Zhao Wu 50024849
Fall 2011 IE 675 Homework No. 1
Due: September 14, 2011
1. Briefly describe one simple story happened to your real life in the past week, which could be
modeled as a game. Specify the key components of this game, including: the players (who),
options/moves (what to do) for each player, sequences of moves (who move first/second; or
simultaneously), objectives/payoffs of each player (and dependence on others), information
(complete vs. incomplete information), time (one-shot or repeated; ending points), and what
is (or should be) equilibrium.
Answer:
In the past week, I was looking for a nice car and I actually found my ideal car but I failed to
get it.
The players: Me(Zhao Wu), the unknown guy X
Objectives: To buy a car in a relatively low price and good condition
Sequences of moves: 1.Zhao Wu 2.X
Information: We both heard some information from the seller and only have a basic test drive,
so we both have incomplete information. We both think it’s might be a good deal.
Options for me:
A.Pay some subscription and ask the seller not to sell the car until I can do a complete check
on this car.
B.Buy the car immediately.
C.Not paying the subscription, just make an appointment that I will do a complete check the
day after tomorrow.
Zhao Wu 50024849
Options for X:
If A: Not paying the subscription, just make an appointment that he will do a complete check
if Zhao Wu don’t buy or give up.
If B: Give up.
If C: Buy the car immediately or Pay some subscription and ask the seller not to sell the car
until I can do a complete check on this car.
Time: Buying is of course an one-shot
Equilibrium: I think equilibrium should be that Zhao Wu should choose A and X choose to
make an appointment.
Well, the real situation is that Zhao Wu chose C and X chose to buy the car immediately
without a complete check.
2. What is your favorite food (or any discrete consumption of goods)? Try to identify and draw
your relative preferences over all possible quantities 0, 1, 2, ..., N, where N is a large
number such that you really hate the Nth consecutive consumption of your favorite food.
Provide what your N equals to.
Answer:
My favorite food would be steamed dumplings.
Blue lines represent the number of steamed dumplings I supposed to have, and red lines
represent my relative preferences over them.
Zhao Wu 50024849
25
20
15
10
Series1
5
Series2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
-5
-10
-15
In my case, when I have about 22 steamed dumplings I would start to hate them.
3. Only using definitions and rationality assumptions and axioms introduced in lectures, prove
the following proposition:
Proposition: If  is rational then:
※> is both irreflective (x > x never holds) and transitive (if x > y and y > z, then x > z).
※∼ is reflective (x ∼ x for all x), transitive (if x ∼ y and y ∼ z, then x ∼ z), and
symmetric (if x ∼ y, then y ∼ x).
※If x > y  z, then x > z.
Answer：
1.1 With the definition of strict preference,
we can know that if x>x holds then we would have the conclustion that x  x but not x
Zhao Wu 50024849
 x, which is obviously self-contradiction.
1.2
With the definition of rationality,
we can know that if x>y then x  y but not y  x; if y>z then y  z but not z  y.
So we can draw the conclusion that x  z but not z  x, and we can use the definition
of strict preference again to deduce that x>z.
2.1
With the definition of indifference relation,
We can know that if x  x and x  x, which obviously holds, then x x holds.
2.2
Still based on the definition of indifference relation, we can see that if x y then x  y
and y  x; if y z then y  z and z  y. So we have x  y  z and z  y  x. Then based
on the transitivity property of rationality,
We can deduce that x  z and z  x. Based on the definition of the indifference
relation, we can conclude that x z.
2.3
Based on the definition of indifference relation, if x y, then x  y and y  x.
Vice Versa, if y  x and x  y then we can deduce that y x.
3.1
We can think about 2 situations
i. if y>z, then we can have x>y>z. According to 1.2 , we can deduce that x>z.
ii. if y z, then we can simply deduce that x>z.
As a conclusion, if x>y  z, then x>z.
Zhao Wu 50024849
4. Using the definition of utility functions, prove the following: For any strictly increasing function f : ℜ → ℜ,v(x)= f (u(x)) is a new utility function representing the same preferences as
u(·).
Answer:
Assume that we have a,b  ℜ and a  b. With the definition of utility functions,
We can conclude that u(a)  u(b). (u(a),u(b)  ℜ)
For v(x)=f(u(x)), we can insert u(a) and u(b), so we can get v(a)=f(u(a) and v(b)=f(u(b).
Because u(a)  u(b), we can deduce that v(a)  v(b).
Finally, we can see that if a,b  ℜ and a  b, then u(a)  u(b) and v(a)  v(b). So we can
prove the assumption in the problem.
