<p>So <span class="math-container">$h$</span> is just some history of the game.
Consider the following game, where Player 1 first decides Heads or Tails, then depending on his choice, a coin is flipped whose outcome and probabilities depends on the choice of Player 1.</p>
<p><a href="https://i.sstatic.net/ySNlw.png" rel="noreferrer"><img src="https://i.sstatic.net/ySNlw.png" alt="enter image description here" /></a></p>
<p>An example of a history <span class="math-container">$h$</span> is <span class="math-container">$h = Heads$</span>, i.e. after heads is chosen.</p>
<p>Consider the very beginning of the game, before Player 1 has chosen anything, it is the empty history <span class="math-container">$h = \emptyset$</span>.</p>
<p>By the definition above, <span class="math-container">$P(\emptyset) = 1$</span>, since player 1 moves there, <span class="math-container">$P$</span> tells us who moves there.</p>
<p>On the other hand, consider <span class="math-container">$h = Tails$</span>, so the node right after Player 1 has chosen Tails.</p>
<p>(i) Only nature moves (NOT a player!) so then <span class="math-container">$P(Tails) = \emptyset$</span>.</p>
<p>(ii) Now, <span class="math-container">$A(h)$</span> tells us what moves are available to whoever is moving at that point. So for our example, <span class="math-container">$A(\emptyset)  = \{Heads,Tails\}$</span>, and <span class="math-container">$A(Heads) = \{1,2\}$</span>.</p>
<p>(iii) Finally, <span class="math-container">$f$</span> just tells us what the probabilities of nature's moves. So <span class="math-container">$f(1 | Heads) = Pr(\text{1 happens | Heads}) = 3/4$</span>.</p>
<p>For completion, <span class="math-container">$f(3 | Tails) = 1/3$</span>.</p>
<p>To summarise, <span class="math-container">$P$</span> tells us who moves at a node. If <span class="math-container">$P(h) = \emptyset$</span>, Nature is moving.</p>
<p><span class="math-container">$A_i(h)$</span> tells us the moves available to Player <span class="math-container">$i$</span> at node <span class="math-container">$h$</span>.
In the case nature is moving, it is just the available outcomes of nature's randomisation, and the randomisation occurs via the probabilities described by <span class="math-container">$f(\cdot | h)$</span>.</p>

