# The order independence of iterated dominance in extensive games

## Author

Listed:
• Micali, Silvio

() (Department of Electrical Engineering and Computer Science, MIT)

• Chen, Jing

() (Institute for Advanced Study, Princeton and Department of Computer Science, Stony Brook University)

## Abstract

Shimoji and Watson (1998) prove that a strategy of an extensive game is rationalizable in the sense of Pearce if and only if it survives the maximal elimination of conditionally dominated strategies. Briefly, this process iteratively eliminates conditionally dominated strategies according to a specific order, which is also the start of an order of elimination of weakly dominated strategies. Since the final set of possible payoff profiles, or terminal nodes, surviving iterated elimination of weakly dominated strategies may be order-dependent, one may suspect that the same holds for conditional dominance. We prove that, although the sets of strategy profiles surviving two arbitrary elimination orders of conditional dominance may be very different from each other, they are equivalent in the following sense: for each player $i$ and each pair of elimination orders, there exists a function $\phi_i$ mapping each strategy of $i$ surviving the first order to a strategy of $i$ surviving the second order, such that, for every strategy profile $s$ surviving the first order, the profile $(\phi_i(s_i))_i$ induces the same {\em terminal node} as $s$ does. To prove our results we put forward a new notion of dominance and an elementary characterization of extensive-form rationalizability (EFR) that may be of independent interest. We also establish connections between EFR and other existing iterated dominance procedures, using our notion of dominance and our characterization of EFR.

## Suggested Citation

• Micali, Silvio & Chen, Jing, 2013. "The order independence of iterated dominance in extensive games," Theoretical Economics, Econometric Society, vol. 8(1), January.
• Handle: RePEc:the:publsh:942
as

File URL: http://econtheory.org/ojs/index.php/te/article/viewFile/20130125/8134/251

## References listed on IDEAS

as
1. Battigalli, Pierpaolo & Friedenberg, Amanda, 2012. "Forward induction reasoning revisited," Theoretical Economics, Econometric Society, vol. 7(1), January.
2. Itzhak Gilboa & E. Kalai & E. Zemel, 1990. "On the order of eliminating dominated strategies," Post-Print hal-00481648, HAL.
3. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, January.
4. Shimoji, Makoto & Watson, Joel, 1998. "Conditional Dominance, Rationalizability, and Game Forms," Journal of Economic Theory, Elsevier, vol. 83(2), pages 161-195, December.
5. Shimoji, Makoto, 2004. "On the equivalence of weak dominance and sequential best response," Games and Economic Behavior, Elsevier, vol. 48(2), pages 385-402, August.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. Arieli, Itai & Aumann, Robert J., 2015. "The logic of backward induction," Journal of Economic Theory, Elsevier, vol. 159(PA), pages 443-464.
2. Rich, Patricia, 2015. "Rethinking common belief, revision, and backward induction," Mathematical Social Sciences, Elsevier, vol. 75(C), pages 102-114.
3. Perea, Andrés, 2014. "Belief in the opponentsʼ future rationality," Games and Economic Behavior, Elsevier, vol. 83(C), pages 231-254.
4. Bonanno, Giacomo, 2014. "A doxastic behavioral characterization of generalized backward induction," Games and Economic Behavior, Elsevier, vol. 88(C), pages 221-241.
5. Müller, Christoph, 2016. "Robust virtual implementation under common strong belief in rationality," Journal of Economic Theory, Elsevier, vol. 162(C), pages 407-450.

### Keywords

Extensive-form rationalizability; dominance; iterative elimination; equivalence;

### JEL classification:

• C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
• C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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