Abstract
Pearce’s (Econometrica 52:1029–1050, 1984) extensive-form rationalizablity (EFR) is a solution concept embodying a best-rationalization principle (Battigalli, Games Econ Behav 13:178–200, 1996; Battigalli and Siniscalchi, J Econ Theory 106:356–391, 2002) for forward-induction reasoning. EFR strategies may hence be distinct from backward-induction (BI) strategies. We provide a direct and transparent proof that, in perfect-information games with no relevant ties, the unique BI outcome is nevertheless identical to the unique EFR outcome, even when the EFR strategy profile and the BI strategy profile are distinct.
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Notes
That is, games where each player’s payoffs are distinct from one another in the leaves which follow each of this player’s decision nodes.
Somewhat of a misnomer—if the player at the root of the game has a dominated move and she chooses it, at subsequent nodes the other players ‘strongly believe she is rational’ by attributing to her up-front irrationality.
Like EFR, also explicable equilibrium is based on some kind of best-rationalization principle. However, the two concepts are different.
To be more precise, the condition in Pearce and Battigalli is equivalent to Bayesian updating, in the sense that it gives the same probability ratios for strategies that are not ruled out by information sets, but it does not require “normalization”.
It can be shown, namely, that whenever a strategy is optimal for a belief vector that strongly believes a set of opponents’ strategy profiles, then it is also optimal for an equivalent belief vector that strongly believes this same set of strategy profiles, and which satisfies Bayesian updating.
Here, a plan of action represents a class of behaviorally equivalent strategies, that is, a class of strategies that reach the same decision nodes for player \(i\), and make the same choices at these decision nodes. Actually, it can be shown that not only the plans of action induced by \(S_{i}^{\infty }\) are the same in our setting as in Pearce’s and Battigalli’s, but also the plans of action induced by \(S_{i}^{k}\), at every step \(k\) of the procedure.
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We would like to thank an associate editor and two referees for very useful comments.
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Heifetz, A., Perea, A. On the outcome equivalence of backward induction and extensive form rationalizability. Int J Game Theory 44, 37–59 (2015). https://doi.org/10.1007/s00182-014-0418-x
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DOI: https://doi.org/10.1007/s00182-014-0418-x