Existence and Generic Stability of Strong Noncooperative Equilibria of Vector-Valued Games

In this paper, we obtain an existence theorem of general strong noncooperative equilibrium point of vector-valued games, in which every player maximizes all goals. We also obtain an existence theorem of strong equilibrium point of vector-valued games with single-leader–multi-follower framework by using the upper semicontinuous of parametric strong noncooperative equilibrium point set of the followers. Moreover, we obtain some results on the generic stability of general strong noncooperative equilibrium point vector-valued games.

Coordination Games on Weighted Directed Graphs

We study strategic games on weighted directed graphs, where each player’s payoff is defined as the sum of the weights on the edges from players who chose the same strategy, augmented by a fixed nonnegative integer bonus for picking a given strategy. These games capture the idea of coordination in the absence of globally common strategies. We identify natural classes of graphs for which finite improvement or coalition-improvement paths of polynomial length always exist, and consequently a (pure) Nash equilibrium or a strong equilibrium can be found in polynomial time. The considered classes of graphs are typical in network topologies: simple cycles correspond to the token ring local area networks, whereas open chains of simple cycles correspond to multiple independent rings topology from the recommendation G.8032v2 on Ethernet ring protection switching. For simple cycles, these results are optimal in the sense that without the imposed conditions on the weights and bonuses, a Nash equilibrium may not even exist. Finally, we prove that determining the existence of a Nash equilibrium or of a strong equilibrium is NP-complete already for unweighted graphs, with no bonuses assumed. This implies that the same problems for polymatrix games are strongly NP-hard.

Projections of H-mode access and edge pedestal in the SPARC tokamak

In order to inform core performance projections and divertor design, the baseline SPARC tokamak plasma discharge is evaluated for its expected H-mode access, pedestal pressure and edge-localized mode (ELM) characteristics. A clear window for H-mode access is predicted for full field DT plasmas, with the available 25 MW of design auxiliary power. Additional alpha heating is likely needed for H-mode sustainment. Pressure pedestal predictions in the developed H-mode are surveyed using the EPED model. The projected SPARC pedestal would be limited dominantly by peeling modes and may achieve pressures in excess of 0.3 MPa at a density of approximately 3 × 1020 m−3. High pedestal pressure is partially enabled by strong equilibrium shaping, which has been increased as part of recent design iterations. Edge-localized modes (ELMs) with >1 MJ of energy are projected, and approaches for reducing the ELM size, and thus the peak energy fluence to divertor surfaces, are under consideration. The high pedestal predicted for SPARC provides ample margin to satisfy its high fusion gain (Q) mission, so that even if ELM mitigation techniques result in a 2× reduction of the pedestal pressure, Q > 2 is still predicted.

Diagrammatic Theory of RNA Structures and Ensembles with Trinucleotide Repeats

ABSTRACTTrinucleotide repeat expansion disorders (TRED) are associated with the overexpansion of (CNG) repeats on the genome. mRNA transcripts of sequences with greater than 60 to 100 (CNG) tandem units have been implicated in TRED pathogenesis. In this paper, we develop a diagrammatic theory to study the structural diversity of these (CNG)n RNA sequences. Representing structural elements on the chain’s conformation by a set of graphs and employing elementary diagrammatic methods, we have formulated a renormalization procedure to resum these graphs and arrive at a closed-form expression for the ensemble partition function. With a simple approximation for the renormalization and applied to extended (CNG)n sequences, this theory can comprehensively capture an infinite set of conformations with any number and any combination of duplexes, hairpins and 2-way junctions. To quantify the diversity of different (CNG)n ensembles, the analytical equations derived from the diagrammatic theory were solved numerically to derive equilibrium estimates for the secondary structural contents of the chains. The results suggest that the structural ensembles of (CNG)n repeat sequence with n ~ 60 are surprisingly diverse, and they are dominated largely by open segments, with only a small fraction of the nucleotides forming base pairs. At the same time, the variance in the secondary-structural contents on the chains is also quite large, indicating that their structures can undergo strong equilibrium fluctuations and are expected to be rather suspectable to perturbations.STATEMENT OF SIGNIFICANCETrinucleotide repeat expansion disorders (TRED) are associated with the overexpansion of (CNG) repeats on the genome. mRNA transcripts of sequences with critical length greater than 60 to 100 (CNG) tandem units have been implicated in TRED pathogenesis, though their structures remain poorly characterized. Conventional view has tacitly assumed that conformations with maximal C:G base pairing dominate at equilibrium, but here we demonstrate that (CNG) repeat sequences are characterized by diverse ensembles of structurally heterogeneous folds and with a large variance of secondary structural contents. These ensembles of structures also undergo strong equilibrium fluctuations, rendering them rather susceptible to perturbations. These results were based on a novel diagrammatic approach to the ensemble partition function.

Strong Strategic Support of Cooperation in Multistage Games

The problem of cooperation in repeated and multistage games is considered. The strong equilibrium (equilibrium stable against deviations of coalitions) with payoffs which can be attained under cooperation is constructed for a wide class of such games. The new solution concept based on solutions of stage games is introduced and in some cases this solution is a subset of the core defined for repeated and multistage games in a classical way. It is also proved that this newly introduced solution concept is strongly time consistent. The strong time consistency of the solution is a very important property since in case it does not take place players in reality in some time instant in subgame on cooperative trajectory may switch from the previously selected optimal solution to any other optimal solution in the subgame and as result realize the solution which will not be optimal in the whole game.

Synchronisation Games on Hypergraphs

We study a strategic game model on hypergraphs where players, modelled by nodes, try to coordinate or anti-coordinate their choices within certain groups of players, modelled by hyperedges. We show this model to be a strict generalisation of symmetric additively separable hedonic games to the hypergraph setting and that such games always have a pure Nash equilibrium, which can be computed in pseudo-polynomial time. Moreover, in the pure coordination setting, we show that a strong equilibrium exists and can be computed in polynomial time when the game possesses a certain acyclic structure.