A game management center linked to at least one remote game terminal, sends the terminals to which it is linked, a message containing a game identifier and a question. The response from a player entered via a remote game terminal is transmitted in a form blinded with the aid of a secret to the game management center so as to guarantee the anonymity of the player. If the blinded response message is received before the instant of broadcasting of the response, the game management center sends an acknowledgement of receipt to said game terminal. The response in answer to the question is subsequently broadcast to all the remote game terminals. If the player has given the right answer, the terminal must, to claim its prize, send a message containing the acknowledgement of receipt and an item proving that it is in possession of the secret used to blind the response.