Incentive Compatible Mechanisms for Group Ticket Allocation in Software Maintenance Services

Transcription

1 14th Asa-Pacfc Software Engneerng Conference Incentve Compatble Mechansms for Group Tcket Allocaton n Software Mantenance Servces Karthk Subban, Ramakrshnan Kannan IBM R Ina Software Lab, EGL D Block, Sxth Floor, Bangalore, Ina. ksubban,rkrshnan@n.bm.com Raghav K Gautam, Y Narahar CSA Department, Inan Insttute of Scence, Bangalore, Ina raghavg,har@csa.sc.ernet.n Abstract A customer reporte problem (or Trouble Tcket) n software mantenance s typcally solve by one or more mantenance engneers. The ecson of allocatng the tcket to one or more engneers s generally taken by the lea, base on customer elvery ealnes an a gue complexty assessment from each mantenance engneer. The key challenge n such a scenaro s two fols, un-truthful (hke up) elctaton of tcket complexty by each engneer to the lea an the ecson of allocatng the tcket to a group of engneers who wll solve the tcket wth n customer ealne. The ecson of allocaton shoul ensure Invual an Coaltonal Ratonalty along wth Coaltonal Stablty. In ths paper we use game theory to examne the ssue of truthful elctaton of tcket complextes by engneers for solvng tcket as a group gven a specfc customer elvery ealne. We formulate ths problem as strategc form game an propose two mechansms, (1) Dvson of Labor (DOL) an (2) Extene Secon Prce (ESP). In the propose mechansms we show that truth tellng by each engneer consttutes a Domnant Strategy Nash Equlbrum of the unerlyng game. Also we analyze the exstence of Invual Ratonalty (IR) an Coaltonal Ratonalty (CR) propertes to motvate voluntary an group partcpaton. We use Core, soluton concept from co-operatve game theory to analyze the stablty of the propose group base on the allocaton an payments. 1 Introucton A trouble tcket (or synonymously a tcket) s a software problem as reporte by a customer to be analyze an fxe by a team of mantenance engneers. A basc moel currently beng followe n software mantenance process s shown n Fgure 1. The problem tcket can come to the organzaton through fferent nterfaces such as web nterface system, call centers, emals etc. The tcket receve through any such nterface wll then be channelze to a lea. The lea n turn takes the responsblty to allocate the tcket to one of the reportng engneers. The complexty of the reporte problem actually propagates from the bottom layer (engneer) to the top (lea) where as the allocaton an the payment happens n the opposte recton. Fgure 1. Typcal Software Mantenance Work flow In ths case an engneer may not fn t n hs best nterest to report the tcket complexty truthfully an hence boost the reporte value of tcket complexty for nvual selfsh benefts, whch may lea to n-effcent tcket allocaton. Hence, the central objectve of the tcket allocaton problem s to ensure that every nvual partcpatng n the allocaton oes not mprove hs payoff by revealng tcket complextes untruthfully. Often the problem tckets are solve n groups rather /07 $ IEEE DOI /ASPEC Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.

