The chances of winning are close enough to zero to safely consider them zero. If it is entertaining to you to buy a ticket and dream about winning then go ahead and buy one ticket. But never buy more than one thinking you are increasing your odds of winning. 5 tickets * 0 is 0. 100 * 0 is still 0 - any practicle number of tickets * 0 is still 0
zero
But if you have a large enough bankroll to do so, you should buy as many tickets as you can get your hands on - assuming #1.) the expected value is positive and #2.) your total investment falls below your Kelly criterion. I believe a city in Texas did this once when the lotto odds were in their favor but my google-fu escapes me.
So, according to your theory, you should buy ONE ticket or ALL the tickets, but nothing in between?
And don't forget to factor split winnings into the calculations. Here in Qld (Australia) the media coverage of lottery prizes starts well before the Kelly criterion tips in your favour, which means prizes are almost always split - there was one case where $20-odd million was split so many ways that the first division prize was less than $750K.
In that situation, even buying ALL the tickets wouldn't help ... you'd need to buy MULTIPLE copies of ALL the tickets, so you would have several shares in the split. I'd like to see the math for that!
There have been syndicates put together to do this: to buy hundreds of thousands of tickets in lotteries were the expected value is positive.
There are a few problems with this approach, though. One is that there is a real risk of split pots, where multiple winners drive the ex post probabilistic payoffs negative.
The larger issue is operational hurdles: it is tough to buy a million tickets, even if one has the coin. The retailers simply aren't setup to handle that sort of flow, and the state lottery officials have setup measures to prevent this sort of things (mostly as a result of being picked off in the past). To them, it is a business, they expect to make money, and they just change the rules relating to ticket sale mechanics to limit/punish outlets that collaborate with the syndicates.
Maybe I'm misreading your post, but the Kelly Criterion says nothing about a single bet. What it tries to maximize is your overall return over a series of bets. The charts in the OP are with regards to buying only a single ticket/draw. Re-running the numbers for more than one ticket would result in an even larger necessary bankroll.
I think there's a reasonable argument for using the Kelly criteria, but leaving that question aside, the author misses a most important factor: the marginal utility of the money. If I am a millionaire, the upside of extra money is low, and the downside of losing all my money is high. If I am choosing between a cup of coffee and a lottery ticket, the upside (even divided by my chance of winning) is high and the downside low.
A agree with the importance of marginal utility but disagree with your 'marginal' maths. The pay off with these mega jackpots needs to be in the 10s of millions to make it positive net value. I'd argue that the marginal value of money starts to taper of considerably at around between $2m and $5M. So the extra say $35M of jackpot money won't really have as much value to you on top of the $5M you might normally win. So in marginal terms you are back to negative expectation.
Of course I think the marginal entertainment value is what gets most people playing.
It's sad to think that some people derive their hope entirely from the lottery. I wish there was more of a fervent drive to discover how to make money rather than just win it.
Although the discussion of the Kelly criterion is an interesting academic exercise, the analysis does not apply to an actual lottery like 6/49, because it is based on a false assumption:
... "assuming that I can avoid picking the same numbers as someone else".
Unfortunately, as the number of tickets purchased increases, the probability that another person chooses the same numbers as you also increases. Thus, when there is a large prize, it becomes very likely that it will be shared.
Hey, if buying one ticket makes sense, then surely buying more than one ticket makes more sense. So you should spend all your $65m buying lottery tickets for the $49m jackpot lottery. If each $2 has an expected $3.50 return, then $65m's worth of tickets will have an expected $113m return! Bingo. Safe as houses.
Or perhaps this is an absurd argument, as proven by the above reductio ad absurdum, and you should keep your money to spend it on things that make sense.
But don't let me stop you. After all, the people who run the lottery need to eat too.
Your comment reminds me of a group which, I apologize for being vague on the details of, planned to buy as many lottery tickets as needed to have a 1/1 chance of winning. I can't remember if that meant just getting enough individual numbers that their odds "went up" or if they were actually planning to purchase all the possible number combinations. Irrelevant I suppose.
Probability math aside, they were able to raise the funds required to purchase the necessary tickets, but were eventually undone in a rather unexpected way... turns out, it's harder to procure 2 million lottery tickets than they had anticipated. The day of, they simply weren't able to get enough tickets to make it happen.
I honestly don't remember if they won or lost, but I remember expecting them to fail through having to split the winnings, throwing their math off... as if it weren't already off anyway.
If I am not mistaken, it happened in Australia, and while they were not able to cover all the numbers, they did win (they had covered 90%+ of combinations).
Hey, don't they realize that us little people (at least in U.S.) don't play because we want to win. Having such big hearts, most of us just like having an additional way to funnel money into state government (since they allow us to pay so little in taxes). [/tongue-cheek]
the reliance on the Kelly criterion doesn't make any sense. the Kelly criterion is based on a game you can play over and over. it calculates an exponential expected growth rate based on how much you're favored.
favorable lotteries don't show up very often. if a favorable lottery that you could play existed every day, then the Kelly criterion makes sense (it sort of makes sense in its original context, the stock market).
