tl; dr: Markets are fundamental: unDutchBookable betting odds—not probability distributions encode our true beliefs
The idea that our beliefs are constrained by the bets that we are willing to take is widely accepting on LessWrong—see the of-quoted adagium: Bet or Update; or perhaps better yet: Kelly Bet or Update. Dutch Book Fundamentalism goes one step further in that it tries to equate our belief with the bets we are willing to take and offer.
That Probability distributions are the right way to quantify uncertainty is often defended by Dutch book arguments (e.g. de Finetti): probability distributions induce betting odds—we’d like them to be resistant to a Dutch book. Logical Induction & especially Shafer-Vovk game-theoretic probability suggests to turn that logic on its head: the Dutch book & betting odds is fundamental and the probability distribution is derived. In particular, Shafer & Vovk derive all classical & advanced probability theory in terms of markets that are resistant to dutch books (like Logical Inductors).
Additional motivation comes from Wentworth’s Generalized Heat Engines. Wentworth convincingly argues that the oft-conjectured analogy between thermodynamics and information theory is not just an analogy but a precise mathematical statement. Moreover, he shows that thermodynamic systems can be understood as special kinds of markets. It remains to given a general formulations of markets and thermodynamic systems.
In a generic prediction market given an event there is not just one price (or probability) but a whole order book. The prediction market contains much more information than just mid-point price ~= probability.
A probability distribution p gives a very simple order book: p(A) equal the buy and the sell price for a ticket on A and the agent has no risk aversion—it plays with all its capital. When we generalize from probability distributions to non-arbitragable betting odds this changes: the buy and the sell price may differ and the agent might not put all of its money on a given price level but as in real market might increase the sell price as it gets bought out.
The stockmarket doesn’t have one price for an asset; rather it has a range of bid and ask prices depending on how much of the asset you want to buy or sell.
If we accept the gospel of Dutch Book & Market Fundamentalism we’d like to formalize markets. How to formalize this exactly is still a little murky to me but I think I have enough puzzle pieces to speculate what might go in here.
How do Prediction Markets generalize Probability Distributions?
Ways in which (prediction) markets generalize probability distributions & statistical models:
Markets generically have a nontrivial bid-ask spread; i.e. markets have both buy & sell prices.
Markets price general (measurable) real-valued functions (“gambles”) that may not be recoverable from the way it prices events.
Markets have finite total capital size
Markets are composed of individual traders
Traders may not be willing (knowledgable enough) to bet on all possible events.
Traders may be risk-averse and not be willing to buy/sell all their holdings at a given price. In other words, there is a limited bet size on bets.
Traders can both offer trades as well as taking trades—i.e. there are limit orders and market orders.
Remark. (Imprecise Probability & InfraBayesianism) Direction 1. & 2. point towards Imprecise Probability (credal sets) and more generally InfraBayesianism.
Remark. (Garrabrants New Thing) 4&5 are likely related to Garrabrants new (as-of-yet unpublished) ideas on partial distributions & multigrained/multi-level distributions.
Remark. (Exponential Families) Wentworth’s analysis of thermodynamic systems as intimately tied to the MaxEnt principle and markets suggest a prominent collection of examples of markets should be families of MaxEnt distributions or as they’re known in the statistics literature: exponential families. Lagrange Multipliers would corresponds to price of various securities.
Additionally
Markets may evolve in time (hence these dynamic markets generalize stochastic processes
We might have multiple connected ‘open’ markets, not necessarily in equilibrium. (generalizing general Bayesian networks, coupled thermodynamic systems and Pearlian causal models).
Remark. Following Shafer-Vovk, probability theory always implictly refers to dynamic processes/ stochastic processes so the general setting of dynamic (and interconnected open) is probably the best level of analysis.
Duality Principles
As a general ‘mathematical heuristic’ I am always on the lookout for duality principles. These usually point toward substantive mathematical content’ and provide evidence that we are engaging with a canonical concept or natural abstraction.
In the context of two bettors/gamblers/traders/markets trading and offering bets on gambles & events: I believe there are three different dualities:
[Long-short] Duality between going long or short on an asset.
[Legendre-Fenchel] Duality between market orders and limit orders
[Advocate/ Adversary] Duality between the bettor and the counterparty
Remark. The first duality shows up in the well-known put-call parity. Incidentally, this is pointing towards European option being perhaps more ‘natural’ than American options.
Remark. The second duality is intimately related to the Legendre(-Fenchel) transform.
Final Thoughts
Many different considerations point towards a coherent & formal notion of prediction market as model of belief. In follow-up posts I hope to flesh out some of these ideas.
