularly sever uncertainty. In this case, a bookmaker may not be willing to quantify with a number his subjective opinion that the event ϕ will happen. He will probably prefer associating to ϕ an interval of probability, choosing a betting odd β for betting on ϕ and a betting odd α β for betting against ϕ. Martina Fedel (UNISI) 21 / 32
31 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets Example Let us suppose to know that the extraction of the bingo are biased. If we don t know how it is biased, the event A great number will be drawn out is a fuzzy event under particularly sever uncertainty. In this case, a bookmaker may not be willing to quantify with a number his subjective opinion that the event ϕ will happen. He will probably prefer associating to ϕ an interval of probability, choosing a betting odd β for betting on ϕ and a betting odd α β for betting against ϕ. Martina Fedel (UNISI) 21 / 32
32 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets Definition A book that contains fuzzy events under particularly severe uncertainty is a finite set of the form = {(ϕ 1, [α 1, β 1 ]),..., (ϕ n, [α n, β n ])}, where for 0 α i β i 1. If α i = β i for all i = 1,..., n we are again in the case of reversible betting game. If α i < β i the conditions are disadvantageous for the bettor: if the bettor bets on φ he can win at the most λ(1 β) and he can loose up to λβ. if the bettor bets against φ he can win at the most λα and he can loose up to λ(1 α). Remark If the bookmaker is a rational being, he wants to protect himself from the risk to loose a huge amount of money. Thus he forbids negative bets because he doesn t want to reverse his role with the bettor. Martina Fedel (UNISI) 22 / 32
33 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets Definition A book that contains fuzzy events under particularly severe uncertainty is a finite set of the form = {(ϕ 1, [α 1, β 1 ]),..., (ϕ n, [α n, β n ])}, where for 0 α i β i 1. If α i = β i for all i = 1,..., n we are again in the case of reversible betting game. If α i < β i the conditions are disadvantageous for the bettor: if the bettor bets on φ he can win at the most λ(1 β) and he can loose up to λβ. if the bettor bets against φ he can win at the most λα and he can loose up to λ(1 α). Remark If the bookmaker is a rational being, he wants to protect himself from the risk to loose a huge amount of money. Thus he forbids negative bets because he doesn t want to reverse his role with the bettor. Martina Fedel (UNISI) 22 / 32
34 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets In order to give the rules for betting in a nonreversible betting game, firstly we replace every element (ϕ, [α, β]) of the book with the pairs (ϕ, β), ( ϕ, 1 α). If a bettor wants to bet on ϕ, he pays λβ with λ > 0 and he will recive λv(ϕ). If he wants to bet against ϕ, he bets on ϕ; hence he pays λ(1 α) with λ > 0 and he will recive λv( ϕ) where v is a valuation in [0, 1]. Definition (Subjective Approach) The upper probability of ϕ is the number β that a rational and nonreversible bookmaker would propose for the preceding bet. The lower probability of ϕ is instead the number α. Martina Fedel (UNISI) 23 / 32
35 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets In order to give the rules for betting in a nonreversible betting game, firstly we replace every element (ϕ, [α, β]) of the book with the pairs (ϕ, β), ( ϕ, 1 α). If a bettor wants to bet on ϕ, he pays λβ with λ > 0 and he will recive λv(ϕ). If he wants to bet against ϕ, he bets on ϕ; hence he pays λ(1 α) with λ > 0 and he will recive λv( ϕ) where v is a valuation in [0, 1]. Definition (Subjective Approach) The upper probability of ϕ is the number β that a rational and nonreversible bookmaker would propose for the preceding bet. The lower probability of ϕ is instead the number α. Martina Fedel (UNISI) 23 / 32
36 Outline Coherent Books and Imprecise Probabilities An adequate coherence criterion 1 Preliminaries 2 A Representation for Imprecise Probabilities States and Imprecise probabilities over MValgebras Tools from functional analysis Imprecise probabilities and sets of states 3 Coherent Books and Imprecise Probabilities An interpretation of imprecise probabilities in terms of bets An adequate coherence criterion Martina Fedel (UNISI) 24 / 32
