Online Parameter Identification of Cost Functions in Generalized Nash Games

This work studies the online parameter identification of cost functions in a generalized Nash game, where each player's cost function is influenced by an observable signal and some unknown parameters. A learner can sequentially observe the equilibria reached by the game system as the observable signal changes, and its goal is to identify the unknown parameters in the cost functions. We recast this problem as an online optimization and introduce a novel online parameter identification algorithm. To be specific, we construct a regularized loss function that balances between conservativeness and correctiveness, where the conservativeness term ensures that the renewed estimates do not deviate significantly from the current estimates, while the correctiveness term is captured by the Karush-Kuhn-Tucker conditions of the generalized Nash equilibrium. For which, the balance is adjusted by a learning rate parameter. We then prove that when the players' cost functions are linear with respect to the unknown parameters and the learning rate satisfies μk ∝ 1/ √ k, along with other assumptions, the regret bound of the proposed algorithm is O(√ K) (k and K represent the current round of online algorithm and the total rounds of games played, respectively). Finally, we conduct numerical simulations on the parameter identification of a Nash-Cournot problem to demonstrate that the performance of our proposed online algorithm is comparable to that of the offline setting.