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You are an oil mogul considering the purchase of drilling rights to an as yet unexplored tract of land.
The well’s expected value to its current owners is uniformly distributed over [$1..$100]. (i.e., a 1% chance it’s worth each value b/w $1..$100, inclusive).
Because you have greater economies of scale than the current owners, the well will actually be worth 50% more to you than to them (but they don’t know this).
The catch: although you must bid on the well before drilling starts (and hence, before the actual yield of the well is known), the current owner can wait until *after* the well’s actual value is ascertained before accepting your bid or not.
What should you bid?
This problem amounts to properly defining the expected value of the well to you.
The following equation does it:
(1%) * [(1.5 – 1)] +
(1%) * [(3 – 2) + (1.50 – 2)] +
(1%) * [(4.5 – 3) + (3 – 3) + (1.5 – 3)] +
…
(1%) * [(150-100) + … + (3-100) + (1.5-100)]
Each line represents your expected value from a bid of 1$, 2$, …, 100$, respectively.
eg, consider line 2 above. if you bid $2…
With 98% probability you won’t win the contract, so your profit is 0. With 1% probability, you will win something worth (150%*1) = 1.5, for which you paid 2$ With 1% probability, you will something worth worth (150%*2) = 3, for which you paid 2$
So, your goal is to maximize the following function of x, where x is your bid.
f(x) = 1% * Sum_{i = 1 to floor(x)}{ x – 1.5*i }
There’s no benefit to non-integer bets, so re-write the maximization function as :
ARGMAX(k) {1% * Sum_{i = 1 to k}{1.5*i – k}}
(=) ARGMAX(k) {Sum_{i=1 to k}{1.5*i – k}} /* 1% isn’t a function of k or i, so toss it */
(=) ARGMAX(k) {Sum_{i=1 to k}{1.5*i} – Sum_{i=1 to k}{k}} /* Split the summation */
(=) ARGMAX(k) {(0.75)(K)(K+1) – K^2 }} /* Closed form for summations */
(=) ARGMAX(k) {(0.75)(k)-(0.25)(K^2)}} /* Algebra */
And that function is maximized at k = 1 and k = 2.
When choosing b/w $1 and $2, you should bid $1 because of time-value, reinvestment risk, etc, of the extra dollar.
(ie, if you don’t have to spend the extra $$ now, don’t)
That’s my solution.