[game] Implemented ELO system

app/Models/Game.php

@@ -13,6 +13,7 @@
  *   type="object",
  *   @OA\Property(type="integer", property="id"),
  *   @OA\Property(type="integer", property="server_id"),
+ *   @OA\Property(type="number", property="elo"),
  *   @OA\Property(type="string", property="created_at", format="date-time"),
  *   @OA\Property(type="string", property="updated_at", format="date-time"),
  * )
@@ -46,7 +47,7 @@ protected static function booted(): void
 
     public function users()
     {
-        return $this->belongsToMany(User::class, 'users_games', 'game_id', 'user_id')->using(UserGame::class)->withPivot("team_id", "rank");
+        return $this->belongsToMany(User::class, 'users_games', 'game_id', 'user_id')->using(UserGame::class)->withPivot("team_id", "rank", "elo");
     }
 
     public function server(): BelongsTo
@@ -61,7 +62,7 @@ public function start()
         $this->save();
     }
 
-    public function end($user_data)
+    private function calculate_ranks($user_data)
     {
         // We first do the initial sorting to remove all oddities
         // that the server could have added
@@ -77,7 +78,9 @@ public function end($user_data)
             $sorted_user_data[] = [
                 "id" => $user["id"],
                 "rank" => $k + 1,
-                "team_id" => $u->pivot->team_id
+                "team_id" => $u->pivot->team_id,
+                "elo" => $u->elo,
+                "game_elo" => 0
             ];
         }
         $user_data = $sorted_user_data;
@@ -104,17 +107,82 @@ public function end($user_data)
             $sorted_user_data = [];
             foreach ($user_data as $k => $user) {
                 $u = $this->users()->findOrFail($user["id"]);
-                $sorted_user_data[] = [
-                    "id" => $user["id"],
-                    "rank" => $k + 1,
-                    "team_id" => $user["team_id"]
-                ];
+                $user["rank"] = $k + 1;
+                $sorted_user_data[] = $user;
             }
             $user_data = $sorted_user_data;
         }
 
+        return $user_data;
+    }
+
+    private function probability_function($users, $user, $D, $N)
+    {
+        // We first calculate the probabilit of winning for the player.
+        // For player A agains all players i, R_x being the current ELO of x
+        // and N the number of players in a game, we have
+        // E_A = { sum from {1 <= i <= N, i <> A} {1 over {1 + 10^{(R_i-R_a) / D}}}} over { {N cdot (N - 1)} over 2 }
+
+        $E_A = 0;
+        foreach($users as $other) {
+            if ($user == $other)
+                continue;
+            
+            $E_A += 1 / (1 + 10**(($other["elo"] - $user["elo"]) / $D));
+        }
+
+        return $E_A / ($N * ($N - 1) / 2);
+    }
+
+    private function score_function($user, $alpha, $N) {
+        // This function calculates the gain from the rank of the player
+        // S^exp_A ( p_A, %alpha ) = { a^{N - p_A} - 1 } over { sum from i = 1 to N {a^{N-i} - 1} }; %alpha in [ 0;+infinity ]
+
+        $top = ($alpha ** ($N - $user["rank"]) - 1);
+
+        $bottom = 0;
+        for($i = 1; $i <= $N; $i++) {
+            $bottom += $alpha ** ($N - $i) - 1;
+        }
+
+        return $top / $bottom;
+    }
+
+    private function calculate_elo($user_data)
+    {
+        $D = 400;
+        $K = 32;
+        $alpha = 2;
+        $N = count($user_data);
+
+        // We first calculate the probabilit of winning for each players.
+        // For player A agains all players i, R_x being the current ELO of x
+        // and N the number of players in a game, we have
+        // E_A = { sum from {1 <= i <= N, i <> A} {1 over {1 + 10^{(R_i-R_a) / D}}}} over { {N cdot (N - 1)} over 2 }
+
+        foreach($user_data as &$user)
+        {
+            $user["E_A"] = $this->probability_function($user_data, $user, $D, $N);
+            $user["new_elo"] = $user["elo"] + $K * ($N - 1) * ($this->score_function($user, $alpha, $N) - $user["E_A"]);
+            $user["game_elo"] = $user["new_elo"] - $user["elo"];
+        }
+        return $user_data;
+    }
+
+    public function end($user_data)
+    {
+        $user_data = $this->calculate_ranks($user_data);
+
+        if ($this->server->official)
+        {
+            $user_data = $this->calculate_elo($user_data);
+        }
+
         foreach ($user_data as $user) {
-            $this->users()->updateExistingPivot($user['id'], ["rank" => $user["rank"]]);
+            $this->users()->updateExistingPivot($user['id'], ["rank" => $user["rank"], "elo" => $user["game_elo"]]);
+            $u = User::findOrFail($user["id"]);
+            $u->elo = $user["new_elo"];
+            $u->save();
         }
 
