Nicolas would like to sell a company that he owns to Marc. The company's true value is an integer between 10 and 12 (including 10 and 12), in millions of dollars. Marc has to make a take-it-or-leave-it offer, and Nicolas has to decide whether to accept the offer or reject it. If Nicolas accepts the offer, the company is sold, Nicolas's payoff is the amount that he got, and Marc's payoff is the difference between the company's true value and the amount that he paid.

If Nicolas rejects the offer, the company is not sold, Nicolas's payoff is the value of the company, and Marc's payoff is 0.

For each one of the following three information structures, describe the situation as a game with incomplete information, and find all the Bayesian equilibria in the corresponding game.

In each case, the description of the situation is common knowledge among the players. In determining Nicolas's action set, note that Nicolas knows what Marc's offer is when he decides whether or not to accept the offer.

(a) Neither Nicolas nor Marc knows the company's true value; both ascribe probability 1/3 to each possible value.

(b) Nicolas knows the company's true value, whereas Marc does not know it, and ascribes probability 1/3 to each possible value.

(c) Marc does not know the company's worth and ascribes probability 1/3 to each possible value. Marc further ascribes probability p to the event that Nicolas knows the value of the company, and probability 1 - p to the event that Nicolas does not know the value of the company, and instead ascribes probability  1/3  to each possible value.