13 #define MAX_PLAYERS 4096
15 float mu[MAX_PLAYERS];
16 float sigma[MAX_PLAYERS];
21 * L(mu_vec, sigma_vec, matches) = product[ L(mu_A, sigma_A, mu_B, sigma_B, score_AB - score_BA) ]
22 * log-likelihood = sum[ log( L(mu_A, sigma_A, mu_B, sigma_B, score_AB - score_BA) ) ]
24 * L(mu1, sigma1, mu2, sigma2, score2 - score1) = sigmoid(mu2 - mu1, sqrt(sigma1² + sigma2²), (score2 - score1))
26 * pdf := 1/(sigma * sqrt(2*Pi)) * exp(-(x - mu)^2 / (2 * sigma^2));
27 * pdfs := subs({ mu = mu1 - mu2, sigma = sqrt(sigma1^2 + sigma2^2) }, pdf);
28 * diff(log(pdfs), mu1);
36 map<int, vector<match> > matches_for_player;
38 void dump_scores(const vector<string> &players, const float *mu, const float *sigma, int num_players)
41 for (int i = 0; i < num_players; ++i) {
42 printf("%s=[%5.1f, %4.1f] ", players[i].c_str(), mu[i], sigma[i]);
46 for (int i = 0; i < num_players; ++i) {
47 printf("%5.1f ", mu[i]);
51 for (int i = 0; i < num_players; ++i) {
52 printf("%f %s\n", mu[i], players[i].c_str());
58 * diff(logL, mu1) = -w * (mu1 - mu2 - x) / sigma_c^2
59 * maximizer for mu1 is given by: sum_i[ (w_i/sigma_c_i)^2 (mu1 - mu2_i - x_i) ] = 0
60 * sum_i[ (w_i/sigma_c_i)^2 mu1 ] = sum_i [ (w_i/sigma_c_i)^2 ( mu2_i + x_i ) ]
61 * mu1 = sum_i [ (w_i/sigma_c_i)^2 ( mu2_i + x_i ) ] / sum_i[ (w_i/sigma_c_i)^2 ]
63 void update_mu(float *mu, float *sigma, int player_num, const vector<match> &matches)
65 if (matches.empty()) {
69 float nom = 0.0f, denom = 0.0f;
70 for (unsigned i = 0; i < matches.size(); ++i) {
71 float sigma1 = sigma[player_num];
72 float sigma2 = sigma[matches[i].other_player];
73 float inv_sigma_c2 = matches[i].weight / (sigma1 * sigma1 + sigma2 * sigma2);
74 float x = matches[i].margin; // / 70.0f;
76 nom += (mu[matches[i].other_player] + x) * inv_sigma_c2;
77 denom += inv_sigma_c2;
79 mu[player_num] = nom / denom;
83 * diff(logL, sigma1) = sigma1 (-sigma1² - sigma2² + (x - mu)²) / sigma_c²
84 * maximizer for sigma1 is given by: sum_i[ (1/sigma_c_i)² sigma1 ((x - mu)² - (sigma1² + sigma2²) ] = 0
85 * sum_i[ (x - mu)² - sigma1² - sigma2² ] = 0 |: sigma1 != 0, sigma2 != 0
86 * sum_i[ (x - mu)² - sigma2² ] = sum[ sigma1² ]
87 * sigma1 = sqrt( sum_i[ (x - mu)² - sigma2² ] / N )
89 void update_sigma(float *mu, float *sigma, int player_num, const vector<match> &matches)
91 if (matches.size() < 2) {
96 for (unsigned i = 0; i < matches.size(); ++i) {
97 float mu1 = mu[player_num];
98 float mu2 = mu[matches[i].other_player];
100 float sigma2 = sigma[matches[i].other_player];
101 float x = matches[i].margin;
103 //fprintf(stderr, "x=%f mu=%f sigma2=%f add %f-%f = %f\n", x, mu, sigma2, (x-mu)*(x-mu), sigma2*sigma2, (x - mu) * (x - mu) - sigma2 * sigma2);
104 sum += (x - mu) * (x - mu) - sigma2 * sigma2;
110 //fprintf(stderr, "sum=%f\n", sum);
111 sigma[player_num] = sqrt(sum / matches.size());
115 * diff(logL, sigma) = w ( (x - mu)² - sigma² ) / sigma³
116 * maximizer for sigma is given by: sum_i[ (w_i/sigma)³ ((x - mu)² - sigma²) ] = 0
117 * sum_i[ w_i ( (x - mu)² - sigma² ) ] = 0 |: sigma != 0
118 * sum_i[ w_i (x - mu)² ] = sum[ w_i sigma² ]
119 * sigma = sqrt( sum_i[ w_i (x - mu)² ] / sum[w_i] )
121 void update_global_sigma(float *mu, float *sigma, int num_players)
123 float nom = 0.0f, denom = 0.0f;
124 for (int i = 0; i < num_players; ++i) {
125 for (unsigned j = 0; j < matches_for_player[i].size(); ++j) {
126 const match& m = matches_for_player[i][j];
128 // Only count each match once.
