1. Consumer Behavior Modeling Linling He, Xiaoyan Liu, and Muhong Zhang April 29, 2005

2. Consumer Behavior Modeling Outlines Choice-Based Model in Revenue Management Literature Review in Inventory Competition Model Choice based inventory competition Model

3. Why Model Consumer Behavior? Usually in RM or inventory control, we start our analysis with some set of assumptions assuming an underlying stochastic or deterministic demand process Usually in RM or inventory control, we start our analysis with some set of assumptions assuming an underlying stochastic or deterministic demand process What if … What if … –assumptions are incorrect –our model parameters are unknown?

4. Why Model Consumer Behavior? (cont) Example: Example: –Two classes of tickets (low/high fares) –Flexible customer, but buys low fare when available –An air-line chooses reserve level for the high- fare tickets based on historical data on sales –What happens if we neglect to account for consumer ’ s choice behavior?  “ Spiral Down Effect ”  “ Spiral Down Effect ”

5. Choice-Based Model in Revenue Management History History Motivation for choice modeling Motivation for choice modeling Single-leg Model Single-leg Model –Dynamic Programming Formulation Network RM Model Network RM Model –Dynamic Programming Formulation –Deterministic LP Formulation –Asymptotic optimality

6. 1.1 What is Revenue Management? Before 1972: controlled booking Before 1972: controlled booking Around 70 ’ s: discount/full fare seat inventory control Around 70 ’ s: discount/full fare seat inventory control Littlewood (1972) marked the beginning of Yield Management. Littlewood (1972) marked the beginning of Yield Management. “… Discount fare booking should be accepted as long as their revenue value exceeded the expected revenue of future full fair bookings …” Single-leg control, segment control, origin- destination control Single-leg control, segment control, origin- destination control

7. 1.1 What is Revenue Management? (cont) Practice of controlling the availability and/or pricing of travel seats in different booking classes with the goal of maximizing expected revenues or profits. Practice of controlling the availability and/or pricing of travel seats in different booking classes with the goal of maximizing expected revenues or profits. Fundamental revenue management decision: whether or not to accept or reject this booking Fundamental revenue management decision: whether or not to accept or reject this booking Key areas of research Key areas of research –Forecasting –Overbooking –Seat inventory control –pricing

8. 1.1 What is Revenue Management (cont) Consumer Behavior and Demand Forecasting Consumer Behavior and Demand Forecasting –Demand volatility –Sensitivity to pricing actions –Demand dependencies btwn different booking classes Control System Control System –Booking lead time –Overbooking –Leg-based, segment- based, or full ODF control Revenue Factors Revenue Factors –Fare values –Frequent flyer redemptions –Cancellation penalties or restrictions Variable Cost Factors Variable Cost Factors –Marginal costs per passenger –Denied boarding penalties Problem Scale Problem Scale –Large airline or airline alliance

9. 1.2 Motivation for Choice Modeling (cont) Choices among fare products Choices among fare products Y $320 Unrestricted Q $189 Nonrefundable Flight AB123

10. 1.2 Motivation for Choice Modeling (cont) Choice among multiple departure times between same OD Choice among multiple departure times between same OD 6:50 AM Open: Y, M, B, Q 9:15 AM Open: Y, MClosed: Q, B 1:10 PM Open: Y, M, BClosed: Q

11. 1.2 Motivation for Choice Modeling (cont) Choice among different routing Choice among different routing Long Beach Oakland New York Q Class: nonstop flight $154 Q class, connecting flight $179

13. 1.2 Motivation for Choice Modeling Traditional model: Traditional model: –Demand is mutually statistically independent and also unaffected by the availability control. –Heuristic correction: “ buy-up ” “ buy-down ” –But, traditional RM techniques are limited! –More over … Competition is fundamentally about choice … Competition is fundamentally about choice … It is natural to consider Consumer Behavior Models! It is natural to consider Consumer Behavior Models!

