[foosball] / foosrank.cpp

#include <stdio.h>
#include <math.h>
#include <assert.h>

#include <vector>
#include <algorithm>

// step sizes
static const double int_step_size = 50.0;
static const double pdf_step_size = 10.0;

// rating constant (see below)
static const double rating_constant = 455.0;

using namespace std;

double prob_score(int k, double a, double rd);
double prob_score_real(int k, double a, double prodai, double kfac, double rd_norm);
double prodai(int k, double a);
double fac(int x);

// Numerical integration using Simpson's rule
template<class T>
double simpson_integrate(const T &evaluator, double from, double to, double step)
{
        int n = int((to - from) / step + 0.5);
        double h = (to - from) / n;
        double sum = evaluator(from);

        for (int i = 1; i < n; i += 2) {
                sum += 4.0 * evaluator(from + i * h);
        }
        for (int i = 2; i < n; i += 2) {
                sum += 2.0 * evaluator(from + i * h);
        }
        sum += evaluator(to);

        return (h/3.0) * sum;
}

// probability of match ending k-a (k>a) when winnerR - loserR = RD
//
//   +inf  
//     / 
//    |
//    | Poisson[lambda1, t](a) * Erlang[lambda2, k](t) dt
//    |
//   /
// -inf
//
// where lambda1 = 1.0, lambda2 = 2^(rd/455)
//
// The constant of 455 is chosen carefully so to match with the
// Glicko/Bradley-Terry assumption that a player rated 400 points over
// his/her opponent will win with a probability of 10/11 =~ 0.90909. 
//
double prob_score(int k, double a, double rd)
{
        return prob_score_real(k, a, prodai(k, a), fac(k-1), rd/rating_constant);
}

// Same, but takes in Product(a+i, i=1..k-1) and (k-1)! as an argument in
// addition to a. Faster if you already have that precomputed, and assumes rd
// is already divided by 455.
double prob_score_real(int k, double a, double prodai, double kfac, double rd_norm)
{
        double nom = prodai * pow(2.0, rd_norm * a); 
        double denom = kfac * pow(1.0 + pow(2.0, rd_norm), k+a);
        return nom/denom;
}

// Calculates Product(a+i, i=1..k-1) (see above).
double prodai(int k, double a)
{
        double prod = 1.0;
        for (int i = 1; i < k; ++i)
                prod *= (a+i);
        return prod;
}

double fac(int x)
{
        double prod = 1.0;
        for (int i = 2; i <= x; ++i)
                prod *= i;
        return prod;
}

// 
// Computes the integral
//
//   +inf
//    /
//    |
//    | ProbScore[a] (r2-r1) Gaussian[mu2, sigma2] (dr2) dr2
//    |
//   /
// -inf
//
// For practical reasons, -inf and +inf are replaced by 0 and 3000, which
// is reasonable in the this context.
//
// The Gaussian is not normalized.
//
// Set the last parameter to 1.0 if player 1 won, or -1.0 if player 2 won.
// In the latter case, ProbScore will be given (r1-r2) instead of (r2-r1).
//
class ProbScoreEvaluator {
private:
        int k;
        double a;
        double prodai_precompute, kfac_precompute, r1, mu2, sigma2, winfac;

public:
        ProbScoreEvaluator(int k, double a, double prodai_precompute, double kfac_precompute, double r1, double mu2, double sigma2, double winfac)
                : k(k), a(a), prodai_precompute(prodai_precompute), kfac_precompute(kfac_precompute), r1(r1), mu2(mu2), sigma2(sigma2), winfac(winfac) {}
        inline double operator() (double x) const
        {
                double probscore = prob_score_real(k, a, prodai_precompute, kfac_precompute, (x - r1)*winfac);
                double z = (x - mu2)/sigma2;
                double gaussian = exp(-(z*z/2.0));
                return probscore * gaussian;
        }
};

double opponent_rating_pdf(int k, double a, double r1, double mu2, double sigma2, double winfac)
{
        double prodai_precompute = prodai(k, a);
        double kfac_precompute = fac(k-1);
        winfac /= rating_constant;

        return simpson_integrate(ProbScoreEvaluator(k, a, prodai_precompute, kfac_precompute, r1, mu2, sigma2, winfac), 0.0, 3000.0, int_step_size);
}

