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..$100, inclusive).
Because you have greater economies of scale than the current owners, the well will actually be worth 50% more to you than to them (but they don’t know this).
The catch: although you must bid on the well before drilling starts (and hence, before the actual yield of the well is known), the current owner can wait until *after* the well’s actual value is ascertained before accepting your bid or not.
What should you bid?
Solution
This problem amounts to properly defining the expected value of the well to you.
The following equation does it:
(1%) * [(1.5 – 1)] +
(1%) * [(3 – 2) + (1.50 – 2)] +
(1%) * [(4.5 – 3) + (3 – 3) + (1.5 – 3)] +
…
(1%) * [(150-100) + … + (3-100) + (1.5-100)]
Each line represents your expected value from a bid of 1$, 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.
2026 Update: Optimization and Decision Problems in Interviews
Optimization puzzles like Oil Mogul — where you must allocate resources optimally under constraints — remain highly relevant. They model real engineering decisions: server capacity planning, budget allocation, and ML training compute scheduling.
The connection to system design in 2026: Resource allocation under constraints is now discussed in system design interviews explicitly — “how do you allocate GPU compute across training jobs with different priorities and deadlines?” is structurally identical to classic optimization puzzles.
What’s new in 2026: Candidates are expected to recognize when a problem reduces to a known optimization class:
- Linear programming (LP) — if the objective and constraints are linear
- Integer programming — if variables must be whole numbers
- Greedy algorithms — if local optimal choices lead to global optimum
- DP — if there are overlapping subproblems
Still asked at (2026): Amazon (operations research), quant firms (portfolio optimization), and any role involving resource allocation or supply chain reasoning.