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.
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.