A Lottery Thought Experiment

In this post, I'm going to lead you through a thought experiment based on a theoretical game called Tomo. Tomo is pretty simple: you buy any number of tickets at a fixed price, and each ticket has a certain chance of winning and paying out a fixed price reward. In Tomo, the chance of winning is not affected by how many tickets have been sold, and neither is the payout per ticket. So it's not like a 50/50 raffle, in which the total number of tickets bought by anyone in the game decreases the chances of any particular ticket winning, and the payout increases as more tickets are bought. It's more like a slot machine, in which each pull of the lever costs a certain amount and has a certain chance of paying out. As a further simplification, Tomo does not have degrees of success on a single ticket. Each individual ticket either wins and pays out or it doesn't, there is no variability in how much a winning ticket can pay out.

Tomo is a game that, aside from these basic rules, can be played many different ways. The person running a game can decide how much a ticket will cost, what it's chance of winning will be, and what it's payout will be if it wins. The people running Tomo games always have enough resources to cover the winnings other people make, regardless of how much that might be, so they can set these figures however they want.

I'm going to say that in a particular game of Tomo, T = the price of a single ticket, P = the probability of a single ticket winning, and W = the winnings collected from a single winning ticket. 

From these values we can figure out some other important details about the game. The expected value of a ticket is the product of P and W. I'll say expected value = V. As an equation, V = PW. The expected value of a ticket represents how much a single ticket is worth after its already been paid for. If I have ten tickets, each of them with a 50% chance of winning five dollars, then I can expect about half of them to win, thus making me twenty-five dollars. Another way of looking at it is that each ticket has an expected value of $2.50, therefore ten of them are worth 10 x $2.50, or $25.

The expected profit of a ticket is the expected value (V) minus the cost of the ticket (T). I'll say expected profit = E. As an equation, E = V - T. Expected profit represents how much profit you can expect to make on the purchase of a ticket. Note that this number can be either positive or negative. A negative expected profit means you will be expecting to lose money on each ticket. If I have ten tickets, each of them with a 50% chance of winning five dollars, and they each cost me two dollars, then V = $2.50, T= $2.00, and E = $2.50 - $2.00, meaning each ticket has an expected profit of fifty cents. This means that each of my ten tickets can be expected to make a fifty cent profit, which amounts to a five dollar profit in total. This makes sense, because the tickets together cost $20, and their combined expected value is $25.

I know this is pretty boring so far, but stick with me here, I promise I have some interesting stuff coming up (well, interesting to me anyway). Imagine a game where T = $1, P = 1%, and W = 100. In this game, V = 100 x 1% = 1. Therefore E = 1 – 1 = 0. The expected profit is nothing. If you buy a bunch of tickets, you will expect to make no money, but lose no money either.

The interesting question, which is at the heart of this thought experiment, is: who would play a game of Tomo under these rules?

The first group you might expect to play such a game would be gamblers who take chances with their money for entertainment. People often travel to Vegas to play games that generally feature a great likelihood of losing money. This game has better odds than any Vegas game, so its easy to imagine gamblers would play it enthusiastically. A key feature of Tomo, however, is that the more tickets you buy, the less random it becomes. This feature arises from the laws of statistics. With a ticket price of only one dollar, this game would appeal only to low-rolling gamblers. Anyone wanting to gamble a substantial amount of money would find the game boring, as your chances of winning or losing any significant amount of money is minor. If you only buy one ticket, randomness is very high. You have a 1% chance of multiplying your investment by 100! That's a lot more exciting than if you spend $1000 to buy 1,000 tickets, and end up with a good likelihood of only winning or losing a few hundred dollars.

Another group that might want to play Tomo would be people who would value $100 more than a hundred times greater than $1. Money is sometimes used as a measure of objective value, but it's important to remember that value is ultimately a subjective measurement. A house right next door to my best friend is of greater value to me than the same house a hundred miles away, even if their market value is the same. If I need a heart transplant, then a compatible heart is of extreme value to me, but of little value to a dead guy. It is conceivable that someone could value $100 as being more than 100 times more valuable than $1.

