Our paper studies the setting of players using no-regret algorithms in various two-player games. We address whether having stronger regret guarantees or playing against an opponent with weaker regret guarantees yields higher utilities for the player in question. We consider a hierarchy of algorithms from weakest to strongest: uniform random play, no-regret, and no-swap-regret. We find, counterintuitively, that in many games, no-swap-regret is a worse choice for players (and gives better utility for their opponents). We find the root cause of this phenomenon to be a difference in effective learning rate between the two algorithms, where the no-swap-regret algorithms learn $N$ times slower than no-regret algorithms. To address this, we attempt to equalize learning rates, leading to closer utility between no-regret and no-swap-regret players. Finally, we show that for certain random games with $7$ actions per player, no-swap-regret algorithms can perform noticeably better than no-regret algorithms in a manner that cannot be explained away by unfairly adjusted learning rates.

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