14. 1.3 Single-leg Model Traditional Model Traditional Model (At most one arrival per period)

15. 1.3 Single-leg Model (cont) Discrete DP Formulation Discrete DP Formulation

16. 1.3 Single-leg Model (cont) Bid price decision rule: Open class j iff … Bid price decision rule: Open class j iff … Open class j if the revenue of class j is greater than or equal to the marginal value of extra capacity

17. 1.3 Single-leg Model (cont) Nested allocation decision rule: Nested allocation decision rule:  i nested protection level for class i or higher aircraft cabin

18. 1.3 Single-leg Model (cont) Traditional model assumes a arrival pattern for class j, does not consider consumer ’ s choice Traditional model assumes a arrival pattern for class j, does not consider consumer ’ s choice In reality, each customer makes a choice among the fare products that are offered (Talluj and van Ryzin) In reality, each customer makes a choice among the fare products that are offered (Talluj and van Ryzin) Control decision now: which products do we make available? Control decision now: which products do we make available? Product21-DayFare YNo$800 MYes$500 QYes$450 No Purchase

19. 1.3 Single-leg Model (cont) Example Example –Discounts: –Customer Segments: ProductSA-Stay21-DayFare YNoNo$800 MNoYes$500 QYesYes$450 Qualify for Restriction Willing to Buy? Segment%Pop SA Stay 21-Day Adv Y Class M Class Business 1 10%NoNoYesYes Business 2 20%NoYesYesYes Leisure 1 20%YesNoNoYes Leisure 2 20%YesYesNoYes Leisure 3 30%YesYesNoNo

20. 1.3 Single-leg Model (cont) Each offer set leads to a different outcome … Each offer set leads to a different outcome … Ex. If we offer the full fare (Y) and the 21- Day advance purchase discount (M), then 10% buy full fare (Y) 40% buy 21-Day discount (M) 50% do not buy Why? Why?

21. 1.3 Single-leg Model (cont) The revised single-leg model The revised single-leg model

22. 1.3 Single-leg Model (cont) Revised choice-based DP Revised choice-based DP ttime remaining xnumber of seats remaining V t (x)value function P j (S)probability of choosing j given offer set S Bellman ’ s Equation: (1)

23. 1.3 Single-leg Model (cont) Efficient Sets: A set T is said to be inefficient if a mixture of other offer sets can be used to generate more revenue for the same (or lower consumption rate. Efficient Sets: A set T is said to be inefficient if a mixture of other offer sets can be used to generate more revenue for the same (or lower consumption rate.Mathematically:  set of convex weights  (S), S  N satisfying  S  (S)=1 and  (S)>=0,  S  N such that R(T)<  S  (S)R(S) Q(T)   S  (S)Q(S) R(S)=expected revenue Q(S)=probability of selling a unit using offer set S

24. 1.3 Single-leg Model (cont) An optimal policy: select a set k * from among the m efficient, ordered sets {S k : k=1, … m}, where S 1  S 2 …  S m that maximizes (1). select a set k * from among the m efficient, ordered sets {S k : k=1, … m}, where S 1  S 2 …  S m that maximizes (1). For a fixed t, the largest optimal index k * is increasing in the remaining capacity x, For a fixed t, the largest optimal index k * is increasing in the remaining capacity x, For any fixed x, k * is decreasing in the remaining time t. For any fixed x, k * is decreasing in the remaining time t.

25. 3 efficient combinations: {Y}, {Y,Q}, {Y,Q,M} These are the only combinations we should consider! An “efficient frontier inefficient offer sets

26. Optimal policy: Use efficient sets in this order as capacity / time change Optimal policy: Use efficient sets in this order as capacity / time change 1.3 Single-Leg Model (cont) More capacity/ Less time Less capacity/ more time Optimal policy: Use efficient sets in this order as capacity/time change

27. 1.3 Single-leg Model (cont) Notice what ’ s “ odd ” in this example! Notice what ’ s “ odd ” in this example! –It can be optimal to offer deep discount (Q) but NOT the moderate discount (M). Why? Some business customers who buy the full fare may “ buy-down ” to M if it is offered Some business customers who buy the full fare may “ buy-down ” to M if it is offered Leisure customers will not buy full fare anyway Leisure customers will not buy full fare anyway –Numerical examples show revenue benefits over standard methods can be very large.

28. 1.3 Single-leg Model (cont) A variety of choice models can be used … A variety of choice models can be used … –MNL (The ratio of the probabilities of any two alternatives is independent from the choice set.) –But there are many other possibilities … Finite mixture logit Finite mixture logit Nested logit Nested logit General random utility General random utility

29. 1.4 Network RM Model

30. 1.4 Network RM Model (cont) m = no. of legs n = no. of products (itinerary x fare-class combo) N = {1,…n}=set of all products c = (c 1,…c m )=initial capacity rj = revenue of fare-product j A = [a ij ] mxn =incidence matrix (a ij =1 if leg i is use for product j) A j = incidence vector for product j A i = incidence vector for leg I = prob. Of an arrival in each time period = prob. Of an arrival in each time period

31. 1.4 Network RM Model (cont) Control: set of products to make available at each point in time, i.e. offer set, S  N P j (S) = prob. of arriving customer chooses fare- product j given offer set S P 0 (S) = prob. Of no purchase  j  S P j (S)+P 0 (S)=1 Decision: find an optimal policy for choosing the offer set S at each time t and giving the remaining capacity x such that their total expected revenue is maximized.

