博弈论(复旦大学 王永钦)复旦大学研究生一年级博弈论课程讲义,英文

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
Cont’d
Firm A’s problem:
A PqA cqA (a qA qB )qA cqA
dA dqA
a 2qA
qB
c
0
qA
a
qB 2
c
d 2 A 2 0 dqA2
Cont’d
By symmetry, firm B’s problem. Figure Illustration: Response Function, Tatonnement Process Exercise: what will happens if there are n identical Cournot
Game TheorΒιβλιοθήκη Baidu (Microeconomic Theory (IV))
Instructor: Yongqin Wang Email: yongqin_wang@yahoo.com.cn School of Economics and CCES, Fudan University
December, 2004
are a NE, if for each player i,
si* is (at least tied for) player i’s best response to the strategies
specified for the n-1 other players,
( s1* , ...,
s* i 1
,
si* ,
s* i 1
,
...,
sn*
)
ui
( s1* , ...,
s* i 1
,
si
,
s* i 1
,
...,
sn* )
Cont’d
Proposition B In the n -player normal form game
G {S1,..., Sn;u1,...,un}
if the strategies (s1* ,..., sn* ) are a NE, then they survive
Hardin(1968) : The Tragedy of Commons
Cont’d
There are n farmers in a village. They all graze their goat on the
village green. Denote the number of goats the ith farmer owns
strategies are the unique NE of the game.
A Formal Definition of NE
In the n-player normal form G {S1,..., Sn ;u1,...,un}
the strategies (s1*,..., sn* )
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.
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 (s1*,..., sn* ) , then these
iterated elimination of strictly dominated strategies.
1.3.2 Existence of NE
Theorem (Nash, 1950): In the n -player normal form game
G {S1,..., Sn;u1,...,un}
goats to own (to choose gi ).
by gi , and the total number of goats in the village by G g1 ... gn
Buying and caring each goat cost c and value to a farmer of
grazing each goat is v(G) .
Cont’d
A maximum number of goats : Gmax : v(G) 0 ,
for G Gmax but v(G) 0 for G Gmax
Also v '(G) 0, v ''(G) 0
The villagers’ problem is simultaneously choosing how many
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 overutilized.
if n is finite and S i is finite for every i , then there exist at
least one NE, possibly involving mixed strategies.
See Fudenberg and Tirole (1991) for a rigorous proof.