Generally, the lottery only pays out 50 cents on each dollar that it takes in, the rest going to fund whatever. So the expected value of a $1 ticket is only $0.50 and that’s obviously a stupid investment.
But what if they paid out all the money they took in? Should you play? Is the lottery still “a tax on stupid people” in that situation?
It wouldn’t change the odds at all, only increase the payoff. The chance that you’d pick the right 6 numbers would still be something like $40 million to 1 (here is California, at least).
Some Lottery Math.
In order to pay out all the money that comes in, they’d have to guarantee that, on average, for each dollar played, one dollar gets paid out. That’s equivalent to saying that every player breaks even. So, you’d still be better off not bothering.
Remove the dollar sign and that sentence might actually make sense.
Playing the lottery isn’t stupid.
Playing the lottery with money you can’t afford to waste is stupid.
Playing the lottery with money you can afford to waste is a voluntary contribution to the state and, maybe, some lucky souls.
It’s like dropping a quarter in the slot machine on your way out of the casino versus sitting for hours and losing the rent money.
Well, for me, $40 million is not worth 40 million times as much as $1. This is an odd concept, but it’s easy to see, because the difference between getting $1 and $0 is a lot more than the difference between getting $40,000,000 and $39,999,999. So in that sense, the lottery would not be worthwhile for me if it paid out all its money. I imagine that I’m not atypical in this regard.
When you get into payoffs/odds as high/long as the lottery, it isn’t necessarily irrational to take a bet with a negative expected value or choose an option among many that isn’t the highest expected value. The distribution of utility on the money curve isn’t linear; the utility of the bet size and the payoff come into play. For instance, if I offered you a 75% chance of doubling your money, it would be stupid not to take the bet right? What if I said the minimum bet was your entire net worth and the value of your house? Now has smart is it? Is it rational to take that bet? Likewise, when the expected utility (likelikhood to significantly alter life) is high, but the expected value is negative (it’s a losing proposition) it can still be considered rational to take the bet. You’d be stupid to take a bet that offered you a 25% chance to double your money for a small sum. But as the gap between the bet size and the payoff widen to life-changing proportions, the negative expected value of the bet can be overcome by the postive utility of the difference in sums.
Of course, The value of a 1 for a homeless person with 5 and a 1 for a person who makes $100,000 dollars a year is significantly different. The utility of the $1 to the beggar is significantly higher and probably can’t overcome even the long odds, let alone the negative expected value.
Emphasis mine. That’s incorrect. You could say that overall, the sum of all the players is to break even but not that every player breaks even. If everyone played an vast number of times the trend would be for everyone to break even but that’s all.
It wouldn’t be a “tax on stupid people” anymore because there would be no tax aspect left in the lottery if it paid out everything it took in.
It depends on your von Neumann-Morgenstern preference function.
I.e. do you think that 1000 dollars is 1000 times as good as one dollar? If you think that $1000 is more than 1000 times as good as $1 it makes sense to play, as the expectance value of your preference function would be greater than the preference function of the ticket price.
For more detail, read Fun and Games by Binmore. It’s written for economists, so it’s fairly understandable, as well as being rather entertaining at times.
I’m wondering how this could happen even theoretically. The paper costs money, the administration costs money. If it’s funded with other dollars, then that comes out of the taxes of the same people. I see this as asking what would it be like if perpetual motion machines really existed.
With that said… there is a large group of player who get $1 of value out of playing by the psychological mood it puts them in, the daydreaming it allows, having some hope. For these players at least, increasing the payoff only makes it a better transaction.
Ok, stupid question here:
Let’s say there is a mega million super lottery drawing of $500,000,000. The odds of winning are 80 million to one, right?
If I went out and bought 80 million lottery tickets, am I guranteed a win?
If there are 80 million different numbers, one of which wins and the rest of which lose, and each of your lottery tickets has a different one of those numbers on it, then yes, you’re guaranteed to win. If each of the tickets has an equal chance to win (1 in 80 million), then it’s not guaranteed.
Think of it this way: the chance that a given ticket loses is (80 million - 1)/ 80 million. If you have two tickets, the chance that they both lose is ((80 million - 1)/ 80 million)^2. If you have 80 million tickets, the chance that they all lose is ((80 million -1)/80 million)^(80 million), which is a very small number but still greater than zero.
Not only would you win the grand prize if you bought one ticket for each possible lottery drawing result, but if for example the drawing was six numbers each from one to fifty, then you’d win fifty second prize awards, 50*50 third prizes, etc.
(Come to think of it, that’s not exactly right; you’d win 49 second prize awards, for example. But I was close.)
Yes, but not a profit. You wouldn’t win all the prize money since everyone else who would have won is still going to win and you have to share with them.
Plus, there’s the tremendous amount of time and manpower required to purchase 80 million tickets. I would imagine this would require a massive team of people, mobilized over days, with a massive logistics and coordination effort.
I’d imagine it’s a massive chore to purchase 1 million non-repeating tickets, let alone 80 million.
Good point. Take the lottery down in stakes. Pretend that your friend is giving a payout of $2, a ticket is $1, and there’s a 50/50 chance of winning. On average, for each dollar you spend, you get $1 payback. You break even. Not dumb, but a waste of time.
Bump up the stakes. 50/50 again. You lose, you lose your house. You win, you get a payout equivalent to the equity you have in your house. Not many people will take this bet.
The difference is risk. In the first case you are essentially risk-neutral, the second you are risk-averse.
In the case of a $1 lottery ticket, you should be close to risk-neutral, but since realistically you’re going to lose that dollar, you’d be a little averse. So, you multiply the payoff by the chances that you’d win that dollar (which will equal $1), but then multiply by a fraction which would be a numeric estimation of your risk aversion. In this case it might be .99, so including your risk aversion, you get a payback of $0.99 for your $1 spent. At least that’s how businesses try to quantify the effects of risk into payback calculations.
Let me come to the defense, somewhat, of all those “stupid people.”
When someone making minimum wage and just getting by goes ahead and splurges $2 on a lottery QuickPick ticket, he is not buying it for the purpose of investing his money for an expected return.
What he is buying is that great feeling that he can carry with him for the next couple of days, until the drawing, that he could be a millionaire next week and not have to worry about all his current financial troubles. (He’d end up worrying about other financial troubles but that’s a different story.)
It seems to me that blowing $2 for the purpose of being more cheerful for a couple days is a reasonable expenditure. Certainly better than blowing it on booze in the form of a bottle of 2-Buck Chuck. 
In Ireland, where I live, there was one famous occasion where a special easter promotion was announced. There are six balls in the lotto, each with 40 numbers. Match six is the jackpot, theres prizes for matching 4 and 5 balls too. This promotion was a double payout for 4 and 5 ball matches. A visiting polish mathematician figured out that if he bought every single ticket possible, he would win big, a few million, I believe. He organized a team of helpers, and many months were spent beforehand filling out the lotto cards. As the draw approached, all around the country, the tickets were put through. Before all of them were in, the system underwent a mysterious “breakdown” possibly do do with the lottery company noticing all the combinations on their screen fill up. In the end he did make a profit though, from all the bonus payouts, the jackpot was mostly used in paying for the millions of tickets. Unfortunately I don’t have the exact maths of the situation, i might try find them. The lottery here has smartened up its act since then…