The Combinatorial Retention Auction Mechanism (CRAM)

Transcription

1 The Combnatoral Retenton Aucton Mechansm (CRAM) By Peter Coughlan 1, Wllam Gates 2, and Noah Myung 3 October 13, 2013 Graduate School of Busness & Publc Polcy Naval Postgraduate School 555 Dyer Road, Monterey, CA Workng Paper Abstract: We propose a reverse unform prce aucton called Combnatoral Retenton Aucton Mechansm (CRAM) that ntegrates both monetary and nonmonetary ncentves (NMIs). CRAM computes the cash bonus and NMIs to a sngle cost parameter, retans the lowest cost employees and provdes them wth compensaton equal to the cost of the frst excluded employee. CRAM s domnant strategy ncentve compatble. We provde optmal bddng strategy, and show that there s cost savng compared a benchmark aucton (monetary retenton aucton). Because CRAM and the benchmark may retan dfferent employees, we provde for whom and under what condtons the utlty may ncrease or decrease by CRAM. Fnally, we show that there s an ncrease n the total socal welfare by utlzng CRAM to the benchmark. Keywords: Combnatoral Aucton; Labor Markets; Compensaton; Defense Economcs. JEL classfcaton: D44, D47, J33, J45 Approved for publc release; dstrbuton s unlmted. 1 Emal: pjcoughl@nps.edu 2 Emal: bgates@nps.edu Phone: Emal: noah.myung@nps.edu Phone: Web: 1

2 0. Introducton We propose a mechansm called the Combnatoral Retenton Aucton Mechansm (CRAM), whch an organzaton can utlze to provde both monetary and nonmonetary ncentves (NMIs) for retenton. CRAM s a reverse unform prce aucton where a sngle employer retans the desred number of employees. The mechansm s smple: CRAM elcts employees reservaton value by askng them how much cash bonus and what set of NMIs they requre to be retaned. The cost of provdng the requested cash bonus and the set of NMIs s calculated and presented as a sngle cost parameter. Then, CRAM selects the preannounced number of the lowest costng employees to retan. Fnally, the beneft each retaned employee receves s determned by the cost of the frst excluded employee. In other words, any retaned employee receves the set of NMIs and the cash bonus he requested. Furthermore, every retaned employee receves an addtonal cash bonus that equals the cost of the frst excluded employee mnus the cost of retanng the partcular retaned employee. Therefore, as wth the standard unform prce aucton, the cost of retanng each employee equals the cost of retanng the frst excluded employee. CRAM s domnant strategy ncentve compatble; t s weakly domnant for every employee to reveal ther true reservaton value by announcng the bonus requred and selectng a set of NMI that maxmzes the dfference between the value of NMIs and the cost of NMIs. We wll work under the retenton framework, although CRAM may be generalzed to desgnng compensaton packages for newly hred employees. A general dscusson about the combnatoral aucton and the applcaton of CRAM to the US Department of Defense (DoD) s retenton problem wll be presented n the followng sectons. Readers wth suffcent background n ether feld may choose to skp over the partcular secton. 2

3 Our contrbutons are as follows: we provde general framework, characterzaton and propertes of CRAM. CRAM provdes a smple and straghtforward way of determnng the retenton cost and the benefts to the employees. Ths process lessens the burden on the auctoneer (the employer), as well as the partcpants (the employees). We provde an optmal bddng strategy for the employee that s domnant strategy ncentve compatble and show that any optmal bddng strategy must take ths form. The employee should reveal ther true reservaton value, select the set of NMIs that maxmzes the total surplus, and specfy how many dollars he needs to receve n addton to NMIs to be retaned. The dollar amount plus the value of the NMIs should equal the reservaton value. Next, we show that the cost of retanng employees va CRAM s (weakly) cheaper than retanng employees purely by monetary retenton aucton. Ths result s drven by the fact that CRAM takes advantage of the surplus generated by provdng NMIs nstead of cash. Because CRAM may retan a dfferent set of ndvduals compared to the monetary aucton, wth a dfferent cut off cost, determnng whom the mechansm benefts s not straghtforward. Therefore, we compare an employee s utlty under CRAM to the monetary retenton aucton and show whch sets of employees are better off and worse off. The employees are broken nto four sets. An employee not retaned under ether mechansm s ndfferent. An employee retaned under the monetary retenton aucton and not CRAM s better off under monetary retenton aucton because he receves hgher than the reservaton utlty when retaned. Smlarly, an employee retaned under CRAM but not under the monetary retenton aucton prefers CRAM. For anyone who s retaned under both mechansms, he may be better or worse off dependng on how much the cutoff cost decreases. An employee wll be better off under CRAM f the cutoff value does not drop more than the gan n surplus from the NMIs. 3

4 Fnally, we compare the socal welfare, the sum of both retaned and not retaned employees utlty mnus employer s cost, and show that CRAM s socal welfare s (weakly) greater than the monetary retenton aucton. I. Combnatoral Auctons Combnatoral auctons generally deal wth bddng on multple objects. What makes combnatoral auctons nterestng and dffcult s the computatonal complexty. Wth n goods ntroduced, there are 2 n 1 possble combnatons of goods that the auctoneer and the partcpants may have to consder. Formally, these problems are consdered to be NP-complete, meanng that typcal computers may have a dffcult tme fndng an optmal soluton. Whle combnatoral auctons have always been of nterest, the feld has seen the greatest growth wth the applcaton of the Federal Communcaton Commsson (FCC) spectrum auctons. Between 1994 and 2003, the FCC has utlzed some form of combnatoral aucton 41 tmes, whch rased over $40 bllon n revenue (Kwasnca, Ledyard, Porter, and DeMartn 2005). Even pror to the major utlzaton by the FCC, combnatoral auctons had been utlzed to enhance market and non-market transactons by publc and prvate enttes. Grether, Isaac, and Plott (1981) were one of the earler proposers of usng an aucton type of desgn to solve arport tme slot allocaton problems for the FAA. Rassent, Smth, and Bulfn (1982) further mproved the use of a computerasssted smart market way of solvng the landng rghts problem. Banks, Olson, Porter, Rassent, and Smth (2003) provded a lst of references analyzng varous combnatoral auctons that have been utlzed to solve complcated government and non-government allocaton problems. These references are: Arzona Energy Exchange for energy tradng, Federal Energy Regulatory Commsson study for gas delvery, payload manfest for Space Shuttle, resource allocaton for Cassn 4

