Revenue Maximization in the Dynamic Knapsack Problem

Transcription

1 Revenue Maximization in the Dynamic Knapsack Problem Deniz Dizar, Alex Gershkov an Benny Molovanu Abstract We characterize the revenue maximizing policy in the ynamic an stochastic knapsack problem here a given capacity nees to be allocate by a given ealine to sequentially arriving agents. Each agent is escribe by a toimensional type that re ects his capacity requirement an his illingness to pay per unit of capacity. Types are private information. We rst characterize implementable policies. Then e solve the revenue maximization problem for the special case here there is private information about per-unit values, but eights are observable. After that e erive to sets of aitional conitions on the joint istribution of values an eights uner hich the revenue maximizing policy for the case ith observable eights is implementable, an thus optimal also for the case ith to-imensional private information. Finally, e analyze a simple policy for hich per-unit prices vary ith requeste eight but o not vary ith time. Its implementation requirements are similar to those of the optimal policy an it turns out to be asymptotically revenue maximizing hen available capacity/ time to the ealine both go to in nity. 1 Introuction The knapsack problem is a classic combinatorial optimization problem ith numerous practical applications: several objects ith given, knon capacity requests (or eights) an given, knon, values must be packe in a "knapsack" of given capacity in orer to maximize the value of the inclue objects. In the ynamic an stochastic version (see Ross an Tsang [22]) objects sequentially arrive over time an their eight/value combination is stochastic but becomes knon to the esigner at arrival times. In the present paper e a incomplete information to the ynamic an stochastic setting: there is a nite number of perios, an at each perio a request for capacity We are grateful to the German Science Founation for nancial support. Dizar, Gershkov, Molovanu: Department of Economics, University of Bonn, Lennestr. 37, Bonn Germany; izar@uni-bonn.e, alex.gershkov@uni-bonn.e, mol@uni-bonn.e 1

2 arrives from an agent that is privately informe about both his valuation per unit of capacity an the neee capacity 1. Each agent erives positive utility if he gets the neee capacity (or more), an zero utility otherise. The esigner accepts or rejects the requests in orer to maximize the revenue obtaine from the allocation. The ynamic an stochastic knapsack problem ith complete information about values an requests has been analyze by Papastavrou, Rajagopalan an Kleyegt [18] an by Kleyegt an Papastavrou [13]. These authors have characterize optimal policies in terms of threshols. Kincai an Darling [11], an Gallego an van Ryzin [7] look at a moel that can be re-interprete as having (one imensional) incomplete information about values, but in their frameorks all requests have the same knon eight 2. In particular, Gallego an van Ryzin sho that revenue is concave in capacity in the case of equal eights. Kleyegt an Papastavrou have examples shoing that total value may not be concave in capacity if the eight requests are heterogeneous. Gershkov an Molovanu [8] generalize the Gallego-van Ryzin moel to incorporate objects ith the same eight but ith several qualities that are equally ranke by all agents, inepenently of their types (hich are also one-imensional). The theory of multiimensional mechanism esign is relatively complex: the main problem is that incentive compatibility - hich in the one-imensional case often reuces to a monotonicity constraint - imposes, besies a monotonicity requirement, an integrability constraint that is not easily inclue in maximization problems (see examples in Rochet [2], Armstrong [2], Jehiel, Molovanu an Stacchetti [1], an the survey of Rochet an Stole [21]). Our implementation problem is special though because useful eviations in the eight imension can only be one-sie (upars). This feature allos us a less cumbersome characterization of implementable policies that can be embee in the ynamic analysis uner certain conitions on the joint istribution of values an eights of the arriving agents. Other multiimensional mechanism esign problems ith restricte eviations in one or more imensions have been stuie by Blackorby an Szalay [4], Iyengar an Kumar [9], Kittsteiner an Molovanu [12], an Pai an Vohra [17]. Our main results characterize the revenue maximizing policy for the knapsack problem in several cases. The logic of the construction is as follos: We rst characterize implementable policies, as explaine above. Then e solve the revenue maximization problem for the special case here there is private information about per-unit values, but eights are observable: uner a stanar monotonicity assumption on virtual values, e sho that this policy is Markovian, eterministic, an has a threshol property. It is important to emphasize that the resulting optimal policy nee not be implementable for the case here both values an eights are unobservable, unless aitional conitions are impose. We then erive to sets of aitional conitions 1 The results are easily extene to the setting here arrivals are stochastic an/or time is continuous. 2 We refer the reaer to the book by Talluri an Van Ryzin [23] for references to the large literature on revenue (or yiel) management that aopts variations on these moels. 2

3 on the joint istribution of values an eights uner hich the revenue maximizing policy for the case ith observable eights is implementable, an thus optimal also for the case ith to-imensional private information. These conitions - hich are satis e in a variety of intuitive settings - involve a form of positive correlation beteen eights an values expresse by a hazar rate orering of conitional values, an a eakening of the rst set of su cient conitions in combination ith the non-primitive assumption of concave revenues respectively. Finally, e analyze a simple policy for hich per-unit prices vary ith requeste eight but not ith time. Its implementation requirements are similar to those of the optimal policy an it turns out to be asymptotically revenue maximizing hen available capacity/ time to the ealine both go to in nity. This is particularly valuable since policies hich lea to prices that sometimes ecrease in time create incentive issues if agents are strategic ith respect to their arrival times. We also point out that a policy that varies ith time but not ith requeste eight (hose asymptotic optimality in the complete information case has been establishe by Lin, Lu an Yao [14]) is usually not optimal uner incomplete information. The paper is organize as follos: In Section 2 e present the ynamic moel an the informational assumptions about values an eights. In Section 3 e characterize incentive compatible allocation policies. In Section 4 e focus on ynamic revenue maximization. We rst characterize the revenue maximizing policy for the case here values are private information but eight requests are observable. We then o er to results that exhibit conitions uner hich the above policy is incentive compatible, an thus optimal also for the case here both values an eights are private information. In Section 5 e introuce a simpler time-inepenent policy as escribe above, analyze the limit case here the capacity an time to ealine become very large an emonstrate asymptotic optimality. 2 The Moel The esigner has a "knapsack" of given capacity C 2 R that he ants to allocate in a revenue-maximizing ay to several agents in at most T < 1 perios. In each perio, an impatient agent arrives ith a eman for capacity characterize by a eight or quantity request an by a per-unit value v 3 : While the vector (; v) is private information to the arriving agent, the esigner is assume to kno the istribution of the ranom vector (; v) hich is given by the joint cumulative istribution function F (; v), ith continuously i erentiable ensity f(; v), e ne on [; 1) 2. Demans are inepenent across i erent perios. In each perio, the esigner ecies ho much capacity to allocate to the arriving agent (possibly none) an on a monetary payment. Type (; v) s utility is given by v p if at price p he is allocate a capacity an by p if he is assigne 3 It is an easy extension to assume that the arrival probability per perio is given by p < 1: 3

