TITLE: Is a fair lottery possible?
QUESTION [16 upvotes]: I'm trying to come up with a scheme for a lottery where each individual has roughly the same chance of becoming the winner, regardless of the number of tickets one holds. So no individual should have an incentive to buy a second ticket, but even if they decide to do so, it should not gain them any significant advantage over any of the other players.
Ofcourse, the simplest solution would be to just refuse to sell them multiple tickets, but since all players are anonymous, I cannot differentiate between ticketholders and tickets, so that's impossible.
A working solution would be to increase the house-edge exponentially (or decrease the jackpot) as soon as a new ticket is sold, so that buying another one lowers the expected profit more than is gained by having an extra ticket. No rational player would want do that, unless they have no previous tickets. But the problem with that is that only a very small amount of tickets can be sold untill the house has 100% chance of winning, so in practice nobody will ever win.
Is there a better formula possible? By dynamicly changing parameters like the ticket-price, jackpot, or win-chance with every new ticket? The lottery does not have to be economicly feasible, its no problem if the organizer always makes a loss from it for example.
I have a feeling I'm asking for the impossible, but since the proposal above would work fine for very small groups, there should be a formula which works in larger groups too.

REPLY [6 votes]: With the various restrictions in place the answer to the question as asked seems to me to be 'no, there is no such system'. 
It was said there is one prize to be distributed (one person to be selected) and it was also said the tickets could even be free. And, the lottery should be fair, so that different individuals have essentially the same chance to win they holding one or more ticktets regardless. And, of course the key point that one cannot distinguish tickets from individuals.
However, the prize can be adjusted in the process.  
So there are tickets $\{1, \dots, n\}$ distributed, and there is a partition of the set $\{1,\dots, n\}$ into disjoint nonempty sets $I_1, \dots, I_k$ corresponding to the tickets one individual holds. 
As described the only thing the organizer of the lottery knows is $n$. But, they have no control over the $I_j$. It is impossible to assign winning propabilties $p(i)$ to the different tickets so that $\sum_{i \in I_j}p(i)$, the probability individual $j$ will win, is about equal for all $j$ and each possible choice of $I_j$, except $p(i)= 0$ or $n\le 2$.
Now, the above assumes that tickets are distributed and then some 'drawing' is done possibly following some complicated system. 
If one does not insist on this there are of course ways to assign in a in some sense fair way one prize in a way that it is pointless or even bad to get more than one ticket (even if it is free). 
The "unique is the winner" was already mentioned, one could also do "first come, first served" (the first ticket sold always wins).
Finally, in OP and in some answers something else is discussed, which in my opinion is not really "fair" in the sense of the question. Merely, these would be systems to make it unreasonable to get more than one ticket but still if somebody would they would be in some sense better of than the others in the same game. Yet, yes, I think there one could (in a certain formalization) show that one needs at least exponential decrease of expected win to make this work.