Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Week 7: Games in Extensive Form Dr Daniel Sgroi Reading: 1. Osborne chapter 5 (and also 6, 7); 2. Snyder & Nicholson, pp. 255{260. With thanks to Peter J. Hammond. EC202, University of Warwick, Term 2 1 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Sequential BoS, I 1 BS XX 2 XXX2 BS @ b BS @ r @@ r @@ 2; 1 0; 0 0; 0 1; 2 r r r r Consider the BoS game, with the key difference that player 1 who prefers (B; B) to (S; S) moves first, and player 2 observes player 1's move before replying. This difference affects the normal or strategic form, because player 2 now has four instead of two strategies, as follows: 1. B always (which we denote by b); 2. S always (which we denote by s); 3. B if B and S if S (which we denote by m for match); 4. S if B and B if S (which we denote by x for cross). EC202, University of Warwick, Term 2 2 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Sequential BoS, II Here is the normal form of sequential BoS, with (multiple) best responses starred as usual. P2 b s m x B 2∗ 0 2∗ 0∗ 1∗ 0 1∗ 0 ∗ ∗ P1 S 0 1 1 0 0 2∗ 2∗ 0 There are three pure strategy Nash equilibria, and many mixed strategy equilibria (for example, B with any mixture of b and m). EC202, University of Warwick, Term 2 3 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Sequential BoS, III P2 b s m x B 2∗ 0 2∗ 0∗ 1∗ 0 1∗ 0 ∗ ∗ P1 S 0 1 1 0 0 2∗ 2∗ 0 Player P2's strategy x is strictly dominated by m, while both b and s are weakly dominated by m. If we eliminate all weakly dominated strategies once, then eliminate 1's strictly dominated strategy, only (B; m) remains. This is a refinement of the set of Nash equilibria. EC202, University of Warwick, Term 2 4 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Sequential BoS, IV An alternative way to justify P2 choosing strategy m considers the two different subgames that arise after player 1 has chosen either B or S. In each of these player 2 can also choose B or S: after B, player 2's best response is B; after S, player 2's best response is S. Thus player 2 matches by choosing m in the full game. Player 1's best response is B. In effect, player 1's move anticipates player 2's response. The result is player 1's favoured outcome, so moving first confers an advantage in this game. Player 1 has a “first-mover advantage". EC202, University of Warwick, Term 2 5 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Sequential Cornot, I Consider the Cournot duopoly model where firms have zero cost, and inverse demand is p = maxf0; 100 − qg, with q = q1 + q2. Firm i's best response, taking the other firm’s qj as fixed, 1 solves maxqi (100 − qi − qj )qi , so qi = 50 − 2 qj . After allowing for the possibility of a corner solution, we derive 1 the best response (BR) function bi (qj ) = maxf0; 50 − 2 qj g. As in sequential BoS, suppose that before firm 2 chooses q2, it will have observed firm 1's choice of q1. Firm 2's strategy space becomes the set of all response functions q1 7! r2(q1) from R+ to R+. A strategy for firm 2 that weakly dominates all others is to choose r2(·) to be the best response function b2(·). When making its move, firm 2 not only has the deterministic belief, but the certain knowledge that q1 has been played. EC202, University of Warwick, Term 2 6 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Sequential Cornot, II Assuming common knowledge of rationality, what should firm 1 do? A deterministic belief about q2 would be na¨ıve, since firm 1 knows that a rational firm 2 1 will choose b2(q1) = maxf0; 50 − 2 q1g in response to its choice of q1. This, in turn, means that a rational firm 1 would replace the “fixed” q2 in its profit function with firm 2's best response b2(q1) to 1's choice of q1. That is, firm 1 now chooses q1 to solve max[100 − q1 − b2(q1)] q1: q1 EC202, University of Warwick, Term 2 7 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Stackelberg, I Firm 1 chooses q1 to solve max[100 − q1 − b2(q1)] q1; q1 1 where b2(q1) = maxf0; 50 − 2 q1g. In the interior case, the maximand 1 1 is [100 − q1 − (50 − 2 q1)] q1 = (50 − 2 q1) q1, with derivative 50 − q1, which is ? 0 as q1 7 50. So, anticipating firm 2's response b2(q1) to q1, firm 1's optimal strategy is q1 = 50. After this firm 2 responds by choosing q2 = b2(50) = 25. The price is 100 − q1 − q2 = 100 − 75 = 25. This is the Stackelberg equilibrium. EC202, University of Warwick, Term 2 8 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Stackelberg, II Given q1 = 50, q2 = 25, and p = 25, the resulting profits (revenues) are π1 = 25 · 50 = 1; 250 and π2 = 25 · 25 = 625. Recall that in the original Cournot equilibrium (with simultaneous moves) 1 the quantities were q1 = q2 = 100=3 = 33 3 , 1 the price was p = 100=3 = 33 3 , 1 and the profits were π1 = π2 = 10; 000=9 = 1; 111 9 . So firm 1 has a first-mover advantage, relative to both firm 2 and the Cournot equilibrium. Total profits, however, are lower. EC202, University of Warwick, Term 2 9 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Extensive Form Games One obvious drawback (sometimes an advantage) of the normal form of a game is its failure to capture time. A player's strategy describes one complete plan for the whole game. Payoff functions represent how players value the consequences of strategy profiles. But how is the order of moves captured? A game in extensive form captures sequential strategic situations when there is a well defined order of moves. Our analyses of sequential BoS and sequential Cournot suggest that the order of moves can affect what we would regard as a reasonable outcome of the game. It illustrates a solution concept that captures an important idea of sequential rationality. EC202, University of Warwick, Term 2 10 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Players and Payoffs The extensive form formally captures situations where: 1. players may move sequentially; 2. what players know, when it is their turn to move, may depend on players' previous choices. Like the normal or strategic form, any extensive form game has two key ingredients in its description: (EF1) The set of players, N (EF2) The players' payoffs (ui (·))i2N as functions of everybody's actions. EC202, University of Warwick, Term 2 11 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Move Order and Feasible Sets To allow sequential play to be represented, we extend the notion of pure strategy by specifying two separate parts: first, as before, what players can do; second, when (and with what information) they can do it. In the Stackelberg example, as in the Cournot game, players can choose any quantity they like. But we also needed to specify that player 1 moves first, and only then does player 2 move (after observing 1's prior choice). Thus, in general we need two extra components in order to model allowable move sequences: (EF3) The order of moves. (EF4) What choices players have when they do move (their feasible sets). EC202, University of Warwick, Term 2 12 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Players' Knowledge Simultaneous moves in the normal form represent the situation where players know nothing about other players' prior moves. In the extensive form, it is not the chronological order of play that matters, but what players know when they choose. In both the Cournot and Stackelberg models of duopoly, firm 1 may indeed move some time before firm 2. But if firm 2 must choose without observing firm 1's choice, then the Cournot model of simultaneous choice still applies. In contrast, if firm 1's choice is revealed before firm 2 moves, then we should use the sequential Stackelberg model. In general, we need to specify how information and knowledge change over time: (EF5) Players' information when making their moves. EC202, University of Warwick, Term 2 13 of 33 Battle of the Sexes Stackelberg Essential Ingredients Game Trees Backward Induction Subgame Perfection Chance Moves Exogenous random events vastly enrich the strategic decision problems we can describe. Example Suppose firm A embarks on a research and development (R&D) project that may succeed or fail. Competing firm B can adapt its strategy to the outcome of firm A's R&D project, by waiting to see if it succeeds or not.