5. Back to our original lottery example...
$10 w.p. 0.4
Option A: $1 vs. Option B:
$0 w.p. 0.6
$10M w.p. 0.4
Option A’: $1M vs. Option B’:
$0 w.p. 0.6
Suppose a decision maker is an expected utility maximizer with each of following four
2
utility functions u1(x)= 1 + 2x, u2(x)= x , u3(x)=
√
x, u4(x)= log(x)
Maybe redesign such that we prefer A but not A’... exponential..
Zhao Wu 50024849
(a) Draw the utility diagram
Answer:
1. u1(x)=1+2x
2. u2(x)=x2
Zhao Wu 50024849
3. u3(x)=
x
4. u4(x)=log(x)
(b) What are the certainty equivalents?
Answer:
1. U1(x)=1+2x
For Option A and B:
U(x=10)=11
U(x=0)=1
EU=11X0.4+1X0.6=5
U(c)=EU  1+2c=5  c=2
For Option A’ and B’:
Zhao Wu 50024849
U’(x=10^7)=1+2X10^7
U’(x=0)=1
EU’=0.4X(1+2X10^7)+0.6X1=0.8X10^7
U(c’)=EU’  1+2c=0.8X10^7  c=0.4X10^7
2. U2(x)= x 2
For Option A and B:
U(x=10)=100
U(x=0)=0
EU=100X0.4+0X0.6=40
U(c)=EU  c 2 =40  c=6.32
For Option A’ and B’:
U’(x=10^7)=10^14
U’(x=0)=0
EU’=0.4X10^14+0.6X0=4X10^13
U(c')=EU’  c 2 =4X10^13  c=6.32X10^6
3. U3(x)= x
For Option A and B:
U(x=10)=3.16
U(x=0)=0
EU=3.16X0.4+0X0.6=1.265
U(c)=EU 
c =1.265  c=1.6
For Option A’ and B’:
U’(x=10^7)=3.16X10^3
Zhao Wu 50024849
U’(x=0)=0
EU’=0.4X3.16X10^3+0.6X0=1.265X10^3
U(c')=EU’  c =1.265X10^3  c=1.6X10^6
4. U4(x)=log(x)
For Option A and B:
U(x=10)=1
U(x=0)=0
EU=1X0.4+0X0.6=0.4
U(c)=EU  log(c)=0.4  c=2.512
For Option A’ and B’:
U’(x=10^7)=7
U’(x=0)=0
EU’=0.4X7+0.6X0=2.8
U(c')=EU’  log(c)=2.8  c=630.96
Zhao Wu 50024849
(c) Should we choose A or B; and A’ or B’, using these four utility functions? Why?
Answer:
1. For U1(x)=1+2x,
For A and B,We can find utility of expected value=8.4, and the utility of $1=3,so we
would choose B.
For A’ and B’, We can find utility of expected value=8M, and the utility of $1M=2M,so we
would choose B.
Zhao Wu 50024849
2. For U2(x)= x 2
For A and B,We can find utility of expected value=40, and the utility of $1=1,so we would
choose B.
For A’ and B’, We can find utility of expected value=4X10^13, and the utility of
$1M=10^12,so we would choose B.
Zhao Wu 50024849
3. For U3(x)=
x,
For A and B, We can find utility of expected value=1.265, and the utility of $1=1,so we
would choose B.
For A’ and B’, We can find utility of expected value=1265, and the utility of $1M=1000,so
we would choose B.
Zhao Wu 50024849
4. For U4(x)=log(x)
For A and B, We can find utility of expected value=0.4, and the utility of $1=0,so we
would choose B.
For A’ and B’, We can find utility of expected value=2.8, and the utility of $1M=6,so we
would choose A.
Zhao Wu 50024849
(d) Which utility function better fit yours? why?
Answer:
I think U4(x) would better fit mine since it is more accurate than the former three.
Because I think I can’t afford the “loss” of 1M even if I would have a chance to win 10M.
Besides, 1M is already enough for me. Maybe I would choose to B’ when I have 5M.
6. Preparing for your project proposal:
(a) Tell me what general research area you want to work on for your project (e.g., supply chain
management, transportation, computer network, homeland security, health care, politics,
communication, etc.)
(b) Identify the top 2 academic journals (or conferences proceedings if more prestigious) in your
specialty.
(c) With the help of UB-subscribed databases (e.g., “ISI Web of Knowledge”), find a total
number of three full papers from of the top 2 journals you identified in (b), related to the general
research area you identified in (a) AND game theory. Provide the full citation for those three
papers. (Look at the “reference” section of those papers for citation styles.)
