Transcript Stackelberg -leader/follower game

Stackelberg -leader/follower game
2 firms choose quantities sequentially
(1) chooses its output; then (2) chooses it output; then the market clears
This changes the game relative to Cournot competition
- (2) can respond to (1)’s output, so that there is no need for a conjecture
- (1) can anticipate (2)’s reaction when it chooses its output
Subgame perfect equilibrium is typically used in this setting
- start at the end of the game and work backwards
- Since firm (1) moves first, take q1 as given, find (2)s best response q2
- then back-up and consider (1)s choice q1
Firm (2):
Max  2   a  q1  q 2  c  q 2
q2
q2 
a  c  q1
2
q1 is known; it is not a conjecture
Stackelberg – first subgame
Firm (1) solves a different problem, since it knows how (2) will react to its choice.
Max  1   a   q1  q 2   c  q1
q1
=

 a  c  q1  
  a  c  q1  
  q1
2



1
 a  c  q1  q1
2
q1 * 
a c
2
then
a c 
a c 

2  a c

q2 * 

2
4
Q *  q1 * q 2 * 
Firms produce different quantities:
P*  a 
3a  c 
4
;
π 1* 
a  c 
8
3a  c 
2
;
π2* 
4
a  c 
16
2
;
Nash Equilibria


Are there NE that yield paths other than a 2 , a 4 ?
 a  q1  c

s 2  q1   
2
a  q1
if q1  qˆ1
0  qˆ1  a  c
 q1 *  qˆ
else
This is a non-credible threat but it satisfies NE conditions:
- the strategies are complete contingent plans
- players are best responding on the equilibrium path
if (2) plays s 2 then (1)'s BR is qˆ1
if (1) plays
qˆ1 then (2)'s BR is
a  q1  c
2
SPNE: requires (2) optimize after all q1 ' s . NE: a continuum of eqm. from 0  qˆ1  a
First-mover advantage
Note: that if we compute the SPNE, the leader is always better off than in the simultaneous
move case (Cournot competition)
Since (2) best responds, (1) can get at least the simultaneous move payoff by choosing the
Cournot q. There is a first-mover advantage in any game with strategic substitutes.
Defn: Strategies are strategic substitutes if best response functions are downward sloping,
and strategies are strategic complements if best response functions are upward sloping.
In games of strategic complements, first movers have a disadvantage
Policy game
Consider a policy game, where we show how politicians can get voters to spend too much.
A politician proposes a policy change and then a voter decides whether to accept or reject the
proposal. We should think of the voter as the median voter; the marginal person.
Suppose that the policy proposal involves setting some number:
- possibly a level of expenditure on education, for example
- normalize so that the voter’s preferred point is 0
- suppose the current level of expenditure is below the voter’s preferred point
A strategy for the politician is a number s 1  
A strategy for the voter is a mapping
status quo
sq
0
s2 :
 reject, accept
Suppose that the politician has to play a pure strategy, but that the voter can randomize.
Payoffs
We are assuming that the politician’s payoff is increasing in the expenditure level
- we do not need to be any more precise than that.
- denote the expenditure level by x
Assume that the voter's preferences are U  x   x 2
(drops off symmetrically)
So x  0 gives the highest utility and as x is further away from 0 the voter is worse off
Look for a SPNE by starting with the voter's BR fn. the status quo is sq U  sq     sq 
2
The voter will
- reject any policy proposal that gives a lower utility (is further away from 0 than sq)
- accept any policy that has a higher utility
- be indifferent between accepting and rejecting any proposal with the same utility as sq,
and therefore might mix
- at sq and -sq any probability of accepting is a best response
The SPNE
reject
accept
accept
sq
accept with prob 
0
reject
s q
accept with prob 
if
reject

if
accept
s2 *  
accept with prob 
accept with prob 

x  sq,
x   sq
x   sq,  sq 
if
x  sq
if
x   sq
Now back up to the politician’s move.
Since the politician wants expenditure to be as high as possible, the politician will
propose the highest level of expenditure that the voter is willing to accept.
For "the highest level" to exist it must be the case that   1.
If   1 , then the politician will choose an x that is as close as possible to - sq.
U  sq   1   U  sq   U  sq    for  small enough
But for any x  sq there is always an x  s.t. x  x   sq
Infinite NE
So the SPNE has
s1   sq
if x  sq x   sq
reject

s2  accept
if x   sq,  sq 
accept with prob  if x  sq

Note: as in the Stackelberg game, there are an infinite # of NE and an infinite # of NE paths
accept if x  x *
s2  x   
reject otherwise
x*   sq,  sq 
s1  x *
Another set of NE use the voter's indifferece between sq and  sq
accept with prob  if x   sq
s2  x   
otherwise
reject
s1   sq