The Unlucky Door

Monte and Conie play the famous Three-Door Game. The quiz-team hides the prize behind one of the doors. Conie, who does not know where the prize was hidden, is asked to choose one of the doors as a first guess. Monte, who saw where the prize was hidden, will reveal then one of the doors which does not conceal the prize, but never the door chosen by Conie. Finally, Conie will be offered to either hold her choice or to switch to another yet unrevealed door. Conie wins if her final choice is the door which hides the prize. See the book by Rosenhouse for history and variations of the problem [4]. The Three-Door-Game in proper sense of the game theory, as interaction of two actors, appeared before in [1], [2], [3]. In this note we shall focus on a combinatorial aspect of the game and a possibility of cooperative play for a certain design with four doors. Let us label the doors 1, 2, 3 and think of four moves in the game. The first move is simple: the quiz-team hides the prize behind door p. On the second move Conie chooses door x. On the third move Monte offers a switch to door y 6= x by revealing a door which is not p. On the fourth move Conie chooses z from x and y: if she decides to hold her initial choice then z = x and if she decides to switch then z = y. She wins if z = p. The strategy of the quiz-team is just the action of hiding the prize. We shall think of this move as a move of nature. What can Conie do? On move two she chooses x, and on move four makes her finial choice z which depends on both x and y. For instance, she may