Linear algebraic structure of zero-determinant strategies in repeated games

Zero-determinant (ZD) strategies, a recently found novel class of strategies in repeated games, has attracted much attention in evolutionary game theory. A ZD strategy unilaterally enforces a linear relation between average payoffs of players. Although existence and evolutional stability of ZD strategies have been studied in simple games, their mathematical properties have not been well-known yet. For example, what happens when more than one players employ ZD strategies have not been clarified. In this paper, we provide a general framework for investigating situations where more than one players employ ZD strategies in terms of linear algebra. First, we theoretically prove that a set of linear relations of average payoffs enforced by ZD strategies always has solutions, which implies that incompatible linear relations are impossible. Second, we prove that linear payoff relations are independent of each other under some conditions. These results hold for general games with public monitoring including perfect-monitoring games. Furthermore, we provide a simple example of a two-player game in which one player can simultaneously enforce two linear relations, that is, simultaneously control her and her opponent's average payoffs. All of these results elucidate general mathematical properties of ZD strategies.Comment: 19 pages, 2 figure

Replica analysis of Franz-Parisi potential for sparse systems

We propose a method for calculating the Franz-Parisi potential for spin glass models on sparse random graphs using the replica method under the replica symmetric ansatz. The resulting self-consistent equations have the solution with the characteristic structure of multi-body overlaps, and the self-consistent equations under this solution are equivalent to the one-step replica symmetry breaking (1RSB) cavity equation with Parisi parameter $x=1$. This method is useful for the evaluation of transition temperatures of the $p$-spin model on regular random graphs under a uniform magnetic field.Comment: 21 pages, 3 figure

Replica symmetry breaking in trajectories of a driven Brownian particle

We study a Brownian particle passively driven by a field obeying the noisy Burgers equation. We demonstrate that the system exhibits replica symmetry breaking in the path ensemble with the initial position of the particle being fixed. The key step of the proof is that the path ensemble with a modified boundary condition can be exactly mapped to the canonical ensemble of directed polymers.Comment: 12 pages, 16 figure

Calculation of 1RSB transition temperature of spin glass models on regular random graphs under the replica symmetric ansatz

We study $p$-spin glass models on regular random graphs. By analyzing the Franz-Parisi potential with a two-body cavity field approximation under the replica symmetric ansatz, we obtain a good approximation of the 1RSB transition temperature for $p=3$. Our calculation method is much easier than the 1RSB cavity method because the result is obtained by solving self-consistent equations with Newton's method.Comment: 21 pages, 1 figur

Application of zero-determinant strategies to particle control in statistical physics

Zero-determinant strategies are a class of strategies in repeated games which unilaterally control payoffs. Zero-determinant strategies have attracted much attention in studies of social dilemma, particularly in the context of evolution of cooperation. So far, not only general properties of zero-determinant strategies have been investigated, but zero-determinant strategies have been applied to control in the fields of information and communications technology and analysis of imitation. Here, we provide another example of application of zero-determinant strategies: control of a particle on a lattice. We first prove that zero-determinant strategies, if exist, can be implemented by some one-dimensional transition probability. Next, we prove that, if a two-player game has a non-trivial potential function, a zero-determinant strategy exists in its repeated version. These two results enable us to apply the concept of zero-determinant strategies to control the expected potential energies of two coordinates of a particle on a two-dimensional lattice.Comment: 9 pages, 2 figure

Numerical calculation of overlap distribution of (2+1)(2+1)-dimensional directed polymer in random media

We investigate $(2+1)$-dimensional discretized directed polymers in Gaussian random media. By numerically calculating the probability distribution function of overlap between two independent and identical systems on a common random potential, we show that there is no replica symmetry breaking. We also show that while the mean-squared displacement of one polymer end exhibits superdiffusive behavior, the mean-squared relative distance of two polymer ends exhibits subdiffusive behavior.Comment: 8 pages, 4 figure

Common knowledge equilibrium of Boolean securities in distributed information market

We investigate common knowledge equilibrium of separable (or parity) and totally symmetric Boolean securities in distributed information market. We theoretically show that clearing price converges to the true value when a common prior probability distribution of information of each player satisfies some conditions.Comment: 14 page

Absolute negative mobility in evolution

We investigate a population-genetic model with a temporally-fluctuating sawtooth fitness landscape. We numerically show that a counter-intuitive behavior occurs where the rate of evolution of the system decreases as selection pressure increases from zero. This phenomenon is understood by analogy with absolute negative mobility in particle flow. A phenomenological explanation about the direction of evolution is also provided