37 Coherent Books and Imprecise Probabilities An adequate coherence criterion Definition Let a fixed book. A winning strategy (resp. a loosing strategy) for the bettor based on consists of a system of bets on a finite subset of events in which ensures to the bettor a strictly positive (resp. strictly negative) payoff indipendently of the valuation. A bad bet (resp. a good bet) is a bet on an element of for which we can find a system of bets on a finite subset of events in which ensures to the bettor a better payoff (resp. a worse payoff) independently of the valuation. Martina Fedel (UNISI) 25 / 32
38 Coherent Books and Imprecise Probabilities An adequate coherence criterion Coherence Criterion (Reversible bettinggame) A book is said to be rational or coherent if there is no winning strategy for the bettor. Theorem Let be a book in a reversible game. The following are equivalent: There is no winning strategy for the bettor. There is no loosing strategy for the bettor. There is no bad bet for the bettor. There is no good bet for the bettor. Martina Fedel (UNISI) 26 / 32
39 Coherent Books and Imprecise Probabilities An adequate coherence criterion Coherence Criterion (Reversible bettinggame) A book is said to be rational or coherent if there is no winning strategy for the bettor. Theorem Let be a book in a reversible game. The following are equivalent: There is no winning strategy for the bettor. There is no loosing strategy for the bettor. There is no bad bet for the bettor. There is no good bet for the bettor. Martina Fedel (UNISI) 26 / 32
40 Coherent Books and Imprecise Probabilities An adequate coherence criterion Example Let Γ = {(ϕ, 1 3 ), ( ϕ, 1), (ψ, 1 3 ), ( ψ, 1), (ϕ ψ, 1), ( (ϕ ψ), 1)} The bettor has no winning strategy for this book because if both ϕ and ψ are false, then he cannot win anything. This book does not look rational: the bookmaker might make it more attractive for the bettor by reducing the betting odd on ϕ ψ to 2 3, without any loss of money if the bettor plays his best strategy. Example Let a book that contains (ϕ, α), ( ϕ, β) with α + β > 1. The bettor has a loosing strategy, namely, betting 1 on both ϕ and ϕ. He has also a good bet, namely, betting nothing. This book cannot be considered irrational: it can be obtained from an element (ϕ, [.30,.60]) that is obviously acceptable. Martina Fedel (UNISI) 27 / 32
41 Coherent Books and Imprecise Probabilities An adequate coherence criterion Example Let Γ = {(ϕ, 1 3 ), ( ϕ, 1), (ψ, 1 3 ), ( ψ, 1), (ϕ ψ, 1), ( (ϕ ψ), 1)} The bettor has no winning strategy for this book because if both ϕ and ψ are false, then he cannot win anything. This book does not look rational: the bookmaker might make it more attractive for the bettor by reducing the betting odd on ϕ ψ to 2 3, without any loss of money if the bettor plays his best strategy. Example Let a book that contains (ϕ, α), ( ϕ, β) with α + β > 1. The bettor has a loosing strategy, namely, betting 1 on both ϕ and ϕ. He has also a good bet, namely, betting nothing. This book cannot be considered irrational: it can be obtained from an element (ϕ, [.30,.60]) that is obviously acceptable. Martina Fedel (UNISI) 27 / 32
42 Coherent Books and Imprecise Probabilities An adequate coherence criterion In the case of imprecise probabilities and nonreversible games, Remark The non existence of a winning strategy for the bettor is a necessary but not sufficient condition for rationality. The non existence of a loosing strategy, or the non existence of a good bet are too strong criteria, in the sense that it is not reasonable to expect them to hold. Coherence Criterion (Nonreversible bettinggame) A book is said to be rational or coherent if there is no bad bet for the bettor. Martina Fedel (UNISI) 28 / 32
43 Coherent Books and Imprecise Probabilities An adequate coherence criterion In the case of imprecise probabilities and nonreversible games, Remark The non existence of a winning strategy for the bettor is a necessary but not sufficient condition for rationality. The non existence of a loosing strategy, or the non existence of a good bet are too strong criteria, in the sense that it is not reasonable to expect them to hold. Coherence Criterion (Nonreversible bettinggame) A book is said to be rational or coherent if there is no bad bet for the bettor. Martina Fedel (UNISI) 28 / 32