2 than by nvual engneers 1. The reason for such collaboraton may be ue to techncal complexty or a specfc customer elvery (tme) ealne. In such cases the tcket has to be allocate to a group of engneers (calle propose group) such that, each engneer n the propose group makes partal contrbuton to solve the tcket. The engneers n the propose group get pa 2 for ther porton of contrbuton to the overall problem. The ecson of ecng the propose group, ther contrbuton an payments s calle as a Group Tcket Allocaton Problem. The key challenge n ths problem s two fol; (1) Ensure truth elctaton by each engneer an (2) Dece the propose group that wll solve the reporte problem an the payments to engneers n the propose group. Along wth the above ecsons we shoul ensure that propose group s stable an all the engneers are motvate to partcpate n the game an the propose group (.e. they shoul not ncur loss by partcpatng n the game or the group). In ths paper we focus our attenton to solve the Group Tcket Allocaton problem an propose two nterestng mechansms. We analyze the followng four esrable propertes for each of the propose mechansm. Incentve Compatblty (IC): Each engneer fns n hs best nterest to reveal truth. Invually Ratonal (IR): Each engneer s not worse off by partcpatng n the game Coaltonally Ratonal (CR): Each engneer s not worse off by partcpatng n the coalton (or group). Stablty: The stablty of the mechansm ensures that any subset of players from the propose coalton wll not have any benefts to break-away from the propose coalton. 1.1 Revew of Relevant Work A few analytcal approaches have been explore to mprove the effcency of the software mantenance process. The work by Kulkarn et al. [2] moels the mantenance process usng queueng theory an entfes the optmal number of engneers to be allocate for the task of mantenance urng a specfc tme pero. The work by Antonol et al. [3] moels the mantenance organzaton as a queueng network to assess staffng, evaluate servce level, an fns the probablty of meetng the mantenance ealnes. Many authors also use statstcal an emprcal technques 1 In our earler work [1] we have aresse the problem of truth elctng mechansms for Invual Tcket Allocaton problem n software mantenance. 2 payment n ths case s not necessarly a country currency, t coul as well be a vrtual utlty or a score whch can be later converte for monetary or other benefts [4],[5] to analyze an mprove the software mantenance process. The problem of tcket allocaton (or bug assgnment) s also been looke upon usng a recommener system base approach. The work of Anvk [6], [7], proposes an recommener system whch reveals a set of possble evelopers to whom a trouble tcket (or bug) mght possbly be assgne, base on the past hstory of tckets resolve. In work by Duggan et al. [8] task allocaton n software constructon s looke upon as mult-objectve optmzaton problem an prove a set of tme an qualty optmal solutons for the ecson maker. The thrust of the above relevant papers has been prmarly on analyzng the mantenance ata for mprovng the mantenance process. All the above papers mplctly make a crucal assumpton, namely, that the ata s truthfully reporte by all the agents an the agents are loyal to the organzaton. However, the ratonalty of the engneers may nuce them to report the complexty of a tcket n an untruthful way so as to ncrease ther payoffs. Ths leas to non-optmal tcket allocaton, ncrease payments an tme to resolve. Ths paper aresses the problem of truthful elctaton of tcket complextes usng a game theoretc approach for Group Tcket Allocaton. 1.2 Contrbuton an Organzaton The paper offers the followng contrbutons: The problem of Group Tcket Allocaton s moelle as that of esgnng an ncentve compatble mechansm, that s a mechansm whch makes truth revelaton an optmal (or best response) strategy for the players. Ths s the subject of Secton 2. We propose a Domnant Strategy Incentve Compatble (DSIC) [9] mechansm calle Dvson of Labor (DOL) for ths problem, whch ensures that truth revelaton s optmal for each player (or engneers) rrespectve of other players reporte type. We show that ths mechansm oes not motvate the group to solve the problem (.e., not CR) even though every engneer s motvate for a voluntary partcpaton (IR). Ths s topc of scusson n Secton 3. We propose a secon mechansm Extene Secon Prce (ESP) whch s DSIC [9] an IR. We show that ths mechansm also satsfes the most esre property of group formaton, CR. We show that the propose allocaton an payment for ths mechansm les n the core of the game. We scuss ths n Secton Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.

