arXiv subjects

Alexander Skopalik

Publications and source records attributed to Alexander Skopalik.

At least 19 recordsLinked to original sources

arXiv2026

An Entropy Potential for Type-Composition Games

Potential functions are a key tool in theoretical computer science with applications ranging from the runtime analysis of algorithms and data structures, through the analysis of the expected behavior of random processes and search heuristics, to proving the existence of equilibrium states in strategic games. Typically, proofs that employ potential functions are short, elegant, and easy to verify, yet very powerful. Moreover, potential functions are essential ingredients for constructive proofs, in particular in algorithmic game theory. There, a key question is the existence of equilibrium states, but the most powerful theorem in the field -- Nash's theorem -- is unfortunately non-constructive. For many strategic games, potential functions come to the rescue by enabling constructive proofs that sometimes even yield efficient algorithms for finding equilibria. We add to this by providing a novel class of entropy-inspired log-multinomial potential functions for natural game-theoretic settings where rational agents of different types strategically choose actions to maximize their utility. In particular, we consider utility functions that are based on the fraction of same- and other-type agents taking the same action. We demonstrate the versatility of the new potential function class by presenting simple equilibrium existence proofs for two recent game-theoretic models, for which only involved technical proofs were previously known. Even better, the new potential function class yields efficient algorithms for constructing equilibria for much more general models. Thereby, we positively resolve several open problems.

cs.GT↗

arXiv2026

Selfish routing games with priority lanes

We study selfish routing games where users can choose between regular and priority service for each network edge on their chosen path. Priority users pay an additional fee, but in turn they may travel the edge prior to non-priority users, hence experiencing potentially less congestion. For this model, we establish existence of equilibria for linear latency functions and prove uniqueness of edge latencies, despite potentially different strategic choices in equilibrium. Our main contribution demonstrates that marginal cost pricing achieves system optimality: When priority fees equal marginal externality costs, the equilibrium flow coincides with the socially optimal flow, hence the price of anarchy equals $1$. This voluntary priority mechanism therefore provides an incentive-compatible alternative to mandatory congestion pricing, whilst achieving the same result. We also discuss the limitations of a uniform pricing scheme for the priority option.

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arXiv2025

Social Welfare in Battery Charging Games

The recent rise of renewable energy produced by many decentralized sources yields interesting market design challenges for electrical grids. Balancing supply and demand in such networks is both a temporal and spatial challenge due to capacity constraints. The recent surge in the number of household-owned batteries, especially in regions with rooftop solar adoption, offers mitigation potential but often acts misaligned with grid-level objectives. In fact, the decision to charge or discharge a household-owned battery is a strategic choice by each battery owner governed by selfish incentives. This calls for an analysis from a game-theoretic point of view. We initiate this timely research direction by considering a game-theoretic setting where selfish agents strategically charge or discharge their batteries to increase their profit. In particular, we study a Stackelberg-like market model where a third party introduces price incentives, aiming to optimize renewable energy utilization while preserving grid feasibility. For this, we study the existence and the quality of equilibria under various pricing strategies. We find that the existence of equilibria crucially depends on the chosen pricing and that the obtained social welfare varies widely. This calls for more sophisticated market models and pricing mechanisms and opens up a rich field for future research in Algorithmic Game Theory on incentives in renewable energy networks.

cs.GT↗

arXiv2025

Playing Snake on a Graph

Snake is a classic computer game, which has been around for decades. Based on this game, we study the game of Snake on arbitrary undirected graphs. A snake forms a simple path that has to move to an apple while avoiding colliding with itself. When the snake reaches the apple, it grows longer, and a new apple appears. A graph on which the snake has a strategy to keep eating apples until it covers all the vertices of the graph is called snake-winnable. We prove that determining whether a graph is snake-winnable is NP-hard, even when restricted to grid graphs. We fully characterize snake-winnable graphs for odd-sized bipartite graphs and graphs with vertex-connectivity 1. While Hamiltonian graphs are always snake-winnable, we show that non-Hamiltonian snake-winnable graphs have a girth of at most 6 and that this bound is tight.

