that (1)is still true; in particular, Player 2 will believe that 1 has played T. But then he will best-respond with d, which breaks the original equilibrium Since we can break the(X, u) equilibrium without using a"transformation"of the game, I suggest that we accept the latter, and think about the possibility of refining away"un- reasonable"sequential equilibria. We shall return to this point in the notes on forward induction Addition/Deletion of a Superfluous Move. In my opinion, this is the crucial transformation- one that I would certainly not call"inessential. " Recall that a move of Player i is superfluous if, roughly speaking, (i) it does not influence payoffs; (i) Player i does not know that the move she is making is superfluous; (iii) no opponent of Player i observes the superfluous move. Thus, adding such a move sounds pretty harmless However, the mere eristence of a superfluous move may change the strategic problem faced by Player i. Here is an extreme example: consider the Entry Deterrence game from Lecture 11(Fig. 1 in the notes ); now add two superfluous moves, labelled f and a, after the Entrant's choice N; both moves lead to the terminal payoff corresponding to N in the original game; finally, let(E)and(N) belong to the same information set of the Incumbent You can check the definiton in OR to verify that the moves f and a added after N are indeed superfluous The game one obtains is ostensibly the normal form of the Entry Deterrence game Clearly, we do not consider the two games equivalent! To put it differently, the transformation applied to the Entry Deterrence game has changed the nature of the strategic problem faced by the Incumbent to a dramatic extent In the original formulation, if the Incumbent was called upon to move, he knew that the Entrant had entered: hence. he was certain that a was the better course of action. After the modification, however the incumbent does not observe the choice of the entrant thus we can construct an equilibrium in which the Incumbent threatens to play f, and the Entrant stays out. If we want to push the "entry " story a bit harder, we can even say that this threat is credible. If the entrant chooses e the incumbent does not observe this and continues to believe that, as specified by the equilibrium, the Entrant has actually choosen N. Since the Incumbent is indifferent between f and a after N, f is a best reply. bea this argument at least suggests that addition/ deletion of a superfluous move may not an "inessential" transformation Perfection and Properness Perfe The idea behind perfect equilibrium is that equilibria should be robust to small"trem bles"of the opponents away from the predicted play. This idea applies equally well to the normal and the extensive formthat (1) is still true; in particular, Player 2 will believe that 1 has played T. But then he will best-respond with d, which breaks the original equilibrium. Since we can break the (X, u) equilibrium without using a “transformation” of the game, I suggest that we accept the latter, and think about the possibility of refining away “un￾reasonable” sequential equilibria. We shall return to this point in the notes on forward induction. Addition/Deletion of a Superfluous Move. In my opinion, this is the crucial transformation— one that I would certainly not call “inessential.” Recall that a move of Player i is superfluous if, roughly speaking, (i) it does not influence payoffs; (ii) Player i does not know that the move she is making is superfluous; (iii) no opponent of Player i observes the superfluous move. Thus, adding such a move sounds pretty harmless. However, the mere existence of a superfluous move may change the strategic problem faced by Player i. Here is an extreme example: consider the Entry Deterrence game from Lecture 11 (Fig. 1 in the notes); now add two superfluous moves, labelled f and a, after the Entrant’s choice N; both moves lead to the terminal payoff corresponding to N in the original game; finally, let (E) and (N) belong to the same information set of the Incumbent. You can check the definiiton in OR to verify that the moves f and a added after N are indeed superfluous. The game one obtains is ostensibly the normal form of the Entry Deterrence game. Clearly, we do not consider the two games equivalent! To put it differently, the transformation applied to the Entry Deterrence game has changed the nature of the strategic problem faced by the Incumbent to a dramatic extent. In the original formulation, if the Incumbent was called upon to move, he knew that the Entrant had entered; hence, he was certain that a was the better course of action. After the modification, however, the Incumbent does not observe the choice of the Entrant; thus, we can construct an equilibrium in which the Incumbent threatens to play f, and the Entrant stays out. [If we want to push the “entry” story a bit harder, we can even say that this threat is credible. If the Entrant chooses E, the Incumbent does not observe this, and continues to believe that, as specified by the equilibrium, the Entrant has actually choosen N. Since the Incumbent is indifferent between f and a after N, f is a best reply.] This argument at least suggests that addition/deletion of a superfluous move may not be an “inessential” transformation. Perfection and Properness Perfection The basic idea behind perfect equilibrium is that equilibria should be robust to small “trem￾bles” of the opponents away from the predicted play. This idea applies equally well to the normal and the extensive form. 4