Divide x into y parts by decreasing amount

Question

If I had $1000(variable) and I want to split that amount up and give it to 20(variable) people, but rather than give it evenly to each person, I want to give more to the 1st person, and the 2nd person, etc.

So the 20th person gets the least, and the 5th person gets the 5th most.

How would I achieve that?

Thanks

Edit:

Formula:

int people = 20;
float prize = 1000;

float k = (2 * prize) / ((people) * (people - 1));
float sum = 0;

for (int i = 1; i < people; ++i)
{
    var personsPrize = i * k;
    sum += personsPrize;
    Console.WriteLine(personsPrize);
}
Console.WriteLine("sum = " + sum);

Answer 1

Adding answer to Ensure prize pool doesn't award tied participants less than participants who scored worse

Two things

1. Incorrect, when assigning prices start from first person to the n-1 one person does't get prize, correct is start from first person to the n

2. Valid basePrize noted with K is

   k = (2 * prize) / ((people) * (people + 1))

Now the answer the other question:

int totalPeople = 20;
float prize = 5000;

List<int> peopleScores = new List<int>();
Random r = new Random();
for (int i = 0; i < totalPeople; ++i)
{
    peopleScores.Add(r.Next(0, 100));
}

var totalPeopleWithScore = peopleScores.Where(x => x > 0).Count();

var groupedScores = peopleScores
    .Where(x => x > 0)
    .ToList()
    .GroupBy(x => x)
    .Select(grp => new
    {
        score = grp.Key,
        peopleScores = grp.ToList()
    })
    .OrderBy(g => g.score)
    .ToList();

var groupCount = groupedScores.Select(x => new { count = x.peopleScores.Count() }).Select(z => z.count).Count();

float basePrizeRate = 2 * prize / totalPeopleWithScore / (totalPeopleWithScore + 1);
Console.WriteLine("Base Price rate: " + basePrizeRate);
float sum = 0;
var leaderboardPosition = 0;
var totalWinners = 0;
foreach (var positionScore in groupedScores)
{
    var countWinner = positionScore.peopleScores.Count();
    Console.WriteLine();
    Console.WriteLine($"On leaderboard position : {groupedScores.Count() - leaderboardPosition} are {countWinner} winners with score: {positionScore.score}");

    float positionPrizePool = 0;
    for (int i = 1; i <= positionScore.peopleScores.Count(); ++i)
    {
        totalWinners++;
        positionPrizePool += totalWinners * basePrizeRate;
    }
    Console.WriteLine("Prize Pool " + positionPrizePool);
    var personPoolPrize = positionPrizePool / positionScore.peopleScores.Count();
    foreach (var x in positionScore.peopleScores)
    {
        Console.WriteLine($"Winner {totalWinners} won: {personPoolPrize}");
        sum += personPoolPrize;
    }
    leaderboardPosition++;
}
Console.WriteLine();

Console.WriteLine("Total Prize: " + sum);
Console.WriteLine("Total Winners: " + totalPeopleWithScore);

Answer 2

You owe me a beer, actually a chug of beer!!

Using Gauss distribution (N*(N+1))/2 you cannot have a slight margin over the competitors ranks, prizes have linear increase https://mathbitsnotebook.com/Algebra2/Sequences/SSGauss.html.

What you need is something that increases exponentially or with a fixed ration, I want to solve your problem with a mathematical approach, so I will use a geometric progression https://mathbitsnotebook.com/Algebra2/Sequences/SSGeometric.html

I won't add any other answer after this one, you have to build your own prize pool system!!

For Geometric Distribution formula is:

rankPrize = ((1 - distributionFactor) / (1 - distributionFactor ^ winners) * distributionFactor ^ (rank - 1)) * prize

where distributionFactor - is between 0 and 1

Play around with distribution Factor and chose the correct value for your system, I've chosen 0.8 because the gap between prizes is not that big!

Here we go, fiddle link: https://dotnetfiddle.net/qmJnYd to play around and implementation:

static void GaussDistributionPricePool()
        {
            Console.WriteLine("________________________________________________________");
            Console.WriteLine("Begin Gauss distribution price pool");

            int totalPeople = 20;
            float prize = 5000;

            List<int> peopleScores = new List<int>();
            Random r = new Random();
            for (int i = 0; i < totalPeople; ++i)
            {
                peopleScores.Add(r.Next(0, 100));
            }

            var totalPeopleWithScore = peopleScores.Where(x => x > 0).Count();

            var groupedScores = peopleScores
                .Where(x => x > 0)
                .ToList()
                .GroupBy(x => x)
                .Select(grp => new
                {
                    score = grp.Key,
                    peopleScores = grp.ToList()
                })
                .OrderBy(g => g.score)
                .ToList();

            var groupCount = groupedScores.Select(x => new { count = x.peopleScores.Count() }).Select(z => z.count).Count();
            // Gauss formula. (N*(N+1))/2
            // https://mathbitsnotebook.com/Algebra2/Sequences/SSGauss.html
            float basePrizeRate = 2 * prize / totalPeopleWithScore / (totalPeopleWithScore + 1);
            Console.WriteLine("Base Price rate: " + basePrizeRate);
            float sum = 0;
            var leaderboardRank = 0;
            var totalWinners = 0;
            foreach (var positionScore in groupedScores)
            {
                var countWinner = positionScore.peopleScores.Count();
                Console.WriteLine();
                Console.WriteLine("On leaderboard rank : " + (groupedScores.Count() - leaderboardRank) + " are " + countWinner + " winners with score: " + positionScore.score);

                float positionPrizePool = 0;
                for (int i = 1; i <= positionScore.peopleScores.Count(); ++i)
                {
                    totalWinners++;
                    positionPrizePool += totalWinners * basePrizeRate;
                }
                Console.WriteLine("Prize Pool " + positionPrizePool);
                var personPoolPrize = positionPrizePool / positionScore.peopleScores.Count();
                foreach (var x in positionScore.peopleScores)
                {
                    Console.WriteLine("Winner " + totalWinners + " won: " + personPoolPrize);
                    sum += personPoolPrize;
                }
                leaderboardRank++;
            }
            Console.WriteLine();

            Console.WriteLine("Total Prize: " + sum);
            Console.WriteLine("Total Winners: " + totalPeopleWithScore);

            Console.WriteLine("End Gauss distribution price pool");
            Console.WriteLine("________________________________________________________");
        }

Answer 3

Formula is correct, needed a little touch.

1. Don't cast float into int, data loss!

2. When going within the for go from the first person to the n-1

   int people = 20;
   float prize = 1000;
   
   float k = (2 * prize) / ((people) * (people - 1));
   float sum = 0;
   
   for (int i = 1; i < people; ++i)
   {
       var personsPrize = i * k;
       sum += personsPrize;
       Console.WriteLine(personsPrize);
   }
   Console.WriteLine("sum = " + sum);