An optimal mechanism charging for priority in a queue
aa r X i v : . [ ec on . T H ] F e b An optimal mechanism charging for priority ina queue
Moshe Haviv ∗ and Eyal Winter † February 18, 2020
Abstract
We derive a revenue-maximizing scheme that charges customerswho are homogeneous with respect to their waiting cost parameter fora random fee in order to become premium customers. This schemeincentivizes all customers to purchase priority, each at his/her drawnprice. We also design a revenue-maximizing scheme for the case wherecustomers are heterogeneous with respect to their waiting cost param-eter. Now lower cost parameter customers are encouraged to join thepremium class at a low price: Given that, those with high cost pa-rameter would be willing to pay even more for this privilege.
There are numerous examples in which service is granted to an order basednot on the standard first-come first-served (FCFS) regime, but rather onpriority levels held by the customers. For example, in an emergency roompatients who need life-saving care are treated before those who complainof a minor fever. Another example is early boarding granted to premiumcustomers by airlines. By paying a priority price these passengers can get ∗ Department of Statistics and Data Science, and the Federmann Center for the Studyof Rationality, the Hebrew University of Jerusalem † The Management School, the University of Lancaster and the Department of Eco-nomics and the Federmann Center for the Study of Rationality, The Hebrew Universityof Jerusalem Cµ -rule: ser-vice is prioritized based on an index each customer possesses. The value ofthe index is the product of the cost per unit of time in the queue (usuallydenoted by C ) and the inverse of the mean service time (denoted by µ ). See,e.g., [8], pp. 72–73. Here, as in the emergency room, those who suffer morefrom waiting get priority over those who suffer less. Similarly, since it issocially better that a long task waits for a short one, rather than the otherway around, short tasks have priority over long ones.The second example shows that firms or monopolies can generate profitby offering priority based on some charge. Customers who compete amongthemselves for priority might be willing to pay more for it. They might bemore willing to do so if they suffer more from waiting and wish to overtakeothers, or if they suffer from the idea of being overtaken by others, and wish toremove the threat. Note that no extra effort or cost is incurred by the profit-maximizer who administrates the priority system but nevertheless generatesmore profit to itself. Also note that if all decide to purchase priority, prioritywill in fact be granted to none (and payment will not be reimbursed).The above examples illustrate the case where two priority levels exist;however, one can administrate any finite or continuous number of levels,each of which comes with its own price.Another question is the implementation related to the administration ofa priority scheme. For example, in order to implement the Cµ -rule, a centralplanner needs to know the cost parameter of each of the arriving customers.Yet, asking customers for their cost parameter C comes with the risk thatcustomers will cheat and represent themselves as having a higher value for C than their actual one. It is shown in [13] and [2] that there exists a price menufor priority levels where purchasing the true priority level across customersis a Nash equilibrium profile. We omit here the exact details of the modelsdescribed there.We also like to note that by administrating a priority scheme one canregulate the arrival rate. Usually customers over-congest a queueing system.It is shown in [5] that if customers who decide to join a queue compete forpriority levels, so that as the more they pay, the higher their priority, they end2p joining the system at the socially optimal rate. The same phenomenonexists in the model presented in [13]. We also like to mention [9] where is itshown how a random priority scheme regulates the system in the sense thatit results in a socially optimal joining rate. For the observable version of thismodel, a menu of prices, one for each priority level, is designed in [3]. Thisresults in customers joining the queue only when it is socially optimal to doso. Finally, the priority scheme described above (where customers possess apriority parameter and the one who holds the highest value is the first to begranted service) is not the only one which exists. For example, [11] and [7]deal with a relative priority model in which the probability that one entersservice next is proportional to one’s priority parameter. Another model isbased on accumulating priority. Here customers increase their priority levelwith the time they are in the