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2021-11-15T17:31:56.773Z
You mean that when $h=\emptyset$, in other words at the starting point, then the players who moves is player $1$ namely $P(\emptyset)=1$?
Hunger Learn
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2021-11-15T18:06:32.847Z
Yessir, indeed.
Walrasian Auctioneer
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2021-11-15T19:15:19.757Z
thank you. In essence, this player function tells us, after histroy $h$ has moved forward, which players still remain in the game. Maybe, there are players that do not go on playing the game infinitely for many reasons. However, this chance move function is an inovvation that I have not seen anywhere else. I can not understand what is the intuition behind this...
Hunger Learn
CC BY-SA 4.0
2021-11-15T21:36:03.670Z
I would say $P$ just tells you who moves at that point in time, it does not tell you about future events. I'm surprised you have never seen chance moves on a game tree, its quite standard.
Walrasian Auctioneer
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2021-11-16T10:34:30.450Z
You either stay home and get 0 payoff or you go for a walk. If you go, it will be rain or sunshine with prob. 1/2 each and payoffs -1 and 1, respectively. That's a "move of nature" or a chance event, and by definition $P(go)=\emptyset$. I think that's very natural and the "intuition behind this" is not mysterious.
VARulle
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2021-11-17T11:21:42.160Z
@WalrasianAuctioneer so we could simply say that $P$ is like a count function. I mean in the fist game all the players may play it. If we say they are $I$ players, then they all play the game. But in the next stage or a future stage maybe a $K$ out of $I$ players $(|K|<|I|)$ do not participate and hence we know form the player function that the rest of the players, namely $|I|-|K|$ are playing the game...right?
Hunger Learn
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2021-11-17T18:23:19.843Z
@HungerLearn sure, you could interpret it that way.
Walrasian Auctioneer
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2021-11-15T16:55:33.853Z
2021-11-15T16:55:33.853Z
Walrasian Auctioneer
<p>In <a href="https://mpra.ub.uni-muenchen.de/25895/1/MPRA_paper_25895.pdf" rel="nofollow noreferrer">Heller et al</a>, they use the Osborne and Rubinstein formal definition for the extensive form games with public information. To some point they refer to the following two properties</p>
<ul>
<li><span class="math-container">$P$</span> is a mapping that assigns to each non-terminal history <span class="math-container">$h$</span> the set of players <span class="math-container">$P(h)\subseteq I$</span> that have to take an action after history <span class="math-container">$h$</span>. If <span class="math-container">$P(h)= \emptyset$</span>, then there is a chance move after history <span class="math-container">$h$</span>.</li>
</ul>
<p><span class="math-container">$\textbf{Question 1:}$</span> What does this prhase mean: ``<span class="math-container">$P(h)= \emptyset$</span>, then there is a chance move after history <span class="math-container">$h$</span>&quot;? Is it like the game is played upon some graph?</p>
<ul>
<li><p><span class="math-container">$A$</span> is a mapping that assigns to every non-terminal history <span class="math-container">$h$</span> such that <span class="math-container">$P(h)\neq ∅$</span>, and to every player <span class="math-container">$i \in P(h)$</span>, a finite set <span class="math-container">$A_i (h)$</span> of actions available to player <span class="math-container">$i$</span> after that history. Let <span class="math-container">$A(h)$</span> be the set of available action-profiles at <span class="math-container">$h$</span>: <span class="math-container">$A(h) = ×i∈P(h)$</span> <span class="math-container">$A_i (h)$</span>. If <span class="math-container">$P(h)=∅$</span> for some non-terminal history <span class="math-container">$h$</span>, then <span class="math-container">$A(h)$</span> is the finite set of chance moves at the history <span class="math-container">$h$</span>.</p>
</li>
<li><p><span class="math-container">$f$</span> is a mapping that assigns to every non-terminal history <span class="math-container">$h$</span> such that <span class="math-container">$P(h)=\emptyset$</span>, a probability distribution <span class="math-container">$f(·|h)$</span> over chance moves <span class="math-container">$A(h)$</span>. That is, when chance has to move after a non-terminal history <span class="math-container">$h$</span>, an action <span class="math-container">$a \in A(h)$</span> is chosen according to the probability distribution <span class="math-container">$f(·| h)$</span>.</p>
</li>
</ul>
<p><span class="math-container">$\textbf{Question $2$:}$</span> Could anybody give an example about how these three properties work for some game?</p>

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2021-11-15T10:36:45.370Z
Prisoner's dilemma and Chicken game are simultaneous move games, so their extensive form would be rather atypical. Also, they don't have chance moves.
VARulle
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2021-11-15T10:45:33.970Z
@VARulle then think of some game.... I will change the question.
Hunger Learn
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2021-11-15T12:43:18.957Z
It's not yet clear what the question really is. These points are not "properties" of a game but the *definition* of an extensive form game with public information.
VARulle
CC BY-SA 4.0
2021-11-15T14:00:17.633Z
@VARulle ok1 Then the question is, show us with an example what each one of these means? The defitiotion has these three parts. Tell as what you understand from them? Can you give an example so as we could understand how the game works too?
Hunger Learn
<mathematical-economics><game-theory><probability><extensive-games>
Understanding the properties of extensive form games
Hunger Learn
# Understanding the properties of extensive form games

(Asked by: username_0 on 2021-11-15)

In [Heller et al](https://mpra.ub.uni-muenchen.de/25895/1/MPRA_paper_25895.pdf), they use the Osborne and Rubinstein formal definition for the extensive form games with public information. To some point they refer to the following two properties

* $P$ is a mapping that assigns to each non-terminal history $h$ the set of players $P(h)\subseteq I$ that have to take an action after history $h$. If $P(h)= \emptyset$, then there is a chance move after history $h$.

$\textbf{Question 1:}$ What does this prhase mean: ``$P(h)= \emptyset$, then there is a chance move after history $h$"? Is it like the game is played upon some graph?

