Bdtyktl

Your military junta has recently overthrown the government of Malta and taken control of the small island nation. As a way to finance your wise and just dictatorship, you have decided to threaten the European government that you will invoke Maxit – Maltese exit from the union – unless they offer you a suitable monetary compensation to convince you to stay. You are currently preparing for the negotiations. In particular, you need to tell your negotiators the smallest acceptable contribution with which a deal can still be made.

You know for a fact that the union has budgeted some maximum amount that they are willing to contribute, and this amount is somewhere between zero and one hundred coffers of euro banknotes. If you ask for more than their budgeted amount, the talks will conclude immediately and no deal will be made. Unfortunately, your spies have not been able to get hold of the budget's details – from your perspective, any number between 0 and 100 is equally likely to be the union's maximum contribution.

Civilized European negotiation rules dictate that once your minimum requirement and the union's budgeted maximum have been presented, your negotiator can get the final price up to exactly halfway between those two numbers. So if your minimum acceptable number is 70 coffers and the union's budget is 80 coffers, the negotiations will eventually lead to a deal of 75 coffers.

Being a dictator, you are completely uninterested in whether the country will actually leave or stay in the union, as you care solely about the financial gain. What number should you give your negotiators as the minimum price? How large is the average contribution you can expect?

You should ask for

$frac{100}3$ coffers of euros.

Suppose

you're asking for $x$ coffers of euros and considering making it $x+delta$ instead. What difference does this make? (To first order, as $deltarightarrow0$.) With probability $delta/100$ you've just gone beyond what the EU was willing to offer, and have lost $x$. And with probability $1-x/100$ you had a deal and still have one, but now you'll get $delta/2$ more. The optimal choice of $x$ will be where the gain and loss are equal, so $x/100=(1-x/100)/2$ or $x=100/3$.

In this case you will get, on average,

zero, 1/3 of the time; and 100/3 plus half the average excess, or 50, 2/3 of the time. That is, on average you will get exactly what you ask for. Perhaps there's an elegant reason why this must obviously be so when you choose optimally?

[EDITED to add:] Discussion with user Excited Raichu in comments to their answer has brought up an important point. Are we to assume that the sums of money here are continuously variable and that the EU's offer will be a uniform random quantity between 0 and 100? Or that they are integer numbers of coffers of euros (with, presumably, the possibility of splitting a coffer in half if the parity of offer and demand are different) and that the EU is equally likely to offer 0, 1, 2, ..., 100 coffers? Or something else? My answer above assumes the first of those options. The second is possibly best handled by direct computation, which (if I've done it right) leads to the conclusion

that you should ask for 34 coffers of euros, and will on average get 33.5 coffers.

Brainless Python code to deal with the discrete version of the question:

def dprofit(demand):

    s = 0

    for offer in range(0, 101):

        if offer >= demand:

            s += 0.5*(offer+demand)

    return s



best = (0,0)

for demand in range(0,101):

    p = dprofit(demand)

    if p>best[0]: best = (p,demand)

after which

best = (3383.5, 34) and 3383.5/101 = 33.5.

We should specify a minimum of

one coffer of euros

This average contribution we can expect is

25.75 coffers

Because

the union has a maximum between 0 and 100, uniformly distributed. Except of course we rule out 0 -- there would be no negotiation in that case, but they are sitting at the table. If 1, we get 1. If 100, we get 50.5, and the average of these is 25.75 (yes, linearity of average works here).

This is so, despite the fact that

the other answer, 33 or 34, gives us a higher average outcome. But the goal is financial gain. The assumption that 'best' means 'highest average' is called 'risk neutrality'. This assumption is valid for small sums. But a coffer of euros sounds awfully large to me. This is immediately post-coup, our military is going to have a cash flow problem and a coffer of euros sounds like a non-negligible chunk of our budget.

The average outcome is still pretty good as it is, and the prospect of increasing this somewhat clearly does not justify the risk of getting nothing at all.

The moral of this story is

to invest more in intelligence, we should have been able to get more information than that for much less than one coffer of euros.

My answer is the same as that of Gareth, but I intuited differently so I thought I'd share my solution as well. This is for the continuous case, by the way, where fractional coffers are allowed.

First, let's assume Malta chooses to demand 50 coffers, then one of two things can happen.

Either 50 coffers is less than or equal to what EU offers, or it's more. If we name Malta's offer $M$ and EU's offer $E$, then
$$mathbf{P}(M lt E) = frac{100-M}{100}$$

If 50 coffers happened to be less than what EU offered then we know that

$$E sim mathcal{U}(50, 100)$$
and
$$mathbf{E}(E) = 75$$

Expected gain in this case will therefore be

$$frac{50 +75}{2} = 62.5$$
but that's assuming EU's offer was greater than 50, and probability of that is $0.5$. Therefore, if Malta offered $50$ coffers, the expected gain is $62.5/2 = 31.25$.

With that settled we can derive a general formula for any $M$.

! With $G$ being Malta's expected gain:

! $$begin{align}
G &= frac{M + frac{M + 100}{2}}{2} cdot frac{100 - M}{100} \
&= frac{(3M + 100)(100 - M)}{400} \
&= -0.0075M^2 + 0.5M + 25
end{align}$$

! As we want to know for what $M$ we get the largest $G$, the final step is to take the first derivative and solve $G^prime = 0$

! $$ begin{align} 0 &= -0.015M + 0.5 \
M &= frac{0.5}{0.015} \
M &= frac{100}{3} end{align}$$

I also wrote a small R script to double check and draw the curve

f <- function(x, y) {

    (x <= y)*(x + y)/2

}



s <- seq(0, 100, by=0.1)

m <- outer(s, s, f)

r <- rowMeans(m)

s[which.max(r)]



plot(r ~ s, type="l")

curve(-0.0075*x^2 + 0.5*x + 25, 0, 100, add=TRUE, lty=3, col=2, lwd=3)

I actually got something a bit different from the other answer. Here's a picture of my work, please let me know if anything is wrong:

EDIT: After the conversation in comments, I redid it using (101-x)/101 and got the same result...