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 * March 23, 2016 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 the Combnatoral Retenton Aucton Mechansm (CRAM) ntegratng both monetary and nonmonetary ncentves (NMIs). CRAM combnes the cash bonus and NMIs each employee demands nto a sngle cost parameter, retans the lowest cost employees, and provdes each wth an ndvdualzed compensaton package whose total cost equals the cost of the frst excluded bd. We provde an optmal bddng strategy that s domnant strategy ncentve compatble, characterze whch employees are better off or worse off, demonstrate the potental cost savngs, and document the ncrease n socal welfare from utlzng CRAM nstead of cash compensaton alone. Keywords: Market Desgn; 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: * Correspondng author: Noah Myung

2 0. Introducton We propose a mechansm called the Combnatoral Retenton Aucton Mechansm (CRAM), whch allows an organzaton to utlze both monetary and non-monetary ncentves (NMIs) for employee retenton, selecton, or promoton. CRAM s a reverse unform-prce mult-tem aucton n whch a sngle employer dentfes the lowest-cost subset of employees to hre, retan, or promote. By ncorporatng a contnuous monetary ncentve nto the tradtonal combnatoral aucton envronment n whch bdders have dscrete, and possbly nterdependent, valuatons for multple non-monetary goods or servces, we are able to desgn a mechansm that not only acheves unform cost outcomes, but whch s also domnant strategy ncentve compatble. The mechansm s mplementaton s straghtforward: CRAM elcts employees reservaton values by askng them the mnmum monetary wage or bonus and set of NMIs they requre to be selected for employment or promoton. For each employee, the cost of provdng the requested cash ncentve n addton to the set of NMIs s calculated and presented as a sngle cost parameter. Then, CRAM selects any specfed number of lowest-cost employees, wth each selected employee recevng the set of NMIs he or she requested along wth a cash ncentve at least as large as the monetary amount requested. In partcular, each selected employee receves a monetary ncentve equal to the total cost of the frst-excluded bd (from the lowest-cost employee not selected) mnus the total cost of the selected employee s set of NMIs receved. Therefore, as would be the case wth a standard unform-prce monetary aucton, the cost of each selected employee equals the cost of the frst excluded employee. CRAM s domnant-strategy ncentve-compatble; t s weakly domnant for each employee to reveal hs or her true reservaton value and true NMI values by (1) choosng the set of NMIs that maxmzes the dfference between the 1

3 employee s value of NMIs and the employer s stated cost of NMIs and (2) announcng the mnmum addtonal monetary ncentve requred for voluntary selecton. In what follows, we wll prmarly dscuss the applcaton of CRAM n the context of employee retenton, although the mechansm can also be appled to the selecton of employees and the desgn of compensaton packages for purposes of hrng or promoton. A general but a bref dscusson about combnatoral auctons and the applcaton of CRAM to the U.S. 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. Our contrbutons are as follows: Frst, we provde the general framework, characterzaton, and propertes of CRAM. CRAM provdes a smple and straghtforward way of determnng the retenton cost of each employee, whch set of employees should be retaned, and whch benefts (monetary and nonmonetary) should be provded to each employee. Ths process lessens the burden on the employer (who acts as the auctoneer under CRAM) as well as the employees (who are the bdders or aucton partcpants). 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. Each employee maxmzes hs welfare or utlty by selectng the set of NMIs that maxmzes hs total surplus (total value mnus total cost for the entre NMI combnaton), and specfyng the addtonal cash compensaton he requres (n addton to hs chosen NMIs) to be retaned. The cash amount bd plus the value of the NMI combnaton chosen should equal the reservaton value for each employee. Next, we show that the cost of retanng employees va CRAM s less than or equal to the cost of retanng employees under even the most effcent and costeffectve cash compensaton system: A monetary retenton aucton. Ths result s 2

