Appendix

Process Model

We define an expert strategy as one that generates long-term aggregate returns significantly greater than those generated by random play.Footnote 2 In horse racing, the simplest form of random play involves probable random horse selection in each horse race with equal probabilities, although other more complex forms of random play are possible.Footnote 3 Any actual aggregate return-on-investment (ROI) (and number of bets in which it was achieved) may be compared to the distribution of ROIs yielded by random play. The probability of observing the actual ROI is correctly interpreted as the probability of making an equal or greater return by random play. The null-hypothesis is that a given observed ROI is drawn from this population of random returns. Rejection of the null hypothesis entails that the observed return would be unlikely to have occurred by chance, and we may then accept the alternate conclusion that the observed strategy is expert.

In order to address this, we consider an imaginary ‘naïve’ player of horse racing, who has no expert knowledge, and who implements the simple strategy of placing a bet of $1 on a randomly selected horse (starter) on each race.Footnote 4 We assume only simple betting on a single horse to win, and do not consider more exotic forms of betting (e.g., win, place or show). Denote the race index as \(k \in \left\{ {1,2, \ldots ,K} \right\}\) and the number of starters in each race as N k . The selection of the nth starter (\(n \in \left\{ {1,2, \ldots ,N} \right\}\)) would entail a raw return for the player on that race of \(r_{k}^{n}\):

$$r_{k}^{n} = \left\{ {\begin{array}{*{20}c} {\left. {r_{k}^{*} } \right|n_{k}^{ + } = n_{k}^{*} } \\ { 0 otherwise} \\ \end{array} } \right.$$

where \(r_{k}^{*}\) is the payoff for selecting the winning horse \(n_{k}^{*}\) (determined by the totalisator or equivalent system), and \(n_{k}^{ + }\) is actual selection made. The expected return of a random player choosing a single starting horse per race to win is

$$E\left( {r_{k} } \right) = \mathop \sum \limits_{n} P\left( {r_{k}^{n} } \right)r_{k}^{n}$$

with \(P\left( {r_{k}^{n} } \right) = P\left( {n_{k}^{ + } = n} \right)\) being the probability of selecting each starter and \(\mathop \sum

olimits P\left( {r_{k}^{n} } \right) = 1\). Since the naïve player selects horses with equal probability, they have a 1/N k probability of choosing the winning horse, other returns are zero, so the expected return is\(E\left( {r_{k} } \right) = N_{k}^{ - 1} r_{k}^{*}\).

Since the bet size is constant $1 for each race, the expected average ROI (R)

$$R^{K} = K^{ - 1} \mathop \sum \limits_{k} r_{k}$$

gives the ratio of total return to total investment over K races. Assuming the totalisator system allocates a standard proportion \(Q \in \left( {0,1} \right)\) of bets to winning players each race, and if the totalisator system were a perfect reflection of the underlying odds of winning, then

$$E\left( {R^{K} } \right) = E\left( {r_{k} } \right) = Q.$$

That is, the expected average return of a random player does not vary with the number of bets, and would be determined by the constant Q (currently approximately .85 in the Australian TAB). Note that this a prior expectation, given no knowledge of the odds of the payoff \(r_{k}^{*}\) of the horse the ultimately won. The posterior expectation \(E\left( {\left. {R^{K} } \right|r_{k}^{*} } \right) > Q\) (i.e. random players earn more) in races where the long-odds horse won, with the converse being true for races in which a short-odds horse won. If expert knowledge is not equally distributed in the population, then the totalisator system would not be a perfect reflection of the underlying odds of winning, and therefore \(E\left( {R^{K} } \right) < Q\), with the discrepancy in expected winnings being distributed to players with more expertise.

A human player picking starters on the basis of expert judgement has the hypothesis that their expertise will provide them with a probability of selecting the winning horse greater than the odds for that horse. We can summarise the effect of expert play as an ‘edge’ J > 0, that will improve their expected returns

$$E\left( {\widehat{R}^{k} } \right) = Q + J$$

which implies they may expect to experience positive returns only if J > 1 − Q, as \(K \to \infty\).

Analysis

A gambler deciding whether to persevere with their strategy in future (and indeed, whether to continue gambling at all) must first make the assumption that past return performance \(\left\{ {\widehat{r}_{1} ,\widehat{r}_{2} ,\widehat{r} \cdots \widehat{r}_{K} } \right\}, \widehat{R}^{K} = K^{ - 1} \sum {\widehat{r}_{K} }\) (using virtual or actual money) is a valid indicator of future performance. Assuming above average returns have been achieved, the next issue is to determine to what degree this play-history is a reliable indicator of underlying expertise (or an effective strategy) yielding: J > 0. Applying a standard frequentist perspective, we invert the problem by evaluating the null hypothesis: Assuming that J = 0, what is the probability of observing \(\widehat{R}^{K}\) (a player’s observed return after K bets) by chance alone? This is equivalent to determining the probability of drawing \(\widehat{R}^{K}\) or greater from the population of random returns \(R^{K}\).

