How to discourage contestants from entering a lottery twice?

Question

Suppose you have a lottery. And you want to prevent participants from buying multiple tickets. What would be the best way to discourage this?

For example, increasing the win-chance for all previously sold ticket-numbers as soon as a new one is bought, would make sure everyone buys only one, because buying more than one puts you at an even larger disadvantage compared to the existing ticket-holders.

But this is far from fool-proof, so is there a better way?

Answer 1

For each player $P_i$, let $\sigma(P_i)=k_i$ be the number of tickets player $i$ bought.

Furthermore let $(\sum\limits_{i=1}^n \sigma(P_i))^2 = H $ by the number of tickets given "to the house".

When the lottery is done, if a house ticket wins then every player loses.

In this scenario a player has no incentive to buy any tickets after the first.

For a players first ticket his chances of winning are $\frac{1}{H+\sqrt{H}}$

However his second ticket gives a probability of $\frac{2}{3\sqrt{H}+H+2}$.

With a bit of calculus it's easy to verify that their probability is no less than before buying the extra ticket. So they have no incentive to change, even if the ticket is free.

Answer 2

As long as the set of tickets is uncountable, only having a finite subset of tickets will almost surely have you lose. http://en.wikipedia.org/wiki/Almost_surely

In which case every person does have an equal chance of winning, with this chance being 0.

Answer 3

If we can keep track of how many tickets have been sold and who has purchased one, then we can charge a price for the second ticket that is always beyond the marginal value of that ticket.

Suppose $N$ tickets have been sold, the lottery is structured so that one ticket must win and each ticket has an equal chance of winning. If Alice wants to buy a second ticket we can charge her a price $P_2 > J \frac{1}{N + 1}$ where $P_2$ is the price of the second ticket and $J$ is the value of the jackpot. This means that Alice's expected value from purchasing the ticket is negative, so a rational self-interested agent would not purchase the second ticket.

Of course, using this reasoning, no one would ever buy a ticket to any of the major lotteries currently run by the state governments in the US. So, as a real-world solution this probably will not work.