5 msson to Saturn, tran schedulng, transportaton servces, polluton markets, and markets to exchange fnancal portfolos. Whle we have a pretty good understandng of sngle-object auctons, combnatoral auctons are faced wth other problems n addton to computatonal complexty. These nclude: exposure problem, threshold problem, and wnner determnaton problem. 4 Consder a combnatoral aucton wth three objects, {a} {b} {c}, and four partcpants. An exposure problem occurs when a partcpant values a combnaton of a good more than the sum of the ndvdual good. If partcpant 1 s value for {a}, {b}, and {ab} are 2, 2, and 6, respectvely, and the hghest bd for {a} and {b} are currently 2 and 2, should partcpant 1 ncrease hs bd for object {a} and {b}? If partcpant 1 can only obtan object {a} and not {b}, then he wll end up payng more than 2 for an object he only values at 2. Threshold problem occurs when the sum of the smaller bdder s valuaton exceeds the larger bdder s valuaton for the package of goods but cannot sngle-handedly outbd the larger bdder. Suppose that partcpant 1 bds 32 for {abc} and that partcpants 2, 3, and 4 bd 10 each for {a}, {b}, and {c}, respectvely. Partcpant 1 should be awarded package {abc} and pay 32 for t. On the other hand, f partcpants 2, 3, and 4 value {a}, {b}, and {c} at 11 each, respectvely, the revenue maxmzng allocaton would be to allocate the goods to the ndvduals and charge 11 per good. However, a sngle ndvdual rasng hs bd to 11 wll not change the wnner and would requre some coordnaton wth the other two bdders to change the allocaton (though colludng s often not allowed n most aucton formats). 4 Jump bddng: although some may state that jump bddng s a problem, as dscussed by Isaac, Salmon, and Zllante (2004), jump bddng s not necessarly a problem and, n fact, can ncrease revenue. 5

6 Smply put, t s extremely dffcult to determne the wnner of a combnatoral aucton. To start wth, the hghest bd on a sngle package s not guaranteed to wn f some alternatve combnaton of bds can generate hgher revenue. Fndng the alternatve combnaton of bds s also not a smple problem. Further problems arse wth te breakng rules. How are tes to be broken? Whch group of people are wnners f dfferent combnatons of goods by dfferent sets of partcpants yeld the same revenue? What f the hghest revenue generatng combnaton does not utlze all possble resources? These problems can cause lower revenue for the auctoneer and neffcent allocaton of resources among the partcpants. Pekeč and Rothkopf (2003) provded an excellent overvew of combnatoral auctons and ts challenges. Wthn the combnatoral aucton famly, the followng are some aucton formats that have drawn consderable attenton: 1. Smultaneous Multple Round Aucton (SMR): Format utlzed by the FCC and often used as a benchmark comparson to other combnatoral auctons. Ths aucton format does not allow for package bddng. 2. Adaptve User Selecton Mechansm (AUSM): Developed by Banks, Ledyard, and Porter (1989), AUSM allows for package bddng n contnuous tme. 3. Resource Allocaton Desgn (RAD): Developed by Kwasnca, at el. (2005), RAD s a hybrd of SMR and AUSM plus an addtonal prcng feature to gude bdders. 4. Combnatoral Clock Aucton (CCA): Developed by Porter, Rassent, Roopnarne, and Smth (2003) CCA uses a clock as a gude for bddng (smlar to an Englsh aucton). 5. Smultaneous Multple Round Package Bddng (SMRPB): Developed by the FCC as a varant of RAD, SMRPB ncludes the ablty to utlze an exclusve OR functon. 6

7 The detals of each aucton are excluded n ths paper but references are provded for nterested readers. Brunner, Goeree, Holt, and Ledyard (2010) summarzed some of the commonly dscussed combnatoral auctons mentoned above and compared ther performance va experments. Brunner et al. (2010) found that, when complementartes are present, package bddng s recommended and that CCA generally yelds the hghest revenue. Due to nsttutonal restrctons, utlzaton of the auctons mentoned above s not a straghtforward applcaton to the retenton framework. The above aucton formats are called forward auctons and prmarly deal wth sellng objects. Procurement auctons, or reverse auctons, are auctons where one s nterested n buyng goods and servces nstead of sellng. Therefore, procurement auctons are closer to retenton auctons. There are many dfferences between procurement auctons and retenton auctons, however, agan due to nsttutonal features. One can procure half of the goods and servces, or splt the award among the multple provders n order to keep the compettors compettve 5 (Chaturved, Bel, and Martnez-de-Albenz 2013). However, n the actve-duty mltary, t s not feasble to retan a porton of a person. Furthermore, NMIs are specfc ncentves that are salent for compensatng employees but may not be salent n procurement or the standard forward auctons. The followng secton wll further dscuss some characterstcs and nsttutonal features that requre changes to the known combnatoral auctons and the reason for developng CRAM. Fnally, t s worth notng that the combnatoral aucton s also an extremely useful tool for aggregatng nformaton, as well as endogenously determnng a market-clearng prce. When the desgner lacks nformaton on whch NMIs may or may not be sub or super modular, t may be best left for the 5 Awardng the procurement to only one vendor may make that vendor a monopoly n the future due to technologcal advancement. 7

