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Last-iterate convergence has received extensive study in two player zero-sum games starting from bilinear, convex-concave up to settings that satisfy the MVI condition. Typical methods that exhibit last-iterate convergence for the aforementioned games include extra-gradient (EG) and optimistic gradient descent ascent (OGDA). However, all the established last-iterate convergence results hold for the restrictive setting where the underlying repeated game does not change over time. Recently, a line of research has focused on regret analysis of OGDA in time-varying games, i.e., games where payoffs evolve with time; the last-iterate behavior of OGDA and EG in time-varying environments remains unclear though. In this paper, we study the last-iterate behavior of various algorithms in two types of unconstrained, time-varying, bilinear zero-sum games: periodic and convergent perturbed games. These models expand upon the usual repeated game formulation and incorporate external environmental factors, such as the seasonal effects on species competition and vanishing external noise. In periodic games, we prove that EG will converge while OGDA and momentum method will diverge. This is quite surprising, as to the best of our knowledge, it is the first result that indicates EG and OGDA have qualitatively different last-iterate behaviors and do not exhibit similar behavior. In convergent perturbed games, we prove all these algorithms converge as long as the game itself stabilizes with a faster rate than $1/t$., Comment: 44 pages, accepted for NeurIPS 2023

Gaitonde and Tardos recently studied a model of queueing networks where queues compete for servers and re-send returned packets in future rounds. They quantify the amount of additional processing power that guarantees a decentralized system's stability, both when the queues adapt their strategies from round to round using no-regret learning algorithms, and when they are patient and evaluate the utility of a strategy over long periods of time. In this paper, we generalize Gaitonde and Tardos's model and consider scenarios where not all servers can serve all queues (i.e., the underlying graph is an incomplete bipartite graphs) and, further, when packets need to go through more than one layer of servers before their completions (i.e., when the underlying graph is a DAG). For the bipartite case, we obtain bounds comparable to those by Gaitonde and Tardos, with the factor slightly worse in the patient queueing model. For the more general multi-layer systems, we show that straightforward generalizations of the utility function and servers' priority rules in Gaitonde and Tardos's model may lead to unbounded gaps between centralized and decentralized systems when the queues use no regret strategies. We define a new utility and a service priority rule that are aware of the queue lengths, and show that these suffice to restore the bounded gap between centralized and decentralized systems observed in bipartite graphs., Comment: Accepted by the 18th Conference on Web and Internet Economics (WINE 2022)