4 an insu cient capacity <. Each agent observes the remaining capacity of the esigner. 4 Finally, e assume strict monotonicity of the conitional virtual values, 1 F (vj) more precisely: for all, ^v(v; ) := v is increasing in v ith strictly f(vj) positive erivative. 3 Incentive Compatible Policies In this section, e characterize incentive compatible allocation policies. Without loss of generality, e restrict attention to irect mechanisms here every agent, upon arrival, reports a type (; v) an here the mechanism then speci es an allocation an a payment. The schemes e evelop also have an obvious an immeiate interpretation as inirect mechanisms, here the esigner sets a time- an capacity-epenent menu of per-unit prices, one for each eight eman. An allocation policy is calle eterministic an Markovian if, at any perio t = 1; :::; T an for any possible type of agent arriving at t, it uses a non-ranom allocation rule that only epens on the arrival time t, on the eclare type of the arriving agent, an on the still available capacity at perio t, enote by c. The restriction to these policies is innocuous as shon in Section 4. We can assume ithout loss of generality that a eterministic Markovian allocation policy for time t ith remaining capacity c has the form c t : [; +1) 2! f1; g here 1 () means that the reporte capacity eman is satis e (not satis e). Inee, it never makes sense to allocate an insu cient quantity < < because iniviually rational agents are not illing to pay for this. On the other han, allocating more capacity than the reporte eman is useless as ell: Such allocations o not further increase agents utility hile they may ecrease continuation values for the esigner. Let q c t : [; +1) 2! R be the associate payment rule. Proposition 1 A eterministic, Markovian allocation policy f c tg t;c is implementable if an only if for every t an every c it hols that: 1. 8 (; v); v v; c t(; v) = 1 ) c t(; v ) = The function p c t() is non-ecreasing in ; here p c t() = inffv = c t(; v) = 1g 5 Proof. See Appenix. The threshol property emboie in conition 1 of the above Proposition is stanar, an is a natural feature of elfare maximizing rules uner complete information. When there is incomplete information in the value imension, this conition imposes 4 Alternatively, e can assume that each agent observes the entire history of the previous allocations. 5 We set p c t() = 1 if the set fv= c t(; v) = 1g is empty. 4

5 limitations on the payments that can be extracte in equilibrium. Conition 2 is ne: it re ects the limitations impose in our moel by the incomplete information in the eight imension. 4 Dynamic Revenue Maximization In this section, e rst emonstrate ho the ynamic revenue maximization problem may be solve if is observable. Hence e rst assume that there is incomplete information only about v. We then ientify a set of conitions ensuring that the corresponing optimal policy is implementable even if is not observable. The logic of the erivation for solving the revenue maximization problem is somehat involve, an e no etail it belo: 1. Without loss of generality, e can restrict attention to Markovian policies. The optimality of Markovian, possibly ranomize, policies is stanar for all moels here, as is the case here, the per-perio rears an transition probabilities are history-inepenent - see for example Theorem in Puterman [19] hich shos that, for any history-epenent policy, there is a Markovian, possibly ranomize, policy ith the same payo. 2. If there is incomplete information about v; but complete information about the eight requirement, then Markovian, eterministic an implementable policies are characterize for each t an c by the threshol property of Conition 1 in Proposition Naturally, in the given revenue maximization problem ith complete information about e nee to restrict attention to interim iniviually-rational policies here no agent ever pays more than the utility obtaine from her actual capacity allocation. It is easy to see that, for any Markov, eterministic an implementable allocation policy c t, the maximal, iniviually-rational payment function hich supports it is given by q c t (; v) = p c t () if c t(; v) = 1 if c t(; v) = here p c t() = inffv = c t(; v) = 1g as e ne in the above section. Otherise, the esigner pays some positive subsiy to the agent, an this cannot be revenuemaximizing. 4. At each perio t, an for each remaining capacity c; the esigner s problem uner complete information about is equivalent to a simpler, one-imensional static problem here a knon capacity nees to be allocate to the arriving agent, an here the seller has a salvage value for each remaining capacity: the 5

6 salvage values in the static problem correspon to the continuation values in the ynamic version. Analogously to the analysis of Myerson [16], each static revenue-maximization problem has a monotone (in the sense of Conition 1 in Proposition 1), non-ranomize solution as long as, for any eight ; the agent s 1 F (vj) f(vj) conitional virtual valuation v is increasing in v. Inee, the expecte revenue R(c; T + 1 t) if per-unit prices are set at p c t() in perio t T ith remaining capacity c an if the optimal Markovian policy is folloe from time t + 1 onars can be ritten as: R(c; T + 1 t) = + p c t() (1 F (p c t()j)) f () [(1 F (p c t()j))r (c ; T t) + F (p c t()j)r (c; T t)] f () ; here f enotes the marginal ensity in, an here R enotes optimal revenues ith R (c; ) = for all c. The rst-orer conitions for the revenuemaximizing unit prices p c t() are given by: p c t() 1 F (p c t()j) f(p c t()j) = R (c; T t) R (c ; T t): 5. By backar inuction, an by the above reasoning, the seller has a Markov, nonranomize optimal policy in the ynamic problem ith complete information about. Note also that, by a simple uplication argument, R (c; T +1 t) must be monotone non-ecreasing in c: Points 1, 4 an 5 above imply that the restriction to the eterministic an Markovian allocation problems is ithout loss of generality. If the above solution satis es the incentive compatibility constraint in the eight imension, i.e. if p c t() happens to be monotone as require by Conition 2 of Proposition 1, then the associate allocation here c t(; v) = 1 if an only if v p c t() is also implementable in the original problem ith incomplete information about both v an. It then constitutes the revenue maximizing scheme that e are after. The next example illustrates that Conition 2 of Proposition 1 can be bining. Example 1 Assume that T = 1. The istribution of the agents types is given by the folloing stochastic process. First, the eight request is realize accoring to an exponential istribution ith parameter. Next, the per-unit value of the agent is sample from the folloing istribution here > an 2 (; c). F (vj) = ( 1 e v if > 1 e v if 6

