Problems Problem 1: A simple extensive game 1j L1 2j A l 1 A r1 A A j A 1 1 A 2 L2 A R2 A A j A 2 2 A 1 l 3 A r3 A A A 0 5 5 0 @ @ R1 @ @ j @ 2 Q A Q A r2Q o2 l2 A QQ A j Q A 1 Q 4 −2 A 3 L3 A R3 −3 A A j A 2 3 A 4 A l4 r4 A A A 6 7 −1 6 (i) How many possible strategies does player 1 have to choose from? (ii) What are all of player 1’s possible strategies? (iii) How many possible strategies does player 2 have to choose from? (iv) For player 1, say which (if any) strategies are equivalent to others. (v) What are the players’ backwards-induction strategies? (vi) What is the backwards-induction outcome? 1 Example: Centipede game Problem 2: The centipede game 1 Take the money (1, 0) Pass 2 Pass 1 Pass 2 Pass 1 Share Take Take Take Take (0, 10) (100, 0) (0, 1000) (10000, 0) (5000, 5000) Exercise: solve this game by backward induction. (i) What is the backwards-induction outcome of this game? (ii) Is this outcome efficient? Would you say that this outcome is reasonable? Problem 3: Stackelberg duopoly Consider two firms, 1 and 2, with profit functions πi (q1 , q2 ) = P (Q)qi − cqi for i = 1, 2, where P (Q) = a − bQ is a market-clearing price for an aggregate quantity Q = q1 + q2 , c is the constant marginal cost (zero fixed costs), and a, b, c > 0. The timing of the game is: (1) firm 1 chooses q1 > 0; (2) firm 2 observes q1 and then chooses q2 > 0; (3) the firms receive payoffs πi . (i) What are the backwards-induction strategies for the firms? (ii) What is the backwards-induction outcome of this game? 2 Answers Problem 1 (i) Player 1 has three decision nodes, and two choices at each node. Thus, the total number of (pure) strategies available to Player 1 is 23 = 8. (ii) All eight of Player 1’s possible (pure) strategies are: L1 L2 L3 L1 L2 R3 L1 R2 L3 L1 R2 R3 R1 L2 L3 R1 L2 R3 R1 R2 L3 R1 R2 R3 (iii) Player 2 has a total of 23 × 3 = 24 possible (pure) strategies. (iv) For Player 1: L1 L2 L3 is equivalent to L1 L2 R3 ; L1 R2 L3 is equivalent to L1 R2 R3 ; R1 L2 L3 is equivalent to R1 R2 L3 ; and R1 L2 R3 is equivalent to R1 R2 R3 . (v) Player 1’s backwards-induction strategy is R1 L2 L3 , and Player 2’s is r1 r2 l3 l4 . 1j L1 2j A A r1 l1 A A j AU 1 1 A 2 L2 A R2 A A j A 2 2 A 1 l 3 A r3 A A A 0 5 5 0 @ @ R1 @ @ j @ R 2 AQQ A rQ l2 2 Qo2 A Q A j Q AU 1 Q 4 −2 A 3 L3 A R3 −3 A A j A 2 3 A 4 l 4 A r4 A A A 6 7 −1 6 (vi) The backwards-induction outcome is (R1 L3 , r2 l4 ). Problem 2 (i) The backwards-induction outcome is for player 1 to take the money in the first node. (ii) This outcome is clearly inefficient: the players obtain payoffs (1, 0) while they could obtain payoffs (5000, 5000) by choosing to always pass the money. Moreover, this outcome may not look reasonable. For player 1 to prefer to take the money rather than pass it, she must expect player 2 to take the money in the subsequent node. Otherwise, if player 1 expected player 2 to choose Pass, player 1 would also choose Pass. Now, if player 1 chooses Pass, would then player 2 choose Pass? Having observed that player 1 chose Pass, which is not consistent with the backwards-induction prediction, player 2 might believe that player 1 is not rational and thus expect player 1 to choose Pass again at her next move, in which case player 2 would benefit from choosing Pass. However, the concept of backwards induction does not allow for this possibility. When using backwards induction, we assume that at any point in the game, the players will assume that the remaining play of the game will be consistent with backwards-induction, even if play up to that point has contradicted this. In other words, any 3 deviation from prescribed play is treated as “an unlikely mistake that is unlikely to happen again.” Problem 3 (i) We first compute firm 2’s reaction function. Having observed q1 , firm 2 solves ( a−bq1 −c if a−c ∗ 2b b > q1 max [a − b (q1 + q2 )] q2 − cq2 ⇒ q2 (q1 ) = q2 0 otherwise Now note that firm 1 can solve for this outcome just as well as firm 2 does. Thus, using firm 2’s reaction function, firm 1 solves a−c a − bq1 − c q1 − cq1 ⇒ q1∗ = max a − b q1 + q1 2b 2b The backwards-induction strategies are q1∗ and q2∗ (q1 ). (ii) The backwards-induction outcome is q1∗ and q2∗ (q1∗ ), where q2∗ (q1∗ ) = 4 a−c 4b .