4 drven by the fact that CRAM takes advantage of the surplus generated by offerng NMIs that employees may value more than the cost to provde. Thus, under CRAM, the employer can potentally provde an employee the same value as a cash ncentve but at a lower cost by ncorporatng NMIs. Note, moreover, that the CRAM approach s actually a more general framework n whch a monetary retenton aucton s smply a specal case of CRAM (n whch the set of NMIs offered s empty). Because CRAM wth NMIs may retan a dfferent set of ndvduals compared to the monetary aucton, determnng whom the mechansm benefts s not straghtforward. Therefore, we also compare an employee s utlty under CRAM to utlty under the monetary retenton aucton and show whch sets of employees are better off and whch are worse off under varous condtons. 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 compensaton greater 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 the amount of change n the cost per employee and the surplus (value mnus cost) the employee generates through the NMIs. An employee retaned under both mechansms wll be better off under CRAM f the cost per employee does not drop by more than hs gan n surplus from the NMIs. Fnally, we compare the socal welfare, the sum of both retaned and not retaned employees' utlty mnus employer s cost, and show that socal welfare s (weakly) greater under CRAM than under the monetary retenton aucton. There are two mportant caveats to keep n mnd. Frst, the approach of our work s n lne wth practcal market desgn (Roth 2002) rather than wth 3

5 optmal mechansm desgn. Second, the desgn of CRAM s prmarly motvated by efforts to mprove the exstng retenton process used by the Department of Defense, whch currently utlzes a strctly monetary bonus system for such purposes. In partcular, we developed CRAM to reduce retenton cost, accurately retan the desred number of servce members, and mprove the effcency of NMI dstrbuton. I. Combnatoral Auctons A. General Characterstcs and Applcatons 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 possble combnatons of goods (ncludng the empty set) 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 dffculty fndng an optmal soluton. Whle combnatoral auctons have always been of nterest, the feld has seen the greatest growth wth ts applcaton to the Federal Communcaton Commsson (FCC) spectrum auctons. Between 1994 and 2003, the FCC utlzed some form of combnatoral aucton 41 tmes, rasng over $40 bllon n revenue (Kwasnca, Ledyard, Porter, and DeMartn 2005). Even pror to ths hghly publczed utlzaton by the FCC, combnatoral auctons had been employed 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) documented varous combnatoral auctons that have been utlzed to solve complcated government and non-government 4

6 allocaton problems, ncludng use for energy tradng by the Arzona Energy Exchange, gas delvery by the Federal Energy Regulatory Commsson, payload manfest for Space Shuttle, resource allocaton for Cassn msson to Saturn, tran schedulng, transportaton servces, polluton markets, and markets to exchange fnancal portfolos. B. Unque Challenges of Combnatoral Auctons Such combnatoral auctons face problems not usually encountered wth sngle-object auctons, ncludng the followng challenges detaled by Pekeč and Rothkopf (2003): 4 The Exposure Problem Frst, mult-tem auctons are subject to the exposure problem, n whch bdders face the rsk of wnnng unwanted tems. For example, consder an aucton of two tems a and b. If these two tems are perfect complements (such that a and b only have postve value when combned) then, f bds on the combnaton {a, b} are not allowed, bdders are exposed to the rsk of wnnng only one (worthless) tem f they must submt separate bds on the ndvdual tems. Alternatvely, f the two tems are perfect substtutes (such that a bdder would lke to wn a or b but not both) then bdders mght be exposed to the rsk of wnnng both tems when only one or the other s wanted. The Communcaton Complexty Problem The exposure problem above suggests that, whenever the tems beng auctoned are complements or substtutes (to any degree), t may be nsuffcent for the aucton to accept only bds on ndvdual tems. Instead, bdders must potentally submt bds on all possble combnatons of tems he or she mght receve. Even wth a relatvely small 4 Pekeč and Rothkopf (2003) also dscuss the threshold problem, but ths concern s not detaled here as t s not partcularly relevant n the present context. Other authors also add jump bddng as a problem for combnatoral auctons but, as dscussed by Isaac, Salmon, and Zllante (2004), jump bddng s not necessarly a dsadvantage from the auctoneer s perspectve and, n fact, can ncrease revenue. 5

