scholarly journals Dynamics of Strategy Distributions in a One-Dimensional Continuous Trait Space for Games with a Quadratic Payoff Function

Games ◽  
2020 ◽  
Vol 11 (1) ◽  
pp. 14
Author(s):  
Georgiy Karev
Keyword(s):  
Broad Class ◽  
Limit State ◽  
Single Point ◽  
Initial Distribution ◽  
Payoff Function ◽  
Quadratic Term ◽  
Uniform Distributions ◽  
Entire Line ◽  
Payoff Functions ◽  
Initial Distributions

Evolution of distribution of strategies in game theory is an interesting question that has been studied only for specific cases. Here I develop a general method to extend analysis of the evolution of continuous strategy distributions given a quadratic payoff function for any initial distribution in order to answer the following question—given the initial distribution of strategies in a game, how will it evolve over time? I look at several specific examples, including normal distribution on the entire line, normal truncated distribution, as well as exponential and uniform distributions. I show that in the case of a negative quadratic term of the payoff function, regardless of the initial distribution, the current distribution of strategies becomes normal, full or truncated, and it tends to a distribution concentrated in a single point so that the limit state of the population is monomorphic. In the case of a positive quadratic term, the limit state of the population may be dimorphic. The developed method can now be applied to a broad class of questions pertaining to evolution of strategies in games with different payoff functions and different initial distributions.

An extension of a convergence theorem for Markov chains arising in population genetics

2016 ◽  
Vol 53 (3) ◽  
pp. 953-956 ◽  
Author(s):  
Martin Möhle ◽  
Morihiro Notohara
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Convergence Theorem ◽  
Transition Matrix ◽  
Initial Distribution ◽  
Generator Matrix ◽  
Finite Dimensional ◽  
Finite State ◽  
Initial Distributions ◽  
Finite State Space

AbstractAn extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (XN(r))r be a Markov chain with the same finite state space S and transition matrix ΠN=I+dNBN, where I is the unit matrix, Q a generator matrix, (BN)N a sequence of matrices, limN℩∞cN= limN→∞dN=0 and limN→∞cN∕dN=0. Suppose that the limits P≔limm→∞(I+dNQ)m and G≔limN→∞PBNP exist. If the sequence of initial distributions PXN(0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (XN([t∕cN))t≥0 converge to those of the Markov process (Xt)t≥0 with initial distribution μ, transition matrix PetG and limN→∞(I+dNQ+cNBN)[t∕cN]

scholarly journals Robust predictions for the large-scale cosmological power deficit from primordial quantum nonequilibrium

2016 ◽  
Vol 25 (06) ◽  
pp. 1650068 ◽  
Author(s):  
Samuel Colin ◽  
Antony Valentini
Keyword(s):  
Large Scale ◽  
Initial Conditions ◽  
Broad Class ◽  
Initial Distribution ◽  
Born Rule ◽  
Nonequilibrium Distribution ◽  
Pilot Wave ◽  
Inflationary Phase ◽  
Power Deficit ◽  
Energy States

The de Broglie–Bohm pilot-wave formulation of quantum theory allows the existence of physical states that violate the Born probability rule. Recent work has shown that in pilot-wave field theory on expanding space relaxation to the Born rule is suppressed for long-wavelength field modes, resulting in a large-scale power deficit [Formula: see text] which for a radiation-dominated expansion is found to have an approximate inverse-tangent dependence on [Formula: see text] (assuming that the width of the initial distribution is smaller than the width of the initial Born-rule distribution and that the initial quantum states are evenly-weighted superpositions of energy states). In this paper, we show that the functional form of [Formula: see text] is robust under changes in the initial nonequilibrium distribution — subject to the limitation of a subquantum width — as well as under the addition of an inflationary era at the end of the radiation-dominated phase. In both cases, the predicted deficit [Formula: see text] remains an inverse-tangent function of [Formula: see text]. Furthermore, with the inflationary phase the dependence of the fitting parameters on the number of superposed pre-inflationary energy states is comparable to that found previously. Our results indicate that, for the assumed broad class of initial conditions, an inverse-tangent power deficit is likely to be a fairly general and robust signature of quantum relaxation in the early universe.