8 decson maker to choose the best set of hs own NMI. When t comes to prce formaton, nstead of exogenously estmatng the prce wth a large room for error, these auctons wll endogenously determne the market-clearng prce. II. Applcaton to the US Department of Defense Wth over 1.4 mllon actve duty and 1.1 mllon Reserve and Natonal Guard servce members n 2013 (DoD 2013), the US mltary s labor force s unque because t s nternally grown. As an example, f an Admral retres and the Navy s n need of a replacement, t cannot smply go to the general labor market and hre a new Admral. The Navy must promote from wthn. Therefore, the DoD and each of the servces must carefully plan ts force structure over the long term. In terms of budget and compensaton, approxmately 51.4 percent of mltary compensaton s cash compensaton, whle 20.5 percent of mltary compensaton nvolves non-cash tems (educaton, health care, etc.), and 28.1 percent of the compensaton s deferred compensaton (retrement pay accrual, etc.) (DoD 2012). Out of the $525 bllon budget for the DoD n 2012, $181 bllon was related to pay and benefts for mltary personnel (Harrson and Montgomery 2011). Wth cuts n the defense budget, the DoD also needs to fnd savngs n pay and benefts. Specal and Incentve (S&I) pays are authorzed by law to provde the mltary servces the flexblty needed for force shapng (OSD Mltary Compensaton 2013). There are currently over 60 authorzed S&I pays. These pays can be sgnfcant. Examples nclude: 1) Selectve Reenlstment Bonus (SRB), whch authorzes the servces to pay up to $90,000 for a mnmum threeyear reenlstment; 2) Surface Warfare Offcer Contnuaton Pay that authorzes the Navy to pay up to $50,000 to elgble offcers for commttng to a Department 8

9 Head tour; 6 and 3) Crtcal Sklls Retenton Bonus (CSRB), whch authorzes up to $200,000 over a servce member s career 7 for a skll-specfc retenton. Some S&I pays are much smaller, such as Demolton Duty Pay--a hazardous duty, whch adds $150 per month for the assgnment s duraton. Of course, these S&I pays are reserved for very select groups of servce members durng a shortage of manpower. To provde perspectve on a servce member s base cash compensaton durng the 2013 calendar year, excludng S&I pays, an average Staff Sergeant n the US Army (pay grade E-6) wth 10 years of total servce and three dependents would get annual cash compensaton 8 of $60, An average Captan n the US Marne Corps (pay grade O-3) wth fve years of total servce and no dependents would get annual cash compensaton of $80, (DFAS 2013). Therefore, these S&I pays can be a sgnfcant porton of the servce member s cash ncome. The CRAM s desgned to support DoD s retenton process. CRAM s developed to mprove control n 1) reducng retenton cost, 2) accurately retanng the proper number of servce members, and 3) mprovng the effectveness of NMIs. The DoD has been lmted to utlzng a posted-prce format for provdng the S&I bonuses mentoned above, ncludng selectve retenton bonuses. 9 Furthermore, these bonuses are provded as purely monetary compensaton, thus forgong any surplus that may be ganed by ncludng NMIs. Coughlan, Gates, and Myung (2013), CGM henceforth, descrbed the addtonal surplus that the 6 Department Head tour s a type of leadershp tour for the Navy s shp drvers. 7 US mltary servce members are typcally elgble for full retrement at 20 years of servce. 8 Cash compensaton s the Basc Pay (salary) plus addtonal cash payments for housng and allowance for subsstence. In addton, there are deferred and unversal compensaton elements, such as health nsurance and tax advantages, but we do not nclude these n computng cash compensaton. 9 Posted-prce format mples that the Servce announces the bonus amount and the market determnes how many servce members accept the announced bonus. Ths method lacks control over the quantty of servce members acceptng the bonus and can be expensve f too many servce members accept. 9

10 DoD can potentally gan by provdng personalzed NMIs. Furthermore, CGM stressed the mportance of utlzng NMIs, the dffculty of provdng a unversal ncentve package 10 of NMIs, and the varablty n preference for NMIs by servce members between and wthn communtes. CGM found that, although none of the NMIs examned provded sgnfcant value to at least 50 percent of the servce members surveyed, approxmately 80 percent of the surveyed servce members expressed a sgnfcant value for at least some NMIs. As wth desgnng any market, the market desgner must consder mportant normatve and postve characterstcs that the customer values. For example, Pekeč and Rothkopf (2003) dscussed that some of the key consderatons of desgnng a combnatoral aucton are allocatve effcency, cost mnmzaton, low transacton cost, farness, falure freeness, and transparency. In addton to the aforementoned consderatons, our market desgn for the DoD emphaszed the followng normatve characterstcs as crtcal features of a combnatoral aucton: 1. Egaltaran: percepton of equalty. The mltary s of a strong mndset that everyone should get the same pay. Meanng, when S&I pay s beng used, everyone under the specfc S&I bonus should get the same bonus. 2. Domnant strategy ncentve compatble mechansm: transparency and ease of usng the mechansm. The mltary prefers a mechansm that s easy to understand and mnmzes strategc gamng by the partcpants. 3. Low transacton cost: mnmum tme requrements for aucton partcpants. Ths consderaton s dfferent from many other combnatoral aucton desgns. Unlke the FCC aucton, whch can take a form of ascendng bd aucton requrng partcpants to observe and nteract for hours or days at a tme, ths s not feasble for the DoD. 10 Unversal Incentve Package means that everyone receves the same set of NMIs. 10

11 Dfferent servce members may be nvolved n operatonal actvtes throughout the world. A submarner may be undersea for an extended perod of tme and only have one chance to submt a sngle bd. An arman may be deployed n a hostle envronment and unable to frequently check the current aucton market status. Therefore, conductng a smultaneous ascendng bd or clock type aucton s not practcal. DoD. Takng these features nto consderaton, we developed CRAM for the III. Outlne We descrbe the general envronment for the CRAM aucton n Secton IV and formally defne CRAM n Secton V. Secton VI dscusses the employee s optmal bddng strategy. Secton VII ntroduces the monetary retenton aucton as a benchmark aganst whch CRAM s characterstcs are compared. Secton VIII compares the employer s cost under CRAM to the monetary retenton aucton, whle Secton IX compares the employees utlty, and Secton X compares socal welfare. We end wth conclusons n Secton XI. IV. The Envronment A. The Retenton Problem Let I be a set of employees currently seekng retenton wth a gven employer. The employer wll retan q I of these employees. The employer offers ts employees both a monetary ncentve, m R, as well as a set of nonmonetary ncentves, N. Each employee ultmately retaned by the employer receves a monetary ncentve as well as some combnaton of NMIs as hs or her retenton package. 11

