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

Consumer Behavior Consumer Behavior ModelingModeling

Linling He, Xiaoyan Liu, and Muhong Linling He, Xiaoyan Liu, and Muhong ZhangZhang

April 29, 2005April 29, 2005


OutlinesOutlinesChoice-Based Model in Revenue Choice-Based Model in Revenue ManagementManagement

Literature Review in Inventory ComLiterature Review in Inventory Competition Modelpetition ModelChoice based inventory competitioChoice based inventory competition Modeln Model

Why Model Consumer Why Model Consumer Behavior?Behavior? Usually in RM or inventory control, we Usually in RM or inventory control, we

start our analysis with some set of start our analysis with some set of assumptions assuming an underlying assumptions assuming an underlying stochastic or deterministic demand stochastic or deterministic demand processprocess

What if…What if…– assumptions are incorrect assumptions are incorrect – our model parameters are unknown?our model parameters are unknown?

Why Model Consumer Why Model Consumer Behavior? (cont)Behavior? (cont) Example: Example:

– Two classes of tickets (low/high fares)Two classes of tickets (low/high fares)– Flexible customer, but buys low fare when Flexible customer, but buys low fare when

availableavailable– An air-line chooses reserve level for the An air-line chooses reserve level for the

high-fare tickets based on historical data high-fare tickets based on historical data on saleson sales

– What happens if we neglect to account for What happens if we neglect to account for consumer’s choice behavior?consumer’s choice behavior?

““Spiral Down Effect”Spiral Down Effect”

Choice-Based Model in Choice-Based Model in Revenue ManagementRevenue Management

HistoryHistory Motivation for choice modelingMotivation for choice modeling Single-leg ModelSingle-leg Model

– Dynamic Programming FormulationDynamic Programming Formulation Network RM ModelNetwork RM Model

– Dynamic Programming FormulationDynamic Programming Formulation– Deterministic LP FormulationDeterministic LP Formulation– Asymptotic optimalityAsymptotic optimality

1.1 What is Revenue 1.1 What is Revenue Management?Management?

Before 1972: controlled bookingBefore 1972: controlled booking Around 70’s: discount/full fare seat Around 70’s: discount/full fare seat

inventory controlinventory control Littlewood (1972) marked the beginning Littlewood (1972) marked the beginning

of of Yield Management.Yield Management.“…“…Discount fare booking should be accepted Discount fare booking should be accepted

as long as their revenue value exceeded the as long as their revenue value exceeded the expected revenue of future full fair expected revenue of future full fair bookings…”bookings…”

Single-leg control, segment control, Single-leg control, segment control, origin-destination controlorigin-destination control

1.1 What is Revenue 1.1 What is Revenue Management? (cont)Management? (cont)

Practice of controlling the Practice of controlling the availabilityavailability and/or and/or pricingpricing of travel seats in of travel seats in different booking classesdifferent booking classes with the goal of with the goal of maximizingmaximizing expected revenuesexpected revenues or or profits. profits.

Fundamental revenue management decision: Fundamental revenue management decision: whether or not to accept or reject this bookingwhether or not to accept or reject this booking

Key areas of researchKey areas of research– ForecastingForecasting– OverbookingOverbooking– Seat inventory controlSeat inventory control– pricingpricing

1.1 What is Revenue 1.1 What is Revenue Management (cont)Management (cont)

Consumer Behavior Consumer Behavior and Demand and Demand ForecastingForecasting– Demand volatilityDemand volatility– Sensitivity to pricing Sensitivity to pricing

actionsactions– Demand dependencies Demand dependencies

btwn different booking btwn different booking classesclasses

Control SystemControl System– Booking lead timeBooking lead time– OverbookingOverbooking– Leg-based, segment-Leg-based, segment-

based, or full ODF controlbased, or full ODF control

Revenue FactorsRevenue Factors– Fare valuesFare values– Frequent flyer Frequent flyer

redemptionsredemptions– Cancellation penalties or Cancellation penalties or

restrictionsrestrictions Variable Cost FactorsVariable Cost Factors

– Marginal costs per Marginal costs per passengerpassenger

– Denied boarding Denied boarding penaltiespenalties

Problem ScaleProblem Scale– Large airline or airline Large airline or airline

alliancealliance

1.2 Motivation for Choice 1.2 Motivation for Choice Modeling (cont) Modeling (cont)

Choices among fare productsChoices among fare products

Y $320 Unrestricted

Q $189 Nonrefundable

Flight AB123

1.2 Motivation for Choice 1.2 Motivation for Choice Modeling (cont)Modeling (cont)

Choice among multiple departure Choice among multiple departure times between same ODtimes between same OD

6:50 AMOpen: Y, M, B, Q

9:15 AMOpen: Y, M Closed: Q, B

1:10 PMOpen: Y, M, B Closed: Q

1.2 Motivation for Choice 1.2 Motivation for Choice Modeling (cont)Modeling (cont)

Choice among different routingChoice among different routing

Long Beach

Oakland

New York

Q Class: nonstop flight $154

Q class, connecting flight $179

1.2 Motivation for Choice 1.2 Motivation for Choice ModelingModeling

Traditional model:Traditional model:– Demand is mutually statistically independent and Demand is mutually statistically independent and

also unaffected by the availability control.also unaffected by the availability control.– Heuristic correction: “buy-up” “buy-down”Heuristic correction: “buy-up” “buy-down”– But, traditional RM techniques are limited!But, traditional RM techniques are limited!– More over…More over…

Competition is fundamentally about choice…Competition is fundamentally about choice…

It is natural to consider It is natural to consider Consumer Behavior Consumer Behavior Models!Models!

1.3 Single-leg Model1.3 Single-leg Model

Traditional ModelTraditional Model

)0(reject or )1(accept

}1,0{

.decision..ect accept/rejan is Control

0...

j class fare of revenue

j) classfor request is P(Arrival

classesfareofset},...,2,1{

periodstime,...1

21

jj

j

n

j

j

uu

u

rrr

r

λ

nN

Tt

(At most one arrival per period)

1.3 Single-leg Model (cont)1.3 Single-leg Model (cont)

Discrete DP FormulationDiscrete DP Formulation

Njtjtjj

tjjtjjjNju

t

t

xVuxVr

xVuxVurxV

xV

x

t

n

)(}))(max{(

)}()1())(({{max)(

equationBellman

function value)(

remaining seats ofnumber

remaining time

11

11}1,0{

capacity of valuemarginal exp.