44 Coherent Books and Imprecise Probabilities An adequate coherence criterion Theorem (Fedel,Keimel,Montagna,Roth) Let be a book in a nonreversible game over a 2divisible MValgebra A of events. Then the following are equivalent: There is no bad bet for the bettor based on. There is an upper probability P + over the MValgebra A of events such that, if (ϕ, α), then P + (ϕ) = α. Martina Fedel (UNISI) 29 / 32
45 Coherent Books and Imprecise Probabilities An adequate coherence criterion Theorem (Fedel,Keimel,Montagna,Roth) Let be a book in a nonreversible game over a 2divisible MValgebra A of events. Then the following are equivalent: There is no good bet for the bettor based on. There is a lower probability P over the MValgebra A of events such that, if (ϕ, α), then P (ϕ) = α. Problem: Why does the nonexistence of a good bet, which is not a rationality criterion, correspond to a significant property, that is, extendability to a lower probability? Remark If betting λ > 0 on ϕ is a good bet in a game in which only positive bets are allowed, then betting λ on ϕ is a bad bet in a game in which only negative bets are allowed. Hence, accordance with a lower probability corresponds to a reasonable rationality criterion. Martina Fedel (UNISI) 30 / 32
46 Coherent Books and Imprecise Probabilities An adequate coherence criterion Theorem (Fedel,Keimel,Montagna,Roth) Let be a book in a nonreversible game over a 2divisible MValgebra A of events. Then the following are equivalent: There is no good bet for the bettor based on. There is a lower probability P over the MValgebra A of events such that, if (ϕ, α), then P (ϕ) = α. Problem: Why does the nonexistence of a good bet, which is not a rationality criterion, correspond to a significant property, that is, extendability to a lower probability? Remark If betting λ > 0 on ϕ is a good bet in a game in which only positive bets are allowed, then betting λ on ϕ is a bad bet in a game in which only negative bets are allowed. Hence, accordance with a lower probability corresponds to a reasonable rationality criterion. Martina Fedel (UNISI) 30 / 32
47 Coherent Books and Imprecise Probabilities An adequate coherence criterion Theorem (Fedel,Keimel,Montagna,Roth) Let be a book in a nonreversible game over a 2divisible MValgebra A of events. Then the following are equivalent: There is no good bet for the bettor based on. There is a lower probability P over the MValgebra A of events such that, if (ϕ, α), then P (ϕ) = α. Problem: Why does the nonexistence of a good bet, which is not a rationality criterion, correspond to a significant property, that is, extendability to a lower probability? Remark If betting λ > 0 on ϕ is a good bet in a game in which only positive bets are allowed, then betting λ on ϕ is a bad bet in a game in which only negative bets are allowed. Hence, accordance with a lower probability corresponds to a reasonable rationality criterion. Martina Fedel (UNISI) 30 / 32
48 Coherent Books and Imprecise Probabilities An adequate coherence criterion Theorem (Fedel,Keimel,Montagna,Roth) Let be a book in a nonreversible game over a 2divisible MValgebra A of events. Then the following are equivalent: There is no good bet and no bad bet based on. There is a state P on A such that, if (ϕ, α), then P(ϕ) = α. Martina Fedel (UNISI) 31 / 32
49 Coherent Books and Imprecise Probabilities An adequate coherence criterion...thanks! Martina Fedel (UNISI) 32 / 32
Imprecise probabilities, bets and functional analytic methods in Lukasiewicz logic
Imprecise probabilities, bets and functional analytic methods in Lukasiewicz logic Martina Fedel, Klaus Keimel, Franco Montagna and Walter Roth February 22, 2011 Abstract In his foundation of probability
More informationNonArchimedean Probability and Conditional Probability; ManyVal2013 Prague 2013. F.Montagna, University of Siena
NonArchimedean Probability and Conditional Probability; ManyVal2013 Prague 2013 F.Montagna, University of Siena 1. De Finetti s approach to probability. De Finetti s definition of probability is in terms
More informationFranco Montagna AN ALGEBRAIC TREATMENT OF IMPRECISE PROBABILITIES
DEMONSTRATIO MATHEMATICA Vol. XLIV No 3 2011 Franco Montagna AN ALGEBRAIC TREATMENT OF IMPRECISE PROBABILITIES Abstract. This is a survey paper about an algebraic approach to imprecise probabilities. In
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationProbability, Fuzzy Logic and bets: foundational issues and some mathematics.
