I will use “lottery” to refer to any game of chance where you pay a small price to have a chance to win big. Mega-Millions, Power-Ball, and your state lottery all fall into this category.

All lotteries, by definition to be financially viable, have a negative expected value. I am not going to dive into the nuances and minor instances where this has not been the case. If you are an MIT professor and have a scheme with a positive payout, more power to you. But the rest of us aren't going to find that loophole, much less have the cash to bankroll such an operation.

Most, if not all, lotteries have information publicly available that shows how much goes to payouts and other expenses. A common expected payout is $0.80 for every $1 played. We will use this for our example, but the important part is that the eighty cents is less than one dollar.

So taking this example further, if I play the lottery everyday for 365 days and buy a $2 ticket, at the end of the year you should expect to win 365 x 2 x 0.8 = $584 after paying $730. So why spend $146?

Because of variance. Quite obviously, this is over simplified. You probably won't loss $146. You are actually more likely to loss all $730. But of course there is a chance you will win millions, but this is unlikely. It would be irrational to believe yourself so lucky that you think you will make money. But there are some rational reasons to play...

Fun

To get it out of the way, and this applies to all gambling, if you enjoy it, and it is not hurting anyone, do it. If a $2 lottery ticket brings you more join (particularly on the days when you win something) then a Starbucks coffee, then playing the lottery is right for you. You might actually end up wasting less money than going to Starbucks everyday.

Lottery Pools

This one always gets me. Especially lottery pools at work. A basic pool works by collecting money from everyone in the office and if you win, you all split the money.

So to dispel a common myth, this does not actually help your odds. While in an office of 100 people, you now have a 100 tickets, you will also only get 1% of the winnings. So your expected winnings is the same (100 times the chance, but 1/100 payout).

However, this does reduce the volatility of your winnings, so you are more likely to have an outcome closer to the expected value. Due to the Law of Large Numbers (beautiful law, but it would be quite a lot to elaborate on in this brief aside), your variance of outcomes will be close to the mean. So winning less but with more consistency doesn't seem like a good reason to play... You're still loosing

But severe FOMO is a valid reason. On the off-chance your office wins, do you want to be the only one left out? All your coworkers could quit and you'll be left with all the work and no support! Imagine you as the only nurse with 100 patients (yes this is extreme, but it could happen).

So to rethink the outcome of a lottery pool it might be helpful to think of it as insurance. I am paying $2 so that if the pool wins, I will be covered.

The Jackpot

To illustrate this idea, I need to first set up a simple example and go into an economics 101 concept – utility.

First, let's imagine the lottery has two outcomes, you win $100,000,000 or you win $0. The lottery usually costs about $2 to play, but I am going to use $1 in my example to simplify the math. To get an expected payout of 80%, you would have a 0.0000008% chance of winning. That's not very good. You could also expand this formula to make for more realistic payouts, but this will suffice for now (for example, lower chance of winning the jackpot, but add a small chance you win a few bucks here or there). No one plays the lottery hoping to just win one of the smaller payouts, so I will ignore for now.

Second, let's talk about utility. Utility is the “pleasure units” derived from having or doing something. So for example, $1 may give me one unit of utility. Logically from that $2 should give me two units of utility. In conventional economic theory, this would continue linearly, so $1,000 creates 1,000 units of utility.

But is this linear assumption appropriate? Does a penny give you 1/100 of a the sense of happiness a dollar gives you? For me, not really. I can't do much with a penny and I haven't taken may change jar to the bank in ten years. I might even say a penny gives me no utility.

Elaborating further, Does $10 give you ten times the utility as $1? Well for $10 I can get a burrito, a beer at the bar, or a a myriad of other things. For a dollar, I can't get anything really, maybe a McDonald's cheese burger. And what about $1,000? I could get a laptop! So in a linear assumption of utility, 1,000 cheeseburgers, 100 burritos and a new laptop should all bring me the same amount of joy. Sorry, I don't buy this. I'm opting for the computer.

Bringing it back to the lottery, particularly those giant jackpots, $100,000,000 brings me much more happiness than 100,000,000 cheeseburgers. I can illustrate this in two ways: logically and pure math.

Logically, $100 million dollars is more money than I can comprehend having. To me, that's basically infinite money. I don't think I could spend it in my lifetime. $100 million might as well be $100 billion. I can buy everything I need and never have to worry. Investing modestly and getting a 5% return, I could spend $5 million a year, forever! Considering the median annual income of around $60,000, I can spend 80 times what the average household makes in a year!

This $100 million does not just mean 100 million cheeseburgers, but it means complete piece of mind and freedom from financial stress. Free from obligatory work. Free from needing to exchange my time for money. I can now have all of my time to whatever I choose to do!

Mathematically, we need to change our expected value equation from dollars to utility. So instead of:

100,000,000 x 0.0000008% + 0 x 99.9999992% = $.80 <$1

I would think about it as:

(infinite money) x 0.0000008% + 0 x 99.9999992% = (infinite money) >$1

This is because anything times infinity is infinity. Perhaps a more realistically, I can think of $100,000,000 would bring me the same pleasure as a $1,000,000,000, perhaps even 10,000,000,000 units of happiness, making the equation:

10,000,000,000 x 0.0000008% + 0 x 99.9999992% = 80 units of happiness

Which is greater than the initial happiness 1 dollar gives me. However, this logic falls apart the more money you put into playing the lottery. While that initial few dollars gambled away don't cost much utility, the later dollars do. I can get Dunkin' instead of Starbucks and recover from a small loss. It's hard to recover from a large loss...

Final Thoughts on the Lottery

After all that, I am still not advocating you play the lottery. While there are a lot of zeros after the $100,000,000 price, there are also a lot of zeroes in front of the 0.0000008% chance of winning. Most of you, if not all of you reading this will never win the lottery. NEVER.

It can be fun and give hope, but many sociologist might attack it for being a tax on the poor. After all, Bill Gates isn't playing to double his money. I will leave this subject to more socially minded treatise.

I personally only ever play when there is a giant jackpot or the office is collecting money for a pool (usually because there is a giant jackpot). I haven't won yet, but there is still time, right? And besides, I'd rather be working for a paycheck than waiting to win the lottery.