)1()()( where 111

xVxVxV ttt


Bid price decision rule: Open class j iff…Bid price decision rule: Open class j iff…

)(1 xVr tj

Open class j if the revenue of class j is greater than or equal to the marginal value of extra capacity


Nested allocation decision rule: Nested allocation decision rule:

i i nested protection level for class i or highernested protection level for class i or higher

2

1

3

aircraft cabin


Traditional model assumes a arrival pattern for Traditional model assumes a arrival pattern for class j, does not consider consumer’s choice class j, does not consider consumer’s choice

In reality, each customer makes a In reality, each customer makes a choicechoice among among the fare products that are offered the fare products that are offered (Talluj and van Ryzin)(Talluj and van Ryzin)

Control decision now: which products do we make Control decision now: which products do we make available? available?

ProductProduct 21-Day21-Day FareFare

YY NoNo $800$800

MM YesYes $500$500

QQ YesYes $450$450

No PurchaseNo Purchase


ExampleExample– Discounts:Discounts:

– Customer Segments:Customer Segments:

ProductProduct SA-SA-StayStay

21-Day21-Day FareFare

YY NoNo NoNo $800$800

MM NoNo YesYes $500$500

QQ YesYes YesYes $450$450

Qualify for RestrictionQualify for Restriction Willing to Buy?Willing to Buy?

SegmentSegment %Pop%Pop SA StaySA Stay 21-Day 21-Day AdvAdv

Y ClassY Class M ClassM Class

Business Business 11

10%10% NoNo NoNo YesYes YesYes

Business Business 22

20%20% NoNo YesYes YesYes YesYes

Leisure 1Leisure 1 20%20% YesYes NoNo NoNo YesYes

Leisure 2Leisure 2 20%20% YesYes YesYes No No YesYes

Leisure 3Leisure 3 30%30% YesYes YesYes NoNo NoNo


Each offer set leads to a different Each offer set leads to a different outcome…outcome…

Ex. If we offer the full fare (Y) and the Ex. If we offer the full fare (Y) and the 21-Day advance purchase discount (M), 21-Day advance purchase discount (M), thenthen

10% buy full fare (Y)10% buy full fare (Y)

40% buy 21-Day discount (M)40% buy 21-Day discount (M)

50% do not buy50% do not buy

Why?Why?


The revised single-leg modelThe revised single-leg model

)S | j choosecustomer ()(

tat time products fareopen ofset

set"...offer "an now is t each timeat Control

0...

j class fare of revenue

period)in arrives P(customer

classesfareofset},...,2,1{

periodstime,...1

nformulatio gprogrammin dynamic timeDiscrete

t

21

PSP

NS

rrr

r

λ

nN

Tt

tj

t

n

j


Revised choice-based DPRevised choice-based DPt time remainingx number of seats remainingVt(x) value functionPj(S) probability of choosing j given offer set S

Bellman’s Equation:

tt

tt

Sjttjtj

NS

tttjtjSj

NSt

xVxVrSP

xVSPxVrSPxV

)())}()(({max

)}()1)(())1()((({max)(

11

101

(1)


Efficient Sets: A set T is said to be inefficient if a Efficient Sets: A set T is said to be inefficient if a mixture of other offer sets can be used to mixture of other offer sets can be used to generate more revenue for the same (or lower generate more revenue for the same (or lower consumption rate.consumption rate.

Mathematically:Mathematically: set of convex weights set of convex weights (S), S(S), SN satisfying N satisfying S S

(S)=1 and (S)=1 and (S)>=0, (S)>=0, S SN such thatN such thatR(T)< R(T)< S S (S)R(S)(S)R(S)Q(T)Q(T) S S (S)Q(S)(S)Q(S)

R(S)=expected revenueR(S)=expected revenueQ(S)=probability of selling a unit using offer set SQ(S)=probability of selling a unit using offer set S


An optimal policy:An optimal policy: select a set select a set kk** from among the from among the mm efficient, efficient,

ordered sets ordered sets {S{Skk: k=1,…m}: k=1,…m}, where , where SS11SS22… … SSm m that maximizes (1). that maximizes (1).

For a fixed t, the largest optimal index For a fixed t, the largest optimal index kk** is is increasing in the remaining capacity increasing in the remaining capacity xx, ,

For any fixed For any fixed xx, , kk** is decreasing in the is decreasing in the remaining time t.remaining time t.

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0 .6 0.7 0 .8 0.9 1

Q(S)

R(S)

{Y,Q}

{Y,Q,M}

{Y}

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0 .6 0.7 0 .8 0.9 1

Q(S)

R(S)

{Y,Q}

{Y,Q,M}

{Y}

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0 .6 0.7 0 .8 0.9 1

Q(S)

R(S)

{Y,Q}

{Y,Q,M}

{Y}

3 efficient combinations: {Y}, {Y,Q}, {Y,Q,M}These are the only combinations we should consider!

An “efficient frontier

inefficient offer sets

Optimal policy: Optimal policy: Use efficient sets in this order Use efficient sets in this order as capacity / time changeas capacity / time change

1.3 Single-Leg Model (cont)1.3 Single-Leg Model (cont)

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0 .6 0.7 0 .8 0.9 1

Q(S)

R(S)

{Y,Q}

{Y,Q,M}

{Y}

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0 .6 0.7 0 .8 0.9 1

Q(S)

R(S)

{Y,Q}

{Y,Q,M}

{Y}

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0 .6 0.7 0 .8 0.9 1

Q(S)

R(S)

{Y,Q}

{Y,Q,M}

{Y}

More capacity/Less time

Less capacity/more time

Optimal policy: Use efficient sets in this order as capacity/time change


Notice what’s “odd” in this example!Notice what’s “odd” in this example!– It can be optimal to offer deep discount (Q) It can be optimal to offer deep discount (Q)

but but NOTNOT the moderate discount (M). Why? the moderate discount (M). Why? Some business customers who buy the full fare Some business customers who buy the full fare

may “buy-down” to M if it is offeredmay “buy-down” to M if it is offered Leisure customers will not buy full fare anywayLeisure customers will not buy full fare anyway

– Numerical examples show revenue Numerical examples show revenue benefits over standard methods can be benefits over standard methods can be very large.very large.


A variety of choice models can be used…A variety of choice models can be used…– MNL (MNL (The ratio of the probabilities of any two The ratio of the probabilities of any two

alternatives is independent from the choice alternatives is independent from the choice setset.).)