Probability, Fuzzy Logic and bets: foundational issues and some mathematics. Logical Models of Reasoning with Vague Information Franco Montagna (Siena) 1. The concept of probability. The concept of probability
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationα α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationThe epistemic structure of de Finetti s betting problem
The epistemic structure of de Finetti s betting problem Tommaso Flaminio 1 and Hykel Hosni 2 1 IIIA  CSIC Campus de la Univ. Autònoma de Barcelona s/n 08193 Bellaterra, Spain. Email: tommaso@iiia.csic.es
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationFinite dimensional topological vector spaces
Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the
More informationCyberSecurity Analysis of State Estimators in Power Systems
CyberSecurity Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTHRoyal Institute
More informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationChapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z
More informationFINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationThe epistemic structure of de Finetti s betting problem
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 www.iiia.csic.es/ tommaso 2 SNS Scuola
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More information3. Prime and maximal ideals. 3.1. Definitions and Examples.
COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationGambling Systems and MultiplicationInvariant Measures
Gambling Systems and MultiplicationInvariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More informationNoBetting Pareto Dominance
NoBetting Pareto Dominance Itzhak Gilboa, Larry Samuelson and David Schmeidler HEC Paris/Tel Aviv Yale Interdisciplinary Center Herzlyia/Tel Aviv/Ohio State May, 2014 I. Introduction I.1 Trade Suppose
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationJeux finiment répétés avec signaux semistandards
Jeux finiment répétés avec signaux semistandards P. ContouCarrère 1, T. Tomala 2 CEPNLAGA, Université Paris 13 7 décembre 2010 1 Université Paris 1, Panthéon Sorbonne 2 HEC, Paris Introduction Repeated
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationProblem Set I: Preferences, W.A.R.P., consumer choice
Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationNonlinear Optimization: Algorithms 3: Interiorpoint methods
Nonlinear Optimization: Algorithms 3: Interiorpoint methods INSEAD, Spring 2006 JeanPhilippe Vert Ecole des Mines de Paris JeanPhilippe.Vert@mines.org Nonlinear optimization c 2006 JeanPhilippe Vert,
More informationA CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT
Bol. Soc. Mat. Mexicana (3) Vol. 12, 2006 A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT F. MONTALVO, A. A. PULGARÍN AND B. REQUEJO Abstract. We present some internal conditions on a locally
More informationGod created the integers and the rest is the work of man. (Leopold Kronecker, in an afterdinner speech at a conference, Berlin, 1886)
Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an afterdinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More informationNash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31
Nash Equilibrium Ichiro Obara UCLA January 11, 2012 Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Best Response and Nash Equilibrium In many games, there is no obvious choice (i.e. dominant action).
