Pushing array into multidimensional array (php) [duplicate]

I've attempted many different ways to push an array into a multidimensional array, including array_push(), $array['index'] = $toPush but I keep being met with quite unexpected results. I have used both var_dump() and print_r() as detailed below in an attempt to debug, but cannot work out the issue.

My reasoning behind is to run a while loop to pull game id's and game names and store these in an assoc. array, and then push them into my main array.

$games_array = array 
(

    "games" => array 
     (
          array("id"=>"1", "game"=>"first game");
          array("id"=>"2", "game"=>"second game");
     )

);
// a while loop would run here and update $game_to_add;
$game_to_add = array("id"=>"$game['id']", "game"=>"$game['title']");

$games_array = array_push($games_array['games'], $game_to_add);

In this example, the while() would update the ID and the Game inside of $game_to_add

But, whenever I attempt this it simply overwrites the array and outputs an integer ( example: int(3) )

I don't understand what the problem is, any explination would be appreciated as I cannot find a question specifically for this.

My actual test code:

$games_array = array( "games" => array(
    array("id" => "1", "name" => "Star feathers"),
    array("id" => "2", "name" => "chung fu")
)
);

$another_game = array("id" => "3", "name" => "some kunt");
$games_array = array_push($games_array["games"], array("id" => "3", "name" 
=>"some game"));
var_dump($games_array);