32. 1.4 Network RM Model (cont) Dynamic Programming formulation Dynamic Programming formulation Boundary Conditions: V t (0)=0t=1,…,T V T+1 (x)=0  x  0

33. 1.4 Network RM Model (cont) DP not solvable for most realistic network due to the large dimensionality of the state space DP not solvable for most realistic network due to the large dimensionality of the state space Use Deterministic Approximation with stochastic quantities replaced by mean values and assume continuous capacities and demand Use Deterministic Approximation with stochastic quantities replaced by mean values and assume continuous capacities and demand

34. 1.4 Network RM Model (cont) R(S) = expected revenue from an arriving customer when S is offered. R(S) = expected revenue from an arriving customer when S is offered. P(S)=vector of purchase probabilities P(S)=vector of purchase probabilities Q i (S) = probability of using a unit of capacity on leg i, i=1, …,m, given S. Q i (S) = probability of using a unit of capacity on leg i, i=1, …,m, given S.

35. 1.4 Network RM Model (cont) Choice-Based Deterministic LP (CDLP) Choice-Based Deterministic LP (CDLP) Max. total revenue Consumption less than capacity Total time sets offered less than horizon length S = subset of available (open) products (the offer set) Decision variables: t(S) = Total time subset S is offered

36. 1.4 Network RM Model (cont) This LP has an exponential number of variables! How do we solve it? This LP has an exponential number of variables! How do we solve it? –Gallego et al. show for special classes of choice models, LP can be solved relatively efficiently using column generation Ex: Customers belong to L segment and each segment l –Has a disjoint consideration set C l –Makes multinomial logit (MNL) choice among products in C l Then columns can be generated by a simple ranking procedure (linear complexity in | C l |)

37. 1.4 Network RM Model (cont) Asymptotic optimality (van Ryzin & Liu 2004 ) Theorem: Consider scaled problem in which capacity and time are both increased by the same factor  …. Let t*(S) denote the optimal solution to the original choice-based LP. Then the solution  t*(S) is asymptotically optimal for the corresponding stochastic dynamic network choice problem (suitably defined). Capacity =  x Time horizon =  T

38. 1.4 Network RM Model (cont) How do we use the choice-based DLP solution? How do we use the choice-based DLP solution? –Directly apply time variables t*(S) (Gallego et al. 2004) –Discard primal solution, but use dual information in a decomposition heuristic and other analysis (van Ryzin & Liu 2004)

39. 1.4 Network RM Model (cont) Dual of CDLP: Dual of CDLP:

40. 1.4 Network RM Model (cont) Prop. 3 A set T is efficient if and only if for some  =(  1, …,  m ) t  0, T is the optimal solution to Prop. 3 A set T is efficient if and only if for some  =(  1, …,  m ) t  0, T is the optimal solution to max S {R(S)-  t Q(S)}. Prop. 4 If t * (T)>0 in the optimal solution to the CDLP, then T is an efficient set. Prop. 4 If t * (T)>0 in the optimal solution to the CDLP, then T is an efficient set. Prop. 5 If Q(S 2 )  Q(S 1 ), then R(S 2 )  R(S 1 ). Prop. 5 If Q(S 2 )  Q(S 1 ), then R(S 2 )  R(S 1 ). –Efficient sets are partially order. (compare to the single-leg problem)

41. 1.4 Network RM Model (cont) Can solution to Single-leg RM problem be applied to Network RM? Can solution to Single-leg RM problem be applied to Network RM? –V t-1 (x)-V t-1 (x-A j ) is not additive in A j –In general, efficient sets are not the only optimal solution to the original DP. –Special case: additivity property holds if each product uses only a single leg –But, by our Asymptotic optimality condition, the efficient sets are asymptotically optimal for the DP.

42. 1.4 Network RM Model (cont) Simulation-based optimization in network RM Basic Idea: 1.Define a parametric class of policies (e.g. booking limit, bid price) 2.Use simulation and sensitivity information to optimize the policy parameters.