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
Another aspect of markets (and thermodynamic systems!) is that they may be open systems: they can have excess demand or supply of goods—and be open to the meddling of outside investors.
So an open market might have input./output nodes where we might have nonzero flows of goods (or particles). A formal mathematical model might make use of ideas from compositionality and applied category theory.
An inflow of a good will—all else equal—lower the price of that good. If we think of forecasting markets this would correspond to evidence against that proposition. By how much a given inflow of a good will lower the price of that good is a characteristic feature of a market. If we think in terms of forecasting/probability theory, a forecasting market might have more or less confidence in a given proposition and inflow of negative evidence might have more or less impact on the probability/price.
Statistical Equilibrium Theory of Markets
A cute paper I think about from time to time is a paper by Foley called “statistical equilibrium theory of markets”. Classical Walrasian economics starts with a collection of market participants endowed with goods and preferences—it then imagines an outside ‘auctioneer’ that determines the market transactions. Walras then proved that this gives an exact equilibrium. In contrast, Foley uses the MaxEnt principle to gives an approximate market equillibrium. In this equilibrium the probability of a given transaction happening is proportional to the number of ways that transaction is possible.
It’s a somewhat natural but perhaps also a little weird idea. The cute thing is that in this formalism average excess demand of a good corresponds to the derivative of the partition function with respect to the price.
Heat Capacity and Elasticity
In the thermodynamic case where price corresponds to a conjugate variable like temperature this derivative would be the average energy.
The second derivative corresponds to the variance of the energy which in turns can be used to define the heat capacity as
On the market side the heat capacity would correspond to price elasticity: how does the demand for a good change as we vary the price.
Negative Heat Capacity and Giffen Goods
Most physical systems exhibit a positive heat capacity; constant-volume and constant-pressure heat capacities, rigorously defined as partial derivatives, are always positive for homogeneous bodies. However, even though it can seem paradoxical at first, there are some systems for which the heat capacity is negative.
In the economic analogy this analogous to goods for which the price elasticity is negative: as the price increases the demand for the good grows. Economists call these “Giffen goods”.
Defining Order Books or Markets All The Way Down
I’d like to think of markets as composed of market participants which themselves might be (open) markets. By this I mean: they have prices and demand/supply for goods.
I am especially interested in forecasting markets, so let’s focus on those. A simple model for a forecaster-market is that it assigns probabilities pA to events A. The price of that event equals the probability. We’ve already argued that instead of a single price we should really be thinking in terms of bid & ask prices. I’d like to go further: a given forecaster-trader-market should have an entire order book.
That is; depending on the amount of shares SA demanded on a proposition A the price ¯¯¯¯PA(S) changes.
The key parameter is the price-elasticity.
Rk. We might not just consider the second derivative with respect of the price of the partition function but arbitrary higher-order derivatives. The partition function can be thought of as a moment generating function, and there under weak assumptions a random variable is determined by its moments. There is probably some exciting connections with physics here that is above my pay-grade.
A forecaster-trader-market F might then be defined by giving for each proposition A an order book P(Sbid,Sask)=(P––(Sbid),¯¯¯¯P(Sask)), which are compatible [how does this compatibility work exactly? Look at nonarbitrageble bets]. A canonical class of these forecasters would be determined by a series of constraints on the derivatives of the partition function determining the demand and (higher) price elasticity for various classes of goods/ propositions.
(This shortform was inspired by the following question from Daniel Murfet: Can you elaborate on why I should care about Kelly betting? I guess I’m looking for an answer of the form “the market is a dynamical process that computes a probability distribution, perhaps the Bayesian posterior, and because of out of equilibrium effects or time lags or X, the information you derive from the market is not the Bayesian posterior and therefore you should bet somehow differently in a way that reflects that”?)
Timing the Market
Those with experience with financial markets know that knowing whether something will happen is only half the game—knowing when it will happen, and moreso when that knowledge is revealed & percolated to the wider market is often just as important. When betting the timing of the resolution is imperative.
Consider Logical Inductors: the market is trying to price the propositions that are gradually revealed. [Actually—the logical part of it is a bit of a red herring: any kind of process that reveals information about events over time can be “inducted upon”; i.e. we consider a market over future events.] Importantly, it is not necessarily specified in what order the events appear!
A shrewd trader that always ‘buys the hype’ and sells just at the peak might outcompete one that had more “foresight” and anticipated much longer in advance.
Holding a position for a long time is an opportunity cost.
