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Iterated Mixed Strategies and Pascal’s Wager

Abstract

Mixed strategies have been used to show that Pascal’s Wager fails to offer sufficient pragmatic reasons for believing in God. Their proponents have argued that, in addition to outright belief in God, rational agents can follow alternatives strategies whose expected utility is infinite as well. One objection that has been raised against this way of blocking Pascal’s Wager is that applying a mixed strategy in Pascal’s case is tantamount to applying an iterated mixed strategy which, properly understood, collapses into the pure strategy of becoming a theist (Monton, Analysis 71:642–645, 2011). I argue that since the assumptions used to develop the iterated mixed strategies response are even more questionable than those the initial objection relies on, this type of response to the mixed strategy objection fails.

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Notes

  1. 1.

    Easwaran and Monton [3, p. 684] concede “a weakness of applying the notion of “mixed strategies” outside the original game-theoretic context for which it was designed”.

  2. 2.

    See Robertson [7, p. 297].

  3. 3.

    I am ignoring here the possibility one has every Friday to override the automatic update configuration. One can imagine that the power to change the configuration has been irreversibly transferred to a system administrator.

  4. 4.

    For instance, the one-shot indirect version of Robertson’s increased-size strategy (make the one-time decision to roll dies with an increasing number of sides), with its \(\frac{1}{2}\) probability of the die eventually landing on #1, would be equivalent to the one-shot direct mixed strategy of flipping a fair coin and believing that God exists if and only if the outcome is heads.

  5. 5.

    This can take place within a finite interval of time if the time interval between two successive die rollings is gradually diminished. It can be cut, for instance, in half.

  6. 6.

    See Black [1] and Thompson [8].

  7. 7.

    The inspiration comes from the method used in the St. Petersburg paradox, which deals with a series of coin flips. Since, in the series of coin flips, for each \(n \ge 2\), the probability of Heads coming up after n attempts is double the probability of Heads coming up after \(n - 1\) attempts, the possible reward must be doubled at each stage in order to offset the diminishing probabilities. This is enough to ensure that the expected utilities of each of the possible outcomes are equal and positive and, consequently, that they add up to infinity.

  8. 8.

    It can be argued that this is a problem for Robertson’s increasing-die-size strategy, but not for the 6-sided-die strategy due to the fact that, in the latter case, the likelihood of not getting a #1 after a finite number of attempts is zero. However, the zero probability on not getting a #1 does not suffice to solve the problem for the same reason that the zero probability of not getting Heads after a finite number of flips does not solve the St. Petersburg paradox.

References

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Acknowledgements

Many thanks to the audience of the 2nd World Congress of Logic and Religion, Warsaw 2017, for useful comments.

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Correspondence to Emil Badici.

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Badici, E. Iterated Mixed Strategies and Pascal’s Wager. Log. Univers. 13, 487–494 (2019). https://doi.org/10.1007/s11787-019-00233-1

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Keywords

  • Pascal’s Wager
  • Mixed strategies
  • Game theory
  • Theism

Mathematics Subject Classification

  • Primary 62C05
  • Secondary 00A05
  • 00A06
  • 03B48
  • 91A30