G = {S1 ,..., S n ; u1 ,..., un }
if the strategies
* ( s1* , ..., s n )
are a NE, then they survive
iterated elimination of strictly dominated strategies.
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
gi
).
Cont’d
His payoff is
giv(g1 +... + gi1 + gi + gi+1 +... + gn ) cgi
In NE
* * ( g1 ,..., g n ) , for each i , gi* must maximize
(1)
(1), given
that other farmers choose * * 1 i 1
Proposition A
In the
n -player normal form game
G = { S1 ,..., S n ; u1 , ..., u n }
if iterated elimination of strictly dominated strategies eliminates all but the strategies
( g ,..., g , g , g )
* i +1
* n
Cont’d
First order condition (FOC):
* * v( gi + g i ) + gi v '( gi + g i ) c = 0
* * * g i ≡ g1 + ... + gi*1 + gi*+1 + ... + g n
≥0;
q1 and then chooses a quantity q2 > 0 ；
(3) The payoff to firm
i is given by the profit function
π i (qi , q j ) = qi [ P (Q ) c]
P(Q) = a Q is the inverse demand function, Q = q1 + q2 , and
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
c is the constant marginal cost of production (fixed cost being zero).
Cont’d
We solve this game with backward induction
q2 ∈ arg max π 2 (q1 , q2 ) = q2 (a q1 q2 c) a q1 c q = R2 (q1 ) = 2
1.3.2 Existence of NE
Theorem (Nash, 1950): In the
n -player normal form game
G = {S1 ,..., S n ; u1 ,..., un }
if
n is finite and S i is finite for every i , then there exist at
Then
a1 ∈ arg max u1 (a1 , R2 (a1 ))
“People think backwards”
2.1.B An example: Stackelberg Model of Duopoly
Two firms quantity compete sequentially. Timing: (1) Firm 1 chooses a quantity q1 (2) Firm 2 observes
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.
Here the information set is not a singleton. Consider following games (1)Players 1 and 2 simultaneously choose actions
a1
and
a2
from feasible sets A1 and A2, respectively. (2) Players 3 and 4 observe the outcome of the first stage ( a1, a2) and then simultaneously choose actions and from feasible sets A3 and A4, respectively. (3) Payoffs are ui ( a1 , a2 , a3 , a4 ) , i = 1, 2,3, 4
( s ,..., s ) , then these
* 1
* n
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
max Gv ( G ) Gc
FOC:
v(G**) + G**v '(G**) c = 0
(4)
Comparing (3) and (4), we can see that
G* > G **
Implications for social and economic systems (Coase Theorem)

n farmers’ FOC and then dividing by n yields
(3)
1 v (G * ) + G * v '( G * ) c = 0 n
Cont’d
In contrast, the social optimum G ** should resolve
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 i th farmer owns by
Cont’d
A maximum number of goats : for Also
Gmax : v(G) > 0
for
,
G < G max
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
* 2
(provided that
q1 < a c
).
Cont’d Now, firm 1’s problem
q1 ∈ arg max π 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.
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 ) .
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: (1) Player 1 chooses an action (2) Player 2 observes the feasible set (3) Payoffs are
a1
from the feasible set