Eco514-Game Theory The Trembling Hand: Normal-Form Analysis and Extensive-Form Implications Marciano siniscalchi January 10, 2000 Introduction: Invariance In their seminal contribution, Von Neumann and Morgenstern argue that the normal form of a game contains all"strategically relevant"information. This view, note well, does not invalidate or trivialize extensive-form analysis; rather, it leads those who embrace it to be uspicious of extensive-form solution concepts which yield different predictions in distinct extensive games sharing the same normal (or reduced normal) form. Solution concepts which do not yield different predictions for such games are called invarian The supposed"strategic sufficiency of the normal form"also motivated the search for normal-form solution concepts which exhibit "nice"properties in every ectensive-form as- sociated with a given normal-form game. The main proponent of this line of research is JF Mertens In my opinion, whether or not the normal form contains all "strategically relevant information depends crucially on the solution concept one wishes to apply. This is actually a rather trivial point, but I am afraid it was overlooked in the debate on the sufficiency of the normal form. For instance, in order to compute the minmax value of a game, one only needs to look at strategies and payoffs associated with strategy profiles; the information conveyed by the extensive form of the game(if such information is at all provided) is irrelevant as far as the minmax value calculation is concerned. Since Von Neumann and Morgenstern were mostly concerned with minmax values, the normal form was indeed sufficient for their purposes. The argument readily extends to Nash equilibrium analysis However, as soon as one wishes to restrict the attention to sequential equilibria, it is clear that the normal form is not sufficient to carry out the analysis. Quite simply, the formal notion of "normal-form game"does not include a specification of information sets This point is more subtle that it appears. You will remember that, given an extensive me r and its normal form G, we defined, for each information set I, the collection ofEco514—Game Theory The Trembling Hand: Normal-Form Analysis and Extensive-Form Implications Marciano Siniscalchi January 10, 2000 Introduction: Invariance In their seminal contribution, Von Neumann and Morgenstern argue that the normal form of a game contains all “strategically relevant” information. This view, note well, does not invalidate or trivialize extensive-form analysis; rather, it leads those who embrace it to be suspicious of extensive-form solution concepts which yield different predictions in distinct extensive games sharing the same normal (or reduced normal) form. Solution concepts which do not yield different predictions for such games are called invariant. The supposed “strategic sufficiency of the normal form” also motivated the search for normal-form solution concepts which exhibit “nice” properties in every extensive-form as￾sociated with a given normal-form game. The main proponent of this line of research is J.F. Mertens. In my opinion, whether or not the normal form contains all “strategically relevant” information depends crucially on the solution concept one wishes to apply. This is actually a rather trivial point, but I am afraid it was overlooked in the debate on the sufficiency of the normal form. For instance, in order to compute the minmax value of a game, one only needs to look at strategies and payoffs associated with strategy profiles; the information conveyed by the extensive form of the game (if such information is at all provided) is irrelevant as far as the minmax value calculation is concerned. Since Von Neumann and Morgenstern were mostly concerned with minmax values, the normal form was indeed sufficient for their purposes. The argument readily extends to Nash equilibrium analysis. However, as soon as one wishes to restrict the attention to sequential equilibria, it is clear that the normal form is not sufficient to carry out the analysis. Quite simply, the formal notion of “normal-form game” does not include a specification of information sets! This point is more subtle that it appears. You will remember that, given an extensive game Γ and its normal form G, we defined, for each information set I, the collection of 1