Pricing and Production Game under Revenue Sharing
and Information Update
Sirong Luo
Metin Çakanyıldırım
School of Management
University of Texas at Dallas
UTDallas.edu/∼luosrong
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Outline
• Introduction and motivation
• Literature review
• Model: 1) Whole Sale Game; 2) Revenue Sharing Game
• Information Update
• Multiperiod Games: 1) Two Period Games; 2) Finitely Repeated Games
• Concluding remarks
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Introduction and Motivation
Industry Initiatives in Supply Chain Management:
• Vendor Managed Inventory System: Dell Inc, Wal Mart Stores Inc, Kraft Inc.
Retailer decides on the price; with less control for inventory, loss of market share,
and increased risk of disruption; sets the higher sale price to make more profit.
Supplier decides on the inventory; with worrying about the excess inventory cost,
produces less inventory.
Information Technology allows Vendor Managed Inventory.
EDI,
Quick
Response(QR),
Collaborative
Planning
Forecasting
and
Replenishment(CPFR), and Just-in-Time Distribution(JITD).
• Revenue Sharing: Blockbuster and Studios
Higher tape price and lower rental price lead to poor video availability.
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Literature review
• N.C. Petruzzi and M. Dada(1999):- Survey: The Newsvendor Pricing.
• A. Federgruen and A. Heching(1999) and Chen and Simchi-Levi(2003):- Dynamic
Pricing, Base Stock List Price Policy. With Fixed Ordering Cost, (s, S, A, p) Policy.
• S.P.Sethi and F.M.Bass(2002):- Optimal pricing with hazard rate model of demand.
• G.P. Cachon and M.A. Lariviere(2001):- Supply Chain Coordination with RevenueSharing Contracts.
• Q. Li and D. Atkins(2002):- Coordinating Relenishment and Pricing, Production
Game and Service Game.
• Iyer and Bergen(1997):- Traditional Channel, Manufacturer doesnot always benefit
from Quick Response.
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Some notation
• p, K: Sale price and production quantity;
• I: Initial inventory;
• pmin: The lower bound for the retailer’s sale price;
• w, c: Whole sale price and production cost;
• λ: Revenue sharing ratio;
• π, Π, Ω: The retailer, supplier and supply chain profit;
• D(y(p), ) = y(p) + : Stochastic demand with mean y(p) = a − bp;
• : Additive demand randomness with IFR distribution.
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Model: Whole Sale Game
Centralized Supply Chain System:
Ω(p, K) = max pS(p, K) − cK
p,K
Expected profit is not joint concave in p and K for most distributions, but the expected
sales is.
Decentralized System: whole sale game where the supplier charges a fixed whole sale
price for every unit sold.
Retailer Profit Function:
π w (p, K) = max(p − w)S(p, K)
p≥w
Supplier Profit Function:
Πw (p, K) = cI + max wS(p, K) − cK
K≥I
The Game has unique Nash Equilibrium[Li and Atkins(2002)].
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Model: Wholesale Game
Observations:
i) The Nash Equilibrium price pw decreases with standard deviation σ.
ii) Both retailer and supplier profits and Nash Equilibrium price pw , quantity K w
decrease with demand sensitivity b.
iii) The double marginalization.
b
5
10
15
20
25
KI
94.78
82.69
71.33
60.29
49.40
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pI
21.95
11.94
8.61
6.94
5.93
ΩI
1590.97
618.21
309.51
166.34
89.20
Kw
50.33
39.83
29.33
18.83
8.33
pw
29.75
15.93
11.32
9.01
7.63
πw
1109.38
341.50
119.85
34.74
4.26
Πw
139.96
108.46
76.96
45.46
13.96
Ωw
1249.34
449.97
196.81
80.21
18.22
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Model: Revenue Sharing Game
Decentralized System: revenue sharing game where the supplier sets a fixed revenue
sharing ratio of channel revenue.
Retailer Profit Function:
π λ(p, K) = max (1 − λ)pS(p, K)
p≥pmin
Response function:
S(p(K), K) − bp(K)F (K − y(p(K))) = 0
Supplier Profit Function:
Πλ(p, K) = cI + max λpS(p, K) − cK
K≥I
Response function:
c
F (K(p) − y(p)) = 1 −
λp
The Game also has unique Nash Equilibrium.
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Model: Revenue Sharing Game
The monotone price property: with uncertain demand, the price decreases with quantity,
if
c 1
cσ
2
−1
pmin φ Φ
.
1−
≥
λ pmin
λb
p
K(p)
p(K)
Case A:
(K*, p*)
pmin
K
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Comparison of Whole sale game and Revenue sharing game
If pmin ≤ pw , the revenue sharing game can achieve pareto improving in terms of
expected profit.
Retailer: 1-λ
Supplier: λ
cKw
cKλ
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The Games with Initial Inventory
When I ≥ 0, the whole sale game and the revenue sharing game have unique Nash
Equilibrium:
∗ ∗
∗
(p , K ) if I ≤ K
.
(p(I), I) otherwise
Retailer and Supplier reaction curves in revenue sharing game:
p
K(p)
K(p)
p(K)
Case A:
(K*,p*)
p(I)
K(p(I))
Case B:
K(pmin), pmin
pmin
K
I
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The Effect of Information Update
• The positive effect:
Better information increases the sales and decrease the excess inventory;
• The negative effect:
Better information increases the retailer price and decreases the supplier production
quantity; the double marginalization is worse off.
• The information update structure:
Forecast ε0
0
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Collect
Information ε1
Update
Forecast ε0 | ε1
Product is delivered
Sale begins
L1
L
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The Effect of Information Update
• Bivariate normal information update mode:
0, 1–BIN(0, 0, σ0, σ1, ρ).