3 2 The Moel The approach that we take for analyzng the tcket allocaton for groups s base on both non-cooperatve an co-operatve game theory. In ths secton we formulate a strategc form game an apply aucton mechansm to nuce esre propertes n the propose game. The moel as shown n Fgure 2 wll nclue a lea an a set of reportng engneers, N = {1, 2,..., n}. The aucton s conucte by lea n whch all engneers n N partcpate. Let the type announce by each engneer N be. Ths type value enotes the amount of ays spent by engneer to solve the problem nepenently.the valuaton of each engneer s v s the amount of value ascertane, f the tcket s allocate to, an have been aske to work for ays. The payment to each engneer s represente by t an the utlty s u = t v. Fgure 3. Workng Moel of Aucton 2.2 Notatons The followng table contans all the notatons that wll be use n further sectons. Fgure 2. The Moel N Θ θ max = {1, 2,..., n}, Set of n engneers Type set of player Actual type reporte by player whch s number of ays requre by to solve the problem nvually, where Θ = max N θ max = θ\θ max 2.1 How Aucton Works? The lea receves the reporte tcket an the number of ays wthn whch the customer emans the tcket to be solve. The lea announces the reporte customer problem to all the engneers n N (an retans the customer elvery ealne as prvate nformaton). In turn the lea receves nvual type values from each engneer n terms of number of ays requre to solve the problem. The lea uses the Optmzaton Problem (efne n Secton 2.3) to etermne the number of ays wthn whch the problem can be solve an the group g that can solve the tcket. Usng an g lea computes the payment for each player. Ths s epcte n Fgure 3. Further n each of the propose mechansm we wll explan the payment rule use an wll prove the esre propertes satsfe by the mechansm. Now we wll explan the notatons that wll be use to state an prove our propostons. max = arg max N N max = N\{ max } x Customer announce elvery ealne n number of ays = 1, enotes presence of player n the propose group; 0 otherwse. g = { : x = 1, N max }, Propose group to whch the problem s allocate g t = g\{}, a subset of group g where player s exclue g Number of ays n whch the problem wll be fxe by propose group Payment mae to player, for solvng the 272 Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.

4 θ g ρ w v u θ c θ c problem n the propose group, g. = max g, max type value n g. = mn N\g { : θ g } aly wage of every player; w = 1 s the value of w n ths paper. = x w, valuaton of player N = t v, N = max c, c g = mn N\c, c g more capable than j, that s 1 1 θ j. Hence payment for all players n group g wll nclue the capablty factor of 1, g. The nvual contrbuton of player n ths case s. All players who are not n group g wll receve zero payment. The problem of ecng the group g wll be solve usng the optmzaton problem propose n secton 2.3. The problem of How much to pay? wll be aresse n ths secton. Now we wll formally efne the payment rule for ths mechansm, t = ρ, g t = 0, / g Where ρ s the lowest b of the player who s not n the group g (See notatons). Now we shall show n the followng propostons that ths payment mechansm s nee DSIC an IR. t vcg ϑ(c) Payment to player usng the Vckrey-Clarke-Grove payment rule. Worth of coalton c, c N 2.3 Optmzaton Problem In ths secton we have lste the optmzaton problem that s epcte n Fgure 3. The optmzaton problem eces the group g an the number of ays requre by the group to solve the tcket. The formulaton ams to mnmze the number of ays gven the gven customer elvery ealne. The factor 1 s the percentage of work engneer woul complete n one ay. Objectve mn Subject To N max x = 1 x {0, 1} 3 Dvson of Labor Mechansm The funamental ea behn ths mechansm s Every one shoul get what they are capable of. We splt the total amount of money base on the amount of contrbuton by each engneer for ays. Even though all players n g mght have worke for same number of ays, player mght be Proposton 1: DOL Mechansm s DSIC. Proof: Let be the true type of player an θ + 3 be the type of player when les. Smlarly, + be the number of ays gven by optmzaton problem for an θ + announcements respectvely. Lkewse g, g + s the propose group gven by optmzaton problem for, θ + type announcements N respectvely. Also we know that θ + an +. The optmzaton soluton ensures that an + wll obey the followng conton for a subset of g an g + players respectvely. = 1 (1) g = 1 (2) We know that, g + + θ + g\{} g + \{} because all people n g other than wll reman n g + because other than none of them le. (2)-(1) gves (3) g + + θ + g + Let us now look at four cases, 1. player s n g an also n g + = 0 (3) 3 we have purposefully chosen θ + as the notaton for false type, as the possble le n software mantenance can be only greater than. The symbol + enotes the ncrease n value compare to 273 Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.

