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Why do we include actions even on nodes which will not be reached if the strategy is followed, while defining the pure strategies of an extensive form game?

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    $\begingroup$ Does the answer to this question (economics.stackexchange.com/q/15747/42) help? $\endgroup$ – Herr K. Apr 28 at 18:33
  • $\begingroup$ Because we care about the behaviour of the agent in every node of the game. We need to know the actions that are off the equilibrium path, even the probability to reach this branch of the tree is zero and these actions are non credible, because it tell as about the agents rationality. In most games we do not have the chance to know all the actions that are possible to be followed by some or all the agents, so we can not have a clear vision about the instrumental rationality in any possible path of the game (even those that are not subgame perfect). $\endgroup$ – Hunger Learn Apr 28 at 18:41
  • $\begingroup$ Think of those actions as threats: "if you deviate and do x then I'll do y." Thus, subgame perfection is in a sense a refinement of NE that formalizes the idea of a credible threat. $\endgroup$ – Mmmmmm Apr 28 at 19:05
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Think about a player (female) that moves first and is considering the profitability of deviating. Let's say that according to the strategy she is supposed to play action $x$ and then the second player (male) is supposed to play action $a$. It will not be correct for the first player to think that if she instead chooses $y$, the second player will necessarily still play $a$. Remember this is a sequential game, so after seeing that she chooses $y$ instead of $x$ player 2 can also deviate to some other action, say $b$. Therefore, by specifying the actions that each player will take in each node (even in the ones that are not supposed to be reached according to the strategy) we completely specify the strategy of each player, and can correctly check that no player has profitable deviations.

In fact, in many games, the only reason why the player that moves first chooses an action $x$ is because of the response of the other player after observing $x$, and the knowledge that choosing something else will also affect the decisions of other players, etc.

In a practical example, if you are playing chess and consider moving your queen to get one step closer to eat an important piece of the other player, you must anticipate they will react to this movement and protect their piece, or attack something more valuable of yours, etc. This is why when thinking about the strategy of the other player you must consider how different they will react depending on which piece you move next and to where.

BTW, chess is really hard to fully specify as a sequential game precisely because the number of possible nodes is massive, and properly specifying a complete strategy for each player has proven extremely hard. However, hopefully it gave you some insight of why actions and strategies are not the same things in an extensive game.

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  • $\begingroup$ Thank you, this clears it. $\endgroup$ – Renee Anthony Apr 29 at 8:22
  • $\begingroup$ Great, in that case, would you mind marking the answer as accepted? $\endgroup$ – Regio Apr 29 at 15:06
  • $\begingroup$ I am sorry, I did not know I was supposed to do that. (New to this). $\endgroup$ – Renee Anthony Apr 29 at 16:47
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    $\begingroup$ No problem :) welcome to the site. $\endgroup$ – Regio Apr 29 at 17:29

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