queue (as in FCFS) but the rate of increase isindividual. Thus, a higher rate reflects a higher priority level. In particular,this model grants priority based both on some priority parameter and onseniority. For more details see [10] and [1] for the cases where the menu ofchoices being continuous or discrete, respectively.This paper considers the profit-maximizer point of view where, as in theairline example, there are two priority levels in a queue, say premium andordinary classes. Here the objective is to design a price mechanism underwhich the highest possible revenue is generated. The primitive assumption isthat once a priority mechanism is introduced, the customers decide, whetheror not to pay the price offered to them for belonging to the premium class. Ofcourse, they decide while minding only their selfish interest. Clearly, for anyprice mechanism, customers are engaged in a noncooperative game amongthemselves and we look for the resulting Nash equilibrium, which in turndetermines the monopoly’s revenue. As we discuss below, the existence ofmultiple equilibria is common and indeed this is the case where a uniformcharge is offered. See [6], pp. 83–85 for further details. In the case of of mul-tiple equilibria, we judge a price mechanism based on the worst equilibriumin terms of the revenue it generates to the operator (a maxmin criterion).We deal with two cases. The first, in Section 3, is when customers arehomogeneous with respect to their cost per unit of time parameter C . Thesecond is when they are not homogeneous and hence C can be looked at as arandom variable. In both cases we design the maxmin revenue mechanism. Inthe homogeneous case, the optimal mechanism charges a random price whosedistribution function is derived in Section 3. In the heterogeneous case the3ptimal charge is cost-parameter dependent. As expected, the higher thecost-parameter, the higher the charge is. Details are given in Section 4.The queueing model and the required preliminaries from queueing theoryare given in Section 2. Section 5 concludes. We consider the memoryless single-server
M/M/ λ . Service timesfollow an exponential distribution with rate denoted by µ . Denote λ/µ by ρ and assume for stability (positive recurrence) that ρ <
1. Customers belongto two priority groups, where premium customers have preemptive priorityover ordinary customers. Within a class, we assume the first-come first-served (FCFS) regime. Suppose that a fraction q of the customers belong tothe priority class, while the rest are ordinary customers. It is well-known, seee.g., [8], p. 75, that the mean waiting time (service inclusive) at the formergroup is W ( q ) = 1 µ (1 − qρ ) , while that of the latter group is W ( q ) = 1 µ (1 − ρ )(1 − qρ ) . It is then easy to see that f ( q ) = W ( q ) − W ( q ) = ρµ (1 − ρ )(1 − qρ ) . (1)As pointed out in [6], p. 84, this function is monotone increasing with q ,0 ≤ q ≤
1. This implies that the larger the group of premium customers, themore valuable it is for an individual customer to belong to this group. Thisis the follow the crowd (FTC) phenomenon. See [6], p. 6, for more details.Seen from the individual customer’s point of view, and measured in unitsof time, the value of priority is bounded between f (0) and f (1): f (0) = ρµ (1 − ρ ) ≤ f ( q ) ≤ ρµ (1 − ρ ) = f (1)4nd, by definition, it is a function of the fraction of customers who belong tothe priority class. On the one hand, f (0) = 1 µ (1 − ρ ) − µ . This is the gain for one who solely gets top priority as opposed to being anaverage customer, for example, one who waits in a FCFS queue. On theother hand, f (1) = 1 µ (1 − ρ ) − µ (1 − ρ ) . (2)This is the gain from being an average customer as opposed of being a standbycustomer (i.e., getting service only when the server would otherwise be idle).Put differently, one can get this gain if one switches from the ordinary classto the premium class, when all are premium customers. This switch is whenpriority is most valuable. See [8], p .64 and pp. 76–77, for more on standbycustomers.In [6], p .83, a decision model, in fact a non-cooperative game, wherecustomers are charged for belonging to the premium class (where the defaultis to become an ordinary customer) is introduced. For a comprehensivereview on the literature on queueing games see [12]. Denote this commoncharge by θ . Due to the monotonicity of f ( q ), it is clear that the solutionto the dilemma of whether or not to purchase priority is simple in the casewhere θ ≤ f (0): priority should be purchased. This is a dominant strategy:regardless of what the others do, one is better off paying θ and joining thepremium class. Of course, the end result is that all follow this strategy andthe resulting queue regime is FCFS. An analogous