Fra\"\i ss\'e-Ehrenfeucht Games and Asymptotics

Fra\"\i ss\'e-Ehrenfeucht games are played on two structures, where a structure might, for example, consist of a unary function mapping a finite set into itself. Via generating series and a Tauberian theorem, it is possible to investigate the asymptotic probability of having a winning strategy for such a game, when it is played using a fixed structure, and a random structure of size $n$, with $n$ going to infinity. Actually for unary functions this gives a convergence law for all properties of the structure which are definable in monadic second order logic. But for most of the talk we will pretend not to know the connection.