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Information, Control and Games
Homework #1 (Due 10/6/2005)
1. (Exercise 5.3 of the textbook, Altruistic preferences)
Person 1 cares about her income and person 2’s income. Precisely, the value she
attaches to each unit of her own income is the same as the value she attaches any
two units of person 2’s income. Fore example, she is indifferent between a
situation in which her income is 1 and person 2’s is 0, and one in which her
income is 0 and person 2’s is 2. How do her preferences order the outcomes
(1,4), (2,1), and (3,0), where the first component in each case is her income and
the second component is person2’s income? Give a payoff function consistent
with her preferences.
2. Apply the optimality conditions to
(a) show that the 2-dimentsional function f(x,y) = (x2 - 4)2 + y2 has twp global
minima and one stationary point, which is neither a local maximum nor a local
minimum and
(b) find all local maxima and all local minima of f(x,y)= sinx + siny + sin(x+y)
with the set {(x,y) | 0< x < 2, 0< y < }.
3. Let f(x) = x12 + x22- x1x2 + x1 + 2x2.
(a) Derive f(x) and 2f(x).
(b) Is 2f(x) positive definite?
4. Consider the problem
x x 
Maximize x1 2 3
Subject x1+ x2+ x3 =1,
xi>=0, i=1,2,3,
10
6
A
5
7
8
6
10
7
8
9
10
7
5
9
5
12
8
6
7
9
6
7
where and  are given positive scalars.
Write down the optimality conditions of this problem.
Figure 1
5. There are two electrical power generators and u1 and u2 are the power generated
respectively. The generation cost functions are
f1(u1) = 0.5u12 + u1 + 1, 0 <= u1 <=3 and
f2(u1) = 0.25 u12 + 2, 0 <= u2 <=4.
Now you have to generate 5 units of powers with the minimum cost.
(a) Write down a mathematical problem formulation of the optimization problem.
(b) Use the Lagrange multiplier method to solve (5.a).
(c) What is the interpretation of the Lagrange multiplier you used in (5.b)?
6. Describe your procedure of finding the shortest path from A to B in Figure 1 by
starting from A (in contrast to the backward procedure starting from B as
described in the class.)
C
10
B
11
D