Given a finite state space and common priors, common knowledge of the identity of an agent with the minimal (or maximal) expectation of a random variable implies ‘consensus’, i.e., common knowledge of common expectations. This ‘extremist’ statistic induces consensus when repeatedly announced, and yet, with n agents, requires at most log2 n bits to broadcast.

Engineers’ love of technology often gets in the way of their being useful. Consider Post-it Notes or, better yet, plain paper notepads. These probably seemed like trivial ideas, but they turned out to be terribly useful. Why? Because the marvel that is the human brain has a horrible short-term memory, which means that dumb-as-dirt memory aids can make people substantially smarter.

In practice, scoring rules elicit good probability estimates from individuals, while betting markets elicit good consensus estimates from groups. Market scoring rules combine these features, eliciting estimates from individuals or groups, with groups costing no more than individuals. Regarding a bet on one event given another event, only logarithmic versions preserve the probability of the given event. Logarithmic versions also preserve the conditional probabilities of other events, and so preserve conditional independence relations. Given logarithmic rules that elicit relative probabilities of (...) base event pairs, it costs no more to elicit estimates on all combinations of these base events. (shrink)

Since utilities and probabilities jointly determine choices, event-dependent utilities complicate the elicitation of subjective event probabilities. However, for the usual purpose of obtaining the information embodied in agent beliefs, it is sufficient to elicit objective probabilities, i.e., probabilities obtained by updating a known common prior with that agent’s further information. Bayesians who play a Nash equilibrium of a certain insurance game before they obtain relevant information will afterward act regarding lottery ticket payments as if they had event-independent risk-neutral utility and (...) a known common prior. Proper scoring rules paid in lottery tickets can then elicit objective probabilities. (shrink)

Given common priors, no agent can publicly estimate a non-zero sign for the difference between his estimate and another agent’s future estimate. Thus rational agents cannot publicly anticipate the direction in which other agents will disagree with them.  2002 Elsevier Science B.V. All rights reserved.