An approach to simple bargaining games and related problems

We determine the fractal dimension df of infinite spherically symmetric random trees (all vertices at distance n from the root have the same degree dn where {dn} are independent random variables). If dn takes the value 3 or 2 with probabilities qn and 1-qn, then df=(log 2) limn nqn+1 a.s. We show how df is closely related to the type of the simple random walks (SRW) on trees. We prove that the SRW is a.s. transient if df>2 a.s. and a.s. recurrent if df<2 a.s. and if df=2 a.s. we obtain a.s. transience or recurrence. We also consider another type of random trees which are corresponding to branching processes in varying environments. In particular, we consider a tree such that the degrees of the vertices at distance n from the root are independent identically distributed (iid) random variables following the distribution of a random variable that takes the value 3 or 2 with probabilities qn and 1-qn respectively. These iid random variables are also independent of the degrees of the vertices of the other generations. We prove that the SRW is a.s. recurrent if and only if df<=2 a.s. We also prove for such trees that df=limn nqn+1 a.s.