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Common Knowledge: A New Problem for Standard Consequentialism

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Abstract

This paper reveals a serious flaw in the consequentialist solution to the inefficacy problem in moral philosophy. The consequentialist solution is based on expected utility theory. In current philosophical literature, the debate focuses on the empirical plausibility of the solution. Most philosophers consider the cases of collective actions as of the same type as a horse-racing game, where expected utility theory is adequate to solve the choice problem. However, these cases should be considered as of the same type as a coordination game, where the assumption of common knowledge is also required. However, the assumption is implausible—it is impossible to obtain common knowledge of rationality in cases such as voting.

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Notes

  1. It is worth noting that there are various other consequentialist strategies for responding to the inefficacy problem, such as the group-based approach proposed by Frank Jackson. See Lawford-smith & Tuckwell (2020). To clarify, the consequentialist solution to the inefficacy problem, which is challenged in this paper, refers to the consequentialist attempt that appeals to the expected utility theory. In some places in this paper, I also call it the “expected-utility-based consequentialist solution”.

  2. The idea that consequentialist maximizers cannot solve the problem of collective actions is not new in.

    economic literature. For example, Widerquist (2003) and Binmore (1994, 1998) have demonstrated that altruism solves neither prisoners’ dilemma nor their coordination problem.

  3. See Nefsky (2019); Barnett (2020).

  4. See Nefsky, (2019, p. 11).

  5. The most optimistic estimate in the literature claims that in a presidential election, an American voter could have as high as a one in 10 million chance of determining the final result, and only if the voter is living in one of three or four ‘swing states’ and votes for a major-party candidate (Edlin et al. 2007). William Riker and Peter Ordeshook (1968) estimated an even lower probability of an individual’s vote being decisive in the outcome of a US presidential election (i.e., p = 10–8). In the literature, it is widely believed that voting from a prudential perspective is rarely rational (Brennan 2016; Goldman 1999).

  6. It can be demonstrated that in most cases, a satisfied voter is indifferent about voting versus not voting.

  7. Compared with the roulette game, since the roulette machine’s behaviour is fixed, each outcome occurs with a positive and equal probability.

  8. An assurance game has two Nash equilibria, but one is socially better than the other. Once society is trapped in a worse equilibrium, it is difficult to move to the better one. To make this move from worse to better, everyone should move together. However, this is often thought to be a difficult task. Agents can attain a better equilibrium only if each has enough certainty that the others are also choosing the appropriate action.

  9. Nash equilibrium has the following property: no one can gain by deviating by themselves. If an individual is the only person deviating from equilibrium, they cannot truly gain.

  10. X is common knowledge if everybody knows X, everybody knows that everybody knows X, and so on. In game theory, if nothing is said to the contrary in a rational analysis of a game, then it is often implicitly assumed that the rationality of all players is common knowledge.

  11. In real-life situations of this kind, the convergence of expectations may be achieved by agreements through communications or legally binding contracts among agents. In large group events, such as national elections, communication might be achieved by scheduling meetings or making announcements on different social media. In some election systems, voting is required by law. Eligible citizens are required to register and vote in elections, and penalties may be imposed on those who fail to do so. However, these devices only work if everyone knows that everyone else is paying attention to them because successful coordination requires the desired equilibrium to be a focal point. The agents’ expectations must converge on this equilibrium: everyone should know that everyone knows that everyone will vote.

  12. According to Lewis, a convention involves the deliberative choice of one regularity R over another regularity R in a situation S in virtue of shared knowledge among a population P. In ideal cases, a regulation R is a convention for P if it is common knowledge among the members of P that they follow R. For Lewis, common knowledge is required for convention. See Lewis (1969). However, not every scholar accepts this stance. For example, Binmore argues that common knowledge is better excluded from the concept of convention. See Binmore (2008). In this paper, I follow Lewis’ definition.

  13. Lewis’ account of common knowledge is often criticized as too weak to be the right concept. Common belief is thought to be a proper concept for Lewis’ account. For more details, please refer to Kraus and Lehman (1988). Yet, this distinction does not make a significant impact on my argument, as my argument shows that common knowledge fails, even in its weaker concept.

  14. The result is finite but incomputable; however, it is much smaller than one in 100 million. For example, if we compute the probability that each person knows that 20,000 people are altruistic voters, the result is 0.9920000 = 5.056988325166235e-88, which is already astronomically small. Thus, 0.991,999.999 should be much lower than this number. For reference, 1e + 88 roughly equals the number of all elementary particles in the universe. It is difficult to fathom that any social policy could have a value as large as or even larger than 1e + 88.

  15. I have presented a simplified version of the analysis, which is sufficient to serve my argument. More specifically, five possible outcomes are jointly exhaustive in this case:

    1. (1)

      Everyone votes;

    2. (2)

      More than 1.6 million but less than 2 million people vote;

    3. (3)

      No one votes;

    4. (4)

      More than one but less than 1.6 million vote;

    5. (5)

      Precisely 1.6 million people (80% of the population) vote.

    For any altruistic individual, (5) is the preferable outcome; (2) is better than (1), but less preferred than (5), as (2) involves inefficient votes; (1) is better than (3), and (3) is better than (4).

  16. In the real world, the election thresholds in all democratic countries are much less than 80% (usually no higher than 10%). However, given that the population in national elections is often large, even with a 1% election threshold, the epistemological probability would still be low.

  17. One of the features of esoteric morality implies that even though the consequentialist believes that acts are right only if they give rise to consequences at least as good as any alternative the agent could have obtained, the consequentialist may need to discourage others from embracing consequentialism in public. See Lazari-Radek and Singer (2010, p. 35).

  18. Harsanyi’s model can also be considered as a quasi-Kantian approach, according to which the reason to vote relies on a general practice of voting, which can be universalized.

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Acknowledgements

I am grateful to two anonymous reviewers’ comments and suggestion. I am also grateful to for Davide Fassio and Jie Gao’s extremely helpful comments on the early version of this article. I thank all the audiences who gave me inspiring feedbacks during my seminars at Zhejiang University.

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  1. Lingnan University, Hong Kong, Hong Kong

    Fei Song

  2. Zhejiang University, Hangzhou, China

    Fei Song

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Song, F. Common Knowledge: A New Problem for Standard Consequentialism. Ethic Theory Moral Prac 25, 299–314 (2022). https://doi.org/10.1007/s10677-022-10267-y

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