Competitive and Cooperative Inventory Policies

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Competitive and Cooperative InventoryPolicies in a Two-Stage Supply ChainGerard P. Cachon Paul H. ZipkinThe Fuqua School of Business, Duke University, Durham, North Carolina 27708

[email protected]@mail.duke.edu

W e investigate a two-stage serial supply chain with stationary stochastic demand andfixed transportation times. Inventory holding costs are charged at each stage, and eachstage may incur a consumer backorder penalty cost, e.g. the upper stage (the supplier) maydislike backord ers at the low er stage (the retailer). We consider two ga me s. In both, the stagesinde pen den tly cho ose base stock policies to minim ize their costs. The games differ in how thefirms track their inventory levels (in one, the firms are committed to tracking echeloninventory; in the other they track local inventory). We compare the policies chosen under thiscompetitive regime to those selected to minimize total supply chain costs, i.e., the optimalsolution. We show that the games (nearly always) have a unique Nash equilibrium, and itdiffers from the optim al solution. Hence, competition reduces efficiency. Furthe rmo re, the twogam es' equilibria are different, so the tracking method infiuences strategic beha vior. We showthat the system optimal solution can be achieved as a Nash equilibrium using simple lineartransfer payments. The value of cooperation is context specific: In some settings competitionincreases total cost by only a fraction of a percent, wh ereas in other settings the cost increaseis enormous. We also discuss Stackelberg equilibria.(Supply Chain; Game Theory; Multiechelon Inventory; Incentive Contracts)

1. IntroductionHow should a supply chain manage inventory? If theme mb ers care only about overall system performance,they should choose policies to minimize total costs,i.e., the optimal solution. While this approach is ap-pealing, it harbors an imp ortant w eakness. Each mem -ber may incur only a portion of the supply chain'scosts, so the optimal solution may not minimize eachme mb er's ow n costs. For example, a supplier may caremore than a retailer about consumer backorders forthe supplier's product, or the retailer's cost to holdinventory m ay be higher than the supplier 's. While thefirms may agree in principal to cooperate, each mayface a temptation to deviate from any agreement, toreduce its own costs. Supposing each firm can antici-pate these temptations, how will the firms behave?

Furthermore, to what extent will these temptalead to supply chain inefficiency?This paper studies the difference between glcooperative and independent/competitive optimtion in a serial supply chain with one supplier andretailer. (We assum e there are two indepen dent fbut the model also applies to independent awithin the same firm.) Consumer demand is stotic, but independent and stationary across perThere are inventory holding costs and consum er order penalty costs, but no ordering costs. Thereconstant transportation time between stages, ansupplier's source has infinite capacity. Inventotracked using either echelon inventory or local in(A firm's local inventory is its on-hand inventoryits echelon inventory is its local inventory plu

0025-1909/99/4507/0936$


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CACHON AND ZIPKINCompetitive and Cooperative Inventory Policies

in the supply chain.) In thethe firms choose base stock policies,

in 3. These policies can be implem ented byor local inventory.

we con-the Echelon Inventory (EI) game and

nvento ry (LI) game. In both gam es the firmsis

and it cannot be modifiedit is announced. The supplier pays holding costs

in its possession or in-transit to theand the retailer pays holding costs on units it

are concerned about consumerthe supplier pays a consumer backorderas does the retailer. This is an important

it allows us to study how theand, in turn, the performance of the system.

3 discusses this modeling issue.)he EI and LI games differ in only one way: In the

both firms are committed to tracking echelonin the LI game both firms track

in each game play a Nash equilibrium. (Aof strategies is a Nash equilibrium, if each firmits own cost assuming the other player

its equilibrium strategy.) Thus, each firman optimal decision given the behavior of the

and therefore neither firm has an incentivethe equilibrium.

in each game there is (usually) aWe compare the games'

to each other and to the optimal solution.is typically nota Nash equilib-so competitive decision making degrades sup-We evaluate the magnitude of

an extensive numerical study.of the cooperative solution requires

the firms eliminate the incentives to deviate, i.e.,so that the optimal

a Nash equilibrium . This goal can beby a contract that specifies linear transfer

on easily verifiable performance mea-and backorders. We develop a set

of linear contracts that meet this objective, and briefiydiscuss other techniques for aligning incentives.In these games neither player dominates the otherand the firms simultaneously choose their strategiesWe also study Stackelberg versions of the games, inwhich one dom inant player chooses its strategy beforthe other.The next section reviews the related literature, and3 formulates the model. Section 4 describes thesystem optimal solution, and 5 analyzes the twogames. Section 6 compares the games' equilibria withthe o ptimal solution. Section 7 describes contracts thamake the system optimal solution a Nash equ ilibriumSection 8 discusses the numerical study. Section 9analyzes the Stackelberg games, and 10 concludes.

2. Literature ReviewThe literature on supply chain inventory managemenmostly assumes policies are set by a central decisiomaker to optimize total supply chain performanceThree exceptions are Lee and Whang (1996), Che(1997), and Porteus (1997).In Lee and Whang (1996), the firms use echelostock policies and all backorder penalties are chargeto the lowest stage. The upper stage incurs holdincosts only. Therefore, with competitive selection opolicies, the upper stage carries no inventory, therebminimizing its own cost. They develop a nonlineatransfer payment contract that induces each firm tchoose the system optimal base stock policies. Oumodel differs from theirs on several dimensions. Wassume the upper stage (the supplier) may care abouconsumer backorders, so it may carry inventory evewhen inventory policies are chosen competitivelyHence, the competitive decisions are nontrivial. Wdistinguish between echelon inventory and local inventory and investigate how these different methodfor tracking inventory infiuence strategic behavioFinally, we develop linear transfer payment contractThey consider a setup cost at the upper echelon, whilwe do not. Porteus (1997) studies a model similar tLee and Whang's model, but he proposes a differencoordination scheme, called responsibility tokens.