7 aucton nvolvng only 7 tems, for example, ths could potental demand as many as = 127 bds from each bdder. Thus, combnatoral bddng s hghly complex n terms of the communcaton, calculaton, and even cognton requred. The Wnner Determnaton Problem It can be much more dffcult to determne wnners n a combnatoral aucton because, unlke a sngle-tem aucton, the hghest bd on a sngle (or package of tems) s not guaranteed to wn that tem (or package). Ths s because some alternatve combnaton of bds mght generate hgher revenue. Fndng ths revenue-maxmzng 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? Dependng upon the aucton desgn, any or all of the above problems mght also be present to some degree n the current context of bddng on combnatons of non-monetary ncentves (NMIs). As we wll dscuss, however, the partcular format of bd submsson we propose can crcumvent both the exposure and communcaton complexty problems whle favorable features of the envronment (and, n some cases, reasonable smplfyng assumptons) greatly mnmze the wnner determnaton problem. C. Establshed Combnatoral Aucton Formats Wthn the combnatoral aucton famly, the followng are some aucton formats that have drawn consderable attenton: 1. Smultaneous Multple Round Aucton (SMR): An aucton format utlzed by the FCC whch does not allow for package bddng, SMR s often used as a benchmark comparson to other combnatoral aucton desgns. 6

8 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 ncorporatng 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 bddng functon. We do not nclude the detals of these aucton mechansms here, but nterested readers may consult the provded references. 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. Wth the wde varety of combnatoral aucton mechansms proposed and employed, t s natural to ask whether the classc Vckrey-Groves-Clarke (VCG) mechansm for sngle-tem aucton desgn (Vckrey 1961, Clarke 1971, Groves 1973) could be generalzed and appled to bddng for combnatons of tems. Such a combnatoral VCG mechansm can ndeed be generated and even retans the desrable features of allocatve effcency and truth revelaton among bdders. As Pekeč and Rothkopf (2003) explan, however, VCG mechansms are both mpractcal and unattractve and therefore rarely used n the combnatoral context, because such mechansms nvolve refundng to bdders the ncrease n value caused by ther bds. Snce these refunds can be sgnfcant, 7

9 the VCG mechansm s revenue defcent. In addton, such mechansms may be subject to manpulaton by ether bdders or bd-takers (va colluson, nsncere bddng, or false name bddng). D. Applcaton to Retenton and Non-Monetary Incentves Due to nsttutonal restrctons, the specfc aucton mechansms mentoned above cannot be drectly or easly adapted and appled to the retenton problem. Frst, the above aucton formats are forward auctons that prmarly deal wth sellng objects. Procurement auctons, or reverse auctons, are auctons where the auctoneer s nterested n buyng goods and servces nstead of sellng. Therefore, retenton auctons are more smlar to such procurement auctons. There are many dfferences between procurement auctons and retenton auctons, however, agan due to nsttutonal features. For example, n procurement auctons, buyers can procure half of the goods and servces, or splt the award among multple provders n order to keep the bdders compettve 5 (Chaturved, Bel, and Martnez-de-Albenz 2013). In the actve-duty mltary, however, 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 next 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 a combnatoral aucton can be 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 5 Awardng the procurement to only one vendor may make that vendor a monopoly n the future due to technologcal advancement. 8

10 or may not be sub or super modular, t may be best left for each bdder to choose the optmal set of hs own NMIs. For example, a bddng employee may value the NMI of geographc stablty hghly, but only n combnaton wth another NMI whch provdes hm the job locaton of hs choce (such that the two NMIs are complements). Another bdder mght assgn hgh value ether to a more flexble work schedule or to the opportunty to telework, but may not ascrbe a relatvely hgh value to the combnaton (such that the two NMIs are substtutes). Allowng each bdder (rather than the bd-taker) to select for hmself the combnaton of NMIs that generates the most value (relatve to cost) greatly smplfes the nformaton aggregaton task of the mechansm. Moreover, 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 U.S. Department of Defense There s no more natural and valuable applcaton for CRAM than to the retenton problem wthn the armed forces. Wth over 1.4 mllon actve duty and 1.1 mllon Reserve and Natonal Guard servce members n 2013 (DoD 2013), the U.S. mltary s labor force s not only dstnctve n ts sze but also because t s nternally grown, wth strctly nternal promotons and no external hres beyond entry-level postons. For 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 from another organzaton. Instead, the Navy must promote from wthn. Therefore, the DoD and each of the servces must carefully plan ts force structure over the long term, wth precse retenton and promoton decsons beng extremely crtcal force-management actvtes. 9

11 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 (such as educaton and health care benefts), and 28.1 percent of the compensaton s deferred compensaton (such as retrement pay accrual) (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, however, the DoD also needs to fnd savngs n ts cost of compensaton. Specal and Incentve (S&I) pays are authorzed by law to provde the mltary servces the flexblty needed for recrutment, retenton, and separaton. (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 three-year 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 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 U.S. Army (pay grade E-6) wth 10 years of total servce and three dependents 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. 10

