Suppose you are helping organize a summit on climate change in Copehagen. Each of m $1$ countries has sent n delegats to the summit, and there is also a delegation of n UN facilitators there, as well. $($So there are mn total people.$)$ The delegates are initially going to be split into n groups for team$-$building exercises where each group contains exactly one person from each of the m delegations $($the facilitators and the m $1$ countries$).$ But, of course, saving the world is never that simple. Each person p has submitted a preference list, giving their rankings of each of the other mn$1$ people at the summit. We need your help assembling a set of groups that are stable in the following sense: Consider two people x and y representing the same country assigned to groups G and G$,$ respectively. The people x and y form an instability if they both strictly prefer every delegate in the other group to the corresponding delegate in their own group. That is$,$ x and y form an instability if$,$ for every delegation D$,$ x prefers D$$s representative in G$$ to D$$s representative in G and y prefers D$$s representative in G to D$$s representative in G$.$

$($Note that this definition precludes an instability formed by two facilitators, since the definition of

an instability relies on the two individuals representing the same country, and the facilitators are not representing any country.$)$

Prove that there exists a stable set of groups for any preference lists by giving an $($efficient$)$

algorithm to find one. As always, prove your algorithm correct and analyze its running time.

Hint: to get started, think about the m $=2$ case, where there are two delegations $($one country, plus the UN$).$ Now can you find a way to include a second country? Notice that this definition of stability is different from that in the stable matchings we have otherwise discussed. Here, an instability is formed by two people of the same type $($from the same country$),$ whereas in the stable$-$matching setting, an instability is formed by two people of different types $($e$.$g$.,$ a professional and a celebrity$).$