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I wish to illustrate two extreme cases of playing the lottery, one which is the obvious and the other which is the less obvious but more common. In the first case I will look at a situation where you can buy all combinations of tickets, while in the second case I will look at the situation where you only buy one ticket. It is worth pointing out that the lottery has predictable odds (albeit small) and a limited downside, so Nassim Taleb (of Black Swan/ Anti-fragile fame) places lotteries into the ludic-fallacy category (where real-life is assumed to be like a game, especially in terms of odds). I can't disagree that the odds are not well-known, nor I can disagree that the downside is limited but then the downside in many real-life situations are also known (and capped). My modification to his comment is when the cost is very low, in context it is low as a percentage of earnings, and ideally you ought to look for situations where the odds are much higher than the lottery (perhaps venture capital).

I'm not sure that there is an obvious section for this article to fall into, some may object (or make too great an inference) if I put this article into the natural place of investing/ trading. Yet I could place it into my 'abstruse' category where I put general articles that don't fit anywhere else (these tend to be more philosophical, and I haven't created additional categories to cater for them). This piece is mostly for illustration in order to understand Black Swan/ Anti-fragile mentality, although I must concede that it is perhaps too close to the ludic fallacy. The importance of this article is the thought processes and the valuation of limits; what costs / odds are we comfortable with?

All combinations

If you can buy every possible combination of numbers then you are guaranteed to be holding the winning ticket (probability of one). In the UK lottery the number of combinations is circa 14 million, and at the current price of £1 this equates to £14 million to buy all possible combinations (NB: price per ticket is soon to change to £2). This obviously makes no sense when the pay-out (the winnings) is expected to be less than or similar to £14 million; there is going to be some number that makes the effort worthwhile. Let's ignore situations when the pay-out is too close to 14m to be unworthy of effort.

In some cases the pay-out has been stated as a guaranteed £20million. This easily covers the 14 million of costs and provides a guaranteed profit of 6 million. In theory at least, I don't know if there is a cap on the total number of combinations that you can buy. This would be on grounds of fairness / competition; however, the loss incurred by the lottery company is capped at the maximum number of winnings minus their fee for running the game (including their profit). The problem would be that if the winner was always a syndicate then perhaps people would stop playing: the game would appear rigged and hence unfair.

One ticket

The next interesting case for me is the situation where you buy a single ticket and accept the 14 million-to-1 odds. This is of course an "unlikely" set of odds to win against but they are not insurmountable. I think the best way to look at this as exposing yourself to (positive) tail-risk; confer: black swans and anti-fragility, where you take advantage of randomness for a positive return.

Yes the chances of you winning are slim and you are most likely to go your entire life without winning but for the small amount of £1 you take an opportunity to benefit from randomness. In the unexpected case that you do win (classic black swan) then the pay-out makes it worth taking the risk. The key to playing this game is to manage risk, to cap risk at a maximum. I suggest £1 per game as this amount is so trivial to your expenditure that it is meaningless. Over the space of an 80-year life (assuming no price increases) then the total amount spent is circa 52 weeks * 80 years * £1 = £4160. Even with price increases along the way the total expenditure over 80 years is small compared to the potential winnings.

On current full-time minimum wage (circa £1k per month), the cost of £4 per month (once per week) is rather small. The problem with playing the lottery is that it is easy to spend far too much on something that is unlikely. While I suggest that benefiting from randomness is good exposure I don't advocate taken excessive/ unnecessary risk to gain such exposure. It makes no sense to spend £100 per month (10% of a 1k pm salary) to gain this exposure as this amount is a significant part of one's earnings. The trick is to keep the cost low and easily manageable: 0.1% of your salary (in this case) is certainly a small amount. The national average salary is somewhere around 30k, so the strategy above equates to roughly 0.033% of your salary.

Consider that 60 working years at a flat rate of £10k per annum is £600k. I was suggesting buying tickets over the space of 80 years (you can assume 60, makes little difference) amounts to £4. The total cost of playing over one's lifespan is small compared to total earnings. Moreover, if one should win then the total winnings are significantly greater than the cumulative salary of one's life. In many cases the total pay-out is greater than £1 million.

If we used the figures for the national average salary then the total cost of playing the lottery is an even smaller percentage of one's cumulative life-time income (~1.8m over 60 years). The bottom-line is that the size of the pay-out justifies taking risk of such an unlikely pay-out if and only if you can cap the total cost of playing. The odds of winning are high enough to be attractive, as is the possible total pay-out.

Conclusion

Therefore there exist some optimal region where pay-out is high enough, cost of playing is low-enough and the odds are favourable enough.

Such activities are found better in places other than the lottery. The problem with the lottery is that the odds are so low that the winning outcome is unlikely (albeit possible - people do win the lottery). The best scenarios are where downside is capped (such as with equity), the cost of entry is low (and capped), and the upside is potentially large (albeit unknown).

Examples (heuristics)

What I'd be less interested in playing is a game where total payout is capped at say £100,000, the cost of entry was £5 and the chances of winning were also 14 million-to-1.

Odds of one trillion-to-one are not something I'm greatly fond of unless the pay-out was £1trillion and my life-time cumulative cost was £1000.