[Paper Review] Emergence of scale invariance and efficiency in a racetrack betting market

Shintaro Mori, Masato Hisakado|ArXiv.org|Nov 17, 2009

Sports Analytics and Performance3 references3 citations

TL;DR

This paper investigates the emergence of scale invariance and market efficiency in Japan Racing Association (JRA) racetrack betting markets through time-series analysis of win bet fractions. Using a Pólya-like voting model in a double scaling limit, it demonstrates that the cumulative distribution of winning horses scales as $ x_1 = x_0^eta $, with $ \beta = 1.81 $, over the full range $ 0 \leq x_0, x_1 \leq 1 $, revealing exact scale invariance driven by self-organized dynamics in collective betting behavior.

ABSTRACT

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 $x_{0}$. $x_{1}\propto x_{0}^α$ 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 $x_{1}=x_{0}^α$ holds over the entire range $0\le x_{0},x_{1}\le 1$.

Motivation & Objective

• To analyze the temporal evolution of win bet fractions in JRA races and identify emergent patterns in horse rankings.
• To investigate the emergence of scale invariance between cumulative distributions of winning and losing horses.
• To explain the observed scale invariance through a simple, stochastic voting model with two types of bettors.
• To explore the conditions under which exact scale invariance $ x_1 = x_0^\alpha $ arises, particularly in the double scaling limit.
• To assess the convergence to market efficiency and the non-monotonic dynamics of vote distribution during the betting process.

Proposed method

• Time-series data from 3,250 JRA races in 2008 were analyzed, focusing on win bet fractions derived from public odds using the transformation $ x^{r}_{i,k} = \frac{0.788}{O^{r}_{i,k}-0.1} $, normalized to sum to one.
• The betting process was mapped to a time variable $ t $ based on the average public win pool $ V_r $, enabling temporal alignment across races.
• A stochastic voting model was introduced where bettors vote proportionally to current vote shares, simulating a Pólya urn-like process with two categories: winning ($ \mu=1 $) and losing ($ \mu=0 $) horses.
• The model assigns relative vote probabilities proportional to parameters $ s_1 $ and $ s_0 $, leading to gamma-distributed vote counts in the thermodynamic limit.
• Exact scale invariance $ x_1 = x_0^\alpha $ was derived in the double scaling limit $ Z_0 \to \infty $, $ s_1, s_0 \to 0 $, with $ \alpha = s_1/s_0 $ fixed.
• The model was shown to be equivalent to a random ball-removal process with weights $ s_1 $ and $ s_0 $, generating exact gradation patterns across $ \alpha $ values.

Experimental results

Research questions

• RQ1 How does the cumulative distribution of winning horses relate to that of losing horses during the evolution of betting in JRA races?
• RQ2 What mechanism underlies the observed power-law scaling $ x_1 \propto x_0^{1.81} $ in the small win bet fraction region?
• RQ3 Under what conditions does exact scale invariance $ x_1 = x_0^\alpha $ hold over the entire interval $ [0,1] $?
• RQ4 Why is the convergence to market efficiency non-monotonic, and what drives the re-concentration of votes on popular horses after initial dispersion?
• RQ5 How does the double scaling limit $ Z_0 \to \infty $, $ s_1, s_0 \to 0 $ with $ \alpha = s_1/s_0 $ fixed enable exact scale invariance?

Key findings

• The cumulative distribution of winning horses $ x_1 $ scales as $ x_1 \propto x_0^{1.81} $ in the small win bet fraction region, indicating strong scale invariance.
• Market efficiency emerges over time, but convergence is non-monotonic, with votes initially spreading across many horses before re-concentrating on favorites.
• In the double scaling limit $ Z_0 \to \infty $, $ s_1, s_0 \to 0 $ with $ \alpha = s_1/s_0 $ fixed, the exact relation $ x_1 = x_0^\alpha $ holds over the entire range $ 0 \leq x_0, x_1 \leq 1 $.
• The scale invariance arises from a mixing process in a two-category Pólya-like voting model, where vote shares follow gamma distributions with shape parameters $ s_1 $ and $ s_0 $.
• The model is equivalent to a random ball-removal process with weights $ s_1 $ and $ s_0 $, which generates exact gradation patterns across varying $ \alpha $.
• The double scaling limit is essential: without $ Z_0 \to \infty $, the system collapses to a single dominant horse; without $ s_1, s_0 \to 0 $, scale invariance breaks.