The price of anarchy as a classifier for mechanism design in a pareto-bayesian-nash context

The complex interaction of fixed incentives for multiple agents can lead to inefficient results. The Price of Anarchy (PoA) is a concept employed in economics and game theory to quantify how, in the worst-case scenario, a system's efficiency diminishes due to the selfish behavior of its actors. It evaluates the outcomes obtained by perfect coordination to that obtained by self-interested players in a Nash equilibrium. This paper investigates the PoA in mechanism design for Pareto-Bayesian-Markov (PBN) games in which the players privately know their information. Our approach is restricted to the development of algorithmic mechanisms. We consider an ergodic finite Markov chain in which players' transitions are independent and decisions are made across a finite number of periods. As a key outcome, we completely characterize the Pareto frontier and calculate an efficient, ex-post incentive-compatible technique. We employ Tikhonov's regularization technique, for assuring that the original problem's solution converges to a unique solution. The ratio equilibria of the PoA is found employing a projection-gradient method, which is our main result. We compute the PoA using an auxiliary variable and present the formulas for retrieving the variables of interest. We show that our theoretical conclusions ensure convergence and we provide a numerical example to demonstrate the method's utility.