Tadelis ‘trading places’ Bayesian game
This is a static Bayesian game. The two players share a common prior over the type space which is characterized by the uniform distribution. For Bayesian games, you have to specify the belief of each player given his/her type, and the belief of each player determines their action. Basically, the strategy is a mapping from one's type to an action through belief specification.  

What is player $i$'s belief about $v_j$? It is independently drawn from the uniform distribution over $[0,1]$, so $E(v_j) = 0.5$ for now. And he will exchange only when $v_i \leqslant \frac{3}{2}v_j$, but he does not know exactly the value of $v_j$, so he takes an expectation over $v_j$, then as long as $v_i \leqslant \frac{3}{2} \times 0.5 = \frac{3}{4}$, he will choose to exchange. 

The same goes for player $j$ because they are symmetric, so player $j$ will only exchange when $v_j \leqslant \frac{3}{4}$.  

Given this, $E(v_j)$ now decreases to $\frac{3}{8}$, player $i$'s expected payoff from exchange decreases to $\frac{3}{2}\times \frac{3}{8} = (\frac{3}{4})^2$. Therefore, player $i$ will only choose exchange when $v_i \leqslant (\frac{3}{4})^2$.

Since $\frac{3}{4} &lt; 1$, this process will go on and on such that no one will choose exchange in the end which is the only equilibrium.