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Rewards and the private provision of public goods on dynamic networks

Abstract

We study an evolutionary model of a public good game with rewards played on a network. Giving rewards to contributors transforms the game but gives rise to a second-order dilemma. By allowing for coevolution of strategies and network structure, the evolutionary dynamics operate on both structure and strategy. Players learn with whom to interact and how to act and can overcome the second-order dilemma. More specifically, the network represents social distance which changes as players interact. Through the change in social distance, players learn with whom to interact, which we model using reinforcement dynamics. We find that, for certain parameter constellations, a social institution, prescribing prosocial behavior and thus solving the second-order dilemma, can emerge from a population of selfish players. Due to the dynamic structure of the network, the institution has an endogenous punishment mechanism ensuring that defectors will be excluded from the benefits of the institution and the public good will be supplied efficiently.

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Notes

  1. See for example Stark and Behrens (2010); Eshel et al. (1998); Nowak and May (1993) and Szolnoki and Perc (2010).

  2. For concrete examples see Durrett and Levin (1994).

  3. In the recent biology and physics literature there has been some work on dynamic graphs, e.g., Pacheco et al. (2006) or Fu et al. (2008). However, these models differ from our model in that they do not model the evolution of the network using reinforcement learning.

  4. All results in Sections 4 and 5 are results from simulations. The parameter values are mentioned in the tables’ captionss. The simulation was programmed in Python and the code is available upon request.

  5. For economists the most familiar way of modeling an individual player’s learning process is belief-based learning. For a detailed survey on modeling economic learning see Brenner (2006). The individual player is assumed to have an internal model of how the world works and beliefs about unknown parameters. New observations enter the player’s state of mind as new pieces of information. The new information is integrated into the player’s beliefs which then govern behavior. Bayesian updating is the most familiar form of belief-based learning.

  6. If payoffs are negative the linear response rule can become problematic and one often adopts the logistic response rule, \(p_{i}=\frac {exp(bw_{i})}{\sum _{s}exp(bw_{s})}\) where b is a learning parameter.

  7. With respect to the Reward stage our model is similar to a helping game (Seinen and Schram 2006) in which the donor decides whether to help the recipient or not.

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Correspondence to Matthias Greiff.

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This paper is builds upon chapter 5 of my dissertation (Greiff 2011) and extends the model presented there. I thank Jason McKenzie Alexander, Wolfram Elsner, Torsten Heinrich, Fabian Paetzel, Hannes Rusch, Stefan Traub and an anonymous referee for helpful comments, discussions and encouragement. The usual disclaimer applies; all remaining errors are my own.

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Greiff, M. Rewards and the private provision of public goods on dynamic networks. J Evol Econ 23, 1001–1021 (2013). https://doi.org/10.1007/s00191-013-0328-2

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Keywords

  • Agent-Based modeling
  • Dynamic networks
  • Evolutionary game theory
  • Public goods
  • Reinforcement learning
  • Social networks

JEL Classifications

  • C 63
  • C 73
  • D 85
  • H 41