Pure Nash Equilibria in Resource Graph Games

This paper studies the existence of pure Nash equilibria in resource graph games, a general class of strategic games succinctly representing the players’ private costs. These games are defined relative to a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function of the load vector of a certain subset of resources. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively. For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players’ strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. This implies in particular that for any other cost structure there is a resource graph game that does not have a pure Nash equilibrium. For weighted games where players have intrinsic weights and the cost of each resource depends on the aggregated weight of its users, pure Nash equilibria are guaranteed to exist if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load. We further discuss the computational complexity of pure Nash equilibria in resource graph games showing that for unweighted games where pure Nash equilibria are guaranteed to exist, it is coNP-complete to decide for a given strategy profile whether it is a pure Nash equilibrium. For general resource graph games, we prove that the decision whether a pure Nash equilibrium exists is Σ p 2 -complete.

Atomic Dynamic Flow Games: Adaptive vs. Nonadaptive Agents

We propose a game model for selfish routing of atomic agents, who compete for use of a network to travel from their origins to a common destination as quickly as possible. We follow a frequently used rule that the latency an agent experiences on each edge is a constant transit time plus a variable waiting time in a queue. A key feature that differentiates our model from related ones is an edge-based tie-breaking rule for prioritizing agents in queueing when they reach an edge at the same time. We study both nonadaptive agents (each choosing a one-off origin–destination path simultaneously at the very beginning) and adaptive ones (each making an online decision at every nonterminal vertex they reach as to which next edge to take). On the one hand, we constructively prove that a (pure) Nash equilibrium (NE) always exists for nonadaptive agents and show that every NE is weakly Pareto optimal and globally first-in first-out. We present efficient algorithms for finding an NE and best responses of nonadaptive agents. On the other hand, we are among the first to consider adaptive atomic agents, for which we show that a subgame perfect equilibrium (SPE) always exists and that each NE outcome for nonadaptive agents is an SPE outcome for adaptive agents but not vice versa.

Shadowing and shielding: Effective heuristics for continuous influence maximisation in the voting dynamics

Influence maximisation, or how to affect the intrinsic opinion dynamics of a social group, is relevant for many applications, such as information campaigns, political competition, or marketing. Previous literature on influence maximisation has mostly explored discrete allocations of influence, i.e. optimally choosing a finite fixed number of nodes to target. Here, we study the generalised problem of continuous influence maximisation where nodes can be targeted with flexible intensity. We focus on optimal influence allocations against a passive opponent and compare the structure of the solutions in the continuous and discrete regimes. We find that, whereas hub allocations play a central role in explaining optimal allocations in the discrete regime, their explanatory power is strongly reduced in the continuous regime. Instead, we find that optimal continuous strategies are very well described by two other patterns: (i) targeting the same nodes as the opponent (shadowing) and (ii) targeting direct neighbours of the opponent (shielding). Finally, we investigate the game-theoretic scenario of two active opponents and show that the unique pure Nash equilibrium is to target all nodes equally. These results expose fundamental differences in the solutions to discrete and continuous regimes and provide novel effective heuristics for continuous influence maximisation.

A Hashing Power Allocation Game with and without Risk-free Asset

Miners in various blockchain-backed cryptocurrency networks compete to maintain the validity of the underlying distributed ledgers to earn the bootstrapped cryptocurrencies. With limited hashing power, each miner needs to decide how to allocate their resource to different cryptocurrencies so as to achieve the best overall payoff. Together all the miners form a hashing power allocation game. We consider two settings of the game, depending on whether each miner can allocate their fund to a risk-free asset or not. We show that this game admits unique pure Nash equilibrium in closed-form for both settings.

The Minimum Tollbooth Problem in Atomic Network Congestion Games with Unsplittable Flows

This work analyzes the minimum tollbooth problem in atomic network congestion games with unsplittable flows. The goal is to place tolls on edges, such that there exists a pure Nash equilibrium in the tolled game that is a social optimum in the untolled one. Additionally, we require the number of tolled edges to be the minimum. This problem has been extensively studied in non-atomic games, however, to the best of our knowledge, it has not been considered for atomic games before. By a reduction from the weighted CNF SAT problem, we show both the NP-hardness of the problem and the W[2]-hardness when parameterizing the problem with the number of tolled edges. On the positive side, we present a polynomial time algorithm for networks on series-parallel graphs that turns any given state of the untolled game into a pure Nash equilibrium of the tolled game with the minimum number of tolled edges.

Pure Nash Equilibria and Best-Response Dynamics in Random Games

In finite games, mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies and the payoffs are independent and identically distributed with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of pure Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a pure Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that various phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

Intermodal Hinterland Network Design Games

Intermodal hinterland transportation is becoming increasingly critical for global container supply chains. Managing intermodal hinterland networks is challenging because multiple actors often interact in practice. The intermodal hinterland network design games that we propose enable assessing the impact of having noncooperative users in intermodal networks. The games fall into the class of network design games but have key distinctive features. We provide some general results as well as an instance without a pure Nash equilibrium for the general case. Subsequently, we focus on the special case with a single intermodal connection available. We show that a pure Nash equilibrium always exists but that this one is not always unique. We additionally identify key structural properties for this single-hub game. These properties enable us to identify all pure Nash equilibria and a system-optimal solution in polynomial time. We illustrate our results with an application related to the development of an extended gate in the Netherlands and derive a series of insights. Overall, the results show that the multiple user feature of intermodal hinterland networks is critical and needs to be accounted for at the network design stage. We believe that this latter statement holds for general network design problems with multiple users.

Nash versus coarse correlation

We run a laboratory experiment to test the concept of coarse correlated equilibrium (Moulin and Vial in Int J Game Theory 7:201–221, 1978), with a two-person game with unique pure Nash equilibrium which is also the solution of iterative elimination of strictly dominated strategies. The subjects are asked to commit to a device that randomly picks one of three symmetric outcomes (including the Nash point) with higher ex-ante expected payoff than the Nash equilibrium payoff. We find that the subjects do not accept this lottery (which is a coarse correlated equilibrium); instead, they choose to play the game and coordinate on the Nash equilibrium. However, given an individual choice between a lottery with equal probabilities of the same outcomes and the sure payoff as in the Nash point, the lottery is chosen by the subjects. This result is robust against a few variations. We explain our result as selecting risk-dominance over payoff dominance in equilibrium.

The Pure Price of Anarchy of Pool Block Withholding Attacks in Bitcoin Mining

Bitcoin, a cryptocurrency built on the blockchain data structure, has generated significant academic and commercial interest. Contrary to prior expectations, recent research has shown that participants of the protocol (the so-called “miners”) are not always incentivized to follow the protocol. We study the game induced by one such attack – the pool block withholding attack – in which mining pools (groups of miners) attack other mining pools. We focus on the case of two pools attacking each other, with potentially other mining power in the system.We show that this game always admits a pure Nash equilibrium, and its pure price of anarchy, which intuitively measures how much computational power can be wasted due to attacks in an equilibrium, is at most 3. We conjecture, and prove in special cases, that it is in fact at most 2. Our simulations provide compelling evidence for this conjecture, and show that players can quickly converge to the equilibrium by following best response strategies.