cs.DM↗

arXiv2025

The Bakers and Millers Game with Restricted Locations

We study strategic location choice by customers and sellers, termed the Bakers and Millers Game in the literature. In our generalized setting, each miller can freely choose any location for setting up a mill, while each baker is restricted in the choice of location for setting up a bakery. For optimal bargaining power, a baker would like to select a location with many millers to buy flour from and with little competition from other bakers. Likewise, a miller aims for a location with many bakers and few competing millers. Thus, both types of agents choose locations to optimize the ratio of agents of opposite type divided by agents of the same type at their chosen location. Originally raised in the context of Fractional Hedonic Games, the Bakers and Millers Game has applications that range from commerce to product design. We study the impact of location restrictions on the properties of the game. While pure Nash equilibria trivially exist in the setting without location restrictions, we show via a sophisticated, efficient algorithm that even the more challenging restricted setting admits equilibria. Moreover, the computed equilibrium approximates the optimal social welfare by a factor of at most $2\left(\frac{e}{e-1}\right)$. Furthermore, we give tight bounds on the price of anarchy/stability. On the conceptual side, the location choice feature adds a new layer to the standard setting of Hedonic Games, in the sense that agents that select the same location form a coalition. This allows to naturally restrict the possible coalitions that can be formed. With this, our model generalizes simple symmetric Fractional Hedonic Games on complete bipartite valuation graphs and also Hedonic Diversity Games with utilities single-peaked at 0. We believe that this generalization is also a very interesting direction for other types of Hedonic Games.

cs.GT↗

arXiv2024

Equilibria in Two-Stage Facility Location with Atomic Clients

We consider competitive facility location as a two-stage multi-agent system with two types of clients. For a given host graph with weighted clients on the vertices, first facility agents strategically select vertices for opening their facilities. Then, the clients strategically select which of the opened facilities in their neighborhood to patronize. Facilities want to attract as much client weight as possible, clients want to minimize congestion on the chosen facility. All recently studied versions of this model assume that clients can split their weight strategically. We consider clients with unsplittable weights but allow mixed strategies. So clients may randomize over which facility to patronize. Besides modeling a natural client behavior, this subtle change yields drastic changes, e.g., for a given facility placement, qualitatively different client equilibria are possible. As our main result, we show that pure subgame perfect equilibria always exist if all client weights are identical. For this, we use a novel potential function argument, employing a hierarchical classification of the clients and sophisticated rounding in each step. In contrast, for non-identical clients, we show that deciding the existence of even approximately stable states is computationally intractable. On the positive side, we give a tight bound of $2$ on the price of anarchy which implies high social welfare of equilibria, if they exist.

cs.GT↗

arXiv2023

Strategic Resource Selection with Homophilic Agents

The strategic selection of resources by selfish agents is a classic research direction, with Resource Selection Games and Congestion Games as prominent examples. In these games, agents select available resources and their utility then depends on the number of agents using the same resources. This implies that there is no distinction between the agents, i.e., they are anonymous. We depart from this very general setting by proposing Resource Selection Games with heterogeneous agents that strive for joint resource usage with similar agents. So, instead of the number of other users of a given resource, our model considers agents with different types and the decisive feature is the fraction of same-type agents among the users. More precisely, similarly to Schelling Games, there is a tolerance threshold $\tau \in [0,1]$ which specifies the agents' desired minimum fraction of same-type agents on a resource. Agents strive to select resources where at least a $\tau$-fraction of those resources' users have the same type as themselves. For $\tau=1$, our model generalizes Hedonic Diversity Games with a peak at $1$. For our general model, we consider the existence and quality of equilibria and the complexity of maximizing social welfare. Additionally, we consider a bounded rationality model, where agents can only estimate the utility of a resource, since they only know the fraction of same-type agents on a given resource, but not the exact numbers. Thus, they cannot know the impact a strategy change would have on a target resource. Interestingly, we show that this type of bounded rationality yields favorable game-theoretic properties and specific equilibria closely approximate equilibria of the full knowledge setting.

cs.GT↗

arXiv2022

Strategic Facility Location with Clients that Minimize Total Waiting Time

We study a non-cooperative two-sided facility location game in which facilities and clients behave strategically. This is in contrast to many other facility location games in which clients simply visit their closest facility. Facility agents select a location on a graph to open a facility to attract as much purchasing power as possible, while client agents choose which facilities to patronize by strategically distributing their purchasing power in order to minimize their total waiting time. Here, the waiting time of a facility depends on its received total purchasing power. We show that our client stage is an atomic splittable congestion game, which implies existence, uniqueness and efficient computation of a client equilibrium. Therefore, facility agents can efficiently predict client behavior and make strategic decisions accordingly. Despite that, we prove that subgame perfect equilibria do not exist in all instances of this game and that their existence is NP-hard to decide. On the positive side, we provide a simple and efficient algorithm to compute 3-approximate subgame perfect equilibria.

