log sum of exponential

The so-called “log sum of exponentials” is a functional form commonly encountered in dynamic discrete choice models in economics. It arises when the choice-specific error terms have a type 1 extreme value distribution–a very common assumption. In these problems there is typically a vector v of length J which represents the non-stochastic payoff associated with each of J choices. In addition to vj, the payoff associated with choice j has a random component εj. If these components are independent and identically distributed across j and follow the type 1 extreme value distribution, then the expected payoff from choosing optimally is