Consider a simple simultaneous-bid poker game. First, nature selects num- bersx1andx2. Assume that these numbers are independently and uniformly distributed between 0 and 1. Player 1 observesx1and player 2 observesx2, but neither player observes the number given to the other player. Simultane- ously and independently, the players choose either to fold or to bid. If both players fold, then they both get the payoff1. If only one player folds, then he obtains1 while the other player gets 1. If both players elected to bid, then each player receives 2 if his number is at least as large as the other player's number; otherwise, he gets2. Compute the Bayesian Nash equilibrium of this game. (Hint: Look for a symmetric equilibrium in which a player bids if and only if his number is greater than some constanta. Your analysis will reveal the equilibrium value ofa.)

Consider the Cournot duopoly game with incomplete information. First, nature chooses a numberx, which is equally likely to be 8 or 4. This number represents whether demand is high (x=8) or low (x=4). Firm 1 observesxbecause this firm has performed market research and knows the demand curve. Firm 2 does not observex. Then the two firms simultaneously select quantities,q1andq2, and the market price is determined byp=xq1q2. Assume that the firms produce at zero cost. Thus, the payoff of firm 1 is (xq1q2)q1,andthepayoffoffirm2is(xq1q2)q2.

1. (a)Note that there are two types of firm 1, the high type (observingx=8) and the low type (observingx=4). LetqH1andqL1denote the quantity choices of the high and low types of firm 1. Calculate the players' best- response functions.
2. (b)Find the Bayesian Nash equilibrium of this game.
3. (c)Doesfirml'sinformationgiveitanadvantageoverfirm2inthisgame?
4. Quantify this.