Superior Beings: They May Be Undecidable

  • Steven J. Brams


It is now time to summarize the copious results that game theory yields when superior beings confront ordinary beings in games (see Appendix). I wish to reiterate that these theoretical results are just that—conceptual guides for thinking about certain religious-theological-philosophical questions but not scientific findings supported by any kind of empirical evidence, even if couched in mathematical language. As I recapitulate the effects of SB’s powers in games, I shall use them as a springboard to discuss what I identified as the “central question” in the first sentence of the Preface of this book: “If there existed a superior being who possessed the supernatural qualities of omniscience, omnipotence, immortality, and incomprehensibility, how would he/she act differently from us, and would these differences be knowable?”


Nash Equilibrium Mixed Strategy Superior Quality Sequential Rule Decidable Game 
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  1. 1.
    A mathematical system is consistent if no two theorems can be derived that contradict each other. For further details, see Ernest Nagel and James R. Newman, Gödel’s Proof (New York: New York University Press, 1968);Google Scholar
  2. 1a.
    Mark Kac and Stanislaw M. Ulam, Mathematics and Logic: Retrospect and Prospects (New York: New American Library, 1969);Google Scholar
  3. 1b.
    Edna E. Kramer, The Nature and Growth of Modern Mathematics, Vol. 2 (Greenwhich, CT: Fawcett, 1970);Google Scholar
  4. 1c.
    Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (New York: Basic, 1969);Google Scholar
  5. 1d.
    and Rudy Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite (Boston: Birkhäuser, 1982).zbMATHGoogle Scholar
  6. 1e.
    For arguments that Gödel’s Theorem is not just a “logician’s trick,” see Gina Kolata, Does Gödel’s Theorem matter to mathematics? Science 218, 4574 (19 November 1982), 779–780.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 1f.
    The connection between Gödel’s Theorem and theology is made in Howard Eves, Great Moments in Mathematics (after 1650) (Washington, DC: Mathematical Association of America, 1981), Lecture 38 (Mathematics as a branch of theology, pp. 200–208); Eves quotes from Frank DeSua, Consistency and completeness—a resume, Am. Math. Monthly 63, 5 (May 1956), 295–300, to the effect that only mathematics among disciplines carries within it—via Gödel’s Theorem— a rigorous demonstration that its foundations rest on a kind of religious faith.For arguments in the political science literature that there is an undecidabil-ity problem in ascertaining the exercise of power, see Political Power: A Reader in Theory and Research, ed. Roderick Bell, David V. Edwards, and R. Harrison Wagner (New York: Free Press, 1969), particularly the two articles by Peter Bachrach and Morton S. Baratz, Two faces of power (pp. 94–99) and Decisions and nondecisions: an analytical framework (pp. 100–109). These articles were originally published in, respectively, Am. Political Sci. Rev. 56, 4 (December 1962), 947–952; and Am. Political Sci. Rev. 57, 3 (September 1963), 632–642. After I completed this book, an article by J. P. Jones, Some undecidable determined games, Int. J. Game Theory 11, 2 (1982), 63–70, appeared. Jones proves that there are rather simple two-person win-lose games with perfect information in which one player has a winning strategy, but it cannot be decided which one; this kind of undecidability is different from determining whether one is playing against a superior being but, in my opinion, equally fascinating and perplexing.zbMATHGoogle Scholar
  8. 2.
    In Section 4.41 called such games—in which the outcome the stronger or quicker player can induce is ranked the same by both players—“fair,” for they give him no special advantage; here, though, I now assume that superiority can be based on any of the three different kinds of power.Google Scholar
  9. 3.
    Martin Shubik, Games of status, Behavioral Sci. 16, 2 (March 1971), 117–129.MathSciNetCrossRefGoogle Scholar
  10. 4.
    Adding to these 24 games the seven fair games with (3, 3) outcomes in which no kind of power is effective brings the total to 31, more than half of the 57 games of conflict without a mutually best (4, 4) outcome. In other words, “power-lessness” on the part of SB is more the rule than the exception in the 2 × 2 ordinal games in which the players disagree about the best outcome. If one equates unde-cidability and powerlessness under the assumption that SB cannot be distinguished in such games—whether he ranks the (common) moving/staying/threat power outcome higher or lower or the same as P-SB’s incognito status is greater than that suggested by the 24 (semi-)undecidable games discussed in the text. Also, if undecidability is defined in terms of powerlessness, it is rendered independent of interpersonal comparisons of rankings by the players, which are viewed as problematic by some theorists.Google Scholar
  11. 5.
    However, this is not to say that SB is intentionally remorseless or vindictive but rather that his behavior, insofar as it is judged to be amoral, is so because of his apparent desultory choices. Moral abandon, in other words, may be as much a product of caprice as cunning and legerdemain that are more transparent in their deviousness.Google Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Steven J. Brams
    • 1
  1. 1.Department of PoliticsNew York UniversityNew YorkUSA

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