Can I find the rationalizable strategies for a game where none of the players has strict dominance but only weak dominance?
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1$\begingroup$ Welcome to Economics:SE. Thank you for your question; please consider revising it to be more in line with our community expectations. Like many other stacks, we expect questions to provide evidence of prior research. That helps us to understand the question, and avoids our repeating work you've already done. Our help center, and other stacks provide additional resources to assist with revisions. $\endgroup$– 1muflon1 ♦Commented Feb 25, 2021 at 17:00
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$\begingroup$ "Can I find the rationalizable strategies..." -- Yes, rationalizable strategies exist as long as the game has a Nash equilibrium. $\endgroup$– Herr K.Commented Feb 25, 2021 at 19:18
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$\begingroup$ What do you mean by a player "has weak dominance"? Strategies can be weakly dominated or weakly dominant. Players "having weak dominance" doesn't make much sense. $\endgroup$– VARulleCommented Feb 26, 2021 at 10:43
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1 Answer
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In 2-player games, the strategies that survive iterated elimination of strictly dominated strategies are called rationalizable. Note that even if no strategy is strictly dominant, there can be strictly dominated strategies. If you cannot eliminate any strategy, then all strategies are rationalizable.
Only if correlation of players' randomization is allowed, all strategies that are rationalizable (not never-a-best response) are also equivalent to those that survive iterated elimination of strictly dominated strategies in games with more players.
In any case, you can always find rationalizable strategies if a best response exists -- independent of whether a (strictly or weakly) dominant strategy exists.
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3$\begingroup$ I think rationalizable strategies are those that survive iterated elimination of never-a-best-response strategies, which are not necessarily the same as the strictly dominated ones. MWG p.245 has section that explains the difference. $\endgroup$– Herr K.Commented Feb 26, 2021 at 3:17
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1$\begingroup$ I see. I had two-player games in minde where both should be the same. I believe for more players both only differ if you restrict the kind of mixed strategies. I will have a look as I am curious for an example. $\endgroup$– BayesianCommented Feb 26, 2021 at 8:49
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2$\begingroup$ With more than two players a strategy is undominated if and only if it is a best reply to some probability distribution of over the strategies of the other players. Since mixed strategies induce independent randomizations over strategies only, the result need not work with mixed strategies instead. There should be counterexamples in textbooks. $\endgroup$ Commented Feb 26, 2021 at 14:13
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$\begingroup$ Great comments! Feel free to edit my post to improve it. This is the point in time where I would become hand-wavy and introduce a public randomization device. $\endgroup$– BayesianCommented Feb 26, 2021 at 15:42