12 Denote by S N a subset of NMIs potentally receved by any retaned employee. Each employee can consume at most one of each of the N NMIs. Therefore, there are 2 N dfferent potental combnatons of NMIs an employee could receve. We assume that each NMI s a non-rvalrous but excludable good (thus, each s a club good). B. Employee Preferences Each employee s endowed wth a utlty for any combnaton of a monetary ncentve (m ) and a non-monetary ncentve package (S ), gven by U ( m,s ) = v S + m. We normalze 0 explctly allow for an employee s valuaton, v = for all I. Note that we v S, of any package of nonmonetary ncentves to be addtve, sub-addtve, super-addtve, or some combnaton thereof. Each employee I s further endowed wth a reservaton value r R, whch reflects the employee s opportunty cost of beng retaned by the employer (or, alternatvely, the employee s wllngness-to-retan or the expected value of the employee s outsde offer or opportunty). If not retaned by the employer, each employee wll enjoy utlty r. Each employee s reservaton value, r, and valuaton of non-monetary ncentves, v, are prvate nformaton. We denote the fnal retenton package consstng of cash and a set of NMIs gven to any retaned employee as P = (m, S ). Employee s utlty for ths fnal retenton package s then gven by U ( P ) = U m, S = v S + m. 12

13 C. Employer Costs For each ndvdual NMI, s n S N, the employer s cost to provde that NMI to any ndvdual employee, cost s n, s publc knowledge (or at least communcated to all employees pror to the retenton decson). Because each NMI s a non-rvalrous club good, provson of each NMI s characterzed by constant margnal cost. Hence, the cost to provde any gven NMI, s n, to any gven number of employees, x, s smply gven by x cost s n. Hence, there are nether economes nor dseconomes of scale n provdng any partcular NMI. We normalze cost ( ) = 0. We further assume that there are nether economes nor dseconomes of scope n provdng any combnaton of NMIs. That s, the total cost to provde any set of NMIs, S, s gven by cost( S) = P cost(sn ). 11 s n S Therefore, the employer s total cost to provde a fnal retenton package ( m, = to any retaned employee s gven by cost ( P ) m cost ( n cost P m cost( s ). = + n s S = + or D. NMI Surplus Wth ths understandng of employee preference and employer cost, t s helpful to defne the employee NMI surplus. Thus, for any bdder and any set of NMIs S, let surplus( v ( cost (, =. 11 We may relax the assumptons of constant margnal cost as well as the economes of scope n provdng NMI combnatons. However, ths adds computatonal complexty wthout addng value to the ntroducton of our mechansm. 13

14 Note that, for a gven set of NMIs S, surplus(,s) R, s not necessarly postve. The followng lemma guarantees, however, that employee NMI surplus wll not be negatve for all sets of NMIs, nor wll NMI surplus be postve for all sets of NMIs. LEMMA 1: For any set of NMIs N and any employee, max S N ( surplus(,s) ) 0 and mn S N surplus(,s) 0. Wth ths noton of NMI surplus, t s nstructve to note that an employee s utlty for the fnal retenton package can now be wrtten as U (P ) = U (m,s ) = v (S ) + m = m + cost(s ) + surplus(,s ) = cost(p ) + surplus(,s ). Hence, employee s utlty for the fnal retenton package s smply the employer s cost to provde that package plus the employee s NMI surplus. Before proceedng, however, t s mportant to dstngush the noton of NMI surplus from total employee surplus (or suppler surplus). Recall that each employee I has a reservaton value r R that reflects the employee s opportunty cost of beng retaned by the employer. Whle an employee s NMI surplus reflects how much that employee values a set of NMIs above and beyond the cost of provdng those NMIs, employee total surplus reflects the utlty for a total compensaton package above and beyond that employee s reservaton value. U P r v S m r =. + Hence, total employee surplus s equal to 14

15 V. Combnatoral Retenton Aucton Mechansm (CRAM) A. Mechansm We frst formally outlne the mechansm n ths secton. Detaled explanaton of the mechansm wll be provded n the subsecton to follow. Frst, defne B = ( m, as the message or strategy beng submtted by the employee to the employer. m s the monetary ncentve and S s the employee s requested set of NMIs. The employer s cost of provdng B s b = cost( B ) = m + cost(. Wthout loss of generalty, let b represent the q + 1 lowest costng bd, or b b q + 1 =. The CRAM mechansm Γ = (B 1,..., B I,g( )) s a collecton of I bds, B,..., 1 B I and an outcome functon g: B1... BI X, where the outcome determnes the retenton and the compensaton package n the followng manner: P = (m,s ) f b < b (0, ) f b b and retaned f b < b where m = m cost(. Therefore, persons wth receve ther reservaton value. B. Employee Bds b b are not retaned and Informally, the CRAM bddng process can be understood as nvolvng two decson elements for each employee: (1) selectng NMIs and (2) submttng a mnmum monetary ncentve (or cash compensaton) requred to retan. For the frst decson element, employees must choose whch NMIs they desre from a menu n whch each NMI has an assocated cost. As we wll detal below, the employer wll add the cost of each NMI selected to the employee s 15

16 monetary ncentve request to determne the cost of retanng that employee. Thus, the NMI cost, and not just ts value to the employee, factors nto the employee s decson regardng whch combnaton of NMIs to select from the menu. The second decson element of the bddng process nvolves requestng a monetary ncentve or cash compensaton ncentve. Because retaned employees receve each and every NMI they chose from the menu as part of the frst bddng decson element, the monetary ncentve bd reflects the mnmum cash amount an employee must receve n order to retan, condtonal on the fact that the retaned employee wll also receve all NMIs selected. Thus, the CRAM bddng process can be formally descrbed as follows. Each employee submts a bd of the form B ( m, =, where m s the monetary ncentve (or cash compensaton) and S s the combnaton of NMIs that employee requests (or demands) to be retaned. Let B = (B 1, B 2,, B I ) be the set of all submtted employee bds. Further, let B - = (B 1, B 2,, B -1, B +1,, B I ) denote the set of bds submtted by all employees other than employee, or employee s competng bd set. C. Employee Cost and Retenton To retan employee who has submtted bd B ( m, =, the employer must provde that employee the set of NMIs, S, and cash compensaton of m (or greater) to retan the employee. Thus, the cost to retan that employee s b = cost B = m + cost S. The employer wll retan the least expensve set of q employees. In other words, the employer wll retan those q employees who submt the q lowest-cost 16

