Dynamic Games Lesson 4: Economic Aplica4ons Universidad Carlos III
Economic Aplica4ons Ø DGPI Ø Sequen4al compe44on in quan44es: Stackelberg Ø Unions and firm: Nego4a4ons Ø Others: Contribu4on to a public good, price compe44on, Ø JDII Ø Vo4ng Games Ø Sincere or strategic vo4ng Ø Agenda manipula4on
DGPI and con4nuous variable Ø Two players. Player 1 must choose x, x 0, and Player 2 must choose z, z 0. Payoffs are: u1(x, z) and u2(x, z) Ø Player 1 chooses first, and her choice will be known by 2 before choosing z. Ø Player 1 has one informa4on set. Her strategy is a number x. Ø Player 2 has infinitely many informa4on sets (one for each possible value of x). We will represent this with an asterisc (*). His strategy must contain infinitely many values. It is a func4on: z = f (x).
Player 1 P2* u1(x, z) u2(x, z)
SPNE Ø By backward induc4on, start with Player 2. He will choose z to maximize his profits given the value of x: Max z u 2 (x, z) z = f (z) where f Reaction function Ø Subs4tute this reac4on func4on by 2 in the expression for 1 s u4lity. Player 1 will choose the value x* that maximizes her profits Max u 1 (x, f (x)) x * x ENPS = (x*, f (x))
Sequen4al compe44on in quan44es Ø 2 firms produce a homogeneous good. Ø Linear demand: P=a- Q. Ø Constant marginal cost equal for both firms: c<a Ø Decision sequence: Ø The leader (Firm 1) decides its produc4on. Ø The follower (Firm 2) observes the leader s output and then decides its own. Ø The difference with Cournot s model is that decisions are made sequen4ally and not simultaneously.
The extensive form: Leader Follower* l (ql, qf) f (ql, qf)
Strategies Ø In a perfect informa4on situa4on, the leader has one informa4on set and the follower has infinitely many. We ll again use an asterisc (*) to indicate that in the picture of the game. Ø There are infinitely many subgames, one for each possible value of. q l Strategies: For the leader, one strategy is a value of. For the follower, a strategy must have infinitely many elements, one for each one of its informa4on sets. Its strategy is thus a func(on. q l
Backwards induc4on Ø To compute the set of SPNE we solve backwards, star4ng with the last subgame where the follower produces the quan4ty that maximizes its profits given the quan4ty previously chosen by the leader : Max q f using the first order condi4on we find the best reply func4on : Π f q = a ql 2q f c = 0 f Π = (P(Q) c) q f f = (a q l q f c) q f q f = MR ( f l q ) = a c ql 2 Ø No4ce that its best reply func4on is iden4cal to the one we obtain in the sta4c Cournot game.
Given the follower s best reply, the leader maximizes its profits (it an4cipates that best reply and takes it into account when solving its own maximiza4on problem). In other words, we are moving backwards to the first stage of the game: Max (P(Q) c) q l = (a q l q l q f c) q l t.q. f q = a c ql 2 Subs4tute the restric4on in the objec4ve func4on and calculate the first order condi4on: Π = (a # q l l a c ql % $ 2 # SPNE = a c 2, a c q & l % ( $ 2 ' & # ( c) q ' l = a c q & l % q $ 2 ' ( l
Prices and profits in the SPNE Q = q l + q f = a c 2 + " a c $ a c # 2 2 P(Q) = a 3 ( 4 a c ) = 1 4 a + 3 4 c Profits for each firm: % ' & = 3 ( 4 a c ) Π = " 1 l $ 4 a + 3 # 4 c c %" ' a c % $ ' = 1 ( &# 2 & 8 a c ) 2 Π = " 1 f $ 4 a + 3 # 4 c c %" ' a c % $ ' = 1 &# 4 & 16 a c ( ) 2 Ø Although they both have the same technology, the leader makes higher profits.
Cournot versus Stackelberg Ø Recall Cournot: q 1 = q 2 = a c 3 Q = 2 # % a c $ 3 2 & ( ' Π = 1 Π = (a c) 2 9 Ø Comparing with Cournot, the leader makes higher profits, and the follower, lower ones. Ø There is a strategic advantage to move first. Intui4on: The leader has the op4on to choose q c, and the follower will reply with q c, as that is its best reply. But the leader can do beber, as it knows that if it produces more, the follower will react by reducing its quan4ty (recall that quan44es ares strategic subs4tutes).
The advantage of moving first or second The advantage of moving first When the other players reac4on moves in the opposite direc4on the the player playing first. For instance: quan4ty compe44on. The advantage of moving second When the other players reac4on moves in the same direc4on the the player playing first. For instance: price compe44on.
