Condorcet Markets

TL;DR

This work analyzes information markets for single binary events through an epistemic social choice lens, linking market equilibrium prices to collective truth-tracking rules. It proves equivalence results: Naive markets implement simple majority via equilibrium prices that correspond to $(1-p)$-quantiles of agent beliefs, while Kelly markets implement weighted-majority rules with weights proportional to $2q_i-1$, with prices equaling the average belief. Extending to taxed Kelly markets, the equilibrium price ratios converge to log-odds sums, enabling weights proportional to $\ln\frac{q_i}{1-q_i}$ and approaching the Grofman optimal weighting in the limit, i.e., a perfect weighted majority. These results suggest a principled, market-based route to robust truth-tracking elections and point to future work in iterated settings, alternative equilibria, and relaxing independence or prior assumptions.

Abstract

The paper studies information markets concerning single events from an epistemic social choice perspective. Within the classical Condorcet error model for collective binary decisions, we establish equivalence results between elections and markets, showing that the alternative that would be selected by weighted majority voting (under specific weighting schemes) corresponds to the alternative with highest price in the equilibrium of the market (under specific assumptions on the market type). This makes it possible in principle to implement specific weighted majority elections, which are known to have superior truth-tracking performance, by means of information markets without needing to elicit voters' competences.

Figures (5)

• Figure 1: Elections and information markets commute.
• Figure 2: Simple majority and Naive markets commute.
• Figure 3: Weighted majority with weights $q_i-0.5$ and Kelly markets commute.
• Figure 4: Left: returns after taxation by $T$ as a function of investment (Equation \ref{['eq:taxed']}). Right: investment strategy (red) approximating $\ln\left(\frac{b_i}{1-b_i}\right)\frac{1}{k}$ (blue) as $k$ grows when price equals $0.5$. Functions plotted for $k \in \{ 0.1,0.2,1,2,10,20 \}$.
• Figure 5: As tax parameter $k \to \infty$, perfect majority and taxed Kelly markets commute.

Theorems & Definitions (30)

• theorem thmcountertheorem: grofman83thirteen
• remark thmcounterremark
• definition thmcounterdefinition: Competitive equilibrium
• remark thmcounterremark: Null price
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof