Each week there is a lottery. X tickets are sold. There are Y people buying those tickets. There are N winners in one week. Buyers of the tickets are identified. If you buy Y tickets as a same person, you can still win only once, but the odds of you winning are Y times bigger. If one identified buyer buys 1 ticket he has $\frac{1}{X}$ odds of winning P dollars where P is the prize money. If one identified buyer buys 2 tickets he will have $\frac{2}{X}$ odds of winning, an the prize money is the same. If I can somehow fake my id and buy 2 tickets as 2 separate persons the odds of winning P dollars with one draw is still $\frac{2}{X}$ and also I have a change of winning more money. Odds of that happening shoul be $\frac{1}{X}$+$\frac{1}{X-1}$. If I'd somehow fake my id J times then odds of me winning the lottery once would be J times bigger, plus I'd now have a change to win up to J*P amount of money.
How to design the lottery so that the buyer wouldn't be incentivised to fake ID's and buy tickets as multiple persons? So that the expected return value (EV) would be same, no matter if he buys J tickets as one person or as J different persons?
I assume I'd have to give more weight to any person holding multiple tickets. So odds of him winning are more than $\frac{J}{X}$. So that if he'd participate in let's say 10000 weeks in a row, it wouldn't matter if he'd buy tickets each week with the same ID or with different ID's. The EV would be the same after those 10000 weeks.
I've formulated it so that it may look like a homework assignment, but honest to god it's not. I am actually designing such a lottery and I've just experimented with different code snippets on how to make it work, but haven't found any good solution yet.
I'll give an example also to perhaps make it more clear:
There is a lottery and there are 9 ticket holders and 100 tickets. 2 of the holders will be drawn as winners. One holder has 20 tickets and 8 holders have 10 tickets. Two of the ten ticket holders are actually the same person falsely acting as two. How to increase the chance of the honest 20 ticket holder winning so much that the EV would be the same as the 10*2 ticket holder? So that the honest guy would win that much more often across multiple lotteries that it would even out the times when 10x2 ticket holder would win twice the amount of money.