* $A$ is a mapping that assigns to every non-terminal history $h$ such that $P(h)\neq ∅$, and to every player $i \in P(h)$, a finite set $A\_i (h)$ of actions available to player $i$ after that history. Let $A(h)$ be the set of available action-profiles at $h$: $A(h) = ×i∈P(h)$ $A\_i (h)$. If $P(h)=∅$ for some non-terminal history $h$, then $A(h)$ is the finite set of chance moves at the history $h$.
* $f$ is a mapping that assigns to every non-terminal history $h$ such that $P(h)=\emptyset$, a probability distribution $f(·|h)$ over chance moves $A(h)$. That is, when chance has to move after a non-terminal history $h$, an action $a \in A(h)$ is chosen according to the probability distribution $f(·| h)$.

$\textbf{Question $2$:}$ Could anybody give an example about how these three properties work for some game?

**Tags:** mathematical-economics, game-theory, probability, extensive-games

> **Comments:**
> **username_1**: Prisoner's dilemma and Chicken game are simultaneous move games, so their extensive form would be rather atypical. Also, they don't have chance moves.  
> **username_0**: @username_1 then think of some game.... I will change the question.  
> **username_1**: It's not yet clear what the question really is. These points are not "properties" of a game but the *definition* of an extensive form game with public information.  
> **username_0**: @username_1 ok1 Then the question is, show us with an example what each one of these means? The defitiotion has these three parts. Tell as what you understand from them? Can you give an example so as we could understand how the game works too?  

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## 1 answers

### Answer by username_2 on 2021-11-15. Score: 5 | Accepted answer

So $h$ is just some history of the game.
Consider the following game, where Player 1 first decides Heads or Tails, then depending on his choice, a coin is flipped whose outcome and probabilities depends on the choice of Player 1.

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

An example of a history $h$ is $h = Heads$, i.e. after heads is chosen.

Consider the very beginning of the game, before Player 1 has chosen anything, it is the empty history $h = \emptyset$.

By the definition above, $P(\emptyset) = 1$, since player 1 moves there, $P$ tells us who moves there.

On the other hand, consider $h = Tails$, so the node right after Player 1 has chosen Tails.

(i) Only nature moves (NOT a player!) so then $P(Tails) = \emptyset$.

(ii) Now, $A(h)$ tells us what moves are available to whoever is moving at that point. So for our example, $A(\emptyset) = \{Heads,Tails\}$, and $A(Heads) = \{1,2\}$.

(iii) Finally, $f$ just tells us what the probabilities of nature's moves. So $f(1 | Heads) = Pr(\text{1 happens | Heads}) = 3/4$.

For completion, $f(3 | Tails) = 1/3$.

To summarise, $P$ tells us who moves at a node. If $P(h) = \emptyset$, Nature is moving.

$A\_i(h)$ tells us the moves available to Player $i$ at node $h$.
In the case nature is moving, it is just the available outcomes of nature's randomisation, and the randomisation occurs via the probabilities described by $f(\cdot | h)$.

> **Comments:**
> **username_0**: You mean that when $h=\emptyset$, in other words at the starting point, then the players who moves is player $1$ namely $P(\emptyset)=1$?  
> **username_2**: Yessir, indeed.  
> **username_0**: thank you. In essence, this player function tells us, after histroy $h$ has moved forward, which players still remain in the game. Maybe, there are players that do not go on playing the game infinitely for many reasons. However, this chance move function is an inovvation that I have not seen anywhere else. I can not understand what is the intuition behind this...  
> **username_2**: I would say $P$ just tells you who moves at that point in time, it does not tell you about future events. I'm surprised you have never seen chance moves on a game tree, its quite standard.  
> **username_1**: You either stay home and get 0 payoff or you go for a walk. If you go, it will be rain or sunshine with prob. 1/2 each and payoffs -1 and 1, respectively. That's a "move of nature" or a chance event, and by definition $P(go)=\emptyset$. I think that's very natural and the "intuition behind this" is not mysterious.  
> **username_0**: @WalrasianAuctioneer so we could simply say that $P$ is like a count function. I mean in the fist game all the players may play it. If we say they are $I$ players, then they all play the game. But in the next stage or a future stage maybe a $K$ out of $I$ players $(|K|<|I|)$ do not participate and hence we know form the player function that the rest of the players, namely $|I|-|K|$ are playing the game...right?  
> **username_2**: @HungerLearn sure, you could interpret it that way.  

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