Finally, consider the sub-subgame r(h"). By construction, slh differs from silh, only at the empty history of r(h"), i.e. it constitutes a one-time deviation. Moreover, it must be a profitable deviation in r(h ), because otherwise we could find a deviation s h in r(h)which differs from sihh at fewer histories than sih. Contradiction. I Subgame Perfection: A Critical Look The prototypical criticism of SPE is based on the Centipede game, depicted in Figure 2. 2A1 2.2 di 0,2 Figure 2: A Three-Legged Centipede The argument goes as follows. According to SPE, both players should go down at each history. However, consider Player 2s predicament: he understands this, and he also un- derstands that Player 1 should understand this. Say that Player 2 is certain that Player 1 is rational; then, if in the course of the game he is called upon to move, how should he interpret the fact that Player 1 chose al despite the fact that(1) she is rational and(2)she understands that Player 2 will go down? If this argument sounds too informal to be right as stated, you are absolutely correct! However, it does contain at least a grain of truth. But consider also the game in Figure 3 which is perhaps even more striking 24 3.3 d1 1,1 0,0 Figure 3: Deviations from SPE?Finally, consider the sub-subgame Γ(h 0 ). By construction, s 0 i |h0 differs from si |h0 only at the empty history of Γ(h 0 ), i.e. it constitutes a one-time deviation. Moreover, it must be a profitable deviation in Γ(h 0 ), because otherwise we could find a deviation s 00 i |h in Γ(h) which differs from si |h at fewer histories than s 0 i |h. Contradiction. Subgame Perfection: A Critical Look The prototypical criticism of SPE is based on the Centipede game, depicted in Figure 2. 2,2 r 1 1,0 d1 a1 r 2 D A 0,2 r 1 d2 a2 3,0 Figure 2: A Three-Legged Centipede The argument goes as follows. According to SPE, both players should go down at each history. However, consider Player 2’s predicament: he understands this, and he also un￾derstands that Player 1 should understand this. Say that Player 2 is certain that Player 1 is rational; then, if in the course of the game he is called upon to move, how should he interpret the fact that Player 1 chose a1 despite the fact that (1) she is rational and (2) she understands that Player 2 will go down? If this argument sounds too informal to be right as stated, you are absolutely correct! However, it does contain at least a grain of truth. But consider also the game in Figure 3, which is perhaps even more striking. 3,3 r 1 2,2 d1 a1 r 2 D A 1,1 r 1 d2 a2 0,0 Figure 3: Deviations from SPE? 4