situation happens in thecase where θ ≥ f (1): nobody purchases priority as this is the dominantstrategy. Here too the resulting queue regime is FCFS.The situation is more involved when f (0) < θ < f (1). Now no dominantstrategy exists. In particular, the best response for an individual dependson the fraction q . Denote by q e the unique value for q that obeys f ( q e ) = θ .Then, if q < q e the best response is not to purchase priority, while if q > q e ,it is to purchase. In the case where q = q e , either purchasing or not, is one’sbest response. In particular, one is indifferent between these two options. Infact, any mixing between the two is also one’s best response.Once a dominant strategy does not exist, one looks for a symmetricNash equilibrium, namely, a strategy that if used by all but one player,5s one’s best response. As shown in [6], p. 83, there are three equilibria when θ ∈ ( f (0) , f (1)): when all purchase, when none do, and when all purchasewith probability q e (namely, a fraction of q e of the customers become pre-mium customers). Which one of these three equilibria will emerge is notclear. In terms of the stability of the equilibrium, it is possible to see thatthe equilibrium that is based on mixing with probabilities q e and 1 − q e isunstable: a small deviation in one direction tilts the balance in the samedirection. For example, if q e + ǫ for ǫ > f (0) backfires as under theworst equilibrium, nobody pays. The optimal flat price is hence f (0). Aswe will show in the next section, there exist random price mechanisms thatgenerate more income. The main purpose of this article is to introduce a price mechanism that leads,under the resulting unique Nash equilibrium in the game among customers,to a gain per customer that is greater than f (0) (but still less than f (1)).Moreover, the suggested mechanism is optimal. The mechanism is basedon a random entry fee. Specifically, each arrival will be asked to draw arandom number from a to-be-determined distribution whose support is theinterval [ f (0) , f (1)]. Accepting this fee will be the unique equilibrium amongcustomers and the resulting queue regime will again be FCFS. Note that inorder to ease the exposition we assume that whenever customers are indif-ferent between purchasing priority or not, they elect for the former option. Theorem 3.1
Suppose that the charge for priority is a random variable with DF F ( p ) , where F ( p ) = p ≤ f (0) , ρ − µ (1 − ρ ) 1 p f (0) ≤ p ≤ f (1) , p ≥ f (1) . (3) If, for some given price p , f (0) ≤ p ≤ f (1) , all who are asked to pay a pricethat is less than p do so, then it is uniquely best for one who is asked to pay p to do so as well, regardless of what those who are asked to pay a price greaterthen p actually do. Note that a tie exists only in the case when all from thelatter group do not pay, and remain ordinary customers. Proof:
Consider a random price mechanism with a cumulative distributionfunction (CDF) F ( p ). For a given value for p , suppose all those who areasked to pay a price less than p do so. Then, under the lowest possible valuefor priority for one who is asked to pay p , namely, when all who are asked topay a price greater than p do not do so, the value of priority, based on (1),equals ρµ (1 − ρ )(1 − F ( p ) ρ ) , f (0) ≤ p ≤ f (1) . Thus, if one is asked to pay p , where p = ρµ (1 − ρ )(1 − F ( p ) ρ ) , (4)then one should still pay. (In practice, one will be asked to pay a little bitless but for the sake of ease of exposition we ignore this issue.) Indeed, one isindifferent between paying or not if and only if all customers who are askedto pay a price greater than p do not do so. Solving (4) for F ( p ) as a func-tion of p shows that (3) holds. In particular, F ( f (0)) = 0 and F ( f (1)) = 1 . Remark.
The mean payment based on the scheme stated in Theorem 3.1equals − log(1 − ρ ) µ (1 − ρ ) . (5)Of course, f (0) = ρµ (1 − ρ ) < − log(1 − ρ ) µ (1 − ρ ) < ρµ (1 − ρ ) = f (1) . emark. A way to implement the above scheme is to announce that thecustomer who will be in the F ( p ) percentile to purchase priority will beoffered to pay p for it, f (0) ≤ p ≤ f (1). Customers can be looked at asbeing in competition over who will be the first to pay. The earlier they do sorelative to others, the less they will have to pay. Another interpretation is asfollows. Suppose that customers are somehow ordered. The first is offeredpriority at the price of f (0) (which he/she accepts and pays), the second isoffered priority at a somewhat higher price (and is willing to pay more thanthe first customer since s/he wishes not to be overtaken by the first), thethird is offered priority at an even higher price, which s/he duly pays (inorder not to be overtaken by the first two is), etc. Remark.