         $this->finished_at = Carbon::now();

app/Models/User.php

@@ -19,6 +19,7 @@
  *   @OA\Property(type="integer", property="id"),
  *   @OA\Property(type="string", property="username"),
  *   @OA\Property(type="integer", property="server_id", nullable=true),
+ *   @OA\Property(type="number", property="elo"),
  *   @OA\Property(type="string", property="created_at", format="date-time"),
  *   @OA\Property(type="string", property="updated_at", format="date-time"),
  * )

database/migrations/2023_05_09_121757_add_users_elo.php

+<?php
+
+use Illuminate\Database\Migrations\Migration;
+use Illuminate\Database\Schema\Blueprint;
+use Illuminate\Support\Facades\Schema;
+
+return new class extends Migration
+{
+    /**
+     * Run the migrations.
+     *
+     * @return void
+     */
+    public function up()
+    {
+        Schema::table('users', function (Blueprint $table) {
+            $table->decimal("elo")->default(1000);
+        });
+    }
+
+    /**
+     * Reverse the migrations.
+     *
+     * @return void
+     */
+    public function down()
+    {
+        Schema::table('users', function (Blueprint $table) {
+            $table->dropColumn("elo");
+        });
+    }
+};

database/migrations/2023_05_09_121918_add_users_games_elo.php

+<?php
+
+use Illuminate\Database\Migrations\Migration;
+use Illuminate\Database\Schema\Blueprint;
+use Illuminate\Support\Facades\Schema;
+
+return new class extends Migration
+{
+    /**
+     * Run the migrations.
+     *
+     * @return void
+     */
+    public function up()
+    {
+        Schema::table('users_games', function (Blueprint $table) {
+            $table->decimal("elo")->default(0);
+        });
+    }
+
+    /**
+     * Reverse the migrations.
+     *
+     * @return void
+     */
+    public function down()
+    {
+        Schema::table('users_games', function (Blueprint $table) {
+            $table->dropColumn("elo");
+        });
+    }
+};

tests/Feature/GameCreationTest.php

@@ -14,7 +14,7 @@
 class GameCreationTest extends TestCase
 {
     use RefreshDatabase;
-
+    
     public function test_good()
     {
         $user1 = User::create([
@@ -34,13 +34,17 @@ public function test_good()
             'password' => Hash::make('test'),
             'owner_id' => $user1->id,
         ]);
+        $server->official = true;
+        $server->save();
 
         $user1->server_id = $server->id;
         $user1->server_state = 'connected';
+        $user1->elo = 1200;
         $user1->save();
 
         $user2->server_id = $server->id;
         $user2->server_state = 'connected';
+        $user1->elo = 1000;
         $user2->save();
 
         $token = Auth::guard('server')->login($server);
@@ -83,8 +87,10 @@ public function test_good()
         ]);
         $response->assertStatus(200);
 
-        $this->assertEquals(Game::first()->users()->findOrFail($user1->id)->pivot->rank, 1);
-        $this->assertEquals(Game::first()->users()->findOrFail($user2->id)->pivot->rank, 2);
+        $this->assertEquals(1, Game::first()->users()->findOrFail($user1->id)->pivot->rank);
+        $this->assertGreaterThan(1200, Game::first()->users()->findOrFail($user1->id)->elo);
+        $this->assertEquals(2, Game::first()->users()->findOrFail($user2->id)->pivot->rank);
+        $this->assertLessThan(1000, Game::first()->users()->findOrFail($user2->id)->elo);
     }
 
     public function test_good_four_players()
@@ -118,21 +124,27 @@ public function test_good_four_players()
             'password' => Hash::make('test'),
             'owner_id' => $user1->id,
         ]);
+        $server->official = true;
+        $server->save();
 
         $user1->server_id = $server->id;
         $user1->server_state = 'connected';
+        $user1->elo = 1200;
         $user1->save();
 
         $user2->server_id = $server->id;
         $user2->server_state = 'connected';
+        $user1->elo = 1400;
         $user2->save();
 
         $user3->server_id = $server->id;
         $user3->server_state = 'connected';
+        $user1->elo = 800;
         $user3->save();
 
         $user4->server_id = $server->id;
         $user4->server_state = 'connected';
+        $user1->elo = 900;
         $user4->save();
 
         $token = Auth::guard('server')->login($server);
@@ -222,6 +234,8 @@ public function test_good_four_players_teams()
             'password' => Hash::make('test'),
             'owner_id' => $user1->id,
         ]);
+        $server->official = true;
+        $server->save();
 
         $user1->server_id = $server->id;
         $user1->server_state = 'connected';