Post on 29-Dec-2015
transcript
Dynamic Competitive Revenue Management with
Forward and Spot Markets
Srinivas Krishnamoorthy
Guillermo Gallego
Columbia University
Robert Phillips
Nomis Solutions
Entrant
1010 pppp
1p 1p
Entrant
0Capacity C
0p 0p
101 ppp
0p
Entrant
0Capacity C
1p 1p
0p
0Capacity C
10 pp
1p
Entrant
1p
Motivation
Incumbent
Demand
D
Buyer
E[D] =
1Capacity C
Example Applications
Buyer
• OEM • Utility company• Tour operator• Freight consolidator• Ad agency
Capacity providers
• Contract manufacturers• Power plants• Airlines• Freight carriers• TV Networks
Related Literature
Competitive Revenue Management & Pricing
• Perakis & Sood (2002, 2003)
• Netessine & Shumsky (2001)
• Li & Oum(1998)• Talluri (2003)
CompetitiveNewsvendor
• Parlar (1988)• Karjalainen (1992) • Lippman & McCardle
(1997)• Mahajan & van Ryzin
(1999)• Rudi & Netessine (2000) • Dana & Petruzzi (2001)
Model
• A buyer faces random demand D• Two providers with capacities C0 and C1
• Entrant offers forward price p0 and spot price p0
• Incumbent offers forward price p1 and spot price p1 • Prices satisfy p0 < p1 < p0 < p1
• Entrant’s decision - offer C0 units forward • Incumbent’s decision - offer C1 units forward• Buyer’s decision - buy forward x units from entrant and y
units from incumbent• Buyer satisfies any excess demand by buying in spot
market
The Buyer’s Problem
• Buyer’s cost = entrant’s revenue + incumbent’s revenue
• Buyer minimizes expected cost
Optimal solution (x*, y*) depends on C0, C1
),(ˆ),(ˆ),( 10 yxyxyxc
],)[(Min),(ˆ 0000 xCyxDEpxpyx
],)[(Min),(ˆ 10111 yCyCDEpypyx
ZyxCyCx
yxc
,,0,0 s.t.
),(min
10
The Providers’ Problems
• Entrant maximizes expected revenue
• Incumbent maximizes expected revenue
)),(),,((ˆ),(where
,0 s.t.
),(max
10*
10*
0100
000
100
CCyCCxCC
ZCCC
CC
)),(),,((ˆ),(where
,0 s.t.
),(max
10*
10*
1101
111
101
CCyCCxCC
ZCCC
CC
Game Between Providers
• Forward and spot prices are fixed.• Entrant and incumbent simultaneously announce
forward capacities C0 and C1 respectively. – Entrant attempts to maximize 0(C0,C1).
– Incumbent attempts to maximize 1(C0,C1).
• Buyer determines forward purchases x*, y* that minimize c(x,y).
• After forward purchasing, she observes total demand D and satisfies any excess demand in the spot market.
Buyer’s Market
C0
C1
= 50
C0 = 50
C1 = 100
(0,10)
(0,41)
(0,0)
(43,11)
(50,0)
Market in Flux
C0
C1
= 100
C0 = 50
C1 = 100
(0,59)
(0,87)
(0,0)
(46,60)
(24,0)
(23,64)
(46,87)
(46,0)
Providers’ Market
C0
C1
= 150
C0 = 50
C1 = 100
(0,0)
The Repeated Game
• The game is now played repeatedly an infinite number of times (e.g. two airlines may compete for passengers daily on a particular route)
• Each provider’s revenue is the present value of the revenue stream from the infinite sequence of stage games
• Can each provider obtain higher revenue then under the single stage Nash equilibrium?
• If so, then what is the strategy to be followed by the providers?
The Different Market Regimes
C0
C1
(0,10)
(0,41)
(0,0)
(43,11)
(50,0)C0
C1
(0,59)
(0,87)
(0,0)
(46,60)
(24,0)
(23,64)
(46,87)
(46,0)
C0
C1
(0,0)
Buyer’s Market ( = 50) Market in Flux ( = 100)
Providers’ Market ( = 150)
C0 = 50
C1 = 100
Feasible Revenues
Feasible revenues are convex combinations of pure strategy revenues.