43. 1.5 Summary Traditional RM Choice-based RM Demand Stochastic demand for products Stochastic choice by customers ControlsAccept/reject Offer sets Forecasts Time series by product Censored data Discrete choice model estimation Unobserved no-purchase Optimization Single leg DP\EMSR heuristics Deterministic LP Choice DP Deterministic choice- based LP Simulation-based Optimization

44. Consumer Behavior Modeling Outlines Choice-Based Model in Revenue Management Literature Review in Inventory Competition Model Choice based inventory competition Model

45. Outline – Literature Review on Inventory Competition Parlar (1988) – 2-firm model Parlar (1988) – 2-firm model  Existence and Uniqueness of NE  Maxmin strategy  Cooperation between players Lippman and McCardle (1994) Lippman and McCardle (1994)  Results on 2-firm model  Summary of N-firm model

46. 2.1 Literature Review in Inventory Competition Model Parlar (1988) - 2-Firm Model  Existence of NE  Uniqueness of NE  Maxmin strategy  Cooperation between players

47. Literature Review in Inventory Competition Model 2-Firm Model Assumptions: 1. Single period, Single product 2. Independent random demand at each firm 3. Substitution of products allowed 4. Deterministic fraction of excess demand substitutes to alternatives 5. No price competition 6. Continuous and strictly increasing CDF

48. Literature Review in Inventory Competition Model Notations: p i : shortage cost/unit for firm i ’ s product p i : shortage cost/unit for firm i ’ s product s i : sales price/unit of firm i ’ s product s i : sales price/unit of firm i ’ s product c i : order cost/unit for firm i ’ s product c i : order cost/unit for firm i ’ s product q i : savage value of firm i ’ s product q i : savage value of firm i ’ s product D i : random demand of firm i ’ s product ~ F i (y) D i : random demand of firm i ’ s product ~ F i (y) b i : fraction of firm i ’ s demand which will switch to firm j ’ s product when it is sold out at firm i b i : fraction of firm i ’ s demand which will switch to firm j ’ s product when it is sold out at firm i x i : order quantity of each firm x i : order quantity of each firm

49. Literature Review in Inventory Competition Model Player i ’ s Expected Profit Function: Sales price Savage value Order cost Penalty cost

50. Comparison Recall the game theory version of the newsvendor model introduced in class in week 8. Recall the game theory version of the newsvendor model introduced in class in week 8. It is a compact version of Parlar ’ s model without two terms: savage value and penalty cost. Also, b i is set to be one. It is a compact version of Parlar ’ s model without two terms: savage value and penalty cost. Also, b i is set to be one. Conclusion on that model: the expected profit function is concave, and by Theorem (Debrea, 1952), there exists at least one pure strategy NE in the game. Conclusion on that model: the expected profit function is concave, and by Theorem (Debrea, 1952), there exists at least one pure strategy NE in the game.

51. Discussion on Nash Equilibrium Nash Equilibrium (x 1 *,x 2 *) satisfies: Best response functions I i are obtained as below : Strict Concavity of profit functions:

52. Discussion on Nash Equilibrium Best response functions I 1 : In more details, Thinking x 2 as a function of x 1 in I 1 (x 1,x 2 )=0, we can obtain:

53. Discussion on Nash Equilibrium x 1,x 2 x 1,x 2 Similarly, thinking x 2 as a function of x 1 in I 2 (x 1,x 2 )=0, we can obtain the following by implicit differentiation on both sides of I 2 (x 1,x 2 )=0 over x 1 :

54. Discussion on Nash Equilibrium Summary:  The expected profit function is strictly concave  The best response curve for each player is strictly decreasing in the (x 1,x 2 ) plane.  Since the derivative of x 2 relative to x1 is negative for player 2, x 2 is upper bounded when x 1 =0, x 2 is lower bounded by some value when x 1 approaches positive infinity.  Since the derivative of x 2 relative to x1 is negative for player 1 as well, x 1 is upper bounded when x 2 =0, x 1 is lower bounded by some value when x 2 approaches positive infinity.

55. Discussion on Nash Equilibrium X1=u X2=v

56. Discussion on Nash Equilibrium Theorem: There exists a unique Nash Equilibrium (x 1 *,x 2 *) Proof: Existence: follow from the properties of the best response functions, or follow from strict concavity of the expected profit function due to Theorem (Debrea, 1952) Uniqueness: can be proven by using strict monotone- decreasing properties of the best response functions and showing the following inequality :

57. Maxmin Strategy of Firms Consider the following Scenario:  Firm 2 acts irrationally to minimize the profit of firm 1 as much as possible.  Firm 1 acts to maximize its profit under the assumption that firm 2 tries to minimize firm 1’s profit as much as possible Since the demands are independent at each firm, the only way for the firm 2 to affect firm 1’s profit negatively is not to lose any customer of its own. => Firm 2 will order infinitely many units.

