How to discourage contestants from entering a lottery twice? 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?

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.