User:Benja/It is easy for Bayesians to sell information

Suppose that in a group of perfect Bayesian rational actors with a common prior and different information partitions, one actor wants to sponsor a prediction market, paying the other actors to reveal their private information to the group. Suppose further -- this is an important limitation of the following, as it stands -- that the full true state of the world, not just the value of one particular random variable, will be revealed to the group at a later time $$t'$$. Can the sponsor make an offer to the other agents that makes it rational for them to share all their information, truthfully? I.e., an offer such that agents cannot try to gain more by manipulating the market, for example?

If the agents have no reasons exogenous to the sponsor's offer for wanting the other agents to be misinformed, and the common prior assigns non-zero probability to every possible state of the world, this is easy. Suppose that for every $$\omega \in \Omega$$, the sponsor is willing to pay $$f(\omega)$$ for being informed that $$\omega$$ is not the true state of the world. Then, for every $$\omega$$, the sponsor offers to the first taker a contract that will pay the taker $$f(\omega)$$ at $$t'$$ if $$\omega$$ is revealed not to be the true state of the world, and will cost the taker $$f(\omega) / p(\omega)$$ at $$t'$$ if $$\omega$$ is revealed to be the true state of the world, where $$p$$ is the prior common to the agents.

Let $$I$$ be the information of a particular agent. Then, the agent's expected gain from taking the contract for a particular $$\omega$$ is


 * $$(1-p(\omega|I))f(\omega) - p(\omega|I)f(\omega)/p(\omega)$$.

Now, if $$\omega \notin I$$, then $$p(\omega|I)=0$$, so the agent has an incentive to take the contracts for all $$\omega$$, $$f(\omega)>0$$, that they definitely know not to be the true state of the world.

If $$\omega \in I$$, then $$p(\omega|I) \geq p(\omega)$$, and therefore, $$p(\omega|I)f(\omega)p(\omega) \geq p(\omega)f(\omega)/p(\omega) = f(\omega) > (1-p(\omega|I))f(\omega)$$; the last inequality is strict because we have required that $$\forall \omega \in \Omega : p(\omega) > 0$$ (which we need because we divide by $$p(\omega)$$). Therefore, if $$\omega \in I$$, the expected value of taking the contract for $$\omega$$ is negative, so the actor does not have an incentive to take the contracts for $$\omega$$'s that they do not definitely know not to be the true state of the world.

Finally, revealing an $$\omega$$ to be false does not let other agents to rule out any $$\omega'$$ that the revealing agent doesn't already know to be false also (XXX verify that claim).

Question: Can any of this be extended if only partial information about $$\omega$$ is revealed at $$t'$$?