HAMILTONIAN POTENTIAL FUNCTIONS FOR DIFFERENTIAL GAMES Hamiltonian Potential Functions for Di¤erential Games

We introduce the concept of Hamiltonian potential functions for noncooperative open-loop di¤erential games and we characterise su¢ cient conditions for their existence. We also identify a class of games admitting a Hamiltonian potential and illustrate the related properties of their dynamic structure. Possible similarities with the theory of quasi-aggregative games are discussed. As an illustration, we consider an asymmetric oligopoly game with process innovation. Keywords: Di¤erential games, Potential function, Optimal control, Quasiaggregative games 1 Introduction Following Monderer and Shapley [15], a relatively large literature has been devoted to investigating potential functions for static games. In a potential game, the information about Nash equilibria is nested into a single real-valued function (the potential function) over the strategy space. The speci…c feature of a potential function de…ned for a given game is that its gradient coincides with the vector of …rst derivatives of the individual payo¤ functions of the original game itself. As stressed by Slade [17], the interest of this line of research is that, in a game admitting a potential function, it is as if players were jointly maximising that single function instead of competing to maximise their respective payo¤s. Department of Economics, University of Bologna, Strada Maggiore 45, 40125 Bologna, Italy yDepartment of Economics, University of Bologna, Strada Maggiore 45, 40125 Bologna, Italy and ENCORE, University of Amsterdam, Roeterstraat 11, WB1018 Amsterdam, The Netherlands. Luca.Lambertini@unibo.it zCollege of Engineering, University of California at Berkeley, Berkeley CA 94720, USA xMEMOTEF, Sapienza University of Rome, Via del Castro Laurenziano 9, 00161 Rome, Italy