// normalize the curve so we know that A ~= 1
void normalize(vector<pair<double, double> > &curve)
{
        double peak = 0.0;
        for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
                peak = max(peak, i->second);
        }

        double invpeak = 1.0 / peak;
        for (vector<pair<double, double> >::iterator i = curve.begin(); i != curve.end(); ++i) {
                i->second *= invpeak;
        }
}

// computes matA * matB
void mat_mul(double *matA, unsigned ah, unsigned aw,
             double *matB, unsigned bh, unsigned bw,
             double *result)
{
        assert(aw == bh);
        for (unsigned y = 0; y < bw; ++y) {
                for (unsigned x = 0; x < ah; ++x) {
                        double sum = 0.0;
                        for (unsigned c = 0; c < aw; ++c) {
                                sum += matA[c*ah + x] * matB[y*bh + c];
                        }
                        result[y*bw + x] = sum;
                }
        }
}
                
// computes matA^T * matB
void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
                   double *matB, unsigned bh, unsigned bw,
                   double *result)
{
        assert(ah == bh);
        for (unsigned y = 0; y < bw; ++y) {
                for (unsigned x = 0; x < aw; ++x) {
                        double sum = 0.0;
                        for (unsigned c = 0; c < ah; ++c) {
                                sum += matA[x*ah + c] * matB[y*bh + c];
                        }
                        result[y*bw + x] = sum;
                }
        }
}

void print3x3(double *M)
{
        printf("%f %f %f\n", M[0], M[3], M[6]);
        printf("%f %f %f\n", M[1], M[4], M[7]);
        printf("%f %f %f\n", M[2], M[5], M[8]);
}

void print3x1(double *M)
{
        printf("%f\n", M[0]);
        printf("%f\n", M[1]);
        printf("%f\n", M[2]);
}

// solves Ax = B by Gauss-Jordan elimination, where A is a 3x3 matrix,
// x is a column vector of length 3 and B is a row vector of length 3.
// Destroys its input in the process.
void solve3x3(double *A, double *x, double *B)
{
        // row 1 -= row 0 * (a1/a0)
        {
                double f = A[1] / A[0];
                A[1] = 0.0;
                A[4] -= A[3] * f;
                A[7] -= A[6] * f;

                B[1] -= B[0] * f;
        }

        // row 2 -= row 0 * (a2/a0)
        {
                double f = A[2] / A[0];
                A[2] = 0.0;
                A[5] -= A[3] * f;
                A[8] -= A[6] * f;

                B[2] -= B[0] * f;
        }

        // row 2 -= row 1 * (a5/a4)
        {
                double f = A[5] / A[4];
                A[5] = 0.0;
                A[8] -= A[7] * f;
                
                B[2] -= B[1] * f;
        }

        // back substitute:

        // row 1 -= row 2 * (a7/a8)
        {
                double f = A[7] / A[8];
                A[7] = 0.0;

                B[1] -= B[2] * f;
        }

        // row 0 -= row 2 * (a6/a8)
        {
                double f = A[6] / A[8];
                A[6] = 0.0;

                B[0] -= B[2] * f;
        }

        // row 0 -= row 1 * (a3/a4)
        {
                double f = A[3] / A[4];
                A[3] = 0.0;

                B[0] -= B[1] * f;
        }

        // normalize
        x[0] = B[0] / A[0];
        x[1] = B[1] / A[4];
        x[2] = B[2] / A[8];
}

// Give an OK starting estimate for the least squares, by numerical integration
// of statistical moments.
void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
{
        double h = (curve.back().first - curve.front().first) / (curve.size() - 1);

        double area = curve.front().second;
        double ex = curve.front().first * curve.front().second;
        double ex2 = curve.front().first * curve.front().first * curve.front().second;

        for (unsigned i = 1; i < curve.size() - 1; i += 2) {
                double x = curve[i].first;
                double y = curve[i].second;
                area += 4.0 * y;
                ex += 4.0 * x * y;
                ex2 += 4.0 * x * x * y;
        }
        for (unsigned i = 2; i < curve.size() - 1; i += 2) {
                double x = curve[i].first;
                double y = curve[i].second;
                area += 2.0 * y;
                ex += 2.0 * x * y;
                ex2 += 2.0 * x * x * y;
        }
        
        area += curve.back().second;
        ex += curve.back().first * curve.back().second;
        ex2 += curve.back().first * curve.back().first * curve.back().second;

        area = (h/3.0) * area;
        ex = (h/3.0) * ex / area;
        ex2 = (h/3.0) * ex2 / area;

        mu_result = ex;
        sigma_result = sqrt(ex2 - ex * ex);
}
        
// Find best fit of the data in curves to a Gaussian pdf, based on the
// given initial estimates. Works by nonlinear least squares, iterating
// until we're below a certain threshold.
//
// Note that the algorithm blows up quite hard if the initial estimate is
// not good enough. Use estimate_musigma to get a reasonable starting
// estimate.
void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
{
        double A = 1.0;
        double mu = mu1;
        double sigma = sigma1;

