Prediction markets and their possible role in logical uncertainty

Computational omnipotence; unlimited computing speed and memory such that if a result is computable, it can be computed.

Computable; There exists an algorithm which can produce the result in a finite number of steps

Uncertainty is a topic I think everyone understands to a certain degree. There is a field of mathematics dedicated to demystifying its nebulous nature. This is, of course, the field of probability theory. There is, however, a limit to probability theory which comes from one of the assumptions that underpins it. Computational omnipotence is assumed when dealing with the calculations. That is to say that given the information you have, you can calculate all of the resulting probabilities that are possible to calculate with the starting information you have.

Your uncertainty is limited to uncertainty about the environment. For example, when a die is rolled you are uncertain of which face will land up. If n dice are thrown you are only uncertain about which faces will be on top, you are not uncertain of the calculation (assuming fair dice) to work out the probability of the sum being a given number.

Logical uncertainty is a field with its roots far closer to computer science in that it discusses the uncertainty you may have due to the limits of your own computation.

For example, the 100th digit of pi is a fixed value. It cannot change, however, you may not know what that value is. I know off the top of my head I can’t tell what it is.

I can, however, get a feel for the answer to the question: “Is the 100th digit of pi 7?”

This is a true or false statement, there is no ambiguity in the value, but if we haven't calculated it, and we are not given the time to calculate it, can we create some sense of confidence in the answer being true?

One might reason about the domain of the question. There are two possible answers, either the 100th digit is 7 or it isn’t. We have no reason (At this stage…) to believe either is more ‘probable’ (using probably loosely here) and so we can assume a uniform distribution over this range. This leads us to a 50% chance that the 100th digit is 7.

Thinking about this question a little more, we do actually have reason to believe the uniform distribution is incorrect. We know for example that there are 10 possible digits that the 100th digit of pi could take, 0 through 9. In this case the domain is a collection of 10 elements, only one of which would make the statement in question true. Here too we have no reason to assume any digit is more probable than any other. We may therefore reasonably assume a uniform prior over this new, extended, collection.

This extra thought about the problem allows us to update our belief that the 100th digit of pi is 7, from 50% down to only 10%.

I think at this point people would consider this a reasonable value of confidence to give that the statement is true. You might have initially jumped to this value, skipping the 50% step.

Realise here that there is no true probability involved here. The 100th digit of pi is 9. The statement is false. We do not need to observe anything more from our environment to get the result of 9. We just need to calculate more, think longer.

This is what makes logical uncertainty different from the environmental uncertainty that probability theory deals with.

This field of study is much less explored than that of pure probability and I find it fascinating. It is so often the case that the questions we come across are theoretically calculable yet computationally intractable. A framework to understanding how to better handle this logical uncertainty in a formalised way would allow a more rigorous study of this kind of unknown.

One proposed method is the prediction market. The idea behind a prediction market is that the more confident agents are of their beliefs the more they are willing to stake on them. It also takes ideas from the idea core to market theory of the invisible hand balancing supply and demand to create a natural equilibrium. This is also talked about in terms of the wisdom of crowds:

In 1906, the great statistician Francis Galton observed a competition to guess the weight of an ox at a country fair. Eight hundred people entered. Galton, being the kind of man he was, ran statistical tests on the numbers. He discovered that the average guess (1,197lb) was extremely close to the actual weight (1,198lb) of the ox.

(There is an extension which is an amusing satire of the finance industry)

The purpose of the market is to create a place where different agents can “vote” on the answers to questions. The “price” of the answer to the decision is then presumed to be the best guess for that answer.

Let us consider again the example of a digit of pi. Say we list a prediction “stock” on the market whose value is $1 if the nth digit of pi is 7. What might be the value of this “stock”?

If we were sure that for the chosen n, the digit was indeed 7, then the value of this “stock” would be $1. However, if we knew in the case of the 100th digit of pi the correct digit to be 9 then the “stock” would be valued at $0.

Initially, we know there is uncertainty over if the answer is true or not. This “stock” might IPO at 50¢. This represents our initial 50% chance, discussed earlier. After a very short amount of time, this may fall to 10¢ as the market participants realise that the domain in which the question sits actually has a cardinality of 10.

The prices might if modelled with the assumption that each digit of pi has to be expanded after the last, look like those shown in the following graph.

The rational and production of this graph is explored in a previous post: here

I was introduced to the idea of prediction markets as a useful method for thinking about logical uncertainty from this paper via this video from Robert Miles. The paper talks about desirable properties a system for dealing with logical uncertainty would have and what that might mean. I hope to do another post going through these.