Marcin Konrad Dziubiński

Network Design and Defence

Infrastructure networks are a key feature of an economy. Their functionality depends on the connectivity and sizes of different components and they face a variety of threats, from natural disasters to intelligent attacks. How should networks be defended and designed to ensure the best functionality?

The Hotelling-Downs Model with Runoff Voting

We consider the Hotelling–Downs model with n⩾3 office-seeking candidates and runoff voting. We show that Nash equilibria in pure strategies always exist and that there are typically multiple equilibria, both convergent (all candidates are located at the median) and divergent (candidates locate at distinct positions), though only divergent equilibria are robust to free entry. Moreover, two-policy equilibria exist under any distribution of votersʼ ideal policies, while equilibria with more than two policies exist generically but under restrictive conditions that we characterize.

Non-symmetric discrete General Lotto games

We study a non-symmetric variant of General Lotto games introduced in Hart (Int J Game Theory 36:441–460, 2008). We provide a complete characterization of optimal strategies for both players in non-symmetric discrete General Lotto games, where one of the players has an advantage over the other. By this we complete the characterization given in Hart (Int J Game Theory 36:441–460, 2008), where the strategies for symmetric case were fully characterized and some of the optimal strategies for the non-symmetric case were obtained. We find a group of completely new atomic strategies, which are used as building components for the optimal strategies. Our results are applicable to discrete variants of all-pay auctions.

Electoral competition in 2-dimensional ideology space with unidimensional commitment

We study a model of electoral competition between two candidates with two orthogonal issues, where candidates are office motivated and committed to a particular position in one of the dimensions, while having the freedom to select (credibly) any position on the other dimension. We analyse two settings: one where both candidates are committed to the same dimension, and the other where each candidate is committed to a different dimension. We focus on characterisation and existence of pure strategy Nash equilibria when the core is empty. We show that if the distribution of voters’ ideal policies is continuously differentiable and has a bounded support, then an equilibrium exists if the candidates are differentiated enough. Our results for the case where the candidates have a common committed issue have implications for the literature on valence.

How do you defend a network

Modern economies rely heavily on their infrastructure networks. These networks face threats ranging from natural disasters to human attacks. As networks are pervasive, the investments needed to protect them are very large; this motivates the study of targeted defence. What are the ‘key’ nodes to defend to maximize functionality of the network? What are the incentives of individual nodes to protect themselves in a networked environment and how do these incentives correspond to collective welfare? We provide a characterization of equilibrium attack and defence in terms of two classical concepts in graph theory – separators and transversals. We use this characterization to study the intensity of conflict (the resources spent on attack and defence) and the prospects of active conflict (when both adversary and defender target nodes for action) in networks. Finally, we show that welfare costs of decentralized defence can be very large

Strategies in Dialogues: A Game-Theoretic Approach

The aim of the paper is to propose a game-theoretic description of strate- gies available to players in dialogues. We show how existing dialogical systems can be formalized as Nash-style games, and how the game-theoretic concept of so- lutions (dominant strategies, Nash equilibrium) can be used to analyse these sys- tems. Our first study, discussed in this article, describes the game DC introduced by Mackenzie.

Individual security and network design

Individuals derive benefits from being connected to others: computer users benefit from sharing content, criminals benefit from cooperating. Connections, however, may transmit external threats. A virus may spread through a computer network. An investigation may dismantle entire criminal organization. Given agents’ individual incentives to protect, which network(s) should be chosen to maximize agents’ welfare? We consider the tension between the value of being connected and the exposure to contagion when a protection technology is available. There are (n + 2) ‘players’. The designer first chooses the network over the n nodes. Given this network, the nodes (simultaneously) choose whether to protect or not; protection is costly. Finally, the adversary chooses a node to attack. If the attacked node is not protected, then this node and all nodes with a path to the attacked node through unprotected nodes are eliminated. Nodes derive benefits from their connectivity: a surviving node gets, as a gross payoff, an equal share of the value of its surviving component. Component value is a convex and increasing function of its size. Node’s net payoffs are equal to its connectivity payoffs less the cost of protection. The designer seeks to maximize the sum of nodes’ payoffs. The adversary aims to minimize connectivity-related payoffs. The first best design and defence profile that a central planner would choose is as follows. For low costs, all nodes is protected and the network is connected. For intermediate costs, a centrally protected star is chosen. The adversary eliminates a spoke. If costs are high, protection is dropped and network is split into several components. The adversary removes a largest one. A number of problems arise for the designer when he cannot control defence decisions. First, a node does not internalize the benefits accruing to others from its own protection. Thus, it is possible that the center-

How to Defend a Network

Modern economies rely heavily on their infrastructure networks. These networks face threats ranging from natural disasters to human attacks. As networks are pervasive, the investments needed to protect them are very large; this motivates the study of targeted defence. What are the ‘key’ nodes to defend to maximize functionality of the network? What are the incentives of individual nodes to protect themselves in a networked environment and how do these incentives correspond to collective welfare? We provide a characterization of equilibrium attack and defence in terms of two classical concepts in graph theory – separators and transversals. We study the welfare costs of decentralized defence. We apply our results to the defence of the US Airport Network and the London Underground.

Contagion Risk and Network Design

Individuals derive benefits from their connections, but these may, at the same time, transmit external threats. Individuals therefore invest in security to protect themselves. However, the incentives to invest in security depend on their network exposures. We study the problem of designing a network that provides the right individual incentives. Motivated by cybersecurity, we first study the situation where the threat to the network comes from an intelligent adversary. We show that, by choosing the right topology, the designer can bound the welfare costs of decentralized protection. Both over-investment as well as under-investment can occur depending on the costs of security. At low costs, over-protection is important: this is addressed by disconnecting the network into two unequal components and sacrificing some nodes. At high costs, under-protection becomes salient: it is addressed by disconnecting the network into equal components. Motivated by epidemiology, we then turn to the study of random attacks. The over-protection problem is no longer present, whereas under-protection problems is mitigated in a diametrically opposite way: namely, by creating dense networks that expose the individuals to the risk of contagion.

Complexity of a theory of collective attitudes in teamwork

Our previous research presents a methodology of cooperative problem solving for BDI systems, based on a complete formal theory. This covers both a static part, defining individual, bilateral and collective agent attitudes, and a dynamic part, describing system reconfiguration in a dynamic, unpredictable environment. In this paper, we investigate the complexity of the satisfiability problem of the static part of our theory, focusing on individual and collective attitudes up to collective intention. Our logics for teamwork are squarely multi-modal, in the sense that different operators are combined and may interfere. One might expect that such a combination is much more complex than the basic multi-agent logic with one operator, but in fact we show that the individual part of our theory of teamwork is PSPACE-complete. The full system, modeling a subtle interplay between individual and group attitudes, turns out to be EXPTlME-complete, and remains so even if propositional dynamic logic is added to it.