scholarly journals Lipschitz Continuity and Approximate Equilibria

Algorithmica ◽  
2020 ◽  
Vol 82 (10) ◽  
pp. 2927-2954
Author(s):  
Argyrios Deligkas ◽  
John Fearnley ◽  
Paul Spirakis
Keyword(s):  
Polynomial Time ◽  
Lipschitz Continuity ◽  
Payoff Function ◽  
Polynomial Algorithms ◽  
Continuous Action ◽  
Lipschitz Continuous ◽  
Payoff Functions ◽  
Action Spaces ◽  
Best Responses ◽  
Strongly Polynomial

Abstract In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in $$L_1$$ L 1 and $$L_2^2$$ L 2 2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

Equilibrium probability distributions for low density highway traffic

1966 ◽  
Vol 3 (1) ◽  
pp. 247-260 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Poisson Process ◽  
Probability Distributions ◽  
Initial Distribution ◽  
Order Effects ◽  
Highway Traffic ◽  
Wide Range ◽  
Headway Distribution ◽  
Equilibrium Distributions ◽  
Initial Distributions ◽  
Special Case

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.

Different Behavioral Types of Distributional Preferences Are Characterized by Distinct Neural Signatures

2021 ◽  
pp. 1-14
Author(s):  
Katharina Koch ◽  
Lorena R. R. Gianotti ◽  
Jan Hausfeld ◽  
Mirjam Studler ◽  
Daria Knoch
Keyword(s):  
Inferior Frontal Gyrus ◽  
Initial Distribution ◽  
Behavioral Types ◽  
Distributional Preferences ◽  
Different Types ◽  
Clustering Approach ◽  
Neural Underpinnings ◽  
The Right ◽  
Initial Distributions ◽  
Neural Signatures

Abstract There are many situations where resources are distributed between two parties and where the deciding party has information about the initial distribution and can change its outcome, for example, the allocation of budget for funds or bonuses, where the deciding party might have self-interested motives. Although the neural underpinnings of distributional preferences of resources have been extensively studied, it remains unclear if there are different types of distributional preferences and if these types underlie different disposing neural signatures. We used source-localized resting EEG in combination with a data-driven clustering approach to participants' behavior in a distribution game in order to disentangle the neural sources of the different types of distributional preferences. Our findings revealed four behavioral types: Maximizing types always changed initial distributions to maximize their personal outcomes, and compliant types always left initial distributions unchanged. Disadvantage-averse types only changed initial distributions if they received less than the other party did, and equalizing types primarily changed initial distributions to fair distributions. These behavioral types differed regarding neural baseline activation in the right inferior frontal gyrus. Maximizing and compliant types showed the highest baseline activation, followed by disadvantage-averse types and equalizing types. Furthermore, maximizing types showed significantly higher baseline activation in the left OFC compared to compliant types. Taken together, our findings show that different types of distributional preferences are characterized by distinct neural signatures, which further imply differences in underlying psychological processes in decision-making.

Growth rate for free-electron lasers through a warm beam layered model

2016 ◽  
Vol 82 (3) ◽  
Author(s):  
E. Peter ◽  
F. B. Rizzato ◽  
A. Endler
Keyword(s):  
Growth Rate ◽  
Free Electron ◽  
Thermal Effects ◽  
Linear Growth ◽  
Initial Distribution ◽  
Linear Growth Rate ◽  
One Dimensional ◽  
Linear Set ◽  
Particle Simulations ◽  
Initial Distributions

In the present work, we describe the linear growth rate of the laser field for a one-dimensional theoretical single-pass free-electron laser, including space-charge and thermal effects, in the hydrodynamical regime. In a recent work (Peter, Endler & Rizzato, Phys. Plasmas, vol. 21, 2014, 113104), the thermal effects were already included for a water-bag initial distribution for the longitudinal velocities of the particles of the beam. Here, we extend the result for different and symmetrical initial distributions, considering that in the hydrodynamical regime, the beam can be thought of as a warm fluid composed of a sum of different fluids with different densities, where the initial distribution of each fluid is a water-bag distribution. The total pressure of the beam is related to the sum of the pressures of these fluids. This approach is much less complicated than the kinetic approach. We compare the results given by the linear set of equations and wave–particle simulations for water-bag and Gaussian initial distributions. The evolution of the particle distribution in the phase space is also shown in order to demonstrate that the assumption of the sum of different fluids reproduces the physics of the system in a reasonable fashion.