– But there are But there are manymany other possibilities… other possibilities… Finite mixture logitFinite mixture logit Nested logitNested logit General random utilityGeneral random utility

weightsoftor vec

price) (e.g. attributes ofr vecto )(

j

Si

y

y

j

y

e

eSP

iT

jT

1.4 Network RM Model1.4 Network RM Model

Figure 1: Two-fareclasses airlinenetwork

1.4 Network RM Model (cont)1.4 Network RM Model (cont)

m = no. of legsm = no. of legsn = no. of products (itinerary x fare-class combo)n = no. of products (itinerary x fare-class combo)N = {1,…n}=set of all productsN = {1,…n}=set of all products

c = (cc = (c11,…c,…cmm)=initial capacity)=initial capacity

rj rj = revenue of fare-product j= revenue of fare-product j

A = [aA = [aijij]]mxnmxn=incidence matrix (a=incidence matrix (aijij=1 if leg i is use for =1 if leg i is use for product j)product j)

AAj j = incidence vector for product j= incidence vector for product j

AAi i = incidence vector for leg I= incidence vector for leg I = prob. Of an arrival in each time period= prob. Of an arrival in each time period


Control: set of products to make available at each point in Control: set of products to make available at each point in time, i.e. offer set, Stime, i.e. offer set, SNN

PPjj(S) = prob. of arriving customer chooses fare-product j (S) = prob. of arriving customer chooses fare-product j given offer set Sgiven offer set S

PP00(S) = prob. Of no purchase(S) = prob. Of no purchase

jjSSPPjj(S)+P(S)+P00(S)=1(S)=1

DecisionDecision: find an optimal policy for choosing : find an optimal policy for choosing the offer set S at each time t and giving the the offer set S at each time t and giving the remaining capacity x such that their total remaining capacity x such that their total expected revenue is maximized.expected revenue is maximized.


Dynamic Programming Dynamic Programming formulationformulation

)()()()(max

)()1)(())()((max

)(

111

101

xVAxVxVrSP

xVSPAxVrSP

functionvaluexV

tSj

jttjjNS

Sjtjtjj

NS

t

Boundary Conditions: Vt(0)=0 t=1,…,TVT+1(x)=0 x0


DP not solvable for most realistic DP not solvable for most realistic network due to the large network due to the large dimensionality of the state spacedimensionality of the state space

Use Use Deterministic ApproximationDeterministic Approximation with stochastic quantities with stochastic quantities replaced by mean values and replaced by mean values and assume continuous capacities and assume continuous capacities and demanddemand

1.4 Network RM Model 1.4 Network RM Model (cont)(cont) R(S) = expected revenuR(S) = expected revenu

e from an arriving custoe from an arriving customer when S is offered.mer when S is offered.

P(S)=vector of purchasP(S)=vector of purchase probabilitiese probabilities

QQii(S) = probability of usi(S) = probability of using a unit of capacity on ng a unit of capacity on leg i, i=1,…,m, given S.leg i, i=1,…,m, given S.

Sj

jj SPrSR )()(

)(

))(),...(()( 1

SAP

SQSQSQt

m

1.4 Network RM Model 1.4 Network RM Model (cont)(cont) Choice-Based Deterministic LP Choice-Based Deterministic LP

(CDLP)(CDLP)

: )(

: )( )( ..

)( )(max

TSt

xStSAPts

StSRV

NS

NS

NS

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

1.4 Network RM Model 1.4 Network RM Model (cont)(cont)

This LP has an exponential number of variables! This LP has an exponential number of variables! How do we solve it?How do we solve it?– Gallego et al. show for special classes of choice models, LP Gallego et al. show for special classes of choice models, LP

can be solved relatively efficiently using can be solved relatively efficiently using column generationcolumn generation

Ex: Customers belong to Ex: Customers belong to LL segment and each segment and each segment segment ll– Has a disjoint Has a disjoint consideration setconsideration set CCll

– Makes multinomial logit (MNL) choice among products in Makes multinomial logit (MNL) choice among products in CCll

Then columns can be generated by a simple Then columns can be generated by a simple ranking procedure (linear complexity in |ranking procedure (linear complexity in |CCll|)|)

? )()(max

SAPSR

NS


Asymptotic optimality Asymptotic optimality (van Ryzin & Liu 2004(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 = xTime horizon = T


How do we use the choice-based DLP How do we use the choice-based DLP solution?solution?

– Directly apply time variables Directly apply time variables t*(S) t*(S) (Gallego et al. 2004)(Gallego et al. 2004)

– Discard primal solution, but use dual Discard primal solution, but use dual information in a decomposition heuristic information in a decomposition heuristic and other analysis and other analysis

(van Ryzin & Liu 2004)(van Ryzin & Liu 2004)


Dual of CDLP:Dual of CDLP:


Prop. 3 A set T is efficient if and only if for Prop. 3 A set T is efficient if and only if for some some =(=(11,…,,…,mm))tt0, T is the optimal solution 0, T is the optimal solution to to

maxmaxSS{R(S)- {R(S)- ttQ(S)}.Q(S)}.

Prop. 4 If tProp. 4 If t**(T)>0 in the optimal solution to (T)>0 in the optimal solution to the CDLP, then T is an efficient set. the CDLP, then T is an efficient set.

Prop. 5 If Q(SProp. 5 If Q(S22))Q(SQ(S11), then R(S), then R(S22) ) R(SR(S11). ). – Efficient sets are partially order. (compare to the Efficient sets are partially order. (compare to the

single-leg problem)single-leg problem)


Can solution to Single-leg RM problem Can solution to Single-leg RM problem be applied to Network RM?be applied to Network RM?– VVt-1t-1(x)-V(x)-Vt-1t-1(x-A(x-Ajj) is not additive in A) is not additive in Ajj

– In general, efficient sets are In general, efficient sets are notnot the only the only optimal solution to the original DP. optimal solution to the original DP.

– Special case: additivity property holds if Special case: additivity property holds if each product uses only a single legeach product uses only a single leg

– But, by our Asymptotic optimality But, by our Asymptotic optimality condition, the efficient sets are condition, the efficient sets are asymptotically optimal for the DP. asymptotically optimal for the DP.


Simulation-based optimization in network Simulation-based optimization in network RMRM

Basic Idea:Basic Idea:

1.1. Define a Define a parametric classparametric class of policies (e.g. of policies (e.g. booking limit, bid price)booking limit, bid price)

2.2. Use simulation and sensitivity information Use simulation and sensitivity information to to optimize the policy parameters.optimize the policy parameters.