Chen (1997) studies a game similar to the populaBeer Gam e (Sterman 1989), except that the dem and s i


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C A C H O N A N D ZIP K INCompetitive and Cooperative Inventory Policies

different periods are independent random variableswith a common distribution that is known to allplayers. Unlike our players, his share the objective ofminimizing total system costs; they have no compet-ing interests. He outlines an accounting scheme thatallows each player to optimize its own costs and yetchoose the system optimal solution. This scheme ismore complex than ours in some ways, though sim-pler in others, as explained in 7. He also studies thebehavior of boundedly rational players, whereas weonly assume rational players.

Several other papers address related issues, yet theirmodels are significantly different. Lippman and Mc-Cardle (1997) study competition between two or m orefirms in a one-period setting, where a consumer mayswitch among firms to find available inventory. Parlar(1988) and Li (1992) also stiidy the role of inventory inthe competition among retailers. In a multiechelonmode l with m ultiple retailers, Muck stadt and Thom as(1980), Hausman and Erkip (1994), and Axsater (1996)investigate a centralized control system that allowseach firm to optimize its own costs and still choose anoutcome desirable to the central planner. The behaviorof a central planner has also been investigated insettings with moral hazard (e.g., Porteus and Whang1991, Kouvelis and Lariviere 1996). Many papers in-vestigate how a supplier can induce a retailer tobehave in a manner that is more favorable to thesupplier (e.g., Donohue 1996, Tsay 1996, Ha 1996, Laiand Staelin 1984, Moses and Seshadri 1996, Na raya nanand Raman 1996, Pasternack 1985). Chen et al. (1997)study competitive selection of inventory policies in amultiechelon model with deterministic demand.

3. Model DescriptionConsider a one-product inventory system with onesupplier and one retailer. The supplier is Stage 2 andthe retailer is Stage 1. Time is divided into an infinitenumber of discrete periods. Consumer demand at theretailer is stochastic, independent across periods andstationary. The following is the sequence of eventsduring a period: (1) shipments arrive at each stage; (2)orders are submitted and shipments are released; (3)consumer demand occurs; (4) holding and backorderp e n a l t y c o s t s a r e c h a r g e d . " ='ii ; / ? T v;>:: ? ; ^-i: . : ? *

There is a lead tim e for shipm ents from the sourthe supplier, L,, and from the supplier to the retaL ,. Each firm may order any nonnegative amoueach period. There is no fixed cost for placinprocessing an order. Each firm pays a constant per unit ordered, so there are no quantity discou

The supplier is charged holding cost h2 per pefor each unit in its stock or en route to the retailerretailer's holding cost is /JJ + /i, per period for unit in its stock. Assume /ij > 0 and /z, > 0.

Unm et dem and s are backlogged, and all backoare ultimately filled. Both the retailer and the supmay incur costs when demand is backordered.retailer is charged ap for each backorder, andsupplier (1 - a)p, 0 ^ a ^ 1. The parameter p itotal system backorder cost, and a specifies howcost is divided among the firms. The parameterexogenous.

These backorder costs have several possible intetations, all standard. They may represent the cosfinancing receivables, if customers pay only uponfulfillment of dem and s. (This requires a discountedmodel to represent exactly, but the approximation is standard in the average-cost context, analogous ttreatment of inventory financing costs.) Alternatithey may be proxies for losses in customer good-which in tum lead to long-run declines in demand. costs need not affect the firms equally, which is whallow the fiexibility to choose a G [0, 1]. Finally,prov ide a crude approx imation to lost sales. (It wo ubetter, of course, to model lost sales directly, butintroduces considerable analytical difficulties. Evenoptimal policy is unknown.)

In period t before demand define the followingstage / : in-transit inventory, IT; echelon inventory ILj,, is all inventory at stage / or lower in the syminus consumer backorders; local inventory leveis inventory at stage i minu s backorders at stage /supplier's backorders are unfilled retailer ordechelon inventory position, IP,,, IP,, = IL^, + IT,,;local inventory position, IP,,, IP,, = IL,, + IT,,.

Each firm uses a base stock policy. Using an echbase stock level, each period the firm orders a scient amount to raise its echelon inventory posplus outstanding orders to that level. A firm's


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s as stage f's echelon base stock level and s, as itsD ^ denote random total demand over T periods,

T periods. Letnd be the density and distrib ution functions of

0, so the same is true of \ T > 0. Furthermore,= 0, so positive demand occurs in each period.[x]* = max{0, x}; [x]~x]; [a, b] is the closed interval from a to b;E[x] is the expected value of x. A prime denotes

System Optimal Solution

h firm chooses a policy that minim izes itsunction. T his section briefiy outlines this m ethod .t G''^(IL^, - D') equal the retailer's charge int, where

t, define G(IP,,) as the retailer'st + L^, where

y that minimizes Gl(y):(1)

In period t charge the supplier G2(IP2f), whereGl(y) = E[Gl(y - D^n

T he suppl ie r ' s op t ima l eche lon base s tock leve l ,minimizes5. Echelon and Local InventoryGamesIn the Echelon Inventory (EI) game, the two stages areindependent firms or players. In the game's onlymove, the players simultaneously choose their strategies, s, G (T = [0, S], wh ere s, equals player i'echelon base stock level, a is player i's strategy spaceand S is a ver y large con stant. (S is sufficiently largthat it never constrains the players.) A joint strategy is a pair (s,, Sj). After their choices, the playerimplement their policies over an infinite horizon. Inaddition, all model parameters are common knowledge.In the Local Inventory (LI) game the supplier andthe retailer choose local base stock levels, s,, s, G aAgain, strategies are chosen simultaneously, the players are committed to their strategies over an infinithorizon, and all parameters are common knowledgeThe players know which game they are playing; thchoice between the EI and LI gam es is not one of theidecisions.

