strategy profiles which reach I, S(I). Now, the latter is a normal-form object: it is simply a set of strategies. The key point is that, in order to define it, we used the information contained in the definition of the extensive game T: the set s(n) is not part of the formal description of the normal-form game GI I Note for the interested reader: Mailath, Samuelson and Swinkels(1993) characteriz the sets of strategy profiles which can correspond to some information set in some extensive game with a given normal form. Their characterization only relies on the properties of the normal-form payoff functions, and is thus purely "normal-form"in nature and inspiration However, a more sophisticated version of the argument given above applies: even granting that a given normal form contains enough information about all potential information set extensive games derived from that game differ in the actual information sets that the players have to take into account in their strategic reasoning. Clearly, this is"strategically relevant information! I usual, I am not going to ask you to subscribe to my point of view. And, in any case. even if one does ne ot "believe in the sufficiency of the normal form, it may still be interesting to investigate the extensive-form implications of normal-form solution concept This is what we shall do in these notes Before that, we will Thompson's and Dalkey's result concerning inessential transforma- ions"of extensive games; I refer you to OR for a formal treatment Inessential transformations As you will remember, Thompson and Dalkey propose four transformations of extensive games which, in their opinion, do not change the strategic problem faced by the players. As may be expected, these transformations are prima facie harmless, but, on closer inspection at least some of them should not be accepted easily The result Thompson and Dalkey prove is striking: if game r can be mutated into game t- by means of a sequence of inessential transformations, then I and r have the same (reduced) normal form. Corollary": the normal form contains all strategically relevant information Of course, the result is correct, but the " Corollary"is not a formal statement: we can only accept it if we accept the transformations proposed by Thompson and Dalkey as irrelevant Let me emphasize a few key points. First, you will recall that I introduced a fifth transformation which entails replacing a non-terminal history where Chance moves, followed by terminal histories only, with a single terminal history; the corresponding payoffs are lotteries over the payoffs attached to the original terminal nodes, with probabilities given by the relevant Chance move probabilities. I mention this only for completeness: in my opinion once we accept that players are Bayesian, this transformation is harmlessstrategy profiles which reach I, S(I). Now, the latter is a normal-form object: it is simply a set of strategies. The key point is that, in order to define it, we used the information contained in the definition of the extensive game Γ: the set S(I) is not part of the formal description of the normal-form game G! [ Note for the interested reader: Mailath, Samuelson and Swinkels (1993) characterize the sets of strategy profiles which can correspond to some information set in some extensive game with a given normal form. Their characterization only relies on the properties of the normal-form payoff functions, and is thus purely “normal-form” in nature and inspiration. However, a more sophisticated version of the argument given above applies: even granting that a given normal form contains enough information about all potential information sets, extensive games derived from that game differ in the actual information sets that the players have to take into account in their strategic reasoning. Clearly, this is “strategically relevant” information! ] As usual, I am not going to ask you to subscribe to my point of view. And, in any case, even if one does not “believe” in the sufficiency of the normal form, it may still be interesting to investigate the extensive-form implications of normal-form solution concepts. This is what we shall do in these notes. Before that, we will Thompson’s and Dalkey’s result concerning“inessential transforma￾tions” of extensive games; I refer you to OR for a formal treatment. Inessential Transformations As you will remember, Thompson and Dalkey propose four transformations of extensive games which, in their opinion, do not change the strategic problem faced by the players. As may be expected, these transformations are prima facie harmless, but, on closer inspection, at least some of them should not be accepted easily. The result Thompson and Dalkey prove is striking: if game Γ 1 can be mutated into game Γ 2 by means of a sequence of inessential transformations, then Γ 1 and Γ 2 have the same (reduced) normal form. “Corollary”: the normal form contains all strategically relevant information! Of course, the result is correct, but the “Corollary” is not a formal statement: we can only accept it if we accept the transformations proposed by Thompson and Dalkey as irrelevant. Let me emphasize a few key points. First, you will recall that I introduced a fifth transformation which entails replacing a non-terminal history where Chance moves, followed by terminal histories only, with a single terminal history; the corresponding payoffs are lotteries over the payoffs attached to the original terminal nodes, with probabilities given by the relevant Chance move probabilities. I mention this only for completeness: in my opinion, once we accept that players are Bayesian, this transformation is harmless. 2