1. Resolution of events might have time delays (and it might not resolve at all!)
2. We might not know the timing of the resolution
3. Even if you know the timing you may still need to hold if nobody is willing to be a counterparty
Dutch Book Fundamentalism
tl; dr: Markets are fundamental: unDutchBookable betting odds—not probability distributions encode our true beliefs
The idea that our beliefs are constrained by the bets that we are willing to take is widely accepting on LessWrong—see the of-quoted adagium: Bet or Update; or perhaps better yet: Kelly Bet or Update. Dutch Book Fundamentalism goes one step further in that it tries to equate our belief with the bets we are willing to take and offer.
That Probability distributions are the right way to quantify uncertainty is often defended by Dutch book arguments (e.g. de Finetti): probability distributions induce betting odds—we’d like them to be resistant to a Dutch book. Logical Induction & especially Shafer-Vovk game-theoretic probability suggests to turn that logic on its head: the Dutch book & betting odds is fundamental and the probability distribution is derived. In particular, Shafer & Vovk derive all classical & advanced probability theory in terms of markets that are resistant to dutch books (like Logical Inductors).
Additional motivation comes from Wentworth’s Generalized Heat Engines. Wentworth convincingly argues that the oft-conjectured analogy between thermodynamics and information theory is not just an analogy but a precise mathematical statement. Moreover, he shows that thermodynamic systems can be understood as special kinds of markets. It remains to given a general formulations of markets and thermodynamic systems.
In a generic prediction market given an event there is not just one price (or probability) but a whole order book. The prediction market contains much more information than just mid-point price ~= probability.
A probability distribution p gives a very simple order book: p(A) equal the buy and the sell price for a ticket on A and the agent has no risk aversion—it plays with all its capital. When we generalize from probability distributions to non-arbitragable betting odds this changes: the buy and the sell price may differ and the agent might not put all of its money on a given price level but as in real market might increase the sell price as it gets bought out.
If we accept the gospel of Dutch Book & Market Fundamentalism we’d like to formalize markets. How to formalize this exactly is still a little murky to me but I think I have enough puzzle pieces to speculate what might go in here.
How do Prediction Markets generalize Probability Distributions?
Ways in which (prediction) markets generalize probability distributions & statistical models:
Markets generically have a nontrivial bid-ask spread; i.e. markets have both buy & sell prices.
Markets price general (measurable) real-valued functions (“gambles”) that may not be recoverable from the way it prices events.
Markets have finite total capital size
Markets are composed of individual traders
Traders may not be willing (knowledgable enough) to bet on all possible events.
Traders may be risk-averse and not be willing to buy/sell all their holdings at a given price. In other words, there is a limited bet size on bets.
Traders can both offer trades as well as taking trades—i.e. there are limit orders and market orders.
Remark. (Imprecise Probability & InfraBayesianism) Direction 1. & 2. point towards Imprecise Probability (credal sets) and more generally InfraBayesianism.
Remark. (Garrabrants New Thing) 4&5 are likely related to Garrabrants new (as-of-yet unpublished) ideas on partial distributions & multigrained/multi-level distributions.
Remark. (Exponential Families) Wentworth’s analysis of thermodynamic systems as intimately tied to the MaxEnt principle and markets suggest a prominent collection of examples of markets should be families of MaxEnt distributions or as they’re known in the statistics literature: exponential families. Lagrange Multipliers would corresponds to price of various securities.
Additionally
Markets may evolve in time (hence these dynamic markets generalize stochastic processes
We might have multiple connected ‘open’ markets, not necessarily in equilibrium. (generalizing general Bayesian networks, coupled thermodynamic systems and Pearlian causal models).
Remark. Following Shafer-Vovk, probability theory always implictly refers to dynamic processes/ stochastic processes so the general setting of dynamic (and interconnected open) is probably the best level of analysis.
Duality Principles
As a general ‘mathematical heuristic’ I am always on the lookout for duality principles. These usually point toward substantive mathematical content’ and provide evidence that we are engaging with a canonical concept or natural abstraction.
In the context of two bettors/gamblers/traders/markets trading and offering bets on gambles & events: I believe there are three different dualities:
[Long-short] Duality between going long or short on an asset.
[Legendre-Fenchel] Duality between market orders and limit orders
[Advocate/ Adversary] Duality between the bettor and the counterparty
Remark. The first duality shows up in the well-known put-call parity. Incidentally, this is pointing towards European option being perhaps more ‘natural’ than American options.
Remark. The second duality is intimately related to the Legendre(-Fenchel) transform.
Final Thoughts
Many different considerations point towards a coherent & formal notion of prediction market as model of belief. In follow-up posts I hope to flesh out some of these ideas.
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
Open Markets
Another aspect of markets (and thermodynamic systems!) is that they may be open systems: they can have excess demand or supply of goods—and be open to the meddling of outside investors.
