The Price of Isolation: an Integrated Study of System Inefficiencies

We present in this paper an integrated study on the selfish behavior of the players participating in the system (modeled by a noncooperative game) and the myopic attitudes of the players (modeled by the ‘rolling-horizon’ type of strategies in the dynamic decision-making process), and introduce a solution termed myopic Nash equilibrium to characterize the equilibrium state when all the players are selfish and myopic. To characterize the loss of system efficiencies caused by such behaviors, we use a (worst possible) ratio between the total social value at the worst myopic Nash equilibrium and the optimal total social value, which is termed the Price of Isolation (PoI) in this paper. We consider some specific game models to bound this ratio. We first consider an extension of the static transportation game, with K players competing on a given set of routes to get their jobs done at minimum cost, over T stages. If the unit cost on each route in different stages are all affine linear in the total demand and the unit cost of the previous stage, then it is shown that the PoI of this game is precisely 4 as T → ∞. We then turn to dynamic profit maximization games, including generalized settings of the Cournot and/or Bertrand competition games, with K producers competing to sell their products in the market over T stages. We prove that the PoI is in the order of 1 KT if the commodities are substitute in the Cournot game, and are complementary in the Bertrand game, when the unit costs of the resources used to manufacture the products and the selling prices of the finished products are assumed to be affine linear functions. Interpretations of these results are also discussed.