A lot of people are probably pretty skeptical of this claim. Allow me to introduce you to my friend Joe. Joe is recently homeless, and he lives in a place that has bitterly cold winters. The city he lives in is rather heartless, and provides no homeless shelters. This city is actually on an island, and he has no way to leave the island without paying to take a boat, which he cannot afford. He is only able to make a dollar a day from begging, which is exactly what it costs him to buy food to keep himself alive. Right now it is autumn, but winter is coming. He realizes that without shelter in the winter, he will almost certainly die on the streets. With $100, he could buy warmer clothing, or perhaps pay for a very inexpensive room, maybe just a warm place to stay the night. As far as Joe is concerned, spending one dollar on a ticket with a small chance of earning him a hundred dollars is a rational decision. If he doesn't get his hands on a hundred dollars, he will die. Even though it is an unlikely gamble, its not really a risk at all. Either he doesn't play and dies, or he plays and has a 1% chance of surviving. In this situation, Joe values $100 as more than a hundred times more valuable than $1. The difference is not just $99; the difference is that between life and death.

Joe is obviously an extreme example. But its easy to see the same sort of principle at work in less extreme cases. If there is something in particular I very badly want and I need a hundred dollars more than I have to get it, then one hundred dollars is more than a hundred times more useful than one dollar. This is especially true if I operate on a very tight budget, and have no ability to save my monthly earnings. In a case like this I can still save windfall money, extra money that I haven't budgeted yet, but this windfall money will only come very rarely. One dollar of windfall money is not all that useful. But if I have a chance of turning that one dollar of windfall money into a hundred dollars of windfall money, it might make sense to take that chance.

This argument can also rationalize playing a game with a negative expected profit. Imagine T = $2, P = 1%, and W = $100. In this scenario, V = $1, and E = $1 - $2, or negative one dollars. If you bought a lot of tickets, you would expect to lose a dollar for each ticket you bought. A small-time gambler might still play this game, but will typically lose money at it. A high-roller will avoid this game, since it offers little excitement and an almost total guarantee of losing money. My friend Joe, however, might still be willing to take this bet (if he could skip a day of eating to afford it). And a guy on a tight budget might find it a reasonable bet too, if he really could use a hundred dollars but has nothing to do with two dollars.

Lets switch it up and imagine a very different sort of scenario. Suppose that someone sets up a game of Tomo where T = $100, P = 10%, and W = $2,000. In this game, V = $200 and E = $100 dollars. This means that each ticket you buy makes, on average, a one hundred dollar profit. It seems obvious that as soon as someone set up such a game, there would be a rush to buy these tickets as fast as possible. Sure, some of them might lose, but it is fairly likely that if you buy enough of them you will end up making considerable amounts of money.

The interesting question to ask here is, who would not buy these tickets?

Some people might avoid these tickets due to extreme risk aversion. They might understand that they will probably make money, but may nevertheless decide that they would rather not take the risk.

Another group that would not buy these tickets would be the people who can't afford it. If you don't have a hundred dollars to spend, you're locked out of this game. This is an especially sad sort of spot to be in, since you'll be surrounded by people making pots of money playing Tomo and you're stuck without any way to even get your foot in the door. Even if you can afford to buy a ticket or two, each ticket alone has only a 10% chance of winning, so most likely you're going to lose all your money trying to play. Even though you have a chance to make a lot of money, there's a 90% chance you'll end up on the street. Nobody likes those odds.

This Tomo scenario is strikingly similar to how our economy often works. A common adage is “you have to spend money to make money.” Another variant of this is you have to risk money to make money. It is a common idea that the greater the risk, the greater the potential payoff should be. This makes intuitive sense. Humans are naturally risk-averse, so we expect a payoff for our risky behavior. However, risk is heavily dependent on your point of view. In the Tomo scenario above, if you have millions of dollars, the game is no risk at all. You can spend your millions and be pretty much guaranteed of making millions more. If you have very little money, then the risk is a lot higher. Your ability to engage in the game is very limited. Even if you could afford a few tickets, most likely you would just lose money and never see any of the profits that the rich guys are raking in. Obviously, the real world economy is far more complicated. This Tomo scenario is, however, a very simple way of seeing how an economy that rewards high-roller risk-taking will naturally lead to the rich becoming richer while the poor sit there unable to participate.