12 would get annual cash compensaton 8 of $60, An average Captan n the U.S. 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 overall 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 effcency and effectveness of dstrbutng NMIs. The DoD has been lmted to utlzng a posted-prce format for provdng the S&I bonuses mentoned above, ncludng selectve reenlstment bonuses. 9 Furthermore, these bonuses are provded as purely monetary compensaton, thus forgong any surplus that may be ganed by ncorporatng NMIs as well. Coughlan, Gates, and Myung (2014), CGM henceforth, descrbed the addtonal surplus that the DoD can potentally gan by provdng personalzed NMI packages. Furthermore, CGM stressed the mportance of utlzng NMIs, the dffculty and neffcency of provdng a unversal ncentve package 10 of NMIs, as well as the extreme varablty n preference for NMIs by servce members across 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 servce members expressed a sgnfcant value for at least some NMIs. 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. 10 Unversal Incentve Package means that everyone receves the same set of NMIs. 11

13 As wth desgnng any market, the market desgner must consder mportant normatve and postve characterstcs that the market user may value. For example, Pekeč and Rothkopf (2003) dscussed that some of the key consderatons when 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 emphaszes the followng normatve characterstcs as crtcal features of any rectenton mechansm: 1. Egaltaran Percepton of equalty: The mltary s of a strong mndset that servce members of equal rank and poston should get equal pay. Hence, n terms of S&I pay, everyone recevng the specfc S&I bonus should get the same bonus amount. 2. Domnant strategy ncentve compatble mechansm Transparency and ease of use: The mltary prefers a mechansm that s easy to understand and mnmzes strategc gamng by the partcpants. 3. Low transacton cost Mnmal tme requred of 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. 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. 12

14 Takng these features nto consderaton, we beleve that CRAM s optmally desgned for the dual problems of retenton and dstrbuton of NMIs wthn the DoD. III. Outlne We descrbe the general envronment for CRAM 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, Secton IX explores employee utlty, and Secton X compares total socal welfare under the alternatve mechansms. We end wth conclusons n Secton XI. IV. The Envronment A. The Retenton Problem Let I be a set of employees currently qualfed for and seekng retenton wth a gven employer. The employer wll retan q I of these employees. Each employee I ultmately retaned by the employer wll receve a monetary ncentve, m R, as well as some combnaton of NMIs n hs or her ultmate compensaton package. Denote by N the set of all NMIs offered by the employer and by S N the subset of NMIs potentally receved by employee I. 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 (ncludng the empty set). We assume that each NMI s a non-rvalrous but excludable good (thus, each s a club good). 13

15 B. Employee Preferences Each employee s endowed wth a NMI valuaton functon v : S N! that calculates a dollar-equvalent value for any combnaton of NMIs. Each employee s utlty for any combnaton of a monetary ncentve (m ) and a non-monetary ncentve package (S ) can then be expressed by a quas-lnear utlty functon U ( m,s ) = v S ( ) + m. We normalze v ( ) = 0 for all I. Note that we also explctly allow for an employee s valuaton, v ( S ), of any combnaton of non-monetary ncentves to be modular, submodular, or supermodular wth respect to the valuatons of the ndvdual NMIs ncluded n that combnaton. In other words, the NMIs wthn a package mght be complements or substtutes 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. 11 Each employee s reservaton value, r, and NMI valuaton functon, v, are prvate nformaton. It should be noted that the productvty of employee s assumed to be ndependent of both hs reservaton value, r, and hs NMI valuaton functon, v. 11 Note that we do allow for the possblty that r < 0, such that the employee would actually be wllng to pay the employer or take a pay cut to be retaned. 14

