Can anyone help me understand how to solve this type of asymmetric information Bayesian game? So the game is a different version of the Tadelis 'trading house games'. It involves 2 players that each own a house. Each player values his own house at $v_i$. The value of player $i$’s house to the other player, i.e. to player $j \ne i$, is $\alpha v_i − c$ where $\alpha > 1$. Each player knows the value $v_i$ of his own house to himself, but not the value of the opponent’s house. Both players know $\alpha$. The values $v_i$ are distributed uniformly on the interval [0, 1] and are independent across players.
I suppose that player $i$ will agree to exchange only if $v_i \leq \alpha v_j - c$ but I don't know how to go from there to find the Bayesian Nash equilibrium and how to assign probabilities. Are we suppose to use the expected value of $v_i$ through the uniform distribution, i.e. $E(v_i)=1/2$ ?
Would be very grateful for some help!!