        // column-major
        double matA[curve.size() * 3];  // N x 3
        double dbeta[curve.size()];     // N x 1

        // A^T * A: 3xN * Nx3 = 3x3
        double matATA[3*3];

        // A^T * dβ: 3xN * Nx1 = 3x1
        double matATdb[3];

        double dlambda[3];

        for ( ;; ) {
                //printf("A=%f mu=%f sigma=%f\n", A, mu, sigma);

                // fill in A (depends only on x_i, A, mu, sigma -- not y_i)
                for (unsigned i = 0; i < curve.size(); ++i) {
                        double x = curve[i].first;

                        // df/dA(x_i)
                        matA[i + 0 * curve.size()] = 
                                exp(-(x-mu)*(x-mu)/(2.0*sigma*sigma));

                        // df/dµ(x_i)
                        matA[i + 1 * curve.size()] = 
                                A * (x-mu)/(sigma*sigma) * matA[i + 0 * curve.size()];

                        // df/dσ(x_i)
                        matA[i + 2 * curve.size()] = 
                                matA[i + 1 * curve.size()] * (x-mu)/sigma;
                }

                // find dβ
                for (unsigned i = 0; i < curve.size(); ++i) {
                        double x = curve[i].first;
                        double y = curve[i].second;

                        dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
                }

                // compute a and b
                mat_mul_trans(matA, curve.size(), 3, matA, curve.size(), 3, matATA);
                mat_mul_trans(matA, curve.size(), 3, dbeta, curve.size(), 1, matATdb);

                // solve
                solve3x3(matATA, dlambda, matATdb);

                A += dlambda[0];
                mu += dlambda[1];
                sigma += dlambda[2];

                // terminate when we're down to three digits
                if (fabs(dlambda[0]) <= 1e-3 && fabs(dlambda[1]) <= 1e-3 && fabs(dlambda[2]) <= 1e-3)
                        break;
        }

        mu_result = mu;
        sigma_result = sigma;
}

void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
{
        vector<pair<double, double> > curve;

        if (score1 > score2) {
                for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
                        double z = (r1 - mu1) / sigma1;
                        double gaussian = exp(-(z*z/2.0));
                        curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score1, score2, r1, mu2, sigma2, 1.0)));
                }
        } else {
                for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
                        double z = (r1 - mu1) / sigma1;
                        double gaussian = exp(-(z*z/2.0));
                        curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score2, score1, r1, mu2, sigma2, -1.0)));
                }
        }

        double mu_est, sigma_est;
        normalize(curve);
        estimate_musigma(curve, mu_est, sigma_est);
        least_squares(curve, mu_est, sigma_est, mu, sigma);
}

int main(int argc, char **argv)
{
        double mu1 = atof(argv[1]);
        double sigma1 = atof(argv[2]);
        double mu2 = atof(argv[3]);
        double sigma2 = atof(argv[4]);

        if (argc > 6) {
                int score1 = atoi(argv[5]);
                int score2 = atoi(argv[6]);
                double mu, sigma;
                compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma);
                printf("%f %f\n", mu, sigma);
        } else {
                int k = atoi(argv[5]);

                // assess all possible scores
                for (int i = 0; i < k; ++i) {
                        double newmu1, newmu2, newsigma1, newsigma2;
                        compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1);
                        compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2);
                        printf("%u-%u,%f,%+f,%+f\n",
                                k, i, prob_score(k, i, mu2-mu1), newmu1-mu1, newmu2-mu2);
                }
                for (int i = k; i --> 0; ) {
                        double newmu1, newmu2, newsigma1, newsigma2;
                        compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, newmu1, newsigma1);
                        compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, newmu2, newsigma2);
                        printf("%u-%u,%f,%+f,%+f\n",
                                i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
                }
        }
}