Namely, Fig. When this investment cost is large enough, a high level of cooperation can be sustained with a large t , i.

The results are in accordance with the theoretical results in Eqs 7 and 8. On the other hand, with sufficiently high values of the cost, one can reach significant cooperation with a rather small threshold for t.

## Game Theory

In panel b and c , the results are scaled by Log Thus, a question arises as to whether it is always the case that a more spreading interference scheme requires a larger budget EC? In other words, should we simply use the smallest possible value of t that leads to the required level of cooperation? When stochastic factors, such as mutation and frequency-dependence dynamics, are absent, clearly that is the case. However, when such stochastic factors are present, the answer is not obvious anymore. Indeed, as shown in Fig. This observation can be explained by looking at the expected number of times of investment i.

Even when the number of C-players in the population is rather large i.

Thus, as observed from results in Fig. We now study how robust this observation is. Indeed, Fig. That is, whenever selection is not too strong, we would expect to find the optimal interference scheme not always making an investment in cooperators, and the smaller the selection strength, the less spreading an investment scheme should be. Indeed, in Fig. In summary, the present work seeks to answer the question of how to interfere in a population of self-interested players in order to promote high cooperation in a cost-effective way.

In particular, the cost of interference is measured as the consumption of certain resources, and the higher impact we want to make, the higher cost we have to pay.

## Game Theory Decisions, Interaction and Evolution by James Webb | eBay

To tackle this problem, we have developed a cost-efficient interference model based on evolutionary game methods in finite populations. In our model, an exogenous decision-maker aims to optimise the cost of interference while guaranteeing a certain minimum level of cooperation, having to decide whether and how much to invest in rewarding cooperative behaviours at each time step.

For a general investment scheme, we have obtained the sufficient conditions for the scheme to achieve a concrete level of cooperation. Moreover, we have provided numerical analyses for a specific investment scheme that makes a fixed investment in a cooperator i. This individual-based scheme can be considered a more general form of the prevalent models of institutional incentive strategies, such as institutional punishment and reward 35 , 36 , 37 , 38 , 42 , 43 , 44 , 45 , 46 , which do not take into account the behavioural composition or state of the population.

Our results have shown that whenever the intensity of selection is not too strong, an intermediate value of the threshold t leads to a minimal total cost of investment while guaranteeing at least a given desired fraction of population cooperation. Furthermore, there is a wide range of the threshold t where individual-based investment is more cost-effective than the above mentioned institutional incentive strategies; and the smaller the intensity of selection, the wider this range is.

### James N. Webb

Note that our work is also different from the existing institutional incentive models 35 , 36 , 37 , 38 , 42 , 43 , 44 , 45 , 46 , as well as the existing literature on the evolution of cooperation 1 , 2 , 3 , 4 , in that, its aim is to minimise the cost of interference while guaranteeing high levels of cooperation, while cost-efficiency is mostly ignored in those works. Similarly, our work also differs from EGT literature on optimal control in networked populations 47 , 48 , 49 , where cost-efficiency is not considered.

Instead, these works on controllability focus on identifying which individuals or nodes are the most important to control i. Moreover, it is important to note that in the context of institutional incentives modelling, a crucial issue is the question of how to maintain the budget of incentives providing.

The problem of who pays or contributes to the budget is a social dilemma itself, and how to escape this dilemma is critical research question. Facilitating solutions include pool incentives with second order punishments 42 , democratic decisions 37 , positive and negative incentives combination 36 and spatial populations 44 , just to name a few. Our work does not address this issue of who to contribute to the budget, but rather focus on how to optimise the spending, given a budget already, which has not been addressed by these works.

However, it would be interesting to study whether and how interference strategies should be customised for different types of incentive providers, which we aim to study in future work. Furthermore, related to our work here is a large body of research on sequential decision-making in Artificial Intelligence and Multi-agent systems, which provide a number of techniques for making a sequence of decisions that lead to optimal behaviour of a system e. Thus, these works failed to exploit the system intrinsic properties, and hence, not able to efficiently achieve a desired outcome and system status e.

Examples of closed systems assumption are classical game theoretical models, see e. Therefore, these works are not suitable to tackle our settings either. Based on the general model we developed here, more efficient interference strategies can be studied. In particular, it would be interesting to consider more adaptive interference strategies, which modify the amount and the frequency of investment dynamically depending on the current state of the system. Moreover, we aim to extend our analysis to systems with other, more complicated scenarios such as structured populations and multi-player games, where more behavioural equilibria 61 , 62 , 63 and structure-dependent 13 , 19 interference strategies might be required to ensure cost-efficiency.

In the former case, interference strategies would need to take into account the structural information in a network such as the cooperative properties in a neighbourhood for the results of cost-efficient interference strategies in square lattice populations, see our recent work in ref. In the latter case, the strategies might need to consider the group size as well as cooperative properties in the group to decide whether to make an investment.

Both the analytical and numerical results obtained here use Evolutionary Game Theory EGT methods for finite populations 34 , 65 , A similar description of the Methods section was used in refs 67 , For convenience of numerical computations, but without affecting analytical results, we assume here small mutation limit 65 , 66 , As such, at most two strategies are present in the population simultaneously, and the behavioural dynamics can thus be described by a Markov Chain, where each state represents a homogeneous population and the transition probabilities between any two states are given by the fixation probability of a single mutant 65 , 66 , The resulting Markov Chain has a stationary distribution, which describes the average time the population spends in an end state.

Having obtained the fixation probabilities between any two states of a Markov chain, we can now describe its stationary distribution. See e. Axelrod, R. The evolution of cooperation. Science , — Nowak, M. Five rules for the evolution of cooperation. Sigmund, K. Perc, M. Statistical physics of human cooperation. Physics Reports , 1—51 Hamilton, W. The genetical evolution of social behaviour.

Journal of Theoretical Biology 7 , 1—16 Traulsen, A.

Evolution of cooperation by multilevel selection. Evolution of indirect reciprocity. Nature Han, T. Corpus-based intention recognition in cooperation dilemmas. Artificial Life journal 18 , — Hilbe, C. Memory-n strategies of direct reciprocity. Proceedings of the National Academy of Sciences , — Van Veelen, M.

## Series: Springer Undergraduate Mathematics Series

Direct reciprocity in structured populations. Zisis, I. Generosity motivated by acceptance-evolutionary analysis of an anticipation game. Scientific reports 5 , Santos, F. Evolutionary dynamics of social dilemmas in structured heterogeneous populations.

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Journal of theoretical biology , — Chen, X. Probabilistic sharing solves the problem of costly punishment. New Journal of Physics 16 , Pinheiro, F.