<p>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.</p>
<p>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.</p>
<p>In any case, you can always find rationalizable strategies if a best response exists -- independent of whether a (strictly or weakly) dominant strategy exists.</p>

CC BY-SA 4.0
2021-02-26T03:17:13.510Z
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.
Herr K.
CC BY-SA 4.0
2021-02-26T08:49:09.273Z
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.
Bayesian
CC BY-SA 4.0
2021-02-26T14:13:18.403Z
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.
Michael Greinecker
CC BY-SA 4.0
2021-02-26T15:42:12.303Z
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.
Bayesian
CC BY-SA 4.0
2021-02-25T23:06:25.470Z
2021-02-26T10:29:36.593Z
2021-02-26T10:29:36.593Z
Bayesian
<p>Can I find the rationalizable strategies for a game where none of the players has strict dominance but only weak dominance?</p>

CC BY-SA 4.0
2021-02-25T17:00:02.273Z
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](https://economics.stackexchange.com/help/how-to-ask). That helps us to understand the question, and avoids our repeating work you've already done. Our [help center](https://economics.stackexchange.com/help), and [other stacks](https://interpersonal.meta.stackexchange.com/questions/3224/how-do-i-write-a-good-question) provide additional resources to assist with revisions.
1muflon1
CC BY-SA 4.0
2021-02-25T19:18:00.597Z
"*Can I find the rationalizable strategies...*" -- Yes, rationalizable strategies exist as long as the game has a Nash equilibrium.
Herr K.
CC BY-SA 4.0
2021-02-26T10:43:52.073Z
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.
VARulle
<game-theory><nash-equilibrium><dominant-strategy><rationality>
Rationalizable strategies and Weak Dominance
Micro Economics
# Rationalizable strategies and Weak Dominance

(Asked by: username_0 on 2021-02-25)

Can I find the rationalizable strategies for a game where none of the players has strict dominance but only weak dominance?

**Tags:** game-theory, nash-equilibrium, dominant-strategy, rationality
> **Comments:**
> **username_1**: 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](https://economics.stackexchange.com/help/how-to-ask). That helps us to understand the question, and avoids our repeating work you've already done. Our [help center](https://economics.stackexchange.com/help), and [other stacks](https://interpersonal.meta.stackexchange.com/questions/3224/how-do-i-write-a-good-question) provide additional resources to assist with revisions.  
> **username_2**: "*Can I find the rationalizable strategies...*" -- Yes, rationalizable strategies exist as long as the game has a Nash equilibrium.  
> **username_3**: 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.  

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## 1 answers

### Answer by username_4 on 2021-02-25. Score: 2

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.

> **Comments:**
> **username_2**: 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.  
> **username_4**: 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.  
> **username_5**: 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.  
> **username_4**: 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.  

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