129 if (m.other_player <= i) {
134 float mu2 = mu[m.other_player];
135 float mu = mu1 - mu2;
139 nom += w * ((x - mu) * (x - mu));
144 float best_sigma = sqrt(nom / denom) / sqrt(2.0f); // Divide evenly between the two players.
145 for (int i = 0; i < num_players; ++i) {
146 sigma[i] = best_sigma;
150 void renormalize(float *mu, float *sigma, int num_players)
153 for (int i = 0; i < num_players; ++i) {
156 float corr = 1500.0f - avg / num_players;
157 for (int i = 0; i < num_players; ++i) {
163 * Compute Hessian matrix of the negative log-likelihood, ie. for each term in logL:
165 * M_ij = D_i D_j (- logL) = -w / sigma² for i != j
166 * w / sigma² for i == j
168 * Note that this does not depend on mu or the margin at all.
170 double hessian[MAX_PLAYERS][MAX_PLAYERS];
171 void construct_hessian(const float *mu, const float *sigma, int num_players)
173 memset(hessian, 0, sizeof(hessian));
175 for (int i = 0; i < num_players; ++i) {
176 double sigma1 = sigma[i];
178 for (unsigned k = 0; k < matches_for_player[i].size(); ++k) {
179 int j = matches_for_player[i][k].other_player;
181 double sigma2 = sigma[j];
182 double sigma_sq = sigma1 * sigma1 + sigma2 * sigma2;
184 float w = matches_for_player[i][k].weight;
186 hessian[i][j] -= w / sigma_sq;
187 hessian[i][i] += w / sigma_sq;
191 for (int i = 0; i < num_players; ++i) {
192 for (int j = 0; j < num_players; ++j) {
193 printf("%.12f ", hessian[i][j]);
199 int main(int argc, char **argv)
202 if (scanf("%d", &num_players) != 1) {
203 fprintf(stderr, "Could't read number of players\n");
207 if (num_players > MAX_PLAYERS) {
208 fprintf(stderr, "Max %d players supported\n", MAX_PLAYERS);
212 vector<string> players;
213 map<string, int> player_map;
215 for (int i = 0; i < num_players; ++i) {
217 if (scanf("%s", buf) != 1) {
218 fprintf(stderr, "Couldn't read player %d\n", i);
222 players.push_back(buf);
228 char pl1[256], pl2[256];
232 if (scanf("%s %s %d %d %f", pl1, pl2, &score1, &score2, &weight) != 5) {
233 //fprintf(stderr, "Read %d matches.\n", num_matches);
239 if (player_map.count(pl1) == 0) {
240 fprintf(stderr, "Unknown player '%s'\n", pl1);
243 if (player_map.count(pl2) == 0) {
244 fprintf(stderr, "Unknown player '%s'\n", pl2);
249 m1.other_player = player_map[pl2];
250 m1.margin = score1 - score2;
252 matches_for_player[player_map[pl1]].push_back(m1);
255 m2.other_player = player_map[pl1];
256 m2.margin = score2 - score1;
258 matches_for_player[player_map[pl2]].push_back(m2);
261 float mu[MAX_PLAYERS];
262 float sigma[MAX_PLAYERS];
264 for (int i = 0; i < num_players; ++i) {
266 sigma[i] = 70.0f / sqrt(2.0f);
268 renormalize(mu, sigma, num_players);
270 for (int j = 0; j < 1000; ++j) {
271 float old_mu[MAX_PLAYERS];
272 float old_sigma[MAX_PLAYERS];
273 memcpy(old_mu, mu, sizeof(mu));
274 memcpy(old_sigma, sigma, sizeof(sigma));
275 for (int i = 0; i < num_players; ++i) {
276 update_mu(mu, sigma, i, matches_for_player[i]);
277 renormalize(mu, sigma, num_players);
279 update_global_sigma(mu, sigma, num_players);
280 /* for (int i = 0; i < num_players; ++i) {
281 update_sigma(mu, sigma, i, matches_for_player[i]);
282 dump_scores(players, mu, sigma, num_players);
285 float sumdiff = 0.0f;
286 for (int i = 0; i < num_players; ++i) {
287 sumdiff += (mu[i] - old_mu[i]) * (mu[i] - old_mu[i]);
288 sumdiff += (sigma[i] - old_sigma[i]) * (sigma[i] - old_sigma[i]);
290 if (sumdiff < EPSILON) {
291 //fprintf(stderr, "Converged after %d iterations. Stopping.\n", j);
292 printf("%d -1\n", j + 1);
296 dump_scores(players, mu, sigma, num_players);
297 //fprintf(stderr, "Optimal sigma: %f (two-player: %f)\n", sigma[0], sigma[0] * sqrt(2.0f));
298 printf("%f -2\n", sigma[0]);
300 // construct_hessian(mu, sigma, num_players);