Problem on EU commission
Solution 1:
This is a very interesting problem! On your theorem: I don't understand why you maximize $z$, which is just a density (up to that point everything seems fine). You want to maximize $P(V\ge A)$. You assume for sufficiently many countries that this can be approximated by a normal distribution, invoking some version of the CLT. Thus, $$P(V\ge A)\approx \int^\infty_A \phi\left(\frac{x-\sum c_ip_i}{\sqrt{\sum c_i^2 p_i(1-p_i})}\right) dx=1-\Phi\left(\frac{A-\sum c_ip_i}{\sqrt{\sum c_i^2 p_i(1-p_i)}}\right).$$ How to maximize this? I am not entirely sure. I remember that the normal distribution is log-concave, and the CDF of log-concave functions is log-concave. So if $log(\Phi(x))$ is concave, then $-log(\Phi(x))$ is convex. But that doesn't help us here..
A few more suggestions:
On assumption 2): Why do you need $M$ if you defined weights $c_i$ already?
In your formulation, you could constrain the weights to $\sum_i c_i=1$, then the condition is $V\ge A$, looks nicer but is not necessary.
On assumption 3): I agree it doesn't make the problem meaningless, but it seems unnecessary - it doesn't change the maximization problem. It just implies that, say, even an equal voting weight distribution would lead to more passes than failures of resolutions. But the problem of $p_i<A$ for some (or even all) countries would still be interesting. Given the normal approximation, there is still a positive probability for the resolution to pass whenever $p_i>0$ for all $i$, but it would be harder. This way you could model "harder resultions".
On assumption 1) and 3): if $n\to\infty$, then assumption 3 guarantees that the resolution passes, as long as you have positive weight $c_i$ on all countries. Because the expected voteshare is above $A$, and asymptotically the expected vote share realizes with probability 1 (some strong law of large numbers). Interesting: if some countries have $p_i>A$ (a positive mass, to be exact) and some $p_i<A$, then as $n\to\infty$ you can guarantee passing of the resolution by giving positive weights to all with $p_i>A$ and none to the others.
Given the previous point, it seems dangerous to talk about "asymptotics" - you just want finitely but many countries so that you can approximate with a normal distribution, but you don't actually want $n\to\infty$ as the problem then is trivial given assumption 3). Maybe this is what the commenter above meant.
Where can you find something similar? I think your best bet might be the finance literature, where you compute the probability that your portfolio investment return is above some threshold $A$. There are some assets which have a similar structure as these votes (e.g., bonds): either the asset pays a positive dividend or the issuer goes bankrupt and the return is zero - just like your random variable $X_i$. Same for a loan portfolio.
- Your comments about "getting a decision right" reminded me of the Condorcet jury theorems. The setting is a special case of yours, where every member has the same voting weight and the majority threshold is $A=1/2$
- It also reminded me of the Feddersen Pesendorfer game theoretic analysis of unanimity voting rules. Neither of the two are directly related to your problem though. Again, finance seems to be your best bet.