16 Hence, the employer s ndfferent over whch employees are actually retaned, as long as q I employees are retaned and that the cost of dong so s mnmzed. 12 We denote the fnal compensaton 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 C. Employer Costs ( P ) = U m *, S ( ) = v S ( ) + m *. For each ndvdual NMI, s n S N wth n { 1,2,..., N }, the employer s cost to provde that partcular NMI to any ndvdual employee, cost s n ( ), s publc knowledge (or at least communcated to all employees pror to the retenton decson). 13 Note that, n the operaton of the mechansm, an NMI s cost ultmately and smply reflects the prce tag charged to any employee who receves that NMI. Therefore, the employer s actually free to publsh (and utlze n CRAM) a cost that s hgher than ts actual estmate of unt cost, perhaps to hedge aganst perceved rsk due to cost or demand uncertanty. The model and mechansm can also easly generalze to the scenaro n whch the cost of each NMI mght vary across employees such that the employer s cost to provde NMI s n to employee would be gven by cost ( s ) n but, for smplcty, we assume unform NMI costs n the dscusson that follows. Because each NMI s a non-rvalrous club good, provson of each NMI s characterzed by constant margnal cost. 14 Hence, the cost to provde any gven 12 A model ncorporatng observable qualty n a retenton aucton framework s ntroduced by Myung (2013). 13 The alternatve bd-method CRAM desgn, descrbed later n Appendx C, does not requre dsclosure of NMI costs. 14 Note that the assumpton of constant margnal cost also mples the absence of quantty constrants, such that each NMI could concevably be provded to all retaned employees. If, 15

17 NMI, n s, to any gven number of employees, x, s smply gven by x cost( s n ). Thus, 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 ). 15 s n S Therefore, the employer s total cost to provde a fnal retenton package ( m *, S ) * = to any retaned employee s gven by cost ( P ) m cost ( S ) ( ) * n cost P m cost( s ). = + n s S D. NMI Surplus = + or Wth ths understandng of employee preference and employer cost, t s helpful to defne the employee NMI surplus (value n excess of cost). For any bdder and any set of NMIs S, let surplus( S ) v ( S ) cost ( S ), =. 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 nether be negatve for all sets of NMIs nor postve for all sets of NMIs. nstead, bndng quantty constrants are present, the publshed cost of any over-demanded NMI could smply be rased and those employees who have selected that partcular NMI could be asked to revse ther NMI selectons based on the new, hgher cost. Ths process could be repeated as necessary untl the quantty of each NMI demanded no longer exceeds the quantty avalable. Alternatvely, the bd-method CRAM desgn descrbed n Appendx C allows for the approprate dstrbuton of quantty-constraned NMIs to be determned wthout repeated bddng by employees. 15 We can relax the assumpton of constant margnal cost and absence of economes of scale or scope n provdng NMIs, however ths adds computatonal complexty wthout addng sgnfcant value to the ntroducton of our mechansm. 16

18 max LEMMA 1: For any set of avalable NMIs N and any employee, ( surplus(,s) ) 0 and mn ( 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. Hence, total employee surplus for any retaned employee s equal to U v ( ) + m * r. S V. Combnatoral Retenton Aucton Mechansm (CRAM) ( ) r = A. Mechansm We formally outlne the mechansm n ths secton, wth detaled explanaton of the mechansm to be provded n the subsectons that follow. We also nclude two examples of the mechansm n Appendx C. Frst, defne B = ( m, S ) as the message or strategy beng submtted by the employee to the employer, where 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( S ). P 17

19 Wthout loss of generalty, let b b j f < j for all, j I and let b * represent * ( ) lowest cost bd, or b b q + 1 the 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 : B 1... B I X, where the outcome determnes the retenton and the compensaton package n the followng manner: P = (m *,S ) f q and b b* (0, ) f > q and b b * and retaned f q where m * = b * cost(s ). Therefore, persons wth bd such that > q are not retaned and receve ther reservaton value. B. Employee Bds Informally, the CRAM bddng process can be separated nto two decson elements for each employee: (1) selectng NMIs and (2) submttng a mnmum monetary ncentve (or cash compensaton) the employer must provde to retan that employee. For the frst decson element, employees must choose whch NMIs they desre from a menu n whch each NMI has an assocated (known) cost. As we wll detal below, the employer wll add the cost of each NMI selected to the employee s 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 have chosen from the menu, the monetary 18

20 ncentve bd reflects the mnmum cash amount an employee must receve n order to reman voluntarly employed, 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, S ) =, where m s the monetary ncentve and S s the combnaton of NMIs that employee requests 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, S ) =, the employer must provde that employee the set of NMIs, S, and cash compensaton of at least m. Thus, the mnmum 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 bds. That sad, note that, for all of the results n ths paper, t s not necessary that q, the number of employees to be retaned, be (1) known by the employees, (2) communcated to the employees, (3) estmable by the employees, or (4) even determned n advance by the employer pror to bd submsson. In fact, gven the nature of the mechansm, the employer could even choose the number of desred retanees after observng the bds submtted and calculatng the total cost of varous levels of retenton. Allowng the employer to do so would have no effect on the optmal bddng strategy, the cost savngs, or the socal welfare mplcatons of the mechansm. 19

