Smart Greedy Distributed Energy Allocation: A Random Games Approach

Consider a network of N providers that each has a certain supply of energy and B consumers that each has a certain demand. The efficiency of transmitting energy between providers and consumers is modeled using a weighted bipartite graph G. Our goal is to maximize the amount of utilized energy using a distributed algorithm that each provider runs locally. We propose a noncooperative energy-allocation game and adopt the best-response dynamics for this game as our distributed algorithm. We prove that the best-response dynamics converge in no more than N steps to one of at most N! pure Nash equilibria (NE) of our game. However, we show that these NE can be very inefficient. Remarkably, our algorithm avoids the inefficient NE and achieves asymptotically (in B) optimal performance in 'almost all' games. The traditional game-theoretic analysis using a potential function does not explain this encouraging finding. To fill this gap, we analyze the best-response dynamics in a random game, where the network is generated using a random model for the graph G. We prove that the ratio between the utilized energy of our algorithm and that of the optimal solution converges to one in probability as B increases (and N is any function of B). Numerical simulations demonstrate that our asymptotic analysis is valid even for B=10 consumers. Our novel random games approach analytically explains why the performance of our algorithm is asymptotically optimal almost always despite the fact that bad NE may exist.