Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy

Subgame perfect equilibria are specific Nash equilibria in perfect information games in extensive form. They are important because they relate to the rationality of the players. They always exist in infinite games with continuous real-valued payoffs, but may fail to exist even in simple games with slightly discontinuous payoffs. This article considers only games whose outcome functions are measurable in the Hausdorff difference hierarchy of the open sets (\textit{i.e.} $\Delta^0_2$ when in the Baire space), and it characterizes the families of linear preferences such that every game using these preferences has a subgame perfect equilibrium: the preferences without infinite ascending chains (of course), and such that for all players $a$ and $b$ and outcomes $x,y,z$ we have $\neg(z <_a y <_a x \,\wedge\, x <_b z <_b y)$. Moreover at each node of the game, the equilibrium constructed for the proof is Pareto-optimal among all the outcomes occurring in the subgame. Additional results for non-linear preferences are presented.Comment: The alternative definition of the difference hierarchy has changed slightl

Infinite sequential Nash equilibrium

In game theory, the concept of Nash equilibrium reflects the collective stability of some individual strategies chosen by selfish agents. The concept pertains to different classes of games, e.g. the sequential games, where the agents play in turn. Two existing results are relevant here: first, all finite such games have a Nash equilibrium (w.r.t. some given preferences) iff all the given preferences are acyclic; second, all infinite such games have a Nash equilibrium, if they involve two agents who compete for victory and if the actual plays making a given agent win (and the opponent lose) form a quasi-Borel set. This article generalises these two results via a single result. More generally, under the axiomatic of Zermelo-Fraenkel plus the axiom of dependent choice (ZF+DC), it proves a transfer theorem for infinite sequential games: if all two-agent win-lose games that are built using a well-behaved class of sets have a Nash equilibrium, then all multi-agent multi-outcome games that are built using the same well-behaved class of sets have a Nash equilibrium, provided that the inverse relations of the agents' preferences are strictly well-founded.Comment: 14 pages, will be published in LMCS-2011-65

Stable states of perturbed Markov chains

Given an infinitesimal perturbation of a discrete-time finite Markov chain, we seek the states that are stable despite the perturbation, \textit{i.e.} the states whose weights in the stationary distributions can be bounded away from $0$ as the noise fades away. Chemists, economists, and computer scientists have been studying irreducible perturbations built with exponential maps. Under these assumptions, Young proved the existence of and computed the stable states in cubic time. We fully drop these assumptions, generalize Young's technique, and show that stability is decidable as long as $f\in O(g)$ is. Furthermore, if the perturbation maps (and their multiplications) satisfy $f\in O(g)$ or $g\in O(f)$, we prove the existence of and compute the stable states and the metastable dynamics at all time scales where some states vanish. Conversely, if the big-$O$ assumption does not hold, we build a perturbation with these maps and no stable state. Our algorithm also runs in cubic time despite the general assumptions and the additional work. Proving the correctness of the algorithm relies on new or rephrased results in Markov chain theory, and on algebraic abstractions thereof

TRIDEnT: Building Decentralized Incentives for Collaborative Security

Sophisticated mass attacks, especially when exploiting zero-day vulnerabilities, have the potential to cause destructive damage to organizations and critical infrastructure. To timely detect and contain such attacks, collaboration among the defenders is critical. By correlating real-time detection information (alerts) from multiple sources (collaborative intrusion detection), defenders can detect attacks and take the appropriate defensive measures in time. However, although the technical tools to facilitate collaboration exist, real-world adoption of such collaborative security mechanisms is still underwhelming. This is largely due to a lack of trust and participation incentives for companies and organizations. This paper proposes TRIDEnT, a novel collaborative platform that aims to enable and incentivize parties to exchange network alert data, thus increasing their overall detection capabilities. TRIDEnT allows parties that may be in a competitive relationship, to selectively advertise, sell and acquire security alerts in the form of (near) real-time peer-to-peer streams. To validate the basic principles behind TRIDEnT, we present an intuitive game-theoretic model of alert sharing, that is of independent interest, and show that collaboration is bound to take place infinitely often. Furthermore, to demonstrate the feasibility of our approach, we instantiate our design in a decentralized manner using Ethereum smart contracts and provide a fully functional prototype.Comment: 28 page

Minkowski Games

We introduce and study Minkowski games. In these games, two players take turns to choose positions in ℝd based on some rules. Variants include boundedness games, where one player wants to keep the positions bounded (while the other wants to escape to infinity), and safety games, where one player wants to stay within a given set (while the other wants to leave it). We provide some general characterizations of which player can win such games, and explore the computational complexity of the associated decision problems. A natural representation of boundedness games yields coNP-completeness, whereas the safety games are undecidable.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

The Brouwer Fixed Point Theorem Revisited

We revisit the investigation of the computational content of the Brouwer Fixed Point Theorem in [7], and answer the two open questions from that work. First, we show that the computational hardness is independent of the dimension, as long as it is greater than 1 (in [7] this was only established for dimension greater than 2). Second, we show that restricting the Brouwer Fixed Point Theorem to L-Lipschitz functions for any L > 1 also does not change the computational strength, which together with prior results establishes a trichotomy for L > 1, L = 1 and L < 1.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

Reduction Techniques for Model Checking and Learning in MDPs

info:eu-repo/semantics/publishe

Dual-band fiber-chip grating coupler in a 300 mm silicon-on-insulator platform and 193 nm deep-UV lithography

4 pags., 5 figs., 1 tab.Surface grating couplers are fundamental building blocks for coupling the light between optical fibers and integrated photonic devices. However, the operational bandwidth of conventional grating couplers is intrinsically limited by their wavelength-dependent radiation angle. The few dual-band grating couplers that have been experimentally demonstrated exhibit low coupling efficiencies and rely on complex fabrication processes. Here we demonstrate for the first time, to the best of our knowledge, the realization of an efficient dual-band grating coupler fabricated using 193 nm deep-ultraviolet lithography for 10 Gbit symmetric passive optical networks. The footprint of the device is 17 × 10 µm. We measured coupling efficiencies of −4.9 and −5.2 dB with a 3-dB bandwidth of 27 and 56 nm at the wavelengths of 1270 and 1577 nm, corresponding to the upstream and downstream channels, respectively.Spanish Ministry of Science, Innovation and Universities (MICINN) (RTI2018-097957-B-C33, TEC2015-71127-C2-1-R with FPI Scholarship BES-2016-077798); Community of Madrid - FEDER funds (S2018/NMT-4326); Horizon 2020 Research and Innovation Program (Marie Sklodowska-Curie 734331); H2020 European Research Council (ERC POPSTAR 647342); European Commission (H2020- ICT-26127-2017 COSMICC 688516); French Industry Ministry (Nano2022 project under IPCEI program); Agence Nationale de la Recherche (ANR-MIRSPEC-17-CE09-0041)