1.5 Summary1.5 Summary

Traditional RMTraditional RM Choice-based RMChoice-based RM

DemandDemand Stochastic demand for Stochastic demand for productsproducts

Stochastic choice by Stochastic choice by customerscustomers

ControlsControls Accept/rejectAccept/reject Offer setsOffer sets

ForecastsForecasts Time series by productTime series by product

Censored dataCensored dataDiscrete choice model Discrete choice model estimationestimation

Unobserved no-Unobserved no-purchasepurchase

OptimizationOptimization Single leg DP\EMSR Single leg DP\EMSR heuristicsheuristics

Deterministic LPDeterministic LP

Choice DPChoice DP

Deterministic choice-Deterministic choice-based LPbased LP

Simulation-based Simulation-based OptimizationOptimization



Literature Review in Inventory ComLiterature Review in Inventory Competition Modelpetition ModelChoice based inventory competitioChoice based inventory competition Modeln Model

Outline – Literature Outline – Literature Review on Inventory Review on Inventory CompetitionCompetition Parlar (1988) – 2-firm modelParlar (1988) – 2-firm model

Existence and Uniqueness of NEExistence and Uniqueness of NE Maxmin strategyMaxmin strategy Cooperation between playersCooperation between players

Lippman and McCardle (1994)Lippman and McCardle (1994) Results on 2-firm modelResults on 2-firm model Summary of N-firm modelSummary of N-firm model

2.1 Literature Review in 2.1 Literature Review in Inventory Competition Inventory Competition ModelModel

Parlar (1988) - Parlar (1988) - 2-Firm Model 2-Firm Model Existence of NEExistence of NE Uniqueness of NEUniqueness of NE Maxmin strategyMaxmin strategy Cooperation between playersCooperation between players

Literature Review in Literature Review in Inventory Competition Inventory Competition ModelModel

2-Firm Model Assumptions: 2-Firm Model Assumptions:

1.1. Single period, Single productSingle period, Single product

2.2. Independent random demand at each Independent random demand at each firmfirm

3.3. Substitution of products allowedSubstitution of products allowed

4.4. Deterministic fraction of excess demand Deterministic fraction of excess demand substitutes to alternativessubstitutes to alternatives

5.5. No price competitionNo price competition

6.6. Continuous and strictly increasing CDFContinuous and strictly increasing CDF

Literature Review in Literature Review in Inventory Competition Inventory Competition ModelModelNotations:Notations: ppii : shortage cost/unit for firm i’s product : shortage cost/unit for firm i’s product ssii: sales price/unit of firm i’s product: sales price/unit of firm i’s product ccii: order cost/unit for firm i’s product: order cost/unit for firm i’s product qq ii: savage value of firm i’s product: savage value of firm i’s product DDii: random demand of firm i’s product ~ F: random demand of firm i’s product ~ Fii(y)(y) bb ii: fraction of firm i’s demand which will switch t: fraction of firm i’s demand which will switch t

o firm j’s product when it is sold out at firm io firm j’s product when it is sold out at firm i xx ii: order quantity of each firm: order quantity of each firm

Literature Review in Literature Review in Inventory Competition Inventory Competition ModelModelPlayer i’s Expected Profit Function:Player i’s Expected Profit Function:

}])(,min{[),( jjjiiiDjii xDbDxsExx

}]0,max{[ iiiiiD xDpxcE

)}])((,0max{[ jjjiiiD xDbDxqE

Sales price

Savage

valueOrder cost

Penalty cost

}])(,min{[),( jjjiiiDjii xDbDxsExx

Comparison Comparison

Recall the game theory version of the Recall the game theory version of the newsvendor model introduced in class newsvendor model introduced in class in week 8.in week 8.

It is a compact version of Parlar’s model It is a compact version of Parlar’s model without two terms: savage value and without two terms: savage value and penalty cost. Also, bpenalty cost. Also, bii is set to be one. is set to be one.

Conclusion on that model: the expected Conclusion on that model: the expected profit function is concave, and by profit function is concave, and by Theorem (Debrea, 1952), there exists at Theorem (Debrea, 1952), there exists at least one pure strategy NE in the game.least one pure strategy NE in the game.

Discussion on Nash Discussion on Nash EquilibriumEquilibrium

Nash Equilibrium (xNash Equilibrium (x11*,x*,x22*) satisfies:*) satisfies:

),(),( *211

*2

*11 xxxx ),(),( 2

*11

*2

*12 xxxx

Best response functions Ii are obtained as below: 0),( 211

1

1

xxIx

0),( 212

2

2

xxIx

0),(

1

2112

1

12

x

xxI

x

0

),(

2

2122

2

22

x

xxI

x

Strict Concavity of profit functions:


1

1 0 111112111

1 )()()(),(x

xdxxfqdxxfpsxxI

x

Best response functions I1 : 0),( 2111

1

xxIx

In more details,

0)()()( 10 1211

1

22

1

cdydxxfyfqs

x

xb

xx

Thinking x2 as a function of x1 in I1(x1,x2)=0, we can obtain:

0)()()(

)()()()(1|

22

120 111

11221111

21

2

11

dxxb

xxfxfqs

xfpxFxfqs

bdx

dxxI


Similarly, thinking x2 as a function of x1 in I2(xx11,,xx22)=0, we can obtain the following by implicit differentiation on both sides of I2(xx11,x,x22)=0 over x1:

0)(])()(

1)()()[(

)()()(

|

22211

210 2

1221122

0 11

21222

1

2

2

2

2

xfpdyx

byx

fyfb

xfxFqs

dyxb

yxfyfqs

dx

dxx

x

I

Discussion on Nash Discussion on Nash EquilibriumEquilibriumSummary:Summary: The expected profit function is strictly concaveThe expected profit function is strictly concave The best response curve for each player is The best response curve for each player is

strictly decreasing in the (xstrictly decreasing in the (x11,x,x22) plane. ) plane. Since the derivative of xSince the derivative of x22 relative to x1 is relative to x1 is

negative for player 2, xnegative for player 2, x22 is upper bounded when is upper bounded when xx11=0, x=0, x22 is lower bounded by some value when is lower bounded by some value when xx11 approaches positive infinity. approaches positive infinity.

Since the derivative of xSince the derivative of x22 relative to x1 is relative to x1 is negative for player 1 as well, xnegative for player 1 as well, x11 is upper is upper bounded when xbounded when x22=0, x=0, x11 is lower bounded by is lower bounded by some value when xsome value when x22 approaches positive infinity. approaches positive infinity.


