1. Dynamic Competitive Revenue Management with Forward and Spot Markets Srinivas Krishnamoorthy Guillermo Gallego Columbia University Robert Phillips Nomis Solutions

2. Entrant Motivation Incumbent Demand D Buyer E[D] =

3. Example Applications Buyer OEM Utility company Tour operator Freight consolidator Ad agency Capacity providers Contract manufacturers Power plants Airlines Freight carriers TV Networks

4. 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)

5. Model A buyer faces random demand D Two providers with capacities C 0 and C 1 Entrant offers forward price p 0 and spot price p 0 Incumbent offers forward price p 1 and spot price p 1 Prices satisfy p 0 < p 1 < p 0 < p 1 Entrant’s decision - offer C 0 units forward Incumbent’s decision - offer C 1 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

6. The Buyer’s Problem Buyer’s cost = entrant’s revenue + incumbent’s revenue Buyer minimizes expected cost Optimal solution (x *, y * ) depends on C 0, C 1

7. The Providers’ Problems Entrant maximizes expected revenue Incumbent maximizes expected revenue

8. Game Between Providers Forward and spot prices are fixed. Entrant and incumbent simultaneously announce forward capacities C 0 and C 1 respectively. –Entrant attempts to maximize  0 ( C 0,C 1 ). –Incumbent attempts to maximize  1 ( C 0,C 1 ). 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.

9. Buyer’s Market C0C0 C1C1 = 50 C 0 = 50 C 1 = 100 (0,10) (0,41) (0,0)(0,0) (43,11) (50,0)

10. Market in Flux C0C0 C1C1 = 100 C 0 = 50 C 1 = 100 (0,59) (0,87) (0,0)(0,0) (46,60) (24,0) (23,64) (46,87) (46,0)

11. Providers’ Market C0C0 C1C1 = 150 C 0 = 50 C 1 = 100 (0,0)(0,0)

12. 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?

13. The Different Market Regimes C0C0 C1C1 (0,10) (0,41) (0,0)(0,0) (43,11) (50,0) C0C0 C1C1 (0,59) (0,87) (0,0)(0,0) (46,6 0) (24,0) (23,64) (46,87) (46,0) C0C0 C1C1 (0,0)(0,0) Buyer’s Market ( = 50)Market in Flux ( = 100) Providers’ Market ( = 150) C 0 = 50 C 1 = 100

14. Feasible Revenues Feasible revenues are convex combinations of pure strategy revenues. Lemma There exists a feasible revenue that yields revenues (z 0, z 1 ) with z 0 > f 0 and z 1 > f 1

15. 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 (z 0, z 1 ) with z 0 > f 0 and z 1 > f 1 Proof From Lemma (previous slide) and Friedman’s Theorem (1971) for repeated games

16. Trigger Strategy If (C z 0, C z 1 ) is the collection of actions that yields (z 0, z 1 ) as the average revenues per stage, then the subgame – perfect equilibrium can be achieved by the following strategy for the entrant (incumbent) : Play C z 0 (C z 1 ) in the first stage. In the t th stage, if the outcome of all the preceding stages has been (C z 0, C z 1 ), then play C z 0 (C z 1 ), otherwise play C f 0 (C f 1 ).

17. Obtaining Higher Revenues in a Buyer’s Market (e 0, e 1 )  (8100, 2165) (z 0, z 1 )  (8920, 2837) (s 0, s 1 )  (9917, 2165) (f 0, f 1 )  (7949, 2176) (i 0, i 1 )  (7923, 3508) (e 0, e 1 ) = 8100, 2165)

18. Numerical Results (Buyer’s Market) (C f 0, C f 1 )(f 0, f 1 )(z 0, z 1 ) ** 30(23, 5)(5124, 0.15)(5500, 450)0.500.891 45(38, 8)(7390, 212)(8100, 836)0.500.903 50(43, 11)(7828, 648)(8598, 1339)0.500.897 54(47, 14)(7954, 1248)(8836, 1910)0.500.903 59(46, 19)(7949, 2176)(8920, 2837)0.500.901

19. Market in Flux The two providers obtain revenues (m 0, m 1 ) at a mixed strategy equilibrium (  0,  1 ) for the stage game Proposition There exists a convex combination of the revenues (s 0, s 1 ) and (i 0, i 1 ) that yields revenues (z 0, z 1 ) with z 0 > m 0 and z 1 > m 1

20. 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 (z 0, z 1 ) with z 0 > m 0 and z 1 > m 1 (The subgame perfect equilibrium can once again be achieved by a trigger strategy similar to the strategy for a Buyer’s Market.)

21. Numerical Results (Market in Flux) (0,C f 0 )  0 (0) (0,C f 1 )  1 (0) (m 0, m 1 )(z 0, z 1 ) ** 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)

22. 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