Dynamic Supervisors in Elixir

game.ex

defmodule Game do
  use GenServer

  def init(game_id) do
    {:ok, %{game_id: game_id}}
  end

  def start_link(game_id) do
    GenServer.start_link(__MODULE__, game_id, name: {:global, "game:#{game_id}"})
  end

  def add_player(pid, player_name) do
    GenServer.call(pid, {:add_player, player_name})
  end

  def handle_call({:add_player, player_name}, _from, %{game_id: game_id} = state) do
    # Uh oh, we started this process but it's not under supervision!
    start_status = Player.start_link({player_name, game_id})
    {:reply, start_status, state}
  end
end

game_supervised.ex

defmodule SupervisedGame do
  # ...
  def handle_call({:add_player, player_name}, _from, %{game_id: game_id} = state) do
    # Now we replace this with supervised management
    start_status = PlayerSupervisor.add_player(player_name, game_id)
    {:reply, start_status, state}
  end
end

player.ex

defmodule Player do
  use GenServer

  def init({player_name, game_id}) do
    {:ok, %{name: player_name, game_id: game_id}}
  end

  def start_link({player_name, game_id}) do
    GenServer.start_link(
      __MODULE__,
      {player_name, game_id},
      name: {:global, "player:#{player_name}"}
    )
  end

  def get(pid) do
    GenServer.call(pid, :get)
  end

  def handle_call(:get, _from, state) do
    {:reply, {:ok, state}, state}
  end
end

player_dynamic_supervisor.ex

defmodule PlayerDynamicSupervisor do
  use DynamicSupervisor

  def start_link(_arg) do
    DynamicSupervisor.start_link(__MODULE__, :ok, name: __MODULE__)
  end

  def init(:ok) do
    DynamicSupervisor.init(strategy: :one_for_one)
  end

  # Start a Player process and add it to supervision
  def add_player(player_name, game_id) do
    # Note that start_child now directly takes in a child_spec.
    child_spec = {Player, {player_name, game_id}}
    # Equivalent to:
    # child_spec = Player.child_spec({player_name, game_id})
    DynamicSupervisor.start_child(__MODULE__, child_spec)
  end

  # Terminate a Player process and remove it from supervision
  def remove_player(player_pid) do
    DynamicSupervisor.terminate_child(__MODULE__, player_pid)
  end

  # Nice utility method to check which processes are under supervision
  def children do
    DynamicSupervisor.which_children(__MODULE__)
  end

  # Nice utility method to check which processes are under supervision
  def count_children do
    DynamicSupervisor.count_children(__MODULE__)
  end
end

player_supervised.ex

defmodule Player do
  # ...

  # Insert this start_link/2 method, which intercepts the extra `[]`
  # argument from the Supervisor and molds it back to correct form.
  def start_link([], {player_name, game_id}) do
    start_link({player_name, game_id})
  end

  def start_link({player_name, game_id}) do
    # Original implementation...
  end
  
  # ...
end

player_supervisor.ex

defmodule PlayerSupervisor do
  use Supervisor

  def start_link([]) do
    Supervisor.start_link(__MODULE__, :ok, name: __MODULE__)
  end

  def init(:ok) do
    Supervisor.init([Player], strategy: :simple_one_for_one)
  end

  # Start a Player process and add it to supervision
  def add_player(player_name, game_id) do
    # Note that the second arg to start_child/2 must be an Enumerable
    Supervisor.start_child(__MODULE__, [{player_name, game_id}])
  end

  # Terminate a Player process and remove it from supervision
  def remove_player(player_pid) do
    Supervisor.terminate_child(__MODULE__, player_pid)
  end

  # Nice utility method to check which processes are under supervision
  def children do
    Supervisor.which_children(__MODULE__)
  end

  # Nice utility method to check which processes are under supervision
  def count_children do
    Supervisor.count_children(__MODULE__)
  end
end

I am not sure, but probably the correct start_link signature in Player module should look like this def start_link([player_name, game_id]). Anyway that was the issue in my case, may be will help somebody.