7 In this case, for an observable eight request, the seller charges the take-it-or-leaveit o er of 1 ( 1 ) per unit if the eight request is smaller (larger) than or equal to : This implies that ( p c if > t() = if. an therefore, p c t() is not monotone. We next procee to ientify conitions on the istribution of types ensuring the monotonicity of p c t(). 4.1 The Hazar Rate Stochastic Orering A key conition guaranteeing implementability is a stochastic orering of the conitional istributions of per-unit values: the conitional istribution given a higher eight shoul be (eakly) statistically higher in the hazar rate orer than the conitional istribution given a loer eight. Theorem 1 For each c; t; an let p c t() enote the solution to the revenue maximizing problem uner complete information about, etermine recursively by the Bellman equation p c t() Assume that the folloing conitions hol: 1 F (p c t()j) = R (c; T t) R (c ; T t): (1) f(p c t()j) 1. For any ; the conitional hazar rate f(vj) 1 F (vj) is non-ecreasing in v6 : 2. For any, an for any v; f(vj) 1 F (vj) f(vj ) 1 F (vj ). Then, p c t() is non-ecreasing in ; an, consequently, the unerlying allocation here c t(; v) = 1 if an only if v p c t() is implementable. In particular, equations (1) characterize the revenue maximizing scheme uner incomplete information about both values an eights. Proof. See Appenix. An important special case for hich the conitions of the above Theorem hol is the one here the istribution of per-unit values is inepenent of the istribution of eights, an has an increasing hazar rate. 6 Note that this conition alreay implies the neee monotonicity in v of the conitional virtual value for all : 7

8 4.2 Concavity of Expecte Revenue in Capacity A major result for the case here capacity comes in iscrete units, an here all eights are equal is that expecte revenue is concave in capacity (see Gallego an van Ryzin [7] for a continuous time frameork ith Poisson arrivals an Bitran an Monschein [3] for a iscrete time setting). This is a very intuitive property since it says that aitional capacity is more valuable to the esigner hen capacity itself is scarce. Due to the more complicate combinatorial nature of the knapsack problem ith heterogenous eights, concavity nee not generally hol (see Papastavrou, Rajagopalan an Kleyegt [18] for examples here concavity of expecte elfare in the frameork ith complete information fails). Our main result in this subsection ienti es a conition on the istribution of types that, together ith concavity of the expecte revenue in the remaining capacity, ensures that, for each t an c, p c t() is increasing, hence renering the unerlying istribution implementable. Afterars e provie conitions on the moels primitives that are su cient for the concavity of the expecte revenue. Theorem 2 Assume that 1. The expecte revenue R (c; T + 1 t) is a concave function of c for all times t: 2. For any ; v 1 F (vj) f(vj) v 1 F ( v j ). f( v j ) For each c; t; an let p c t() enote the solution to the revenue maximizing problem uner complete information about ; etermine recursively by equations (1). Then p c t() is non-ecreasing in ; an hence the unerlying allocation here c t(; v) = 1 if an only if v p c t() is implementable. In particular, equation (1) characterizes the revenue maximizing scheme uner incomplete information about both values an eights. Proof. See Appenix. Remark 1 The su cient conitions for implementability use in Theorem 1 are, taken together, stronger than Conition 2 in Theorem 2. To see this, assume that, for any ; the conitional hazar rate is increasing in v; an that for any an for all v, v f(vj) f(vj ) 1 F (vj) 1 F (vj ) 1 F (vj) f(vj) f(vj) 1 F (vj). This yiels: v 1 F ( v j) f( v v 1 F ( v j ) j) f( v j ) here the rst inequality follos by the monotonicity of the hazar rate, an the secon by the stochastic orer assumption. Theorem 2 an in particular its Conition 2 ill also be useful hen iscussing the implementability of the simple policy in Section 5.2. Note also that Conition 2 of are non- Theorem 2 can be formulate as requiring that the functions v ecreasing in. 8 1 F (vj ) f(vj )

9 Our next result ienti es conitions on the joint istribution F (; v) that imply concavity of expecte revenue ith respect to c for all perios, as require by the above Theorem. It is convenient to introuce the joint istribution of eight an total valuation u = v, hich e enote by G(; u) ith ensity g(; u). By means of a transformation of variables, the ensities f an g are relate by g(; v) = f(; v): In particular, marginal ensities in coincie, i.e. f () = f(; v) v = g(; u) u = g (): An increasing virtual value implies that the virtual total value is increasing in u ith strictly positive erivative for any given : ^u(u; ) := u 1 G(uj) g(uj) = v 1 F (vj) f(vj)= = ^v(v; ) We rite ^u 1 (^u; ) for the inverse of ^u(u; ) ith respect to u an e ne a istribution ^G(^u; ) by both ^G(^uj) := G(^u 1 (^u; )j) for all an ^g () := g (). On the level of ^v, this correspons to ^F (^vj) = F (^v 1 (^v; )j) an ^f () = f (). Theorem 3 Assume that the conitional istribution ^G(j^u) is concave in for all ^u; that both ^g(j^u) an ^g(j^u) are boune, an that the total virtual value ^u has a nite mean. Then, in the revenue maximization problem here the esigner has complete information about ; the expecte revenue R (c; T + 1 t) is concave as a function of c for all times t: Proof. See Appenix. Example 2 A simple example here the conitions of Theorem 2 are satis e is obtaine by assuming that G(u; ) is such that u an are inepenent, u u(u) 1 G g u(u) is i erentiable ith strictly positive erivative, an G is concave 7. Conition 1 in that Theorem is satis e by Theorem 3, hile Conition 2 is satis e since by inepenence v ^v( ; ) = ^u(v; ) = ^u(v; ) = ^v(v; ) an by assumption. 5 Simple, Asymptotically Optimal Policies The optimal policy characterize above seems too sensitive to be use in practice since it requires price ajustments in every perio, an for any quantity request. Our main result in the present section suggests that hile exploiting epenency beteen an v - if there is any - may be important for revenue maximization, carefully chosen ynamics are super uous if both capacity an time go to in nity. As above, e start by focusing on the case of observable eights. We then sho that the su cient conitions 7 We also assume that the other very mil technical conitions of Theorem 3 are satis e. 9

