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**Summary::**The price of a house is uniformly distributed between 0 and 1000 but we do not know its exact value. If we place a bid higher than the value, then we obtain the house, but if our bid is lower then we get nothing. If we know we can sell the house on to another person (guaranteed) for ## nV ## (e.g. ## n = 1.5 ##) where ## V ## is the value of the house, then should we bid for the house? If so, how much?

**Question:**The price of a house is uniformly distributed between 0 and 1000 but we do not know its exact value. If we place a bid higher than the value, then we obtain the house, but if our bid is lower then we get nothing. If we know we can sell the house on to another person (guaranteed) for ## nV ## (e.g. ## n = 1.5 ##) where ## V ## is the value of the house, then should we bid for the house? If so, how much?

**My attempt:**

We can consider the expected value of our profit ##P = nV - B ## from a deal. If we win the auction (## B > V ##), then we have a guaranteed pay-off of ## nV - B ##, otherwise we gain nothing. I am struggling to represent this mathematically and extract a useful answer from this problem.

We can define indicator variables for when we win (1) or not (2). Then:

$$ P = X_1 + X_2 $$

$$ E[P] = E[X_1] + E[X_2] $$

we know that [itex] E[X_2] = 0 [/itex] (the scenario where we don't get the house) and thus by linearity of expectation:

[tex] E[P] = E[X_1] = E[nV - B] = nE[V] - E[ B ] [/tex]

We want to make a profit and therefore want ## E[P] > 0 ##:

$$ nE[V] - E[ B ] > 0 $$

If the distribution of the house's value is uniform then ## E[V] = 500 ## and therefore, we want to bet less than ## 500 n ##. In the case where ## n = 1.5 ##, then we would bet (just) less than 750.

Is this correct logic? I have seen other versions of this problem on the internet and I have seen some responses to those variants say that we should not be betting, but don't provide much explanation to back up that claim.

Any help is greatly appreciated.