5 2. player s not n g an n g + 3. player s n g an not n g + 4. player s not n g an not n g + Let us show that n all four cases, player cannot better off by revealng theta untruthfully as θ +. More formally, u u +, g. Case 1: Player s n g an also n g +. Now let us take a look at the utlty of player n both g an g+ propose groups. ( ) u = ( u + = ρ + θ + ρ + ) + The ncrease[( n utlty ) by( le can )] be epcte as, u + u = ρ + ( + ) + θ + ρ If s the player who les, an the player s n both g an g + propose groups then only one case s possble where g = g + an ρ = ρ +. We also know, u + u = [ ] ( + θ + )ρ + + (4) In ths case all players nclung reman n both g an g + hence +. Hence the secon component of equaton (4), + 0. We can prove that there s no rase n utlty for player n ths case, f the frst component s also less than zero. [ ] u + u = ( + θ + )ρ + + But we know that, (from equaton (3)) + θ + = 0 g θ + g + From whch we can say, ( + ) = = j g g + j g θ j + θ j Ths clearly shows that for Case 1, j g + \g u + u 0 + θ + j g + j + θ j 0 Case 2: Player s not n g an n g +. Ths case s not possble as player can never enter g + by sayng le, when he coul not have entere g (when he reveals truth), when all other players other than contnue to reveal same type value. Case 3: Player s n g an not n g +. The utlty of player when he s n the group g s u 0 an when he s not n the group g + s u + = 0. Hence, ncrease n utlty by lyng for player s, u + u 0 Case 4: player s not n g an not n g +. In ths case the ncrease n utlty of player (by a le) s zero, because the utlty of player whle revealng truth an le s zero. Formally, u + u = 0 Thus we have shown n all the above four cases, DOL mechansm s DSIC. Proposton 2: DOL mechansm s Invually Ratonal. Proof: We nee to show that every player N by partcpatng n the game gets more or the same as aganst not partcpatng n the game. We nee to show, u 0, N. For all players g,u = ρ For all player / g, u = 0 By efnton we know, ρ >, g. we can see, u 0, g. From whch Proposton 3: DOL mechansm s not CR. Proof: The DOL mechansm s Coaltonally Ratonal can be countere wth the followng example. The bs place by 1, 2, 3 an 4 are 14, 28, 29 an 30 respectvely. Say customer elvery ealne s 8 ays. The players 1, 2 an 3 collaborate to solve the problem n 7.06 ays. The payments are 15.13, 7.56, 7.30 an 0 for engneers 1, 2, 3 an 4 respectvely. The utltes for players 1,2,3 an 4 are 8.06, 0.5, 0.24 an 0 respectvely. If player 1 woul have solve the tcket separately he woul obtane payment of 28 an an utlty of 14. Here by collaboratng 1 receves utlty of 8.06 nstea of 14. Hence DOL mechansm s not CR. Now we shall analyze the stablty of the propose allocaton an payment scheme. We use core [10] the most funamental soluton concept from cooperatve game the- 274 Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.

6 ory for ths purpose. The core of the game, s efne as, { core(n, ϑ) = (t ) N : t =ϑ(n), N } t ϑ(c), c N C We efne the characterstc functon ϑ(.) n the followng manner, 0 : f mn < max θ j N\C j C ϑ(c) = mn : f mn max θ j N\C N\C j C For the case, when N = C, we efne mn = θ max. N\C The valuaton of coalton C, ϑ(c), s the amount of value that the coalton can generate by knowng the true b value of every other player n the coalton. By knowng every other players true valuaton, they can b the maxmum true b value known wth-n the coalton n orer to gan maxmum avantage, as the payment to the group wll be the secon hghest b value (f they wn, zero otherwse). Proposton 4: DOL mechansm s not n core. Proof: The DOL mechansm s not CR from proposton 3. The Coaltonal Ratonalty s an necessary conton as seen from the efnton of core. Snce, DOL mechansm oes not have Coaltonal Ratonalty, we can comfortably conclue the allocaton an payment by ths mechansm oes not le n the core of the game, an hence not stable. We have so far shown that DOL mechansm has IR an DSIC propertes an oes not hol core an CR propertes that are very essental for coaltonal stablty. In orer to acheve core an CR propertes we exten our scusson to our next mechansm, Extene Secon Prce (ESP). 