# When action and strategy differ in game theory

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?

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:

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

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.

• This was extremely helpful. Thank you very much! – Frank Swanton Mar 17 '17 at 19:22
• 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 Mar 17 '17 at 19:28
• 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 Apr 8 '17 at 1:19
• 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. Apr 8 '17 at 15:55