58. Maxmin Strategy of Firms Results:  Based on our assumption, firm 1 will know firm 2 orders infinitely amount of units. Hence, firm 1 will maximize its own profit assuming no exogenous demand from firm 2 => traditional newsvendor model with no product substitution substitution  Using maxmin strategy, each firm will obtain lower expected profits since it loses the chance to make profit from the other firm’s demand. => any strategy other than the one trying to damage each other could bring more expected profits => any strategy other than the one trying to damage each other could bring more expected profits

59. Discussion on Nash Equilibrium X1=u X2=v

60. Cooperation between Firms Consider the following scenario:  The two firms decide to cooperate with each other so that it does not incur penalty cost if the competitor satisfies the corresponding demand.  The objective is trying to maximize the total profits from both firms Since only the penalty cost is affected, we can easily modify the objective function:

61. Cooperation between Firms Expected Profit Function: Sales price Savage value Order cost Penalty cost where Extra term

62. Cooperation between Firms Results:  The total profit under cooperation is shown to be larger than the sum of profits obtained without cooperation. => increase the social welfare by cooperation  Finding the optimal solution (x 1 *,x 2 *) can be done by solving a mathematical programming.

63. Weakness of this Model  The assumption of the independent demand is not reasonable sometimes. This motivates the paper by Lippman and McCardle (1994)

64. 2.2 Another model Lippman and McCardle (1994) Instead of the individual demand, aggregate demand is a random variable with a continuous distribution, thus isolating the pure impact of competition for consideration. Instead of the individual demand, aggregate demand is a random variable with a continuous distribution, thus isolating the pure impact of competition for consideration. Consider the allocation of aggregate industry demand to each firm, thus allowing the individual demand to be correlated. Consider the allocation of aggregate industry demand to each firm, thus allowing the individual demand to be correlated.

65. Model Assumptions No price competition No price competition Industry demand does not change with the number of firms, and is allocated to each firm based on some initial allocation rules, which is known to all firms before the competition starts. Industry demand does not change with the number of firms, and is allocated to each firm based on some initial allocation rules, which is known to all firms before the competition starts. => allow demand to be correlated. Deterministic fraction of excess demand is reallocated to other firms based on some reallocation rules, through which a firm ’ s decision impacts the other firms in the industry. Deterministic fraction of excess demand is reallocated to other firms based on some reallocation rules, through which a firm ’ s decision impacts the other firms in the industry.

66. Initial Allocation Rules Rule 1: Deterministic splitting Rule 1: Deterministic splitting Rule 2: Simple random splitting Rule 2: Simple random splitting Rule 3: Incremental random splitting Rule 3: Incremental random splitting Rule 4: Independent random demands Rule 4: Independent random demands For simplicity of discussion, the aggregate industry demand D is assumed to be uniformly distributed on [0,1] from now on unless specified otherwise.

67. Deterministic Splitting Example 1 – linear splitting: Example 1 – linear splitting: D 1 =  D, D 2 =(1-  )D (where 0=<  <=1) D 1 =  D, D 2 =(1-  )D (where 0=<  <=1)  This spitting rule is called nondecreasing, since each firm ’ s share is nondecreasing in total industry demand.  D 1 has a continuous density provided D does. Example 2 – discontinuous splitting: Example 2 – discontinuous splitting: D 1 =D when D 0.5 D 1 =D when D 0.5  The distribution of D 1 has discontinuity

68. Simple Random Splitting Example: Example: Consider flipping a fair coin, P(D 1 =0)=P(D 1 =D)=0.5  The probabilistic structure of D 1 is a mixture of 0 and a uniform random variable on [0,1], hence, D 1 doesn ’ t necessarily have a density when D does

69. Incremental Random Splitting Incremental Random Splitting can be considered as applying the simple random splitting rule to each atom of the whole demand D Incremental Random Splitting can be considered as applying the simple random splitting rule to each atom of the whole demand D It is shown to be equivalent to the linear deterministic splitting rule It is shown to be equivalent to the linear deterministic splitting rule

70. Duopoly Analysis Note: Due to these splitting rules used to allocate the demand, the individual demand distribution function might not have a density, or equivalently, CDF is not continuous. Notation: Notation:  X j : player j ’ s strategy space   j (x 1,x 2 ): payoff of player j  X=X 1* X 2 : the strategy space of the game   (x 1,x 2 )=(  1 (x 1,x 2 ),  2 (x 1,x 2 )): the joint payoff function of the game