Conclusions 1. The leader produces more than in the simultaneous game so that the follower reacts producing less. 2. The leader can choose the output level that maximizes prfits, given the follower s reac4on. This means that, by the defini4on of the problem, the leader will have at least the same profits as in the simultaneous game. 3. The problem is to have credibility to become a leader: install capacity, have a market image, etc. In general, make some sunk investment to become a leader.
Colec4ve negocia4on Let s have an economy with a Union and a Firm. The Union is the only provider of labor, and has exclusive power over the salary w. The Firm, on the other hand, decides how much labor to hire L The Union s objec4ve is to maximize total wage income: wl. The Firm uses only labor as an input. Its goal is to L to maximize profits Max L F(L) wl, sea F(L) = 8L 2 L 2 The game has the following 4ming: 1. The Union chooses w. 2. The Firm observes w and then chooses L.
SPNE Solve by backwards induc4on. Start with the Firm. Max 8L L 2 L 2 wl cpo :8 L w = 0 L(w) = 8 w Subs4tute the Firm s reac4on func4on in the Union s objec4ve func4on: Max wl w w( 8 w) w* = SPNE = (4, 8- w) Equilibrium path : w = 4, L = 4 Equilibrium payoffs : U = 16, = 8 4
DGII: vota4on games Ø Suppose that in a parliamentary commibee there are three proposals: A, B and C Ø The commibee has three members: 1, 2 y 3. Ø The vo4ng rules are as follows. First, the commibee votes between A and B. Then, the vote is between the winner in the previus vota4on and C. The overall winner is the op4on with most votes in this second vota4on. Ø The preferences of the commibee members are: Member 1 : Member 2 : Member 3 : A C B B A C C A B Ø To simplify the problem, assume that the best op4on gives a u4lity of 3, the second best, 2, and the last, 1.
Subgames and strategies Ø We are dealing with a DGII with 9 subgames: the 8 subgames that start ajer the first stage (one ajer each possible combina4on of votes: AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB) and the whole game. Ø A strategy for each player must have 9 elements: it must show what to do in stage 1 and what to do in each subgame. Ø Find first the NE in the 8 subgames of the second stage. Replace the subgames with the payoffs in the NE and compute the NE of the resul4ng game.
Subgames There are two types: subgames ajer A wins and subgames ajer B wins A wins ajer AAA, AAB, ABA, BAA B wins ajer ABB, BAB, BBA, BBB NE in the subgames In subgames ajer A wins, voters have two ac4ons: vote A or C. Assume players don t use weakly dominated strategies, then the NE is: (A,A,C). (Note that (A,A,A) is also an NE, but that Player 3 is using a weakly dominated strategy.) U(EN) = (u 1 (A), u 2 (A),u 3 (A)) = (3, 2, 2) In subgames ajer B wins, voters have to acc4ons: vote for B or C. Assume again that in case of indifference, they vote their most preferred op4on, the NE is: (C,B,C). U(NE) = (2, 1, 3)
Replace subgames with EN payoffs
Strategic vo4ng Ø Let s see that (AAB) is a NE of the game u 1 (AAB) = 3 > u 1 (BAB) = 2 (B wins, and in B vs. C, C wins) u 2 (AAB) = 2 > u 2 (ABB) = 1 (B wins, and in B vs. C, C wins) u 3 (AAB) = 2 = u 3 (AAA) = 2 (A wins, and in A vs. C, A wins) Ø Note that Player 2 votes A in the first stage even thought he prefers B: Strategic vo(ng Ø The next strategies cons4tute a SPNE: s 1 = (A, A if A wins, C if B wins), s 2 = (A, A if A wins, B if B wins), S 3 = (B, C if A wins, C if B wins).
Agenda manipula4on Consider the same vo4ng problem, but now with preferences Player 1 : Player 2 : Player 3 : A B C B C A C A B If Player 1 has the agenda, what can she do? Compare star4ng with A vs. B and with A vs. C. In both cases, the sincere vo4ng will follow at t=2.
Order 1) A vs. B In the first stage either A wins or B wins: C If A wins, the second stage is A vs. C with (A, C, C) U = (1, 2, 3) If A wins, the second stage is A vs. C with (B, B, C) U = (2, 3, 1) By backwards induc4on, votes at t=1 are (B,B,A) B wins B
Order 2) A vs. C In the first stage either A wins or C wins: A If A wins, the second stage is A vs. B with (A, B, A) U = (3, 1, 2) If C wins, the second stage is C vs. B with (B, B, C) U = (2, 3, 1) By backwards induc4on, votes at t=1 are (A,C,A) A wins There are incen4ves to manipulate the agenda and pick the second order of votes. B