The position of the particle moving in space at time t ≥ 0 is r(t) = (2 + 2 cos(t))i − 2 sin(t)j + (3 − t π )k. Find the first t
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Answer:
a) attached below
b) ( T,T )
c) The Pure-strategy Nash equilibria are : ( N,E ) and ( E,N )
d) The mixed-strategy Nash equilibrium for Firm 1 = ( 1/3 , 0, 2/3 )
while the mixed -strategy Nash equilibrium for Firm 2 = ( 1/3 , 0, 2/3 )
Step-by-step explanation:
A) write down the game in matrix form
let: E = exit at the industry immediately
T = exit at the end of the quarter
N = exit at the end of the next quarter
matrix is attached below
B) weakly dominated strategies is ( T,T )
C) Find the pure-strategy Nash equilibria
The Pure-strategy Nash equilibria are : ( N,E ) and ( E,N )
D ) Find the unique mixed-strategy Nash equilibrium
The mixed-strategy Nash equilibrium for Firm 1 = ( 1/3 , 0, 2/3 )
while the mixed -strategy Nash equilibrium for Firm 2 = ( 1/3 , 0, 2/3 ) since T is weakly dominated then the mixed strategy will be NE
Assume that P is the probability of firm 1 exiting immediately ( E )
and q is the probability of firm 1 staying till next term ( N ) ∴ q = 1 - P.
hence the expected utility of firm 2 choosing E = 0 while the expected utility of choosing N = 4p - 2q .
The expected utilities of E and N to firm 2 =
0 = 4p - 2q = 4p - 2 ( 1-p) = 6p -2 which means : p = 1/3 , q = 2/3