71. Notations Notations: p i : shortage cost/unit for firm i ’ s product p i : shortage cost/unit for firm i ’ s product s i : sales price/unit of firm i ’ s product s i : sales price/unit of firm i ’ s product c i : order cost/unit for firm i ’ s product c i : order cost/unit for firm i ’ s product q i : savage value of firm i ’ s product q i : savage value of firm i ’ s product D: aggregate industry demand ~unif (0,1) D: aggregate industry demand ~unif (0,1) D i : demand allocated to firm i based on splitting rules on D D i : demand allocated to firm i based on splitting rules on D R i : effective demand of firm i ’ s product after considering the reallocation of demand R i : effective demand of firm i ’ s product after considering the reallocation of demand b i : fraction of firm i ’ s demand which will switch to firm j ’ s product when it is sold out at firm i b i : fraction of firm i ’ s demand which will switch to firm j ’ s product when it is sold out at firm i x i : order quantity/inventory levels of each firm x i : order quantity/inventory levels of each firm R i = D i + b j (D j -x j ) + R i = D i + b j (D j -x j ) +

72. Nash Equilibrium in Duopoly game Existence Theorem: A pure-strategy Nash equilibrium in inventory levels (x 1,x 2 ) exists. Existence Theorem: A pure-strategy Nash equilibrium in inventory levels (x 1,x 2 ) exists. Proof by Topkis theorem after showing: Proof by Topkis theorem after showing: –Strategy space is a complete lattice –Joint payoff function is upper semi- continuous –Each firm ’ s payoff function is supermodular

73. Duopoly Analysis Theorem 3.1 Topkis (1979) Theorem 3.1 Topkis (1979) If the strategy space of X is a complete lattice (i.e. a nonempty partially ordered set in which every nonempty subset has a supremum and infimum), the joint payoff function is upper- semicontinuous and each player ’ s payoff function is supermodular, then there exists a pure strategy Nash equilibrium.

74. Duopoly Analysis One way to show supermodularity: One way to show supermodularity: if f(x,z)-f(y,z) is nondecreasing in z for all x>y, then f is supermodular, i.e. f has nondecreasing difference. if f(x,z)-f(y,z) is nondecreasing in z for all x>y, then f is supermodular, i.e. f has nondecreasing difference.

75. Nash Equilibrium in Duopoly game Show each firm ’ s payoff function is supermodular (for simplicity, let the penalty cost and savage value be zero, the sales price s and the order cost c are the same for both firms): Show each firm ’ s payoff function is supermodular (for simplicity, let the penalty cost and savage value be zero, the sales price s and the order cost c are the same for both firms): Let ’ s show that  1 (x 1,x 2 )-  1 (y 1,x 2 ) is nondecreasing in x 2 for all x 1 >= y 1

76. Nash Equilibrium in Duopoly Game Example (to gain some intuition for next theorem): Example (to gain some intuition for next theorem):  Assume initial firm demands D 1 and D 2 are independently, and each is uniformly distributed on [0,1]  c=1,s=2, and b 1 =b 2 =1  Monopolist ’ s optimal inventory level x*: P(D x*)=0.5) P(D x*)=0.5) => P(D 1 + D 2 P(D 1 + D 2 <x*) =0.5 => x*=1 => x*=1  In order to achieve the same stockout probability when we consider two competitive firms, we have P(D 1 + (D 2 -x 2 ) + <x 1 )=0.5 P(D 1 + (D 2 -x 2 ) + <x 1 )=0.5 => (1-x 1 )+(1-x 2 ) 2 /2=0.5 => (1-x 1 )+(1-x 2 ) 2 /2=0.5 => (1-x 2 )+(1-x 1 ) 2 /2=0.5 (due to symmetry) => (1-x 2 )+(1-x 1 ) 2 /2=0.5 (due to symmetry)

77. Nash Equilibrium in Duopoly Game Example (cont ’ d) Example (cont ’ d)  P(D 1 + (D 2 -x 2 ) + >x 1 )=0.5 => (1-x 1 )+(1-x 2 ) 2 /2=0.5 => (1-x 1 )+(1-x 2 ) 2 /2=0.5 => (1-x 2 )+(1-x 1 ) 2 /2=0.5 => (1-x 2 )+(1-x 1 ) 2 /2=0.5  We observe that the above leads to the unique equilibrium inventory levels: x*=2-2 1/2 (Note: x*>x*=1) x 1 *= x 2 *=2-2 1/2 (Note: x 1 *+ x 2 *>x*=1)  Games where the above procedure converges to this game ’ s unique equilibrium, are called dominance solvable  Supermodular games possessing a unique pure strategy nash equilibrium are dominance solvable.

