Multi-Portfolio Optimization: A Potential Game Approach

In modern asset management, portfolio managers address the multi-account investment decision problem by optimizing each account's portfolio separately based on the trading requirements and portfolio constraints of the individual clients. However, trades associated with the individual accounts are usually pooled together for execution, therefore amplifying the level of the so-called market impact on all accounts. If this aggregate market impact is not considered when each account is individually optimized, the actual market impact can be severely under-estimated. Multi-portfolio optimization aims at finding the optimal rebalancing of the multiple accounts by considering their joint effects while adhering to account-specific constraints. In this paper, we first model this phenomenon as a Nash Equilibrium problem (NEP) and thereafter consider a generalized NEP (GNEP) for the case where there are global constraints imposed on all accounts, adopting as a desirable outcome the concept of Nash Equilibrium (NE). For both game problems, we give a complete characterization of the NE, including its existence and uniqueness, and devise various distributed algorithms with provable convergence. Interestingly, the proposed methodology heavily hinges on a number of well-known and important signal processing techniques.