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1. Randomized Cup Game Algorithms Against Strong Adversaries

   https://doi.org/https://doi.org/10.1145/3313276.3316342  

   Bender, M; Kuszmaul, W. (January 2021, Proceedings of the annual ACMSIAM symposium on discrete algorithms)

   In each step of the cup game on n cups, a filler distributes up to 1" water among the cups, and then an emptier removes 1 unit of water from a single cup. The emptier’s goal is to minimize the height of the fullest cup, also known as the backlog. The cup emptying game has found extensive applications to processor scheduling, network- switch buffer management, quality of service guarantees, and data- structure deamortization. The greedy emptying algorithm (i.e., always remove from the fullest cup) is known to achieve backlog O(log n) and to be the optimal deterministic algorithm. Randomized algorithms can do significantly better, achieving backlog O(log log n) with high probability, as long as " is not too small. In order to achieve these improvements, the known randomized algorithms require that the filler is an oblivious adversary, unaware of which cups the emptier chooses to empty out of at each step. Such randomized guarantees are known to be impossible against fully adaptive fillers. We show that, even when the filler is just slightly non- adaptive, randomized emptying algorithms can still guarantee a backlog of O(log log n). In particular, we give randomized ran- domized algorithms against an elevated adaptive filler, which is an adaptive filler that can see the precise fills of every cup containing more than 3 units of water, but not of the cups containing less than 3 units. 

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2. Hardness amplification for entangled games via anchoring

   https://doi.org/10.1145/3055399.3055433  

   Bavarian, Mohammad; Vidick, Thomas; Yuen, Henry (June 2017, Proceedings of the annual ACM Symposium on Theory of Computing)

   We study the parallel repetition of one-round games involving players that can use quantum entanglement. A major open question in this area is whether parallel repetition reduces the entangled value of a game at an exponential rate --- in other words, does an analogue of Raz's parallel repetition theorem hold for games with players sharing quantum entanglement? Previous results only apply to special classes of games. We introduce a class of games we call anchored. We then introduce a simple transformation on games called \emph{anchoring}, inspired in part by the Feige-Kilian transformation, that turns \emph{any} (multiplayer) game into an anchored game. Unlike the Feige-Kilian transformation, our anchoring transformation is completeness preserving. We prove an exponential-decay parallel repetition theorem for anchored games that involve any number of entangled players. We also prove a threshold version of our parallel repetition theorem for anchored games. Together, our parallel repetition theorems and anchoring transformation provide the first hardness amplification techniques for general entangled games. We give an application to the games version of the Quantum PCP Conjecture. 

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3. Steering No-Regret Learners to a Desired Equilibrium

   Zhang, B; Farina, G; Anagostides, I; Cacciamani, F; McAleer, S; Haupt, A; Celli, A; Gatti, N; Conitzer, V; Sandholm, T (July 2024, EC24)

   A mediator observes no-regret learners playing an extensive-form game repeatedly across T rounds. The mediator attempts to steer players toward some desirable predetermined equilibrium by giving (nonnegative) payments to players. We call this the steering problem. The steering problem captures problems several problems of interest, among them equilibrium selection and information design (persuasion). If the mediator’s budget is unbounded, steering is trivial because the mediator can simply pay the players to play desirable actions. We study two bounds on the mediator’s payments: a total budget and a per-round budget. If the mediator’s total budget does not grow with T, we show that steering is impossible. However, we show that it is enough for the total budget to grow sublinearly with T, that is, for the average payment to vanish. When players’ full strategies are observed at each round, we show that constant per-round budgets permit steering. In the more challenging setting where only trajectories through the game tree are observable, we show that steering is impossible with constant per-round budgets in general extensive-form games, but possible in normal-form games or if the per-round budget may itself depend on T. We also show how our results can be generalized to the case when the equilibrium is being computed online while steering is happening. We supplement our theoretical positive results with experiments highlighting the efficacy of steering in large games. 

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4. Parallel Repetition via Fortification: Analytic View and The Quantum Case

   Bavarian, Mohammad; Vidick, Thomas; Yuen, Henry (January 2017, Innovations in Theoretical Computer Science)

   In a recent work, Moshkovitz [FOCS '14] presented a transformation on two-player games called ``fortification'', and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an \emph{analytic reformulation} of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings. First, we show \emph{any} game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, two problems which have recently received much attention. An important component of our work is a variant of the fortification transformation, called ``ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting. 

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5. Simple Uncoupled No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium

   Farina, G.; Celli, A.; Marchesi, A.; Gatti, N. (January 2021, ArXivorg)

   null (Ed.)

   The existence of simple uncoupled no-regret learning dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and presence of private information in the game, correlation in extensive-form games possesses significantly different properties than its counterpart in normal-form games, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device must keep into account the evolution of beliefs of each player as they make observations throughout the game. Due to that significant added complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics that converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play is proven to be a O(T^-0.5)-approximate EFCE with high probability after T game repetitions, and an EFCE almost surely in the limit. 

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