win_loss_simulation.m

% win_loss_simulation.m
% Simulates skill rating change over many games in Overwatch

function  win_loss_simulation(starting_rank, n_games, seed, distribution, mu, sigma)

% controls the general variance of cdf distributions, ignored for coin-flip
if nargin < 6
    sigma = 500;
end

% controls the average SR for cdf distributions, ignored for coin-flip
if nargin < 5
   mu = 2500;
end

% which distribution to use
if nargin < 4
   distribution = 'cdf'; %or 'coin-flip'
end

% random number generator seed
if nargin < 3
   seed = 73;
end

% number of games
if nargin < 2
   n_games = 1000;
end

% starting SR
if nargin < 1
    starting_rank = 2500;
end

rng(seed)
change_per_win = 25;
rank_at_game = zeros(1,n_games);
result_at_game = zeros(1,n_games);
streaks = zeros(1,n_games);
rank_at_game(1) = starting_rank;
streak_count = 1;
last_result = true;

pd = makedist('Normal', mu, sigma);

if strcmp(distribution, 'cdf')
    label = 'CDF Statistics';
else
    if strcmp(distribution, 'coin-flip')
        label = 'Coin-Flip Statistics';
    else
        ME = MException('win_loss_simulation:badParameter', 'Unrecognized distribution type');
        throw (ME);
    end
end

for n = 2:n_games
    if strcmp(distribution, 'cdf')
        win =  rand >= cdf(pd, rank_at_game(n-1));
    else
        win = rand >=0.5;
    end

    if win
        change = change_per_win;
        result_at_game(n) = 1;
    else
        change = -change_per_win;
        result_at_game(n) = -1;
    end

    if (win == last_result)
        streak_count = streak_count+1;
    else
        streaks(streak_count) = streaks(streak_count)+1;
        streak_count = 1;
    end
    last_result = win;

    rank_at_game(n) = rank_at_game(n-1) + change;
end

figure
plot(rank_at_game);
xlabel('Game Number');
ylabel('Skill Rating');
title(strcat("Skill Rating vs Game, ",label));
figure
errorbar(streaks/n_games, sqrt(streaks)/n_games);
xlabel('Streak Length');
ylabel('Number of Streaks / Number of Games');
title(strcat("Streak Frequency, ", label));
axis([0 25 0 0.5])

dist = zeros(1,5000);
for i=1:5000
    if strcmp(distribution, 'cdf')
        dist(i) = 1 - cdf(pd, i);
    else
        dist(i) = 0.5;
    end
end
figure
plot(dist);
xlabel('Skill Rating');
ylabel('Win Probability');
title(strcat("Win Probability vs Skill Rating, ", label));
axis([0 5000 0 1]);
if strcmp(distribution, 'cdf')
    newline = sprintf('\n');
    txt = ['mu = ', num2str(mu,4), newline, 'sigma = ', num2str(sigma,3)];
    text(500, .1, txt);
end

%auto correlation function calculation
ac_length = 10;
ac_count = zeros(1, ac_length*2+1);
ac_error = zeros(1, ac_length*2+1);
ac_func = zeros(1, ac_length*2+1);
ac_func_sq = zeros(1, ac_length*2+1);
for n = 1:n_games
    for m = n-ac_length:n+ac_length
        if m > 0 && m <= n_games
            if(result_at_game(n) == 0)
                continue;
            end
            ac_count(m - n +ac_length+1) = ac_count(m - n +ac_length+1) + 1;
            if(result_at_game(n) == 1)
                ac_func(m - n + ac_length + 1) = ac_func(m-n+ac_length+1) + result_at_game(m);
            else
                ac_func(m - n + ac_length + 1) = ac_func(m-n+ac_length+1) - result_at_game(m);
            end
            ac_func_sq(m-n+ac_length+1) =  ac_func_sq(m-n+ac_length+1) + result_at_game(m)*result_at_game(m);

        end
    end
end

for n = 1:ac_length*2+1
   ac_error(n) = sqrt(ac_func_sq(n) / (ac_count(n) - 1) - ac_func(n)*ac_func(n)/(ac_count(n) * (ac_count(n)-1))) / sqrt(ac_count(n));
   ac_func(n) = ac_func(n) / ac_count(n);
end

figure
ac_func_x = -ac_length:ac_length;
errorbar(ac_func_x, ac_func, ac_error);
xlabel('Game Differential');
ylabel('Auto Correlation');
title(strcat("Win/Loss Auto-correlation function, ", label));

end