<p>I rely on the definitions from Chapter 2 of the Handbook of Game Theory, Volume 1, by Sergiu Hart.</p>

<p>If I understand you correctly, the proposition can be re-written as</p>

<p><strong>Proposition</strong>. For, every finite extensive form game $\Gamma^E$, there exists a single
 strategic form representation $\Gamma^N =[I^N,\{S^N_i\},\{u^N_i(\cdot)\}]$ (up to relabeling of the agents) such that</p>

<ol>
<li>$I^N = I^E$,</li>
</ol>

<p>and for all $i\in I^N = I^E$,</p>

<ol start="2">
<li><p>$S^N_i = \{$ pure strategies of $i$ in $\Gamma^E$ $\}$ ,</p></li>
<li><p>and $u^N_i(s) = u^E_i(c(s))$, where $c(s) $ associates every profile of pure strategy in $\Gamma^E$  with a terminal node of  $\Gamma^E$ resulting from the pure strategy profile $s$.</p></li>
</ol>

<p>I think 1. and 2. are obvious. There only remains to show 3, which is equivalent to proving that $c(s)$ is a function, i.e. every profile of pure strategies is associated with one and only one terminal node in $\Gamma^E$. This follows directly from the fact that the pure strategy of some player $i$ is a function selecting one and only one possible action from every information set. </p>

<ul>
<li>Start from the root node, $r_0$. </li>
<li>By definition of a game in extensive form, the nodes are partitioned between the agents.</li>
<li>Thus $r_0$ belongs to some agent $i_0$.</li>
<li>By definition of the game in extensive form, the nodes of $i_0$ are partitioned into information sets.</li>
<li>Thus $r_0$ belongs to some information set of $i_0$, say $H_i^0$.</li>
<li>By definition of the pure strategy $s$, for any node in  $H_i^0$, $i_0$ always choses a <em>single</em> successor, among the possible successors at $H_i^0$.</li>
<li>Let this successor be $r_1$.</li>
<li><p>Then $r_1$ must be the successor of $r_0$, and the next node on the path.</p></li>
<li><p>Now consider $r_1$.</p></li>
<li>By definition of agame in extensive form, the nodes are partitioned between the agents.</li>
<li>Thus $r_1$ belong to some agent $i_1$ (where $i_1 = i_0$ is allowed).</li>
<li>$\vdots$</li>
<li>Repeating the argument as many time as needed, because the game is finite, we necessarily reach a single terminal node $r_* = c(s)$.</li>
</ul>

CC BY-SA 3.0
2015-01-28T06:03:08.073Z
2016-01-07T05:05:52.027Z
2016-01-07T05:05:52.027Z
Martin Van der Linden
<blockquote>
  <p><strong>Proposition.</strong> Every finite extensive form game is associated with a unique strategic form representation.</p>
</blockquote>

<p>I think this proposition is true. But how do we prove it rigorously?</p>

CC BY-SA 3.0
2014-12-15T00:42:59.093Z
This naturally depends on the definition of an extensive form you use.
Michael Greinecker
CC BY-SA 3.0
2015-01-28T04:24:24.057Z
And on what you mean by "is associated with".
Martin Van der Linden
<game-theory>
Strategic form representation of extensive form games
Herr K.
# Strategic form representation of extensive form games

(Asked by: username_0 on 2014-12-04)

> **Proposition.** Every finite extensive form game is associated with a unique strategic form representation.

I think this proposition is true. But how do we prove it rigorously?

**Tags:** game-theory

> **Comments:**
> **username_1**: This naturally depends on the definition of an extensive form you use.  
> **username_2**: And on what you mean by "is associated with".  

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## 1 answers

### Answer by username_2 on 2015-01-28. Score: 3 | Accepted answer

I rely on the definitions from Chapter 2 of the Handbook of Game Theory, Volume 1, by Sergiu Hart.

If I understand you correctly, the proposition can be re-written as

**Proposition**. For, every finite extensive form game $\Gamma^E$, there exists a single
strategic form representation $\Gamma^N =[I^N,\{S^N\_i\},\{u^N\_i(\cdot)\}]$ (up to relabeling of the agents) such that

1. $I^N = I^E$,

and for all $i\in I^N = I^E$,

2. $S^N\_i = \{$ pure strategies of $i$ in $\Gamma^E$ $\}$ ,
3. and $u^N\_i(s) = u^E\_i(c(s))$, where $c(s) $ associates every profile of pure strategy in $\Gamma^E$ with a terminal node of $\Gamma^E$ resulting from the pure strategy profile $s$.

I think 1. and 2. are obvious. There only remains to show 3, which is equivalent to proving that $c(s)$ is a function, i.e. every profile of pure strategies is associated with one and only one terminal node in $\Gamma^E$. This follows directly from the fact that the pure strategy of some player $i$ is a function selecting one and only one possible action from every information set.

* Start from the root node, $r\_0$.
* By definition of a game in extensive form, the nodes are partitioned between the agents.
* Thus $r\_0$ belongs to some agent $i\_0$.
* By definition of the game in extensive form, the nodes of $i\_0$ are partitioned into information sets.
* Thus $r\_0$ belongs to some information set of $i\_0$, say $H\_i^0$.
* By definition of the pure strategy $s$, for any node in $H\_i^0$, $i\_0$ always choses a *single* successor, among the possible successors at $H\_i^0$.
* Let this successor be $r\_1$.
* Then $r\_1$ must be the successor of $r\_0$, and the next node on the path.
* Now consider $r\_1$.
* By definition of agame in extensive form, the nodes are partitioned between the agents.
* Thus $r\_1$ belong to some agent $i\_1$ (where $i\_1 = i\_0$ is allowed).
* $\vdots$
* Repeating the argument as many time as needed, because the game is finite, we necessarily reach a single terminal node $r\_\* = c(s)$.


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