Ronald and Jimmy are betting on the result of a coin toss. Ronald ascribes probability 1/3 to the event that the coin shows heads, while Jimmy ascribes probability 3/4 to that event. The betting rules are as follows: Each of the two players writes on a slip of paper "heads" or "tails," with neither player knowing what the other player is writing.
After they are done writing, they show each other what they have written. If both players wrote heads, or both players wrote tails, each of them receives a payoff of 0. If they have made mutually conflicting predictions, they toss the coin. The player who has made the correct prediction regarding the result of the coin toss receives $1 from the player who has made the incorrect prediction. This description is common knowledge among the two players.
(a) Depict this situation as a game with incomplete information.
(b) Are the beliefs of Ronald and Jimmy consistent? Justify your answer.
(c) If you answered the above question positively, find the common prior.
(d) Find a Bayesian equilibrium in this game (whether or not the beliefs of the players are consistent).