博弈论(复旦大学中国经济研究中心)

u1 (a1 , a2 ) and u2 (a1, a2 ) .
Cont’d
Backward Induction:
a2 arg max u2 (a1 , a2 )
Then
a1 arg max u1 (a1, R2 (a1 ))
“People think backwards”
2.1.B An example: Stackelberg Model of Duopoly
* * * gi g1 ... gi*1 gi*1 ... gn
(2)
(where
)
Summing up all
n
farmers’ FOC and then dividing by
n yields
1 * v (G*) G v '(G*) c 0 n
(3)
Cont’d
Proposition A
In the
n -player normal form game
G {S1 ,..., Sn ; u1 ,..., un }
if iterated elimination of strictly dominated strategies
eliminates all but the strategies
1.Static Game of Complete Information

1.3 Further Discussion on Nash Equilibrium (NE) 1.3.1 NE versus Iterated Elimination of Strict Dominance Strategies
n -player normal form game
G {S1 ,..., Sn ; u1 ,..., un }
if
n is finite and S i is finite for every i , then there exist at
least one NE, possibly involving mixed strategies.
Implications for social and economic systems (Coase Theorem)
2. Dynamic Games of Complete Information


2.1 Dynamic Games of Complete and Perfect Information 2.1.A Theory: Backward Induction Example: The Trust Game General features:
utilized.
Hardin(1968) : The Tragedy of Commons
Cont’d
There are
by
n
farmers in a village. They all graze their goat on the
village green. Denote the number of goats the i th farmer owns
competing firms? (Convergence to Competitive Equilibrium)
1.4.2 The problem of Commons
David Hume (1739): if people respond only to private incentives, public goods will be underprovided and public resources over-
q1 q2 , and
Cont’d
We solve this game with backward induction
q2 arg max 2 (q1 , q2 ) q2 (a q1 q2 c) a q1 c q R2 (q1 ) 2
* 2
(provided that
In contrast, the social optimum G ** should resolve
max Gv(G) Gc
FOC:
v(G **) G **v '(G **) c 0
(4)
Comparing (3) and (4), we can see that
G* G **
Having more information may be a bad thing
Exercise: Extend the analysis to
n firm case.
2.2 Two stage games of complete but imperfect information 2.2.A Theory: Sub-Game Perfection
Cont’d
Firm A’s problem:
A Pq A cq A (a q A qB )q A cq A d A a 2q A qB c 0 dq A a qB c qA 2 d 2 A 2 0 2 dq A
Cont’d
Cont’d
A maximum number of goats :
for Also
Gmax : v(G) 0
for
,
G Gmax
but
v(G ) 0
G Gmax
v '(G) 0, v ''(G) 0
The villagers’ problem is simultaneously choosing how many
goats to own (to choose
gi
).
Cont’d
His payoff is
gi v( g1 ... gi1 gi gi1 ... gn ) cgi
In NE
* * ( g1 ,..., gn ) , for each i , gi* must maximize
See Fudenberg and Tirole (1991) for a rigorous proof.
1.4 Applications 1.4.1 Cournot Model
Two firms A and B quantity compete.
Inverse demand function P a Q, a 0 They have the same constant marginal cost, and there is no fixed cost.
(1)
(1), given
that other farmers choose * * 1 i 1
* ( g ,..., g , gi*1, gn )
Cont’d
First order condition (FOC):
* * v( gi gi ) gi v '( gi gi ) c 0
Two firms quantity compete sequentially. Timing: (1) Firm 1 chooses a quantity q1 (2) Firm 2 observes (3) The payoff to firm
0 ;
q1 and then chooses a quantity q2 0 ；
i is given by the profit function
i (qi , q j ) qi [P(Q) c]
P(Q) a Q
is the inverse demand function, Q
c is the constant marginal cost of production (fiபைடு நூலகம்ed cost being zero).
q1 a c
).
Cont’d
Now, firm 1’s problem
q1 arg max 1 (q1 , R2 (q1 )) q1[a q1 R2 (q1 ) c] ac q 2
* 1
so,
ac q 4
* 2
.
Cont’d
Compare with the Cournot model.
(1) Player 1 chooses an action (2) Player 2 observes
a1
from the feasible set
A1
.
the feasible set
(3) Payoffs are
A2 .
a1 and then chooses an action a2 from
* * * * (s1 ,..., si*1, si* , si*1,..., sn ) ui (s1 ,..., si*1, si , si*1,..., sn )
Cont’d
Proposition B In the n -player normal form game
G {S1 ,..., Sn ; u1 ,..., un }
gi
, and the total number of goats in the village by
G g1 ... gn
Buying and caring each goat cost grazing each goat is
c and value to a farmer of
v(G ) .

Here the information set is not a singleton. Consider following games (1)Players 1 and 2 simultaneously choose actions