Cameron L. Jones

Centre for Mathematical Modelling, School of Mathematical Sciences

Swinburne University of Technology

P.O. Box 218, Hawthorn 3122, Australia

Fax: +613 9819 0821, Email: cameron.jones@swin.edu.au

Racetrack gambling is an example of a complex feedback system between the expectation of winning and the probability of losing. This paper examines one aspect of the economic process of gambling, by looking at how humans choose to wager money on the outcome of three different race types: thoroughbred, harness, and greyhounds. A relationship called the Golden section (1:1.618) can then be used to model the division between the amount bet to Win or Place. This information is provided by the Totaliser and is available online, and therefore represents the set of discrete decision events that contribute towards the market. The results show that for races involving a human element (a human rider) for thoroughbred and harness races, the public select to divide wagers into win and place ratios reflecting emergent global decision-making towards Golden section scaling. For greyhound races (with no human rider), the betting public do not follow Golden section statistics in deciding bet size proportionality. This finding is important, since race wagering markets can be modelled as complex systems of interacting agents. This study also reveals the nonlinear signature of human decision making under risk-reward. Further, deciding to bet in the average 0.618 of ones’ money to win demonstrates attractor behavior that results in emergent self-organization.

Keywords: self-organization, gambling markets, golden section, emergence

INTRODUCTION

For centuries a special ratio called the Golden Section has captivated artists, scientists, architects, mathematicians and non-mathematicians alike. The Golden Section or Phi j , describes the division of a segment into two parts called extreme and mean ratio, approaching, 1:1.618. Natural examples include the logarithmic spiral in chambered nautilus shells, whorl patterns in sunflowers, pineapples and pinecones, leaf arrangements, and branching patterns in plants, rivers, and diffusion limited aggregates. In turn, the Golden section also features prominently in architecture, music and painting.

Figure 1. The Golden section ratio that is used to model the amount of money wagered to Win or to Place.

If we consider a line segment of length ab, then point c divides ab into extreme and mean ratio if and only if ac or cb is the mean proportional of ab and the other length. If c divides ab into extreme and mean ratio, then ab/ac=ac/cb or ab/cb=cb/ac. If length ab=1, then length ac=0.618, with the equivalent proportionality for ab/ac=1.618. This ratio is used to examine how the amount of money wagered on different types of races (thoroughbred, harness, greyhound) and on different bet types (Win versus Place) is distributed. Interesting questions include whether the amount of money wagered on each race type and each bet type shows a normal distribution, or whether the market shows emergence towards an attractor, that is often a characteristic of fractal and chaotic phenomena.

The gambling market is an economic system composed of sets of interacting agents. The generator mechanism consists of the totaliser, the different types of runners and various probability measures. Such systems possess perpetual novelty since there are many possible configurations. The dynamics and regularities of the gambling market are likely to display a mix of persistent, antipersistent and random behavior. Each race is a game with each runner having a market price and a natural price that is set by agent demand. The market price is reflected in the totaliser odds, and is one measure of probability. Agents that participate in such games seek to profit when there is equivalence between the expected outcome (market price) and the actual outcome (natural price). Agent interactions therefore assess risk and reward in order to participate in a game-based economy. Since the market grows in size in the approach to each race start, various probability values are continuously communicated throughout the market. This information is generated and displayed hierarchically, and can self-regulate and act as generators for higher levels of organisation.

Racetrack betting and stock market investment share several properties in common. In both cases, future earnings are uncertain, there are a large number of participants, and extensive information is available concerning investment variables [1, 2, 3, 4, 5, 6]. Wagering on race outcome is commonly done through a Pari-Mutual system or totaliser. The tote screen displays win and place dividends for each runner in a given race which reflect the public's odds preferences [7]. Wagers for a particular set of runners in a particular race form the betting pool, from which the track take is first deducted from each wager. Totaliser dividends are updated periodically, and in Australia show the return to Win and to Place for each runner. Although the process of wagering by the public is a continuous event up until race jump, tote dividend changes occur at discrete intervals. Therefore, the tote display as a whole represents the closed set of discrete update events that reflect market opinion of each runner's chances of winning or placing.

The totaliser sets prices for win or place for each runner in a given race. These prices or dividends fluctuate, according to how confident the betting public (the market), perceives each runners probability of winning or placing. The tote is therefore a good example of an iterative feedback system, where information from the public is introduced at discrete steps in time. Such systems are called discrete dynamical systems. Although the way in which money is wagered on the outcome of a particular race may appear frenzied and continuous, the important point is that the tote displays this information in discrete time steps. In Australia, Totaliser information is presented in the form of dividends (for $1) to win or to place. In the lead up to post time, or race jump, it is common for the dividends to fluctuate, often with remarkably large swings. Predicting the outcome of such games for profit involves placing a bet on one or more runners in advance.

