## Some stuff on lotteries

Let’s say you have a population of *n* (ex ante) identical individuals each making an income of *w*. Say you now decide to set up a simple voluntary tax-transfer type scheme, where all individuals who choose to participate are required to pay an amount (/tax), *t*. The contribution/tax t is used to finance a transfer T, which is equal to n*t (the sum of all contributions, i.e. there’s no administration or anything like that to start with). Each individual has a 1/n probability of receiving the transfer T, so that the expected payoff of this scheme is equal to the probability of receiving the transfer times the size of the transfer minus the contribution, or 1/n*(n*t)-t. Which is equal to 0 and independent of both t and n. The expected income of an agent participating in the scheme is w + EP = w.

A risk averse individual will always choose not to participate. A risk neutral is indifferent between participating and not participating given that the reservation utility is 0. Note that even if the expected payoff of the scheme is ‘mathematically’ zero, the way most people think about a scheme like this (..out of context at least, when talking pure math) is that you’re most likely to lose if you participate, especially if n is sufficiently high. If a million people participate and there’s one transfer each month, then the likelihood that you’ll have gotten your money from the contributions back after a year is not very big.

It’s probably even lower than you realize, if you’re not familiar with statistics. To illustrate why this is, let’s get a little more technical. There’s one transfer T each timeperiod. There are n people who participate in the scheme. Now assume that your likelihood of getting a transfer next period does not depend on who got it last period. You can think of it as an assumption stating that an individual can receive several transfers if he or she is very lucky. This assumption is important, but I also think it’s justified in the empirical framework I’m attempting to apply this to – it would be completely justified if the scheme was mandatory, but regarding lotteries we know that a) at least some lottery winners play on after they’ve won anyway and, far more important, b) that the number of participants in real world lotteries is pretty much independent of the behaviour of the winners ex post (1 marginal lottery winner does not translate to one less lottery participant in general) [where ‘behaviour’ here relates only to the decision as to whether to participate in future lotteries or not]. If you don’t like to think of it as an assumption about past winners playing along after they’ve won, you can think of it as new people entering the scheme after past winners decide to exit, keeping the probability of winning constant over time.

Now perhaps a not uncommon way to misunderstand how this works is for people who don’t know statistics to think/assume that if you have 52 participants and 52 weeks of contributions/transfers, then the probability that you receive a transfer is equal to 1 after one year. It’s not, it’s lower than that, because some lucky guy might win 2 times and get the transfer instead of you. The only case where you can be certain to have won after a year is in the case where nobody can win more than once. In that case, the conditional probability of winning is increasing over time – the chance of winning the first lottery is 1/52, if you don’t win the first lottery you have a 1 in 51 chance of winning the next lottery, ect. I’d like to instead look only at the case where the conditional probability of winning is constant over time.

The probability that an individual *i* will receive a transfer before period k, where k is equal to 1,2,3…, follows in that case what is called a geometric distribution, which is itself a negative binomial distribution (I know I’ve linked to that one not long ago here on the blog) with r = 1. The cumulative distribution function, which in this specific case can be thought of as a function telling us how likely we are to have gotten a transfer by the time we reach period k, is equal to 1 – (1 – p)^k. To make this a bit easier, think of throws with a die. After one attempt, the likelihood of rolling a 6 is 1 – (1 – 1/6)^1 = 1 – 5/6 = 1/6 (we knew that!). The likelihood of rolling a 6 after exactly two attempts is equal to: 1 – (1 – 1/6)^2 = 1 – (5/6)^2 = 1 – 25/36 = 11/36. Note that this is smaller than 1/3 (or 12/36) for reasons already mentioned; when outcomes are independent, you can’t just add the probabilities to get your estimate. Also note that the probability of getting that damn 6 is of course increasing in the number of attempts. Now what’s the probability that you will *not* have rolled a 6 after 10 throws? Probably higher than most people think: 1 – [1 – (1 – 1/6)^10] = 0,1615, which is a tiny bit lower than the probability of rolling a 6 in the first attempt. Note that here I take advantage of the fact that there are only two outcomes [roll 6 or don’t roll 6] and that the probability of not rolling 6 in a sequence is equal to 1 minus the probability of doing it (mutually exclusive & collective exhaustive and all that..).

Now if we have a lottery with 1 million people participating (p = 1/1.000.000) and one transfer handed out each week, what’s the probability that you’ve *not* gotten a transfer after 10 years of participation (k=520, 52 weeks in one year…)? Putting in the numbers we get 1 – {1 – (1 – 1/1.000.000)^520} = 0,99948 = 99,948%. The funny thing here is also that the transfer is uncertain but the contributions are not, so if you assume weekly contributions of value $5 over the 10 year period, the certain costs are $5 * 520 weeks = $2.600. So if you play along in this lottery, you pay $2,600 and get nothing with 99,9% certainty. The *expected payout* from the lottery is of course the same as the amount you pay, as the transfer is $5.000.000 and and the probability of getting the transfer each period is one in a million, so that expected payout is 520/1.000.000*5.000.000= 520*5 = 2600 and the expected total payoff is 0.

