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Math Help - 10 buyers and 10 sellers problem

  1. #1
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    10 buyers and 10 sellers problem

    Hello,

    I'm trying to work my way through the following problem:

    Suppose you have ten buyers, we'll call them b_1, b_2,...b_{10} and ten sellers, s_1, s_2,...s_{10}.

    For each i\in[1, 10], let that buyer, b_i have $i available to him/her. Similarly, each seller s_i has the item the buyers want for sale at $i (they all want the same item).

    We put all twenty participants in a room- if a buyer meets a seller and the buyer has enough money to purchase the item, both the buyer and seller make the sale and leave the room.

    What is the expected number of sales?

    ----------

    My immediate problem is even modeling the problem, yet alone computing the probability distribution. I've started by trying to build the model with just two buyers and sellers, since we're in the simple case of just two permutations.

    So, if N = 2, then if X represents the number of sales, we have

    P(0) = 0,
    P(1) = 0.5,
    P(2) = 0.5

    so E(X) = 1.5

    However, the problem quickly expands combinatorially with increased buyers/sellers, and the problem isn't _really_ recursive, that is, the specifics of which buyers and sellers stay in the room is relevant.

    Does anyone have any ideas on how they might approach this problem?
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  2. #2
    MHF Contributor
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    Re: 10 buyers and 10 sellers problem

    Hey Tuufless.

    You might want to consider an indicator function that becomes 1 if an exchange is made and 0 otherwise.

    In terms of the number of sales, this is just the sum of these indicator variables.

    In terms of the indicator variables if you have a distribution of money for the buyer (call it B) and a distribution of the sellers prices (call it P) then the indicator I will be given by I = 1 if (P-B) >= 0 and 0 if P-B < 0.

    So now you need to find a random variable where P-B >= 0 and if you have the joint distribution of P and B, then you can find the region of integration where this is met and create a random variable from this sub-set and find its expectation.
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