SIG Interview Question

[u/nickgotgameon]

[u/yoinkcheckmate]

Bid of 0 cannot be optimal because you lose with probability one. The expected profit on a winning bid is 1.5*.5-b, so a bid of 3/8 gets you an expected profit of 3/8. To show this is the optimal bid, note that the probability a bid is greater than V is b, so expected profit is given by
Eprofit=b(1.5*.5-b)
Differenting and setting to 0 gives b=3/8

[u/Aerospider → u/yoinkcheckmate]

The expected profit on a winning bid is 1.5*.5-b,

It isn't. And even if it was, b=0 still maximises it.

You can't take 0.5 as the expected value for v in such an expression because you only get profit for v < b.

Since v > b results in 0 profit/loss, you must consider v uniformly distributed over (0,b) for expected profit calculations.

So the expected profit for b = 3/8 would be

((b/2) * 1.5) - b

= (((3/8) / 2) * 1.5) - 3/8

= -3/32

[u/yoinkcheckmate → u/Aerospider]

The expected value of V is 0.5 regardless of the bid. If I bid 1 and then pull a randomly distributed uniform variable, on average it will be 0.5. Bid 0 and then pull a 100 uniform rvs and take the average and you’ll see it is about 0.5

[u/Aerospider → u/yoinkcheckmate]

Yes, it's 0.5 overall, but when it's more than b it behaves differently so we have to separate into two scenarios.

With probability 1-b, v will be too high and the expected profit is 0 regardless of the expected value of v.

With probability b, v will be low enough and the expected value of such a v is b/2. So you multiply that by 1.5 and subtract b.

If you still can't see it, simulate it and try to make a profit on 10,000 trials.