It is possible to see that if instead of F ( p ) one implements anotherCDF G ( p ) with G ( p ) ≥ F ( p ) for p , the same customers’ behavior will beinduced. Yet, a reduction in the payment per customer will occur. Thiscan easily be argued from the fact that if a nonnegative random variablecomes with a CDF G ( p ), then its mean value equals R ∞ p =0 (1 − G ( p )) dp . See,e.g., [8], p .3. The following theorem says even more: the scheme suggestedin Theorem 3.1 is revenue maximizer. Theorem 3.2
Among all random price mechanisms that lead to the “allpay” profile being the unique equilibrium, the one suggested in Theorem 3.1is optimal. Moreover, if a random mechanism comes with a CDF G ( p ) inwhich “all pay” is the unique equilibrium, then G ( p ) ≥ F ( p ) for all p ∈ R . Proof:
Aiming for a contradiction, let G ( p ) be another mechanism thatcomes with “all pay” as an equilibrium profile whose profit is higher thanit is under F ( p ). Let p ′ = inf p { G ( p ) < F ( p ) } . Loosely speaking, p ′ is the“first” charge where the probability of paying this value or below it is smallerunder G ( · ) than it is under F ( · ). This is the first point where, at least locally,this mechanism becomes more expensive. Note that our assumption on G ( · )implies the existence of p ′ . (Note that the option where p ′ = 0 is not ruledout.) This implies that those charged p ′ are asked to pay more than neededunder the assumption that only a fraction of F ( p ′ ) will pay. They, as wellas those who are charge more, may now consider not paying. In such a case,it is possible to see that under this mechanism, the profile where all thosewho are charged up to p ′ pay, while the rest do not, is a Nash equilibriumoutcome, making it an additional equilibrium to the assumed “all pay” one.In particular, there is no unique equilibrium under this charging scheme.8 .1 The discrete approach A possible mechanism is to discretize the random variable whose CDF is F ( p ), in such a way that each discrete value gets all the probability mass ofthose values that are greater than it but are still less than the next discretevalue. For example, for some choice of an integer n , denote 1 /n by ǫ . Then,define the following n + 1 values for p : p i = F − ( iǫ ), 0 ≤ i ≤ n . Note that as F − ( y ) = ρ/µ (1 − ρ )(1 − ρy ), we get in fact that p i = ρ/µ (1 − ρ )(1 − ρiǫ ). Inparticular, p = f (0) while p n = f (1). By this construction and by denotingby P the random charge based on the original scheme, it is possible to see that P ( p i ≤ P ≤ p i +1 ) = F ( p i +1 ) − F ( p i ) = ( i + 1) ǫ − iǫ = ǫ , 0 ≤ i ≤ n −
1. Definenow a uniform discrete random variable X whose support is composed of the n values p i , with identical probabilities, i.e., P ( X = p i ) = ǫ , 0 ≤ i ≤ n − P is stochastically dominated by X .By considering a discrete random charge for priority based on X , it issomewhat easier to see the rationale leading to all paying the prescribedprice as its unique equilibrium. The argument in fact goes by induction.Specifically, those who are asked to pay p = f (0) will certainly do so: theyall are willing to pay that regardless of what others do. This fact is realizedby those who are asked to pay p . Knowing that a fraction of ǫ have alreadypaid, they are willing to pay p , regardless of what those who are asked topay more do. Hence, they pay what they are asked to. This argument goeson until all pay. It is based on the idea that for any ǫ > X , given all pay their realized X .Note that E ( X ) = ǫ ρµ (1 − ρ ) n − X i =0 − ρiǫ . (6)Recalling that ǫ = 1 /n and taking the limit when n goes to infinity in (6),we get (5), as expected. Remark.