Since modelling the distribution of the population of random returns \(R^{K}\) is critical to providing a rigorous basis for evaluating a given player’s return \(\widehat{R}^{K}\), it is worth considering its distributional properties. First, we note that it is parameterised broadly K and Q, but more exactly by the set of paired race parameters \(\left\{ {r_{k}^{*} ,N_{k} } \right\}KNN = 2N = 10R^{K}\), which determine the two point distribution of win (\(r_{k} = r_{k}^{*}\)) loss (\(r_{k} = 0\)) events for player picking a random starter in each race. In order to generalise, we must assume that each pair of race parameters in \(\left\{ {r_{k}^{*} ,N_{k} } \right\}\) are drawn from some joint distribution \(G\left( {r^{*} ,N} \right)\) with properties of independence and stationarity. Thus, \(\Pr \left( {R^{K} } \right) = F\left( {r^{*} ,N,K} \right)\), with \(G\left( {r^{*} ,N} \right)\) capturing the distribution of relevant race-characteristics: the number of starters and the spread of long-odds versus short-odds wins. It is not necessary that the data used to approximate \(G\left( {r^{*} ,N} \right)\) is the same as that that used to calculate \(\widehat{R}^{K}\) (i.e. \(\left\{ {r_{k}^{*} ,N_{k} } \right\}\)), as long as we can made the assumption that the distributional properties of \(\left\{ {r_{k}^{*} ,N_{k} } \right\}\) are similar to \(G\left( {r^{*} ,N} \right)\). In other words, we require the observed races to be a representative sample of the population of races of interest, in terms of their distribution of winners with respect to their starting odds. This provides the potential to generate standard reference tables for homogenous race populations (e.g., the Australian TAB), allowing practitioners to do hypothesis testing for a particular \(\widehat{R}^{K}\), without having to recreate the null distribution of returns. To combine some of the notation above, we restate the goal of the analysis as the following: calculate \(\Pr \left( {R^{K} > \widehat{R}^{K} } \right)\), assuming \(\widehat{R}^{K} \sim R^{K} = F\left( {r^{*} ,N,K} \right)\).

Classical theory does not provide much practical help in estimating F. The law of large numbers leads us to expect that \(R^{K}\) converges to \(E\left( {R^{K} } \right) = Q\) as \(K \to \infty\). However this tells us little regarding the dispersion or shape of observed returns around the mean of Q for realistic bet-run lengths K. From our description of the generating process, and assuming no prior information about the race, the distribution of \(R^{K = 1}\) is directly given by the distribution\(G\left( {r^{*} ,N} \right)\): it is a mixture, with majority probability concentrated at zero, with a very long-tail of non-zero returns following an inverse relationship with \(\Pr \left( {R^{K} } \right) \cong 1/R^{K}\). As K becomes very large, understanding the shape of F is also straight-forward. As we are considering aggregate return, and given the distribution of \(R^{K}\) is modelled as a sum of independent and identically distributed race returns, the central limit theorem states that the distribution of \(R^{K}\) converges to the normal distribution as \(K \to \infty .\)

For intermediate values of K, the shape of F may display significant deviations from normality. Even considering simplified approximations \(G\left( Q \right) \cong G\left( {r^{*} ,N} \right)\), deriving closed-form expressions for F(Q,K) does not appear to be a practical option, as the number of horses and distribution of odds varies randomly from race to race. We avoid such difficulties by simply adopting a non-parametric approach, and implement a Monte-Carlo sampling method for estimating the distribution of \(R^{K} \sim F\) for various bet-run-lengths K by repeatedly re-sampling a representative and homogenous sample of races with a constant Q expected return.

An efficient approach to sampling an empirical distribution of \(R^{K}\) from \(K = 1,2, \ldots ,K^{\hbox{max} }\) is via Monte Carlo simulation (Mun 2006; Pitt et al. 2012); i.e. to simulate a large set of J play histories, and to record the ROI after each bet, at each replicate: \(r_{j,K}\). Each history of raw returns \(\widehat{r}_{j,K}\) amounts to a Markov chain with transition probabilities determined by \(\left\{ {r_{m}^{*} ,N_{m} } \right\}\) (assumed to be representative of \(G\left( {r^{*} ,N} \right)\)). The distribution of progressive returns is used to generate statistics of interest, such as the .9, .95, and .99 ‰. Though moderately computationally intensive with contemporary computational resources, Algorithm 1 is conceptually very simple, and as noted above, may be pre-computed in reference tables for practical use in varying gambling contexts. Algorithm 1 is written in terms of the intuitive process of (a) selecting a random race from the ‘library’ of races, and then (b) selecting a random horse, at each step. However, programmers should be aware that it may be written and implemented most efficiently by bootstrap resampling from the full return distribution \(\left\{ {r_{m}^{n} } \right\}\) at each Markov transition.

All analyses and plotting was undertaken in the statistical programming environment ‘R’ (Team 2011). Code is available on request.