The standard answer to this question is a resounding “No”. This is usually based on the idea that the expected return on any ‘investment’ is negative. You may buy a lottery ticket for $1, but since the chance of winning the lottery is so small, you can only expect to receive e.g. $0.50 back on average.
Utility Function
The above argument may not be the full story since money by itself is not a sufficiently meaningful metric to base decisions on. Instead, we need a utility function. A utility function is a way to map dollars to value, it’s based on the idea that one dollar for you and one dollar for me may mean different things. So, we should really ask ourselves what is the expected return of utility when playing the lottery.
One useful property of utility functions is that they are concave. This means that each incremental dollar is worth less to me than the dollar before it. Put another way, if I’m starving and homeless, a dollar is worth a lot whereas if I’m a multi-millionair, that same dollar is worth a lot less. This is called the the law of diminishing marginal utility.
These two ideas (that we should look at utility rather than money, and that the marginal utility decreases) seem to strengthen the argument against playing the lottery. We already know that we will, on average, lose money when playing the lottery. Now we’re also saying that the millions of dollars that we may win aren’t even worth as much as we had hoped they were. The expected return of utility is then even worse than for money.
Utility Function for Playing the Lottery
While it is generally the case that utility functions are concave (marginal utility decreases), there seems to be situations where this is not the case. Consider, as a toy example, a universe with precisely two products: chewing gum and sports cars. Suppose that each piece of gum costs $1 and that the sports car costs $100,000. Consider two agents in this universe, one with a net worth of $100 and one with a net worth of $99,999. The traditional argument of decreasing marginal utility would tell us that a dollar is worth a lot more to the person who is worth $100 than to the person who is worth $99,999. However, in this arguably contrived universe, the only thing the person worth $100 can do with that additional dollar is buy one more piece of gum. The rather wealthy individual, who can currently only purchase 99,999 pieces of gum, can, with the additional dollar, now instead purchase the sports car. It would not be too far fetched to suppose that the wealthy individual would gain more marginal utility for that dollar than the less wealthy individual would. So, in this universe we see that marginal utility is not necessarily decreasing.
It should be pointed out that in the toy example above, the marginal utility is still locally decreasing. To someone who is worth $200, the extra dollar will not be worth as much as it would be to the person worth $100. It’s just at certain transition points that we see discrete jumps.
Let’s go back to the lottery. Presumably the reason why people play the lottery is not so that they can buy more of the things that they already own. Instead they hope that with the additional money they could do things that were simply not possible before. Perhaps they can quit their job, buy a house, or rise out of poverty. It seems reasonable that these types of transitions are similar to the transition in the toy example above. While each player’s utility function may still be locally concave, it may still exhibit large discrete jumps. Taking these into account, it is conceivable that the overall increase in utility in winning the lottery may in fact exceed what would naively be expected based on a purely monetary analysis. The expected return of utility may then be positive, in which case playing the lottery would in fact be a logical thing to do.
Mind Bowling 