It is possible to argue for the resulting Nash equilibrium by theprocess of elimination of weakly dominated strategies. Specifically, as notpurchasing priority is weakly dominated by purchasing for those who areoffered the lowest price, the former strategy can be eliminated. Given that,the same is hence the case for one who is offered the second lowest price andone’s not purchasing strategy can now be eliminated. This goes on until theone who is offered the highest price. 9 .2 When homogeneous customers select their prioritylevel
A related but different decision model is suggested in [4]. See also [6], p.103. Now it is the customers who decide (implicitly) on their priority leveland have to pay accordingly: the more they pay, the higher is their prioritylevel. Ties are broken randomly. The options are all the nonnegative realnumbers. Modifying the analysis given in [4] for the non-preemptive case, itcan shown that there exists a the unique equilibrium payment profile whichcomes with a random payment whose CDF, denoted by B ( y ) equals B ( y ) = 1 − ρ + 1 ρ ( 1(1 − ρ ) − µy ) − , ≤ y ≤ µ (1 − ρ ) − µ . (7)Note that here customers are ex-ante identical but ex-post, due to the use ofa mixed strategy, they are not.Note that maximal possible payment equals the value for becoming a‘sole child’ who gets top priority for one who would otherwise be a standbycustomer. In other words, it is the value of moving from one extreme situation(the worse) to the other (the best). Indeed, being a standby customer is thefate of the one who pays nothing when all pay something.The mean payment per customer is then Z ∞ y =0 (1 − B ( y )) dy = 1 ρ Z µ (1 − ρ )2 − µ y =0 (1 − ( 1(1 − ρ ) − yµ ) − ) dy = ρµ (1 − ρ ) . (8)Recall that this value equals f (1) (see (2)). It is possible to see thatthe ratio between the upper bound in (7) and (8) equals (only) 2 − ρ whichindicates that most of the mass of this distribution is at the upper end.Indeed, it is possible to see that B ′ ( y ), which is the corresponding density ismonotone increasing, going up from (1 − ρ ) / (2 µ ) at 0 to 1 / (2 µ ) at µ (1 − ρ ) − µ .One needs to compare (5) with (8) (the latter is larger) but note that the twomodels differ not only in who determines the price mechanism but, crucially,in the fact that in the former model there exist only two priority levels whilein the latter model there are continuously many.10 Profit maximization: Heterogeneous cus-tomers
Assume now that customers are heterogeneous and they vary in terms oftheir waiting costs. Specifically, a type C customer suffers a cost of C perunit of time in the system (service inclusive). This heterogeneity is modeledby assuming that for each individual C is a nonnegative random variablewith a density function g ( c ) and a CDF G ( c ), G ( c ) = R cy =0 g ( y ) dy .A profit maximizer wishes to charge as much as possible for priority.Suppose that it is possible to discriminate between customers and chargethem for priority an amount that depends on their C value, denoted by c . Itmakes sense to charge a price that is monotone in c as high-cost customerswill be willing to pay more in order to save themselves the same amountof waiting. Moreover, a good profit-maximizing scheme encourages, by wayof a relatively small charge, customers with a lower value for c to purchasepriority (and to make it clear that this is what they do) so as to makehigh-cost customers willing to pay even more for priority when they realizethat the group of those who pay (and hence belong to the premium class) isincreasing in size. Theorem 4.1
The price mechanism that asks a customer with waiting costparameter c to pay ρcµ (1 − ρ )(1 − G ( c ) ρ ) (9) for priority leads to “all pay” being the unique Nash equilibrium for cus-tomers. Moreover, this is optimal among all price mechanisms that lead to aunique equilibrium from the customers’ side. Proof:
Assume that all customers with a cost–parameter C that is smallerthan c belong to the premium class. Then, a customer whose cost–parameterequals c agrees to pay (at least) what is prescribed in (9) as this is the lowestpossible value for priority for him/her (since all those with a higher value for C do not pay for priority). See (1). A similar argument to the one given inthe proof of Theorem 3.2 for the optimality of the mechanism holds here tooand hence will not be repeated.The following corollary now follows.11 orollary 4.2 The optimal profit equals ρµ (1 − ρ ) Z ∞ c =0 c − G ( c ) ρ g ( c ) dc. (10) It is bounded from below and from above by ρ E ( C ) µ (1 − ρ ) and ρ E ( C ) µ (1 − ρ ) , respectively. Remark.
This mechanism can be looked at as bribing customers with a lowvalue for C to join the premium class at a low charge, making those with ahigh value for C face a tougher situation in case they stay behind in the racefor priority. This will make them willing to pay even more than what mightat first sight be considered as a reasonable price. A related but different decision model is suggested in [4]. See also [6], p .103.Here it is the customers who decide how much to pay for priority, having theirown cost parameter as their private information. Moreover, the more onepays, the higher one’s priority level is (breaking ties randomly). Assumingthat the random variable C comes with a continuous non-zero density alongits support, it was shown there that in the unique equilibrium profile, thehigher one’s cost parameter is, the higher one’s payment (and hence one’spriority level) is. Modifying the analysis given in [4] for the non-preemptivecase, we get that if one’s cost per unit of time is c , one’s payment equals2 ρµ Z cy =0 − (1 − G ( y ) ρ ) g ( y ) dy. The mean payment per customer is then of course2 ρµ Z ∞ c =0 Z cy =0 − (1 − G ( y ) ρ ) g ( y ) dyg ( c ) dc. (11)One needs to compare (10) with (11) as both assume that the set ofpriority levels is continuous. The disclaimer we made on such comparisonsat the end of Section 3 holds here too.12 cknowledgment The first author was patially supported by an Israel Research Fund, Grantno. 511/15.
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