# A card game is played by three players.8 Each player uses cards of two colors, red (R) and black…

A card game is played by three players.8 Each player uses cards of two colors, red (R) and black (B). At any point in time, each player has a card on the table in front of him with its color showing. The player then uses a specific rule (known only to this player) for replacing this card by another one. We assume that the card replacement for all three players is done simultaneously. (a) Suppose the players use the following rules: Player 1: Plays R if and only if both he and Player 2 show R; Player 2: Plays the color opposite to that shown by Player 1; Player 3: Plays R if and only if both he and Player 2 show B. Model the game as an automaton, where the only event is “everybody plays.” What is the state space X? Can any of the states be visited more than once during the game? (b) Suppose Player 2 changes his strategy as follows. Play the color opposite to that shown by Player 1, unless BRR is showing on the table. How are the state space and state transition function of the automaton affected? Are any of the states revisited in this case? (c) Now suppose further that Player 3 also decides to modify his strategy as follows. He determines the next card to play by tossing a fair coin. The outcome of the coin now represents an event. What are X (state space) and E (event set) in this case? Construct the new state transition function, and identify which states can be revisited? (d) Suppose you are an outside observer and in each of the above three cases you know the players’ strategies. You are allowed to look at the cards initially showing on the table and then bet on what the next state will be. Can you guarantee yourself a continuous win in each case (a), (b) and (c)? In case (c), if initially RRR is showing, are there any states you should never bet on?

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