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Optimal Prices in the Presence of Discounts: A New Economic Application of Choquet Integrals

Hung T. Nguyen1,2 and Vladik Kreinovich3

1Mathematics, New Mexico State University, USA 2Economics, Chaing Mai University, Thailand 3Computer Science, University of Texas at El Paso

hunguyen@nmsu.edu, vladik@utep.edu

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 12 Go Back Full Screen Close Quit

1. Practical Situation: Discounts

• In many real-life situations, there is a “package deal”:

customers who buy two or more items get a discount.

• When planning a trip, a deal including airfare and hotel

costs less than the airfare and hotel on their own.

• There are often also special deals when we combine

airfare with car rental.

• There are deals that cover all three items plus tickets

to local attractions, etc.

• When a customer buys a house, there are often package

deals to also buy furniture and/or car.

• In fast food restaurants, such group discounts are a

norm.

• Usually, there are several such deals with different com-

binations and different discounts.

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 12 Go Back Full Screen Close Quit

2. How to Describe Discounts in Precise Terms

• Let us denote, by n, the total number of different items,

and let us denote these items by x1, . . . , xn.

• Then, the set of all the items is X = {x1, . . . , xn}.
• Let v(xi) denote the price of each individual item.
• A discount means that for some sets of items S ⊆ X,

the overall price v(S) is v(S) <

xi∈S

v(xi).

• For example, if x1 is the airfare and x2 is the hotel, then

a package deal means that v({x1, x2}) < v(x1) + v(x2).

• Not all combinations lead to package deals.
• For some combinations S, the cheapest price v(S) is

exactly the sum

xi∈S

v(xi) of individual prices.

• We assume that for every set S ⊆ X, we know the

cheapest price v(S) that we have to pay for this set S.

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 12 Go Back Full Screen Close Quit

3. Natural Assumption: the Larger the Group, the Better the Discount

• Natural assumption: the larger the group, the better

the discount.

• Situation: we want to buy items from two sets S and S′.
• Option: buy them separately and pay the sum

v(S) + v(S′).

• Cheaper option:

– to buy the whole group S ∪ S′ – and then – buy additionally all the duplicate items S ∩ S′.

• Formal description: for all sets S and S′, we have

v(S) + v(S′) ≥ v(S ∪ S′) + v(S ∩ S′).

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 12 Go Back Full Screen Close Quit

4. Formulation of the Practical Problem

• What if we want to buy several items of each type, i.e.,

we want to buy:

• d1 items of type x1,
• d2 items of type x2, etc.
• Example: we want to plan a group tour in which:
• some tourists want to rent car,
• some want to visit certain attractions, etc.
• Problem: What is the best way to use all available

discounts?

• Comment:
• prices v(S) correspond to sets (di = 0 or di = 1);
• we want to extend to bags (multisets), when

di = 0, 1, 2, . . .

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 12 Go Back Full Screen Close Quit

5. Towards a Precise Formulation: What Is Given

• Each way to use the discounts consists of:
• selecting discounts – i.e., sets S1, . . . , Sm – and
• selecting how many times t1, . . . , tm we use each of

the discounts.

• Totally, we should get exactly d1 objects of type x1,

d2 objects of type x2, etc.

• Each set Si ⊆ X can be identified with its characteris-

tic function, for which:

• χSj(xi) = 1 if the item xi is in the set Sj and
• χSj(xi) = 0 otherwise.
• For each selection of sets Si and times ti, the overall

price is equal to t1 · v(S1) + . . . + tm · v(Sm).

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 12 Go Back Full Screen Close Quit

6. Towards a Precise Formulation: What We Want Let X = {x1, . . . , xn} be a finite set.

• By a discount function, we mean v : 2X → R+

0 that

maps subsets S ⊆ X into non-negative real numbers s.t. v(S) + v(S′) ≥ v(S ∪ S′) + v(S ∩ S′).

• A task is a tuple d = (d1, . . . , dn) of natural numbers.
• A purchasing plan is a pair P = (S1, . . . , Sm), (t1, . . . , tm),

with Sj ⊆ X and tj ∈ Z.