1
(d) For each of the three papers, (quantitively and/or qualitatively) identify the key component of
the games:
2
.
Players (who?)
.
Options/moves (what to do?)
.
Sequences of moves (who move first/second? simultaneously?)
.
Objectives/Payoffs (and dependence on others)
.
Information (complete vs. incomplete information)
.
Time (repeated? Ending point?)
.
Equilibrium (solution to the game; how to calculate?)
(e) Suppose you were going to do a project extending any of the above three papers. Briefly
describe what do you plan to do, and how you would do it.
Zhao Wu 50024849
Hint: During your searching process, you may use some key words like “game
1
theory/game-theoretic” “equilibrium/equilibria,” “collaboration/collaborative,” “interaction/interactive,”
“competition/competitive,” “multi-agent,” “multi-player,” “signaling,” “screening.” Be smart and good luck!!
You can focus on one game for each paper if there are multiple games in one paper.
2
Anwer:
(a).I would like to focus on the area of homeland security.
(b).
Defence and Peace Economics
ANNALS OF OPERATIONS RESEARCH
(c). In fact , there is one full paper that I am interested in but it’s not on the listed journals in (b).
I think it would be helpful for my proposal so I still put it here.
From:Economica
The Interplay Between Preemptive and Defensive Counterterrorism
Measures: A Two-stage Game
By SUBHAYU BANDYOPADHYAY and TODD SANDLER
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From:Annals of Operations Research
A Stackelberg game model for resource allocation
in cargo container security
By Niyazi Onur Bakır
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(d).
The Interplay Between Preemptive and Defensive Counterterrorism
Measures: A Two-stage Game
Players: Homeland (H) and Foreign Country (F)
Background: H and F are the potential targets of a terrorist group
Options: H and F can choose to be preemptive or defensive
Sequences: It is the problem.
Objectives: To protect homeland at an optimal cost.
Information: Both have incomplete information because no country would show the
homeland security to others.
Time: Once
Equilibrium:
There are five key determinants: the countries’ relative defensive costs, their relative
preemption costs, their relative assets abroad, their relative damage assessment at home,
and terrorists’ attack preferences.
The high-cost defender will often provide preemption that benefits both targeted countries.
In addition, the prime-target country is prone to preempt in order to reduce its subsequent
defence spending.
SECRECY AND DECEPTION AT EQUILIBRIUM, WITH APPLICATIONS
TO ANTI--TERRORISM RESOURCE ALLOCATION
Players: the attacker(A) and the defender(D)
Options: For D, she can choose from secrecy, truthful disclosure, and deceptive disclosure.
Sequences: first, nature chooses the types of the attacker and defender (θA and θD),
according to the probability distributions pA(θA) and pD(θD). (Recall that the realization
of the random variable θA is observable only to the attacker, and the realization of θD is
observable only to the defender.) Second, a defender of type θD chooses a (possibly mixed)
strategy σD(θD) and a signal s(θD). Finally, an attacker of type θA responds to the
observed signal s by choosing a (possibly mixed) attacker effort σA(s,θA), leading to
attacker and defender total utilities given by UA[σA(s,θA), σD(θD),θA,θD]) and
UD[σA(s,θA), σD(θD),s(θD),θA,θD]), respectively. The attacker’s response σA(s,θA) is
determined endogenously in this model.
FIGURE 1
Sequence of actions for defender and attacker with private information
Zhao Wu 50024849
Objectives: A wants to maximize the probability of a successful attack. D wants to minimize
the expected attack consequences.
Information: Both have incomplete information.
Time: Repeated.
Equilibrium:
For general situation,
Zhao Wu 50024849
A Stackelberg game model for resource allocation in cargo container security
Players: defender(D) and attacker(A)
Options: There are multiple container routes for A to choose to transport weapons and D
has to allocate money and human resources on observing these routes.
Sequences: D allocates first and A decides where to attack later.
Objectives: A wants to maximize the probability of a successful attack. D wants to minimize
the sum of the total cost and expected attack consequences.
Information: Both are incomplete.
Time: Repeated.
Equilibrium:
(e).I would like to do a project extending A Stackelberg game model for resource
allocation in cargo container security.
I am thinking about what if we let attackers move first and defenders react later. As long
as the reaction is taken without delay, defenders can also protect their homeland and
reduce the loss to a minimum.
I plan to read more papers and try to fully understand what they say and what the
function mean because I just have a basic idea now and I need more specialist knowledge
to support my project.
I also plan to learn some cases about homeland security in order to be familiar with all
variables involved.
.