<p>If you happen to know that the "normal form" is sometimes also referred to as the <em>strategic form</em>, you'd most likely not be surprised that it doesn't help you distinguish between the notions of action and strategy. The normal form is precisely used to represent strategies (not actions) in a game. To appreciate the difference between strategies and actions, it's probably best to consider another form of game representation: the extensive form.</p>

<p>Suppose 2 players move sequentially, and that Player 2 observes Player 1's choice before making his decision. The scenario can be represented using a game tree as follows: 
<a href="https://i.sstatic.net/LnMpD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LnMpD.png" alt="enter image description here"></a></p>

<p>Here we can call $\{A,B\}$ Player 2's actions (actually the correct jargon here is <em>behavior strategies</em>, but I sense that this level of nuance is unnecessary at this point). However, they are not his <em>strategies</em>, for strategies in this case is a "full contingent plan" that specifies an action at each information set the player moves. Since Player 2 moves at two information sets (one after $L$ and the other after $R$), so one of his strategies would be </p>

<blockquote>
  <p>"choose <strong>A</strong> if Player 1 chooses L and choose <strong>A</strong> if Player 1 chooses R", or simply denoted as <strong>AA</strong>.</p>
</blockquote>

<p>So, Player 2 has four (pure) strategies: $\{AA, AB, BA, BB\}$. In comparison, since Player 1 only moves at one information set (at the beginning), her actions and strategies are identical: $\{L,R\}$. </p>

<p>To represent this game using the normal/strategic form, we'd have </p>

<p><a href="https://i.sstatic.net/Uflbr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Uflbr.png" alt="enter image description here"></a> </p>

<p>It's not a 2-by-2 game, but 2-by-4, precisely because strategies and actions are not the same. </p>

<p>Lastly, a note on Prisoners' Dilemma (PD) and other similar one-shot simultaneous-move games. If you consider the extensive form of PD, you'd see that each player moves at only one information set, and so their set of actions and set of strategies coincide (just like Player 1 in the above example). This is why you can use the two terms interchangeably in those games. </p>

CC BY-SA 3.0
2017-03-17T19:22:55.803Z
This was extremely helpful. Thank you very much!
Frank Swanton
CC BY-SA 3.0
2017-03-17T19:28:54.147Z
So the way I think about now regarding action and strategy in extensive form game is looking at the number of information sets. In your example, Pl.2 having two information set implies the "potential scenario" of pl.2 being called upon to act, so although in actuality pl.2 will end up in one and only one of her information sets, the "contingency plan", which is strategy requires her to lay out the roadmap for every possible situation she is summoned to make a move. So it makes sense to me now that the distinction is very visible in the extensive form game.
Frank Swanton
CC BY-SA 3.0
2017-03-17T20:07:19.513Z
@FrankSwanton: Yes, your understanding is right on. I'm glad my answer was helpful.
Herr K.
CC BY-SA 3.0
2017-04-08T01:19:05.457Z
I have one question regarding the term 'information set'. When you have a situation where Pl.2 knows exactly where she ends up in her decision node, it is proper to refer them as information set as well? My understanding is that they are called decision nodes. But I have read papers referring them as information set? For example, when converting extensive form to normal form representation, I read some authors say if you have 3 information sets where Pl.1 can be called upon to move and 2 possible actions at each information set, then pl.1 has $3^2$ pure strategies...
Frank Swanton
CC BY-SA 3.0
2017-04-08T15:55:35.393Z
If a player knows exactly which node she's at while deciding, that node is called a *trivial/singleton information set*, as it still represents an informational state: i.e. the player knows exactly what happened previously. As for the number of strategies, it should be the product of the number of strategies at each information set. So if a player moves at $n$ information sets and each information set has $m_i$ ($i=1,\dots,n$) actions, then she'd have $\prod_{i=1}^nm_i$ strategies. In the example you gave, it should be $2\times2\times2=2^3$ strategies.
Herr K.
CC BY-SA 3.0
2017-04-10T00:56:11.973Z
Hey thanks for the reply. The reason I ask is because of the sequential game of lemon market with used cars. I would make a separate question but since this is so relevant to our discussion, I am just commenting, so please understand. Consider a situation nature decides the state of car being good or bad(lemon) and seller knows the state but buyers doesn't. Seller's action is offer price {High, Low} and Buyer's action is to either {accept, reject}. A simple dynamic incomplete game for lemons.
Frank Swanton
CC BY-SA 3.0
2017-04-10T01:01:01.963Z
Since the strategy encompasses the "full contingent plan", seller is called upon to move in two scenarios, used car good or bad. So it would look like ${(H,h),(H,l),(L,h),(L,l)}$. Buyer doesn't know the state of car but observes what she is offered by the seller, so corresponding two information sets, we can describe the buyer's strategy space as ${(A,a),(A,r),(R,a),(R,r)}$. Now here is my question. Suppose we get rid of incomplete information for a moment. Consider everybody knows the car is good. Then, what does the strategy space looks like?
Frank Swanton
CC BY-SA 3.0
2017-04-10T01:03:51.510Z
In that case, I thought it is same with the tree you drew, because now the nature chose the state to be good with probability 1 all the time, so the seller is called upon only to move at "good" scenario, so her contingent plan boils down to just offer price {High, Low}. As for buyer, wouldn't her strategy space, as you illustrated your example, still {(A,a),(A,r),(R,a),(R,r)}, because the "contingency" for buyer is to be offered in high scenario or low scenario for prices? But the normal form representation I see in textbook is just {Accept, Reject}. Which I was puzzled of.
Frank Swanton
CC BY-SA 3.0
2017-04-10T02:59:20.593Z
The long story short, does removing nature (uncertainty in state of used car) make the game structure static and complete information or does it not make sense to make the game structure static? If we are truly in a situation where you are negotiating with a used car dealer, then even if both buyer and seller know the used car is in a good shape, the original structure of dynamic game (i.e. seller offers a price, buyer observes then moves) shouldn't be altered in my view.
Frank Swanton
CC BY-SA 3.0
2017-04-10T15:05:50.927Z
@FrankSwanton: From what you've described (all cars are good), I think the buyer's strategy space should have four strategies. Which textbook are you using? Unless there are some assumptions that says something like both the buyer and the seller move simultaneously, I'm inclined to think that the book is mistaken on this.
Herr K.
CC BY-SA 3.0
2017-04-10T18:37:28.143Z
Yes. It wasn't a mistake. I realized the information given was just to help us draw the game tree for dynamic incomplete game! Haha thanks.
Frank Swanton
CC BY-SA 3.0
2017-03-14T06:24:56.420Z
2017-03-14T06:24:56.420Z
Herr K.
<p>It looks like in a static, simultaneous move game, action is used interchangeably with strategy.</p>