78. Nash Equilibrium in Duopoly Game Theorem: Theorem: The pair (x*) of inventory levels is a Nash equilibrium if each firm stocks out with probability c/s: The pair (x 1 *, x 2 *) of inventory levels is a Nash equilibrium if each firm stocks out with probability c/s: P(D i + (D j -x j *) + >x i *)=c/s If the CDF of D i is continuous, then this condition is also necessary. Moreover, when b 1 =b 2 =1, then x*>=x*: competition never leads to a decline in industry inventory If the CDF of D i is continuous, then this condition is also necessary. Moreover, when b 1 =b 2 =1, then x 1 *+ x 2 *>=x*: competition never leads to a decline in industry inventory

79. Nash Equilibrium in Duopoly Game Uniqueness is not guaranteed by the above theorem Uniqueness is not guaranteed by the above theorem Example: Example:  D is uniform on [0,1]  b 1 =b 2 =1, c=4, s=9  Deterministic splitting rule D 1 = D/2, if D€[0,4/9) D 1 = D/2, if D€[0,4/9) D 1 = D, if D€[4/9, 5/9) D 1 = D, if D€[4/9, 5/9) D 1 = 0, if D€[5/9,1] D 1 = 0, if D€[5/9,1]  D 1 does not have a continuous CDF though D does  Any pair (x, 2/3-x) with x €[2/9, 4/9) satisfies the previous theorem, so results in a continuum of equilibria

80. Nash Equilibrium in Duopoly Game Uniqueness Theorem: Uniqueness Theorem: If the distribution of D is strictly increasing and continuous and if the initial allocation is effected by a strictly increasing deterministic split, then there is a unique equilibrium and x*=x*. If the distribution of D is strictly increasing and continuous and if the initial allocation is effected by a strictly increasing deterministic split, then there is a unique equilibrium and x 1 *+ x 2 *=x*. The key determinant of the level of industry inventory is the correlation between D 1 and D 2. The greater the negative correlation, the greater the increase in industry inventory. The key determinant of the level of industry inventory is the correlation between D 1 and D 2. The greater the negative correlation, the greater the increase in industry inventory.

81. N-Firm Case Summary of Results There exists a pure strategy equilibrium in the n- firm competitive newsboy. There exists a pure strategy equilibrium in the n- firm competitive newsboy. With identical firms and continuous distributions for the effective demand R i = D i + (D j -x j *) +, there is a unique equilibrium and it is symmetric. With identical firms and continuous distributions for the effective demand R i = D i + (D j -x j *) +, there is a unique equilibrium and it is symmetric. If the distribution of D is strictly increasing and continuous, and if all demand is allocated deterministically via a strictly increasing splitting rules, then there is a unique equilibrium and industry inventory is x*. If the distribution of D is strictly increasing and continuous, and if all demand is allocated deterministically via a strictly increasing splitting rules, then there is a unique equilibrium and industry inventory is x*. If demand is allocated and reallocated via simple random splitting, then expected industry profit converges to zero as the number of firms increases If demand is allocated and reallocated via simple random splitting, then expected industry profit converges to zero as the number of firms increases

82. Consumer Behavior Modeling Outlines Choice-Based Model in Revenue Management Literature Review in Inventory Competition Model Choice Based Inventory Competition Model

83. Summary 2-Firm Model 2-Firm Model n-Firm with particular rules n-Firm with particular rules  Existence of NE  Uniqueness of NE  Extreme case

84. Choice Based Inventory Competition Model 1 2 5 36 Q=4 40 Q=3 1 3 2 3

85. Choice based inventory competition Model n firm Model Assumption: 1. Single-Period Inventory Model 2. N Firms, each stock a single goods 3. Customers arrive sequentially 4. Perfect Information 5. No salvage value

86. Choice Based Inventory Competition Model Notations: T: Number of customers; T: Number of customers; p j : selling price of Firm j; p j : selling price of Firm j; c j : procurement cost; c j : procurement cost; x j : initial inventory level of Firm j; x j : initial inventory level of Firm j; U t j : Utility of customer t to goods j; U t j : Utility of customer t to goods j; Q t : Quantity of goods required by cusomter t Q t : Quantity of goods required by cusomter t x t : (x t 1,…, x t n ) inventory observed by x t : (x t 1,…, x t n ) inventory observed by customer t; customer t;

87. Choice Based Inventory Competition Model Utility:Quantity: Sample Path Assumption (cont.), C is a constant, C is a constant

88. Choice Based Inventory Competition Model Utility Function: Example: Only one alternative good Only one alternative good Prefer to buy something Prefer to buy something

89. Choice Based Inventory Competition Model : number of sales of good j : number of sales of good j Sample Path Profit Function of Firm j Expected Profit

90. Choice Based Inventory Competition Model Questions: 1. Does there exist NE? 2. If so, if the NE is unique? 3. What ’ s the extreme case properties?

