Dynamic competitive revenue management with forward and spot markets
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Dynamic Competitive Revenue Management with Forward and Spot Markets. Srinivas Krishnamoorthy Guillermo Gallego Columbia University Robert Phillips Nomis Solutions. Entrant. Entrant. Entrant. Entrant. Motivation. Demand. Buyer. D. E[ D ] = . Incumbent. Buyer OEM

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Dynamic Competitive Revenue Management with Forward and Spot Markets

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Dynamic competitive revenue management with forward and spot markets

Dynamic Competitive Revenue Management with Forward and Spot Markets

Srinivas Krishnamoorthy

Guillermo Gallego

Columbia University

Robert Phillips

Nomis Solutions


Motivation

Entrant

Entrant

Entrant

Entrant

Motivation

Demand

Buyer

D

E[D] = 

Incumbent


Example applications

Buyer

OEM

Utility company

Tour operator

Freight consolidator

Ad agency

Capacity providers

Contract manufacturers

Power plants

Airlines

Freight carriers

TV Networks

Example Applications


Related literature

Competitive Revenue Management & Pricing

Perakis & Sood (2002, 2003)

Netessine & Shumsky (2001)

Li & Oum(1998)

Talluri (2003)

Competitive

Newsvendor

Parlar (1988)

Karjalainen (1992)

Lippman & McCardle (1997)

Mahajan & van Ryzin (1999)

Rudi & Netessine (2000)

Dana & Petruzzi (2001)

Related Literature


Model

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 pricep1

  • 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

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


The providers problems

The Providers’ Problems

  • Entrant maximizes expected revenue

  • Incumbent maximizes expected revenue


Game between providers

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

Buyer’s Market

(0,41)

  • = 50

  • C0 = 50

  • C1 = 100

C1

(0,10)

(43,11)

(0,0)

(50,0)

C0


Market in flux

Market in Flux

(0,87)

(46,87)

  • = 100

  • C0 = 50

  • C1 = 100

(23,64)

(0,59)

(46,60)

C1

(0,0)

(46,0)

(24,0)

C0


Providers market

Providers’ Market

  • = 150

  • C0 = 50

  • C1 = 100

C1

(0,0)

C0


The repeated game

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

(0,87)

(46,87)

  • C0 = 50

  • C1 = 100

(0,41)

(23,64)

(46,60)

(0,59)

C1

C1

C1

(43,11)

(0,10)

(0,0)

(0,0)

(0,0)

(50,0)

(46,0)

(24,0)

C0

C0

C0

The Different Market Regimes

Buyer’s Market ( = 50)

Market in Flux ( = 100)

Providers’ Market ( = 150)


Feasible revenues

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


Subgame perfect nash equilibrium

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

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).


Obtaining higher revenues in a buyer s market

(z0, z1) (8920, 2837)

Obtaining Higher Revenues in a Buyer’s Market

(i0, i1) (7923, 3508)

(f0, f1) 

(7949, 2176)

(s0, s1) 

(9917, 2165)

(e0, e1) (8100, 2165)

(e0, e1) = 8100, 2165)


Numerical results buyer s market

Numerical Results (Buyer’s Market)


Market in flux1

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


Subgame perfect nash equilibrium market in flux

Subgame – Perfect Nash Equilibrium(Market in Flux)

Theorem

For 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

Numerical Results (Market in Flux)

( = 0.80)


Concluding remarks

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


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