Backward induction is not robust: The parity problem and the uncertainty problem

Steven J. Brams, D. Marc Kilgour

    Research output: Contribution to journalArticlepeer-review


    A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed.

    Original languageEnglish (US)
    Pages (from-to)263-289
    Number of pages27
    JournalTheory and Decision
    Issue number3
    StatePublished - 1998


    • Backward induction
    • Bounded rationality
    • Continuation probability
    • Infinite horizon
    • Parity
    • Uncertainty

    ASJC Scopus subject areas

    • General Decision Sciences
    • Developmental and Educational Psychology
    • Arts and Humanities (miscellaneous)
    • General Economics, Econometrics and Finance
    • Applied Psychology
    • Computer Science Applications
    • General Social Sciences


    Dive into the research topics of 'Backward induction is not robust: The parity problem and the uncertainty problem'. Together they form a unique fingerprint.

    Cite this