Author: Dr John Fry

The United Kingdom is a nation of gamblers. It is thought that nearly half of all the adults in the United Kingdom gamble at least once a month. Further, even excluding the national lottery, the gambling industry in the United Kingdom currently employs over 100,000 people [1]. Against this wider backdrop, there are particular concerns that online gambling may be especially addictive [2].

Some people will bet on almost anything. One of the recent growth areas has been in virtual horse racing – based on nothing more equine than a pseudo random number generator [3]. This has been accompanied with a recent growth in financial spread betting, see e.g. [4]. One of the things that has recently become popular are binary options on stock markets whereby participants are invited to bet on whether or not the underlying stock price will rise or fall over a short time interval (say e.g. 10 minutes or less). Such has been the fascination with both online betting and financial betting that binary options have even been created for pseudo-random stock markets, complete with simulated support and resistance lines. These support and resistance lines define a de facto minimum value and a de facto maximum value that prices are supposed to obey. However, some temporary deviation outside of these limits is possible even if the support and resistance levels have been correctly defined.

One such popular pseudo random stock market game offers a return of 80%. This means that if we correctly guess whether the price will go up or down a bet of £*x* would return 1.8*x*.

A mathematical note of caution. Suppose that we have £1 available and we place a bet of *x* and keep 1-*x* in reserve. This means that if guesses are correct with probability 0.5, then the expected value of the gamble is 1/2(1.8*x*+1-*x*)+1/2(1-*x*)=1-0.1*x*. Since this is decreasing in* x*, this would suggest that we should leave well alone! Repeated play of the game should result in a 10% return for the market maker – a result that appears to be roughly in line with bookmaker margins in other contexts – and would mean that anybody playing the game for long enough would eventually lose everything.

When would placing a bet in such a game actually be worthwhile? Suppose that we are able to correctly judge the direction of future price movements with probability* p*. In this case we have that the expected value of the gamble is given by 1+(1.8*p*-1)x. This is increasing in *x* only if 1.8*p*>1, i.e. if *p*>5/9. (Note that even if *p*=5/9 then the gambler faces eventual ruin upon repeated play of the game). The implications of this simple result are striking. Firstly, it is perhaps to be expected that online gambling is potentially so addictive when playing successfully and depends on one’s ability to determine if our probability of correctly guessing the direction of future price movements is greater than 5/9 or just equal to 1/2. How easy would it be to tell the difference in the heat of the moment? Secondly, we can also be right more often than we are wrong in our guesses of the direction of future price movements and still lose money! The similarity of this result to the absence of arbitrage in classical financial models is striking.

How can we determine whether or not *p*>5/9? This is nothing more than the classical question of inverse probability originally studied by Thomas Bayes. Suppose that under the Bayesian paradigm [5] we have a *U*[0, 1] prior for *p* and we play the game n times. We can update the posterior probability distribution for *p* given the observed data. We have that the posterior distribution for *p* is Beta(*y*+1, *n*–*y*+1) where *y* denotes the number of correct guesses. The probability that *p*>5/9 can then be calculated by integrating over the posterior probability distribution. The final result is given by 1-I5/9(*y*+1, *n*–*y*+1), where Ix denotes the regularised incomplete beta function.

How many consecutive correct guesses would we need in order to be confident that playing the game really was profitable? Suppose that we guess correctly on our first *n* plays of the game. In this case it follows that the posterior distribution for *p* is Beta(*n*+1, 1) using *y*=*n* in the above formula. It is a simple exercise in integration to show that in this case* Pr*(*p* less than or equal to 5/9)=(5/9)*n*+1. In this case, three consecutive correct guesses suggests that *Pr*(*p* less than or equal to 5/9)=0.095. Four consecutive correct guesses gives *Pr*(*p* less than or equal to 5/9)= 0.053, whilst five consecutive correct guesses gives *Pr*(*p* less than or equal to 5/9)=0.029. So, using the standard interpretation of probability values, three consecutive correct guesses would be needed to suggest that playing the game might be in our favour and we would need five consecutive correct guesses to be more confident. (Note that there is a 1/8 chance of 3 consecutive correct guesses by chance alone so that these simple rules are, in themselves, far from definitive.) Continuing in this vein, seven consecutive guesses would be needed for this probability to be less than 0.01. Eleven consecutive guesses would be needed for this probability to be less than 0.001.

The random patterns formed by stock markets and pseudo-random stock markets have a dangerous fascination about them. There is an entire field of behavioural finance that describes how psychological factors leave human beings prone to making poor financial decisions [6]. Many trading books also contain whole chapters on the psychological pitfalls of trading [7]. Gambling panders to evolutionary stable human preferences for risk, return and greed. Online trading games are also likely to be highly addictive – especially if, as in real markets, the margins between success and failure are likely to be very thin. The effect is likely to be intensified by the relative ease with which bets can be placed on such markets.

Amidst all this uncertainty, one central message has remained true since time immemorial. Please gamble responsibly.

**References**

[1] Gambling Commission: Facts and figures.

http://www.gamblingcommission.gov.uk/pdf/Facts-and-figures-2016.pdf

[2] http://www.techaddiction.ca/online-gambling-casino.html

[3] Finley, B. (2016) People will bet on anything. ESPN Horse Racing.

http://www.espn.com/horse/columns/finley_bill/1519329.html

[4] Burns, R. (2010) The naked trader’s guide to spread betting: A guide to making money from up and down shares in financial markets. Harriman House, Petersfield, Hampshire.

[5] Lee, P. M. (2012) *Bayesian statistics: an introduction*, Fourth edition. Wiley.

[6] Forbes, W. (2009) *Behavioural finance*. Wiley, Chichester.

[7] Burns, R. (2014) The naked trader, fourth edition. Harriman House, Petersfield, Hampshire.