Define H,(s,, s^) as player i's expected per-periocost when players use echelon base stock levels (sSj). When S2~S2 + s, an d s, = s,, the local base stocpair (S], S2) is equivalent to (s,, Sj) in the sense thaH,(Sj, S2) = Hi(s,, Sj + s,). Since any echelon basstock pair can be converted into an equivalent locapair, there is no need to define distinct cost functionwith local argum ents . We will frequently switch a paof base stock levels from one tracking method tanother to facilitate comparisons. Although there little operational d istinction betwee n echelon and locabase stock policies, we later show that they diffestrategically. (However, the operational equivalencof echelon and local base stocks does depend on thassumption of stationary demand. In a nonstationardemand environment, it may not be possible to ruthe system optimally with local base stock policies.)


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For the EI game the best reply mapping for firm i isa set-valued relationship associating each strategy S:, ji" i, with a subset of a according to the followingrules:,, S2) = min , S2)}

si) = {S2 E a I H2(Si, S2) = min H2(Si, x)}.Likewise, for the LI gam e, the best reply ma pping s a rerSi) = {Si e o-1 Hi(si, S2 + Sl) = min H^(x, Sj + x)}

xScr''2(si) = {2 G o-1 H2 (si, S2 + Sl) = m in H2(si , x + Si)}.

A pure strategy Nash equilibrium is a pair ofechelon base stock levels, (s\, sJ), in the EI game, orlocal base stock levels, (s[, s'j), in the LI game, suchthat each player chooses a best reply to the otherplayer's equilibrium base stock level:

(We do not consider mixed strategies. We generallyfind a unique pure strategy equilibrium.)5.1. Actual Cost FunctionsIn each peri od, the retailer is charged /z, + /I2 per unitheld in inventory and ap per unit backordered. DefineG i(/Li, - D ') a s the sum of these costs in period t.

Define G i(/P],) as the retailer's expected cost in periodt + L,,

(x -

Define sJ as the value that minimizes this function,that is, the base stock level that minimizes the retail-

er 's costs, assuming retailer orders are shipped imdiately, ' Sl = arg min G,(y ). . .

Differentiation verifies that G, is strictly convex, is determ ined by G'j(sJ) = 0, .. .ap

"1 + "2 + PThe retailer's true expected cost depends on boow n base stock as well as the sup plier 's base stockuse a standard derivation. After the firms place orders in period t - L^, the supplier 's echelon intory position equals S2. After inventory arriveperiod t, but before period t demand, the suppechelon inventory level equals S2 - D^\ Hence, - D^^] is the supply chain's expected inventory(average supply chain inventory m inus average borders). W hen S2 - D '" > s,, the supplie r completely fill the retailer's period t order, so= S]. When S2 - D*" < Sj, the sup pli er cann ot fof the retailer's order, and 7P,, = S2 - D''\ Hen

H i(s ,, S2) = E[Gi(min{s2 -

,(s2 - x)dx.1

the supplier 's actual peefine 62(^^1/t backorder cost.

and G2(/P],) as the supplier's expected period t backorder cost.

Definei, x) = m\n{x, 0

soi , S 2 - S , -


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C A C H O N AND ZIPKINCompetitive and Cooperative Inventory Policies

- X)dx.

is the expected holding cost forto the retailer (from Little's Law),

is the expected cost for inventory heldsupplier and the final two terms are the ex-to the supplier.

the operational equivalence ofand echelon base stock policies wh en s, = s, ands, -I- Sj. However, the change in player i's cost

to a shift in player j's strategy depends on theFor example, holding Sj

the supplier's expected on-hand inventory isof s,, but when s^ stays constant, the

as s, increases. Further-the total system inventory depends on Sj only.

s^ fixed, the retailer's Sj only influences theof inventory between the supplier and theSj fixed, the retailer can

by raising s,.JEchelon Inventory Game Equilibria withShared Backorder Costs

we assume that each firm incurs somei.e., 0 < a < 1. (We subsequently

the extreme cases a = 0 and a = 1.) We beginon the players' cost

and best reply mappings.1. Assuming a < 1, H^{s,, s ) is strictly

in s^, s^ > 0, and Hîs^, s ) is quasiconvex in s,.Fix D'-' and s,. Consider the following

of Sjis, - + ^ sJ).are convex, while the second term is

in the interval Sj [D^\ D'-' + s,].the expectation over D^\ The first term,

- s, - V-y], is convex, and strictly convex

for S2 s: s,. The second term, E[G2(min|s2 - D^\s,))], is convex, and strictly convex when Sj ^ 0 andSj > 0. Hence, H2(s,, Sj) is strictly convex in Sj s: 0.Consider H,. When s, > Sj, H, is constant withrespect to s,. A ssume s, < Sj and differentiate Hj,

When Sj < s, H, is decreasing for s, < s^ andconstant for s, > s . When s^ > s", Hj is decreasingfor s, < s", increasing for s" < Sj :S Sj, and constanfor Sj > Sj. Hence, H, is quasiconvex in s,. aThe following lemma characterizes the supplier's

best reply mapping.L E M M A 2. Assuming a < 1, r^Csi) is a function, r-îs^

> s,, and 0 < r^(s,) < 1.PROOF. From Lemma 1, Hj is strictly convex in Sso T-jCSi) is a function (i.e., Hj has a unique minimumand is determined by the first-order condition

dH2 - X)dx = 0

This condi t ion cannot ho ld at S2 < s, because the^ ' ' ( S 2 - Si) = 0 and G'î}/) < 0. Therefore, s^ , = r^{s> s,. Giv e n s^ > Sj, from the implicit functioth e o r e m.

-, (2

where

bft


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The cross par t ia l of Hj is nega t ive because G'2 < 0< h^ and '-'{s2 - s,) > 0 for s > s,. Since G ^ > 0,0 < r'îs^) < 1. n

Although in the El game the supplier does notalways fill the retailer's orders immediately, the sup-plier 's echelon base stock level has little influence overthe retailer's strategy.

L E M M A 3. For the El game, the retailer's best replymapping is

2t2, S] S 2 < S r

P R O O F . Recall that G,(i /) is str ict ly convex andm i n i m i z e d by y = sJ. Let x = D \ W h e n s, - x < s",s , > S2 minimizes G, (min{s2 x, s,)). W h e n s, - x> s, only s, = s" m in im izes Gi (m in i s2 - x, s,)).

