  In order to harvest the business potential of device-to-device (D2D)
communication, direct communication between devices subscribed to different
mobile operators should be supported. This would also support meeting
requirements resulting from D2D relevant scenarios, like vehicle-to-vehicle
communication. In this paper, we propose to allocate the multi-operator D2D
communication over dedicated cellular spectral resources contributed from both
operators. Ideally, the operators should negotiate about the amount of spectrum
to contribute, without revealing proprietary information to each other and/or
to other parties. One possible way to do that is to use the sequence of
operators' best responses, i.e., the operators make offers about the amount of
spectrum to contribute using a sequential updating procedure until reaching
consensus. Besides spectrum allocation, we need a mode selection scheme for the
multi-operator D2D users. We use a stochastic geometry framework to capture the
impact of mode selection on the distribution of D2D users and assess the
performance of the best response iteration algorithm. With the performance
metrics considered in the paper, we show that the best response iteration has a
unique Nash equilibrium that can be reached from any initial strategy. In
general, asymmetric operators would contribute unequal amounts of spectrum for
multi-operator D2D communication. Provided that the multi-operator D2D density
is not negligible, we show that both operators may experience significant
performance gains as compared to the scheme without spectrum sharing.
