Self-Embedded Agent's Shortform

Alexander Gietelink Oldenziel

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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 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 ... (read more)

1Alexander Gietelink Oldenziel4y

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 v

1Alexander Gietelink Oldenziel4y

(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.

Self-Embedded Agent's Shortform

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Self-Embedded Agent's Shortform

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Dutch Book Fundamentalism

How do Prediction Markets generalize Probability Distributions?

Duality Principles

Final Thoughts

Belief and Disbelief