using namespace std;
-double prob_score(double a, double rd);
-double prob_score_real(double a, double prodai, double rd_norm);
-double prodai(double a);
+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);
-// probability of match ending 10-a when winnerR - loserR = RD
+ 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, 10](t) dt
+// | Poisson[lambda1, t](a) * Erlang[lambda2, k](t) dt
// |
// /
// -inf
// 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(double a, double rd)
+double prob_score(int k, double a, double rd)
{
- return prob_score_real(a, prodai(a), rd/rating_constant);
+ return prob_score_real(k, a, prodai(k, a), fac(k-1), rd/rating_constant);
}
-// Same, but takes in Product(a+i, i=1..9) 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(double a, double prodai, double rd_norm)
+// 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 =
- pow(2.0, -a*rd_norm) * pow(2.0, 10.0*rd_norm) * pow(pow(2.0, -rd_norm) + 1.0, -a)
- * prodai;
- double denom = 362880 * pow(1.0 + pow(2.0, rd_norm), 10.0);
+ 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..9) (see above).
-double prodai(double a)
+// 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)
{
- return (a+1)*(a+2)*(a+3)*(a+4)*(a+5)*(a+6)*(a+7)*(a+8)*(a+9);
+ double prod = 1.0;
+ for (int i = 2; i <= x; ++i)
+ prod *= i;
+ return prod;
}
//
// 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).
//
-static inline double evaluate_int_point(double a, double prodai_precompute, double r1, double mu2, double sigma2, double winfac, double x);
+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(double a, double r1, double mu2, double sigma2, double winfac)
+double opponent_rating_pdf(int k, double a, double r1, double mu2, double sigma2, double winfac)
{
- double prodai_precompute = prodai(a);
+ double prodai_precompute = prodai(k, a);
+ double kfac_precompute = fac(k-1);
winfac /= rating_constant;
- int n = int(3000.0 / int_step_size + 0.5);
- double h = 3000.0 / double(n);
- double sum = evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, 0.0);
-
- for (int i = 1; i < n; i += 2) {
- sum += 4.0 * evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, i * h);
- }
- for (int i = 2; i < n; i += 2) {
- sum += 2.0 * evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, i * h);
- }
- sum += evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, 3000.0);
-
- return (h/3.0) * sum;
-}
-
-static inline double evaluate_int_point(double a, double prodai_precompute, double r1, double mu2, double sigma2, double winfac, double x)
-{
- double probscore = prob_score_real(a, prodai_precompute, (r1 - x)*winfac);
- double z = (x - mu2)/sigma2;
- double gaussian = exp(-(z*z/2.0));
- return probscore * gaussian;
+ 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
{
vector<pair<double, double> > curve;
- if (score1 == 10) {
+ 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(score2, r1, mu2, sigma2, 1.0)));
+ 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(score1, r1, mu2, sigma2, -1.0)));
+ curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score2, score1, r1, mu2, sigma2, -1.0)));
}
}
double mu2 = atof(argv[3]);
double sigma2 = atof(argv[4]);
- if (argc > 5) {
+ 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 <= 9; ++i) {
+ for (int i = 0; i < k; ++i) {
double newmu1, newmu2, newsigma1, newsigma2;
- compute_new_rating(mu1, sigma1, mu2, sigma2, 10, i, newmu1, newsigma1);
- compute_new_rating(mu2, sigma2, mu1, sigma1, i, 10, newmu2, newsigma2);
- printf("10-%u,%f,%+f,%+f\n",
- i, prob_score(i, mu1-mu2), newmu1-mu1, newmu2-mu2);
+ 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 = 10; i --> 0; ) {
+ for (int i = k; i --> 0; ) {
double newmu1, newmu2, newsigma1, newsigma2;
- compute_new_rating(mu1, sigma1, mu2, sigma2, i, 10, newmu1, newsigma1);
- compute_new_rating(mu2, sigma2, mu1, sigma1, 10, i, newmu2, newsigma2);
- printf("%u-10,%f,%+f,%+f\n",
- i, prob_score(i, mu2-mu1), newmu1-mu1, newmu2-mu2);
+ 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);
}
}
}
projects / foosball / blobdiff