// A ballot supports a bogeyman over a coalition if there are any candidates in the coalition one likes worse than the bogeyman.
// A ballot remains neutral between a bogeyman and a coalition if:
// (a) it considers the bogeyman equivalent to any of the coalition-membmers, or
// (b) There exists a smaller coalition consisting solely of candidates one prefers to those of the coalition which beats the bogeyman.
// Oterwise, the ballot supports the coalition over the bogeyman.
private int GetSupportForCoalition(RankedBallot ballot, ulong coalition, int bogeyman)
{
var ranking = ballot.Ranking;
var bogeymanTier = ranking.IndexesWhere(tier => tier.Contains(bogeyman)).Single();
// Detract support for any coalition containing someone we like less than the bogeyman.
if ((GetCoalition(ranking.Skip(bogeymanTier + 1).SelectMany(tier => tier)) & coalition) > 0ul)
{
return(-1);
}
// Remain neutral on any coalition containing someone we like the same as the bogeyman.
if ((GetCoalition(ranking[bogeymanTier]) & coalition) > 0ul)
{
return(0);
}
var preferredCoalition = 0ul;
foreach (var tier in ranking.Take(bogeymanTier))
{
// Support each coalition which is a subset of our preferred candidates.
preferredCoalition |= GetCoalition(tier);
if ((preferredCoalition | coalition) == preferredCoalition)
{
return(1);
}
// Remain neutral on the coalition if there exists a sub-coalition (without cycles) consisting of only preferred candidates which will work as well
if (Beats(preferredCoalition, bogeyman))
{
return(0);
}
}
throw new InvalidOperationException("There is a bug: We ought to have found a coalition which is a subset by now.");
}
private static (List <(int candidate, ulong?preferred)> Candidates, ulong Bogeyman) GetApprovedCandidates(CoalitionBeatMatrix beatMatrix, RankedBallot ballot)
{
var candidates = new List <(int candidate, ulong?preferred)>();
var ranking = ballot.Ranking;
// Approve of the each candidate `c` which is a first choice.
foreach (var c in ranking[0])
{
candidates.Add((c, null));
}
// With fewer than three ranks, no bogeymen can chance one's preferences.
if (ranking.Count < 3)
{
return(candidates, 0ul);
}
// Then, for each candidate `b` (the "bogeyman") ranked third or below,
// compute the smallest "coalition" of candidates `c` which can beat each potential "bogeyman" `b`,
// or includes all candidates one likes better than `b`, whichever comes first.
// Approve of each member of the smallest coalition to beat all bogeymen.
var coalition = GetCoalition(ranking[0]);
var potentialBogeymen = ranking
.Skip(2)
.SelectMany(b => b)
.Where(b => beatMatrix.Beats(coalition, b) != true)
.ToList();
if (!potentialBogeymen.Any())
{
return(candidates, 0ul);
}
foreach (var tier in ranking.Skip(1).Take(ranking.Count - 2))
{
var greaterBogeymen = potentialBogeymen.Where(b => !tier.Contains(b)).ToList();
// If this tier contains all the bogeymen, the previous tier is where our approval stops.
if (!greaterBogeymen.Any())
{
break;
}
// Otherwise, enlist the help of the lesser bogeymen to beat the greater bogeymen.
foreach (var c in tier)
{
candidates.Add((c, coalition));
}
coalition |= GetCoalition(tier);
var remainingBogeymen = greaterBogeymen
.Where(b => beatMatrix.Beats(coalition, b))
.ToList();
// Huzzah! We have beaten all the bogeymen!
if (!remainingBogeymen.Any())
{
break;
}
// Otherwise, check the next teir for potential coalition-mates.
potentialBogeymen = remainingBogeymen;
}
return(candidates, GetCoalition(potentialBogeymen));
}