So an open market might have input./output nodes where we might have nonzero flows of goods (or particles). A formal mathematical model might make use of ideas from compositionality and applied category theory.
An inflow of a good will—all else equal—lower the price of that good. If we think of forecasting markets this would correspond to evidence against that proposition. By how much a given inflow of a good will lower the price of that good is a characteristic feature of a market. If we think in terms of forecasting/probability theory, a forecasting market might have more or less confidence in a given proposition and inflow of negative evidence might have more or less impact on the probability/price.
Statistical Equilibrium Theory of Markets
A cute paper I think about from time to time is a paper by Foley called “statistical equilibrium theory of markets”. Classical Walrasian economics starts with a collection of market participants endowed with goods and preferences—it then imagines an outside ‘auctioneer’ that determines the market transactions. Walras then proved that this gives an exact equilibrium. In contrast, Foley uses the MaxEnt principle to gives an approximate market equillibrium. In this equilibrium the probability of a given transaction happening is proportional to the number of ways that transaction is possible.
It’s a somewhat natural but perhaps also a little weird idea. The cute thing is that in this formalism average excess demand of a good corresponds to the derivative of the partition function with respect to the price.
Heat Capacity and Elasticity
In the thermodynamic case where price corresponds to a conjugate variable like temperature this derivative would be the average energy.
The second derivative corresponds to the variance of the energy which in turns can be used to define the heat capacity as
On the market side the heat capacity would correspond to price elasticity: how does the demand for a good change as we vary the price.
Negative Heat Capacity and Giffen Goods
Most physical systems exhibit a positive heat capacity; constant-volume and constant-pressure heat capacities, rigorously defined as partial derivatives, are always positive for homogeneous bodies. However, even though it can seem paradoxical at first, there are some systems for which the heat capacity is negative.
In the economic analogy this analogous to goods for which the price elasticity is negative: as the price increases the demand for the good grows. Economists call these “Giffen goods”.
Defining Order Books or Markets All The Way Down
I’d like to think of markets as composed of market participants which themselves might be (open) markets. By this I mean: they have prices and demand/supply for goods.
I am especially interested in forecasting markets, so let’s focus on those. A simple model for a forecaster-market is that it assigns probabilities pA to events A. The price of that event equals the probability. We’ve already argued that instead of a single price we should really be thinking in terms of bid & ask prices. I’d like to go further: a given forecaster-trader-market should have an entire order book.
That is; depending on the amount of shares SA demanded on a proposition A the price ¯¯¯¯PA(S) changes.
The key parameter is the price-elasticity.
Rk. We might not just consider the second derivative with respect of the price of the partition function but arbitrary higher-order derivatives. The partition function can be thought of as a moment generating function, and there under weak assumptions a random variable is determined by its moments. There is probably some exciting connections with physics here that is above my pay-grade.
A forecaster-trader-market F might then be defined by giving for each proposition A an order book P(Sbid,Sask)=(P––(Sbid),¯¯¯¯P(Sask)), which are compatible [how does this compatibility work exactly? Look at nonarbitrageble bets]. A canonical class of these forecasters would be determined by a series of constraints on the derivatives of the partition function determining the demand and (higher) price elasticity for various classes of goods/ propositions.
(This shortform was inspired by the following question from Daniel Murfet: Can you elaborate on why I should care about Kelly betting? I guess I’m looking for an answer of the form “the market is a dynamical process that computes a probability distribution, perhaps the Bayesian posterior, and because of out of equilibrium effects or time lags or X, the information you derive from the market is not the Bayesian posterior and therefore you should bet somehow differently in a way that reflects that”?)
Timing the Market
Those with experience with financial markets know that knowing whether something will happen is only half the game—knowing when it will happen, and moreso when that knowledge is revealed & percolated to the wider market is often just as important. When betting the timing of the resolution is imperative.
Consider Logical Inductors: the market is trying to price the propositions that are gradually revealed. [Actually—the logical part of it is a bit of a red herring: any kind of process that reveals information about events over time can be “inducted upon”; i.e. we consider a market over future events.] Importantly, it is not necessarily specified in what order the events appear!
A shrewd trader that always ‘buys the hype’ and sells just at the peak might outcompete one that had more “foresight” and anticipated much longer in advance.
Holding a position for a long time is an opportunity cost.
1. Resolution of events might have time delays (and it might not resolve at all!)
2. We might not know the timing of the resolution
3. Even if you know the timing you may still need to hold if nobody is willing to be a counterparty
4. We face opportunity costs.
5. Finite betsize.