21 Wthout loss of generalty, let the employees be labeled and ordered such that b bj f < j for all, j I. The employer wll then retan employee f and only f q. Any employee wth > q wll not be retaned and thus receves hs reservaton value, r. Note that a te at the margn s possble, such that there exst more than one set of q lowest-cost bds. Whenever ths occurs, t means that bq = b q + 1 and, qute possbly, that other employees have also submtted bds that all have the q th lowest cost. 16 For our analyss, we wll randomly break tes, wth some employees who submt a bd wth the q th lowest cost beng retaned, whle others who submt bds of the same cost not beng retaned. Note that many of our results wll hold wth weak nequalty because we are allowng for tes at b q and b q+1. If we do not allow for tes, our results would be stronger and hold wth strct nequalty. D. Compensaton for Retaned Employees Because CRAM s a unform-prce aucton mechansm, all retaned employees wll receve a total retenton 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. Even though each retaned employee wll receve a package of the same cost, however, the cash compensaton and NMIs ncluded n such packages could dffer sgnfcantly across retanees. Moreover, the utlty that retaned employees enjoy from ther compensaton packages could also dffer. 16 Note that tes n bd costs above b q+1 or below b q wll not change the unqueness of the set of q lowest-cost bds. 20

22 Gven our constructon that b b f < j, note that the frst-excluded bd j s the bd submtted by agent (q+1). We shall refer to the cost of ths frstexcluded bd as the cutoff cost and wll denote ths cost by * b b q + 1 =. The employer wll provde compensaton package P ( * m, S) = to any retaned employee, and the cost of ths compensaton package s gven by * ( ) ( ) cost P = m + cost S. Because we have specfed that the compensaton cost P package for any retaned employee must satsfy ( ) * ( ) * * = b, we have that m + cost S = b or that the cash compensaton provded any retaned employee s gven by m * = b * cost( S ). 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 hs or her 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 from the package requested or bd. LEMMA 3: For any employee retaned under CRAM, U ( P) U ( B). 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. 21

23 Dfferent employees may have submtted dfferent bds, B ( m, S ) dfferent NMI combnatons. B =, requestng Hence, f employees and j are both retaned wth B ( m, S ) ( m, S ) j j j = and =, these employees wll receve dfferent NMI packages whenever S S j. Furthermore, f cost ( S) cost ( S j), these two retaned employees wll * * also receve dfferent cash compensaton, wth m b cost ( S ) * * j ( j) m = b cost S. = and In addton, even f two retaned employees and j do receve the exact same retenton package, the utlty enjoyed by these two employees wll not necessarly be the same. Suppose, for example, that we have P ( * Pj m, S) these two employees. If v ( S) v ( S), then U P j = = for ( ) = v ( S) + m * U j P j ( ) = v j ( S) + m *, and employees and j wll experence dfferent levels of utlty despte recevng dentcal compensaton packages. 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 probably have dfferent reservaton values. Formally speakng, even f U( P) Uj( Pj) U( P) r Uj( Pj) rj =, so long as r r j we wll have and, therefore, the two employees wll receve dfferent 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 22

24 employees may dffer, (3) the utlty enjoyed by retaned employees may dffer, and (4) the surplus receved by retaned employees may dffer. F. Summary Flowchart of the Mechansm Fgure 1 summarzes the overall nputs and calculatons of CRAM. Inputs from the employer are shown n the black boxes, nputs from the employees are ndcated n the gray boxes, calculatons performed by the mechansm are llustrated n the whte boxes, whle transfers of nformaton from one stage to another stage are ndcated by the arrows. In addton, two examples are ncluded n Appendx B. Fgure 1: Summary of CRAM Inputs and Calculatons VI. Optmal Bddng Strategy Havng fully descrbed CRAM and even begun characterzng outcomes under ths mechansm, we now turn to dervng the optmal bddng strategy for 23