Age-Dependent Payoffs and Assortative Matching by Age in a Market with Search

2017 ◽  
Vol 9 (2) ◽  
pp. 263-294
Author(s):  
Anja Sautmann
Keyword(s):  
Sufficient Conditions ◽  
Payoff Function ◽  
Assortative Matching ◽  
Constant Flow ◽  
Age At Marriage ◽  
Matching Market ◽  
Age Dependent ◽  
Finite Period ◽  
Payoff Functions ◽  
Increasing Functions

This paper considers a matching market with two-sided search and transferable utility where match payoffs depend on age at marriage (time until match) and search is finite. We define and prove existence of equilibrium, and provide sufficient conditions for positive assortative matching that build on restricting the slope and curvature of the marriage payoff function to generate single-peaked preferences in age and therefore convex matching sets. Payoff functions that are incompatible with positive sorting by age include all strictly increasing functions and constant flow payoffs enjoyed for some finite period. (JEL C78, D83, J12)

Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

1991 ◽  
Vol 23 (4) ◽  
pp. 683-700 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
State Space ◽  
Spectral Representation ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Death Process ◽  
Finite Support ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Initial Distributions ◽  
Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

ON A CLASS OF NASH EQUILIBRIA WITH MEMORY STRATEGIES FOR NONZERO-SUM DIFFERENTIAL GAMES

2000 ◽  
Vol 02 (02n03) ◽  
pp. 173-192 ◽  
Author(s):  
JEAN MICHEL COULOMB ◽  
VLADIMIR GAITSGORY
Keyword(s):  
Nash Equilibrium ◽  
Feedback Control ◽  
Differential Games ◽  
Differential Game ◽  
Nash Equilibria ◽  
Payoff Function ◽  
Feedback Controls ◽  
Memory Strategies ◽  
Payoff Functions ◽  
With Memory

A two-player nonzero-sum differential game is considered. Given a pair of threat payoff functions, we characterise a set of pairs of acceptable feedback controls. Any such pair induces a history-dependent Nash δ-equilibrium as follows: the players agree to use the acceptable controls unless one of them deviates. If this happens, a feedback control punishment is implemented. The problem of finding a pair of "acceptable" controls is significantly simpler than the problem of finding a feedback control Nash equilibrium. Moreover, the former may have a solution in case the latter does not. In addition, if there is a feedback control Nash equilibrium, then our technique gives a subgame perfect Nash δ-equilibrium that might improve the payoff function for at least one player.

scholarly journals Ensemble filter based estimation of spatially distributed parameters in a mesoscale dust model: experiments with simulated and real data

2013 ◽  
Vol 13 (6) ◽  
pp. 3481-3500 ◽  
Author(s):  
V. M. Khade ◽  
J. A. Hansen ◽  
J. S. Reid ◽  
D. L. Westphal
Keyword(s):  
Surface Wind ◽  
Real Data ◽  
Initial Distribution ◽  
Observation System ◽  
Test Bed ◽  
Aerosol Model ◽  
Deep Blue ◽  
Surface Wind Stress ◽  
Grid Points ◽  
Initial Distributions

Abstract. The ensemble adjustment Kalman filter (EAKF) is used to estimate the erodibility fraction parameter field in a coupled meteorology and dust aerosol model (Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS)) over the Sahara desert. Erodibility is often employed as the key parameter to map dust source. It is used along with surface winds (or surface wind stress) to calculate dust emissions. Using the Saharan desert as a test bed, a perfect model Observation System Simulation Experiments (OSSEs) with 40 ensemble members, and observations of aerosol optical depth (AOD), the EAKF is shown to recover correct values of erodibility at about 80% of the points in the domain. It is found that dust advected from upstream grid points acts as noise and complicates erodibility estimation. It is also found that the rate of convergence is significantly impacted by the structure of the initial distribution of erodibility estimates; isotropic initial distributions exhibit slow convergence, while initial distributions with geographically localized structure converge more quickly. Experiments using observations of Deep Blue AOD retrievals from the MODIS satellite sensor result in erodibility estimates that are considerably lower than the values used operationally. Verification shows that the use of the tuned erodibility field results in better predictions of AOD over the west Sahara and the Arabian Peninsula.

Sign in / Sign up

Export Citation Format

Share Document