Dynamic Games. Econ 400. University of Notre Dame. Econ 400 (ND) Dynamic Games 1 / 18

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1 Dynamic Games Econ 400 University of Notre Dame Econ 400 (ND) Dynamic Games 1 / 18

2 Dynamic Games A dynamic game of complete information is: A set of players, i = 1,2,...,N A payoff function for each player that describes his payoff as a function of the decisions of all the players A description of the timing of the game: Which player is allowed to move, given the previous decisions of all the players? A set of feasible actions for each player, given the previous decisions of all the players. Econ 400 (ND) Dynamic Games 2 / 18

3 Extensive Forms Econ 400 (ND) Dynamic Games 3 / 18

4 Behavior Strategies Definition A behavior strategy for player i is a rule that gives that player s strategy at all of his decision points in the extensive form. For example, Player 1 has two behavior strategies: L or R. Player 2, however, has many behavior strategies that take the form, If Player 1 just chose L, then x, if Player 1 just chose R, then y. Econ 400 (ND) Dynamic Games 4 / 18

5 Throwing away the timing Suppose we just ignored the timing. What happens? We use the set of behavior strategies as the set of pure strategies for each player, and get a strategy form: ac ad bc bd L 1,0 1,0 2,3 2,3 R 0,1-1,0 0,1-1,0 Econ 400 (ND) Dynamic Games 5 / 18

6 Throwing away the timing Suppose we just ignored the timing. What happens? We use the set of behavior strategies as the set of pure strategies for each player, and get a strategy form: ac ad bc bd L 1,0 1,0 2,3 2,3 R 0,1-1,0 0,1-1,0 So we can convert any dynamic game into a static game. This one has two Nash equilibria: (L,bc) and (L,bd). But is (L,bd) a reasonable prediction? Econ 400 (ND) Dynamic Games 5 / 18

7 Every James Bond movie There is a Evil Rational Scientist, who is a perfect sociopath (maximizes his own payoff with complete disregard to the welfare of others). He threatens the United Nations by claiming he will detonate a nuclear weapon, destroying the earth, including himself, unless he gets a billion dollars. Econ 400 (ND) Dynamic Games 6 / 18

8 Every James Bond movie There is a Evil Rational Scientist, who is a perfect sociopath (maximizes his own payoff with complete disregard to the welfare of others). He threatens the United Nations by claiming he will detonate a nuclear weapon, destroying the earth, including himself, unless he gets a billion dollars. Econ 400 (ND) Dynamic Games 6 / 18

9 (The Entry Game) A related, less silly game goes like this: Kmart is deciding on entering a market where Walmart is the incumbent. Kmart can enter or stay out. Walmart, upon entry, can fight the entrant by charging very low prices and hurt them both, or accomodate the entrant and split the market. (Here, the Evil Rational Scientist is Walmart) Econ 400 (ND) Dynamic Games 7 / 18

10 (The Entry Game) A related, less silly game goes like this: Kmart is deciding on entering a market where Walmart is the incumbent. Kmart can enter or stay out. Walmart, upon entry, can fight the entrant by charging very low prices and hurt them both, or accomodate the entrant and split the market. (Here, the Evil Rational Scientist is Walmart) Econ 400 (ND) Dynamic Games 7 / 18

11 Every James Bond movie Let s look at the equilibria of the static game: NN ND DN DD P -, -, -1,1-1,1 DP -, 0,0 -, 0,0 Econ 400 (ND) Dynamic Games 8 / 18

12 Every James Bond movie Let s look at the equilibria of the static game: NN ND DN DD P -, -, -1,1-1,1 DP -, 0,0 -, 0,0 So we ve got three pure-strategy Nash equilibria: (P, DN), (PD, ND), and (DP, DD). But are these all reasonable, given that the Evil Rational Scientist is a perfect sociopath? Econ 400 (ND) Dynamic Games 8 / 18

13 Subgames Definition A subgame is a set of decision points, actions, and payoffs, which all follow from a single decision point to the end of the game. Econ 400 (ND) Dynamic Games 9 / 18

14 Subgames Definition A subgame is a set of decision points, actions, and payoffs, which all follow from a single decision point to the end of the game. To construct a subgame: Pick a single decision point, and circle all the subsequent things that can happen, from that point to the end of the game. Econ 400 (ND) Dynamic Games 9 / 18

15 Subgame Perfect Nash Equilibrium Definition A Nash equilibrium s = (s1,s 2,...,s N ) is subgame perfect if it is a Nash equilibrium in every subgame. Econ 400 (ND) Dynamic Games 10 / 18