25 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 a beneft of v ( S ) f retaned, the revsed opportunty cost of beng retaned s gven by r = r v ( ). In the lemma that follows, we show that employee 's S optmal bddng strategy nvolves submttng a monetary bd that truthfully reveals ths revsed opportunty cost. S LEMMA 4: Gven any reservaton value r R and any set of NMIs N, bd B = ( m, S ) 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 S 24

26 S N. In ths sub-secton, we characterze the optmal non-monetary bddng 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( S ) v ( S ) cost ( S ), =. LEMMA 5: For any reservaton value r R and any monetary bd m R, employee s utlty from beng retaned under CRAM, U maxmzed f and only f he submts a bd B ( m, S) S argmax ( surplus(, S) ). ( ), wll be P = where Lemma 5 essentally says that an employee maxmzes the utlty of hs retenton package f and only f he selects a set of NMIs S that maxmzes hs NMI surplus. Wth our frst theorem, we show that selectng such a set of NMIs, whle submttng the optmal monetary bd descrbed n Lemma 4, s the only utlty-maxmzng bddng strategy. THEOREM 1: (Domnant Strategy Incentve Compatblty Theorem) Gven any reservaton value r R, bd B ( m, S ) = maxmzes employee 's utlty under CRAM for any possble sets of competng bds, B, f and only f m = r = r v ( S ) and S argmax ( surplus(, S) ). 25

27 Wth ths theorem, we have now proven that submttng a bd B = ( m, S ) ( ) wth S argmax surplus(, S) and m r v ( S ) domnant bddng strategy under CRAM. 17 = s the unque weakly C. Costs and Utlty under the Optmal Bddng Strategy Immedately followng from Theorem 1, we have two corollares that characterze the equlbrum employee cost-to-retan and retenton utlty under CRAM. COROLLARY 1: In the domnant strategy equlbrum n Theorem 1, the cost-to-retan assocated wth any employee under CRAM s gven by ( ( )) b = r max surplus, S. Corollary 1 ndcates that, the greater the maxmum potental NMI surplus for any employee, the lower the employee s cost-to-retan and, hence, the more lkely ths cost wll be below the cutoff cost and therefore the more lkely that the employee wll be retaned. COROLLARY 2: In the domnant strategy equlbrum n Theorem 1, any employee wll receve a retenton package P generatng utlty U ( P )= b * + max ( surplus(,s) ) f retaned. 17 Because we have allowed for the possblty that r < v (S ), there are scenaros n whch employee s optmal monetary bd, m = r v (S ), wll be negatve. Ths has no mpact on the results that follow. If, however, negatve monetary bds were dsallowed as a matter of polcy, ths would mpact the optmal bddng strategy for any employee for whom the optmal monetary bd would otherwse be negatve. For example, f r < 0 but negatve monetary bds are not permtted, employee would be better off choosng no NMIs and submttng a monetary bd of zero, as ths would be the lowest cost bd possble. Nonetheless, even n ths scenaro, CRAM would stll retan the lowest cost employees and, so long as the cutoff cost b * was suffcently postve that there would be no changes at the margn (.e., the set of ndvduals submttng the q lowest cost bds was unchanged), CRAM would also stll retan the set of employees most wllng to retan and the socal welfare results presented later n the paper would also stll hold. In other words, f negatve monetary bds were dsallowed, dffcultes would arse only f a sgnfcant number of employees were wllng to retan wthout any bonus whatsoever. Under such crcumstances, however, there would be no need for retenton ncentves n the frst place, renderng the entre problem addressed by ths paper moot. 26

28 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 (due to lower cost-to-retan) and the better off the employee wll be (due to hgher utlty) f, n fact, retaned. VII. Benchmark Mechansm: Monetary Retenton Aucton To evaluate the relatve performance of CRAM, we compare t to the tradtonal method of motvatng retenton by offerng 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 more cost-effectve and welfare enhancng approach to settng a monetary retenton ncentve (and determnng whch employees to retan). Therefore, let us formally descrbe the benchmark alternatve to CRAM as a monetary retenton aucton. Furthermore, for consstency, we wll consder the unform-prce aucton format. Such a monetary retenton aucton s actually a specal varaton of CRAM, n whch the set of NMIs N = { }, each employee smply 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 the unform-prcng rule, each retaned employee receves the same monetary retenton ncentve, whch s set equal to the (q+1) st -lowest bd. Let us denote the amount of ths unform monetary retenton ncentve or, alternatvely, the amount of the (q+1) st -lowest bd by It s well establshed that the domnant strategy n such a unform-prce monetary retenton aucton s for any employee to truthfully-reveal hs or her * ˆm. 27

