Our work

Prediction market prices are widely read as event probabilities—pure expectations with no risk premium. Whether a wedge between price and probability reflects compensation for risk or a distortion of beliefs cannot be settled in ordinary markets, where the physical probability is unobserved and every test joins a pricing model to a belief model. Event contracts loosen that constraint: each resolved contract records a 0/1 outcome, so repeated resolutions estimate the price–outcome calibration relationship among listed, resolved contracts even though no single contract's latent probability is observed. Summarizing that relationship by a scalar λ requires a maintained parameterization; I use the Wang transform, which arises exactly when the market-selected pricing measure is a one-parameter exponential tilt of a latent Gaussian threshold factor, p(mkt) = Φ(Φ⁻¹(p*) + λ), with favorite-longshot bias as a corollary. The data are 291,309 resolved contracts from eight source samples on six platforms. In the three exchange samples (N = 287,118), the fitted λ is positive on Polymarket (0.166) and Kalshi (0.187) and negative on Manifold (−0.218), the one play-money exchange; the three no-money forecasting tournaments are positive (0.287, 0.570, 0.635). The Manifold-versus-real-money contrast is descriptive: capital status varies with venue design, contract mix, aggregation, and participant composition, and the contrast does not identify a stakes channel. In the baseline Polymarket sample the wedge falls with volume (−0.072 per log unit, SE 0.005), rises with duration (+0.143 per log unit, SE 0.012), and declines over contract life. Contract returns are essentially uncorrelated with the S&P 500; under the maintained mapping, attributing the wedge to aggregate-risk compensation would require a portfolio beta near 139, more than forty times the upper confidence bound of the estimated beta. The data do not distinguish the remaining mechanisms—local risk-bearing, beliefs, market design, or omitted factors—but they do show that the tested linear S&P covariance channel is quantitatively too small to reproduce the fitted wedge.

A collector polled the Kalshi and Polymarket game-winner contracts every 15–30 seconds through the April 4, 2026 semifinal (657 and 857 observations), alongside ESPN play-by-play for a score-based benchmark. Both venues opened Illinois near 55% and diverged by up to 25 percentage points in fast-moving stretches, with Polymarket consistently pricing Illinois higher. A naive tick-level Wang-transform estimate comes out three times too large — an artifact of the probit transform's slope near 0 and 1. Restricted to the competitive range, the intra-game wedge is about 0.13 on both venues, in line with the paper's cross-sectional estimate. Against ESPN's in-game model, Kalshi's residual was indistinguishable from zero while Polymarket's was small but significant. One game, but the methodological lesson generalizes: intra-game wedge estimates must filter extreme probabilities.