Price and Non-Price Competition in Oligopoly – An Analysis ...
On the Value of using Group Discounts under Price Competition
26
On the Value of using Group Discounts under Price Competition Reshef Meir, Tyler Lu, Moshe Tennenholtz and Craig Boutilier
description
On the Value of using Group Discounts under Price Competition. Reshef Meir, Tyler Lu, Moshe Tennenholtz and Craig Boutilier. Example. ( 3 , 8 ). Base price : 5$ Price for two clients or more : 2 $. Base price : 4$. Example. ( 3 , 8 ). u1 = 3 – 5 = -2. Base price : 5$ - PowerPoint PPT Presentation
Transcript of On the Value of using Group Discounts under Price Competition
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?
