Schmidt’s game on fractals

We construct (α, β) and α-winning sets in the sense of Schmidt’s game, played on the support of certain measures (absolutely friendly) and show how to compute the Hausdorff dimension for some. In particular, we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of ℝ ^N satisfying the open set condition, (the Cantor’s ternary set, Koch’s curve and Sierpinski’s gasket to name a few known examples), then for any countable collection of non-singular affine transformations, Δ _i : ℝ ^N → ℝ ^N ,

Pre-commitment or post-payment: Which worsens a line-sitting firm’s revenue?

Recently, Cui et al. (2018) investigated the economics of line-sitting, where a customer could hire a line-sitter to wait for her, and the line-sitting fee is paid at the completion of service. Usually, it is believed that the line-sitting firm benefits in the pre-commitment payment method, in which the customer still pays a pre-specified amount of fee for service even if the actual waiting time is less than the pre-committed wait. By comparing two payment mechanisms, we find that, surprisingly, under the optimal line-sitting rate, the revenue of firm worsens under pre-commitment case, but a better customer welfare can be obtained.     

An attrition game on a network ruled by Lanchester’s square law

We consider two-person zero-sum attrition games in which an attacker and a defender are in combat with each other on a network. The attacker marches from a starting node to a destination node, hoping that the initial members survive the march. The defender deploys his forces on arcs in order to intercept the attacker. If the attacker encounters the defender on an arc, the attacker incurs casualties according to Lanchester’s square law. We consider two models: a one-shot game in which the two players have no information about their opponents, and a two-stage game in which both players have some information about their opponents. For both games, the payoff is defined as the number of survivors for the attacker. The attacker’s strategy is to choose a path, and the defender’s is to deploy the defending forces on arcs. We propose a numerical algorithm, in which nonlinear programming is embedded, to derive the equilibrium of the game.     

Considering individual satisfaction levels enhances cooperation in a spatial prisoner’s dilemma game

Based on the observance in human society, the satisfaction level of an individual as a result of an obtained payoff depends on personal tendency to some extent; we establish a new model for spatial prisoner’s dilemma games. We describe individual satisfaction as a stochastically deviated value around each of the four payoffs stipulated by a payoff matrix, which is maintained throughout the life of a certain agent. When strategy updating, an agent who refers to his own satisfaction level cannot see neighbors’ satisfaction levels but can only observe neighbors’ accumulated payoffs. By varying the update rule and underlying topology, we perform numerical simulations that reveal cooperation is significantly enhanced by this change. We argue that this enhancement of cooperation is analogous to a stochastic resonance effect, like the payoff noise effects Perc (2006).     

A bijection for covered maps, or a shortcut between Harer-Zagier?s and Jackson?s formulas

We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g) with a marked unicellular spanning submap (which can have any genus in {0,1,…,g}). Our main result is a bijection between covered maps with n edges and genus g and pairs made of a plane tree with n edges and a unicellular bipartite map of genus g with n+1 edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in Bernardi (2007) [4].Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps.We also show that the bijection of Bouttier, Di Francesco and Guitter (2004) [8] (which generalizes a previous bijection by Schaeffer, 1998 [33]) between bipartite maps and so-called well-labeled mobiles can be obtained as a special case of our bijection.     

Staircase rook polynomials and Cayley’s game of Mousetrap

First introduced by Arthur Cayley in the 1850’s, the game of Mousetrap involves removing cards from a deck according to a certain rule. In this paper we find the rook polynomial for the number of Mousetrap decks in which at least two specified cards are removed. We also find a new expression for the rook polynomial for the number of decks in which exactly one specified card is removed and give expressions for counts of two kinds of Mousetrap decks in terms of other known combinatorial numbers.     

An optical model for implementing Parrondo’s game and designing stochastic game with long-term memory

An optical model for a photon propagating through a designed array of beam splitters is developed to give a physical implementation of Parrondo’s game and Parrondo’s history-dependent game. The winner in this optical model is a photon passed the beam splitter. The loser is a photon being reflected by the beam splitter. The optical beam splitter is the coin-tosser. We designed new games with long-term memory by using this optical diagram method. The optical output of the combined game of two losing games could be a win, or a loss, or an oscillation between win and loss. The modern technology to implement this optical model is well developed. A circularly polarized photon is a possible candidate for this physical implementation in laboratory.     