Figure 1 ReactJon Functions, a = 0.30,p = 5, ft, = /i j =

W h e n Sj < sJ, Sj - D'-' < sj, so = [s^, S]. W h e nS2 > s", i = Si minimizes G,(nain{s2 - fo ral l X, so r,(s2) = s". n

The retailer's best reply is not necessarily unique,but there is only oneNash equilibrium.

T H E O R E M 4. Assuming 0 < a < 1, in the El game (s{= s", S2 = ?'2(Si)) is the unique Nash equilibrium.

PROOF. From T heorem 1.2 in Fudenberg and Tirole(1991), a pure strategy Nash equilibrium exists if (1)each player's strategy space is a nonempty, compactconvex subset of a Euclidean space, and (2) player i'scost function is continuous in s and quasiconvex in s,.By the assumptions and Lemma 1, these conditionsare met, so there is at least one equilibrium. FromLemma 2 in any equilibrium, {s{, sQ, s^ = rîs^) > s\.If s'2 ^ si, Lemma 3 implies s\ s s^,a contradiction.Hence Sj > s,, but from Lemma 3, this implies sJ = sJ.Since r2 is a function, there is only one Sj = rîs").Therefore, the equilibrium is unique, D

Figures 1 and 2 plot the firms' reaction functionsand the resulting equilibrium for two examples.5.3. Local Inventory Game Equilibria with Shared

Backorder CostsThe analysis of the LI game also begins by character-izing the cost functions and the best reply mappings.

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Figure 2 Reaction Functions, a = 0.90, p = 5, /?, =3.803.703.603.503.40

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LEMMA 5. H2(Si, s, + S2) is strictly convex in I2H , ( s , , s, + S2) is strictly convex in s,.

PROOF. SetS2 = s, + S2 ands, = s,. Differentiof H2(s,, s, + S2) reveals that

/ ^ r,(52) + . 2 OOpiiD EIequiA Uequi

S2 ) i , S2)

From Lemma 1, Hîs.^, S2) is strictly convex in SH2(s,, s, + S2) is strictly convex in S2. DifferenH , ( S , , Si + S2 ), . ::-.:, :: f . . ; . _


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C A C H O N A N D ZIPKINCompetitive an d Cooperative Inventory Policies

; , , S] + S2)

S2 )

+(4)

, wh ich mean s that H , is strictly convex in s,. D

6. Assuming a < I, ^,0; and - 1 < f'îs,) < 0. + Sj =For the supplier ^^(s,) + s, = r^{s^),

S2 = s,Sj and s, = Sj. From Lemma 2, rîs^) > s,, which

rîs^) > 0. From the same lemma, 0r[(s,) = r'^{s,) - 1, so - 1 < f'îs,)0. D

7. Assuming a > 0, r,(s2) > sj. When 21 < r',(s2) < 0; and when S2 = 0, r'jCsj) = - 1 .The retailer's best reply is determined by

When s, < s", G;(S,) < 0, and therefore dH^(s,, s,

S 2 ) S 2 )

- x)dx 'S2 > 0. Since ^''(Sj)G'[{s,) > 0, the numer-

ond term in the denom inator, - 1 < f\(S2) < 0.

When s, = 0, $'^'(s2)G';(Si) = 0 and therefore r'îs^= - 1 . a .h .a;~:-^5-When 0 < a < 1, there is a unique N ash equilibriumin the LI game.THEOREM 8. Assuming 0 < a < 1, (s',, Sj) is th eunique Nash equilibrium.PROOF. Lemma 5 confirms the required conditionsfor the existence of an equilibrium (in the proof ofTheorem 4). First, from Lemm a 6, r,(s ,) = rj(s,) - s> 0, so S2 > 0. Now, suppose there are two equilibria(s[, S2) and (s*, s*). Withou t loss of gene rality, a ssume2 < s*. From Lemm a 7, this implies that s* < s',. From

the same lemma, > - 1 , so s* = s^ + s* > s+ 2 = Sj. But from Lemma 2, r2 is increasing, so s> S2 implies that s* > s',, a contradiction. Hen ce, theris a uniqu e equilibrium, nFigures 1 and 2 also display the reaction functions ithe LI game as well as the Nash equilibrium.5.4. Equilibria Under Extreme Backorder Cost

AllocationsSuppose the retailer is charged all of the backordecosts, i.e., a = 1. In this situation, the Nash equilibriumin the El game is no longer unique.THEOREM 9. For a = 1, in the El game the Na

equilibria are (s\ e [Sj, S], Sj e [0, s"])-PROOF. The existence proof in Theorem 4 applieeven when a = 1, so a pure strategy equilibriumexists. When a = 1, the supplier incurs no backordecosts, only holding costs. Hence, the supplier picks S< s,, i.e., r2(s,) = [0, s,]. Suppose (s*, s*) is aequilibrium, where s* > s'. From Lemma 3, ri(s2= s", but an equilibrium only occurs when s* < s*, ss* > sJ cannot be an equilibrium. Suppose s* ^ sFrom Lemma 3, ri(s2) = [s*, S], so for any s* ^ s", {E [s*, S], s*) is an equilibrium, aIn the LI game there is a unique equilibrium evewhen the retailer incurs all of the backorder cost.THEOREM 10. Assum ing a - I, in the LI game {

= ri(O), 2 = 0) is the unique Nash equilibrium.PROOF. When a = 1 the supplier chooses 2 = 0Since r,(s2) is a function, r,(0 ) is un iqu e, n m imWhen the supplier incurs all backorder costs, ther


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CACHON AND ZIPKINCompetitive an d Cooperative Inventory Policies

is a unique equilibrium in both games and they areidentical. dH-,THEOREM 11 . Assuming a = 0, {s \ = 0, s' =

is the unique Nash equilibrium in the El gam e, and {s[= s ', 2 = S2 - s'l) is the unique Nash equilibrium of theI game.