X1=u

X2=v


Theorem: There exists a unique Nash Equilibrium (xTheorem: There exists a unique Nash Equilibrium (x11*,*,xx22*)*)

Proof:Proof:Existence: follow from the properties of the best respExistence: follow from the properties of the best resp

onse functions, or follow from strict concavity of thonse functions, or follow from strict concavity of the expected profit function due to e expected profit function due to Theorem Theorem (Debrea, 1952)(Debrea, 1952)

Uniqueness: can be proven by using strict monotone-Uniqueness: can be proven by using strict monotone-decreasing properties of the best response functiondecreasing properties of the best response functions and showing the following inequality :s and showing the following inequality :

21||

1

2

1

2II dx

dx

dx

dx

Maxmin Strategy of FirmsMaxmin Strategy of Firms

Consider the following Scenario:Consider the following Scenario: Firm 2 acts irrationally to minimize the profit of Firm 2 acts irrationally to minimize the profit of

firm 1 as much as possible.firm 1 as much as possible. Firm 1 acts to maximize its profit under the Firm 1 acts to maximize its profit under the

assumption that firm 2 tries to minimize firm 1’s assumption that firm 2 tries to minimize firm 1’s profit as much as possibleprofit as much as possible

Since the demands are independent at each firm, Since the demands are independent at each firm, the only way for the firm 2 to affect firm 1’s profit the only way for the firm 2 to affect firm 1’s profit negatively is not to lose any customer of its own. negatively is not to lose any customer of its own. => Firm 2 will order infinitely many units. => Firm 2 will order infinitely many units.

Maxmin Strategy of FirmsMaxmin Strategy of Firms

Results:Results: Based on our assumption, firm 1 will know firm 2 orders iBased on our assumption, firm 1 will know firm 2 orders i

nfinitely amount of units. Hence, firm 1 will maximize its nfinitely amount of units. Hence, firm 1 will maximize its own profit assuming no exogenous demand from firm 2 own profit assuming no exogenous demand from firm 2

=> traditional newsvendor model with no product => traditional newsvendor model with no product substitutionsubstitution Using maxmin strategy, each firm will obtain lower expecUsing maxmin strategy, each firm will obtain lower expec

ted profits since it loses the chance to make profit from thted profits since it loses the chance to make profit from the other firm’s demand. e other firm’s demand.

=> any strategy other than the one trying to => any strategy other than the one trying to damage each other could bring more expected damage each other could bring more expected profits profits


X1=u

X2=v

Cooperation between Cooperation between FirmsFirms

Consider the following scenario:Consider the following scenario: The two firms decide to cooperate with The two firms decide to cooperate with

each other so that it does not incur each other so that it does not incur penalty cost if the competitor satisfies penalty cost if the competitor satisfies the corresponding demand. the corresponding demand.

The objective is trying to maximize the The objective is trying to maximize the total profits from both firmstotal profits from both firms

Since only the penalty cost is affected, we Since only the penalty cost is affected, we can easily modify the objective function:can easily modify the objective function:

Cooperation between Cooperation between FirmsFirms

Expected Profit Function: Expected Profit Function:

}])(,min{[),(' jjjiiiDjii xDbDxsExx

}]0),)(1max{([ iiiiiiD xDbpxcE

)}])((,0max{[ jjjiiiD xDbDxqE

Sales price

Savage

value

Order cost

Penalty cost

}]0,)()(max{[ jjiiiiD DxxDbpE

),(),(),( ''jijjiijic xxxxxx

where

Extra term

Cooperation between Cooperation between FirmsFirmsResults:Results: The total profit under cooperation is shown to The total profit under cooperation is shown to

be larger than the sum of profits obtained be larger than the sum of profits obtained without cooperation. without cooperation.

=> increase the social welfare by cooperation=> increase the social welfare by cooperation Finding the optimal solution (xFinding the optimal solution (x11*,x*,x22*) can be *) can be

done by solving a mathematical done by solving a mathematical programming.programming.

),(max0,0

jicxx

xxji

Weakness of this Weakness of this ModelModel The assumption of the independent demanThe assumption of the independent deman

d is not reasonable sometimes.d is not reasonable sometimes.

This motivates the paper by Lippman and McThis motivates the paper by Lippman and McCardle (1994)Cardle (1994)

2.2 Another model2.2 Another model

Lippman and McCardle (1994)Lippman and McCardle (1994) Instead of the individual demand, aggregatInstead of the individual demand, aggregat

e demand is a random variable with a contie demand is a random variable with a continuous distribution, thus isolating the pure inuous distribution, thus isolating the pure impact of competition for consideration.mpact of competition for consideration.

Consider the allocation of aggregate industConsider the allocation of aggregate industry demand to each firm, thus allowing the iry demand to each firm, thus allowing the individual demand to be correlated.ndividual demand to be correlated.

Model AssumptionsModel Assumptions

No price competitionNo price competition Industry demand does not change with the Industry demand does not change with the

number of firms, and is allocated to each firm number of firms, and is allocated to each firm based on some initial allocation rules, which is based on some initial allocation rules, which is known to all firms before the competition starts. known to all firms before the competition starts.

=> allow demand to be correlated.=> allow demand to be correlated. Deterministic fraction of excess demand is Deterministic fraction of excess demand is

reallocated to other firms based on some reallocated to other firms based on some reallocation rules, through which a firm’s reallocation rules, through which a firm’s decision impacts the other firms in the industry.decision impacts the other firms in the industry.

Initial Allocation RulesInitial Allocation Rules

Rule 1: Deterministic splittingRule 1: Deterministic splitting Rule 2: Simple random splittingRule 2: Simple random splitting Rule 3: Incremental random splittingRule 3: Incremental random splitting Rule 4: Independent random demandsRule 4: Independent random demands

For simplicity of discussion, the For simplicity of discussion, the aggregate industry demand D is aggregate industry demand D is assumed to be uniformly distributed on assumed to be uniformly distributed on [0,1] from now on unless specified [0,1] from now on unless specified otherwise.otherwise.

Deterministic SplittingDeterministic Splitting

Example 1 – linear splitting:Example 1 – linear splitting:

DD11==D, DD, D22=(1- =(1- )D (where 0=< )D (where 0=< <=1) <=1) This spitting rule is called nondecreasing, since This spitting rule is called nondecreasing, since

each firm’s share is nondecreasing in total each firm’s share is nondecreasing in total industry demand. industry demand.

