A Monte Carlo Analysis of Hypothetical Multi-Line Slot Machine Play

Abstract

Behavioural research into slot machine gambling tends to focus on characteristics of the gambler or on qualitative aspects of the slot machine such as audiovisual displays and bonus features. In this paper we take a different approach by using Monte Carlo simulation to relate hypothetical slot machine gambling behaviour to the statistical characteristics of the slot machines themselves. The measures we use – expected monetary win, volatility of payouts, and the probability that any single play returns a winning result – have the advantage that they are mathematically precise and can be linked to psychological risk and return criteria that people may look to as they decide both whether to gamble or not and how to play.

Notes

1. Casino revenue in South Africa in 2006 amounted to US$1.4 billion, the next biggest gambling revenue coming from the state lottery (US$314 million) and horserace betting (US$168 million) (National Gambling Statistics 2005/2006, http://www.ngb.org.za/home.asp?pid = 138). Taxable revenue from casinos in Nevada, USA totalled US$11 billion for the year 2004–2005 (State of Nevada Gaming Revenue Report 2005, http://www.gaming.nv.gov/documents/pdf/1 g 05oct.pdf). Research in Dowling et al. (2005) indicates that over half of all gambling expenditure is lost on slot machines, and the 2006 National Survey on Gambling and Problem Gambling in South Africa (Collins & Barr, 2006) indicates that average monthly expenditure on slots is US$111, far greater than the state lottery (US$12) or horserace betting (U$75). The casinos in South Africa, in terms of the configuration and type of their slot machines follow very much the US casino model. That is, the floor is dominated by modern video slot machines manufactured typically by the large USA manufacturers such as IGT, Ballys and Williams. To a lesser extent there are machines from the German manufacturer Atronic and Australian manufacturer Aristocrat.

2. It is worth noting that the volatility (of payout) for a particular machine is generally defined for the case of 1 line play of denomination equal to 1. This core volatility of a machine is a key factor in determining the playing experience of a machine. For example, if a machine has a potential bonus high payout (occurring clearly with low probability) then the volatility of the machine will be high compared to a machine that has a configuration of a lower bonus payout (probably occurring with higher probability). Thus, in the industry volatility of a machine is often associated with the potential excitement value of a machine and thus often its attractiveness to players. An important point of this paper is to clarify how players, by selecting the number of lines and quantum of bet for some machine with a given core volatility, will have exercised control over the effective volatility of payout for the spin in question; selecting a larger bet simply raises the volatility proportionately while selecting many lines (and keeping the bet per line the same) raises the volatility less than proportionately and massively increases the hit percentage. Thus while machines may be selected by casinos according to their core volatility and hence perceived excitement potential, the focus here is on the player's own decision process to self-select some number of lines and quantum of bet given this core volatility, in order to maximise their own player experience.

3. In many instances it is useful to also refer to the expected win per unit bet i.e. E[W _a,b /ab], which when expressed as a percentage gives a single summary of a particular machine's propensity to pay out. The casino industry itself makes uses of the rather more obscure measure 100 × (1 + E[W _a,b /ab])%, apparently without irony called the ‘win percentage’. A player putting US$1 into a machine with a win percentage of 95% thus ‘wins’ 95 cents (for a net loss of 5 cents). In this paper we generally use the expected win measure E[W _a,b ], on the basis that it is consistent with the general literature on choice between risky lotteries (e.g. Diecidue, Schmidt & Wakker, 2004).

4. The lines in Figure 1(a) are almost always line 1, 2 and 3 of a real-world slot machine. Lines 2 and 3 in Figure 1(c) are typical of higher-numbered lines. The configuration in Figure 1(b), with lines overlapping completely, is never encountered.

5. The expected win results given here are all theoretical. Expected wins obtained from the Monte Carlo simulations are guaranteed to converge to these values in the limit, and after 5,000,000 simulations typically only differ by 1–2% from the theoretical expected win value (e.g. for a = 25 on machine C, a simulated expected win of − 0.565 was obtained). Any differences merely obscure the substantive issues. Also note that the expected win figures given here are the absolute expected returns E[W _a,b ] and have not been adjusted for the increasing amounts bet when more lines are played. The expected win percentage 100 × E[W _a,b /ab] is constant at 2.3% for any number of lines played (or amount bet per line). The volatilities have been adjusted for the amount bet, following the earlier discussion on this point.