A Game-Theoretic Approach to Parallel Trade Through the Price of Anarchy

In recent years, the concept of price of anarchy has emerged as a tool to measure the efficiency of Nash equilibria in noncooperative games. In this paper we apply an adaptation of this concept to subgame-perfect Nash equilibria arising in three different dynamic noncooperative game-theoretic models for the parallel trade of pharmaceuticals, where parallel trade refers to the arbitrage opportunity created when the same drug is sold in two countries, with differences in price and/or reimbursement regulations. More specifically, for the three different expressions of the global welfare of two countries, we find in closed form its optimal value, which is obtained by solving a suitable quadratic/maximin optimization problem modeling the decision-making process of a global planner. Then, we evaluate for such expressions the price of anarchy associated with the subgame-perfect Nash equilibria of three dynamic noncooperative games modeling the interaction between a manufacturer in the first country and a distributor in the second country, respectively in the case of parallel trade banning for the distributor, parallel trade threat from the distributor (but no occurrence of parallel trade at equilibrium), and actual parallel trade occurrence at equilibrium. Finally, we evaluate the dependence of such prices of anarchy on the relative market size of the exporting country with respect to the importing one, and on the per-unit parallel trade cost. Differently from other models of parallel trade, our analysis takes into account the total fixed cost of production associated with research and development, which makes it possible to include in the analysis the decision by the manufacturer whether to do or not to do the research and development. Extensions of the methodology to the noncooperative game-theoretic models of parallel trade of pharmaceuticals are discussed