On the expected number of equilibria in a multi-player multi-strategy evolutionary game

In this paper, we analyze the mean number $E(n,d)$ of internal equilibria in a general $d$-player $n$-strategy evolutionary game where the agents' payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next we characterize the asymptotic behavior of $E(2,d)$, estimating its lower and upper bounds as $d$ increases. Two important consequences are obtained from this analysis. On the one hand, we show that in both cases the probability of seeing the maximal possible number of equilibria tends to zero when $d$ or $n$ respectively goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger $n$ and $d$, numerical results are provided and discussed.Comment: 26 pages, 1 figure, 1 table. revised versio

Analysis of the expected density of internal equilibria in random evolutionary multi-player multi-strategy games

In this paper, we study the distribution and behaviour of internal equilibria in a d-player n-strategy random evolutionary game where the game payoff matrix is generated from normal distributions. The study of this paper reveals and exploits interesting connections between evolutionary game theory and random polynomial theory. The main contributions of the paper are some qualitative and quantitative results on the expected density, fn,dfn,d, and the expected number, E(n, d), of (stable) internal equilibria. Firstly, we show that in multi-player two-strategy games, they behave asymptotically as √d−1 as d is sufficiently large. Secondly, we prove that they are monotone functions of d. We also make a conjecture for games with more than two strategies. Thirdly, we provide numerical simulations for our analytical results and to support the conjecture. As consequences of our analysis, some qualitative and quantitative results on the distribution of zeros of a random Bernstein polynomial are also obtained

On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games

In this paper, we study the number of equilibria of the replicator-mutator dynamics for both deterministic and random multi-player two-strategy evolutionary games. For deterministic games, using Decartes' rule of signs, we provide a formula to compute the number of equilibria in multi-player games via the number of change of signs in the coefficients of a polynomial. For two-player social dilemmas (namely, the Prisoner's Dilemma, Snowdrift, Stag Hunt, and Harmony), we characterize (stable) equilibrium points and analytically calculate the probability of having a certain number of equilibria when the payoff entries are uniformly distributed. For multi-player random games whose payoffs are independently distributed according to a normal distribution, by employing techniques from random polynomial theory, we compute the expected or average number of internal equilibria. In addition, we perform extensive simulations by sampling and averaging over a large number of possible payoff matrices to compare with and illustrate analytical results. Numerical simulations also suggest several interesting behaviour of the average number of equilibria when the number of players is sufficiently large or when the mutation is sufficiently small. In general, we observe that introducing mutation results in a larger average number of internal equilibria than when mutation is absent, implying that mutation leads to larger behavioural diversity in dynamical systems. Interestingly, this number is largest when mutation is rare rather than when it is frequent.Comment: 23 page

Smithsonian Folklife Festival 2007 and the Presentation of Cultural Diversity in America

This thesis examines the way in which "cultural diversity" in America was constructed and presented at the 2007 Smithsonian Folklife Festival (SFF) and how American Studies scholars and the museum professionals assess the presentation of diversity at the SFF. I examine the unique museum genre of the SFF and the ways in which the folklore cultures are featured and the grassroots participants are self represented at the Festival. I argue that the Festival has created a civic and social space for cultural diversity in America to be represented and dialogued. I use my double cultural traditions as an outsider from Vietnam and an insider in an American Studies graduate program to provide perspective on the conceptualizations of "cultural diversity" undertaken by Smithsonian Institution museum professionals and those of American Studies scholars. I analyze their thoughts about the presentation of diversity at the SFF and compare the latter with treatments of cultural diversity in American Studies approaches in the United States

The effect of employee empowerment on service quality in foodservice firms

The aim of this study was to examine the effect employee empowerment has on service quality in food service firms. The theoretical background explained in detail how service quality is defined and managed in food service firms, the meaning of employee empowerment and its various forms, and the theoretical effect employee empowerment has on service quality. It also suggested strategies derived from academic sources that were considered efficient for implementing a successful empowerment culture in food service firms. Exploratory research, a qualitative method and semi-structured interviews were employed to collect the required information for this study. In addition, purposive sampling and convenience sampling were used to select the interviewees who were relevant to the research questions and were willing to participate at the time of the visit, in this case, restaurant managers. The results revealed that restaurant managers hold different perspectives on service as well as the way they kept track of it. “Employee empowerment” was still a quite new concept to the majority of the interviewees although there already were some empowerment initiatives exercised in these organisations. Besides, the result showed that employee empowerment seemed to add to service quality more positive effects than negative ones, which was complementary to the literature review in the thesis. Finally, the researcher also recommended some effective methods for restaurant managers to implement a successful empowerment culture in order to improve their service quality

Cost optimisation of individual-based institutional reward incentives for promoting cooperation in finite populations

In this paper, we study the problem of cost optimisation of individual-based institutional incentives (reward, punishment, and hybrid) for guaranteeing a certain minimal level of cooperative behaviour in a well-mixed, finite population. In this scheme, the individuals in the population interact via cooperation dilemmas (Donation Game or Public Goods Game) in which institutional reward is carried out only if cooperation is not abundant enough (i.e., the number of cooperators is below a threshold 1≤t≤N−1, where N is the population size); and similarly, institutional punishment is carried out only when defection is too abundant. We study analytically the cases t=1 for the reward incentive under the small mutation limit assumption and two different initial states, showing that the cost function is always non-decreasing. We derive the neutral drift and strong selection limits when the intensity of selection tends to zero and infinity, respectively. We numerically investigate the problem for other values of t and for population dynamics with arbitrary mutation rates

Evolutionary mechanisms that promote cooperation may not promote social welfare

Understanding the emergence of prosocial behaviours among self-interested individuals is an important problem in many scientific disciplines. Various mechanisms have been proposed to explain the evolution of such behaviours, primarily seeking the conditions under which a given mechanism can induce highest levels of cooperation. As these mechanisms usually involve costs that alter individual payoffs, it is however possible that aiming for highest levels of cooperation might be detrimental for social welfare -- the later broadly defined as the total population payoff, taking into account all costs involved for inducing increased prosocial behaviours. Herein, by comparatively analysing the social welfare and cooperation levels obtained from stochastic evolutionary models of two well-established mechanisms of prosocial behaviour, namely, peer and institutional incentives, we demonstrate exactly that. We show that the objectives of maximising cooperation levels and the objectives of maximising social welfare are often misaligned. We argue for the need of adopting social welfare as the main optimisation objective when designing and implementing evolutionary mechanisms for social and collective goods