  This paper studies self-sustained dynamic multiagent systems (MAS) for
decentralized resource allocation operating at a competitive equilibrium over a
finite horizon. The utility of resource consumption, along with the income from
resource exchange, forms each agent's payoff which is aimed to be maximized.
Each utility function is parameterized by individual preferences which can be
designed by agents independently. By shaping these preferences and proposing a
set of utility functions, we can guarantee that the optimal resource price at
the competitive equilibrium always remains socially acceptable, i.e., it never
violates a given threshold that indicates affordability. First, we show this
problem is solvable at the conceptual level under some convexity assumptions.
Then, as a benchmark case, we consider quadratic MAS and formulate the
associated social shaping problem as a multi-agent LQR problem which enables us
to propose explicit utility sets using quadratic programming and dynamic
programming. Finally, a numerical algorithm is presented for calculating the
range of the preference function parameters which guarantee a socially accepted
price. Some illustrative examples are given to examine the effectiveness of the
proposed methods.
