Try this:
devise_scope :user do
post 'registrations' => 'api/v1/registrations#create', :as => 'register'
post 'sessions' => 'api/v1/sessions#create', :as => 'login'
delete 'sessions' => 'api/v1/sessions#destroy', :as => 'logout'
end
namespace :api do
namespace :v1 do
get 'tournaments' => 'tournaments#index', :as => 'tournaments'
end
end
Be sure to restart the server after modifying the routes.rb
Ensure your controllers are in the directory api/v1
(use only small letters)