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I am super confused about on and off equilibrium path equilibria. Is it safe to say Nash equilibria can be on or off the equilibrium path due to the fact that one player could be irrational and threaten the other player. What about subgame perfect nash equilibria? Are they always on the equilibrium path?

I am assuming a dynamic game here with perfect information. Any help would be greatly appreciated!!

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  • $\begingroup$ "Is it safe to say Nash equilibria can be on or off the equilibrium path due to the fact that one player could be irrational and threaten the other player." How do you define whether an equilibrium is on a path? $\endgroup$
    – Giskard
    Commented Mar 31 at 17:09
  • $\begingroup$ Say there is a game where player 1 moves A or B, then player 2 moves X or Y. Suppose the players choose the equilibrium strategy (A,(X,Y)). Then the outcome is that player 1 plays A, and player 2 follows this by playing X. Was the equilibrium on this path A,X or not? $\endgroup$
    – Giskard
    Commented Mar 31 at 17:13

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I do not think it is standard to call equilibria on or off path. Here are the standard definitions:

A path is the set of histories induced by some strategy profile (graphically, think of one particular way to go down the game tree, the nodes visited along the way are the induced histories). So any equilibrium, insofar as it selects a particular strategy profile, is on a path, namely on the equilibrium path (graphically, this is the way to go down the game tree as dictated by the equilibrium strategies). A strategy is optimal on the path if it is a (sequential) best response at all histories on the equilibrium path. A strategy is optimal off the path if it is a (sequential) best response at all other histories. Hence, NE requires only on path optimality, whereas SPNE requires both on and off path optimality. This is why SPNE is a stronger equilibrium concept than NE.

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