Belief aggregation


Belief aggregation, also known as judgement aggregation (or merging) refers to approaches to combining beliefs where there are multiple parties with different beliefs. It is a collective intelligence technique.

Belief aggregation commonly happens in courtrooms (is the suspect guilty?) and when planning (where's the best place to build a wall?).

Belief aggregation techniques used include voting, betting, forming a committee, and so on.

Beliefs are sometimes mixed together with values. For example, disagreement over whether smoking causes cancer have historically been correlated with whether the people being polled are tobacco company employees. Here we will be discussing beliefs only. If it helps, the associated value can be considered to be assumed to be: find the truth.


One fairly general observation is that it helps if the parties involved are able to cooperate and collaborate. For a range of questions what is needed to find the truth is research - and research can require a team effort. Teamwork allows specialization and division of labor to operate. Teamwork typically requires an information network, a matter network, a medium of exchange and the ability to enforce contracts.

Collaboration is important. It seems fair to say that most questions where people are prepared to invest in learning the answer are likely to involve some kind of collaboration. Collaboration itself involves lots of different techniques. Hiring people, firing people, project management - and so on. Collaboration is important - but it is also a complicated topic. It is broadly equivalent to creating an organization or company whose mission is answering the relevant question. There's no shortage of discussion about how to do this. As a result, I'm not going to discuss it further here.

Belief marketplaces

Belief marketplaces are a useful belief aggregation technique with a proven track record. They can easily be subsidised - so that putting more money in results in better quality answers. They are largely based on competition - rather than cooperation. However they normally permit collaboration. Participants are usually at liberty to form teams and groups - when they think that doing so will be beneficial.

Belief marketplaces do have some issues though. Participation is usually anonymous (so that successful traders can't simply be tracked and copied). This makes it harder to use reputation systems and means that anonymous manipulation is easier. Markets naturally resist manipulation: sucker bets whose purpose is to distort market prices add to the value of the market, attracting corrections from those who are happy to accept the sucker's money. However this mechanism is far from perfect.

Another problem with marketplaces is monopolies. If the market can be monopolized by a consortium of traders, they don't need to bother with finding the truth. They have a monopoly - they can't lose. They will simply collect any incentives associated with whatever question they are asked. Monopolies are probably not too important in large public markets.

Another problem with marketplaces is that they often run into local laws against gambling. The best solution to this problem is not clear. In practice, some establishements are hosted by more permissive establishemnts overseas.

Despite these disadvantages, belief marketplaces are an important belief aggregation tool. They work even when The rest of this article will build on them.

Machine intelligence

Intelligent machines ought to be good at belief aggregation. The basic approach would be: gather as much relevant data as possible, feed it to the intelligent machine and then ask its opinion on the topic in question.

One way of doing this would be to run a belief marketplace - and then make use of data which is not publicly available to make predictions. In marketplaces, betting histories are not normally made public in order to avoid successful traders from being simply copied. The possibility of their bets being copied by "leeches" could easily destroy the incentive to participate of well-informed traders. However, the owners of the marketplace normally would have access to this information - and they could plausibly make use of it. In particular:

  • Traders with a successful history are more likely to be right again;
  • Trading times (and login information) can indicate stale bets that don't reflect current information;

It seems plausible that such information could help to identify market manipulators, foolish traders who haven't yet lost all of their money and traders who have become out-of-touch. With sufficient training data, an intelligent machine could be used to make use of this information with maximal efficiency.

While it goes beyond pure belief aggregation, it might also be possible to gather data experimentally. Natural experiments can sometimes influence market prices. A machine with inside access to the belief marketplace and a comprehensive world model complete with external news feeds may be able to detect which traders are responsive to real world events - and give more weight to their bets. Experimental interventions may also be possible. For example, synthesizing rumours relating to the topic and watching the response of the traders may also be practical.


What should belief marketplace site owners do with such non-public information and an intelligent machine? Probably, they should reflect it in their own market prices in order to decrease the payouts they make, and to increase the reputation of their marketplace for accuracy. This can usually be done without destroying the incentive of successful traders to participate in the market.

This would necessarily involve markets of some beliefs subsidizing markets of other ones. It could also potentially introduce some possibility of the market owners losing money and going bust - if their machine intelligence is too stupid.

The idea of using machine intelligence to aggregate market prices would be possible on stock exchanges. There, some trades are made public, but the exchange itself generally has much more information than is made publicly available. The exchange would be well advised to feed all this information into an intelligent machine, which could then use it to set stock prices.


Tim Tyler | Contact |