Price competition.. Firm Behavior under Profit Maximization Monopoly Bertrand Price Competition.
On the Value of using Group Discounts under Price Competition
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Transcript of On the Value of using Group Discounts under Price Competition
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On the Value of using Group
Discounts under Price Competition
Reshef Meir, Tyler Lu, Moshe Tennenholtz and Craig Boutilier
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Example
Base price : 5$Price for two clients or more : 2 $
Base price : 4$
(3,8)
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Example
Base price : 5$Price for two clients or more : 2 $
Base price : 4$
(3,8)
u1 = 3 – 5 = -2
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Example
Base price : 5$Price for two clients or more : 2 $
Base price : 4$
(3,8)
u1 = 8 – 4 = 4u1 = 3 – 5 = -2
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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
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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
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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
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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)
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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)
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What prices should the Red vendor post?
Vendors as players
Base price : 4$
(3,8)
(7,0)
(5,5)
?
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What prices should the Red vendor post?
Vendors as players
Base price : 4$
(3,8)
(7,0)
(5,5)
Base price : 5$
Revenue = 5$
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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$
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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
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Uncertainty models
Bayesian uncertainty Strict Uncertainty
A common distribution D over all buyers’ types
A set A of possible buyers’ types
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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
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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
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“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
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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.
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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
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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
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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.
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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
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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.
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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
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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
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Thank you!Questions?