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 per

Answer 1

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 peopleScores = new List();
            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("________________________________________________________");
        }