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We consider a number of questions related to tradeoffs between reward andregret in repeated gameplay between two agents. To facilitate this, weintroduce a notion of $\textit{generalized equilibrium}$ which allows forasymmetric regret constraints, and yields polytopes of feasible values for eachagent and pair of regret constraints, where we show that any such equilibriumis reachable by a pair of algorithms which maintain their regret guaranteesagainst arbitrary opponents. As a central example, we highlight the case oneagent is no-swap and the other's regret is unconstrained. We show that thiscaptures an extension of $\textit{Stackelberg}$ equilibria with a matchingoptimal value, and that there exists a wide class of games where a player cansignificantly increase their utility by deviating from a no-swap-regretalgorithm against a no-swap learner (in fact, almost any game without pure Nashequilibria is of this form). Additionally, we make use of generalizedequilibria to consider tradeoffs in terms of the opponent's algorithm choice.We give a tight characterization for the maximal reward obtainable against$\textit{some}$ no-regret learner, yet we also show a class of games in whichthis is bounded away from the value obtainable against the class of common"mean-based" no-regret algorithms. Finally, we consider the question oflearning reward-optimal strategies via repeated play with a no-regret agentwhen the game is initially unknown. Again we show tradeoffs depending on theopponent's learning algorithm: the Stackelberg strategy is learnable inexponential time with any no-regret agent (and in polynomial time with anyno-$\textit{adaptive}$-regret agent) for any game where it is learnable viaqueries, and there are games where it is learnable in polynomial time againstany no-swap-regret agent but requires exponential time against a mean-basedno-regret agent.

Is Learning in Games Good for the Learners?