<p>Consider the following extensive form:
<a href="https://i.sstatic.net/sWQL5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/sWQL5.png" alt="enter image description here" /></a></p>
<p>Here, player <span class="math-container">$1$</span> has two information sets and case (i) holds. Take the strategy profile <span class="math-container">$(a,ce)$</span>, which leads to terminal node <span class="math-container">$t$</span>. But changing player <span class="math-container">$1$</span>'s action at information set <span class="math-container">$I_2$</span> form <span class="math-container">$e$</span> to <span class="math-container">$f$</span> results in the strategy profile <span class="math-container">$(a,cf)$</span>, which still leads to terminal node <span class="math-container">$t$</span>. Thus, in the normal form, one has <span class="math-container">$u(a,ce)=u(a,cf)=(1,2)$</span>, contradicting the assumption that all payoff vectors are distinct.</p>

CC BY-SA 4.0
2024-01-29T19:30:02.647Z
2024-01-29T19:30:02.647Z
VARulle
<p>I'm trying to prove the following statement:</p>
<blockquote>
<p>Prove that if in a 2X2 normal-form game all payoff vectors are
distinct, then up to relabeling of actions <strong>the game has exactly two
extensive forms in which every decision node has at least two actions
and there are no moves of nature.</strong> Prove that this is true for any
finite two-player normal-form game in which all payoff vectors are
distinct (with at least two strategies for each player).</p>
</blockquote>
<p>Can someone please help me understand the answer? Here is the answer:</p>
<p><a href="https://i.sstatic.net/HeXbl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HeXbl.png" alt="enter image description here" /></a></p>
<p>I struggle to understand the case (i). What does it mean by &quot;changing the action corresponding to the information set <span class="math-container">$I_2$</span> for player 1&quot;? Can someone please draw a game tree to illustrate this?</p>

<game-theory>
Trying to understand the relationship between the normal form and the extensive form
Ludwig Gershwin
# Trying to understand the relationship between the normal form and the extensive form

(Asked by: username_0 on 2024-01-29)

I'm trying to prove the following statement:

> Prove that if in a 2X2 normal-form game all payoff vectors are
> distinct, then up to relabeling of actions **the game has exactly two
> extensive forms in which every decision node has at least two actions
> and there are no moves of nature.** Prove that this is true for any
> finite two-player normal-form game in which all payoff vectors are
> distinct (with at least two strategies for each player).

Can someone please help me understand the answer? Here is the answer:

[![enter image description here](https://i.sstatic.net/HeXbl.png)](https://i.sstatic.net/HeXbl.png)

I struggle to understand the case (i). What does it mean by "changing the action corresponding to the information set $I\_2$ for player 1"? Can someone please draw a game tree to illustrate this?

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2024-01-29. Score: 2 | Accepted answer

Consider the following extensive form:
[![enter image description here](https://i.sstatic.net/sWQL5.png)](https://i.sstatic.net/sWQL5.png)

Here, player $1$ has two information sets and case (i) holds. Take the strategy profile $(a,ce)$, which leads to terminal node $t$. But changing player $1$'s action at information set $I\_2$ form $e$ to $f$ results in the strategy profile $(a,cf)$, which still leads to terminal node $t$. Thus, in the normal form, one has $u(a,ce)=u(a,cf)=(1,2)$, contradicting the assumption that all payoff vectors are distinct.


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