The idea of becoming rich from one day to another attracts many of us to buy a lottery ticket, an investment with negative expected return.

An article analyzing the notion of lottery mathematics (http://simplexify.net/blog/2012/5/6/i-am-a-statistician-and-i-buy-lottery-tickets.html) has just come across my table. DC Woods, the author, is a pro-lottery statistician, based on the article. Despite being in favor of buying the lottery, he himself claims after his analysis that buying a lottery has a negative expected return.

The reason why I don’t question whether his calculations are correct, is because I have another, nuclear solution to this that makes every other calculation unnecessary. And that is: Lotteries operate from the revenue of sales. So, take the revenue of all tickets sold, and split it: one fraction goes to the operation cost of the lottery firm, which is significant, and only the rest is redistributed among the winners. Thank God, lottery firms are usually non-profit, so they don’t reduce the pool with their profit, at least. A business model like this, inherently entails that all buyers, regarded as an aggregate, make a loss, not profit. And unless you think you can guess better than others, you are just one of those buyers, and will lose in the long run.

However, you might not be like everyone else. No, I am not referring to the fact that, like everything, draw can be manipulated as well. Neither do I refer to the fact that some claim to tell the future, which exceeds the scope of this blog. There is, however, one single argument that I accept when you say it worth to do lottery, even from pure financial reasoning. As explained above, probability does depend on certain factors, and although – in my opinion – a business model, like the one the lottery is based on, can never offer a positive sum game to the buyers, there might be an advantageous combination of lottery ticket price, winner’s price and probability.

To determine which are those combinations, his calculations could be extended with regards on the size of the headline price. As mentioned, expected return depends on three factors: the prize amount for the divisions, the lottery ticket price and the corresponding probabilities. To calculate the return, we only need the ratio between the prizes and the ticket price. Since probability of winning depends on the number of tickets sold (first price is split among winners, and with more tickets sold, the chance of splitting increases), and number of tickets sold increases exponentially with the price (based on his analysis of 52 draws), the overall expected return depends on the headline price.

We can have three possible outcomes:

First outcome would be, that it is more profitable to invest when headline price is low, due to the low probability of splitting in case of winning.

Second outcome: it is more profitable to invest when headline prize is high. The huge payout outweighs the negative impacts of the increased probability of splitting.

Third, there is maximum somewhere, and if headline price is higher or lower, we are losing expected profit.

Feel free to calculate it for your own local lottery and for the given circumstances, and if you find that there is a loophole in the system, go for it and play the lottery. Other than that, we have no option but to say that lottery makes no sense from pure financial reasoning. There is certain pleasure that it gives to the people, however. The pleasure of daydreaming how they would live if they won.

I have serious problems with this, however. First of all, the probability that someone will throw $100M through your window, and thus you becoming rich, is also a positive number. So, not buying a lottery actually allows you to further fantasizes about being rich without doing anything. And that without a negative expected return.

Secondly, would you prefer $100M with x probability, or $10M with 10x probability? Consistent lottery buyers would have to prefer the previous, because the fact that they are buying lottery means that they have increasing marginal utility. But I am quite sure their marginal utility is diminishing. If it was increasing, or, even if it would be linear, it would mean that they life sucks 100 times more if they have $1M wealth compared to a situation with $100M wealth. Well, this is hell unlikely. It never ceases to amaze me why they wouldn’t settle for less win but with proportionately more probability.

(Plus, I think – I think, building an imaginary world distracts people from reality)

If the idea of winning is so precious to you that you want to keep it at all cost, then you could go the stock market and buy some hazardous investment. Any investment with huge leverage would do it. It offers best of both worlds: the possibility, even if negligible, of winning a fortune, and positive expected return. This is because stock exchanges, unlike the lottery, are positive sum games. The money that flows into the network is harvested by someone else who knows how to leverage the potential value of the capital in the present, and flows back to the network with more value. Value is created in this process, as businesses strive and create prosperity, thank to your trust in them.

Well this day has come as well. I am praising stock exchanges, me, who an advocate of destroying stock exchanges. I really have to hate lotteries if that made me write like this about stock exchanges…

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That clickbaiting-copy title though. Aww. Even here, in the heaven of truth and honesty. I must admit that this is continuous.