17 bds. WLOG, let b b j f < j for all, j I. The employer wll then retan employee f and only f q or, alternatvely, f and only f b bq. Note that a te s possble, n whch there exsts more than one set of q lowest-cost bds. Whenever tes occur, multple employees wll have submtted bds that all have the q th lowest cost. For smplcty, we assume that tes are not possble. An alternatve s to break tes randomly. 12 However, ths mples that only q employees wll be retaned, and therefore some employees who submtted a bd wth the q th lowest cost wll be retaned whle others who submtted bds of the same cost wll not be retaned. Note that, whenever a te occurs, we wll have (at a mnmum) that bq = b q + 1. For our purpose, t s suffcent to assume that there are no tes between b q and b q + 1. D. Compensaton for Retaned Employees Because CRAM s a unform-prce aucton mechansm, all retaned employees wll receve a total retenton (or compensaton) package of unform cost to the employer. In partcular, each retaned employee wll receve a retenton package whose total cost s equal to the cost of the frst-excluded bd, whch s the lowest-cost bd submtted among those employees not retaned. But each employee may not receve the same cash compensaton. Gven our constructon that b b j f < j, the frst-excluded bd s the bd submtted by agent (q+1). We shall refer to the cost of ths frst-excluded bd as the cutoff cost and wll denote ths cost by b b q + 1 =. 12 Acceptng every ted bd may exceed the employer s budget constrant and encourage bdder colluson. 17

18 Recall that the employer wll provde compensaton package P ( = m, S) to any retaned employee, and that the cost of ths compensaton package s gven by cost ( P ) m cost ( = +. Because we have specfed that the cost P compensaton package for any retaned employee must satsfy have that = b, we m + cost S = b or that the cash compensaton provded any retaned employee s gven by m b cost ( =. Hence, for each q, employee s retenton package s gven by (, ) (, ) P = m S = b cost S S. As the followng lemma formalzes, t s mportant to recognze that every retaned employee receves a monetary ncentve greater than or equal to the amount requested n ther bd. LEMMA 2: For any employee retaned under CRAM, m m. Moreover, because each employee receves the exact set of NMIs requested, t s also the case that every retaned employee s utlty for the fnal retenton package receved wll be greater than or equal to ther utlty for ther package bd. LEMMA 3: For any employee retaned under CRAM, U ( P) U ( B). 18

19 E. Dfferences across Retaned Employees Although the cost to the employer s exactly the same for every retaned employee, not every retaned employee receves the same compensaton package. Dfferent employees may have submtted dfferent bds, B ( m, (and ultmately recevng) dfferent NMI combnatons. B S =, requestng Hence, f employees and j are both retaned wth B ( m, ( m, j j j = and =, these employees wll receve dfferent NMI packages whenever S. Furthermore, f cost ( S) cost ( S j), these two retaned employees wll also receve dfferent cash compensaton, wth m b cost ( j ( j) m = b cost S. = and Furthermore, even f two retaned employees and j do receve the exact same retenton package, the utlty enjoyed by these two retaned employees wll not necessarly be the same. Suppose we have P ( Pj m, S) = = for these two employees, they each receve the same cash compensaton and same set of NMIs. But they may value these NMIs dfferently, so they wll not necessarly receve the same utlty despte dentcal compensaton packages. More formally, f v S v S, then j U P = v S + m U P = v S + m. j j j Fnally, even f two retaned employees do receve the same utlty from ther respectve compensaton packages, they do not necessarly enjoy the same total employee surplus, because they lkely have dfferent reservaton values. Formally speakng, even f U( P) Uj( Pj) =, so long as r r j we wll have 19

20 U P r U P r and, therefore, the two employees wll receve dfferent j j j employee surpluses. In sum, even though the cost of all compensaton packages provded to retaned employees wll be the same under CRAM, (1) the NMIs receved by retaned employees may dffer, (2) the cash compensaton receved by retaned employees may dffer, (3) the utlty enjoyed by retaned employees may dffer, and (4) the surplus receved by retaned employees may dffer. VI. Optmal Bddng Strategy Havng fully descrbed the Combnatoral Retenton Aucton Mechansm (CRAM) and even begun characterzng outcomes under ths mechansm, we now turn to dervng the optmal bddng strategy for employees under CRAM. We conduct ths dervaton n two stages, frst dentfyng the optmal monetary bddng strategy and then dentfyng the optmal non-monetary bddng strategy. A. Optmal Monetary Bddng Strategy To understand an employee s optmal strategy for the monetary porton ( m ) of a CRAM bd, t s helpful to recall that the reservaton value r reflects employee s opportunty cost of beng retaned by the employer wthout any of the NMIs the employer has offered. Havng selected a set of NMIs ( S ) as part of hs CRAM bd, however, employee wll receve precsely those NMIs f retaned by the employer. Therefore, when determnng the optmal monetary porton ( m ) of a CRAM bd, employee must consder the revsed opportunty cost of beng retaned wth the chosen set of NMIs ( S ). Snce these NMIs provde employee 20

