### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 263-289 |

Number of pages | 27 |

Journal | Theory and Decision |

Volume | 45 |

Issue number | 3 |

State | Published - 1998 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Economics and Econometrics
- Social Sciences (miscellaneous)

### Cite this

*Theory and Decision*,

*45*(3), 263-289.

**Backward induction is not robust : The parity problem and the uncertainty problem.** / Brams, Steven; Kilgour, D. Marc.

Research output: Contribution to journal › Article

*Theory and Decision*, vol. 45, no. 3, pp. 263-289.

}

TY - JOUR

T1 - Backward induction is not robust

T2 - The parity problem and the uncertainty problem

AU - Brams, Steven

AU - Kilgour, D. Marc

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

KW - Backward induction

KW - Bounded rationality

KW - Continuation probability

KW - Infinite horizon

KW - Parity

KW - Uncertainty

UR - http://www.scopus.com/inward/record.url?scp=0000290076&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000290076&partnerID=8YFLogxK

M3 - Article

VL - 45

SP - 263

EP - 289

JO - Theory and Decision

JF - Theory and Decision

SN - 0040-5833

IS - 3

ER -