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.