If I assign a probability p to an event and my friend assign a probability q to an event at what odds “should” we bet?
It seems that while there are a number of fairly natural suggestions there isn’t ″ one canonical answer to rule them all”. I think the key observation here is that what bet gets made is underdetermined from just the probabilities.
Belief and Disbelief
We need to add more information to the beliefs of me and my friend to resolve this ambiguity. As mentioned above there is a duality between market order (order by number of shares bought or sold) and a limit order (order by bid/ask price desired). This has something to do with Legendre-Fenchel duality.
A trader-forecaster-market can do two things: offer prices on assets, and participate in the market by buying and selling shares. When we give a price P of an asset/proposition A this encodes our belief that at this price we cannot be exploited. On the other hand, when buying shares SA on an asset/proposition A at price q this manifests our skepticism that the price q is ‘right’- i.e. that it cannot be exploited.
That is offering prices (limit orders) and stating probabilities is about defeasible belief while taking up offers and buying shares (market orders) is about skepticism vis-a-vis belief. Probability is about defeasible belief—buying shares is about trying to prove defeasible beliefs wrong.
Imprecise Probability Recap
In imprecise probability (and infraBayesianism) there are three ways to define an ‘imprecise probability distribution’. Let Ω be a sample space, we suppress the sigma algebra structure. Let D(Ω) denote the set of probability distributions on Ω
Convex closed set of probability distributions {pc}c∈C⊂D(Ω)
a) A concave, monotonic [extra condition] lower expectation functional E––:C(X)→R b) A convex, monotonic [extra condition] lower expectation functional ¯¯¯¯E:C(X)→R
A positive convex cone [satisfying conditions] of ‘desirable gambles’ B⊂C(X)
Rk. Note that in the third presentation the positive cone B encodes a preference relation (partial order) on C(X) by f≥g if and only if f−g∈B.
Rk. Note that
We’d like to define open market-trader-forecaster as
If I assign a probability p to an event and my friend assign a probability q to an event at what odds “should” we bet?
It seems that while there are a number of fairly natural suggestions there isn’t ″ one canonical answer to rule them all”. I think the key observation here is that what bet gets made is underdetermined from just the probabilities.
Belief and Disbelief
We need to add more information to the beliefs of me and my friend to resolve this ambiguity. As mentioned above there is a duality between market order (order by number of shares bought or sold) and a limit order (order by bid/ask price desired). This has something to do with Legendre-Fenchel duality.
A trader-forecaster-market can do two things: offer prices on assets, and participate in the market by buying and selling shares. When we give a price P of an asset/proposition A this encodes our belief that at this price we cannot be exploited. On the other hand, when buying shares SA on an asset/proposition A at price q this manifests our skepticism that the price q is ‘right’- i.e. that it cannot be exploited.
That is offering prices (limit orders) and stating probabilities is about defeasible belief while taking up offers and buying shares (market orders) is about skepticism vis-a-vis belief. Probability is about defeasible belief—buying shares is about trying to prove defeasible beliefs wrong.
Imprecise Probability Recap
In imprecise probability (and infraBayesianism) there are three ways to define an ‘imprecise probability distribution’. Let Ω be a sample space, we suppress the sigma algebra structure. Let D(Ω) denote the set of probability distributions on Ω
Convex closed set of probability distributions {pc}c∈C⊂D(Ω)
a) A concave, monotonic [extra condition] lower expectation functional E––:C(X)→R
b) A convex, monotonic [extra condition] lower expectation functional ¯¯¯¯E:C(X)→R
A positive convex cone [satisfying conditions] of ‘desirable gambles’ B⊂C(X)
Rk. Note that in the third presentation the positive cone B encodes a preference relation (partial order) on C(X) by f≥g if and only if f−g∈B.
Rk. Note that
We’d like to define open market-trader-forecaster as