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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?

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  • $\begingroup$ Are the claims and offers always a whole number of coffers, or can we operate with fractions of coffers? $\endgroup$ – jarnbjo Nov 16 '18 at 15:11
  • $\begingroup$ @jarnbjo I think with fractions allowed it fits the "mathematics" tag a little better, since otherwise there's a clear brute-force solution available. Should have made that clear in the description... That said it's probably not fair to change it at this point when people have already answered both variations. $\endgroup$ – jafe Nov 17 '18 at 9:57
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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 $\delta\rightarrow0$.) 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.

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  • $\begingroup$ Is that still true now that I have treated the continuous and discrete versions of the question separately? $\endgroup$ – Gareth McCaughan♦ Nov 16 '18 at 19:03
  • $\begingroup$ No, sorry. I read the question as if it was talking about a number of indivisible coffers, got a different answer than in your first part, wrote my previous comment here, realized where the mismatch was and then commented on the question asking what is actually ment. $\endgroup$ – jarnbjo Nov 16 '18 at 20:25
  • $\begingroup$ Not sure how much is in a coffer, but we can rule out rot13(gung gur havba unf n znk bs mreb, vs fb gurl jbhyqa'g obgure artbgvngvat) $\endgroup$ – deep thought Nov 16 '18 at 20:34
  • $\begingroup$ A 2 minute spreadsheet would confirm your computation (spoilers): i.imgur.com/aPiBcvE.png. Again assuming you can negotiate half of a coffer, rather than it always being rounded down e.g. $\endgroup$ – Mark Peters Nov 16 '18 at 20:49
  • $\begingroup$ @MarkPeters Yup. The Python code also took about two minutes to write. $\endgroup$ – Gareth McCaughan♦ Nov 16 '18 at 21:46
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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.

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    $\begingroup$ Really good approach, I love it how you are dealing explicitly with some common but unrealistic assumptions in optimization. It actually can be extended a step further, it would be optimal to ask one coffer less than what you suggest. If going that low is not permitted, then the whole negotiation is unfair in the sense that there can be a situation where your smallest allowed bid will be higher than the maximum budget - a situation which happens in many realistic negotiations, but somehow feels unsatisfying in the context of the puzzle. $\endgroup$ – rumtscho Nov 16 '18 at 22:44
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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)

enter image description here

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    $\begingroup$ Obviously I have not quite figured out this spoiler thing $\endgroup$ – AkselA Nov 17 '18 at 14:11
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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:

enter image description here

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

enter image description here

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  • $\begingroup$ I think you want 100-x where you have 101-x. E.g., if x=100 then the probability should be 0, not 1%. $\endgroup$ – Gareth McCaughan♦ Nov 16 '18 at 15:08
  • $\begingroup$ (Even more obviously: if x=0 then you've got a probability that's >1.) $\endgroup$ – Gareth McCaughan♦ Nov 16 '18 at 15:09
  • $\begingroup$ @GarethMcCaughan wait, then we may be both wrong- there are 101 possibilities (0 to 100 inclusive)- shouldn’t it be (101-x)/101 then? $\endgroup$ – Excited Raichu Nov 16 '18 at 15:13
  • $\begingroup$ I was assuming a continuously variable offer from the EU. If it's guaranteed to be an integer number of coffers then indeed things may be a bit different. $\endgroup$ – Gareth McCaughan♦ Nov 16 '18 at 15:14
  • $\begingroup$ @GarethMcCaughan actually, with (101-x)/101 my answer is the same- so it actually does depend on whether the offer must be an integer or not. $\endgroup$ – Excited Raichu Nov 16 '18 at 15:18

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