# Lottery with coupon collecting - what prices are fair?

Assume there is a lottery where you can buy lots for 1\$each. To win the grand price you have to collect$n$different coupons$C_1, \ldots, C_n$where$C_i$occurs with probability$p_i$. You may assume that there are "infinitely" many lots, i.e. the$p_i$do not change over time and successive drawings are independant. And of course$\sum p_i\le 1$. I specifically want to consider the case where the$p_i$are far from being equal. Q1: What would the grand prize be worth if the lottery is fair? Q2: What would be a fair price to sell a coupon of type$C_i$to other players? The obvious answer$1\over p_i$seems to be wrong because in order to collect all other coupons one has to buy so many lots anyway that it is likely to find a$C_i$while doing that (unless$p_i\ll p_j$for$j\ne i$) Q3: Assume two players have collected subsets$A$,$B$of$\mathcal C=\{C_1, \ldots, C_n\}$such that$A\cup B=\mathcal C$. If they cooperate, what would be a fair method to share the grand prize? - In commercial applications, often one of the coupons is quite rare, while the rest are common. The assumption then is that everybody has all the common ones. In that case the common ones have zero value and all the value is on the rare one. – Ross Millikan Sep 9 '12 at 17:45 Yes, the question was inspired by such commercial variants - although you can never be sure that the one coupon you still don't have is really rarer, after all there is always a last one and Murphy's law :) – Hagen von Eitzen Sep 9 '12 at 19:50 ## 1 Answer To follow Ross Millikan's comment that "the common ones have zero value and all the value is on the rare one" if trading is allowed, then (Q1) the grand prize should be worth$1/p_{min}$where$p_{min}$is the lowest of the$ p_i $. (Q2a) If there is a single type with that probability, then its value is arbitrarily close to the value of the grand prize, and every other type has value arbitrarily close to zero. (Q2b) If there a$k$types with the same minimum probability then their values are arbitrarily close to the value of the grand prize divided by$k$. (Q3) They should share the prize in proportion to the number of minimum probability coupons they hold. -  Can this be exactified? And maybe I misworded "far from being equal" - I actually wanted to exclude special cases like$p_i=\frac1n$. What if$p_1=a p_2\ge\ldots$with$a=2$or$a=1+\varepsilon$instead of$a\gg 1$? – Hagen von Eitzen Sep 9 '12 at 19:56 It is exact, assuming that there are "infinitely" many lots and that there is trading. – Henry Sep 9 '12 at 21:21 I find this argument unconvincing. Can you justify your answers a bit better? (In particular, Ross says "the assumption is" that everyone has all the common ones - but how did they get them? when is the assumption wrong?) – Ben Millwood Sep 9 '12 at 21:49 If there is trading then, as there are more common coupons in the market than are needed, supply exceeds demand at any positive price so the price of common coupons keeps falling towards zero; as that happens the value of the least common coupon rises towards the prize value. – Henry Sep 9 '12 at 21:59 I think thius works only if not only infinitely many lots are available but are actually sold. If only$N$tickets have been sold at the momen I try to trade a common lot, someone else may own the rare ticket but not the common one (e.g. because he sold them all happily for$\epsilon>0$cents before he got the rare ticket). He may thus either buy about$\frac1{p_1}$lottery tickets from the bank or alternatively be willing to give me about$\frac1{p_1}$\$ for mine ... – Hagen von Eitzen Sep 10 '12 at 21:04
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