Now here’s a twist some of the people who participate in schemes such as these probably don’t fully understand: Assume you try to buy two lottery tickets instead of one to increase your chances of winning. How does that affect the expected payoff? We already know. By assumption it doesn’t, because EP = 0 in our model (see the beginning and above). It also doesn’t matter how many times (weeks) you play, you can’t increase your chances in expected terms by playing for a longer period. Another thing is that in the real world the expected payoff of participating in a lottery is of course always negative – because it takes work and effort to make lotteries work, the contributions need to cover the costs of selling the lottery tickets, tv ads, tax compliance and administration, ect. In the real world, when you buy another lottery ticket, your expected payoff goes down. So to return to our model, if you think it would be mad to participate in a lottery where you pay $2,600 ove a decade and end up with nothing with 99,9% certainty, you should be aware of the fact that *these odds are better than the ones offered by real-world lotteries*. In the real world, the deal offered is even worse.

People who claim to be in favour of income distribution from rich to poor who also participate in lotteries are kind of funny. They say they want one thing from the political system, then they voluntarily decide to participate in a redistribution mechanism which will always have the exact opposite result. When you have a lottery where the winner takes all or most of the money, you redistribute from everybody to one (/soon to be) very rich guy. I know that lotteries hand out both large transfers and small, but on net most of the small transfers probably cancel out because that’s part of what keep people playing.

…

(smbc)

Fine analysis, and a fine way to drive home the fact that most economic theories are preceded by a laundry list of assumptions, among which always always “rational economic agents” and “perfect information” loom large. I think the “people are strange” label should actually read “people have an average IQ of 100, which is not flattering at all”, but that’s just me.

That said, I have no problem with people taking advantage of other people, as long as there is no fraud or coercion. Every business deal does that. Where I do have a problem is where the gov’t reserves this right for itself, and excludes others. If a gov’t sponsored lottery were to compete with privately organized ones, the issue would largely go away – the competition would erode the abnormal profits. Not that this is out of character for gov’ts – they set up and prop up monopolies all the time – but I think it’s particularly egregious when the monopoly they enforce is their own, for the distribution of a “public good” that hardly even meets the definition of a “good”, let alone “public”. A lottery is neither non-rival, nor non-excludable by any stretch of the imagination.

Comment by Plamus | June 1, 2011 |

I used to think a lot about model assumptions – you tend to do that when you start out – and in a way I still do, in particular because they are always a main driver of the results you get (GIGO + you usually can figure out what’s supposed to come out of a model once you know the assumptions, even if the model is very technical). But when you get used to modelling stuff it gets different (as I’m sure you know), you view model assumptions in a different light. I used to think the standard approach to modelling was to start out by coming up with a bunch of assumptions and then see where that’d get you. It’s not. When writing this post I didn’t start out by thinking about which assumptions should be made, rather I decided on a standard basic framework (the ‘tax transfer type scheme’) and then additional assumptions sort of turned up along the way when I decided that I wanted to illustrate this point or that and needed stuff to be simple enough to make an argument that made sense. I think this is part of what people not used to working with models don’t get; that models tend to make stuff approachable; simple and understandable – not the opposite. That’s why they are simple.

Hmm, “people have an average IQ of 100, which is not flattering at all”. Or maybe: “most people aren’t very good at maths” (for obvious reasons. And no, training this stuff beforehand doesn’t work.). But people

arestrange and this category is easily applied to a wide variety of contexts, making it more likely that I’ll use it repeatedly. I used to categorize much more specifically than I do now, and I don’t think that was a better way to do it, probably quite the opposite.There are incidentally utility function specifications for which the individuals participating are not ‘taken advantage of’ in any way, shape or form. Some risk lovers are probably ok with the deal offered by a lottery, private or not, just as some risk lovers get a positive utility from a mountain climb involving high costs and significant risk of death with no monetary upside.

That said, it might be argued that lottery participation rarely, perhaps never, can be justified in an investment good framework – though I’m unsure about that last one because it’s difficult to come up with competing investment schemes with a similar risk/return profile. It’s hard to come up with another investment type that has the potential to give you your money back 100.000 times, which some lottery coupons do. Also, even though lottery participants aren’t maximizing expected payoff, they might be optimizing along the (expected payoff, payoff variance)-set available to them. However even if you make that argument, it doesn’t change that lottery participation can probably be justified when thinking of it in terms of a consumption good (..like mountain climbing), instead of in terms of an investment good. It’s fun to daydream about how you’re going to spend all that money, even though you’ll probably never win.

It’s easy to argue the math, it’s hard to argue the preferences.

Comment by US | June 1, 2011 |