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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?

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    Yes, $t$he 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 :)2012-09-09

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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.

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    This raises an ineresting followup: With only finitely lost sold, the probabilities are not reflected perfectly. How do I know that my only coupon is is common and not rare? According to Bayes the probybility for rare should be $\frac{p_n}{p_1+\ldots+\p_n}$, but then the price should be at least $\frac{p_n}{p_1+\ldots+\p_n}$ of the prize value.2012-09-10