Game Theory and Econometrics Problems

This assignment comprises several problems related to game theory and econometrics. The first section involves analyzing a game with two repair shops, determining Nash equilibrium, dominant strategies, and whether it represents a prisoner's dilemma. The second section presents a game with multiple players and strategies, requiring the identification of maximin strategies, dominated strategies, Nash equilibria (pure and mixed), and sub-game perfect Nash equilibria for sequential scenarios. The third section focuses on econometrics, requiring the application of an AR(1) autoregression model to GDP data (both current dollar and chained 2009 dollars), comparing model fits based on standard error, log-likelihood, and AIC, and predicting GDP for 2017. An extra credit section involves another simultaneous game, requiring the identification of Nash equilibria and an iterated deletion sequence. Finally, a problem explores the correlation between income per person and population health, requiring explanations of how higher income might cause better health and vice-versa, and the implications for public policy.

Problem:
A town has two repair shops, Blue and Red. Each shop must decide whether it will remain open on
Sunday or be closed on that day. The payoffs, which are in dollars, are provided in the normal form
below. This is a static, or one-shot, game.
Red
Open Closed
Sunday Sunday
Open Sunday 8,000 7,000
Blue 4,000 7,000
Closed Sunday 10,000 12,000
3,000 6,000
Assuming the shops make their decisions simultaneously:
(A) Which shop is the most profitable in the Nash equilibrium?
First let’s find out the Nash Equilibrium using Best Response Function
i) BRblue(Open) = Open
ii) BRBlue(Closed) = Open
iii) BRRED(Open) = Open
iv) BRRED(Closed) = Closed
Therefore, the nash equilibrium in this case is (Open,Open) and RED Shop is the most profitable.
(B) Does the shop you identified in part (A) have a dominant strategy? If so, what is it?
Yes, Red shop has dominant strategy. As it can be seen in the first part no matter what strategy the RED
shop follows (i.e. Open or Closed) the Blue will always choose Open strategy.
(C) What should this shop do? Explain your answer.
Considering the nash equilibrium and dominant strategy, we can say that the RED shop must Open its
shop on Sunday as the Blue shop will be opened on Sunday in any case. To get the maximum payoff the
RED must open its shop on Sunday to receive the highest payoff in this particular case i.e. 8000.
(D) Is this an example of a prisoners’ dilemma.
Yes, it is an example of prisoners’ dilemma.

Problem:
Player
2
L C R
U 8 7 9
7 9 9
Player
1 M 5 6 8
9 6 7
D 8 6 7
10 8 8
The payoffs are in dollars and the game is static. For parts (A) through (C) this is a simultaneous move
game.
(A) List the outcome from each player choosing his/her maximin strategy.
(B) List any dominated strategies.
(C) List the Nash equilibria (pure and mixed strategy).
(D) List the sub-game perfect Nash equilibrium for the sequential game with player 1 moving first.
(E) List the sub-game perfect Nash equilibrium for the sequential game with player 2 moving first.
Problem:
The data for this problem (Current dollar and real GDP) is available in BlackBoard in the Assignment tab.
(A) Run an autoregression [AR(1)] on the annual GDP data using the current dollars.
AR_Current_Dollar=arima(Time_Current_Dollars, order=c(1,0,0))
AR_Current_Dollar
(B) Run an autoregression [AR(1)] on the annual GDP data using the chained 2009 dollars.
AR_Chained_Dollar=arima(Time_Chained_Dollars, order=c(1,0,0))
AR_Chained_Dollar
(C) Which of the autoregression models has a better fit? What information are you using to make
your selection?
Result of Current Dollar
Call:
arima(x = Time_Chained_Dollars, order = c(1, 0, 0))
Coefficients:
ar1 intercept
0.8018 42.1427
s.e. 0.0638 7.9521
sigma^2 estimated as 235.9: log likelihood = -365.77, aic = 737.54

Result of Chained Dollar
Call:
arima(x = Time_Current_Dollars, order = c(1, 0, 0))
Coefficients:
ar1 Intercept
0.7721 42.8144
s.e. 0.0669 7.2969
sigma^2 estimated as 260.3: log likelihood = -370.03, aic = 746.07
It can be seen that Result of Current Dollars Auto Regression Model has better fit. It is said based on
3 parameters:
I) Standard Error Coefficients – The S.E. coefficient of Current dollar is lower than coefficient of
Chained Dollar
II) Log Likelihood – Higher the value of log likelihood better is the model. On this basis the
current dollar model has better fit
III) AIC = The value of AIC is lower in the current dollar model therefore this model is better fit
(D) What do your models predict GDP will be for 2017 (provide predictions for both current dollar
and chained 2009 dollars)?
Current Model Predicts 2017 GDP – 22.15
Chained Model Predicts 2017 GDP – 24.72
Extra credit (6 points):
This is a simultaneous game.
Player
2
X Y Z
A 3 0 6
0 0 6
Player
1 B 3 3 0
0 3 6
C 0 6 6
3 3 0
(A) List the Nash Equilibria (pure and mixed strategy) for this game.
(B) Provide the iterated deletion sequence to arrive at outcome (B; Y).

First we can strike off the row A of player 1 because its payoff is lesser or equal to payoff of row B of
player 1. We can now strike off column Z of player 2 since its payoff is either lesser or equal to payoff or
column Y of player 2. Again, we can now strike off column X of player 2 since its payoff is either lesser or
equal to payoff or column Y of player 2. Now we are left with B: Y and C: Y. Since Player 2 payoff in C: Y is
higher than B: Y we can say that C: Y is the final outcome.
Problem:
International data show a positive correlation between income per person and the health of the
population.
a- explain how higher income might cause better health outcomes.
Higher Income means better health care facilities, more nutritious food. Individuals with higher
income will have access to purified water, clean vegetables, medical care and healthy life.
b-explain how better health outcomes might cause higher income.
Better health means better lifestyle, healthy body and fresh mind which could lead to higher
productivity. Weaker or sick people are unable to work with their full capacity and organizations don’t
like to hire or retain weak employees.
c-how might the relative importance of your two hypotheses be relevant for public policy?
The importance of hypothesis will be relevant for public policy considering an overall and complete
public health programs or policies will enhance the total economic efficiency and performance. Also, the
total cost involved in these policies will not be alone borne by the government; a small share of it will be
self born by the positive self induced impacts.