<p>In game theory, we often say that a strategy (or game) might admit equivalent representations. For example, some games can be equivalently represented either in extensive form (using a tree diagram) or in normal form. Similarly, there are cases when strategies can be equivalently be represented either as mixed or behavioural strategies.</p>
<p>Can I ask how one goes about proving such equivalences? Intuitively, this seems to be a matter of showing that there is a bijection between two classes of game/strategy.</p>
<p>To provide some context, I use a non-standard representation of strategies in one of my papers. It is clear how to convert strategies from the standard representation to my representation, or to do the reverse. But I am not sure that I have proven the 'equivalence' in a sufficiently formal manner.</p>
<p>Thank you in advance for any ideas.</p>

CC BY-SA 4.0
2022-10-13T14:31:02.047Z
"*It is clear how to convert strategies*" If "clear" means an algorithm that anyone can follow, you are good.
Giskard
CC BY-SA 4.0
2022-10-13T18:17:55.967Z
I don't think bijections are what you are looking for. Many different extensive form games can have the same normal form, and many mixed strategies can induce the same behavior strategy.
Michael Greinecker
CC BY-SA 4.0
2022-10-13T21:37:20.787Z
@MichaelGreinecker Yes for sure, but I was imagining restricting the class of games so that a bijection does obtain.
afreelunch
CC BY-SA 4.0
2022-10-14T06:48:53.930Z
@afreelunch That is at least very different from the examples. Also, the equivalence between mixed and behavior strategies in games of perfect recall is something that can be clearly formalized and proven, but whether the normal form is strategically equivalent to an extensive form it represents is a philosophical issue on which man papers have been written. In the current form, it is not really clear what you are asking for.
Michael Greinecker
<game-theory>
Equivalent representations in game theory
afreelunch
# Equivalent representations in game theory

(Asked by: username_0 on 2022-10-13)

In game theory, we often say that a strategy (or game) might admit equivalent representations. For example, some games can be equivalently represented either in extensive form (using a tree diagram) or in normal form. Similarly, there are cases when strategies can be equivalently be represented either as mixed or behavioural strategies.

Can I ask how one goes about proving such equivalences? Intuitively, this seems to be a matter of showing that there is a bijection between two classes of game/strategy.

To provide some context, I use a non-standard representation of strategies in one of my papers. It is clear how to convert strategies from the standard representation to my representation, or to do the reverse. But I am not sure that I have proven the 'equivalence' in a sufficiently formal manner.

Thank you in advance for any ideas.

**Tags:** game-theory

> **Comments:**
> **username_1**: "*It is clear how to convert strategies*" If "clear" means an algorithm that anyone can follow, you are good.  
> **username_2**: I don't think bijections are what you are looking for. Many different extensive form games can have the same normal form, and many mixed strategies can induce the same behavior strategy.  
> **username_0**: @username_2 Yes for sure, but I was imagining restricting the class of games so that a bijection does obtain.  
> **username_2**: @username_0 That is at least very different from the examples. Also, the equivalence between mixed and behavior strategies in games of perfect recall is something that can be clearly formalized and proven, but whether the normal form is strategically equivalent to an extensive form it represents is a philosophical issue on which man papers have been written. In the current form, it is not really clear what you are asking for.  

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