  People often interact repeatedly: with relatives, through file sharing, in
politics, etc. Many such interactions are reciprocal: reacting to the actions
of the other. In order to facilitate decisions regarding reciprocal
interactions, we analyze the development of reciprocation over time. To this
end, we propose a model for such interactions that is simple enough to enable
formal analysis, but is sufficient to predict how such interactions will
evolve. Inspired by existing models of international interactions and arguments
between spouses, we suggest a model with two reciprocating attitudes where an
agent's action is a weighted combination of the others' last actions (reacting)
and either i) her innate kindness, or ii) her own last action (inertia). We
analyze a network of repeatedly interacting agents, each having one of these
attitudes, and prove that their actions converge to specific limits.
Convergence means that the interaction stabilizes, and the limits indicate the
behavior after the stabilization. For two agents, we describe the interaction
process and find the limit values. For a general connected network, we find
these limit values if all the agents employ the second attitude, and show that
the agents' actions then all become equal. In the other cases, we study the
limit values using simulations. We discuss how these results predict the
development of the interaction and can be used to help agents decide on their
behavior.