DD1 1 has a continuous density provided D does. has a continuous density provided D does.

Example 2 – discontinuous splitting:Example 2 – discontinuous splitting:

DD11=D when D<=0.5 and D=D when D<=0.5 and D11=0 when D>0.5=0 when D>0.5 The distribution of DThe distribution of D11 has discontinuity has discontinuity

Simple Random Simple Random SplittingSplitting Example:Example:

Consider flipping a fair coin, Consider flipping a fair coin, P(DP(D11=0)=P(D=0)=P(D11=D)=0.5=D)=0.5 The probabilistic structure of DThe probabilistic structure of D11 is is

a mixture of 0 and a uniform random a mixture of 0 and a uniform random variable on [0,1], hence, Dvariable on [0,1], hence, D11 doesn’t doesn’t necessarily have a density when D necessarily have a density when D doesdoes

Incremental Random Incremental Random SplittingSplitting Incremental Random Splitting can Incremental Random Splitting can

be considered as applying the be considered as applying the simple random splitting rule to simple random splitting rule to each atom of the whole demand each atom of the whole demand DD

It is shown to be equivalent to the It is shown to be equivalent to the linear deterministic splitting rulelinear deterministic splitting rule

Duopoly Analysis Duopoly Analysis

Note: Note: Due to these splitting rules used to allocate the Due to these splitting rules used to allocate the

demand, the individual demand distribution demand, the individual demand distribution function might not have a density, or equivalently, function might not have a density, or equivalently, CDF is not continuous. CDF is not continuous.

Notation: Notation: XXjj : player j’s strategy space : player j’s strategy space jj(x(x11,x,x22): payoff of player j): payoff of player j X=XX=X1*1*XX22: the strategy space of the game: the strategy space of the game (x(x11,x,x22)=()=(11(x(x11,x,x22), ), 22(x(x11,x,x22)): the joint payoff function of )): the joint payoff function of

the gamethe game

NotationsNotations

Notations:Notations: ppii : shortage cost/unit for firm i’s product : shortage cost/unit for firm i’s product ssii: sales price/unit of firm i’s product: sales price/unit of firm i’s product ccii: order cost/unit for firm i’s product: order cost/unit for firm i’s product qq ii: savage value of firm i’s product: savage value of firm i’s product D: aggregate industry demand ~unif (0,1)D: aggregate industry demand ~unif (0,1) DDii: demand allocated to firm i based on splitting rul: demand allocated to firm i based on splitting rul

es on Des on D RRii: effective demand of firm i’s product after consi: effective demand of firm i’s product after consi

dering the reallocation of demanddering the reallocation of demand bb ii: fraction of firm i’s demand which will switch t: fraction of firm i’s demand which will switch t

o firm j’s product when it is sold out at firm io firm j’s product when it is sold out at firm i xx ii: order quantity/inventory levels of each firm: order quantity/inventory levels of each firm RRii= = DDii + + bb j j(D(Djj-x-xjj))++

Nash Equilibrium in Nash Equilibrium in Duopoly gameDuopoly game Existence Theorem: A pure-strategy Existence Theorem: A pure-strategy

Nash equilibrium in inventory levels Nash equilibrium in inventory levels (x(x11,x,x22) exists.) exists.

Proof by Topkis theorem after showing:Proof by Topkis theorem after showing:– Strategy space is a complete latticeStrategy space is a complete lattice– Joint payoff function is upper semi-Joint payoff function is upper semi-

continuouscontinuous– Each firm’s payoff function is Each firm’s payoff function is

supermodularsupermodular

Duopoly Analysis Duopoly Analysis

Theorem 3.1 Topkis (1979)Theorem 3.1 Topkis (1979)

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

Duopoly AnalysisDuopoly Analysis

One way to show One way to show supermodularity:supermodularity:

if f(x,z)-f(y,z) is nondecreasing in z if f(x,z)-f(y,z) is nondecreasing in z for all x>y, then f is for all x>y, then f is supermodular, i.e. f has supermodular, i.e. f has nondecreasing difference. nondecreasing difference.

Nash Equilibrium in Nash Equilibrium in Duopoly gameDuopoly game Show each firm’s payoff function is Show each firm’s payoff function is

supermodular (for simplicity, let the penalty cost supermodular (for simplicity, let the penalty cost and savage value be zero, the sales price s and and savage value be zero, the sales price s and the order cost c are the same for both firms):the order cost c are the same for both firms):

ijjjiiDjii cxxDbDxsExx }])(,[min{),(

Let’s show that 1(x1,x2)-1(y1,x2) is nondecreasing in x2 for all x1 >= y1

Nash Equilibrium in Nash Equilibrium in Duopoly GameDuopoly Game

Example (to gain some intuition for next theorem): Example (to gain some intuition for next theorem): Assume initial firm demands DAssume initial firm demands D11 and D and D22 are independently, are independently,

and each is uniformly distributed on [0,1]and each is uniformly distributed on [0,1] c=1,s=2, and bc=1,s=2, and b11=b=b22=1=1 Monopolist’s optimal inventory level x*: Monopolist’s optimal inventory level x*: P(D<x*) =(s-c)/s =0.5 (stockout P(D<x*) =(s-c)/s =0.5 (stockout

probability=p(D>x*)=0.5)probability=p(D>x*)=0.5)

=> P(D=> P(D11 + D + D22 <x*) =0.5 <x*) =0.5 => x*=1=> x*=1 In order to achieve the same stockout probability when we In order to achieve the same stockout probability when we

consider two competitive firms, we haveconsider two competitive firms, we have

P(DP(D11 + (D + (D22-x-x22))++<x<x11)=0.5)=0.5

=> (1-x=> (1-x11)+(1-x)+(1-x22))22/2=0.5/2=0.5

=> (1-x=> (1-x22)+(1-x)+(1-x11))22/2=0.5 (due to symmetry)/2=0.5 (due to symmetry)

Nash Equilibrium in Nash Equilibrium in Duopoly GameDuopoly Game Example (cont’d)Example (cont’d)

P(DP(D11 + (D + (D22-x-x22))++>x>x11)=0.5)=0.5 => (1-x=> (1-x11)+(1-x)+(1-x22))22/2=0.5/2=0.5 => (1-x=> (1-x22)+(1-x)+(1-x11))22/2=0.5 /2=0.5 We observe that the above leads to the We observe that the above leads to the

unique equilibrium inventory levels:unique equilibrium inventory levels: xx1*=x2*=2-2*=2-21/2 1/2 (Note: (Note:

xx1*+x2*>x*=1)*>x*=1)Games where the above procedure Games where the above procedure

converges to this game’s unique equilibrium, converges to this game’s unique equilibrium, are called dominance solvableare called dominance solvable

Supermodular games possessing a unique Supermodular games possessing a unique pure strategy nash equilibrium are pure strategy nash equilibrium are dominance solvable.dominance solvable.

