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Mathematical Framework to Quantify Social Dilemmas

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Part of the Evolutionary Economics and Social Complexity Science book series (EESCS,volume 28)


The context of social dilemma is of great interest in evolutionary game theory because of its importance in explaining the evolution of cooperation in biological systems. The dilemma strength parameters have been used to numerically characterize the social dilemma to understand how much dilemma existing in a game. However, these parameters cannot be explicitly determined in general. To understand the presence of social dilemma associated with more general game systems, in this chapter, we introduce a new index, “social efficiency deficit” quantifying the payoff difference between social optimum and Nash equilibrium. Whereas the dilemma strength provides a numerical measure of how much dilemma prevailing in a game, the social efficiency deficit depicts the payoff shortfall at Nash equilibrium compared with social optimum. We deliberately delineate the relationship between these two parameters for several game classes such as pairwise Prisoner’s dilemma, Chicken game, etc. Notably, Prisoner’s dilemma possesses an inverse relationship between them. Nevertheless, we also explore the consistency of the new parameter presuming the situation if a certain social viscosity (e.g., network reciprocity) is included into a dilemma game. Additionally, the pertinence of social efficiency deficit for revealing the presence of social dilemma in vaccination game and traffic flow analysis is illustrated at the end.


  • Evolutionary game
  • Symmetric two-player & two-strategy game
  • Social dilemma
  • Dilemma strength
  • Social efficiency deficit

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  • DOI: 10.1007/978-981-19-0937-5_6
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  1. 1.

    To learn in detail about GID & RAD and D g and D r as well as the replicator dynamics, you should consult:

    Tanimoto (2021).

  2. 2.

    Nowak (2006).

  3. 3.

    Hamilton (1963).

  4. 4.

    Concerning R reciprocity and ST reciprocity games, you should refer to:

    Wakiyama and Tanimoto (2011).

  5. 5.

    Taylor and Nowak (2007).

  6. 6.

    Ohtsuki et al. (2006).

  7. 7.

    Ohtsuki and Nowak (2006).

  8. 8.

    Wang et al. (2015).

  9. 9.

    This is because for DS, D g′ and Dr′ can be identified in advance of actual analysis. Whenever one knows the payoff matrix, DS can be evaluated.

  10. 10.

    We can reproduce any game model as a form of MAS irrespective of game type, regardless of whether it is two-payer or multi-player, two-strategy or multi-strategy, complex or simple, or realistic or ideal. As long as a MAS model is established, we can explore Nash equilibrium as well as the socially optimal state, at least numerically.

  11. 11.

    With a vaccination coverage above a threshold level, an individual with no vaccination can hardly be infected. Commitment to vaccination from the social majority enables those non-vaccinating individuals to be protected from infection. This is called “herd immunity” and discussed in Chap. 3.


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Correspondence to Jun Tanimoto .

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Tanimoto, J. (2022). Mathematical Framework to Quantify Social Dilemmas. In: Aruka, Y. (eds) Digital Designs for Money, Markets, and Social Dilemmas. Evolutionary Economics and Social Complexity Science, vol 28. Springer, Singapore.

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