4 ESP - Extene Secon Prce In ths mechansm, we propose a payment rule, whch has two components unlke the prevous mechansm whch has only one. The frst component s motvate from the famous VCG payment rule [11], [9] an the secon term s nherte from DOL mechansm. The payment rule we propose s as follows, t = { t vcg + ρ, g 0, / g Proposton 5: ESP Mechansm s DSIC. Proof: We know that the frst component n the payment equaton of ths mechansm maps to Secon Prce Aucton, whch s DSIC [11]. The secon component s the payment of DOL mechansm, whch s also DSIC. Hence unerbng or a le of player wll not ncrease the utlty of through ether of the components. Hence, ths mechansm s DSIC. Proposton 6: ESP Mechansm s Coaltonally Ratonal. Proof: Let us efne u g, for the utlty of player f he partcpates n the propose group, an u g, f s not partcpatng the propose group. For all players / g, o not have any mpact to group g as they are aske not to partcpate anyway. So, only for players g t matters to partcpate or not n the propose group g. If u g u g then partcpatng n the group as propose s always a goo choce for player an t s Coaltonally Ratonal. Here we wll show that g,u g u g. We know that, So u g u g, ff, t vcg u g = u u g = t vcg + ρ t vcg ρ ( ρ ) 0 ρ ρ ρ But, we know by efnton, ρ, hence we prove ESP mechansm s CR. Proposton 7: ESP Mechansm s Invually Ratonal. Proof: We nee to show that u 0, N. We know that / g, u = 0. But, g, u = t vcg + ρ. We know that, t vcg ( 0 ) ρ 1 0 Hence, we can say, g, u 0. Proposton 8: The soluton of ESP mechansm s n core. Proof: Wth out loss of generalty, let us assume the players are arrange n the ncreasng magntue of ther b values. Hence player 1 has the lowest b an player n has the hghest b. Now, the characterstc functon ϑ, can smply be state as, ϑ(p) > 0, p P, where P = {(1,...k) : k = 2,..., n}, whereas for all other coaltons, not n P, the characterstc value s zero. Ths s because at least one player whose not n the coalton wll outb the coalton b (see efnton of characterstc functon). 275 Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.

7 We essentally nee to show that the core conton s true for all coaltons c P. For rest of the coaltons c / P, the sum over payments of players n c, may be greater than equal to zero, because the mechansm s IR. For the coaltons c P, we can say, t = t t k, k = 2,..., n c ) ( ) = ρ + t vcg = c = ϑ(c) + t vcg 1 ϑ(c) ( c ρ + t vcg 1 Hence, we show that the propose payments an allocaton n ESP mechansm le n core of the game. Now after showng the exstence of esre propertes for these two mechansms we further procee to expermentally show the effect of IC an CR propertes usng the ESP mechansm for a selecte set of engneers. 5 Expermental Evaluaton the utlty of player 1 has ncrease when player 2 reveals a le. Ths s just the case of ths experment an not always true. But our mechansm guarantees that the utlty of player 1 wll never ecrease (not at loss) because of untruthful revelaton of player 2. In our secon experment (Core Fgure 4. Truth Elctaton Analyss 5.1 The Setup We have consere a game consstng of 10 engneers N={1,2,...,10}, who wll announce ther bs to a lea, who eces the allocaton an payments base on the socal choce functon efne for the ESP mechansm. The type sets for each engneer s ranomly generate from a unform strbuton over the nterval [0,20]. For every unt tme we have generate a ranom tcket an agents b for the tcket, an an allocaton happens. For the next new tcket, whch arrves n the next unt tme the tcket s auctone. We assume the nterval between any two tckets arrvng nto the system are reasonably hgh, such that every engneer wll have no tcket n hs queue whle placng b for a new tcket. Conton Analyss), we group n-1 ranom players from the system, an apply our allocaton an payment rules from ESP mechansm. We compare the utlty of player 1 an 2 when the groupng of n-1 players s base on () some (ranom) ratonal ecson by the lea (curves cru n Fgure 5) versus () the groupng s base on our propose optmzaton functon n Secton 2 (curves cu n Fgure 5). We fn that t s always best to group the players as propose by the optmzaton functon, as t proves ncrease utlty for every player N than any ranom groupng. We also foun from the experment that t s essental to have the frst (n-1) players n group n orer to acheve a core conton, uner ESP payment scheme. 5.2 Experment We perform two experments to analyze the performance of our mechansm. The frst experment (Truth Elctaton Analyss) s to compare the utlty of players when one of the players reveals ther b untruthfully (a le). For the purpose of experment we have consere player 2 wll be the lar. We epct ths n Fgure 4, where the curve clu, represents the cumulatve utlty (utlty accrue over every tcket solve) of player when θ 2 s a le. The curve cu, represents the utlty of player when all players n the system reveal truth. For the purpose of clarty (of the graph), we show only the utlty curves for player 1 an 2 n Fgure 4. From the fgure t s event that player 2 gets lower utlty when he les, as the ESP mechansm s ncentve compatble. Ths mples that t s always best for every player to reveal truth uner ths mechansm. As a se effect, we see Fgure 5. Core Conton Analyss 276 Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.

