Pareto-Improving Incentive Mechanism for Noncooperative Dynamical Systems Under Sustainable Budget Constraint

Abstract

A Pareto-improving incentive mechanism to improve the weighted social welfare and achieve continual Pareto improvement for a pseudo-gradient-based noncooperative dynamical system is developed. In the proposed approach, the system manager remodels agents' dynamical decision making by collecting taxes from some agents and giving some of the collected taxes to other agents as subsidies with a sustainable budget constraint. Sufficient conditions are derived under which agents' state converges towards the socially maximum state associated with a weighted social welfare function depending on the priority ratio of the agents and the initial state. We present several illustrative numerical examples to illustrate the efficacy of our results and reveal the fact that the potentialization of the payoff structure has a very strong relation to generating Pareto-improving system trajectories.

Incorporation of Likely Future Actions of Agents Into Pseudo-Gradient Dynamics of Noncooperative Games

“…Except for this manner, it is possible to use other types of recursive reasoning such as [31] for characterizing proactive decision-making with Level-k thinking in noncooperative system if the agents are hesitating in how much proactive they should be. APPENDIX 1) Derivation of (31), (32): The derivation of ( 31) is a direct consequence of ( 17), (19) along with…”

“…□ Remark 11 The condition (35) with a i ij replaced by ∂ 2 Ji(x * ) ∂xi∂xj , i, j ∈ N is capable to guarantee locally asymptotic stability of the Nash equilibrium x * for the noncooperative systems with nonquadratic payoff functions. Remark 12 Even though constructing incentive mechanisms is not the focus of this paper, the results shown in Theorems 1, 2, and Corollaries 1, 2 indicate that it is certainly possible for the system manager who has all the information of the payoff functions J i (x), i ∈ N , to design a incentive rule (such as the tax/subsidy approach in [20] and the Pareto-improving incentives with a sustainable budget in [19]) to ensure stability of a target state x * for the agents with Level-k thinking.…”

“…For example, Alpcan et al [18] designed pricing mechanisms to achieve the highest social welfare with the selfish agents driven by the pseudo-gradient dynamics. Yan and Hayakawa studied how to construct incentive mechanisms to ensure Pareto-improving trajectories [19] and Nash equilibrium stabilization [20].…”