 One might think that a corollary to Proposition 3.11 is that in a game of perfect information, common knowledge of the game and of rationality implies the backwards induction solution. This is the classical argument for the backwards induction solution. Many game theorists continue to accept the classical argument, but in recent years, the argument has come under strong challenge, led by the work of Reny (1988, 1992), Binmore (1987) and Bicchieri (1989, 1993). The basic idea underlying their criticisms of backwards induction can be illustrated with the Figure 1.2 game. According to the classical argument, if Alan and Fiona have common knowledge of rationality and the game, then each will predict that the other will follow her end of the backwards induction solution, to which his end of the backwards induction solution is his unique best response. However, what if Fiona reconsiders what to do if she finds herself at the information set \(I^{22}\)? If the information set \(I^{22}\) is reached, then Alan has of course not followed the backwards induction solution. If we assume that at \(I^{22}\), Fiona knows only what is stated in (iii), then she can explain her being at \(I^{22}\) as a failure of either \(\mathbf{K}_1\mathbf{K}_2\mathbf{K}_1 (R_2)\) or \(\mathbf{K}_1\mathbf{K}_2\mathbf{K}_1\mathbf{K}_2 (\Gamma)\) at \(I^{11}\). In this case, Fiona’s thinking that either \(\neg\mathbf{K}_1\mathbf{K}_2\mathbf{K}_1 (R_2)\) or \(\neg\mathbf{K}_1\mathbf{K}_2\mathbf{K}_1\mathbf{K}_2(\Gamma)\) at \(I^{11}\) is compatible with what Alan in fact does know at \(I^{11}\), so Fiona should not necessarily be surprised to find herself at \(I^{22}\), and given that what she knows there is characterized by (iii), following the backwards induction solution is her best strategy. But if rationality and the game are common knowledge, or even if Fiona and Alan both have just have mutual knowledge of the statements characterized by (iii) and (iv), then at \(I^{22}\), Fiona knows that \(\mathbf{K}_1\mathbf{K}_2\mathbf{K}_1 (R_2)\) or \(\mathbf{K}_1\mathbf{K}_2\mathbf{K}_1\mathbf{K}_2 (\Gamma)\) at \(I^{11}\). Hence given this much mutual knowledge, Fiona no longer can explain why Alan has deviated from the backwards induction solution, since this deviation contradicts part of what is their mutual knowledge. So if she finds herself at \(I^{22}\), Fiona does not necessarily have good reason to think that Alan will follow the backwards induction solution of the subgame beginning at \(I^{22}\), and hence she might not have good reason to follow the backwards induction solution, either. Bicchieri (1993), who along with Binmore (1987) and Reny (1988, 1992) extends this argument to games of perfect information with arbitrary length, draws a startling conclusion: If agents have strictly too few or strictly too many levels of mutual knowledge of rationality and the game relative to the number of potential moves, one cannot predict that they will follow the backwards induction solution. This would undermine the central role backwards induction has played in the analysis of extensive form games. For why should the number of levels of mutual knowledge the agents have depend upon the length of the game?