Cooperation in spatial prisoner’s dilemma game with delayed decisions

Phenomena that time delays of information lead to delayed decisions are extensive in reality. The effect of delayed decisions on the evolution of cooperation in the spatial prisoner’s dilemma game is explored in this work. Players with memory are located on a two dimensional square lattice, and they can keep the payoff information of his neighbors and his own in every historic generation in memory. Every player uses the payoff information in some generation from his memory and the strategy information in current generation to determine which strategy to choose in next generation. The time interval between two generations is set by the parameter m. For the payoff information is used to determine the role model for the focal player when changing strategies, the focal player’s decision to learn from which neighbor is delayed by m generations. Simulations show that cooperation can be enhanced with the increase of m. In addition, just like the original evolutionary game model (m = 0), pretty dynamic fractal patterns featuring symmetry can be obtained when m > 0 if we simulate the invasion of a single defector in world of cooperators on square lattice.     

Generalizations of a parameterized Jordan-type inequality,Janous’s inequality and Tsintsifas’s inequality

In this work, we first prove a generalized version of a parameterized Jordan-type inequality. We then use it to prove the generalized versions of Janous’s inequality and Tsintsifas’s inequality which reduce to two inequalities conjectured by Janous and Tsintsifas as special cases.     

Modelling ‘contracts’ between a terrorist group and a government in a sequential game

In this paper, we apply a sequential game to study the possibility of ‘contracts’ (or at least mutually beneficial arrangements) between a government and a terrorist group. We find equilibrium solutions for complete and incomplete information models, where the government defends and/or provides positive rent, and the terrorist group attacks. We also study the sensitivities of equilibria as a function of both players’ target valuations and preferences for rent. The contract option, if successful, may achieve (partial) attack deterrence, and significantly increase the payoffs not only for the government, but also for some types of terrorist groups. Our work thus provides some novel insights in combating terrorism.     

Newton’s method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation

We consider the generalized Nash equilibrium problem (GNEP), where not only the players’ cost functions but also their strategy spaces depend on the rivals’ decision variables. Existence results for GNEPs are typically shown by using a fixed point argument for a certain set-valued function. Here we use a regularization of this set-valued function in order to obtain a single-valued function that is easier to deal with from a numerical point of view. We show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria. This fixed point formulation is then used to develop a nonsmooth Newton method for computing a normalized equilibrium. The method uses a so-called computable generalized Jacobian that is much easier to compute than Clarke generalized Jacobian or B-subdifferential. We establish local superlinear/quadratic convergence of the method under the constant rank constraint qualification, which is weaker than the frequently used linear independence constraint qualification, and a suitable second-order condition. Some numerical results are presented to illustrate the performance of the method.     

Cooperation in an evolutionary prisoner’s dilemma game with probabilistic strategies

In this work, we investigate an evolutionary prisoner’s dilemma game in structured populations with probabilistic strategies instead of the pure strategies of cooperation and defection. We explore the model in details by considering different strategy update rules and different population structures. We find that the distribution of probabilistic strategies patterns is dependent on both the interaction structures and the updating rules. We also find that, when an individual updates her strategy by increasing or decreasing her probabilistic strategy a certain amount towards that of her opponent, there exists an optimal increment of the probabilistic strategy at which the cooperator frequency reaches its maximum.     

Strategy changing penalty promotes cooperation in spatial prisoner’s dilemma game

Many classical studies suggest that punishment is a useful way to promote cooperation in the well-mixed public goods game, whereas relative evidence in the research of spatial prisoner’s dilemma game is absent. To address this issue, we introduce a mechanism of strategy changing penalty, combining memory and penalty during the update process, into spatial prisoner’s dilemma game. We find that increasing penalty rate or memory length is able to promote the evolution of cooperation monotonously. Compared with traditional version, recorded penalty could facilitate cooperation better. Moreover, through examining the process of evolution, we provide an interpretation for this promotion phenomenon, namely, the effect of promotion can be warranted by an evolution resonance of standard deviation of fitness coefficient. Finally, we validate our results by studying the impact of uncertainty within strategy adoptions on the evolution of cooperation. We hope that our work may shed light on the understanding of the cooperative behavior in the society.