Agent Latents

The key questions seem to be:

1. Do we want the “latent agents” to be precisely-defined trading strategies from a simple class of strategies — or should they be fuzzy (e.g. defined by prompts)?
2. How do we handle “slightly different” agents? E.g. both agents are bullish on crypto, but one agent is intelligently so while another is batshit crazy? How do we decide what a “uniform prior” is?
3. How do we interpret investments in agents as probabilities?
4. How do we deal with “adversarial attacks” – e.g. if there is a “bullish on crypto agent”, what if someone just generates questions like “will Satoshi Nakamoto be elected president?” or “will WW-III be fought over bitcoin?”

Approaches:

1. Agent latents
2. Mutual information
3. Gated probabilities

Recall that the LMSR subsidy parameter \(\beta\) on a prediction market for \(X\) is the “price of information on \(X\)”, in dollars-per-bit (well actually dollars-per-nat) — i.e. if your subjective belief has entropy $H(\mathbf{p}) and the market belief has entropy \(H(\mathbf{p}_0)\), then your expected return (according to your beliefs) is \(\beta(H(\mathbf{p})-H(\mathbf{p}))\). In particular what this means is that the price you should be willing to pay for some other piece of information \(Y\) is proportional to the mutual information: \(\beta(H(X\mid Y)-H(X))=\beta I(X;Y)\).

This gives us a way to “update” the liquidity of a market based on the liquidities of other related markets. Perhaps the “inherent” value of information on \(Y\) is \(\beta_Y H(Y)\), but it also gives information on \(X\), which gives us \(\beta_Y H(Y) + \beta_X I(X;Y)\).

This can be extended to any number of markets: say you have \(N\) markets \(X_1\dots X_n\) with liquidities \(\beta_1\dots \beta_N\). If you know the mutual information between these markets, you can get the updated liquidities as:

\[\beta' = \mathbf{I}\beta\]

Where:

\[\mathbf{I} =\begin{bmatrix} 1 & \frac{I(X_2;X_1)}{I(X_1)} & \frac{I(X_3;X_1)}{I(X_1)} & \dots & \frac{I(X_n;X_1)}{I(X_1)} \\ \frac{I(X_1;X_2)}{I(X_2)} & 1 & \frac{I(X_3;X_2)}{I(X_2)} & \dots & \frac{I(X_n;X_2)}{I(X_2)} \\ \frac{I(X_1;X_3)}{I(X_3)} & \frac{I(X_2;X_3)}{I(X_3)} & 1 & \dots & \frac{I(X_n;X_3)}{I(X_3)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{I(X_1;X_n)}{I(X_n)} & \frac{I(X_2;X_n)}{I(X_n)} & \frac{I(X_3;X_n)}{I(X_n)} & \dots & 1 \end{bmatrix}\]

Right?

Wrong. Usually in economics when two goods are circularly interdependent, you don’t just apply this transformation once, but recursively until an equilibrium is reached. I.e. it’s not that \(\beta'_Y=\beta_Y+\beta_X\frac{I(X;Y)}{I(Y)}\), but rather:

\[\beta'_Y=\beta_Y+\beta'_X\frac{I(X;Y)}{I(Y)}\] \[\beta'_X=\beta_X+\beta'_Y\frac{I(Y;X)}{I(X)}\]

Which can be solved as a system of linear equations. More generally we have:

\[\beta'_XI(X)=\beta_XI(X)+\sum_{Y}\beta'_YI(X;Y)\]

(note that mutual information is symmetric i.e. \(I(X;Y)=I(Y;X)\)). This gives us:

\[\mathbf{J}\beta'=\beta\]

Where:

\[\mathbf{J} = \begin{bmatrix} I(X_1) & -I(X_1;X_2) & -I(X_1;X_3) & \dots & -I(X_1;X_n) \\ -I(X_2;X_1) & I(X_2) & -I(X_2;X_3) & \dots & -I(X_2;X_n) \\ -I(X_3;X_1) & -I(X_3;X_2) & I(X_3) & \dots & -I(X_3;X_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -I(X_n;X_1) & -I(X_n;X_2) & -I(X_n;X_3) & \dots & I(X_n) \end{bmatrix} \]

Which is the correct expression. Given the vector of “inherent liquidities” \(\beta\), you can calculate the correct liquidities \(\beta'\) by inverting this matrix.

Then there is the question of invertibility. I believe (though have not proven) that the matrix will be invertible as long as there is no “redundancy” in the questions.

(This can be seen in the two-question case: the matrix would be redundant if and only if \(I(X_1)I(X_2)=I(X_1;X_2)^2\) which, since \(I(X_1;X_2)\le I(X_1)\) and also \(\le I(X_2)\), can only occur when \(I(X_1)=I(X_2)=I(X_1;X_2)\).)

Note: to get these mutual informations, you need markets on each conditional question:

\[\begin{align} I(X;Y) &= H(X) - H(X\mid Y) \\ &= - \sum_x {P(X=x)\log P(X=x)} + \sum_{x}P(X=x)\sum_{y}P(Y=y\mid X=x)\log P(Y=y\mid X=x)\end{align}\]

For the conditional questions you do not have markets for, it is fair to just assume the mutual information is 0.