PROOF. Since the retailer incurs no backorder costJ = sJ = 0. The supplier's best reply m appin g is ain either g am e, so Sj = '"2(0)- Furthe rm ore, Sj

= si - s ' n

. Comparing Equilibrias to each other as well as to the optimal solution.

(s\, s^), into the equivalent pair of eche-(s^, Sj), wh ere sj = s[ and Sj = s\+ S-.

Competitive Equilibria

T H E O R E M 12 . Assuming 0 < a < 1, the base s tockfor both f irms are higher in the LI game equil ibr iumin the El gam e equilibrium , i.e., Sj > S2 and s\ > s[.

2 =PROOF. The equilibrium in the El game is {s [ = s",

J)). From Lemma 7, r,(s2) > s,, which impliess\ > sj = s. From Lemma 2, r^is^) is increasing

r2(s[) > r2{s'^) = si aThe retailer's cost in the LI game equilibrium can be

erical study confirms this.) How ever, the supplier

T H E O R E M 13. Assum ing 0 < a < I, the supplier 'sgame equi libr ium .

PROOF. In the El game the supplier chooses r2(Si).- s^ as its

r2{s^). Differentiate the supplier's costS2 = r2(Si): ' ' -'

5 s,

since dHjis^, r2{s^))/dS2 = 0. From Lemma 2, r2> s, , so '-'(S2 - s,) > 0, an d G j < 0, so dH2< 0. Thus, the supplie r's cost declines as Sj increaSince s[ > s[, H2 is lower at s[. aWhy does the supplier prefer the LI game equirium? The supplier always prefers the retailerincrease its base stock, thereby increa sing the retailinventory and decreasing the supplier's backocosts. The retailer alwa ys chooses a lower ba se stocthe El game than it does in the LI game, hence supplier is always better off in the LI game.6.2. Competitive Equilibria and the OptimalSolutionIn the El game the retailer's base stock level is lothan in the optimal solution.

T H E O R E M 14 . In an El game equilibrium, the retaibase stock leve l is lower than in the optim al solution.

PROOF. Note that Gi'ix) < G\ix), for all x. HeG ;'(y) < G',(y), for all y. Since both G r ( y ) an d Gare increasing in y, s'l > s^ = s\. a

In the LI game either sj > sJ or sJ < Sj is possi(In Figure 1 the retailer chooses s[ < sJ, but in Fig2 s'l > s.) However, when backorder costs charged to the supplier, the supplier's base stock leis lower than in the system optimal solution in bgames.T H E O R E M 15 . Assum ing a < 1, the supp lier's

stock level in bo th the LI and the El equilibria is lower in the sys tem op t imal so lut ion.

PROOF. In any equilibrium Sj > Sj, so it is sufficto show that sj > s'2. For x < - s , .

dx x) = - ( 1 - a)pand


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CACHON AND ZIPKINCompetitive and Cooperative Inven tory Policies

- s , < X < 0

- ( 1 -a)p,O]Gl'ix) = -p. For x > 0, = h^ and

It is possible that s[ = s',' because (p'-^*'-'*'' stochastcally dominates *'- '^' and p / ( / i , + ^2 + p) < {h^+ h2 + p). For the supplier S2= s' l when G2= 0. This occurs precisely when

for x < s,. So in all cases G2'(^)with strict inequality for -s , < x < sJ.GjX^) - dH2/dx, with strict inequality for

S). So, Sj > S2. Dthe supplier's echelon base stock deter-

the supply chain's average inventory level.15 it follows that in either game'sthe supply chain's average inventory

be lower than in the optimal solution,to lower the

The numericalthe supplier incurs no backorder costs, the

is no greater than in the

16. Assuming a = 1, Sj < s^ , and si ^ s^.In the El game Sj = sj < s" < s" < s",

< Sj. When a = 1, in the LI game Sj = s[, soof Theorem 15 demonstrates s^ ^ s^. a

17. Assuming a < 1, the system optimalis not a Nash equilibrium.

From Theorem 15, s ^ si < Sj, so theis not a Nash equilibrium in either

nthe supplier incurs no backorder costs, thecan be a Nash equilibrium

a very special condition.18. Assuming a = I, the system optimal

is a Nash equilibrium in the LI game only when

When a = 1, the LI game Nash equilibriumS2 = s[). S o l v i n g f o r s[.

In that case, Sj = sJ = sJ = s^. a

7. Cooperative Inventory PoliciesAccording to Theorem 17, the optimal solution isvirtually never a Nash equilibrium. Hence, the firmscan lower total costs by acting cooperatively.There are several methods that enable the firms tominim ize total costs and still remain confident that theother firm will not deviate from this agreement. Forinstance, the firms could contract to choose (s, S2) astheir base stock levels. But, since each firm has anincentive to deviate from this contract (because it isnot a Nash equ ilibrium), the contract must also specifya penalty for deviations. Such stipulations are hard toenforce.Alternatively, the firms could write a contract thaspecifies transfer pay m ents wh ich eliminate incentiveto deviate from the optimal solution. There are severaschemes to achieve this goal: a per period fee for thsupplier's backorder; a per unit fee for each unit thsupplier does not ship imm ediately; a per unit fee peconsumer backorder; or a subsidy for each unit oinventory in the system. (Nonlinear payment schedules, as in Lee and Whang (1996), could also bconsidered, but these are necessarily more cumbersome. For instance, the induced penalty function G"nonlinear.) Transfer payments can also be imposed othe retailer, e.g. a subsidy on retailer inventories, or subsidy on backorders.

Payments could also be based on the inventory anbackorder levels that would have occurred had thsupplier performed certain actions. Chen (1997) usethis approach. Define accounting inventory and accounting backorders as the inventory and backordelevels, assuming the supplier fills retailer orders im


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CACHON AND ZIPKINCompetitive and Cooperative Inventor\/ Policies

mediately. Suppose the supplier pays all of the retail-er's actual costs, and the supplier charges the retailerft, per un it of accoun ting inv ento ry and /12 "" P peraccounting backorder. Then, the retailer will chooses . Since the supplier incurs all actual costs, and theretailer chooses s, the supplier chooses S2. With thisscheme, the retailer's decision is independent of thesupplie r's; there is no strategic interaction betw een thefirms. However, this approach creates a challengingaccounting problem.

