Prediction markets and lotteries (and my one simple trick for winning the lottery)

 A short and slightly mad post today, inspired by weird thoughts I was having when I awoke from my sleep this morning (yes, you are all thinking, a completely unwanted insight into the life of Rob). Clickbait is at the end.

There has recently been an uptick in the world of prediction markets. Although not new, and certainly in the UK political betting in a market based environment has been around for yonks (Betfair exchange*, take a bow) the recent entrance of Kalshi** and Polymarket** has spiced things up somewhat. 

* I'm a sometime customer but otherwise unconnected.

** No connection, not a customer, and god knows certainly no endorsement.

Now, as Matt Levine will tell you, to an extent these markets are mostly regulatory arbitrages around the fact that gambling on sports is heavily constrained in the US; Kalshi has taken advantage of an onshore loophole during a friendly political environment, whilst Polymarket is offshore crypto and hence can do whatever it wants. But at least in theory they do help to achieve the economists wet dream of a complete market which would allow you to bet on any outcome and therefore perfectly hedge yourself.

Anyway my mad thought was this; what if I could bet on whether I will win the lottery? 

There is one thing AI is good for. One thing.

In many ways that is a straightforward bet; I could specify the numbers I was going to pick and the conditions of the bet. For example, does only the jackpot count, or subsidiary prizes? For the purposes of this discussion it's simpler to assume a lottery with just one jackpot prize which is shared if multiple numbers win. And unlike certain other bets in these markets there is no debate about whether the bet will pay off.

But this is also quite mad; as there is a much easier way to make this bet, which is to actually play the lottery! Furthermore, it would extremely difficult to find someone to take the other side of this bet. Lottery ticket type bets are massively positively skewed; the person taking the other side must be willing to accept a huge negative skew of a tiny profit if they win or a huge loss if they lose. 

Of course they could hedge this risk by buying a lottery ticket. But then for this to make sense they would have to give the buyer of the bet worse odds than the lottery itself, effectively selling the ticket at a small mark up. This can make sense, again as a regulatory arbitrage if you want to 'play' the gigantic US lotteries but don't actually live there; and some bookmakers do offer this service, but not as far as I know in a market environment where you can choose to sell the bet, only as a standard bookie:client arrangement.

Lotteries only work because there is one counterparty who is willing to take all the bets - the 'promoter'. Because they are taking all the bets, and these bets are by nature perfectly uncorrelated (in the example of the simple lottery jackpot only lottery with prize sharing we are discussing), and they set the size of the prize fund based on ticket sales, they are in a no lose situation. In fact they normally have a substantial edge; even ignoring the possibility of nobody winning. 

(In many lotteries if there are no jackpot winners the prize fund 'rolls over' to the following draw and isn't taken by the promoter. We will return to this possibility later.)

In the UK national lottery this edge is set by law and reduces the prize fund by half before it's paid out, but lottery winnings are tax free. In US state lotteries where the top line reduction is lower the prize is also subject to income tax. Broadly speaking in both cases the net prize fund is around half of the total. Buying lottery tickets, as is well known, is a subsantial -ve EV bet. It is gambling, in the pejorative sense of the word. And nonsense like this, ought to be banned.

However let's imagine a world in which there are people who are willing to take the other side of this bet. One way of doing this is to split the bets into tiny chunks. For example I could take on 0.01 pennies of risk that a given set of numbers will not win a $1 stake lottery with a $1 million dollar prize; if the market is priced perfectly then if I lose I will lose 0.01*1 million = $10,000 if the numbers come up. And I could make 1 million of these bets to cover all possible outcomes and stand to win $10,000. As the market is perfectly priced then assuming everyone buys every combination of numbers I will not make or lose anything. With a million people like me betting against lottery wins, and a million people betting on lottery wins, we'd effectively have a lottery.

Importantly though, this lottery would have no house edge. Instead of 50% of the winnings being drained away by the promoter, 0% of the winnings would be. And of course those selling lottery 'tickets' by betting against possible winning combinations would have the option of doing so at a slightly lower probability. For example, in a 1:million odds lottery offering 900,000:1 on winning. The buyers of lottery tickets are facing a 10% house edge here, but that is still much better than the 50% loss of a normal lottery. There would probably be some cost to the prediction market operating, but even if it was 5% that is still a gain for both buyers and sellers.

Note: Just as in a normal lottery people bet on 'lucky' numbers it might be that sellers of lottery 'tickets' do not bet against all possible numbers coming up, or vary the price they offer on certain numbers. Absurd numbers of people buy birthdays or sequences of numbers. If this happened here then the 'price' of those 'tickets' would be bid up versus totally random numbers, allowing the sellers of those 'tickets' to make higher profits. A risk premium in lotteries!

In any normal world this wouldn't be allowed to happen. Effectively the parallel lottery is free riding on the actual lottery. Governments tend to be very protective of their gambling monopolies! Of course there is nothing to stop the prediction market setting up their own 'lottery', a trusted counterparty who will draw some numbers and post them on the internet every week. But perhaps this is too complicated. After all, lottery type payoffs are already widely available on the internet in the form of dubious crypto coins masquerading as investments. Why overcomplicate things? 

Perhaps a more interesting bet with some chance of being implemented is betting on nobody winning the lottery jackpot. Remember earlier I said many lotteries have a rollover function. Take the Euromillions for example. Odds of winning the jackpot are 1 in 139 million. But typically only 20 to 30 million tickets are purchased. Hence in any given draw the chances are the jackpot will not be won. For example in October and November last year there was no winner for 13 draws and the jackpot reached 156 million Euros from a starting value of 14.8 million. 

If you calculate the exact probability assuming everyone picks their numbers randomly then you get what I would call 'reasonable' odds, so the skew on the bet isn't absurd and it isn't mad to think people would want to bet on this. 

The maths here isn't simple as the longer these stretches of no jackpot go on, the more tickets are sold as the jackpot gets more attractive (in the aforementioned period of 13 draws average sales nearly doubled from 20 million to almost 40 million by the end of the run). So you would need to be able to predict ticket sales and have a good understanding of both the rules of the lottery and betting patterns. Ironically then there will be more skill required here than just betting on random numbers.

I will finish with - as promised- my own lottery winning strategy: If you are going go bet on the lottery, normally a -EV bet as discussed, then doing so when the jackpot is very high after lots of rollovers isn't a bad idea. The retained money from earlier draws makes it a slightly +ve EV bet. Whilst this doesn't guarantee a win, it does mean that in all possible lifetimes you will come out ahead. So that's nice.