29 reservaton value r by bddng mˆ = r. The monetary retenton ncentve, ˆm *, receved by any retaned employee wll then be at least as large as the amount bd by that employee. In other words, ˆm * ˆm = r. Hence, each retaned employee enjoys a surplus of ˆm * r, whle the total retenton cost for the employer s equal to * qm ˆ. In the sectons that follow, we wll compare CRAM s performance to the just-descrbed alternatve of a monetary retenton aucton, focusng on employer cost, employee surplus, and overall socal welfare. Please recognze, however, that the cost and welfare advantages of CRAM llustrated n what follows actually underestmate ts true advantages. Ths s because we are comparng CRAM not to retenton methods actually used (a posted-prce format wth the prce that s mperfectly estmated), but to a more effcent and less costly cash-only sealed-bd aucton method. Incdentally, because the CRAM mechansm actually addresses two human resources challenges at once namely, (1) employee retenton and (2) NMI allocaton one mght ask f there s another approprate benchmark mechansm aganst whch we can compare CRAM n terms of the second challenge of NMI allocaton. The actual method for dstrbutng NMIs n common practce, however, s a one-sze fts all approach: Each non-monetary ncentve s provded ether to all employees or to none. As explaned n Coughlan, Gates, and Myung (2014), however, ths approach s horrbly neffcent n that most non-monetary ncentves (n fact, all NMIs we nvestgated) are valued above cost by only a mnorty of the populaton to whch they may be provded. The only effcent soluton s to provde each NMI, or combnaton of NMIs, to only those ndvduals who actually value t above cost. Ths s acheved n all varatons of CRAM descrbed n Appendx C and, therefore, we could arguably compare the 28

30 descrbed varatons, for example, however that would seem to be outsde the scope of the current paper. Fnally, n the sprt of market desgn, we wll compare CRAM to the mechansm that s currently used: Cash-based compensaton. VIII. Employer Cost Frst we compare CRAM s performance to our benchmark alternatve, a monetary retenton aucton, n terms of overall employer cost. LEMMA 6: For any I and any set of NMIs N, the employer s cost to satsfy employee 's optmal bd under CRAM s less than or equal to the cost to satsfy employee 's optmal bd under a unform-prce monetary retenton aucton. In other words, b mˆ. The above lemma ndcates that all employees wll submt weakly lowercost bds under CRAM than under a unform-prce monetary retenton aucton. Moreover, from the logc of the proof, we can say the followng: As long as an employee values some set of NMIs greater than the cost to provde that set of 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 combnaton of NMIs provdes value greater than the cost to provde that combnaton of NMIs. Knowng that employees wll optmally submt weakly 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 29

31 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. Also, recall that such a unform-prce retenton aucton tself weakly outperforms other cash-only retenton mechansm, such as the posted-prce approach currently employed by the U.S. mltary. Furthermore, CRAM also offers employers the practcal advantage of beng able to mtgate rsk n the presence of uncertanty by rasng the cost or prce tags for NMIs. As noted prevously, the employer could set hgh costs for each NMI, perhaps well above actual costs, n whch case only employees wth exceedngly hgh valuatons would chose NMIs. For all other employees, CRAM would be equvalent to the benchmark monetary aucton. At the extreme, f NMI costs were set hgher than all employee valuatons, CRAM smply converts to a monetary retenton aucton for all partes. Whle Theorem 2 states that CRAM wll always cost no more than the monetary aucton, t s mportant to note that there are many scenaros n whch CRAM wll ndeed cost strctly less than the monetary aucton. Example 1 n Appendx B llustrates one such scenaro. Nonetheless, whle an employer wll always (weakly) prefer CRAM, Example 1 also llustrates that the employees are not necessarly better off under CRAM. Ths queston of employee utlty s the topc of the next secton. IX. Employee Utlty In ths secton, we derve the condtons under whch CRAM wll ncrease or decrease employees utlty relatve to the benchmark unform-prce monetary aucton. Frst of all, Example 1 of Appendx B showed that CRAM could produce both lower combned employee surplus (40 vs. 60) and lower total utlty 30