<p>But when do they differ, say in a more complicated game structure? I am trying to understand the difference between the two and try to keep in mind that I should, in general, NOT equate these two.</p>

<p>Even when I think about the normal form game, prisoners' dilemma, it is not all clear why in this case, action is equivalent to strategy and many authors choose to use two terms interchangeably.</p>

<p>When criminal is called upon to act, she can either cooperate or defect. But the strategy set is</p>

<p>$S=S_1$x$S_2=\{(U,U),(U,D),(D,U),(D,D)\}$</p>

<p>Here, how do I see that criminal 1's strategy is equal to her action which is $S_1\{U,D\}$?</p>

<p>And when do these two notions depart from each other?</p>

<microeconomics><game-theory>
When action and strategy differ in game theory
Frank Swanton
# When action and strategy differ in game theory

(Asked by: username_0 on 2017-03-14)

It looks like in a static, simultaneous move game, action is used interchangeably with strategy.

But when do they differ, say in a more complicated game structure? I am trying to understand the difference between the two and try to keep in mind that I should, in general, NOT equate these two.

Even when I think about the normal form game, prisoners' dilemma, it is not all clear why in this case, action is equivalent to strategy and many authors choose to use two terms interchangeably.

When criminal is called upon to act, she can either cooperate or defect. But the strategy set is

$S=S\_1$x$S\_2=\{(U,U),(U,D),(D,U),(D,D)\}$

Here, how do I see that criminal 1's strategy is equal to her action which is $S\_1\{U,D\}$?

And when do these two notions depart from each other?

**Tags:** microeconomics, game-theory


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## 1 answers

### Answer by username_1 on 2017-03-14. Score: 4 | Accepted answer

If you happen to know that the "normal form" is sometimes also referred to as the *strategic form*, you'd most likely not be surprised that it doesn't help you distinguish between the notions of action and strategy. The normal form is precisely used to represent strategies (not actions) in a game. To appreciate the difference between strategies and actions, it's probably best to consider another form of game representation: the extensive form.

Suppose 2 players move sequentially, and that Player 2 observes Player 1's choice before making his decision. The scenario can be represented using a game tree as follows:
[![enter image description here](https://i.sstatic.net/LnMpD.png)](https://i.sstatic.net/LnMpD.png)

Here we can call $\{A,B\}$ Player 2's actions (actually the correct jargon here is *behavior strategies*, but I sense that this level of nuance is unnecessary at this point). However, they are not his *strategies*, for strategies in this case is a "full contingent plan" that specifies an action at each information set the player moves. Since Player 2 moves at two information sets (one after $L$ and the other after $R$), so one of his strategies would be

> "choose **A** if Player 1 chooses L and choose **A** if Player 1 chooses R", or simply denoted as **AA**.

