Estimation of point-identified Dynamic Discrete Choice models with moment inequalities
Function $Q$ has to be minimized wrt $(\theta,\alpha)$.
The parameters $(\theta,\alpha)$ compatible with Nash equilibrium and rationalizing (or generating) the data have to satisfy $$g(x;\theta,\alpha)\leq0.$$ 
In the case where, for given $x$, $$g(x;\theta,\alpha)&gt;0,$$ the parameter values have to be changed in order to possibly reverse the strict inequality and minimize $Q$. 
If $$g(x;\theta,\alpha)\leq0$$ for some $(\theta,\alpha)$, then this should not contribute to increase $Q$, because it is fully compatible with the data and model. Hence the term $\min\{g(x;\theta,\alpha),0\}$ in the expression of $Q$.