Pseudo-games, or abstract economies [4], are optimization problems that are closely related to min-max Stackelberg games, but which are technically not games, as noted by Facchinei and Kanzow [26, 27], because each player's strategy set is not fixed at the outset (i.e., before they have to make a decision), but instead depends on the other players' choices. In this appendix, we formally define two-player, zero-sum pseudo-games,10 and discuss how they differ from min-max Stackelberg games. We also define the equilibrium concept par excellence of pseudo-games, namely generalized Nash equilibrium, and juxtapose its definition with vanilla Nash equilibrium. A two-player, zero-sum pseudo-game comprises two players, with respective payoff functions f(x,y)and f(x,y), and respective strategy spaces given by the correspondences X: Y X and Y: X Y, i.e., set valued mappings that depend on the choice the other player takes. Pseudo-games are closely related to min-max Stackelberg games, as they both comprise agents with the same objectives and the same space of feasible strategy profiles, namely {(x,y) 2 X Y |8 k 2 [K],gk(x,y) 0}.

As a central example, we highlight the case one agent is no-swap and the other's regret is

Zero-sum stochastic games have found important applications in a variety of fields, from machine learning to economics. Work on this model has primarily focused on the computation of Nash equilibrium due to its effectiveness in solving adversarial board and video games. Unfortunately, a Nash equilibrium is not guaranteed to exist in zero-sum stochastic games when the payoffs at each state are not convex-concave in the players' actions. A Stackelberg equilibrium, however, is guaranteed to exist. Consequently, in this paper, we study zero-sum stochastic Stackelberg games. Going beyond known existence results for (non-stationary) Stackelberg equilibria, we prove the existence of recursive (i.e., Markov perfect) Stackelberg equilibria (recSE) in these games, provide necessary and sufficient conditions for a policy profile to be a recSE, and show that recSE can be computed in (weakly) polynomial time via value iteration. Finally, we show that zero-sum stochastic Stackelberg games can model the problem of pricing and allocating goods across agents and time. More specifically, we propose a zero-sum stochastic Stackelberg game whose recSE correspond to the recursive competitive equilibria of a large class of stochastic Fisher markets. We close with a series of experiments that showcase how our methodology can be used to solve the consumption-savings problem in stochastic Fisher markets.

As a central example, we highlight the case one agent is no-swap and the other's regret is

Real world applications such as economics and policy making often involve solving multi-agent games with two unique features: (1) The agents are inherently asymmetric and partitioned into leaders and followers; (2) The agents have different reward functions, thus the game is general-sum . The majority of existing results in this field focuses on either symmetric solution concepts (e.g.

Real world applications such as economics and policy making often involve solving multi-agent games with two unique features: (1) The agents are inherently asymmetric and partitioned into leaders and followers; (2) The agents have different reward functions, thus the game is general-sum . The majority of existing results in this field focuses on either symmetric solution concepts (e.g.