Lemma
There exists a feasible revenue that yields revenues (z0, z1) with z0 > f0 and z1 > f1
)0,(strategy pure with therevenues ),(
),0(strategy pure with therevenues ),(
)0,0(strategy pure with therevenues ),(
),(strategy pure with therevenues ),(
0
1
10
10
10
10
10
f
f
ff
Cee
Cii
ss
CCff
Subgame – Perfect Nash Equilibrium
Theorem
For discount rates sufficiently close to 1 there exists a subgame-perfect Nash equilibrium for the infinite game that achieves average revenues
(z0, z1) with z0 > f0 and z1 > f1
Proof
From Lemma (previous slide) and Friedman’s Theorem (1971) for repeated games
Trigger Strategy
If (Cz0, Cz1
) is the collection of actions that yields (z0, z1) as the average revenues per stage, then the subgame – perfect equilibrium can be achieved by the following strategy for the entrant (incumbent) :
Play Cz0 (Cz1) in the first stage. In the tth stage, if the
outcome of all the preceding stages has been (Cz0, Cz1
),
then play Cz0 (Cz1), otherwise play Cf0 (Cf1
).
),)(1(),(),( 101010 iisszz
Obtaining Higher Revenues in a Buyer’s Market
(e0, e1) (8100, 2165)2000
2200
2400
2600
2800
3000
3200
3400
3600
7500 8000 8500 9000 9500 10000
Entrant
Incumbent
(z0, z1) (8920, 2837)
(s0, s1) (9917, 2165)
(f0, f1) (7949, 2176)
(i0, i1) (7923, 3508)
100
50
59
1
0
C
C
(e0, e1) = 8100, 2165)
Numerical Results (Buyer’s Market)
(Cf0, Cf1
) (f0, f1) (z0, z1) *
30 (23, 5) (5124, 0.15) (5500, 450) 0.50 0.891
45 (38, 8) (7390, 212) (8100, 836) 0.50 0.903
50 (43, 11) (7828, 648) (8598, 1339) 0.50 0.897
54 (47, 14) (7954, 1248) (8836, 1910) 0.50 0.903
59 (46, 19) (7949, 2176) (8920, 2837) 0.50 0.901
Market in Flux
The two providers obtain revenues (m0, m1) at a mixed strategy equilibrium (0, 1 ) for the stage game
Proposition
There exists a convex combination of the revenues (s0, s1) and (i0, i1) that yields revenues (z0, z1) with z0 > m0 and z1 > m1
))(),0(())(),0((10 111000 ff CC
Subgame – Perfect Nash Equilibrium(Market in Flux)
TheoremFor discount factors sufficiently close to 1 there exists a subgame perfect Nash equilibrium for the infinite game that achieves average revenues
(z0, z1) with z0 > m0 and z1 > m1
(The subgame perfect equilibrium can once again be achieved by a trigger strategy similar to the strategy for a Buyer’s Market.)
Numerical Results (Market in Flux)
(0,Cf0) 0(0) (0,Cf1
) 1(0) (m0, m1) (z0, z1) *
60 (0,46) 0.002 (0, 50) 0.763 (8074, 2373) (8362, 3699) 0.80
75 (0.47) 0.099 (0, 64) 0.751 (8080, 5751) (8455, 6905) 0.65
90 (0,46) 0.153 (0, 78) 0.748 (8106, 9200) (8497, 10168) 0.63
100 (0,46) 0.299 (0, 87) 0.741 (8102, 11500) (8536, 12332) 0.56
115 (0,46) 0.450 (0, 100) 0.726 (8103, 14949) (8614, 15559) 0.44
( = 0.80)
Concluding Remarks
• We have analyzed a revenue management game with two providers selling in a forward and a spot market to a single buyer making bulk purchases
• Competitive considerations can motivate capacity providers to sell in a discounted forward market even when buyers’ willingness-to-pay is the same in both the forward and the spot market
• For the static game there are three market regimes: Buyer’s Market (Low Demand)Market in Flux (Moderate Demand)Providers’ Market (High Demand)
• The two providers can increase their average revenues above their static Nash equilibrium revenues by implicit collusion when the game is played repeatedly