8 6 Concluson Group Tcket Allocaton s one of the crtcal problems face by many managers/leas n software mantenance. The ecsons mae by the leas/managers are purely base on ther ratonalty of farness an are not far to all players, also not motvatng for many partcpatng engneers. Ths results n loss of prouctvty n many mantenance organzaton. In ths paper we aress ths ssue by proposng two Incentve Compatble mechansms for solvng group tcket allocaton problem wth customer elvery ealnes. We showe that these mechansms motvate engneers nvually an also n groups. We have also establshe that the secon mechansm ESP s superor to DOL n terms of CR an core propertes. Thus, the total cost ncurre n ESP s hgher than DOL, n orer to satsfy these atonal propertes. The ecson of choosng one of these mechansms s bascally makng a trae-off between total cost vs ratonalty/stablty. We leave ths choce to the screton of mplementng mantenance organzatons. One can also exten ths moel for herarchcal organzatons, where the groupng spans herarches. We are also currently workng on software nfrastructure to facltate such auctons, groupng an payments for software mantenance servces. 4. [5] I.Coman an A.Slltt, An emprcal exploratory stuy on nferrng evelopers actvtes from lowlevel ata, Proceengs of the 19th Internatonal Conference on Software Engneerng an Knowlege Engneerng, pp , [6] J.Anvk, Automatng bug report assgnment, Proceeng of the 28th nternatonal conference on Software engneerng, pp , [7] J.Anvk, L.Hew, an G.C.Murphy, Who shoul fx ths bug?, ICSE 06: Proceeng of the 28th nternatonal conference on Software engneerng, pp , [8] J.Duggan, J.Byrne, an G.J.Lyons, A task allocaton optmzer for software constructon, IEEE Software, vol. 21, pp , [9] R.B.Myerson, Game Theory: Analyss of Conflct, Harvar Unversty Press, Cambrge, Massachusetts, [10] P.D.Straffn, Game Theory an Strategy, Mathematcal Assocaton of Amerca, New York, [11] D.Garg an Y.Narahar, Founatons of mechansm esgn, Techncal Report, Department of Computer Scence an Automaton, IISc, Bangalore, Ina, References [1] K.Subban an Y.Narahar, Truth elctng mechansms for trouble tcket allocaton n software mantenance servces, Proceengs of the 19th Internatonal Conference on Software Engneerng an Knowlege Engneerng, pp , [2] V.G.Kulkarn an S.P.Seth, Optmal allocaton of effort to software mantenance: A queung theory approach, Workng Paper, The School of Management, Unversty of Texas, TX, USA., [3] G.Antonol, G.Casazza, G.A.Lucca, M.D.Penta, an F.Rago, A queue theory-base approach to staff software mantenance centers, Proceengs of the IEEE Internatonal Conference on Software Mantenance, pp , [4] M.Lee an T.L.Jefferson, An emprcal stuy of software mantenance of a web-base java applcaton, Proceengs of the IEEE Internatonal Conference on Software Mantenance, pp , Other company, prouct, or servce names may be traemarks or servce marks of others. 277 Authorze lcense use lmte to: INDIAN INSTITUTE OF SCIENCE. Downloae on November 18, 2008 at 04:45 from IEEE Xplore. Restrctons apply.