MAST30022 Decision Making - Assignment 3: Pareto order
MAST30022 Decision Making Assignment 3
1. (a) Let θ be a binary relation on a set A which is not necessarily transitive.
Define a binary relation θ∗ on A as follows: for any a,b ∈ A, aθ∗ b if and only if there exists a sequence a1, a2, . . . , ak ∈ A, where k ≥ 2 is an integer (which relies onaandb),suchthata=a1,b=ak andaiθai+1 foralli=1,2,...,k−1.
θ∗ is called the transitive closure of θ. Prove that θ∗ is a transitive relation on A.
(b) Let (T, r) be a rooted tree. Denote the set of nodes of (T, r) by V .
Let θ be the binary relation defined by
θ = {(a,b)|a,b∈V, aisaparentofb}.
(i) Show that θ is not transitive.
(ii) Let θ∗ be the transitive closure of θ and a, b ∈ V . Describe what relation holds between a and b if aθ∗b transitivity, reflexivity, comparability, symmetry, asymmetry, antisymmety.
(ii) Which property(ies) are gained/lost if “even” is replaced by “odd” in θ, and if ̄
Carefully explain your answers by providing proofs or counterexamples. Answer (a)(i) in the box below.
2. Define the two binary relations θ and θ on Z × Z by (a) θ = {(a, b) : a1a2 − b1b2 is even},
(b) θ={(a,b):a1 >b1 ora2 >b2}.
(i) Verify for each of the two relations which of the following properties are satisfied:
“a1 >b1 ora2 >b2”isreplacedby“a1 ≥b1 ora2 ≥b2”inθ?
Continue your answer to (a)(i) in the box below.
Answer (a)(ii) in the box below.
Answer (b)(i) in the box below.
Continue your answer to (b)(i) in the box below.
Answer (b)(ii) in the box below.
3. Let A = {(3,2,−2),(3,1,−1),(−4,2,1),(3,−1,1),(3,−1,−1),(4,−2,1),(−1,1,1)}.
(a) List the lexicographic order of A, and find the greatest and least elements of A.
(b) For the Pareto order on A, use the Boolean matrix representation to find the Pareto-maximal and Pareto-minimal element sets Pmax(A) and Pmin(A), and the Pareto greatest and least elements (if any) of A.
(c) Let f : R3 → R3 be defined by
f(x)=(x1 +x2,x1 +x3,x2 +x3) 3
for all x = (x1,x2,x3)∈R.
Denote the lexicographic order on R3 by L.
You may freely use the results from the lecture that L is reflexive, transitive, antisymmetric, and comparable.
Define θL by
θL ={(a,b)|a,b∈A,f(a)Lf(b)}.
(i) Determine whether θL satisfies the properties of reflexivity, transitivity, anti- symmetry, and comparability.
Continue your answer to (c)(i) in the box below.
(ii) Determine all maximal/minimal elements and greatest/least elements in A with respect to θ .
4. For the upcoming planting season, Farmer Q has four options:
a1: Plant corn;
a2: Plant wheat;
a3: Plant soybeans;
a4: Use the land for grazing.
The profits associated with these actions are influenced by the amount of rainfall, which could be one of four states:
θ1: Heavy rainfall;
θ2: Moderate rainfall; θ3: Light rainfall;
θ4: Drought season.
The profit matrix in (thousands of dollars) is estimated as
θ1 θ2 θ3 θ4
a1 −20 60 30 −5
a2 40 50 35 0
a3 −50 100 45 −10
a4 12 15 15 10
(a) Which course of action should the farmer take if he uses
(i) Wald’s maximin criterion;
(ii) Hurwicz’s maximax criterion;
(iii) Savage’s minimax regret criterion;
(iv) Laplace’s criterion?
Continue your answer to (a) in the box below.
(b) For each α ∈ [0,1] determine the action(s) that is/are imposed by Hurwicz’s α-criterion.
Continue your answer to (b) in the box below.
(c) Use the decision table above to show that Wald’s minimax criterion does not satisfy the axiom of independence of addition of a constant to a column.
