  This paper focuses on analyzing the free-riding behavior of self-interested
users in online communities. Hence, traditional optimization methods for
communities composed of compliant users such as network utility maximization
cannot be applied here. In our prior work, we show how social reciprocation
protocols can be designed in online communities which have populations
consisting of a continuum of users and are stationary under stochastic
permutations. Under these assumptions, we are able to prove that users
voluntarily comply with the pre-determined social norms and cooperate with
other users in the community by providing their services. In this paper, we
generalize the study by analyzing the interactions of self-interested users in
online communities with finite populations and are not stationary. To optimize
their long-term performance based on their knowledge, users adapt their
strategies to play their best response by solving individual stochastic control
problems. The best-response dynamic introduces a stochastic dynamic process in
the community, in which the strategies of users evolve over time. We then
investigate the long-term evolution of a community, and prove that the
community will converge to stochastically stable equilibria which are stable
against stochastic permutations. Understanding the evolution of a community
provides protocol designers with guidelines for designing social norms in which
no user has incentives to adapt its strategy and deviate from the prescribed
protocol, thereby ensuring that the adopted protocol will enable the community
to achieve the optimal social welfare.
