On the Value of using Group

Discounts under Price Competition

Reshef Meir, Tyler Lu, Moshe Tennenholtz and Craig Boutilier

Example

Base price : 5$Price for two clients or more : 2 $

Base price : 4$

(3,8)

Example

Base price : 5$Price for two clients or more : 2 $

Base price : 4$

(3,8)

u1 = 3 – 5 = -2

Example

Base price : 5$Price for two clients or more : 2 $

Base price : 4$

(3,8)

u1 = 8 – 4 = 4u1 = 3 – 5 = -2

Example

Base price : 5$Price for two clients or more : 2 $

Base price : 4$

(3,8)u1 = 4

(3,0)u2 = 0

(6,3)u3 = 0

Example

Base price : 5$Price for two clients or more : 2 $

Base price : 4$

(3,8)u1 = 4

(3,0)u2 = 0

(6,3)u3 = 1

Example

Base price : 5$Price for two clients or more : 2 $

Base price : 4$

(3,8)u1 = 4

(3,0)u2 = 1

(6,3)u3 = 1 4

No buyer wants to switch vendor

The LB model (Lu and Boutilier, EC’12)

Every buyer i has value vij for each vendor Every vendor posts a schedule

pj = (pj(1), pj(2),…, pj(n)) If k buyers (including i) select j, the utility of i is

ui = vij - pj(k) A game instance is given by

V=(vij)ij P= (p1, p2,…, pm)

p1 = (8,…,6,..,2,2)p2 = (9,7,…,3) p3 = (6,6,…,6)

(3,8,5)(6,2,5)

(1,8,4)(3,4,7)

(0,0,9)(5,5,5)

(12,7,7)

The LB model (Lu and Boutilier, EC’12) Lu and Boutilier showed that for any V,P

there is always a Stable Buyer Partition (SBP) Denoted by S(V,P) Maybe more than one SBP S(V,P) is selected by some coordination

mechanism Pareto-optimal TU / NTU

p1 = (8,…,6,..,2,2)p2 = (9,7,…,3) p3 = (6,6,…,6)

What prices should the Red vendor post?

Vendors as players

Base price : 4$

(3,8)

(7,0)

(5,5)

?

What prices should the Red vendor post?

Vendors as players

Base price : 4$

(3,8)

(7,0)

(5,5)

Base price : 5$

Revenue = 5$

What prices should the Red vendor post?

No need for discounts!

Vendors as players

Base price : 4$

(3,8)

(7,0)

(5,5)

Base price : 5$Price for two buyers: 3$Revenue = 6$

Complete information

Theorem I: with complete information, vendors have no reason to use group discounts.

This corroborates similar findings in other models (e.g. Anand & Aron’03).

Why would vendors use discounts? Economies of scale (low marginal production costs) Marketing effect Uncertainty over buyers’ valuations

Uncertainty models

Bayesian uncertainty Strict Uncertainty

A common distribution D over all buyers’ types

A set A of possible buyers’ types

Uncertainty models

Bayesian uncertainty Strict Uncertainty

A common distribution D over all buyers’ types

A set A of possible buyers’ types

A vendor’s utility in a given discount profile P is taken in expectation over all realizations

Vendors maximize expected utility

Uncertainty models

Bayesian uncertainty Strict Uncertainty

A common distribution D over all buyers’ types

A set A of possible buyers’ types

A vendor’s utility in a given discount profile P is taken in expectation over all realizations

A vendor’s Max-Regret* in P is the largest profit it could make by posting some p’ (over all V )

Vendors maximize expected utility Vendors minimize Max-Regret

“Groupon competition” Vendors post price vectors P =(p1, p2,…, pm) Buyers’ types V are set

The stable partition S(V,P) is formed Utilities are realized

By sampling from D By arbitrary selection from A

What is the best strategy for vendor j, given p-j ?In particular, would discounts help?

By the LB model

orMost important

slide

Bayesian model

Theorem II. suppose that:a) Buyers’ preferences are symmetricb) Buyers’ preferences are independentc) Other vendors use fixed pricesThen vendor j has no reason to use discounts.

No longer true if we relax any of these conditions

D is i.i

.d.

Bayesian model (cont.)

Proof outline:

- 1 vendor, 1 buyer

- 1 vendor, n i.i.d. buyers

- m vendors, n i.i.d. buyers

Simulate the n-1 other buyers by sampling from D

V

V

VCreate a new i.i.d distribution D’ for vendor 1:

A distribution on A distribution on

Consider the following (non-i.i.d) example

Suppose that Then Best fixed price is Can do better by posting

Bayesian model (cont.)

a prefers vendor 1 0 1

b prefe

rs v

endo

r 2 1

0

Bayesian model

Theorem II. suppose that:a) Buyers’ preferences are symmetricb) Buyers’ preferences are independentc) Other vendors use fixed priceThen vendor j has no reason to use discounts.

No longer true if we relax any of these conditions

D is i.i

.d.

Consider the following (non-i.i.d) example

Bayesian model (cont.)

a prefers vendor 1 0 1

b prefe

rs v

endo

r 2 1

0

Bayesian model

Theorem II. suppose that:a) Buyers’ preferences are symmetricb) Buyers’ preferences are independentc) Other vendors use fixed priceThen vendor j has no reason to use discounts.

No longer true if we relax any of these conditions

D is i.i

.d.

Strict uncertainty model We have a similar result: If buyers are selected from the same set of

types, then there is no reason to use discounts

However, if buyers are essentially different, discounts can be useful

Future work Suppose buyers are correlated

(E.g. by a signal on product quality) How much can a vendor gain by using discounts? How to compute the best discount schedule?

Equilibrium analysis With or without discounts

Thank you!Questions?