Some people who buy lottery tickets argue that a lottery ticket is a small price to pay for the chance of being a millionaire.

While the expected return of the lottery ticket is negative, they place an extra value on the chance of being a millionaire, in addition to the expected return.

For comparison, suppose there is another lottery with the same negative expected return, but the maximum you can win is $5 (corresponding with a much higher probability of winning so that the expected value is the same). Then players will be less interested -- because you've taken away the value of the chance of winning a huge sum of money.

Is it irrational to place extra value on the chance of winning more money than you could ever make in a lifetime? Perhaps not - sometimes greater success of several orders of magnitude is only possible via a gamble with a negative expected return.