Voting theory, decision theory �and �multi-criteria decision analysis (MCDA)

Conjoint measurement

aka

multi-Criteria Decision Aid

multi-Criteria Decision Making (MCDM)

Multi-Criteria Decision Analysis (MCDA)

A DM wants to rank the alternatives (or identify the best one).

Many methods in Management Science and Operations Research:

MAUT, MAVT, WSM, ELECTRE, AHP, ANP, DEMATEL, VIKOR, COPRAS, MULTIMOORA, TOPSIS, fuzzy TOPSIS, PROMETHEE, TACTIC, …

MAVT

How to choose the parameters v_i and w_i ?

• Direct rating
• Elicitation techniques with fictitious alternatives.

E.g. midvalue splitting: find s < 70 on criterion 1 and t > 0.03 on criterion 3 s.t.

MAVT

How to choose the parameters v_i and w_i ?

• Direct rating
• Elicitation techniques with fictitious alternatives.

E.g. midvalue splitting: find s < 70 on criterion 1 and t > 0.03 on criterion 3 s.t.

( 70 , good , 0.03) ~ ( s , good , t )

MAVT

How to choose the parameters v_i and w_i ?

• Direct rating
• Elicitation techniques with fictitious alternatives.

E.g. midvalue splitting: find s < 70 on criterion 1 and t > 0.03 on criterion 3 s.t.

( 70 , good , 0.03) ~ ( s , good , t ) AND ( s , good , 0.03) ~ ( 30 , good , t )

MAVT

How to choose the parameters v_i and w_i ?

• Direct rating
• Elicitation techniques with fictitious alternatives.

E.g. midvalue splitting: find s < 70 on criterion 1 and t > 0.03 on criterion 3 s.t.

( 70 , good , 0.03) ~ ( s , good , t ) AND ( s , good , 0.03) ~ ( 30 , good , t )

ELECTRE (simplified)

Step 1. One preference relation per criterion: ≿_i (≻_i / ~_i)

x ≻₁ y ≻₁ (z ~₁ w)

w ≻₂ (x ~₂ z) ≻₂ y

w ≻₃ (x ~₃ y) ≻₃ z

​

3 parameters (indifference thresholds)

q₁ = 2, q₂ = 0 , q₃ = 0

ELECTRE (simplified)

Step 1. One preference relation per criterion: ≿_i (≻_i / ~_i)

x ≻₁ y ≻₁ (z ~₁ w)

w ≻₂ (x ~₂ z) ≻₂ y

w ≻₂ (x ~₂ y) ≻₂ z

​

Step 2. Weighted majority relation: S

a S b if sum of weights supporting a is larger than λ.

3 parameters: w₁= 0.25, w₂= 0.25, w₃= 0.5, λ = 2/3

3 parameters (indifference thresholds)

q₁ = 2, q₂ = 0 , q₃ = 0

ELECTRE (simplified)

Step 1. One preference relation per criterion: ≿_i (≻_i / ~_i)

x ≻₁ y ≻₁ (z ~₁ w)

w ≻₂ (x ~₂ z) ≻₂ y

w ≻₂ (x ~₂ y) ≻₂ z

​

Step 2. Weighted majority relation: S

a S b if sum of weights supporting a is larger than λ.

3 parameters: w₁= 0.25, w₂= 0.25, w₃= 0.5, λ = 2/3

y vs z : 0.25+0.5 > 2/3 ⇒ y S z and ¬(z S y)

3 parameters (indifference thresholds)

q₁ = 2, q₂ = 0 , q₃ = 0

ELECTRE (simplified)

Step 1. One preference relation per criterion: ≿_i (≻_i / ~_i)

x ≻₁ y ≻₁ (z ~₁ w)

w ≻₂ (x ~₂ z) ≻₂ y

w ≻₂ (x ~₂ y) ≻₂ z

​

Step 2. Weighted majority relation: S

a S b if sum of weights supporting a is larger than λ.

3 parameters: w₁= 0.25, w₂= 0.25, w₃= 0.5, λ = 2/3

y vs z : 0.25+0.5 > 2/3 ⇒ y S z and ¬(z S y)

x

y

w

z

Step 3. Compute a ‘tournament’ solution (Copeland score or …)

3 parameters (indifference thresholds)

q₁ = 2, q₂ = 0 , q₃ = 0

ELECTRE (simplified)

How to choose the parameters q₁ , q₂ , q₃ , w₁, w₂, w₃, λ ?

• Direct rating
• Elicitation techniques with fictitious alternatives

Axiomatic characterizations of MCDA methods

Mostly in the framework of

​

• Voting theory
• Conjoint measurement

Voting theory

Primitives

• Set of candidates X = { x, y, z, … }
• Set of voters V = {1, 2, …, n}
• Profile of preferences p = (p₁, p₁, …, p_n)

MCDA

​

set of alternatives

set of criteria

performance matrix

Partial holistic preferences ⊵

Conjoint measurement

Primitives

• Attribute i = {x₁, y₁, …, z₁}
• Set of alternatives X = X₁× X₂ × … × X_n
• One preference relation ≿ over X

MCDA

• Set of alternatives is a sparse subset of X₁× X₂ × … × X_n
• What is ≿ ?
• The partial holistic preferences ⊵?
• The final preference relation to be built?
• Both (if final preferences ⊇ holistic preferences)

MCDA in conjoint measurement (CM) framework

Primitives

• Set of alternatives X = X₁× X₂ × … × X_n
• ≿ over X represents final preferences s.t. ⊇ ⊵

Typical result in conjoint measurement

• If ≿ satisfies some properties P then ≿ can be represented in model M.
• The representation is in some sense unique.

