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

Institutional Incentives for the Evolution of Committed Cooperation: Ensuring Participation is as Important as Enhancing Compliance

Both conventional wisdom and empirical evidence suggests that arranging a prior commitment or agreement before an interaction enhances the chance of reaching mutual cooperation. Yet it is not clear what mechanisms can promote the participation in and compliance with such a commitment, especially when the former is costly and deviating from the latter is profitable. Prior work either considers regimented commitments where compensation is assumed enforceable from dishonest committers, or assume implicit commitments from every individual (so they are all in and thus being treated as such). Here we develop a theory of participation and compliance with respect to an explicit prior commitment under institutional incentives where individuals, at first, decide whether or not to join a cooperative agreement to play a one-shot social dilemma game. Using a mathematical model, we determine when participating in a costly commitment and complying with it, is an evolutionary stable strategy (ESS) when playing against all other possible strategies, and results in high levels of cooperation in the population. We show that, given a sufficient budget for providing incentives, reward of commitment compliant behaviours better promotes cooperation than punishment of non-compliant ones. Moreover, by sparing part of this budget for rewarding those who are willing to participate in a commitment, the overall frequency of cooperation can be significantly enhanced, for both reward and punishment. Finally, we find that, surprisingly, the presence of errors in a participation decision favours evolutionary stability of commitment compliant strategies and higher levels of cooperation

Emergence of Social Punishment and Cooperation through Prior Commitments

Social punishment, whereby cooperators punish defectors, has been suggested as an important mechanism that promotes the emergence of cooperation or maintenance of social norms in the context of the one-shot (i.e. non-repeated) interaction. However, whenever antisocial punishment, whereby defectors punish cooperators, is available, this antisocial behavior outperforms social punishment, leading to the destruction of cooperation. In this paper, we use evolutionary game theory to show that this antisocial behavior can be efficiently restrained by relying on prior commitments, wherein agents can arrange, prior to an interaction, agreements regarding posterior compensation by those who dishonor the agreements. We show that, although the commitment mechanism by itself can guarantee a notable level of cooperation, a significantly higher level is achieved when both mechanisms, those of proposing prior commitments and of punishment, are available in co-presence. Interestingly, social punishment prevails and dominates in this system as it can take advantage of the commitment mechanism to cope with antisocial behaviors. That is, establishment of a commitment system helps to pave the way for the evolution of social punishment and abundant cooperation, even in the presence of antisocial punishment

Words are not Wind -- How Joint Commitment and Reputation Solve Social Dilemmas, without Repeated Interactions or Enforcement by Third Parties

Joint commitment was argued to "make our social world" (Gilbert, 2014) and to separate us from other primates. 'Joint' entails that neither of us promises anything, unless the other promises as well. When we need to coordinate for the best mutual outcome, any commitment is beneficial. However, when we are tempted to free-ride (i.e. in social dilemmas), commitment serves no obvious purpose. We show that a reputation system, which judges action in social dilemmas only after joint commitment, can prevent free-riding. Keeping commitments builds trust. We can selectively enter joint commitments with trustworthy individuals to ensure their cooperation (since they will now be judged). We simply do not commit to cooperate with those we do not trust, and hence can freely defect without losing the trust of others. This principle might be the reason for pointedly public joint commitments, such as marriage. It is especially relevant to our evolutionary past, in which no mechanisms existed to enforce commitments reliably and impartially (e.g. via a powerful and accountable government). Much research from anthropology, philosophy and psychology made the assumption that past collaborations were mutually beneficial and had little possibilities to free-ride, for which there is little support. Our evolutionary game theory approach proves that this assumption is not necessary, because free-riding could have been dealt with joint commitments and reputation.Comment: 13 pages (without ref and supp), 8 figure

The efficient interaction of costly punishment and commitment.

To ensure cooperation in the Prisoner's Dilemma, agents may require prior commitments from others, subject to compensations when defecting after agreeing to commit. Alternatively, agents may prefer to behave reactively, without arranging prior commitments, by simply punishing those who misbehave. These two mechanisms have been shown to promote the emergence of cooperation, yet are complementary in the way they aim to instigate cooperation. In this work, using Evolutionary Game Theory, we describe a computational model showing that there is a wide range of parameters where the combined strategy is better than either strategy by itself, leading to a significantly higher level of cooperation. Interestingly, the improvement is most significant when the cost of arranging commitments is sufficiently high and the penalty reaches a certain threshold, thereby overcoming the weaknesses of both strategies.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

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