scholarly journals Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games

2020 ◽  
Vol 13 ◽  
pp. 218-243
Author(s):  
Nikolay A. Krasovskii ◽  
◽  
Alexander M. Tarasyev ◽  
◽  
Keyword(s):  
Nash Equilibrium ◽  
Viscosity Solutions ◽  
Dynamic Equilibrium ◽  
Evolutionary Game ◽  
Value Functions ◽  
Bimatrix Games ◽  
Qualitative Properties ◽  
Minimax Solutions ◽  
Jacobi Equations

In the paper, constructions of the generalized method of characteristics are applied for calculating the generalized minimax (viscosity) solutions of Hamilton-Jacobi equations in dynamic bimatrix games. The structure of the game presumes interactions of two players in the framework of the evolutionary game model. Stochastic contacts between players occur according to the dynamic process, which can be interpreted as a system of Kolmogorov's differential equations with controls instead of probability parameters. It is assumed that control parameters are not fixed and can be constructed by the feedback principle. Two types of payoff functions are considered: short-term payoffs are determined in the current moments of time, and long-term payoffs are determined as limit functionals on the infinite time horizon. The notion of dynamic Nash equilibrium in the class of controlled feedbacks is considered for the long-term payoffs. In the framework of constructions of dynamic equilibrium, the solutions are designed on the basis of maximization of guaranteed payoffs. Such guaranteeing strategies are built in the framework of the theory of minimax (viscosity) solutions of Hamilton-Jacobi equations. The analytical formulas are obtained for the value functions in the cases of different orientations for the “zigzags” (broken lines) of acceptable situations in the static game. The equilibrium trajectories generated by the minimax solutions shift the system in the direction of cooperative Pareto points. The proposed approach provides new qualitative properties of the equilibrium trajectories in the dynamic bimatrix games which guarantee better results of payoffs for both players than static Nash equilibria. As an example, interactions of two firms on the market of innovative electronic devices are examined within the proposed approach for treating dynamic bimatrix games.

Game Theory

2016 ◽  
Author(s):  
Cristina Bicchieri ◽  
Giacomo Sillari
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Evolutionary Game Theory ◽  
Repeated Games ◽  
Evolutionary Game ◽  
Complete Information ◽  
Decision Makers ◽  
Correlated Equilibrium ◽  
Extensive Form

Game theory aims to understand situations in which decision-makers interact strategically. Chess is an example, as are firms competing for business, politicians competing for votes, animals fighting over prey, bidders competing in auctions, threats and punishments in long-term relationships, and so on. In such situations, the outcome depends on what the parties do jointly. Decision-makers may be people, organizations, animals, or even genes. In this chapter, the authors review fundamental notions of game theory and their application to philosophy of science. In particular, Section 1 looks at games of complete information through normal and extensive form representations, introduce the notion of Nash equilibrium and its refinements. Section 2 touches on epistemic foundations and correlated equilibrium, and Section 3 examines repeated games and their importance for the analysis of altruism and cooperation. Section 4 deals with evolutionary game theory.

scholarly journals VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

2007 ◽  
Vol 04 (04) ◽  
pp. 771-795 ◽  
Author(s):  
GIUSEPPE MARIA COCLITE ◽  
NILS HENRIK RISEBRO
Keyword(s):  
Cauchy Problem ◽  
Viscosity Solution ◽  
Viscosity Solutions ◽  
Front Tracking ◽  
Discontinuous Coefficients ◽  
Comparison Result ◽  
The Cauchy Problem ◽  
Temporal Location ◽  
Additional Regularity ◽  
Jacobi Equations

We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main result is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to "internal boundaries". The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by "viscosity solution" we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.

scholarly journals Long-term persistence of crAss-like phage crAss001 is associated with phase variation in Bacteroides intestinalis

2020 ◽  
Author(s):  
Andrey N Shkoporov ◽  
Ekaterina V Khokhlova ◽  
Niamh Stephens ◽  
Cara Hueston ◽  
Samuel Seymour ◽  
...  
Keyword(s):  
Dynamic Equilibrium ◽  
Phase Variation ◽  
Carrier State ◽  
Human Gut ◽  
Rapid Phase ◽  
Host Interaction ◽  
Infected Cells ◽  
Phage Sensitivity

The crAss-like phages are ubiquitous and highly abundant members of the human gut virome that infect commensal bacteria of the order Bacteroidales. Although incapable of classical lysogeny, these viruses demonstrate unexplained long-term persistence in the human gut microbiome, dominating the virome in some individuals. Here we demonstrate that rapid phase variation of alternate capsular polysaccharides plays an important role in dynamic equilibrium between phage sensitivity and resistance in B. intestinalis cultures, allowing phage and bacteria to multiply in parallel. The data also suggests the role of concomitant phage persistence mechanisms associated with delayed lysis of infected cells, such as carrier state infection. From an ecological and evolutionary standpoint this type of phage-host interaction is consistent with the Piggyback-the-Winner model, which suggests a preference towards lysogenic or other 'benign' forms of phage infection when the host is stably present at high abundance.

scholarly journals Semidefinite Programming and Nash Equilibria in Bimatrix Games

2020 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Jeffrey Zhang
Keyword(s):  
Nash Equilibrium ◽  
Semidefinite Programming ◽  
Nash Equilibria ◽  
Valid Inequalities ◽  
Bimatrix Games ◽  
Competitive Game ◽  
Approximate Nash Equilibria ◽  
Hard Problems ◽  
Rank 2

We explore the power of semidefinite programming (SDP) for finding additive ɛ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact Nash equilibrium can be recovered. We show that, for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on the iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most 2 and ɛ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a [Formula: see text]-Nash equilibrium can be recovered for any game, or a [Formula: see text]-Nash equilibrium for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any Nash equilibrium, and testing whether there exists a Nash equilibrium where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.

scholarly journals Integration of Indonesian Coffee Markets with World Coffee Markets

2021 ◽  
Vol 2 (1) ◽  
pp. 33
Author(s):  
Haposan Orlando Napitupulu ◽  
Ana Arifatus Sa'diyah ◽  
Farah Mutiara
Keyword(s):  
Time Series Data ◽  
Dynamic Equilibrium ◽  
Secondary Data ◽  
Series Data ◽  
Short Term ◽  
Robusta Coffee ◽  
Long Run ◽  
Relationship Of ◽  
Coffee Markets

This study aims to analyze the integration of the Arabica and Robusta coffee markets in Indonesia with world coffee prices. The study uses secondary data in the form of annual time series data during the period 1985 - 2015. The study uses the VECM analysis method. This method explains the relationship of long-term dynamic equilibrium and short-term equilibrium in a system of equations. The analysis shows that Indonesian and world Arabica coffee is not integrated in the long term or the short term. In Robusta coffee VECM estimation analysis shows that there is a significant value at the 10% level in a long-term relationship with a value of 0.08579, which means that there is a short-term relationship between world Robusta coffee prices and domestic Robusta coffee prices in the previous year, but no relationship in the long run.

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