Abstract We study the problem of generating paths on a graph that satisfy a collection of $$\omega $$ ω -regular objectives. We propose a decoupled framework in which each objective is assigned to an independent agent that selects a local policy, while a scheduler—oblivious to the graph and objective—dynamically composes these policies into a single path. We ask when such a composition satisfies all objectives, assuming their conjunction is realizable. The framework enables modular policy design but raises fundamental compositional challenges. We show that even extremely fair deterministic schedulers do not ensure correctness, and that stochastic schedulers, while necessary, are insufficient without coordination. For safety objectives, we demonstrate that fully decentralized implementations are impossible, and we introduce a protocol for synchronizing on maximal safe actions. For non-safety objectives, we introduce conventions —simple, a priori restrictions agreed upon before the graph or objectives are revealed—that guarantee satisfaction of all objectives when followed by all agents. We characterize minimally restrictive conventions for major subclasses of $$\omega $$ ω -regular objectives. In particular, Büchi objectives admit universal composition of finite-memory policies without scheduler communication; co-Büchi objectives require only knowledge of whether the agent was scheduled; and parity objectives additionally require knowledge of which agent was scheduled.

A \emph{bidding} game is played on a graph as follows. A token is placed on an initial vertex and both players are allocated budgets. In each turn, the players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder. We focus on \emph{discrete}-bidding, which are motivated by practical applications and restrict the granularity of the players' bids, e.g, bids must be given in cents. We study, for the first time, discrete-bidding games with {\em mean-payoff} and {\em energy} objectives. In contrast, mean-payoff {\em continuous}-bidding games (i.e., no granularity restrictions) are understood and exhibit a rich mathematical structure. The {\em threshold} budget is a necessary and sufficient initial budget for winning an energy game or guaranteeing a target payoff in a mean-payoff game. We first establish existence of threshold budgets; a non-trivial property due to the concurrent moves of the players. Moreover, we identify the structure of the thresholds, which is key in obtaining compact strategies, and in turn, showing that finding threshold is in \NP~and \coNP even in succinctly-represented games.

Abstract Sequential decision-making tasks often require satisfaction of multiple, partially-contradictory objectives. Existing approaches are monolithic, where a singlepolicyfulfills all objectives. We presentauction-based scheduling, adecentralizedframework for multi-objective sequential decision making. Each objective is fulfilled using a separate and independent policy. Composition of policies is performed at runtime, where at each step, the policies simultaneously bid from pre-allocated budgets for the privilege of choosing the next action. The framework allows policies to be independently created, modified, and replaced. We study path planning problems on finite graphs with two temporal objectives and present algorithms to synthesize policies together with bidding policies in a decentralized manner. We consider three categories of decentralized synthesis problems, parameterized by the assumptions that the policies make on each other. We identify a class of assumptions calledassume-admissiblefor which synthesis is always possible for graphs whose every vertex has at most two outgoing edges.

We consider bidding games, a class of two-player zero-sum graph games. The game proceeds as follows. Both players have bounded budgets. A token is placed on a vertex of a graph, in each turn the players simultaneously submit bids, and the higher bidder moves the token, where we break bidding ties in favor of Player 1. Player 1 wins the game iff the token visits a designated target vertex. We consider, for the first time, poorman discrete-bidding in which the granularity of the bids is restricted and the higher bid is paid to the bank. Previous work either did not impose granularity restrictions or considered Richman bidding (bids are paid to the opponent). While the latter mechanisms are technically more accessible, the former is more appealing from a practical standpoint. Our study focuses on threshold budgets, which is the necessary and sufficient initial budget required for Player 1 to ensure winning against a given Player 2 budget. We first show existence of thresholds. In DAGs, we show that threshold budgets can be approximated with error bounds by thresholds under continuous-bidding and that they exhibit a periodic behavior. We identify closed-form solutions in special cases. We implement and experiment with an algorithm to find threshold budgets.

We consider the question of whether, and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider games where each player's weight is small. We prove that when the number of players goes to infinity and their weights to zero, the random flows in all (mixed) Nash equilibria for the finite games converge in distribution to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider an increasing number of players with a unit weight that participate in the game with a decreasingly small probability. In this case, the Nash equilibrium flows converge in total variation towards Poisson random variables whose expected values are Wardrop equilibria of a different nonatomic game with suitably-defined costs. The latter can be viewed as symmetric equilibria in a Poisson game in the sense of Myerson, establishing a plausible connection between the Wardrop model for routing games and the stochastic fluctuations observed in real traffic. In both settings we provide explicit approximation bounds, and we study the convergence of the price of anarchy. Beyond the case of congestion games, we prove a general result on the convergence of large games with random players towards Poisson games.

In a two-player zero-sum graph game, the players move a token throughout the graph to produce an infinite play, which determines the winner of the game. Bidding games are graph games in which in each turn, an auction (bidding) determines which player moves the token: the players have budgets, and in each turn, both players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder. We distinguish between continuous- and discrete-bidding games. In the latter, the granularity of the players' bids is restricted, e.g., bids must be given in cents. Continuous-bidding games are well understood, however, from a practical standpoint, discrete-bidding games are more appealing. In this paper we focus on discrete-bidding games. We study the problem of finding threshold budgets; namely, a necessary and sufficient initial budget for winning the game. Previously, the properties of threshold budgets were only studied for reachability games. For parity discrete-bidding games, thresholds were known to exist, but their structure was not understood. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first algorithm is a fixed-point algorithm, and it reveals the structure of the threshold budgets in these games. Second, we show that the problem of finding threshold budgets is in NP and coNP for parity discrete-bidding games. Previously, only exponential-time algorithms where known for reachability and parity objectives. A corollary of this proof is a construction of strategies that use polynomial-size memory.

Congestion games are a classical type of games studied in game theory, in which n players choose a resource, and their individual cost increases with the number of other players choosing the same resource. In network congestion games (NCGs), the resources correspond to simple paths in a graph, e.g. representing routing options from a source to a target. In this paper, we introduce a variant of NCGs, referred to as dynamic NCGs: in this setting, players take transitions synchronously, they select their next transitions dynamically, and they are charged a cost that depends on the number of players simultaneously using the same transition. We study, from a complexity perspective, standard concepts of game theory in dynamic NCGs: social optima, Nash equilibria, and subgame perfect equilibria. Our contributions are the following: the existence of a strategy profile with social cost bounded by a constant is in PSPACE and NP-hard. (Pure) Nash equilibria always exist in dynamic NCGs; the existence of a Nash equilibrium with bounded cost can be decided in EXPSPACE, and computing a witnessing strategy profile can be done in doubly-exponential time. The existence of a subgame perfect equilibrium with bounded cost can be decided in 2EXPSPACE, and a witnessing strategy profile can be computed in triply-exponential time.