My question concerns the following problem: two players, 1 and 2, each own a house. Each player i values his own house at $v_{i}$. The value of player i's house to the other play, i.e., to player j$\neq$i, is $\frac{3}{2}v_{i}$. Each player i knows the value $v_{i}$ of his own house to himself, but not the value of the other player's house. The values $v_{i}$ are drawn independently from the interval [0,1] with uniform distribution.
Payoffs and actions are defined as follows: each player announces simultaneously whether they want to exchange their houses. If both players agree to an exchange, the exchange takes place. Otherwise no exchange takes place.
As far as finding a Bayesina equilibrium is concerned, I have reached the following stage:
Given j exchanges, i exchanges so long as $v_{i}\leq\frac{3}{2}v_{j}$ and and given i exchanges, j exchanges so long as $v_{j}\leq\frac{3}{2}v_{i}$. This means that j's expected utility from exchanging (given i exchanges) is $\frac{9}{8}v_{j}$ and i's is $\frac{9}{8}v_{i}$. From this I do not know how to proceed.
Due to the symmetry, we can expect a symmetric equilibrium. Let $t$ be the common threshold of the players. At the threshold, the players must be indifferent between exchanging and not exchanging. Not exchanging keeps the value $t$ of the own house, so exchanging must also yield expected value $t$. Conditional on the exchange occurring, the value of the other house to its current owner is uniformly distributed on $[0,t]$, so the expected value to the new owner is $\frac32\cdot\frac t2=\frac34t$. This is only equal to $t$ for $t=0$, so the players will never exchange (even though always exchanging would increase the total value of their houses by a factor $\frac32$).