We study linear transfer payments based on actualinventory and backorder levels. While our approachavoids the problem of tracking accounting inventoryand backorders, Chen's method uses only cost param-eters. Ours also requires a demand parameter. Thus,his technique may be easier to implement in somesituations, ours in others.7.1. Linear ContractsSuppose the firms track local inventory and theyadopt a transfer payment contract with constant pa-rameters ((, ^2' /3i)- This contract specifies that theperiod t transfer payment from the supplier to theretailer is

+ 2, +where /,, is the retailer's on-hand inventory, and B,, isstage i 's backorders, all measured at the end of theperiod. There are no a priori sign restrictions on theseparameters, e.g. i, > 0 represents a holding costsubsidy to the retailer and i, < 0 represents a holdingfee. (We later impose some restrictions on the param-eters.) We also assume the optimal solution is com-mon knowledge. ;,Define Ti(/P,,) as the expected transfer payment in

period f + L, due to retailer inventory and back-orders, whereUy) = E[t,[y - D'^^T +

Define T(s,, 2) as the expected per period transferpayment from the supplier to the retailer.

min{ 0 ,

x)dx.Note that s, influences the retailer inventory backorders, but not the supplier's backorders.H'{s^, Sj + S2) be player i's costs after accountingthe transfer payment,

H\(su s, + S2) = H,(s,, s, + S2) - r (S i, S2),s,, s, + S2) = ,, s, + S2) + r ( s , , S2) .

We wish to de term ine th e set of contracts, (i,, /32such that {s\, 2) is a Nash equilibrium for the functions H'',{s.^, s, + 2), w he re sJ = s", and s\= Sj. With these contrac ts the firms can choose {s\thereby minimizing total costs, and also be assthat no player has an incentive to deviate.To find the desired set of contracts, first assum eH ] is strictly convex in s, , given that player; cho

s'^, j ^ i. Then determine the contracts in whicsatisfies player i's first order condition, thereby mmizing player i's cost. Finally, determine the subsethese contracts that also satisfy the original sconvexity assumption.The following are the first order conditions:

- x)]dx;

0 5 2


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C A C H O N AND ZIPKINCompetitive an d Cooperative Inventory Policies

' +T\{Si + S2-x)]dx. (6)= ^'^'(s" - sj)- (This is the supplier's

0 = - p + (p +X

- si) + (h,

- x)dx.

(7)

ee un know ns.(8)

/ j ,

h-,= ^ 72 (9)t remains to ensure that the costs functions areH E O R E M 19 . When the firms choose ( i , , ^2 / Pi) ^

(8) an d (9), and the following additional restrictions

(i )(ii)

> 1 ^ 0

(iii) ap >/3, s - ( 1 - a)p,

PROOF. When the following second order condi-H ; is stiictly convex in s,, assum ing

+ S2 - x))dx > 0;

- x))dx>0. .: ; u lThe first inequality reduces to

hi + h2+ ap - Li~ Pi>0.Substituting (8) yields i, 0. nThese are quite reasonable conditions: The firrequires that the retailer's inventory subsidy not eliminate retailer holding costs; the second stipulates ththe supplier be penalized for its backorders; and ththird states that the supplier should not fully reimburse the retailer's backorder costs, and the retailshould not overcompensate the supplier's backordcosts.To help interpret these results, consider the threextrem e contracts wh ere one of the param eters is set

zero:= 0

ii) I, =

/3 , = - ( 1 - a)p= 0

iii) t, = (1 -a) J , =0(Of these three contracts, the second does no t meet thcond itions in Theorem 19, because the su pplier fucompensates the retailer for all of its costs. The reta


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CACHON AND ZIPKINCompetitive a nd Cooperative Inventory Policies

With the first contract the retailer fully reimburses

lty for its local back orde rs. Witha > 0). In

P2 = 72''2/(l ~ 72)-Incidentally, (8) and (9) can be written

ap - /3i pft , + +

1 - ft , + ft:= 72-

sider the first identity. The quantity a p - |3, is the

written m ore simply as ft, + ft2 - i, . So thed side is the critical ratio for the total system costs

ing cost ft2 is mu ltiplied by the factor (1 - i-,/(ft,

Additional Contracting Issues

unique equilibrium. Nevertheless, even if there

ing to the optimal solution Pareto dominates aother. Hence, the players can coordinate on this eqlibrium. (There is experimental evidence that playcoordinate on a Pareto dominant equilibrium whthey are able to converse before playing the gam e, eCooper et al. 1989, Cachon and Camerer 1996.)

Although total costs decline when the firms coornate, one firm's cost may increase. This firm willunwilling to participate in the contract unless it ceives an additional transfer payment. To maintain strategic balance of the contract, this payment shoube independent of all other costs and actions. Fexample, the firms could transfer a fixed fee eaperiod. Alternatively, one could seek a contract (i,,Pi) such that each firm's cost is no greater than in original Nash equilibrium.

Finally, this analysis assumes the firms use lobase stock levels. In this context local policies haseveral adv anta ges over echelon policies. Recall thahas no influence on ^2^2 supplier's backorpenalty, when firms use local base stock levels. Hoever, with echelon stock base stock levels, s, dinfluence P2B2/. ho ldin g S2 constant. This can creaperverse incentive. Suppose P2 is large. By increass, > S2, the retailer can make B2, arbitrarily la(assuming S, the limit on s,, is large too). The i32transfer paym ent could easily dominate the additioretailer inventory cost. (Furthermore, once s, > S2,retailer can increase s, without increasing its invtory.) There is a solution to this problem. The transpayment could assume that the retailer chooses s^ ^Hence, the retailer would receive no additional beneby raising s, above sJ'. Clearly this increases complexity of the contract. Local measurements avothis problem altogether.

