Game theory - Definition problem - Strategy vs Action

Question

A one definition is given by :

Strategy is one of the possible actions of a player

But I'm having a trouble. When working with games, We specify a strategy set $S$ and a strategy $s \epsilon S$. For instance consider a typical normal form game, battle of sexes for instance. We have $2$ strategies for each player so the set $S$ has two elements. According to the definition here what would be the actions? What is a "possible action"?

I'm asking this because in the definition of Bayesian games we are given an Action set instead of a strategy set. Can anyone clear things up to me?

Answer 1

I had the same question few weeks ago. I attach the link below that may help you, but here is the approach I took to solidify the difference.

When action and strategy differ in game theory

Don't associate the distinction between action and strategy with whether the game is in normal form or extensive form or sequential or not. This can cause more confusion.

When you want to define strategy, always think of "contingency".

Action is a set of possible things you can do when you are called upon to move. For example, suppose you want to sell a used car as dealer. As seller, what is it that you CAN DO when you are "called upon to move"? Simple: offer high price or low price.

But you immediately realize things can get complicated if, say, dealer has more information on the car's condition. You might suspect that car behind the dealer is a good used card or lemon. Although you still get to offer high price or low price at the end of the day, as player of the game, you must set a contingency plan when the car is good or lemon.

Harsanyi introduces "nature" to do exactly this. The condition of car is simple: good or lemon. This is described as the state space of the game. Both players, seller and buyer, understand the ex-ante probability distribution over this state space, meaning you and potential buyer know the probability you get good car or lemon. Therefore, as dealer, you would have action space $A_1=\{H,L\}$ indicating offering high or low price. However, your strategy space would be $S_1=\{(H,h),(H,l),(L,h),(L,l)\}$ where first entry of the pair is your action when the car is good and second entry is your action when the car is lemon. It is a game plan laid out for you in every possible scenario.

One good exercise may be to see if you can construct the strategy set for buyer if she doesn't observe the condition (i.e. state) of the car the nature draws.

Answer 2

The definition "Strategy is one of the possible actions of a player" is very poorly phrased. As you note there is an important distinction between action sets and strategy sets in some games, e.g. sequantial and Bayesian games.

A strategy selects an action in each information set of the player. If there is only one information set, e.g. simultaneous move games, e.g. battle of the sexes, then the action set of that player coincides with her strategy set.