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

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

On the distribution of the number of internal equilibria in random evolutionary games

In this paper, we study the distribution of the number of internal equilibria of a multi-player two-strategy random evolutionary game. Using techniques from the random polynomial theory, we obtain a closed formula for the probability that the game has a certain number of internal equilibria. In addition, by employing Descartes' rule of signs and combinatorial methods, we provide useful estimates for this probability. Finally, we also compare our analytical results with those obtained from samplings.Comment: 31 pages, comments are welcome. arXiv admin note: substantial text overlap with arXiv:1708.0167

Eliciting patients’ health concerns in consulting rooms and wards in Vietnamese public hospitals

This article examines the doctor’s elicitation of the patient’s presenting health concern in two clinical settings in the Vietnamese public hospital system: the consulting room and the ward. The data were taken from 66 audio-recorded consultations. Our analysis shows that the elicitors used by the doctor in the consulting room often communicate a weak epistemic stance towards the patient’s health issue, while those used in the ward tend to signal a strong epistemic stance. In addition, this contrast between the elicitors employed in the consulting room and the ward is evident in our data regardless of whether the consultation is a first visit or a same follow-up (in which the doctor is the same one that treated the patient on their last visit), though the contrast is less clear for different follow-ups (in which the doctor has not treated the patient before). An additional finding is that the clinical setting has some bearing on the use of inappropriate elicitation formats (in which the doctor opens the visit with an elicitor which is more appropriate for another type of visit). The precise way in which each of the consulting room and the ward operates is, of course, a feature of the Vietnamese public hospital system itself. Hence, the overall contrast between the elicitors and elicitation formats used in these two settings illustrates how, on a more general level, the institutional context can have an impact on doctor-patient communication