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The effects of heterogeneous interaction and risk attitude adaptation on the evolution of cooperation

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Abstract

This paper addresses the evolution of cooperation in a multi-agent system with agents interacting heterogeneously with each other based on the iterated prisoner’s dilemma (IPD) game. The heterogeneity of interaction is defined in two models. First, agents in a network are restricted to interacting with only their neighbors (local interaction). Second, agents are allowed to adopt different IPD strategies against different opponents (discriminative interaction). These two heterogeneous interaction scenarios are different to the classical evolutionary game, in which each agent interacts with every other agent in the population by adopting the same strategy against all opponents. Moreover, agents adapt their risk attitudes while engaging in interactions. Agents with payoffs above (or below) their aspirations will become more risk averse (or risk seeking) in subsequent interactions, wherein risk is defined as the standard deviation of one-move payoffs in the IPD game. In simulation experiments with agents using only own historical payoffs as aspirations (historical comparison), we find that the whole population can achieve a high level of cooperation via the risk attitude adaptation mechanism, in the cases of either local or discriminative interaction models. Meanwhile, when agents use the population’s average payoff as aspirations (social comparison) for adapting risk attitudes, the high level of cooperation can only be sustained in a portion of the population (i.e., partial cooperation). This finding also holds true in both of the heterogeneous scenarios. Considering that payoffs cannot be precisely estimated in a realistic IPD game, simulation experiments are also conducted with a Gaussian disturbance added to the game payoffs. The results reveal that partial cooperation in the population under social comparison is more robust to the variation in payoffs than the global cooperation under historical comparison.

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Notes

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  3. http://www.cnet.com/au/news/samsung-display-lg-display-end-war-and-eye-partnership/

  4. http://www.airchinagroup.com/

  5. http://www.csair.com/en/

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  7. http://www.airchinagroup.com/cnah/zhgl/dsj/index.shtml

  8. http://www.china-airlines.com/en/about/about-1-1.htm

  9. http://bk.travelsky.com/bkair/

  10. http://www.ch.com/

  11. http://www.juneyaoair.com/

  12. The utility function (2) is used to define how agents evaluate average payoff and risk in the IPD game. This differs from the payoff–utility mappings in economics (Kahneman and Tversky 1979), which define the utilities of risk-seeking (or risk-averse) individuals as increasing (or decreasing) with risk. When α is set as 0, an agent will tend to maximize his payoff in each PD move while completely ignoring the income stream risk in the IPD game. In other words, an agent will accept any variance of payoffs, i.e., any income stream risk, along with the payoff maximization. Thus, we use α = 0 to define completely risk-seeking agents. Based on our previous work in Zeng et al. (2016a), ignoring income stream risk makes the risk-seeking agents tend to defect in the IPD game. As a result, highly risk-seeking agents (with α → 0) may benefit from slightly unilateral exploitation when playing against highly risk-averse opponents (with α → 1), and thus they can achieve an above-R average payoff. Meanwhile, two highly risk-seeking agents will be stuck in more defective moves when they play against each other. Therefore, their average payoffs will fall far below R. These cases all result in a high income stream risk for the highly risk-seeking agents. Thus, ignoring income stream risk makes both one-move payoffs and average payoffs more uncertain for agents in the IPD game, which may bring them lower average payoffs.

  13. In our experiments, the IPD game between each pair of agents is repeated 20 times to control for the effect of random pre-game histories. Thus, the average payoff or the average utility of an agent in an IPD game is averaged over 20 repetitions of the game.

  14. In addition to local clustering, which is the main characteristic of ring and grid networks, a small-world network allows a few long-range connections between agents. It was demonstrated that many real-world interactions have small-world properties when some individuals belong to quite distinct clusters (Watts and Strogatz 1998; Uzzi et al. 2007). In particular, successful individuals always possess extensive connections with others, which results in a power-law degree distribution of the interaction network. In this case, the small-world network is also described as a scale-free network (Klemm and Eguiluz 2002).

  15. Both the ring and grid networks are toroidal (Nowak and May 1992; Seo et al. 2000; Shutters 2012).

  16. Risk attitude adaptation is absent in the initial generation if β < 1, because no historical payoffs exist in this case.

  17. The Gaussian disturbance is applied for the consideration of agents increasing (or decreasing) risk attitudes by different ratios. Certainly, the scale of the Gaussian disturbance cannot be too large, since large disturbances will degenerate the agents’ risk attitude adaptation into random changes. The value of v is set as 0.2 to ensure that most of the agents will adapt their risk attitudes either upwards or downwards based on the prospect theory. Experiments demonstrate that the evolutionary outcomes are robust to the change in v over a wide range.

  18. The simulation results are qualitatively identical with N in a wide range of values. Moreover, the evolutionary dynamics becomes stabilized after about the 2,000th generation. Thus, results with N = 256 and G = 10,000 are shown for the LI game model. In addition, the computational complexity of the DI experiment is O(N 2). Thus, we further reduce the population size N to 64 and the number of generations G to 5000 in the DI experiment to lower the computational cost. The experimental results also show that this reduction has no significant influence on the evolution of cooperation.

  19. According to our previous work (Zeng et al. 2016b), the value of γ indicates the sensitivity of agents’ risk attitudes with respect to their performances in the evolutionary IPD game. A too-small value of γ, e.g., γ = 0 (or a too-large value, e.g., γ = 0.5), means that agents will change their risk attitudes too quickly (or too slowly) in response to the game outcomes. The inappropriate adjustment speeds will prevent agents from adapting to the game environment, and thus will have a disruptive impact on the evolutionary outcomes. Thus, the value of γ is set as 0.15 in this study. Meanwhile, comparable results are obtained for different specifications of r up and r down . Thus, we use results with r up  = r down  = 0.2 for illustrative purposes.

  20. In the experiments, the average result over 20 runs is not statistically different from the average results over 30, 40, or 50 runs (with a 95% confidence level).

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Acknowledgments

This study was supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 70925005), the General Program of the National Science Foundation of China (No.71371135), the Key Program of the National Science Foundation of China (No. 71131007), the Science Foundation of Hainan Province, China (No. 20167244), and the Scientific Research Foundation of Hainan University, China (No. kyqd1529). It was also supported by the Program for Changjiang Scholars and Innovative Research Teams in Universities of China (PCSIRT). The authors would like to thank The High Performance Computing Centre (HPCC) of Tianjin University for providing computing support. Authors are very grateful to the editor and all anonymous reviewers whose invaluable comments and suggestions substantially helped improve the quality of the manuscript.

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Authors and Affiliations

  1. College of Management and Economics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Weijun Zeng, Minqiang Li & Nan Feng

  2. School of Economics and Management, Hainan University, Haikou, 570228, People’s Republic of China

    Weijun Zeng

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  1. Weijun Zeng
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Correspondence to Minqiang Li.

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This study was supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 70925005), the General Program of the National Science Foundation of China (No.71371135), the Key Program of the National Science Foundation of China (No. 71131007), the Science Foundation of Hainan Province, China (No. 20167244), and the Scientific Research Foundation of Hainan University, China (No. kyqd1529).

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Zeng, W., Li, M. & Feng, N. The effects of heterogeneous interaction and risk attitude adaptation on the evolution of cooperation. J Evol Econ 27, 435–459 (2017). https://doi.org/10.1007/s00191-016-0489-x

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