91. Choice Based Inventory Competition Model Nash Equilibrium : Global Optimal :

92. Choice Based Inventory Competition Model Ordered Statistic: Rank Inventory Function

93. Choice Based Inventory Competition Model Recursive Sales-to-go Function: Initial conditions:

94. Choice Based Inventory Competition Model Lipschitz Function: Lemma: Function is Lipschitz with modulus, where and is such that. is such that. Homework: Prove the Lemma

95. Choice Based Inventory Competition Model Theorem: There exists a pure strategy NE to n firm inventory game. Proof: First, is continuous; Second, is concave in. Second, is concave in. NE exists (Fudenberg and Tirole). NE exists (Fudenberg and Tirole).

96. Choice Based Inventory Competition Model Lemma: If c.d.f. ’ s of are continuous and, then for all,, then for all, (1) Gradient exists with p.b. 1, (1) Gradient exists with p.b. 1, (2). (2). Lemma: The partial derivatives satisfy, and, and,

97. Choice Based Inventory Competition Model Comparison of NE and Joint Optimal Theorem: If, then at any NE point,, while at any joint optimum point, while at any joint optimum point,

98. Choice Based Inventory Competition Model Uniqueness of NE Definition: Goods are symmetric if the n random T-vectors are exchangeable. Assumption: Identical prices and costs. Incomplete Demand Diversion (IDD): Nontrivial Demand Diversion (NDD):

99. Choice Based Inventory Competition Model Lemma 4: Suppose IDD, is absolutely continuous, and Then. Lemma 5: Suppose NDD, then

100. Choice Based Inventory Competition Model Theorem: By the assumptions, there exists a unique NE to the symmetric game. Proof: First-order Conditions By Lemma 5, no other symmetric equilibria other than. By Lemma 5, no other symmetric equilibria other than. By Lemma 4, no asymmetric equilibria exists. By Lemma 4, no asymmetric equilibria exists.

101. Choice Based Inventory Competition Model Extreme cases: Assumption: Symmetric Game; Customer prefer to buy something. Customer prefer to buy something. Theorem: Under the assumptions. Let be the equilibrium. When,. is the inventory level with zero profits.

102. Choice Based Inventory Competition Model

103. Sample Path Gradient Algorithm 1. Initialize outer looper j 2. Initialize inner looper k 3. For firm j at iteration k Generate a new sample pathGenerate a new sample path Calculate sample path gradientCalculate sample path gradient Update current inventory levelUpdate current inventory level 4. Check convergence or return

104. Choice Based Inventory Competition Model Conclusion: NE exists under mild regularity conditions; NE exists under mild regularity conditions; NE is unique in case of symmetric firms; NE is unique in case of symmetric firms; NE solution is overstocking; NE solution is overstocking; The industry becomes competitive as number of firms increase. The industry becomes competitive as number of firms increase.

105. Choice Based Inventory Competition Model Open Questions: 1. What if without assumption that customers prefer to buy something? 2. Include price and cost factors? 3. What if arrival time is a factor? 4. Analysis cooperative game.

106. Reference: Lippman and McCardle, 1997, The Competitive Newsboy, Operations Research, Vol. 45, No. 1, 54- 65 Lippman and McCardle, 1997, The Competitive Newsboy, Operations Research, Vol. 45, No. 1, 54- 65 Topkis, 1979, Equilibrium Points in Nonzero-sum n- Person Submodular Games, SIAM J. Control and Optimization, Vol. 17, 773-787 Topkis, 1979, Equilibrium Points in Nonzero-sum n- Person Submodular Games, SIAM J. Control and Optimization, Vol. 17, 773-787 Parlar, 1988, Game Theoretic Analysis of the Substitutable Product Inventory Problem with Random Demands, Naval Research Logistics, Vol. 35, 397-409 Parlar, 1988, Game Theoretic Analysis of the Substitutable Product Inventory Problem with Random Demands, Naval Research Logistics, Vol. 35, 397-409 Mahajan and Ryzin, Inventory Competition under Dynamic Consumer choice, Operations Research, Vol. 49, 646-657 Mahajan and Ryzin, Inventory Competition under Dynamic Consumer choice, Operations Research, Vol. 49, 646-657