Nash Equilibrium in Nash Equilibrium in Duopoly GameDuopoly Game Theorem:Theorem:

The pair (xThe pair (x1*,x2*) of inventory levels is a Nash *) of inventory levels is a Nash equilibrium if each firm stocks out with equilibrium if each firm stocks out with probability c/s:probability c/s:

P(DP(Dii + (D + (Djj-x-xjj*)*)++>x>xii*)=c/s*)=c/s

If the CDF of DIf the CDF of Dii is continuous, then this is continuous, then this condition is also necessary. Moreover, condition is also necessary. Moreover, when bwhen b11=b=b22=1, then x=1, then x1*+x2*>=x*: *>=x*: competition never leads to a decline in competition never leads to a decline in industry inventoryindustry inventory

Nash Equilibrium in Nash Equilibrium in Duopoly Game Duopoly Game Uniqueness is not guaranteed by the Uniqueness is not guaranteed by the

above theoremabove theorem Example:Example:

D is uniform on [0,1]D is uniform on [0,1]bb11=b=b22=1, c=4, s=9=1, c=4, s=9Deterministic splitting ruleDeterministic splitting rule DD11 = D/2, if D€[0,4/9) = D/2, if D€[0,4/9) DD11 = D, if D€[4/9, 5/9) = D, if D€[4/9, 5/9)

DD11 = 0, if D€[5/9,1] = 0, if D€[5/9,1]DD11 does not have a continuous CDF though D does not have a continuous CDF though D

doesdoesAny pair (x, 2/3-x) with x €[2/9, 4/9) satisfies Any pair (x, 2/3-x) with x €[2/9, 4/9) satisfies

the previous theorem, so results in a the previous theorem, so results in a continuum of equilibriacontinuum of equilibria

Nash Equilibrium in Nash Equilibrium in Duopoly GameDuopoly Game Uniqueness Theorem:Uniqueness Theorem:

If the distribution of D is strictly increasing If the distribution of D is strictly increasing and continuous and if the initial allocation and continuous and if the initial allocation is effected by a strictly increasing is effected by a strictly increasing deterministic split, then there is a unique deterministic split, then there is a unique equilibrium and xequilibrium and x1*+x2*=x*.*=x*.

The key determinant of the level of The key determinant of the level of industry inventory is the correlation industry inventory is the correlation between Dbetween D11 and D and D22. The greater the . The greater the negative correlation, the greater the negative correlation, the greater the increase in industry inventory.increase in industry inventory.

N-Firm CaseN-Firm Case

Summary of ResultsSummary of Results There exists a pure strategy equilibrium in the There exists a pure strategy equilibrium in the

n-firm competitive newsboy.n-firm competitive newsboy. With identical firms and continuous distributions With identical firms and continuous distributions

for the effective demand Rfor the effective demand Rii= D= Dii + (D + (Djj-x-xjj*)*)++, there , there is a unique equilibrium and it is symmetric.is a unique equilibrium and it is symmetric.

If the distribution of D is strictly increasing and If the distribution of D is strictly increasing and continuous, and if all demand is allocated continuous, and if all demand is allocated deterministically via a strictly increasing deterministically via a strictly increasing splitting rules, then there is a unique splitting rules, then there is a unique equilibrium and industry inventory is x*.equilibrium and industry inventory is x*.

If demand is allocated and reallocated via If demand is allocated and reallocated via simple random splitting, then expected industry simple random splitting, then expected industry profit converges to zero as the number of firms profit converges to zero as the number of firms increasesincreases



Literature Review in Inventory ComLiterature Review in Inventory Competition Modelpetition ModelChoice Based Inventory CompetitioChoice Based Inventory Competition Modeln Model

Choice Based Choice Based Inventory Inventory Competition ModelCompetition Model

SummarySummary 2-Firm Model2-Firm Model n-Firm with particular rulesn-Firm with particular rules

Existence of NEExistence of NE Uniqueness of NEUniqueness of NE Extreme caseExtreme case

Choice Based Choice Based Inventory Competition Inventory Competition ModelModel

1 2 -1

5 3 6

Q=4

4 0

Q=3

1 3 2

3

Choice based Choice based inventory competition inventory competition ModelModeln firm Modeln firm Model

Assumption: Assumption:

1.1. Single-Period Inventory ModelSingle-Period Inventory Model

2.2. N Firms, each stock a single goodsN Firms, each stock a single goods

3.3. Customers arrive sequentiallyCustomers arrive sequentially

4.4. Perfect InformationPerfect Information

5.5. No salvage valueNo salvage value

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelNotations:Notations: T: Number of customers;T: Number of customers; pp jj: selling price of Firm j;: selling price of Firm j; cc jj: procurement cost;: procurement cost; xx jj: initial inventory level of Firm j;: initial inventory level of Firm j; UUtt

jj: Utility of customer t to goods j;: Utility of customer t to goods j; QQ tt: Quantity of goods required by cusomter t: Quantity of goods required by cusomter t xx tt : (x : (x tt

11,…, x,…, x ttnn) inventory observed by ) inventory observed by

customer t;customer t;

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelUtility:Utility:

Quantity: Quantity:

Sample PathSample Path

Assumption (cont.)Assumption (cont.)

