Prediction markets are cool — they work to get you the best possible estimate for any question you might have — but they have one key limitation: they only work for questions where ground truth will be directly revealed in future — or more generally, for questions that are either verifiable or falsifiable.

(Well I did try my hand in generalizing this to first-order (or generally hyperarithmetical) logic sentences: Betting on what is not verifiable or falsifiable (arXiv), but this barely scratches the surface of the realm of sentences we have beliefs about.)

Take e.g. historical questions (“Did Jeffrey Epstein kill himself?”), imprecise questions (“Is journalism dying?”), or generally questions about the noumenal world (“What is the true casualty count of the Israel-Hamas war?”) — all of which are questions which intelligent agents routinely ponder. Why do we regard such sentences as meaningful? Why do (unrelated, independent) agents often converge on mental models in which such sentences are meaningful? What makes them useful?

The reason that intelligent agents are interested in “subjective sentences” is that they capture correlations between more concrete, directly observable sentences. “Did Bob commit the murder?” is a “principal component” that captures correlations between questions like “Will crime rates increase if we release Bob?”, “Will evidence emerge of Bob being the murderer?”, “Will Bob write in his autobiography that he did the murder?”.

Thus “subjective sentences” are best thought of as “latent space variables”. They are agents’ internal representations, learned because they are useful in downstream tasks. One proposal for “latent variable prediction markets” comes from Latent variable prediction markets.

\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

  % Nodes X1, X2, ..., Xn vertically spaced and in a straight line, with circles
  \node[circle, draw] (X1) at (3,1.5) {$X_1$};
  \node[circle, draw] (X2) at (3,0) {$X_2$};
  % Manually adjust "\vdots" position to align with the midpoint if necessary
  \node (dots) at (3,-1.5) {$\vdots$};
  \node[circle, draw] (Xn) at (3,-3) {$X_n$};

  % Y node centered to the X nodes, with a circle
  \node[circle, draw] (Y) at (0,-0.75) {$Y$};
  % Label for Y, adjust position if needed to not overlap with arrow
  \node at (0,-1.5) {$\scriptstyle P(Y)$};

  % Arrows from Y to X1, X2, ... , Xn, with smaller P(...) labels
  \draw[->] (Y) -- (X1) node[midway, above, sloped] {$\scriptstyle P(X_1|Y)$};
  \draw[->] (Y) -- (X2) node[midway, above, sloped] {$\scriptstyle P(X_2|Y)$};
  \draw[->] (Y) -- (Xn) node[midway, above, sloped] {$\scriptstyle P(X_n|Y)$};
  
\end{tikzpicture}
\end{document}

I think a TL;DR of his idea is something like this: instead of letting traders directly bet on the joint distribution \(P(X_1,\dots X_n)\), we let them bet on \(P(Y)\) and any \(P(X_i|Y)\) and score them on the calculated joint distribution.

He seems to be stressing on natural language descriptions of \(Y\), but I think you should rather let the latents be a hidden variable layer and have the market discover what it means. The natural language description of \(Y\) should hopefully be completely redundant: instead the market will find natural choices for latents, where the “meaning” of a latent \(Y\) is determined entirely by its conditional distributions \(P(X_i|Y)\).

The idea of intelligence arising from markets has appeared in e.g. 

[MI3] classifier systems, e.g. in the Hayek machine and references therein

[PM3] logical uncertainty, e.g. Garrabrant Induction

[PM4] bounded rationality, e.g. Oesterheld et al, “A Theory of Bounded Inductive Rationality”.

The first of these — in which a market of relatively simple programmatic agents solve Tower of Hanoi — is particularly interesting, because it specifically addresses how intelligence emerges from a market of relatively dumb programs (which is important if you want markets to be anything more than an instrument for “choosing the best performers” out of an ensemble of already intelligent programs).

The Hayek machine looks like this: at each stage, an agent holds “write access to the world”, and it performs some action and obtains its reward. Then “write access to the world” is auctioned to the highest bidder, who pays its bid to our agent and becomes the new owner of the world. Well, we can think of this as a linear supply chain:

graph LR
    A((a_1)) -- world_1 --> B((a_2)) -- world_2 --> C((a_3)) -- world_3 --> D((a_4)) -- world_4 --> ANDSOON[...]
    style ANDSOON  fill-opacity:0, stroke-opacity:0;

(Arrows for rewards generated at each point in time are suppressed)

\usetikzlibrary{automata, positioning, arrows, shapes}

\begin{document}
\begin{tikzpicture}[->, >=stealth, auto, semithick, node distance=3cm]
\tikzstyle{agent}=[circle, thick, draw=black, fill=gray!20, minimum size=1cm]
\tikzstyle{utility}=[diamond, draw=black, fill=blue!20, minimum size=1cm, align=center]

% agents
\node[agent] (S1) {$\alpha_1$};
\node[agent] (S2) [right of=S1] {$\alpha_2$};
\node[agent] (S3) [right of=S2] {$\alpha_3$};
\node[agent] (S4) [right of=S3] {$\alpha_4$};
\node (ASO) [right of= S4] {$\dots$};

% utilitys
\node[utility] (O1) [below right = 1cm and 0.5cm of S1] {$U_1$};
\node[utility] (O2) [right of = O1] {$U_2$};
\node[utility] (O3) [right of = O2] {$U_3$};
\node[utility] (O4) [right of = O3] {$U_4$};

% Paths
\path (S1) edge node {world$_1$} (S2)
      (S2) edge node {world$_2$} (S3)
      (S3) edge node {world$_3$} (S4)
      (S4) edge node {world$_4$} (ASO);

% utility links
\draw[dashed] (S1) -- (O1);
\draw[dashed] (S2) -- (O2);
\draw[dashed] (S3) -- (O3);
\draw[dashed] (S4) -- (O4);

\end{tikzpicture}
\end{document}

With perfect competition, equilibrium looks like: * Each agent bids its estimate for the value of the world after its action (i.e. the total reward that will be generated by the world) (price efficiency) * The agent that can generate most value wins the auction (allocative efficiency) * Each agent is incentivized to take the most value-adding action, as that earns it the greatest sale value (productive efficiency)

… because any agent that underbid would be beaten by an identical agent that bid the correct value.

(In this simple set up, each agent in the chain makes zero profit at equilibrium: all value created is earned by the original owner of the “write access to the world” resource. This is because in our setting “write access to the world” is the only scarce resource, while the agents’ labour is in infinite supply — the agents have nothing else to do with their labour; there is no opportunity cost that paying them would account for.)

Observe that: 1. the depth of the chain is crucial to its “emergent intelligence” property: it lets different agents just figure out at what point they should bid to join the chain/when they have value to add, rather than having a whole bunch of agents just compete to complete the entire task. 2. the depth of the chain isn’t fixed, but rather depends on when agents stop bidding (e.g. because there is no longer any reward left to be extracted). In other words, at least to an extent the model “hyperparameters” are also determined by market mechanisms.

Let us specify the model fully.

[DP1] Viroli & McLachlan (2017), Deep Gaussian Mixture Models. http://arxiv.org/abs/1711.06929

[DP2] Bishop (1994), Mixture density networks. https://publications.aston.ac.uk/id/eprint/373/1/NCRG_94_004.pdf