10 ienti e in Theorem 1 are also applicable here, ensuring the implementability of the obtaine policy. Instea of solving the stochastic problem, e rst solve a simpler, suitably chosen eterministic maximization problem. The revenue obtaine in the solution to that problem provies an upper boun for the optimal expecte revenue of the stochastic problem, an suggests the use of per-unit prices that epen on, but that are constant in time. We next sho that the erive policy is asymptotically optimal also in the original stochastic problem here both capacity an time go to in nity: the ratio of expecte revenue from folloing the consiere policy over expecte revenue from the optimal Markovian policy converges to one. Moreover, there are various ays to quantify this ratio for moerately large capacities an time horizons. The basic logic hence follos a suggestion mae by Gallego an van Ryzin [7]. Hoever, our knapsack problem ith a general istribution F (; v) is substantially more complex than the moel tackle in their paper. Let us rst recall some assumptions, an introuce further notation. The marginal ensity f () an the conitional ensities f(vj) pin on the istribution of (inepenent) arriving types ( t ; v t ) T. Given, the emane per-unit price p an the probability of a request being accepte are relate by (p) = 1 F (pj). Let p () be the inverse of, an note that this is ell e ne on (; 1]. Because of monotonicity of conitional virtual values, the instantaneous (expecte) per-unit revenue functions r () := p () are strictly concave, an each one attains a unique interior maximum. Inee, p () = F (j) 1 (1 ) an hence r () = p 1 () f(p ()j) = 1 F (p ()j) p () f(p ()j) 2 r () ^v (p 1 (); ) f(p ()j) < : = ^v(p (); ); Consequently, r is strictly concave, strictly increasing up to the ; that satis es ^v(p ( ; ); ) = an strictly ecreasing from there on. 5.1 The Deterministic Problem We no formulate an auxiliary eterministic problem. Let Cap : (; 1)! (; 1) ; 7! Cap() be a measurable function. Consier the problem: Z! 1 TX max max r ( Cap() ( t ) t ) f () ; (2) ;:::;T subject to TX t f () Cap() a.s. an In ors, e analyze a problem here: 1 Cap() C: (3)

11 1. The capacity C nees to be ivie into capacities Cap(), one for each : 2. In each - subproblem, a eterministic quantity request of f () arrives in each perio, an t etermines a share (not a probability!) of this request that is accepte an sol at per-unit price p ( t ). 3. In each sub-problem, the allocate capacity over time cannot excee Cap(), an total allocate capacity in all sub-problems R 1 Cap(), cannot excee C. 4. The esigner s goal is to maximize total revenue. We call the revenue at the solution R (C; T ). As r is strictly concave an increasing up to ;, it is straightforar to verify that, given a choice Cap(); the solution to the - subproblem,! TX max r ( ( t ) t ) TX f () such that t f () Cap() ;:::;T is given by: t ; := ( ; Cap() T f () if ; else Cap() T f () Accoringly, the revenue in the -subproblem is r ( ; )T f (). (4) Proposition 2 The solution to the eterministic problem given by (2) an (3) is characterize by : 1. ^v(p ( ; ); ) = (C; T ) = const 2. t = ; = Cap() 3. T f (), R 1 Cap() = min(c; T R 1 ; f () ) Proof. See Appenix. To get an intuition for the above result, observe that the marginal increase of the optimal revenue for the -subproblem from marginally increasing Cap() is: Cap() r T f () = ^v(p ( ; ); ) if ; > Cap() T f () ; an else. Proposition 2 says that, optimally, the capacity shoul be split in such a ay that the marginal revenue from increasing Cap() is the same for all. Actually solving the problem amounts to the simple static exercise of etermining the constant (C; T ) in accorance ith the integral feasibility constraint. 11

12 The above construction is justi e by the folloing result, shoing that the optimal revenue in the eterministic problem bouns from above the optimal revenue in our original stochastic problem. Since e assume here that eights are observable, a Markovian policy for the original stochastic problem is characterize by the acceptance probabilities t t [c t ] contingent on current time t, remaining capacity c t an eight request t. Expecte revenue from policy at the beginning of perio t (i.e. hen there are (T t + 1) perios left) ith remaining capacity c t is given by: R (c t ; T t + 1) = E s=t s p s ( s s [c s ]) I fvsp s ( s s [cs])g s:t: TX s=t s I fvsp s ( s s [cs])g c t : Here, the constraint must hol almost surely hen folloing. As before, e rite R (c t ; T t+1) for the optimal revenue, i.e. the supremum of expecte revenues taken over all feasible Markovian policies. Theorem 4 For any capacity C an ealine T, it hols that R (C; T ) R (C; T ). Proof. See Appenix. 5.2 A Simple Policy for The Stochastic Problem Theorem 4 above suggests that a -contingent yet time-inepenent pricing policy may be able to yiel close to optimal revenues in the stochastic problem. To construct such a Markovian time-inepenent policy for the stochastic problem, T I ; e procee as follos: 1. Given C an T, solve the eterministic problem to obtain (C; T ), ; an thus p ; := p ( ; ) = ^v 1 ((C; T ); ). 2. In the stochastic problem charge these eight-contingent prices p ; for the entire time horizon, provie that the quantity request oes not excee the remaining capacity. Else, charge a price equal to +1 (i.e., reject the request). An important observation is that, uner the conitions of Theorem 1, the time inepenent policy T I e ne above is implementable also for the case that interests us, here eights are not observable. This follos immeiately by recalling that the eight-contingent prices p ; satisfy the equation ^v(p ( ; ); ) = (C; T ): Inee, uner the conitions of Theorem 1, the solution to this equation is monotonic in ; an hence p ; is also monotonic in ; as require for implementability. Moreover, 12

