I am studying a paper (https://www.jstor.org/stable/1804124) which discusses a two period revenue optimization problem as follows
We are trying the sell a single unit of product over two periods, i.e. if the product does not sell in the first period then it can be sold in the second period for another prize , the problem is to find the optimal pricing policy
Data:
R1 and R2 (Decision Variables) are the prices that you set in period 1 and 2 respectively
N : Number of customers that a arrive in each period
f(V)/F(V) : pdf and cdf of the customer's offer prize i.e. customer will buy the product if (V < Ri) (Assumed to be uniform in this case)
P : probability that the customer is a "window shopper" , this kind of customers do not buy the in any case
So the the problem can be formulated as
$$ Max_{R1 ,R2} R1*(Prob. sale in 1)+ R2*(Posterior prob. sale in 2) * (Prob. no sale in 1). $$
Please help me understand the authors explanation and implementation of bayes rule from here on:
Now, the probability of a sale in period 1 is (1 - F(R1))(1 - P^N), because the probability that every customer is a shopper is P N so that 1 - P^N is the probability of encounter- ing at least one buyer. It only requires one buyer to make the sale as long as R1 < V. Similarly, the posterior probability of a sale in period 2 is (1- F2(R2))(1 -P^N) and the probability of no sale in period 1 is 1 - [(1 - F(R ))( 1- P^N)].
It is now necessary to derive F2(V). Bayes' Theorem states that the posterior probability is proportional to the probability of the sam- ple, given the parameter, times the prior probability of the parameter. The sample in this case is the observation that no one bought during period 1. For V < R1, the probability of no purchase is 1. For V> R1, there is only one reason why the good did not sell during period 1 and that is that all customers were shoppers. This happens with probability p^N. It is easy to show that the normalization required to make the integral of the density function equal to 1 is 1/[R1(- P^N)+ P^N] so that the density is given by
$$ f_2(V) = \frac{1}{R_1(1-P^N) + P^N} for V < R1 $$ $$ f_2(V) = \frac{P^N}{R_1(1-P^N) + P^N} for V > R1 $$
please help me understand the formulation, like where did the the prior probability was function go?