This paper only considers how the win and place pool markets evolve over time, and how the amount of money split between the two bet choices might generate Golden section statistics. This experiment considers only how the total amount bet (ab) on a given race, divided into a win pool component (ac) and a place pool component (cb) is distributed. Commonly, higher dividend returns are paid on successful win bets compared with lower return place bets. Many totalisers such as www.tabcorp.com.au post data to the Internet. As more people enter the wager market, the amount bet to win and place increases following a cumulative power law.

A market is efficient if its prices always fully reflect available information. Most empirical work on market equilibrium has shown such systems to be a function of expected returns. Returns are then quantified on (i) historical results, (ii) other publicly available information, and (iii) inside information. Access to and action taken with available information therefore generates markets that are weak, semi-strong or strongly efficient [8]. The detection of market inefficiency denotes that a security (or bet) is overvalued or undervalued [9]. As well, the process of supply and demand is an example of a feedback system where the demand for a good such as a win or place bet is regulated by supply and the value function between the market price and the natural price [10].

Figure 2. A feedback scheme illustrating the self-organizing process of supply c and demand r having a market price m, and a natural price n.

METHODS

Data was obtained over several months for thoroughbred, harness and greyhound betting markets for races run nationally, and consisted of both metropolitan and country meetings. Data covered 3729 races in Australia in 1999. The totaliser commonly posts updates of the win and place pools at timed intervals approaching race start. All available data posted to the web was used for analysis by preparing histograms of the proportionality between win + place/win that was equivalent to ac+cb/ac. This data was extracted from a Microsoft Access database.

The Golden section ratio is closely related to the Fibonacci sequence defined as: F₁=F₂=1, F_n+1=F_n+F_n-1, n³ 2. Each successive number is the sum of the two preceding numbers, generating the sequence pattern: 1, 1, 2, 3, 5, 8, 13… The ratio of successive pairs of Fibonacci integers to their preceding value generates a proportionality series: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8… Each successive fraction approaches the Golden section with increasing precision.

A histogram was prepared for each of the three race sets using staggered bins to define the region of interest clustered around 1.61. The following bins were used: 0-1, 1.1, 1.21, 1.31, 1.41, 1.49, 1.55, 1.67, 1.75, 1.85, 1.95, 2.05. Therefore the bin which overlaps the Golden section at 1.618 falls at row 1.67 (see Tables in the Results section). This bin contains all data between 1.55-1.67 and was chosen to include data that overlaps this range.

Wager markets for thoroughbreds covered 961 races with a total of 13,199 update events spanning tote open for each race until tote close to allow for late bets to be included (i.e. bets placed just prior to race start) that are sometimes delayed in being posted. Respectively, harness wager markets covered 834 races and 13,312 update events. For greyhounds, a total of 1934 races and 32,603 update events were evaluated. All bettor behavior was accounted for by including Totaliser information posted >-1minute after tote close, but before the race had finished. This reflects late bets being collected from oncourse and all off-course outlets within the state of Victoria. Final totaliser information for each runner was not included since this last update occurs after the race is over and reflects track take after ranking against dividend payouts.

Notably, the Pari-Mutual system offers many possibilities to study the dynamics of market fluctuations. For example, does the totaliser reflect how members choose when to act? The datasets described here are premised on the identification of how the public’s collective decisions cluster together. This behavior is common also to how money managers tend to choose portfolios on the basis of how other money managers or traders bias their forecasts towards those forecasts previously made by other analysts [11]. Indeed, wagering markets are subject to psychological factors that influence bettor decisions [12].

A comparison between the tables in Figure 3, 4 and 5 shows that the cumulative percentage of money wagered to Win or to Place is apportioned towards the Golden section more strongly for thoroughbred horse racing and harness racing, than for greyhound racing. It is notable that for horse and harness racing, the frequency is highest for the histogram bin containing the Golden section proportionality (highlighted up to 1.67 in Tables 1-3). The histograms for each database (DB) were used to evaluate the proportionality for (Win + Place) / Win for each race market.

Figure 3. Frequency distribution, cumulative percentage and histogram for the Harness racing market.

Figure 4. Frequency distribution, cumulative percentage and histogram for the Horse racing market.

Figure 5. Frequency distribution, cumulative percentage and histogram for the Greyhound racing market.

The major finding of this work showed that for races involving a human element, that is a human rider for thoroughbred and harness races, the public select to divide wagers into win and place ratios reflecting emergent global decision-making towards Golden section scaling. Interestingly, for greyhound races (with no human rider/involvement), the betting public do not follow the Golden section in deciding bet size proportionality. This finding is important, since race wagering markets can be modelled as complex systems of interacting agents. Furthermore, deciding to bet in the average 0.618 of ones money to win reveals scaling information concerning how individuals assess risk and reward in wagering markets. These results may impact on other emergent economic or social markets that share similarities with totalisers. It is suggested that race wagering markets self-organize towards the Golden section to optimize resource allocation of money. These findings may also be useful in predicting the social and economic impact of problem gambling by looking for regional differences in Totaliser statistics. Cluster activity that shows Golden section behavior may be a useful and sensitive indicator of a well-developed market.

REFERENCES

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ACKNOWLEGEMENTS