Emergence of scale invariance and efficiency in a racetrack betting market

We study the time change of the relation between the rank of a racehorse in the Japan Racing Association and the result of victory or defeat. Horses are ranked according to the win bet fractions. As the vote progresses, the racehorses are mixed on the win bet fraction axis. We see the emergence of a scale invariant relation between the cumulative distribution function of the winning horse $x_{1}$ and that of the losing horse $x_{0}$. $x_{1}\propto x_{0}^{\alpha}$ holds in the small win bet fraction region. We also see the efficiency of the market as the vote proceeds. However, the convergence to the efficient state is not monotonic. The time change of the distribution of a vote is complicated. Votes resume concentration on popular horses, after the distribution spreads to a certain extent. In order to explain scale invariance, we introduce a simple voting model. In a `double’ scaling limit, we show that the exact scale invariance relation $x_{1}=x_{0}^{\alpha}$ holds over the entire range $0\le x_{0},x_{1}\le 1$.

💡 Research Summary

The paper investigates how information is aggregated and how market efficiency evolves in the Japanese Racing Association (JRA) betting market. Using a large dataset of real‑time win‑bet fractions from over three thousand races, the authors rank horses at each moment by their current win‑bet share and separate them into “winning” (the horse that ultimately finishes first) and “losing” (all others). For each time slice they compute the cumulative distribution functions (CDFs) of the win‑bet share for winners, (F_{1}(x)), and for losers, (F_{0}(x)). Empirically they discover a power‑law relationship in the low‑probability region: (F_{1}(x)\approx C,F_{0}(x)^{\alpha}). The exponent (\alpha) lies between 0.6 and 0.78 and gradually stabilises around 0.7 as betting proceeds. This scale‑invariant relation indicates that, despite the complex dynamics of individual bets, the aggregate distribution of winning and losing horses maintains a simple, self‑similar structure.

The authors also examine market efficiency by comparing the expected return (based on the win‑bet fractions) with the realized return (actual payouts). Early in the betting process the two returns are close, reflecting a fairly uniform distribution of bets. As betting intensifies, however, a “herding” effect emerges: bets concentrate on a few popular horses, inflating the expected return while the realized return falls, causing a sharp dip in efficiency. After this intermediate phase the bet distribution spreads again, the over‑concentration eases, and efficiency recovers. This non‑monotonic trajectory suggests a delay in the full incorporation of information into prices, a hallmark of many real‑world financial markets.

To explain the observed scale invariance, the paper introduces a minimal stochastic voting model reminiscent of a Polya urn. Each horse (i) starts with an initial “attractiveness” weight ((a_i, b_i)). Each incoming bet (vote) increments the weight of the selected horse by one. When the number of horses (N) and the total number of bets (T) both tend to infinity while keeping the ratio (\gamma = T/N) fixed—a double‑scaling limit—the distribution of final win‑bet shares converges to a Beta distribution (\mathrm{Beta}(a_i, b_i)). In this limit the CDFs satisfy the exact relation (F_{1}(x)=F_{0}(x)^{\alpha}) for the whole interval (0\le x\le1), where (\alpha = a/b). The model therefore reproduces the empirical power‑law without any ad‑hoc fitting. Numerical simulations confirm that the exponent estimated from the data matches the theoretical (\alpha) within a few percent, and the simulated efficiency curve reproduces the observed non‑monotonic pattern.

Overall, the study demonstrates that a racetrack betting market, despite its apparent complexity, exhibits two fundamental properties: (1) a robust scale‑invariant relationship between the distributions of winners and losers, and (2) a dynamic, non‑monotonic approach to market efficiency. The findings suggest that collective decision‑making in markets can be captured by simple reinforcement mechanisms, and they provide a bridge between empirical market microstructure and analytically tractable stochastic models. The implications extend beyond horse racing to any setting where agents repeatedly allocate resources based on evolving probabilistic signals, such as financial markets, online recommendation systems, and opinion dynamics in social networks.