8. Numerical StudyThe system optimal solution is virtually never a N aequilibrium, but how large is the difference betwetheir costs? To answer this question, we conductednumerical study.

One period demand is normally distributed wmea n 1 and stand ard deviation 1/4. (There is onltiny probability of negative demand.) The remaini


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are chosen from the 2625 possible combi-of the following:

G {0, 0. 1, 0.3, 0.5 , 0.7, 0.9, 1} p {1, 5, 25}2, 4, 8, 16} fti e {0.1, 0.3, 0 .5, 0.7, 0.9}L 2 G { 1 , 2 , 4, 8, 16} ft2 = l - f t , .ftJ +ft2= 1 in all of the problems ; therefore,

is constant.(1)

e; and (3) the Nash equilibrium of the LIcan be obtained from http://

a = 1 there are mul-Na sh equilibria in the El game; we choose the onethe largest s,.)

in costthe Nash equilibrium over the system optimal

We call this percentage thecompetition pen-Several results areevident from the table. First,the players care about backorder costs equally

a = 0.5), the Nash equilibrium is close to theIn the El and LI games the

is 6% and 3%, respec-y, and the m aximum is13% and 8%, respectively.

The competition penalty increases as the back-order cost allocation becomes more asymmetric. Inthe El game the med ian percentage is84% wh en theretailer incurs no backorder penalty (i.e., a = 0) and483% when the retailer incurs all of the backorderpenalty (i.e., a = 1). The competitive outcome ispoor when a = 0 because th e retailer chooses s, = s,= 0, so all consumer demands are backordered.When a = 1, the competition penalty is substantialbecause the supplier refuses to carry inventory,thereby hampering the retailer's effort to mitigateconsumer backorders.

The LI game's response to changes in a is slightlydifferent. The performance of the competitive solutiondeteriorates rapidly as the retailer incurs lower backorder cost (i.e., a declines). However, as the retailerincurs higher backorder cost (i.e., a increases), themaximum competition penalty increases rapidlywhile the minimum and median penalties do notConsider the extreme case where all backorder costsare allocated to the retailer. In the com petitive solutionthe supplier carries no inventory. Nevertheless, thisbehavior is not always harmful. Look at Figure 3which displays the percentage increase in total systemcosts as a function of ,

Table 1 The Competition Penalty Under Different Allocations of Backorder Costs

EchelonInventory

Game

LocalInventory

G a m e

a00.10.30.50.70.91

00.10.30.50.70.91

Competition

Minimum107%

5%2%1%1%1%2%

107%5%2%1%0%0%0%

Penalty: Percentage Increase inSystem Optimal

5th Percentile

117%9%4%2%1%1%

15%117%8%3%

1%0%0%0%

Median604%41%12%8%4%8%

483%804%37%

' -' -'im'' - ; i ; 1 %

1 *1%

Cost of the Nash EquilibriumSolution

95th Percentile5,930%

97%21 %11 %12%40%

9,271%5,930%

,V 96%19%

0 %: 4% ;:;: . 17%

34%

Over the

Maximum10,939%

120%27%13%19%66%

34,493%10,939%

119%26%

" '[ j f i f c -45%

116%


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CACHON AND ZIPKINCompetitive an d Cooperative Inventory Policies

Figure 3 Competition Penalty When the Retailer Incurs AJI BackorderC o s t s ( a = 1 ) , ,. . ,.; . ,, , : . . ,.,

- . 140%

0.5 1.5

L , + L2 (10)

What does (10) measure? When L^/{L^ + L2) is large,the supplier's lead time is a small fraction of the totalsystem lead time, so the supplier's decisions have asmall impact. When ft2/(ft, +ft2) s large, there is littlebenefit to holding inventory at the supplier rather thanthe retailer. As (10) increases, the supplier's self serv-ing behavior does little dam age, therefore the compet-itive solution's performance is nearly as good as theoptimal solution's. Overall, the competition penalty ishigh when one of the firms has a substantial influenceover a major portion of total system costs, but littleincentive to help manage that cost.

Table 2 presents data on the percentage change inaverage supply chain inventory in the two equilibriarelative to the optimal solution. Averag e inventories inthe competitive solutions are generally lower than inthe optimal solution, except in some cases when theretailer cares little about backorders (a is small).Nevertheless, competition raises supply chain inven-tory by at most 4%. ; ; , ;^:;:.:v. . v .

9. One Dominant PlayerIn the LI and El games the players choose tpolicies simultaneously. In the Stackelberg versioeither game one of the players chooses its base slevel first, announces its choice to the other player,then the other p layer c hooses its base stock level. Ath e El and LI games, a player is committed tochoice, i.e., the first player cannot change its deciafter observing the second player's. We seek sub-gperfect equilibria, i.e., the second player chooseoptimal response to the first player's strategy andfirst player (correctly) anticipates this behavior.Stackelberg version represents a situation whereplayer is the dominant member of the supply c(e.g., WalMart, Intel.)9.1. Supplier Stackelberg GamesThe Stackelberg game with the supplier leadincalled either the Echelon Inventory Supplier g(EIS) or the Local Inventory Supplier game (Ldepending on the inventory tracking method. InEIS (LIS) game the supplier chooses S2 (S2) to mmize its cost, given that it anticipates the retailer choose r,(s2) ('^,(S2))- According to the next theothere is little difference between the EIS and El ga

T H E O R E M 20. When a < I, in the EIS gamer2(sj)) is the unique Stackelberg equilibrium. When a (s , G [S2, S], S2 G [0, sJ]} are the Stackelberg equili

PROOF. The proof of Theorem 13 shows that isupplier could choose s,, it would choose s, as largpossible. When a < 1, r,(s2) ^ s,, so the suppshould anticipate s, ^ s\. Therefore, the suppchooses Sj = r2(s,'), and the retailer chooses s, =When a = 1, the supplier wishes to carry no invenso it chooses S2 G [0, s], since then the retailer choose s, ^ S2. The supplier cannot choose S2 ^because then the retailer chooses s, = s, leavingsupplier with some expected inventory, n