cs.GT↗

arXiv2021

Two-Stage Facility Location Games with Strategic Clients and Facilities

We consider non-cooperative facility location games where both facilities and clients act strategically and heavily influence each other. This contrasts established game-theoretic facility location models with non-strategic clients that simply select the closest opened facility. In our model, every facility location has a set of attracted clients and each client has a set of shopping locations and a weight that corresponds to her spending capacity. Facility agents selfishly select a location for opening their facility to maximize the attracted total spending capacity, whereas clients strategically decide how to distribute their spending capacity among the opened facilities in their shopping range. We focus on a natural client behavior similar to classical load balancing: our selfish clients aim for a distribution that minimizes their maximum waiting times for getting serviced, where a facility's waiting time corresponds to its total attracted client weight. We show that subgame perfect equilibria exist and give almost tight constant bounds on the Price of Anarchy and the Price of Stability, which even hold for a broader class of games with arbitrary client behavior. Since facilities and clients influence each other, it is crucial for the facilities to anticipate the selfish clients' behavior when selecting their location. For this, we provide an efficient algorithm that also implies an efficient check for equilibrium. Finally, we show that computing a socially optimal facility placement is NP-hard and that this result holds for all feasible client weight distributions.

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arXiv2020

Improving approximate pure Nash equilibria in congestion games

Congestion games constitute an important class of games to model resource allocation by different users. As computing an exact or even an approximate pure Nash equilibrium is in general PLS-complete, Caragiannis et al. (2011) present a polynomial-time algorithm that computes a ($2 + ε$)-approximate pure Nash equilibria for games with linear cost functions and further results for polynomial cost functions. We show that this factor can be improved to $(1.61+ε)$ and further improved results for polynomial cost functions, by a seemingly simple modification to their algorithm by allowing for the cost functions used during the best response dynamics be different from the overall objective function. Interestingly, our modification to the algorithm also extends to efficiently computing improved approximate pure Nash equilibria in games with arbitrary non-decreasing resource cost functions. Additionally, our analysis exhibits an interesting method to optimally compute universal load dependent taxes and using linear programming duality prove tight bounds on PoA under universal taxation, e.g, 2.012 for linear congestion games and further results for polynomial cost functions. Although our approach yield weaker results than that in Bilò and Vinci (2016), we remark that our cost functions are locally computable and in contrast to Bilò and Vinci (2016) are independent of the actual instance of the game.

cs.GT↗

arXiv2019

Network Investment Game with Wardrop Followers

We study a two-sided network investment game consisting of two sets of players, called providers and users. The game is set in two stages. In the first stage, providers aim to maximize their profit by investing in bandwidth of cloud computing services. The investments of the providers yield a set of usable services for the users. In the second stage, each user wants to process a task and therefore selects a bundle of services so as to minimize the total processing time. We assume the total processing time to be separable over the chosen services and the processing time of each service to depend on the utilization of the service and the installed bandwidth. We provide insights on how competition between providers affects the total costs of the users and show that every game on a series-parallel graph can be reduced to an equivalent single edge game when analyzing the set of subgame perfect Nash equilibria.

cs.GT↗

arXiv2019

From Hotelling to Load Balancing: Approximation and the Principle of Minimum Differentiation

Competing firms tend to select similar locations for their stores. This phenomenon, called the principle of minimum differentiation, was captured by Hotelling with a landmark model of spatial competition but is still the object of an ongoing scientific debate. Although consistently observed in practice, many more realistic variants of Hotelling's model fail to support minimum differentiation or do not have pure equilibria at all. In particular, it was recently proven for a generalized model which incorporates negative network externalities and which contains Hotelling's model and classical selfish load balancing as special cases, that the unique equilibria do not adhere to minimum differentiation. Furthermore, it was shown that for a significant parameter range pure equilibria do not exist. We derive a sharp contrast to these previous results by investigating Hotelling's model with negative network externalities from an entirely new angle: approximate pure subgame perfect equilibria. This approach allows us to prove analytically and via agent-based simulations that approximate equilibria having good approximation guarantees and that adhere to minimum differentiation exist for the full parameter range of the model. Moreover, we show that the obtained approximate equilibria have high social welfare.