13 implementability is even satis e uner the strictly eaker Conition 2 of Theorem 2, since setting all virtual valuation threshols equal to a constant is like setting them optimally for linear an hence concave salvage values. We no etermine ho ell the time-inepenent policy constructe above performs compare to the optimal Markovian policy. We o this by comparing its expecte revenue, R T I (C; T ); ith the optimal revenue in the eterministic problem, R (C; T ); hich, as e kno by Theorem 4, provies an upper boun for the optimal revenue in the stochastic problem, R (C; T ). Theorem 5 1. For any joint istribution of values an eights, R T I (C; T ) lim C;T!1; C T =const R (C; T ) = 1 2. Assume that an v are inepenent. Then, R T I (C; T ) R (C; T ) 1 In particular, lim min(c;t )!1 R T I (C;T ) R (C;T ) = 1 p E[2 ]=E[] 2 p min(c; E[]T ) Proof. See Appenix. We have chosen to focus on these to general limit results. Various others coul be proven by similar techniques at the expense of slightly more technical e ort an possibly some further assumptions on F. As e inicate in the introuction, an interesting remark is that, since the policy T I is stationary, it oes not generate incentives to postpone arrivals even in a more complex moel here buyers are patient an can choose their arrival time. Remark 2 In a complete information knapsack moel, Lin, Lu an Yao [14] stuy policies hich start by accepting only high value requests, an then sitch-over to accepting also loer values as time goes by. They establish asymptotic optimality of such policies (ith carefully chosen sitch-over times) as available capacity an time go to in nity. In other ors, their prices are time-epenent but o not conition on the eight request. It is easy to sho that, in our incomplete information moel such policies are, in general, suboptimal. Consier rst a one-perio example here the seller has capacity 2, an here the arriving agent has either a eight request of 1 or 2 (equally likely). If the eight request is 1(2), the agent s per-unit value istributes uniformly beteen an 1 (beteen 1 an 2). The optimal mechanism in this case is as follos: if the buyer requests one unit, the seller sells it for a price of.5, an if the buyer requests to units, the seller sells each unit at a price of 1. Note that this policy is implementable since the requeste per-unit price is monotonically increasing in the eight request. The expecte revenue is 9/8. If, hoever, the seller is force to sell 13! :

14 all units at the same per-unit price ithout conitioning on the eight request, he ill charge the price of 1 for each unit, yieling an expecte revenue of 1, an thus loose 1/8 versus the optimal policy. Replicate no this problem so that there are T perios an capacity C=2T. Then, the expecte revenue from the optimal mechanism is 9/8T, hile the expecte revenue from the constraine mechanism is only T. Obviously, the constraine mechanism is not asymptotically optimal. 6 Appenix Proof of Proposition 1. an e ne for any t; c: I) =) So assume that conitions 1 an 2 are satis e q c t (; v) = ( p c t () if c t(; v) = 1 if c t(; v) = Consier then an arrival of type (; v) in perio t ith remaining capacity c: There are to cases: a) c t(; v) = 1: In particular, v p c t(). Then, truth-telling yiels utility (v p c t()). Assume that the agent reports instea ( b; bv): If c t( b; bv) = ; then the agent s utility is zero an the eviation is not pro table. Assume then that c t( b; bv) = 1: By the form of the utility function, a report of b < is never pro table. But, for b, the agent s utility is v bp c t( b) (v p c t()), here e use conition 2. Therefore, such a eviation is also not pro table. b) c t(; v) = : In particular, v p c t(). Truth-telling yiels here utility of zero. Assume that the agent reports instea ( b; bv): If c t( b; bv) = ; then the agent s utility remains zero, an the eviation is not pro table. Assume then that c t( b; bv) = 1: By the form of the utility function, a report of b < is never pro table. Thus, consier the case here b. In this case, the agent s utility is v bp c t( b) (v p c t()) ; here e use conition 2. Therefore, such a eviation is also not pro table. II) (= Consier no an implementable, eterministic an Markovian allocation policy f c tg t;c. Assume rst, by contraiction, that conition 1 in the statement of the Proposition is not satis e. Then, there exist (; v) an (; v ) such that v > v, c t(; v) = 1 an c t(; v ) = : We obtain the chain of inequalities v qt c (; v) > v qt c (; v) qt c (; v ) here the secon inequality follos by incentive compatibility for type (; v). This shos that a eviation to a report (; v) is pro table for type (; v ); a contraiction to implementability. Therefore, conition 1 must hol. In particular, note that for any to types ho have the same eight request, (; v) an (; v ), if both are accepte, i.e. c t(; v) = c t(; v ) = 1; the payment must be the same (otherise the type hich nees to make the higher payment oul eviate an report the other type). Denote this payment by rt(). c Note also that any to types (; v) an ( ; v ) such that c t(; v) = c t( ; v ) = must also make the same 14

15 payment (otherise the type that nees to make the higher payment oul eviate an report the other type) an enote this payment by s. Assume no, by contraiction, that conition 2 oes not hol. Then there exist an such that > but p c t( ) < p c t(): In particular, p c t( ) < 1; an therefore p c t( ) < 1: Assume rst that p c t() < 1. We have p c t( ) rt( c ) = p c t() rt() c = s because, by incentive compatibility, both types (; p c t()) an ( ; p c t( )) must be ini erent beteen getting their request an not getting it. Since by assumption p c t( ) < p c t(); e obtain that rt( c ) < rt(): c Consier no a type (; v) for hich v > p c t(). By reporting truthfully, this type gets utility v rt(); c hile by eviating to ( ; v) he gets utility v rt( c ) > v rt(); c a contraiction to incentive compatibility. Assume no that p c t() is in nite, an therefore p c t() is in nite. Consier a type ( ; v) here v > p c t( ): The utility of this type is v rt( c ) > p c t( ) rt( c ) = s. In particular, rt( c ) must be nite. By reporting truthfully, a type (; v) gets utility s, hile by eviating to a report of ( ; v) he gets v rt( c ): For v large enough, e obtain v rt( c ) > s; a contraiction to implementability. Thus, conition 2 must hol an, in particular, the payment rt() c is monotonic in : Proof of Theorem 1. Let <. We nee to sho that p c t() p c t( ) : If p c t() p c t( ) the result is clear. Assume then that p c t() > p c t( ): We obtain the folloing chain of inequalities: 1 F (p c t ()j) f(p c t()j) 1 F (p c t ()j) f(p c t()j) 1 F (p c t ( )j) f(p c t( )j) 1 F (p c t ( )j ) f(p c t( )j ) 1 F (p c t( )j ) f(p c t( )j ) 1 F (p c t( )j ) f(p c t( )j ) ; here the secon inequality follos by the monotonicity of the hazar rate, an the thir by the hazar rate orering conition. Since R (c ; T t) is monotonically ecreasing in, e obtain that p c t() 1 F (p c t()j) p c f(p t( 1 F (p c ) t( )j ), c t()j) f(p c t( )j ) 1 F (p p c t() p c t( c ) t ()j) 1 F (p c t ( )j ) f(p c t()j) f(p c t( )j ) here the last inequality follos by the erivation above. Hence p c t() p c t( ) as esire. 15

