Find all the Nash Equilibrium (Pure and Mixed Strategy) - Management Assignment Help

Assignment Task:

Task:

Question 1
Players 1 and 2 are bargaining over how to split $10. Each player i names an amount, si , between 0 and 10 for herself. These numbers do not have to be in whole dollar units. The choices are made simultaneously. Each player's payo is equal to her own money payo. We will consider this game under two dierent rules. In both cases, if s1 + s2 = 10 then the players get the amounts that they named (and the remainder, if any, is destroyed).

1

(a) [2] In the rst case, ifs1 + s2 > 10 then both players get zero and the money is
destroyed. What are the pure strategy Nash Equilibria of this game?
(b) [2] In the second case, if s1 + s2 > 10 and the amounts named are dierent, then the person who names the smaller amount gets that amount and the other person gets the remaining money. If s1 + s2 > 10 and s1 = s2, then both players get $5. What are the pure strategy Nash Equilibria of this game?
(c) [1] Is there a rst mover advantage in this game?
Question 2
Find all the Nash equilibrium (pure amd mixed strategy) in the following games

a) [2]

column
left middle right
row up 4,3 3,1 1,2
down 3,0 1,0 0,8

b) [1]

column
left right
row up 1,1 0,-1
down 0,-1 1, 1

c) [2]

column left centre right

row
up 5,4 8,3 5,5
middle 4,5 8,0 6,3
down 3,8 8,8 0,8

Question 3

Consider the following game between two players. Each player i has an initial endow- ment of 5 cans of beer. This beer can hidden in their private account (xi) or put in a public account, the fridge (yi). The happiness/payo (πi) of each player depends on the beer in her private account and the beer in the public account and takes the following

form:

πi = 2√ xi + 1 2 (y1 + y2) (0.1)

(assume only whole cans of beer can be allocated to each account.)
a) [1] What is the happiness function for player i = 1?
b) [2] What are the Nash equilibrium contributions for each player? And what is each player's happiness?
c) [1] Is there a set of contributions that gives each player a higher level of happiness than under the Nash Equilibrium? If not explain why not.
d) [1] What is the socially optimal allocation that maximizes total happiness?
Question 4
Alan and Bella are two shop owners that rent bicycles. They rent out identical bicycles and each need to set the price for their bicycles. Alan's price is denoted Pa and the number of bicycles rented at that price is Qa. Bella's price is denoted Pb and the number of bicycles rented at that price is Qb. It costs each Alan and Bella $8 to rent out a bike, assume they can always nd a bike to rent out if a customer requests it.
a) [3] An initial market survey shows that the relationship between prices and quantities are Qa = 16 − 2Pa + Pb and Qb = 24 − Pb + 2Pa. Alan and Bella choose their prices simultaneously. What is the Nash equilibrium?

b) [2] Consider the situation when Alan and Bella do not choose their price simulta-neously. If Bella chooses her price rst, what are the prices for Alan (Pa) and Bella (Pb)?

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