cs.GT↗

arXiv2018

Multi-unit Bilateral Trade

We characterise the set of dominant strategy incentive compatible (DSIC), strongly budget balanced (SBB), and ex-post individually rational (IR) mechanisms for the multi-unit bilateral trade setting. In such a setting there is a single buyer and a single seller who holds a finite number k of identical items. The mechanism has to decide how many units of the item are transferred from the seller to the buyer and how much money is transferred from the buyer to the seller. We consider two classes of valuation functions for the buyer and seller: Valuations that are increasing in the number of units in possession, and the more specific class of valuations that are increasing and submodular. Furthermore, we present some approximation results about the performance of certain such mechanisms, in terms of social welfare: For increasing submodular valuation functions, we show the existence of a deterministic 2-approximation mechanism and a randomised e/(1-e) approximation mechanism, matching the best known bounds for the single-item setting.

cs.GT↗

arXiv2017

Computing Approximate Pure Nash Equilibria in Shapley Value Weighted Congestion Games

We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply sampling techniques which allow us to achieve polynomial running time. The algorithm builds on the algorithmic ideas of [7], however, to the best of our knowledge, this is the first algorithmic result on computation of approximate equilibria using other than proportional shares as player costs in this setting. We present a novel relation that approximates the Shapley value of a player by her proportional share and vice versa. As side results, we upper bound the approximate price of anarchy of such games and significantly improve the best known factor for computing approximate pure Nash equilibria in weighted congestion games of [7].

cs.GT↗

arXiv2017

Hiring Secretaries over Time: The Benefit of Concurrent Employment

We consider a stochastic online problem where $n$ applicants arrive over time, one per time step. Upon arrival of each applicant their cost per time step is revealed, and we have to fix the duration of employment, starting immediately. This decision is irrevocable, i.e., we can neither extend a contract nor dismiss a candidate once hired. In every time step, at least one candidate needs to be under contract, and our goal is to minimize the total hiring cost, which is the sum of the applicants' costs multiplied with their respective employment durations. We provide a competitive online algorithm for the case that the applicants' costs are drawn independently from a known distribution. Specifically, the algorithm achieves a competitive ratio of 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants' costs is unknown; (iii) the total number $n$ of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than $Ω(\sqrt{n} / \log n)$ if concurrent employment is forbidden.

cs.DS↗

arXiv2017

Congestion Games with Complementarities

We study a model of selfish resource allocation that seeks to incorporate dependencies among resources as they exist in modern networked environments. Our model is inspired by utility functions with constant elasticity of substitution (CES) which is a well-studied model in economics. We consider congestion games with different aggregation functions. In particular, we study $L_p$ norms and analyze the existence and complexity of (approximate) pure Nash equilibria. Additionally, we give an almost tight characterization based on monotonicity properties to describe the set of aggregation functions that guarantee the existence of pure Nash equilibria.

cs.GT↗

arXiv2016

Congestion Games with Mixed Objectives

We study a new class of games which generalizes congestion games and its bottleneck variant. We introduce congestion games with mixed objectives to model network scenarios in which players seek to optimize for latency and bandwidths alike. We characterize the existence of pure Nash equilibria (PNE) and the convergence of improvement dynamics. For games that do not possess PNE we give bounds on the approximation ratio of approximate pure Nash equilibria.

cs.GT↗

arXiv2015

On Existence and Properties of Approximate Pure Nash Equilibria in Bandwidth Allocation Games

In \emph{bandwidth allocation games} (BAGs), the strategy of a player consists of various demands on different resources. The player's utility is at most the sum of these demands, provided they are fully satisfied. Every resource has a limited capacity and if it is exceeded by the total demand, it has to be split between the players. Since these games generally do not have pure Nash equilibria, we consider approximate pure Nash equilibria, in which no player can improve her utility by more than some fixed factor $α$ through unilateral strategy changes. There is a threshold $α_δ$ (where $δ$ is a parameter that limits the demand of each player on a specific resource) such that $α$-approximate pure Nash equilibria always exist for $α\geq α_δ$, but not for $α< α_δ$. We give both upper and lower bounds on this threshold $α_δ$ and show that the corresponding decision problem is ${\sf NP}$-hard. We also show that the $α$-approximate price of anarchy for BAGs is $α+1$. For a restricted version of the game, where demands of players only differ slightly from each other (e.g. symmetric games), we show that approximate Nash equilibria can be reached (and thus also be computed) in polynomial time using the best-response dynamic. Finally, we show that a broader class of utility-maximization games (which includes BAGs) converges quickly towards states whose social welfare is close to the optimum.

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