So, Player 2 has four (pure) strategies: $\{AA, AB, BA, BB\}$. In comparison, since Player 1 only moves at one information set (at the beginning), her actions and strategies are identical: $\{L,R\}$.

To represent this game using the normal/strategic form, we'd have

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

It's not a 2-by-2 game, but 2-by-4, precisely because strategies and actions are not the same.

Lastly, a note on Prisoners' Dilemma (PD) and other similar one-shot simultaneous-move games. If you consider the extensive form of PD, you'd see that each player moves at only one information set, and so their set of actions and set of strategies coincide (just like Player 1 in the above example). This is why you can use the two terms interchangeably in those games.

> **Comments:**
> **username_0**: This was extremely helpful. Thank you very much!  
> **username_0**: So the way I think about now regarding action and strategy in extensive form game is looking at the number of information sets. In your example, Pl.2 having two information set implies the "potential scenario" of pl.2 being called upon to act, so although in actuality pl.2 will end up in one and only one of her information sets, the "contingency plan", which is strategy requires her to lay out the roadmap for every possible situation she is summoned to make a move. So it makes sense to me now that the distinction is very visible in the extensive form game.  
> **username_1**: @username_0: Yes, your understanding is right on. I'm glad my answer was helpful.  
> **username_0**: I have one question regarding the term 'information set'. When you have a situation where Pl.2 knows exactly where she ends up in her decision node, it is proper to refer them as information set as well? My understanding is that they are called decision nodes. But I have read papers referring them as information set? For example, when converting extensive form to normal form representation, I read some authors say if you have 3 information sets where Pl.1 can be called upon to move and 2 possible actions at each information set, then pl.1 has $3^2$ pure strategies...  
> **username_1**: If a player knows exactly which node she's at while deciding, that node is called a *trivial/singleton information set*, as it still represents an informational state: i.e. the player knows exactly what happened previously. As for the number of strategies, it should be the product of the number of strategies at each information set. So if a player moves at $n$ information sets and each information set has $m_i$ ($i=1,\dots,n$) actions, then she'd have $\prod_{i=1}^nm_i$ strategies. In the example you gave, it should be $2\times2\times2=2^3$ strategies.  
> **username_0**: Hey thanks for the reply. The reason I ask is because of the sequential game of lemon market with used cars. I would make a separate question but since this is so relevant to our discussion, I am just commenting, so please understand. Consider a situation nature decides the state of car being good or bad(lemon) and seller knows the state but buyers doesn't. Seller's action is offer price {High, Low} and Buyer's action is to either {accept, reject}. A simple dynamic incomplete game for lemons.  
> **username_0**: Since the strategy encompasses the "full contingent plan", seller is called upon to move in two scenarios, used car good or bad. So it would look like ${(H,h),(H,l),(L,h),(L,l)}$. Buyer doesn't know the state of car but observes what she is offered by the seller, so corresponding two information sets, we can describe the buyer's strategy space as ${(A,a),(A,r),(R,a),(R,r)}$. Now here is my question. Suppose we get rid of incomplete information for a moment. Consider everybody knows the car is good. Then, what does the strategy space looks like?  
> **username_0**: In that case, I thought it is same with the tree you drew, because now the nature chose the state to be good with probability 1 all the time, so the seller is called upon only to move at "good" scenario, so her contingent plan boils down to just offer price {High, Low}. As for buyer, wouldn't her strategy space, as you illustrated your example, still {(A,a),(A,r),(R,a),(R,r)}, because the "contingency" for buyer is to be offered in high scenario or low scenario for prices? But the normal form representation I see in textbook is just {Accept, Reject}. Which I was puzzled of.  
> **username_0**: The long story short, does removing nature (uncertainty in state of used car) make the game structure static and complete information or does it not make sense to make the game structure static? If we are truly in a situation where you are negotiating with a used car dealer, then even if both buyer and seller know the used car is in a good shape, the original structure of dynamic game (i.e. seller offers a price, buyer observes then moves) shouldn't be altered in my view.  
> **username_1**: @username_0: From what you've described (all cars are good), I think the buyer's strategy space should have four strategies. Which textbook are you using? Unless there are some assumptions that says something like both the buyer and the seller move simultaneously, I'm inclined to think that the book is mistaken on this.  
> **username_0**: Yes. It wasn't a mistake. I realized the information given was just to help us draw the game tree for dynamic incomplete game! Haha thanks.  

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