• The updated demand forecast distribution H(0|1) has mean and standard deviation:
µ=ρ
σ0
1
σ1
p
σ = σ0 1 − ρ 2
• 1: the market information related to the product, the higher 1, the more demand.
ρ: the information quality. ρ is bigger, the variance is smaller, the information is
better.
In VMI system, the retailer and supplier are better off from information update in
both whole sale game and revenue sharing game. Their expected profits increase with
information quality ρ.
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Two Period Games with perfect information ρ = 1
• Sales begins in the first period;
• At the second period, the demand is certain and equal to y(p) +
σ0
σ1 1 ;
• The sequence of events ;
Forecast ε0
Sale begins
0
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Sales
Information ε1
Uncertainty Realized
σ0
ε1
σ1
L1
Sale is over, Payoffs
L
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Two Period Games with perfect information
• At the second period, the Nash Equilibrium is: (p2, y(p2) + σσ01 1);
σ
where p2 = max{
a+ σ0 1
1
2b
, pmin}
• At the second period, the supplier has three production states: no production, partial
production, or total production.
• At the first period;
i) For given K1, the retailer’s expected profit is concave in p1. For given p1, the
supplier’s expected profit is concave in K1;
ii) There exists unique subgame perfect Nash Equilibrium.
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Finitely Repeated Games
• Assumptions:
i) The initial inventory level is zero.
ii) The demand is nonnegative and stationary;
iii) The unsatisfied demand is lost;
iiii) At last period, the leftover inventory can be salvaged at value c.
• Inventory costs:
h: per unit per time holding cost,
v: per unit per time shortage cost,
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Finitely Repeated Games
Retailer profit function:
λ
(In)
πnλ(In−1) = max (1 − λ)pnSn(Kn(In−1), pn) + αEπn+1
pn ≥pmin
(1)
Supplier profit function:
Πλn(In−1) = cIn−1 + max λpnSn(Kn, pn(In−1)) − cKn − hE(Kn − Dn)+
Kn ≥In−1
−vE(Kn − Dn)− + αEΠλn+1(In)
(2)
where In = Kn(In−1) − Dn(pn) ∨ 0.
There exists is a myopic Nash Equilibrium (K ∗, p∗) in every period.
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Numerical Study
• Base cases:
a = 200, b = 10, c = 4, w = 7, σ = 5.
• Revenue sharing gap (RG):
Ωλ − Ω w
RG = I
Ω − Ωw
• Whole sale price (w) effect:
• Demand uncertainty (σ) and sensitivity(b) effect:
• Information Quality (ρ) effect.
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The Effect of Wholesale Price: w ∈ (5, 6, 7, 8, 9)
700
ΩI
600
Ωλ
500
450
πλ
400
350
πw
300
Ωw
400
250
Πλ
200
300
Πw
150
200
100
100
50
0
0
5
6
7
8
9
5
6
Supply Chain Profit
7
8
9
Retailer and Supplier Profit
90
0.90
80
KI
0.80
70
Kλ
0.70
60
RG
0.60
50
Kw
40
0.50
λ
0.40
30
0.30
pw
pλ
pI
20
10
0
0.20
0.10
0.00
5
6
7
8
Equilibrium Price and Quantity
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5
6
7
8
9
Revenue sharing ratio and Gap
19
The Effect of Demand Sensitivity: b ∈ (5, 10, 15, 20, 25)
1800
1600
ΩI
1400
Ωλ
1400
1200
1000
1200
Ωw
1000
800
800
600
πλ
πw
600
400
400
200
200
0
0
5
10
15
20
25
Πλ
Πw
5
Supply Chain Profit
60
50
40
30
20
20
25
1.20
KI
80
70
15
Retailer and Supplier Profit
100
90
10
Kλ
1.00
RG
0.80
λ
0.60
Kw
pw
pλ
pI
0.40
0.20
10
0.00
0
5
10
15
20
Equilibrium Price and Quantity
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5
10
15
20
25
Revenue sharing ratio and Gap
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The Effect of Demand Uncertainty: σ ∈ (1, 5, 15, 25, 35)
700
600
500
500
ΩI
450
Ωλ
πλ
400
350
Ωw
πw
300
400
250
300
200
200
100
150
Πλ
100
Πw
50
0
0
1
5
15
25
1
35
5
Supply Chain Profit
15
25
35
Retailer and Supplier Profit
0.70
120
KI
100
RG
0.60
0.50
80
Kλ
λ
0.40
60
Kw
40
0.30
0.20
pw
pλ
I
p
20
0
1
5
15
25
Equilibrium Price and Quantity
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0.10
0.00
1
5
15
25
35
Revenue sharing ratio and Gap
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The Effect of Information Update: ρ ∈ (0, 0.2, 0.4, 0.6, 0.8, 1.0)
ΩI
700
600
Ωλ
500
Ωw
400
500
πλ
450
400
350
πw
300
250
300
Πλ
200
150
200
100
Πw
100
50
0
0
0
0.2
0.4
0.6
0.8
1
Supply Chain Profit
0
0.2
0.4
0.6
0.8
1
Retailer and Supplier Profit
pλ
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Concluding remarks and future research
Conclusion:
• Revenue Sharing: Pareto Improving.
• VMI benefit from information update.
• Subgame perfect N.E exists in two period game with perfect information
Myopic N.E exists in finitely repeated games.
Future Research:
• Two period model with nonperfect information update.
• Nonlinear demand model and multiplicative uncertainty.
• Coordination Mechanism.
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