, C is a constant, C is a constant

),,,( 10 ntttt UUUU

tQ

)},(,),,(),,(,{ 2211 TT QUQUQUT

1}{ CTP

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelUtility Function: Utility Function:

Example:Example: Only one alternative good Only one alternative good

Prefer to buy somethingPrefer to buy something

1}2|}0,:{{| 0 jUUjP j

1}0,{ 0 jUUP j

Choice Based Choice Based Inventory Competition Inventory Competition ModelModel : number of sales of good j: number of sales of good j

Sample Path Profit Function of Firm jSample Path Profit Function of Firm j

Expected Profit Expected Profit

),( xj

jjjjj xcxpx ),(),(

),()( xEx jj

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelQuestions:Questions:

1.1. Does there exist NE?Does there exist NE?

2.2. If so, if the NE is unique?If so, if the NE is unique?

3.3. What’ s the extreme case What’ s the extreme case properties?properties?

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelNash Equilibrium : Nash Equilibrium :

Global Optimal :Global Optimal :

x

Njxxxx jjj

x

jjj

j

),(),( max0

*x

n

j

j

x

xx10

* )()( max

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelOrdered Statistic:Ordered Statistic:

Rank Rank

Inventory Function Inventory Function

]1[]1[0][]1[ nt

mtt

mtt UUUUU

][,)( kt

jt UUifkjb

),,(1 tttjj

t QUxfx

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelRecursive Sales-to-go Function:Recursive Sales-to-go Function:

Initial conditions:Initial conditions:

),(),,(),( 11 tj

ttttjj

ttj

t xQUxfxx

),,(1 tttt QUxfx

NjxTj

T ,0),( 11

xx 1

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelLipschitz Function:Lipschitz Function:

Lemma: Function is Lipschitz with modLemma: Function is Lipschitz with modulus , where andulus , where and

is such that .is such that .

Homework: Prove the LemmaHomework: Prove the Lemma

SyyyyKyhyh h 211212 ,||||||)()(||

),( xj

11 CK j)1(

1 2 nCC1}{ CTPC

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelTheorem: There exists a pure strategy NE to n Theorem: There exists a pure strategy NE to n

firm inventory game.firm inventory game.

Proof: First, is continuous;Proof: First, is continuous;

Second, is concave in . Second, is concave in .

NE exists (Fudenberg and Tirole). NE exists (Fudenberg and Tirole).

)],,([ jjj xxE

),( jjj xx jx

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelLemma: If c.d.f.’s of are continuous and 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 satisfyLemma: The partial derivatives satisfy

, and , and

, ,

tQ

1}{ CTP x

),( x

)],([)],([ xExE

}1,0{),( t

jtj

t

xx

}0,1{),( t

jti

t

xx

ji

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelComparison of NE and Joint OptimalComparison of NE and Joint Optimal

Theorem: If , then at any NE point ,Theorem: If , then at any NE point ,

,, while at any joint optimum point , while at any joint optimum point ,

0)( ijxAP x

Njx

xj

0

)(

*x

Njx

xj

j

0

)( *

}1),(

:{

i

jijx x

xA

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelUniqueness of NEUniqueness of NE

Definition: Goods are Definition: Goods are symmetricsymmetric if the n if the n random T-vectors are random T-vectors are exchangeable. exchangeable.

Assumption: Identical prices and costs.Assumption: Identical prices and costs.

Incomplete Demand Diversion (IDD): Incomplete Demand Diversion (IDD):

Nontrivial Demand Diversion (NDD):Nontrivial Demand Diversion (NDD):

},...,1:{ TtU jt

ijjijj xxMexMP 0,0}),(),({

xyxMyMP jjjj ,0)},(),({

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelLemma 4: Suppose IDD, is Lemma 4: Suppose IDD, is

absolutely continuous, and absolutely continuous, and Then Then

..

Lemma 5: Suppose NDD, then Lemma 5: Suppose NDD, then

rxxMP jjj }),({ ),( jM

jirxexMP jijj ,}),({

xyxxMPyyMP jjjjjj },),({}),({

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelTheorem: By the assumptions, there exists a uTheorem: By the assumptions, there exists a u

nique NE to the symmetric game.nique NE to the symmetric game.Proof: First-order ConditionsProof: First-order Conditions

By Lemma 5, no other symmetric equilibriBy Lemma 5, no other symmetric equilibria other than .a other than .

By Lemma 4, no asymmetric equilibria exiBy Lemma 4, no asymmetric equilibria exists. sts.

NjxxMP pcjjj ,}),({

'x x

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelExtreme cases: Extreme cases:

Assumption: Symmetric Game;Assumption: Symmetric Game;

Customer prefer to buy Customer prefer to buy something.something.

Theorem: Under the assumptions. Let Theorem: Under the assumptions. Let be the equilibrium. be the equilibrium. When , . is the When , . is the inventory level with zero profits. inventory level with zero profits.

n

xn 0xx 0x

Choice Based Choice Based Inventory Competition Inventory Competition ModelModel

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelSample Path Gradient AlgorithmSample Path Gradient Algorithm1.1. Initialize outer looper jInitialize outer looper j2.2. Initialize inner looper kInitialize inner looper k3.3. For firm j at iteration kFor firm j at iteration k

• Generate a new sample path Generate a new sample path • Calculate sample path gradientCalculate sample path gradient• Update current inventory levelUpdate current inventory level

4.4. Check convergence or returnCheck convergence or return

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelConclusion:Conclusion: NE exists under mild regularity NE exists under mild regularity

conditions;conditions; NE is unique in case of symmetric NE is unique in case of symmetric

firms;firms; NE solution is overstocking;NE solution is overstocking; The industry becomes competitive The industry becomes competitive

as number of firms increase.as number of firms increase.

Choice Based Choice Based Inventory Competition Inventory Competition ModelModelOpen Questions: Open Questions:

1.1. What if without assumption that What if without assumption that customers prefer to buy customers prefer to buy something?something?

2.2. Include price and cost factors?Include price and cost factors?

3.3. What if arrival time is a factor?What if arrival time is a factor?

4.4. Analysis cooperative game.Analysis cooperative game.

Reference:Reference:

Lippman and McCardle, 1997, The Lippman and McCardle, 1997, The Competitive Newsboy, Competitive Newsboy, Operations ResearchOperations Research, , Vol. 45, No. 1, 54-65Vol. 45, No. 1, 54-65

Topkis, 1979, Equilibrium Points in Nonzero-Topkis, 1979, Equilibrium Points in Nonzero-sum n-Person Submodular Games, sum n-Person Submodular Games, SIAM J. SIAM J. Control and OptimizationControl and Optimization, Vol. 17, 773-787, Vol. 17, 773-787

Parlar, 1988, Game Theoretic Analysis of the Parlar, 1988, Game Theoretic Analysis of the Substitutable Product Inventory Problem with Substitutable Product Inventory Problem with Random Demands, Random Demands, Naval Research LogisticsNaval Research Logistics, , Vol. 35, 397-409Vol. 35, 397-409

Mahajan and Ryzin, Inventory Competition Mahajan and Ryzin, Inventory Competition under Dynamic Consumer choice, under Dynamic Consumer choice, Operations Operations ResearchResearch, Vol. 49, 646-657, Vol. 49, 646-657

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