16 Subgame Perfect Nash Equilibrium Definition A Nash equilibrium s = (s1,s 2,...,s N ) is subgame perfect if it is a Nash equilibrium in every subgame. Econ 400 (ND) Dynamic Games 10 / 18

17 Backwards Induction How do we solve for subgame perfect Nash equilibrium (SPNE)? Econ 400 (ND) Dynamic Games 11 / 18

18 Backwards Induction How do we solve for subgame perfect Nash equilibrium (SPNE)? Definition Backwards Induction is the process of pruning the game tree described by the following algorithm: Step 1: Start at each of the final subgames in the game. Solve for the players equilibrium behavior. Remove that subgame and replace it with the payoffs that arise from players following their optimal course of action. Step 2: Repeat step 1 until you arrive at the first node in the extensive form. Econ 400 (ND) Dynamic Games 11 / 18

19 Backwards Induction How do we solve for subgame perfect Nash equilibrium (SPNE)? Definition Backwards Induction is the process of pruning the game tree described by the following algorithm: Step 1: Start at each of the final subgames in the game. Solve for the players equilibrium behavior. Remove that subgame and replace it with the payoffs that arise from players following their optimal course of action. Step 2: Repeat step 1 until you arrive at the first node in the extensive form. Theorem The set of strategies constructed by backwards induction is a subgame perfect Nash equilibrium. Econ 400 (ND) Dynamic Games 11 / 18

20 Example 1 Solve for all Nash and subgame perfect Nash equilibria. Are there are any Nash equilibria that aren t subgame perfect? Econ 400 (ND) Dynamic Games 12 / 18

21 Example 2 Solve for all Nash and subgame perfect Nash equilibria. Are there are any Nash equilibria that aren t subgame perfect? Econ 400 (ND) Dynamic Games 13 / 18

22 The ERS Game Solve for all Nash and subgame perfect Nash equilibria. Are there are any Nash equilibria that aren t subgame perfect? Econ 400 (ND) Dynamic Games 14 / 18

23 The Entry Game Solve for all Nash and subgame perfect Nash equilibria. Are there are any Nash equilibria that aren t subgame perfect? Econ 400 (ND) Dynamic Games 15 / 18

24 The Stackelberg Game There are two firms, a Leader, L, and a Follower, F. The price in the market is determined by the quantity chosen by the leader q f and the follower q l as p(q f,q l ) = A q l q f ; the firms have the same total costs, C(q) = cq. The leader chooses his quantity first, which the follower observes, and then the follower chooses his quantity. What is the subgame perfect Nash equilibrium of the game? Is consumer welfare greater or less than it would be in a standard Cournot game? Compare the players strategies to the monopoly and perfectly competitive outcomes. Econ 400 (ND) Dynamic Games 16 / 18

25 Resale Price Maintenance There is a manufacturer M and a retailer R. The manufacturer sells goods to the retailer, who then sells goods to customers (think Best Buy selling Sony t.v. s at retail stores [but not franchises like JCrew that handle of the retailing through franchised stores]). The price in the market is p(q) = A q, and the manufacturer sells units of the good to the retailer at a price h per unit. The manufacturer s total costs are C(q) = cq. What is the subgame perfect Nash equilibrium of the game? Is the price to consumers higher or lower than it would be if the two firms vertically integrated into a single monopoly? Econ 400 (ND) Dynamic Games 17 / 18

26 Strategic Voting There is an unpopular bill coming up for a vote which some politicians prefer pass (like the budget bills associated with super-committee or the fiscal cliff), but prefer not to vote for (they want to appear tough and uncompromising): If all the politicians vote yes, the bill passes and they get a payoff of 0 each. If all the politicians vote no, the bill fails and they get a payoff of 1 each. If two politicians vote yes and the third votes no, they get a payoff of 0 each, and the third gets a payoff of 1 If two politicians vote no and the third votes yes, they get a payoff of 1 each, and the third gets a payoff of 2 Econ 400 (ND) Dynamic Games 18 / 18

27 Strategic Voting There is an unpopular bill coming up for a vote which some politicians prefer pass (like the budget bills associated with super-committee or the fiscal cliff), but prefer not to vote for (they want to appear tough and uncompromising): If all the politicians vote yes, the bill passes and they get a payoff of 0 each. If all the politicians vote no, the bill fails and they get a payoff of 1 each. If two politicians vote yes and the third votes no, they get a payoff of 0 each, and the third gets a payoff of 1 If two politicians vote no and the third votes yes, they get a payoff of 1 each, and the third gets a payoff of 2 Draw out an extensive form and solve for a subgame perfect Nash equilibrium where politician 1 votes first, politician 2 votes second, and politician 3 votes third; what is the probability the bill fails? Econ 400 (ND) Dynamic Games 18 / 18