• A plan P satisfies the task d if for every xi, we have

di =

m

• j=1

tj · χSj(xi).

• A price v(P) of the plan is

v(P)

def

= t1 · v(S1) + . . . + tm · v(Sm).

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 12 Go Back Full Screen Close Quit

7. Precise Optimization Formulation of the Practical Problem

• Given: a discount function v : 2X → R+

0 and a task

d = (d1, . . . , dn),

• Among: all the purchasing plans

P = (S1, . . . , Sm), (t1, . . . , tm) which are consistent with the task d, i.e., for which di =

m

• j=1

tj · χSj(xi),

• Find: the purchasing plan P with the smallest price

v(d) = min Pv(P), where v(P) = t1 · v(S1) + . . . + tm · v(Sm).

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 12 Go Back Full Screen Close Quit

8. Main result

• Theorem. For every discount function v and for every

task d, if we order the values di in the increasing order d(1) ≤ d(2) ≤ . . . ≤ d(n) and order the items accordingly, then v(d) = d(1) · v({x(1), x(2) . . . , x(n)})+ (d(2) − d(1)) · v({x(2), . . . , x(n)}) + . . . + (d(i) − d(i−1)) · v({x(i), x(i+1), . . . , x(n)}) + . . . + (d(n) − d(n−1)) · v({x(n)}). Comments.

• The above expression is exactly (discrete) Choquet in-
• tegral. Thus, the optimal price is the Choquet integral.
• There are other economic applications of Choquet in-

tegral – to decision making.

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 12 Go Back Full Screen Close Quit

9. Proof: Part I v(d) = d(1) · v({x(1), x(2) . . . , x(n)})+ (d(2) − d(1)) · v({x(2), . . . , x(n)}) + . . . + (d(i) − d(i−1)) · v({x(i), x(i+1), . . . , x(n)}) + . . . + (d(n) − d(n−1)) · v({x(n)}). This value is attained when we take the following purchas- ing plan P0:

• d(1) copies of the set {x(1), x(2) . . . , x(n)},
• d(2) − d(1) copies of the set {x(2), . . . , x(n)},
• . . . ,
• d(i) − d(i−1) copies of the set {x(i), x(i+1), . . . , x(n)},
• . . . , and
• d(n) − d(n−1) copies of the set {x(n)}.

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 12 Go Back Full Screen Close Quit

10. Proof: Part II – Main Idea

• We need to prove: that for every purchasing plan

P = (S1, . . . , Sm), (t1, . . . , tm) we have v(P) ≥ v(P0).

• Idea: use v(S1) + v(S2) ≥ v(S1 ∪ S2) + v(S1 ∩ S2).
• Every time we have two sets with S1 ⊆ S2 and S2 ⊆ S1,

replace S1 and S2 with S1 ∪ S2 and S1 ∩ S2.

• Result: a plan P ′ w/v(P ′) ≤ v(P) in which sets S′

i are

totally ordered & can be sorted: S′

1 ⊂ S′ 2 ⊂ . . . ⊂ S′ m.

• In P ′, let each discount S′

j be taken t′ j times.

• For items xi ∈ S′

j \S′ j+1, we get di = t′ 1+. . .+t′ j copies.

• The larger j, the larger this sum.
• Thus, S′

j is the set of all the elements with sufficiently

large di, i.e., S′

j = {x(j), x(j+1), . . . , x(n)}.

• Here, t′

j = (t′ 1 + . . . + t′ j−1 + t′ j) − (t′ 1 + . . . + t′ j−1), so

t′

j = d(j) − d(j−1), hence P ′ = P0 and v(P0) ≤ v(P).

Practical Situation: . . . How to Describe . . . Formulation of the . . . Towards a Precise . . . Towards a Precise . . . Precise Optimization . . . Main result Proof: Part I Proof: Part II – Main Idea Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 12 Go Back Full Screen Close Quit

11. Acknowledgments This work was supported in part by:

• the National Science Foundation grants:
• grant HRD-0734825, and
• grant DUE-0926721;
• by the National Institutes of Health grant:
• Grant 1 T36 GM078000-01.