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An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, Fourier transformation of cooperative games becomes meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context.
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The spreading dynamics of an epidemic and the collective behavioral pattern of the population over which it spreads are deeply intertwined and the latter can critically shape the outcome of the former. Motivated by this, we design a parsimonious game-theoretic behavioral--epidemic model, in which an interplay of realistic factors shapes the co-evolution of individual decision-making and epidemics on a network. Although such a co-evolution is deeply intertwined in the real-world, existing models schematize population behavior as instantaneously reactive, thus being unable to capture human behavior in the long term. Our model offers a unified framework to model and predict complex emergent phenomena, including successful collective responses, periodic oscillations, and resurgent epidemic outbreaks. The framework also allows to assess the effectiveness of different policy interventions on ensuring a collective response that successfully eradicates the outbreak. Two case studies, inspired by real-world diseases, are presented to illustrate the potentialities of the proposed model.
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Low transaction throughput and poor scalability are significant issues in public blockchain consensus protocols such as Bitcoins. Recent research efforts in this direction have proposed shard-based consensus protocols where the key idea is to split the transactions among multiple committees (or shards), which then process these shards or set of transactions in parallel. Such a parallel processing of disjoint sets of transactions or shards by multiple committees significantly improves the overall scalability and transaction throughout of the system. However, one significant research gap is a lack of understanding of the strategic behavior of rational processors within committees in such shard-based consensus protocols. Such an understanding is critical for designing appropriate incentives that will foster cooperation within committees and prevent free-riding. In this paper, we address this research gap by analyzing the behavior of processors using a game-theoretic model, where each processor aims at maximizing its reward at a minimum cost of participating in the protocol. We first analyze the Nash equilibria in an N-player static game model of the sharding protocol. We show that depending on the reward sharing approach employed, processors can potentially increase their payoff by unilaterally behaving in a defective fashion, thus resulting in a social dilemma. In order to overcome this social dilemma, we propose a novel incentive-compatible reward sharing mechanism to promote cooperation among processors. Our numerical results show that achieving a majority of cooperating processors (required to ensure a healthy state of the blockchain network) is easier to achieve with the proposed incentive-compatible reward sharing mechanism than with other reward sharing mechanisms.
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