16 Proof of Theorem 2. For any concave function, an for any x < y < z in its omain, the ell knon "Three Chor Lemma" asserts that (y) y (x) x (z) z (x) x (z) z (y) y Consier then < an let x = c < y = c < z = c: For the case of a concave revenue, the Lemma yiels then: R (c ; T t) R (c ; T t) We obtain in particular R (c; T t) R (c ; T t) R (c; T t) R (c ; T t) : hich yiels p c t( ) 1 F (p c t( )j ) f(p c t( )j ) R (c; T t) R (c ; T t) = R (c; T t) R (c ; T t) = p c t() 1 F (p c t()j) ; f(p c t()j) p c t( ) 1 F (p c t( )j ) f(p c t( )j ) p c t() 1 F (p c t()j) f(p c t()j) pc t() 1 F ( p c t()j ) f( p c t()j ) here the last inequality follos by the conition in the statement of the Theorem. Since virtual values are increasing, this yiels p c t( ) p c t(), p c t( ) p c t() as esire. For the proof of Theorem 3, e rst nee a Lemma on maximization of expecte elfare uner complete information. The result appears (ithout proof) in Papastavrou, Rajagopalan an Kleyegt [18]. Lemma 1. Assume that the total value u has nite mean, an that both g(ju) an g(ju) are boune an continuous. Consier the allocation policy that maximizes expecte elfare uner complete information (i..e, upon arrival the agent s type is reveale to the esigner). If G(ju) is concave in for all u; then the optimal expecte elfare, enote Ut c is tice continuously i erentiable an concave in the remaining capacity c for all perios t T. Proof. Note that, for notational convenience throughout this proof, e inex optimal expecte elfare by the current time t an not by perios remaining to ealine. By stanar arguments, the optimal policy for this unconstraine ynamic optimization problem is eterministic an Markovian, an Ut c is non-ecreasing in remaining capacity c by a simple strategy uplication argument. Moreover, the optimal policy can be characterize by eight threshols t(u) c c : If c remains at time t an a 16

17 request hose acceptance oul generate value u arrives, then it is accepte if an only if c t(u). If U c t+1 u, then the eight threshol must satisfy the ini erence conition u = U c t+1 U c c t (u) t+1 : (5) Otherise, e have t(u) c = c. We no prove the Lemma by backar inuction. At time t = T, i.e. in the ealine perio, it hols that U c T = G(cju)u g u (u) u: This is concave in c because all G(cju) are concave by assumption, because u g u (u) is positive, an because the istribution of u has a nite mean. Since both g(ju) an g(ju) are boune an continuous, U c T is also tice continuously i erentiable. Suppose no that the Lemma has been proven on to time t + 1. The optimal expecte elfare at t provie that capacity c remains may be ritten as: " U c t = ug( c t(u)ju) + Z c t (u) U c t+1 g(ju) + (1 G( c t(u)ju))u c t+1 g u (u) u: (6) We procee to sho concavity ith respect to c of the term in brackets, for all u. This in turn implies concavity of Ut c an hence, ith a short aitional argument for i erentiability, is su cient to conclue the inuction step. We istinguish the cases u > Ut+1 c for hich the ini erence conition (5) oes not hol, an u Ut+1 c for hich it oes. For both cases, e emonstrate that the secon erivative (one-sie if necessary) of the bracket term ith respect to c is non-positive, an thus establish global concavity. Case 1: u > Ut+1. c The bracket term becomes ug(cju) + R c U c t+1 g(ju) + (1 G(cju))Ut+1. c By continuity of Ut+1, c this representation also hols in a small interval aroun c. We n c ug(cju) + = ug(cju) + U c t+1 g(ju) + (1 G(cju))U c t+1 c U t+1 c g(ju) + Ut+1 g(cju) g(cju) U c t+1 + (1 G(cju)) c U c t+1 = (u U c t+1)g(cju) + c U t+1 c g(ju) + (1 G(cju)) c U t+1 c 17

18 an 2 ug(cju) + c 2 U c t+1 g(ju) + (1 G(cju))U c t+1 = (u U c t+1)g (cju) g(cju) c U c t U t+1 2 c U c 2 t+1 g(ju) = g(cju) g(cju) c U c t+1 + (1 G(cju)) 2 c 2 U c t+1: (7) The last term is non-positive by the concavity of U c t+1, the rst term is non-positive because u > U c t+1 an because G(cju) has a non-increasing ensity by assumption. In aition, g(cju) c U c t+1 is non-negative, an hence (7) is boune from above by 2 c U c 2 t+1 g(ju) + g(cju) U t+1 = But R c 2 U c c 2 t+1 g(ju) may be boune from above by g(cju) R c of the ecreasing ensity an because 2 U c c 2 t+1. Thus, 2 ug(cju) + U c c 2 t+1 g(ju) + (1 G(cju))Ut+1 c 2 g(cju) c U c 2 t+1 + U t+1 2 = g(cju) U c 2 t+1 + U t+1 = = c U t+1 c : 2 U c c 2 c U t+1 c c U t+1 c t+1 because = : (8) Case 2: u U c t+1. Here u = U c t+1 U c c t (u) t+1. Consequently, the bracket term in (6) becomes U c t+1 U c c t (u) t+1 G( c t(u)ju) + Z c t (u) U c t+1 g(ju) : (9) Before computing its rst an secon erivatives, e i erentiate the ientity u = Ut+1 c U c c t (u) t+1 to obtain an expression for c c t(u) (erivative from the right if u = U c t+1): = c U c t+1 U t+1 =c c t (u) 1 c c t(u) : Since inee U t+1 > in our setup ith strictly positive ensities, this implies c c t(u) = U t+1 U c =c c t (u) c t+1 : (1) U t+1 =c By concavity of U c t+1, its erivative is non-increasing an hence the ientity (1) yiels c t (u) 18