Can we check that ≿ satisfies properties P?

No, but we can impose it (normative properties).

How to construct ≿ containing ⊵ and satisfying P?

No answer in CM. It characterizes relations representable in model M.

It does not characterize an aggregation procedure.

There are many relations ≿ containing ⊵ and satisfying P!

Other problems with conjoint measurement

It is impossible to state properties like

• Neutrality
• Independence of irrelevant alternatives

New framework for MCDA

Primitives

• Set of alternatives X = { x, y, z, … }
• Set of criteria C = {1, 2, …, k}
• One assessment structure on each criterion

New framework for MCDA

Primitives

• Set of alternatives X = { x, y, z, … }
• Set of criteria C = {1, 2, …, k}
• One assessment structure on each criterion

• a weak order on X
• a mapping from X to ℝ
• a mapping from X to {excellent, good, fair, poor}
• …

New framework for MCDA

Primitives

• Set of alternatives X = { x, y, z, … }
• Set of criteria C = {1, 2, …, k}
• One assessment structure on each criterion
• E_i = set of possible assessment structures on criterion i
• A profile of assessment structures p = (p₁, p₁, …, p_n) ∊ E₁× E₂ × … × E_n
• Holistic preferences: ⊵ (binary relation on X, belonging to H)
• Exogeneous weights: w ∊ W = (ℝ₊)^k \ { (0,…,0) }

​

Aggregation method:

Mapping ≿ from E₁× E₂ × … × E_n × H × W to ℛ (reflexive relations).

​

​

Constraints on assessment structures

• Each assessment structure p_i defines a reflexive preference relation on criterion i.

p_i → δ_i(p_i)

• If π is a permutation of X, there is a natural way to define π(p_i) , � a permutation of p_i

(clean definition uses homomorphism).

​

• If Y is a subset of X with #Y ≥2, there is a natural way to define p_i|_Y , � the restriction of p_i to Y.

• If δ_i(p_i)|_Y = δ_i(q_i)|_Y , there exists p’, q’ : δ_i(p_i’) = δ_i(p_i) and δ_i(q_i’) = δ_i(q_i)

and p_i’|_Y = q_i’|_Y.

​

A result without ⊵

• Independence of Holistic Preferences: ≿(p, ⊵₁, w) = ≿(p, ⊵₂, w)
• Completeness: for all x, y, we have x ≿(p, ⊵, w) y or y ≿(p, ⊵, w) x
• Monotonicity : x ≿(p, ⊵, w) y and x ≿(p, ⊵, w’) y ⇒ x ≿(p, ⊵, w+w’) y

≿ ≻ ≻

• Archimedean: x ≻(p, ⊵, w) y ⇒ ∃ β: for α > β, x ≻(p, ⊵, αw+w’) y
• IIA: p|_xy = q|_xy ⇒ ≿(p, ⊵, w)|_xy = ≿(q, ⊵, w)|_xy
• Ordinality: δ_i(p_i) = δ_i(q_i) for all i ⇒ ≿(p, ⊵, w) = ≿(q, ⊵, w)
• Neutrality: ≿(π(p), π(⊵}, w) = π(≿(p, ⊵, w) )
• Weighted Anonymity: permuting p and w correspondingly has no effect
• IIC: If p and q differ on criteria with weights = 0, then ≿(p, ⊵, w) = ≿(q, ⊵, w)

A result without ⊵

Theorem (building on Myerson, 1995)

Let δ_i(p_i) be complete for all p_i ∊ E_i and each criterion i. The only aggregation procedure satisfying all axioms of previous slide is the simple weighted majority, that is

x ≿(p, ⊵, w) y iff

A result with ⊵

• Ind. of Weights: ≿(p, ⊵, w) = ≿(p, ⊵, w’)
• Unanimity: x δ_i(p_i) y and ¬ y δ_i(p_i) x for all i, ⇒ x ≻(p, ⊵, w) y
• Non-negative responsiveness:

∀ i, [¬ y δ_i(p_i) x ⇒ ¬ y δ_i(q_i) x ] and [ x δ_i(p_i) y ⇒ x δ_i(q_i) y ],

then ≿(p, ⊵, w) ⇒ ≿(q, ⊵, w)

• Strong duality: m profiles p¹, …, p^m

∀ i, number of profiles s.t. x δ_i(p_i^j) y and ¬ y δ_i(p_i^j) x

= number of profiles s.t. y δ_i(p_i^j) x and ¬ x δ_i(p_i^j) y

Then x ≻(p^j, ⊵, w) y and y ≻(p^k, ⊵, w) x for some j, k.

• Weak IIA: p|_xy = q|_xy and ⊵|_xy = ⊵|_xy ⇒ ≿(p, ⊵, w)|_xy = ≿(q, ⊵, w)|_xy

​

A result with ⊵

Theorem (building on Fishburn, 1973)

Let δ_i(p_i) be complete for all p_i ∊ E_i and each criterion i. If ≿ satisfies

• Completeness, Ordinality,
• Unanimity, Non-negative responsiveness, Strong Duality, IIW, Extended IIA

then it is the simple weighted majority, that is,

for all ⊵ ∊ H, there are non-negative real numbers c_i^⊵ such that

x ≿(p, ⊵, w) y iff

x ≿(p, ⊵, w) y iff

x ≿(p, ⊵, w) y iff