The Paradox of Omniscience and the Theory of Moves

  • Steven J. Brams


In Chapter 3 I indicated how one can calculate, from players’ payoffs, probabilities that indicate thresholds at which the cooperative solution in Prisoners’ Dilemma can be rendered stable and the dilemma thereby circumvented. In the fourth Knowability Game, this resolution depended on the predictive abilities of P as well as SB, which it may be unreasonable to assume. On the other hand, in the less ethereal world of international politics, the presumption that the superpower arms race is a Prisoners’ Dilemma, and that both sides have these predictive abilities—based on their intelligence capabilities, supported by reconnaissance satellites and other detection equipment—seems quite reasonable.1


Nash Equilibrium Final Outcome Rational Choice Dominant Strategy Strategy Choice 
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  1. 1.
    See citations in note 14, Chapter 3.Google Scholar
  2. 3.
    Steven J. Brams, Game Theory and Politics (New York: Free Press, 1975), Chap. 1; and Brams, Paradoxes in Politics: An Introduction to the Nonobvious in Political Science (New York: Free Press, 1976), Chap. 5.Google Scholar
  3. 4.
    See Steven J. Brams, Deception in 2 × 2 games, J. Peace Sci. 2 (Spring 1977), 177–203, where I argue that a game other than Chicken better models the Cuban missile crisis.Google Scholar
  4. 5.
    For other examples in which additional information may hurt, see Y. C. Ho and I. Blau, A simple example on informativeness and competitiveness, J. Optimization Theory Appl. 11, 4 (April 1973), 437–440;MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5a.
    Ariel Rubinstein, A Note on the duty of disclosure, Economic Lett. 4 (1979), 7–11;CrossRefGoogle Scholar
  6. 5b.
    Martin Shubik, Game Theory in the Social Sciences: Concepts and Solutions (Cambridge, MA: MIT Press, 1982), p. 274; and Richard Engelbrecht-Wiggans and Robert J. Weber, Notes on a sequential auction involving asymmetrically-informed bidders, Int. J. Game Theory (forthcoming). The paradox of omniscience, while unfortunate for SB, may be regarded as a price he pays for information about the future. Such foreknowledge, as Vladimir Lefebvre pointed out to me (personal communication, March 25, 1980), is of value itself, reaffirming the economist’s adage that everything has a price. Or, to put it slightly differently, even SB must make trade-offs, though new rules of the game may save him, as I shall presently show.zbMATHGoogle Scholar
  7. 6.
    This section is based largely on Steven J. Brams, A resolution of the paradox of omniscience; in Reason and Decision, Bowling Green Studies in Applied Philosophy, vol. III-1981, ed. Michael Bradie and Kenneth Sayre (Bowling Green, OH: Applied Philosophy Program, Bowling Green State University, 1982), pp. 17–30.Google Scholar
  8. 7.
    By “strategy” I mean a course of action that can lead to any of the outcomes associated with it, depending on the strategy choice of the other player; the strategy choices of both players define an outcome at the intersection of their two strategies. While the subsequent moves and countermoves of players could also be incorporated into the definition of a strategy—meaning a complete plan of responses by a player to whatever choices the other player makes in the sequential game—this would make the normal (matrix) form of the game unduly complicated and difficult to analyze. Hence, I use “strategy” to mean the choices of players that lead to an initial outcome, and “moves” and “countermoves” to refer to their subsequent sequential choices, as allowed by rules II–IV.Google Scholar
  9. 8.
    Steven J. Brams and Donald Wittman, Nonmyopic equilibria in 2 × 2 games, Conflict Management Peace Sci. 6, 1 (1983); see also Marc D. Kilgour, Equilibria for far-sighted players, Theory and Decision (forthcoming), for an extension of the concept of nonmyopic equilibrium. To ensure that a final outcome is reached, either at the start or before there is cycling back to the initial outcome, the definition of a nonmyopic equilibrium also includes a termination condition. This condition specifies that there exists a node in the game tree such that the player with the next move can ensure his best outcome by staying at it. This condition is satisfied by the “cooperative” (3, 3) outcome in both Chicken and Prisoners’ Dilemma (discussed in Section 4.5).Google Scholar
  10. 10.
    Of the 78 structurally distinct 2 × 2 ordinal games—in which each player can strictly rank the four outcomes from best to worst, and no interchange of rows, columns, or players can transform one of these games into any other— these five games plus Chicken are the only games in which neither player has a dominant strategy, associated with the outcome induced by SB’s omniscience, that is best for P (column player in Fig. 4.4) and inferior for SB (row player). Accordingly, SB’s omniscience, and P’s awareness of it, are required to guarantee its choice. However, there are two other 2 × 2 games in which P can force SB to choose an outcome of lower rank than his; but this outcome, (3, 2), is the next-best, not best, for P. Furthermore, in each of these games, there is another outcome, (3, 4), better for both players but, paradoxically, unattainable if SB is omniscient and P knows this. For complete listings of the 78 2 × 2 games, see Anatol Rapoport and Melvin Guyer, A taxonomy of 2 × 2 games, General Systems: Yearbook of the Society for General Systems Research 11 (1966), 203–214; and Brams, Deception in 2 × 2 games. The six games vulnerable to the paradox of omniscience, and the two in which P can beat SB but not obtain his best outcome, are identified by their numbers in each listing in Brams, Mathematics and theology, p. 281.Google Scholar
  11. 11.
    The (single) final outcomes along the main diagonal reflect the fact that the player not receiving his best outcome can move the process “through” an off-diagonal inferior outcome—for both players—to the other diagonal outcome best for himself. However, one might take a different view of what constitute final outcomes along the diagonal, which the previous game-tree analysis does not show because it terminates before outcomes are repeated.More specifically, the player initially receiving his best outcome can always trigger a series of moves from this outcome that will bring the process back to it. Moreover, he would be motivated to do so to prevent the other player from moving the process to an outcome not best for him. Thus, a two-sided analysis of these five games suggests that both players would have an incentive to move first from the diagonal outcomes (if these are the initial outcomes), thereby transforming them into joint pairs as final outcomes.However, this view assumes that a player would be motivated to move from an initial outcome simply to ensure that it returns to this starting point—but now as a final outcome. Yet this kind of calculation, though consistent with the earlier rules, seems a bit artificial. In any event, whether the final outcomes along the diagonal are considered single outcomes (as in Fig. 4.4) or joint pairs, it would be a joint pair that would be the final outcome of the game under either interpretation. Hence, the alternative view on final outcomes along the diagonal—that they are joint pairs—would have no effect on the resolution of the paradox: because all outcomes would be the same in the final-outcome matrix under the alternative interpretation, any choice of strategies by the players would yield the dominant-strategy final outcomes circled in Fig. 4.4.Google Scholar
  12. 12.
    This is one of six games vulnerable to the second paradox of omniscience (to be discussed below); it and the other five games are identified in Steven J. Brams, Omniscience and omnipotence: how they may help—or hurt—in a game, Inquiry 25, 2 (June 1982), 217–231.Google Scholar
  13. 13.
    Steven J. Brams and Marek P. Hessel, Absorbing outcomes in 2 × 2 games, Behavioral Sci. 27, 4 (October 1982), 393–401.MathSciNetCrossRefGoogle 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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