19 in particular c c t(u). We no compute the erivatives of (9): " Ut+1 c U c c t (u) t+1 G( c c t(u)ju) + = c U c t+1 U t+1 =c c t (u) Z c t (u) U c 1 + U c c t (u) t+1 g( c t(u)ju) c c t(u) + (1) = c U c t+1 c U t+1 c G(t(u)ju) c + = c U c t+1(1 G( c t(u)ju)) + Thus, " 2 U c c 2 t+1 U c c t (u) t+1 G(t(u)ju) c + Z c t (u) Z c t (u) t+1 g(ju) c c t(u) Z c t (u) Z c t (u) U c G( c t(u)ju) U c c t (u) t+1 g( c t(u)ju) c c t(u) c U t+1 c g(ju) c U t+1 c g(ju) c U t+1 c g(ju) : t+1 g(ju) = 2 c U t+1(1 c G( c t(u)ju)) 2 c U t+1 c g(t(u)ju) c c c t(u) + U t+1 =c c t (u) g(c t(u)ju) Z c t (u) 2 c c t(u) + c U c 2 t+1 g(ju) g(t(u)ju) c Z c c t(u) U t+1 c =c c t (u) c U t+1 c t (u) 2 + U c 2 t+1 g(ju) : For the nal inequality e use concavity of Ut+1, c as ell as 2 t+1 = 2 t+1. Noting that (1) implies that c c t(u) 1 an once more using concavity of Ut+1, c e may boun the rst term from above. Since g(ju) is non-increasing in, e can also boun the secon term to obtain: " 2 Z c U c c 2 t+1 U c c t (u) t+1 G(t(u)ju) c t (u) + Ut+1 c g(ju) (11) g( c t(u)ju) U t+1 =c c t (u) Z c c U t+1 c t (u) + 2 c 2 U c 2 U c t+1! 2 U c = : Taken together, (8) an (11) establish concavity of the integran in (6) ith respect to c. This implies that Ut c is concave. Having a secon look at the computations just one reveals that the integran in (6) has a kink in the secon erivative at u = Ut+1. c Hoever, this event has measure zero for any given c, so that e also get that Ut c is tice continuously i erentiable. This completes the inuction step. Proof of Theorem 3. The main iea of the proof is to translate the problem of setting revenue-maximizing prices hen is observable into the problem of maximizing 19

20 elfare ith respect to virtual values (rather than the values themselves), an then to use Lemma 1. To begin ith, note that there is a ual ay to escribe the policy that maximizes expecte elfare uner complete information. In the proof of Lemma 1, e characterize it by optimal eight threshols t(u). c Alternatively, given any requeste quantity, (not greater than the remaining c) e may set a valuation per unit threshol vt c (). Requests above this valuation are accepte, those belo are not. Optimal such threshols are characterize by the Bellman-type conition: v c t () = U c t+1 U c t+1 : (12) Thus, one ay of riting the optimal expecte elfare uner complete information is: U c t = + vf(vj) v f () vt c() (1 F (v c t ()j))ut+1 c + F (vt c ()j)ut+1 c f () : (13) In contrast, the optimal expecte revenue ith complete information about but incomplete information about v satis es: R (c; T + 1 t) = + p c t() (1 F (p c t()j)) f () (14) [(1 F (p c t()j))r (c ; T t) + F (p c t()j)r (c; T t)] f () ; here p c t() are the per-unit prices from (1). We rephrase this in terms of ^F, hose e nition require monotonicity of virtual values. Setting ^v c t () := ^v(p c t(); ) e have on the one han: F (p c t()j) = ^F (^v c t ()j): On the other han: p c t() (1 F (p c t()j)) = = = p c t () [v f(vj) (1 F (vj))] v ^v(v; ) ^f(^v(v; )j) ^v(v; ) v p c t () v ^v c t () ^v ^f(^vj) ^v: Plugging this an the ientities for the marginal ensities in into (14), e obtain: R (c; T + 1 t) = ^v ^f(^vj) ^v ^f () ^v t () 1 h + (1 ^F (^v c t ()j))r (c ; T t) + ^F (^v t c ()j)r (c; T t)i ^f () : 2

21 Comparing this ith (13), it follos that maximizing expecte revenue hen is observable is equivalent to maximizing expecte elfare ith respect to the istribution of eight an conitional virtual valuation (note the ientical zero bounary values at T + 1). Invoking Lemma 1 applie to ^G, e see that R (c; T + 1 t) is concave ith respect to c for all t (note that the fact that the support of virtual valuations contains also negative numbers oes not matter for the argument of Lemma 1). Proof of Proposition 2. The Proposition is an immeiate consequence of the characterization (4) of optimal solutions for the -subproblems given Cap(), an of a straightforar variational argument ensuring that marginal revenues from marginal increase of Cap() must be constant almost surely in. Proof of Theorem 4. We nee to istinguish to cases: Case 1: Assume that C > T R 1 ; f (). In this case, (C; T ) = an R (C; T ) = T R 1 r ( ; )f (). We also kno that R (C; T ) R (+1; T ); here R (+1; T ) enotes the optimal expecte revenue from a stochastic problem ithout any capacity constraint. But, for such a problem, the optimal Markovian policy maximizes at each perio the instantaneous expecte revenue upon observing t, t r t (): That is, the optimal policy sets t [+1] = ;. Thus, R (C; T ) R (+1; T ) = T t r ( ; ) f () = R (C; T ): Case 2: Assume no that C T R 1 ; f (). For, consier the unconstraine maximization problem max Cap() r Cap() T f () The Euler-Lagrange equation is T f () + r Cap() T f () C Cap() =. Hence, if e rite R (C; T; ) for the optimal value of the above problem, an if e let = (C; T ) here (C; T ) is the constant from Proposition 2, then the solution equals the one of the constraine eterministic problem. In particular R 1 Cap() = C, an R (C; T; (C; T )) = R (C; T ). Recall that for the stochastic problem, an for any Markovian policy e have an e ne R (C; T ) = E t p t ( t t [c t ]) I fvtp t( t t [c t])g R (C; T; (C; T )) = R (C; T ) + (C; T ) C E t I fvtp t( t t [c t])g Since for any policy that is amissible in the original problem, it hols that TX t I fvtp t( t t [c t])g C a:s:; 21 ;! :

