Consider the following game, which we will call the "truth game." There are two players, called 1 and 2, and a game-master. The game master has a coin that is bent in such a way that, flipped randomly, the coin will come up "heads" 80% of the time. (The bias of this coin is known to both players.) The game-master flips this coin, and the outcome of the coin flip is shown to player 1. Player 1 then makes an announcement to player 2 about the results of the coin flip; player 1 is allowed to say either "heads" or, "tails" (and nothing else). Player 2, having heard what player 1 says but not having seen the results of the coin flip then must guess what the result of the coin flip was - either "heads" or "tails." That ends the game. Payoffs are made as follows. For player 2 things are quite simple; player 2 gets $1 if his guess matches the actual results of the coin flip, and he gets $0 otherwise. For player 1 things are more complex. She gets $2 if player 2's guess is that the coin came up "heads", and $0 if player 2 guesses "tails'', regardless of how the coin came up. In addition to this, player 1 gets $1 (more) if what-she (player 1) says to player 2 matches the results of the coin flip, while she gets $0 more if her message to player 2 is different from the result of the coin flip.
Draw an extensive form representation of this game. Can you draw more than one extensive form representation? Then convert this extensive form representation to a normal form representation. (If you wish to test your intuition, answer: If you were playing this game as player 2, what strategy would you select? If you were player 1, what would you do? Record your reasons, and keep them available; later in the book we will analyze this game according to game-theoretic procedures.)