Two-Player Dynamic Potential LQ Games with Sequentially Revealed Costs

Abstract: We investigate a novel finite-horizon linear-quadratic (LQ) feedback dynamic potential game with a priori unknown cost matrices played between two players. The cost matrices are revealed to the players sequentially, with the potential for future values to be previewed over a short time window. We propose an algorithm that enables the players to predict and track a feedback Nash equilibrium trajectory, and we measure the quality of their resulting decisions by introducing the concept of \emph{price of uncertainty}. We show that under the proposed algorithm, the price of uncertainty is bounded by horizon-invariant constants. The constants are the sum of three terms; the first and second terms decay exponentially as the preview window grows, and another depends on the magnitude of the differences between the cost matrices for each player. Through simulations, we illustrate that the resulting price of uncertainty initially decays at an exponential rate as the preview window lengthens, then remains constant for large time horizons.