21 a beneft of gven by r r v ( v S f retaned, the revsed opportunty cost of beng retaned s =. In the lemma that follows, we show that employee s optmal bddng strategy nvolves submttng a monetary bd that truthfully reveals ths revsed opportunty cost. LEMMA 4: Gven any reservaton value r R, any set of competng bds B-, and any set of NMIs S B N, employee s utlty under CRAM from bd = ( m, wll be maxmzed f m satsfes m r r v ( = =. Lemma 4 essentally says that submttng a monetary bd of m = r = r v S s an optmal bddng strategy. Wth the next lemma, we show that such a monetary bd s, n fact, the only optmal bddng strategy. LEMMA 5: Gven any reservaton value r R and any set of NMIs S B ( m, N, bd = maxmzes employee s utlty under CRAM for any set of competng bds, B-, f and only f m r r v ( = =. B. Optmal Non-Monetary Bddng Strategy In the prevous sub-secton, we demonstrated that the unque optmal monetary bd under CRAM s m r r v ( S = =, for any gven set of NMIs N. In ths sub-secton, we characterze the optmal non-monetary bddng 21

22 strategy to accompany the now-establshed optmal monetary bddng strategy. In partcular, we show that the optmal non-monetary bddng strategy s to select a set of NMIs S that maxmzes employee s NMI surplus, whch, recall, s gven by surplus( v ( cost (, =. LEMMA 6: For any reservaton value r R, submttng bd B ( m, m = r v S and S argmax S N surplus,s under CRAM for any set of competng bds B-. = where ( ) maxmzes employee s utlty Lemma 6 essentally says that selectng a set of NMIs S that maxmzes employee s NMI surplus whle submttng a monetary bd of m = r v( s an optmal bddng strategy. Wth our frst theorem, we show that ths bddng strategy s, n fact, the only optmal bddng strategy. THEOREM 1: Gven any reservaton value r R, bd B ( m, = maxmzes employee s utlty under CRAM for any set of competng bds, B-, f and only f = = and S argmax surplus(, S) m r r v S S N. As for the proof of Theorem 1, the f porton of ths theorem was already covered n Lemma 6 and the only f porton of ths porton of ths theorem was partally covered n Lemma 5, wth respect to the monetary bd m. Thus, we must only prove the only f porton for the NMI bd S. In other 22

23 words, we must show that a bd B ( m, S) and m r v( S) = wth S argmax surplus(, S) S N = does not maxmze utlty under CRAM for all sets of competng bds B-. S We have now proven that submttng a bd B ( m, argmax ( surplus(, S) ) and m r v( S) S N = wth = s the unque weakly domnant bddng strategy under CRAM. C. Costs and Utlty Under the Optmal Bddng Strategy Immedately followng from Theorem 1, we have two corollares that characterze the equlbrum employee costs-to-retan and retenton utlty under CRAM. COROLLARY 1: The cost-to-retan assocated wth the optmal bd of any employee under CRAM s gven by b r max surplus(, S) =. S N Corollary 1 ndcates that, the greater the maxmum potental NMI surplus for any employee, the lower the employee s cost-to-retan and the more lkely that the employee wll be retaned. COROLLARY 2: Any employee who submts the optmal bd under CRAM wll receve a retenton package P generatng utlty max (, ) U P = b + surplus S f retaned. S N 23

24 Corollary 2 ndcates that, the greater the maxmum potental NMI surplus for any employee, the greater the employee s utlty f retaned. In combnaton, these two corollares tell us that, the greater an employee s maxmum NMI surplus, the more lkely t s that employee wll be retaned under CRAM and the better off the employee wll be f, n fact, retaned. VII. Benchmark Mechansm: Monetary Retenton Aucton To evaluate the relatve performance of the Combnatoral Retenton Aucton Mechansm, we must compare CRAM to ts logcal alternatve as a benchmark. In ths case, the tradtonal method of motvatng retenton s to offer a unform monetary ncentve to all potental retanees. In practce, the amount of any such monetary retenton ncentve s determned usng some mperfect estmaton method. However, an aucton s the most cost-effectve and welfare maxmzng approach to settng a monetary retenton ncentve (and determnng whch employees to retan). Therefore, let us formally descrbe the logcal best alternatve to CRAM as a monetary retenton aucton. Furthermore, for consstency, we wll consder the unform-prce aucton format. One can consder the monetary retenton aucton as a subset of CRAM, where the set of NMIs N = { }. To dstngush the two aucton mechansms, we use the followng notatons for the monetary retenton aucton. Consder a monetary retenton aucton n whch each employee submts a sngle monetary bd m ˆ and the q employees retaned by the employer are those who submt the q lowest monetary bds. Wth a unform-prcng rule, each retaned employee receves the same monetary retenton ncentve, whch s set equal to the (q+1) th lowest bd. Let us denote the amount of ths unform 24

25 monetary retenton ncentve (or, alternatvely, the amount of the (q+1) th lowest bd) by ˆm. The domnant strategy n a unform-prce monetary retenton aucton s for each bdder to truthfully-reveal ther reservaton value r by bddng mˆ = r. Thus, each retaned employee receves a monetary retenton ncentve equal to ˆ m r q + 1 = and enjoys a surplus of for the employer s equal to qm ˆ. ˆ q+ 1 m r = r r. Further, the total retenton cost In the sectons that follow, we wll compare CRAM s performance to the just-descrbed alternatve of a monetary retenton aucton, consderng employer cost, employee surplus, and overall socal welfare. VIII. Employer Cost Frst we compare CRAM s performance to a monetary retenton aucton n terms of overall employer cost. LEMMA 7: For any I and any set of NMIs N, the employer s cost to satsfy employee s optmal bd under CRAM (.e., the mnmum cost to retan employee ) s less than or equal to the cost to satsfy employee s optmal bd under a unform-prce monetary retenton aucton: b mˆ. The above lemma ndcates that all employees wll submt weakly lowercost bds under CRAM than under a unform-prce monetary retenton aucton. One can consder CRAM as a monetary retenton aucton wth more flexblty, whch can therefore outperform the monetary aucton. More specfcally, from the logc of the proof, we can say the followng: as long as an employee values some 25

