We consider networks evolving over time within an infinite-horizon dynamic setting. Transitions from one network to another are given by a stationary transition probability matrix. We study the problem of fairly and efficiently allocating the value of a network at any point in time among its participants, assuming that agents discount the future by some common discount factor. An allocation rule is called component efficient if it distributes the total value of a connected network among its participants and it is called expected fair if for every direct connection both participants expect to loose or gain the same amount in the future from breaking this connection at time zero. Our main result is that for every transition probability matrix and for almost every discount factor there exists a unique allocation rule which is component efficient and expected fair. We provide a formula to compute this allocation rule. In general, this allocation rule is different from a stage-wise application of the Myerson value. We also provide a sufficient condition on the transition probability matrix such that the component efficient and expected fair allocation rule is equal to the Myerson value.