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationBetting on Fuzzy and Manyvalued Propositions
Betting on Fuzzy and Manyvalued Propositions Peter Milne 1 Introduction In a 1968 article, Probability Measures of Fuzzy Events, Lotfi Zadeh proposed accounts of absolute and conditional probability for
More informationA HennessyMilner Property for ManyValued Modal Logics
A HennessyMilner Property for ManyValued Modal Logics Michel Marti 1 Institute of Computer Science and Applied Mathematics, University of Bern Neubrückstrasse 10, CH3012 Bern, Switzerland mmarti@iam.unibe.ch
More information1 Portfolio mean and variance
Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a oneperiod investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring
More informationDIVISORS AND LINE BUNDLES
DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function
More informationA NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION
1 A NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION Dimitri Bertsekas M.I.T. FEBRUARY 2003 2 OUTLINE Convexity issues in optimization Historical remarks Our treatment of the subject Three unifying lines of
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationCHAPTER 7 GENERAL PROOF SYSTEMS
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationTopics in weak convergence of probability measures
This version: February 24, 1999 To be cited as: M. Merkle, Topics in weak convergence of probability measures, Zb. radova Mat. Inst. Beograd, 9(17) (2000), 235274 Topics in weak convergence of probability
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More information6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium
6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline PricingCongestion Game Example Existence of a Mixed
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationThe effect of exchange rates on (Statistical) decisions. Teddy Seidenfeld Mark J. Schervish Joseph B. (Jay) Kadane Carnegie Mellon University
The effect of exchange rates on (Statistical) decisions Philosophy of Science, 80 (2013): 504532 Teddy Seidenfeld Mark J. Schervish Joseph B. (Jay) Kadane Carnegie Mellon University 1 Part 1: What do
More informationfg = f g. 3.1.1. Ideals. An ideal of R is a nonempty ksubspace I R closed under multiplication by elements of R:
30 3. RINGS, IDEALS, AND GRÖBNER BASES 3.1. Polynomial rings and ideals The main object of study in this section is a polynomial ring in a finite number of variables R = k[x 1,..., x n ], where k is an
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10725/36725
Duality in General Programs Ryan Tibshirani Convex Optimization 10725/36725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationAlgebraic and Transcendental Numbers
Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationRacetrack Betting and Consensus of Subjective Probabilities
Racetrack Betting and Consensus of Subective Probabilities Lawrence D. Brown 1 and Yi Lin 2 University of Pennsylvania and University of Wisconsin, Madison Abstract In this paper we consider the dynamic
More informationCPC/CPA Hybrid Bidding in a Second Price Auction
CPC/CPA Hybrid Bidding in a Second Price Auction Benjamin Edelman Hoan Soo Lee Working Paper 09074 Copyright 2008 by Benjamin Edelman and Hoan Soo Lee Working papers are in draft form. This working paper
More informationThe Kelly criterion for spread bets
IMA Journal of Applied Mathematics 2007 72,43 51 doi:10.1093/imamat/hxl027 Advance Access publication on December 5, 2006 The Kelly criterion for spread bets S. J. CHAPMAN Oxford Centre for Industrial
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More informationNondeterministic Semantics and the Undecidability of Boolean BI
1 Nondeterministic Semantics and the Undecidability of Boolean BI DOMINIQUE LARCHEYWENDLING, LORIA CNRS, Nancy, France DIDIER GALMICHE, LORIA Université Henri Poincaré, Nancy, France We solve the open
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 11: Choice Under Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 11: Choice Under Uncertainty Tuesday, October 21, 2008 Last class we wrapped up consumption over time. Today we
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationLecture Notes on Measure Theory and Functional Analysis
Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents
More informationThe "Dutch Book" argument, tracing back to independent work by. F.Ramsey (1926) and B.deFinetti (1937), offers prudential grounds for
The "Dutch Book" argument, tracing back to independent work by F.Ramsey (1926) and B.deFinetti (1937), offers prudential grounds for action in conformity with personal probability. Under several structural
More information