In the LIS game the supplier anticipates thatretailer will choose r,(s2). Hence, the supplie r's co2 )

Since this is continuous in 2, there exists st such


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Table 2 Change inSupply Chain InventoryPercentage Change inAverage Supply Chain Inventory Inthe Nash Equilibrium Relative

to the Optimal So lution ' '"Minimum 5th Percentile Median 95th Percentile Maximum

EchelonInventory

Game

LocalInventory

Game

00.10.30.60.70.9100.10.30.50.70.91

- 1 8 %- 1 0 %

- 8 %- 1 3 %- 2 2 %- 3 7 %- 7 3 %- 1 8 %- 1 0 %

- 7 %- 1 0 %- 1 8 %- 3 4 %- 3 8 %

- 1 2 %-7%-S %- 9 %

- 1 5 %- 2 7 %- 5 6 %- 1 2 %- 7 %-5%-7%

- 1 1 %- 1 5 %- 1 6 %

-2%-3%-2 %-3%-4 %-7%

- 1 7 %-2 %-2 %-2 %-2 %-2 %-2 %- 1 %

2%1%0%

- 1 %- 1 %- 1 %- 3 %

2%1%0%

- 1 %- 1 %

0%0%

4%3%0%0%0%

- 1 %- 2 %

4%3%1%0%0%0%0%

= infsêio.s] ^2(82). Hence there exists a Stack-Let |sf, sl'l be an equi l ib r ium, and

{sf, sl'l be the equiva len t pa ir of echelon base s tocki.e., sf = sf and s ' = sf + s^

21. When a < 1, in the LIS game thechooses a base stock level lower than in the LI game,

Sj < 2 and S2 < Sj.Assuming s, = r,(s2), differentiate the

lier's cost function with respect to 2

s,

9.2. Retailer Stackelberg GamesIn the Echelon Inventory Retailer (EIR) and Local Inventory Retailer (LIR) games the retailer anticipates thsupplier will choose r2(Sj) and r2(s,), respectively. Sincrîsi) = -s, + r2(s,) wh en s, = s,, the retailer's cost for anbase stock level is the same in the two games. Hencewh e n the retailer is dominant , it is immater ial whethethe firms use echelon or local inventory m easurementsExistence of an equilibr ium is straightforward.

T H E O R E M 22. In the EIR and LIR games the retailechooses a base stock level that is higher than in the El gam(s{), but lower than in the LI game (s[). , . : - , . . . ;

PROOF. The retailer anticipates that the supplier wchoose r2(s,). Differentiate the retailer's cost function:

6, r',(s2) < 0. When a < 1, G'2{y) < 0.{dH,/ds,)r\{s,) > 0. When s^ s r2(s,), aH2/dSj0. Hence, for 2 ^ ^(s,), dH2/ds2 > 0. Therefore,< f2{s[) = s . Since r,(s2) + S2 is decreasing in S2,

tit f,.

ûrn


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CACHON AND ZIPKINCom petitive and Coop erative Inventory Policies

Since r'2(Si) > 0, the above is negative for all s,= sJ, hence the o ptima l s, is larger than s,. Since r'2< 1, and in the LI game equilibrium.'uS[ h)

it holds thatdH,{s[,

ds,Hence the retailer chooses a base stock level lower

10. ConclusionWhen both players care about consumer backorders,the supply chain optimal solution is never a Nashequilibrium, so competitive selection of inventorypolicies decreases efficiency. Although the playersmay agree to cooperate and choose supply chainoptin\al policies, at least one of them has a privateincentive to deviate from the agreement. Furthermore,there is a unique Nash equilibrium in either the Elgame or the LI game, and these equilibria differ.Hence, while there is little operational distinctionbetw een tracking echelon inventory or local inventory(since we assume stationary demand), there is a sig-nificant strategic difference. The supplier prefers localinventory, but the retailer's preference d epen ds on theparameters of the game.

In the games w e study , competition generally lowerssupply chain inventory relative to the optimal solution.In other words, if firms cooperate and choose the opti-mal solution, they will tend to increase inventory. This isa surprising result, since many authors suggest theopposite (e.g., Buzzell and Ortmeyer 1995, Kumar 1996).The rationale is that inventory is a public good: Eachfirm benefits from more inventory, but each wants theother to invest in it. It is well known that participantstend to underinve st in the provision of public goods (seeKreps 1990). In other settings, cooperation may lead tolower inventory. For instance, cooperative firms could

share sales information, and this might enable bpolicies than those available to competitive firms. ertheless, inventory rem ains a public good even heresuspect the re is alway s a strong tendency for comp)efirms to choose lower inventory than in the opsolution.

Should the players wish to choose the optsolution cooperatively, we characterize a set of silinear contracts which eliminate each player's intive to deviate. These contracts are based on ainventories and backorders. Implementation of tcontracts will not provide dramatic improvemwhen the players have similar preferences for reing consumer backorders. We draw this conclufrom a sample of 2625 problems. Eor each problemmeasured the competition penalty, the percentacrease in total cost of the Nash equilibrium oveoptimal solution. When the players view consbac kord ers as equally costly (i.e., a = 0.5), the m ecompetition penalty in the El game is only 6% anthe LI game it is 3%. How ever, when the players divergent backorder costs, the competition pecan be huge. For instance, when the supplier is iferent to consumer backorders, the median comtion penalty in the El game is 483% . These rehighlight an important lesson for managers: Whillack of cooperation/coordination implies the sywill not perform at its best efficiency, the m agn ituthe efficiency loss is context specific'' The authors w ould like to thank the sem inar participaRochester University, the University of Michigan, the UniversChicago, and the 1997 Multi-Echelon Inven tory Conference aYork University. The helpful comments of Eric Anderson, Lariviere, the referees, and the associate editor are also gracacknowledged.ReferencesAxsater, S. 1996. A framework for decentralized multi-e

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Competitive and Cooperative Inventory Policies

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