22 e have R (C; T ) R (C; T; (C; T )). We ill sho belo that, for arbitrary (hich satis es the capacity constraint or not), it hols that: R (C; T; (C; T )) R (C; T; (C; T )): (15) This yiels for any that is amissible in the original problem: R (C; T ) R (C; T; (C; T )) R (C; T; (C; T )) = R (C; T ): Taking the supremum over conclues then the proof for the secon case. It remains to prove (15). The argument uses the ltration ff t g T of - algebras containing information prior to time t ( in particular the value of c t ) an in aition the currently observe t. R (C; T; (C; T )) = E t (p t ( t t [c t ]) (C; T )) I fvtp t( t t [c t])g = E E h t (p t ( t t [c t ]) (C; T )) I fvtp t( t t = E t (p t ( t t [c t ]) (C; T )) E hi fvtp t( t = E t (r t ( t t [c t ]) (C; T ) t t [c t ]) t + (C; T ) C i [c t])gjf t i [c t])gjf t + (C; T ) C + (C; T ) C + (C; T ) C E t r t ( t; ) (C; T ) t; + (C; T ) C = E (t) T t r t ( t; ) (C; T ) t; + (C; T ) C = T (r ( ; ) (C; T ) ; ) f () + (C; T ) C = R (C; T; (C; T )): For the inequality, e have use that ; maximizes r () (C; T ). For the proof of Theorem 5, e rst nee a Lemma: Lemma 2 Let R T I (C; T ) be the revenue obtaine form the stationary policy T I : Let ( e t ; ev t ) T be an inepenent copy of the process ( t ; v t ) T. It hols that: 1. R T I (C; T ) = E (t) T r t ( t; ) t 1 P " t 1 X s=1 e s I fevsp es; g > C t! (16) : 22

23 2. R T I (C; T ) R (C; T ) 1 P T R 1 r ( ; ) min 1; (17) (t 1) 2 I ((T t+1) ) 2 f(t t+1) g + I f>(t t+1) g f () T R 1 r ( ; )f () here := min(c;t R 1 ; f () ), an here 2 T := E[2 I fvp g] R 2 ; = 1 2 ; f () 2. Proof. 1. R T I (C; T ) may be ritten as: R T I (C; T ) = E (t;vt) T = E (t) T r t ( t; ) t E (t;vt) T p t; t I fvtp t ; g I f P t 1 s=1 s I fvsp s; g C tg p t; t I fvtp t ; g I f P t 1 s=1 s I fvsp s; g > C tg In orer to simplify the secon term, e use the fact that v t an ( s ; v s ) t s=1 1 are inepenent conitional on t : p t; t I fvtp t ; g I P f t 1 s=1 s I fvsp s; g > C tg h i = E (t;vt) T E p t; t I fvtp t ; g I P f t 1 s=1 s I fvsp s; g > C tg j t E (t;v t) T = E (t;vt) T = E (t;vt) T p t; t t; P = E (t) T r t ( t; ) t P This establishes equation (16). p t; t E I fvtp t ; gj t E hi f P t 1 s=1 s I fvsp s; g > C tg j t " t 1 X s=1 " t 1 e s I fevsp es; g > C t X e s I fevsp es; g > C t : s=1 2. Recall that R (C; T ) = T R 1 r ( ; )f (). Observe furthermore that ; epens on C an T only through the ratio Ce, here T Ce = min(c; T R 1 via E[I fvp ; g] = R 1 ; f () = Ce P " t 1 X s=1 = P e s I fevsp es; g > C t P " t 1 X s=1 = T. Observe rst that " Xt 1 e s I fevsp es; g > T t s=1 e s I fevsp es; g (t 1) > (T t + 1) t 23 : i ; f () ),

24 We trivially boun the last expression by 1 if (T t + 1) t, an otherise use Chebychev s inequality to euce " t 1 X P e s I fevsp es; g (t 1) > (T t + 1) t s=1 2 P 4 E Xt 1 s=1! 2 e s I fevsp es; g (t 1) > ((T t + 1) t ) 2 5 h Pt 1 s=1 e s I fevsp es; g (t 1) 2 i ((T t + 1) t ) 2 = 3 (t 1) 2 ((T t + 1) t ) 2 : As e are bouning a probability, e can replace this estimate by the trivial boun 1 again henever this is better, i.e. if t is smaller than but close to (T t + 1). Thus, E (t;vt) T TX p t; t I fvtp t ; g I f P t 1 s=1 s I fvsp s; g > C tg r ( ; ) (t 1) 2 min 1; I ((T t + 1) ) 2 f(t t+1) g + I f>(t t+1) g Finally, iviing by R (C; T ) yiels the esire estimate. Proof of Theorem The starting point is the estimate from (17). Note that r ( ; )f () is an integrable upper boun for r ( ; (t 1) 2 ) min 1; I ((T t + 1) ) 2 f(t t+1) g + I f>(t t+1) g f (): Moreover, for xe, for arbitrary 2 (; 1) an for t T e have < (1 eventually as T; C! 1, C = const. Moreover, T (t 1) 2 ((T t + 1) ) T 2! ; as T! 1: 2 ((1 )T ) 2 )T f () : The Dominate Convergence Theorem implies then that r ( ; (t 1) 2 ) min 1; I ((T t + 1) ) 2 f(t t+1) g + I f>(t t+1) g f ()! ; in the consiere limit, for arbitrary 2 (; 1) an for t T. Consequently, also the term that is subtracte in the estimate (17) converges to zero. 2. If an v are inepenent, all the ; for i erent coincie, as o the ;. Call them an, respectively. We have then "! TX R T I (C; T ) = p( )E min C; t I fvtp( )g 2 TX = p( )E 4 t I fvtp( )g 24 TX t I fvtp( )g C! + 3 5

25 We use no the folloing estimate for E[(X k) + ], here X is a ranom variable ith mean m an variance 2 an here k is a constant: p E[(X k) (k m) ] 2 (k m) : 2 Note that by inepenence E t I fvtp( )g V ar t I fvtp( )g If T E[] > C an hence if = R CP (C; T ) R (C; T ) p( ) = E[]T ; = T E[( I fvp( )g )2 ] E[] 2 ( ) 2 = T E[ 2 ] E[] 2 ( ) 2 C this yiels: T E[] q T E[ 2 ] 2 = R (C; T ) 1 If T E[] C an hence if =, then C E PT + E t I fvtp( )g C p 2. Thus, 2 R T I (C; T ) R (C; T ) p( ) p T E( 2 ) 2 h PT = R (C; T ) 1 p! E[2 ]=E[] 2 p : C i t I fvtp( )g, so that p! E[2 ]=E[] 2 p : E()T Hence, e can conclue that: R T I (C; T ) R (C; T ) 1 p E[2 ]=E[] 2 p min(c; T E[]))! : References [1] Albright S.C. (1974): "Optimal Sequential Assignments ith Ranom Arrival Times", Management Science 21 (1), [2] Armstrong, M. (1996): Multiprouct Nonlinear Pricing, Econometrica 64, [3] Bitran, G.R. an Monschein, S.V. (1997): "Perioic Pricing of Seasonal Proucts in Retailing", Management Science 43 (1),