26 NMI (or combnaton of NMIs) greater than the cost to provde that NMI (or those NMIs), the employee wll submt a strctly lower-cost bd under CRAM than under a unform-prce monetary retenton aucton. An employee wll never (optmally) submt a hgher-cost bd under CRAM than under the monetary aucton, and the only scenaro n whch an employee would submt bds of dentcal cost under each mechansm s when no NMI (or combnaton of NMIs) provdes value greater than the cost to provde that NMI (or combnaton of NMIs). Knowng that employees wll optmally submt weekly lower-cost bds under CRAM than under the monetary retenton aucton, t s not surprsng that the actual total retenton cost under CRAM s less than the cost under a monetary aucton. The followng Theorem formalzes ths result. THEOREM 2: Gven any set of employees I, any number of retanees q I, and any set of NMIs N, the cost-per-retanee under CRAM s less than or equal to the cost-per-retanee under a monetary retenton aucton. In other words, b mˆ. Theorem 2 ndcates that CRAM wll weakly outperform a unform-prce monetary retenton aucton n terms of mnmzng employer cost. Whle the theorem only demonstrates that CRAM wll cost no more than the monetary aucton n ths dmenson, t s mportant to note that there are many scenaros n whch CRAM wll strctly outperform the monetary aucton n terms of mnmzng cost. Example 1 n Appendx B llustrates one such scenaro. Furthermore, Example 1 llustrates that the employees are not necessarly better off under CRAM. 26

27 IX. Employee Utlty In ths secton, we llustrate that CRAM may ncrease or decrease employees utlty relatve to the benchmark unform-prce monetary aucton. Example 1 showed that CRAM can produce lower combned employee surplus (40 vs. 60) and utlty (160 vs. 180) than a monetary retenton aucton. Example 2 of Appendx B demonstrates that the opposte may be true. In other words, CRAM may strctly ncrease total employee utlty and surplus relatve to the benchmark unform-prce monetary retenton aucton under some condtons. We now compare the utlty level for all the employees under CRAM to the monetary retenton aucton n a general framework. For a gven set of NMIs N, employees I, and number of retanees q I, denote C= Ib< b as {, } the set of employees retaned under CRAM and ˆ ˆ M= { Im, < m} as the set of employees retaned under the monetary retenton aucton. There are four possble retenton condtons: employees not retaned under ether CRAM or a monetary aucton; employees retaned under ether a monetary aucton or CRAM, but not under both; and employees retaned under both auctons. We consder all four cases to completely contrast employee utlty as well as a Paretomprovement condton. Frst, for employees who are not retaned under ether mechansm, / M C, they receve ther reservaton utlty, r, under both auctons. For anyone retaned under a monetary aucton but not retaned under CRAM, M C c, utlty wll be lower under CRAM than the monetary aucton: U ( ˆm ) > r. Employees not retaned under CRAM receve ther reservaton utlty, but receve utlty greater than ther reservaton utlty f retaned under a monetary aucton. For anyone retaned under CRAM but not under a monetary 27

28 aucton, M c C, utlty wll be hgher under CRAM than the monetary aucton: U (P ) > r. These employees only receve ther reservaton value under a monetary aucton, but receve utlty greater than ther reservaton utlty under CRAM. Fnally, consder employees who are retaned under both mechansms, M C. CRAM s frst excluded bd cost s weakly smaller than the monetary aucton s frst excluded bd: b ˆm. Therefore, utlty may ncrease or decrease between CRAM and monetary aucton for employees retaned under both mechansms. The result s stated n the followng proposton: PROPOSITION 1: In order to have a Pareto-mprovement for the employees M C when swtchng from the monetary retenton aucton to CRAM, the followng condton must hold: mn max surplus(, S) mˆ b MI C S N wth at least one person havng ˆ surplus(, S) > m b. Proposton 1 states that the employees retaned under both auctons wll do at least as well by swtchng from monetary retenton aucton to CRAM f the employee wth the smallest maxmum surplus from the NMIs s at least as large as the decrease n the frst excluded bd. Furthermore, the change would be a Paretomprovement f at least one employee has an NMI surplus greater than the decrease n the frst excluded bd. The followng corollary provdes a condton n whch there wll be a Pareto-mprovement for all employees and the employer. 28

29 COROLLARY 3: If b = ˆm, then C = M. Furthermore, there s a Paretomprovement by all retaned employees, unretaned employees, and the employer f at least one retaned employee has surplus(, > 0. Corollary 3 states that f the frst excluded CRAM bd equals the frst excluded monetary retenton aucton bd, then both mechansms must retan the same set of employees. Furthermore, f at least one of the retaned employees has a strctly postve value from the NMIs receved, ths wll ensure a Paretomprovement, not just among a subset of people but over all employees (retaned and unretaned) and the employer. Table 2 summarzes the result from ths secton. By swtchng from the monetary retenton aucton to CRAM: Table 2. Changes n employee s utlty by swtchng from Monetary Retenton Aucton to CRAM M C M C c M c C M C U : no change U : decreases U : ncreases U : can ncrease or decrease. See Proposton 1. 29

30 X. Socal Welfare In the prevous two sectons, we demonstrated that CRAM generates lower employer costs than a monetary retenton aucton, but also that CRAM may generate lower employee utlty. The crtcal remanng queston, therefore, s whether CRAM maxmzes socal welfare. In partcular, are CRAM s cost savngs greater than or equal to any reducton n employee utlty? In ths secton, we prove that the answer to ths queston s Yes. Snce we compare socal welfare under CRAM to socal welfare under the monetary retenton aucton, we wll contnue to use the notaton for people retaned under CRAM and the monetary retenton aucton as C= { Ib, < b} and ˆ ˆ M= { Im, < m}, respectvely. Frst, we defne socal welfare n ths envronment as total employee utlty (both retaned and unretaned) mnus total employer costs: C( U (P ) b ) + ( r ) for CRAM; and C ( U C ( ˆm ) ˆm ) + ( r M ) M C for the monetary retenton aucton. Ths defnton recognzes that we have explctly defned utlty functons for the employees, but we have not done so for the employer. We have only sad that the employer s objectve s to retan q employees at the lowest possble cost. Therefore, t s natural to measure socal welfare as utlty mnus cost n ths context. To prove that socal welfare s hgher under CRAM than under a monetary retenton aucton, we compare socal welfare for each of the four dfferent sets of employees defned above. Our frst lemma nvestgates the set of employees who are retaned under both CRAM and a monetary retenton aucton, M C. 30

31 LEMMA 8: For any set of employees I, quantty of retanees q, and set of NMIs N, f M C, then U ( P) b U ( mˆ ) mˆ for M C. Furthermore, U (P ) b M C U ( ˆm ) ˆm M C. Lemma 8 states that socal welfare (employee utlty mnus employer cost) s hgher under CRAM for any employee retaned under both mechansms. Because ths s true for each ndvdual employee belongng to the set M C, t s also true for the total socal welfare assocated wth the entre set. Ths result s dfferent from Proposton 1 because Lemma 8 states that the value of socal welfare s at least as large under CRAM as under a monetary retenton aucton n all cases. For the sets C and M, t s not necessarly true that the socal welfare assocated wth ether of these two sets ndvdually s hgher under CRAM. On the other hand, the followng lemma tells that the total socal welfare assocated wth the two sets combned s, n fact, hgher under CRAM. LEMMA 9: Suppose M C C and j M C C. Then, for any I, q I, and N, the socal welfare for, j, and the employer are hgher under CRAM than under the monetary retenton aucton: U (P ) b + r j U j ( ˆm ) ˆm + r. Furthermore, ( ) + ( ( ˆ ) ˆ ) + U P b r U m m r. C C j C j C M I C j MI C j MI C M I C Lemma 9 says that, for any par of employees, one who s only retaned under CRAM and one who s only retaned under the monetary aucton, the total socal welfare assocated wth ths par combned wll always be hgher under CRAM. Because there are an equal number of employees who fall nto each 31

32 category (.e., M C C = M C C ), the entrety of both sets can be broken nto such pars, each of whch has hgher socal welfare under CRAM. Consequently, the socal welfare assocated wth the combned set of employees belongng to ether CRAM only or monetary retenton aucton only, M C C ( M C ) C, s also hgher under CRAM. Wth these two lemmas n place, the followng theorem becomes straghtforward and shows the ncrease n socal effcency. THEOREM 3: For any I, q I, and N, the total socal welfare for all employees I and the employer s weakly hgher under CRAM than monetary retenton aucton: ( U ) ( ˆ ˆ ( P) b + C r U( m ) m ) + C( r). C C M M Theorem 3 thus ndcates that, not only does CRAM reduce employer cost, t also ncreases total socal welfare. Hence, whle there are some condtons n whch CRAM mght lower employee utlty (relatve to the monetary retenton aucton) as stated n Secton IX, n net, the gan n socal welfare outweghs the loss n the welfare. Moreover, there are many condtons (such as Example 2 n the prevous secton) n whch CRAM wll both reduce employer cost and ncrease employee utlty. XI. Summary and Issues for Further Research Employers often have an opportunty to offer employees non-monetary compensaton that employees value well n excess of the employer s cost of provson. However, employee preferences across NMIs are dverse. What s 32

33 valuable to some has lttle or no value to others. As stated earler, surveys of mltary servce members llustrate the dffculty of dentfyng any NMI that has sgnfcant value for even 50 percent of the servce members surveyed, but also show that approxmately 80 percent of the surveyed servce members expressed a sgnfcant value for at least one NMI. These surveys show that employers could reduce compensaton costs by relyng more heavly on NMIs. However, the key to explotng ths potental s personalzng the employees NMI packages to reflect ther ndvdual preferences. CRAM provdes a mechansm to accomplsh ths objectve when settng employee retenton bonuses, though t can easly be extended to voluntary separaton ncentves and other areas of employee compensaton. CRAM s a reverse unform prce aucton that combnes monetary compensaton wth the costs of an ndvdualzed set of NMIs to create a sngle total retenton cost parameter. CRAM retans the least expensve total cost employees, provdng each a compensaton package wth a cost equal to the cost of the frst excluded bd. Each employee receves ther requested NMIs and a cash bonus equal to the total cost of the frst excluded bd mnus the total cost of that employee s package of NMIs. Ths paper has demonstrated that CRAM s a domnant strategy ncentve compatble mechansm. The weakly optmal strategy for any employee s to select the set of NMIs that maxmze surplus value (the employee s value mnus the total provson costs) and nclude a cash request so that the bd s total value to the employee equals the employee s reservaton value of employment. Compared to a reverse unform prce monetary aucton, CRAM s never more expensve than the purely monetary compensaton, and often less expensve. Furthermore, CRAM provdes at least as much, and often greater, total socal welfare compared to a monetary aucton. 33

34 However, the employee outcomes under CRAM are more complcated. Ths s most obvous consderng that potentally dfferent sets of employees are retaned under CRAM and a monetary aucton. In fact, some employees wll be better off under CRAM, ncludng those retaned under CRAM but not retaned n a monetary aucton; some employees are better off under a monetary aucton, ncludng those retaned under a monetary aucton but not under CRAM; some employees are ndfferent, ncludng those not retaned under ether aucton; and some may be worse off or better off dependng on how much the cost of frst excluded bd has changed, ncludng those who were obtaned n both CRAM and monetary aucton. Consderng the expected reducton n employer cost and ncrease n total socal welfare, n conjuncton wth the truth-revealng attrbutes CRAM offers, CRAM appears to be an attractve approach to settng retenton compensaton n the mltary personnel system, and provdes potental for a much broader range of applcatons. Ths s partcularly mportant when there s an ncrease n pressure on the mltary budget. One concern regardng both the current posted-prce mltary retenton process and CRAM or a smple monetary aucton, observes that all three process retan the least expensve employees (most wllng to serve or work). There may be cases where an employer would pay a premum to retan hgher qualty employees or to ncrease the flexblty of the type of employees retaned. Qualty Adjusted Unform Prce Aucton (QUAD) (Myung 2013), s a mechansm developed precsely to control for qualty of employees retaned. QUAD mproves the employer s ablty to control cost and the number of employees retaned, and also the qualty of employees retaned whle stll beng a domnant strategy ncentve compatble mechansm. Myung (2013) argued that, for the DoD s retenton and separaton problem, there are three mportant postve 34