Tournaments, Risk Taking, and the Role of Carrots and Sticks. Scott M. Gilpatric

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1 Tournaments, Rsk Takng, and the Role of Carrots and Stcks Scott M. Glpatrc Department of Economcs Unversty of Tennessee 505A Stokely Management Center Knoxvlle, TN phone: fax: May 7, 004 Abstract We study a Lazear-Rosen tournament n whch players choose both work effort, whch determnes the mean of ther output dstrbuton, and the varance of output. The varance can be ncreased above ts natural level but t s costly to do so. We show that tournaments nvolvng more than two players generate ncentves for rsk-neutral players to pursue hgh-rsk projects. However, the ncorporaton of a penalty for the player who fnshes last n addton to a prze to the player who fnshes frst can elmnate ths rsk-seekng and retreve effcency n the tournament. More generally, the model llustrates that the balance of carrots and stcks n rankdependent compensaton wll nfluence workers atttude toward rsk takng. Keywords: tournaments, rsk-seekng (JEL J33, C7)

2 Introducton When workers compete for promoton, bonuses, or other rewards ted to ther relatve performance rankng vs-à-vs colleagues, competton s lkely to drve not just work effort but, where mperfect montorng permts, other choces at the workers dscreton. In ths paper we ask how rank-order tournament compensaton nfluences workers atttudes toward rsk. Unlke prevous lterature, we assume that t s costly for workers to ndulge any dscreton they have to alter the varance of ther output dstrbuton. We fnd that, despte the cost, workers may ndeed engage n wasteful rsk-seekng, but only when they face multple compettors and tournament payoffs are asymmetrc wth the reward for rankng frst not balanced by a penalty for rankng last. Snce the poneerng work of Lazear & Rosen (98), Nalebuff & Stgltz (983) and others, rank-order tournaments have been used to model worker ncentves n a varety of labor market settngs. In partcular executve compensaton has frequently been modeled as a tournament n whch low-level executves are motvated by the prospect of clmbng the corporate ladder and obtanng the large salares attendant to top-rankng executves (Rosen, 986; Bognanno, 00). Smlarly, the behavor of nvestment fund managers who compete for top fund-rankngs has been modeled as a tournament (Brown, Harlow and Starks, 996). The dstngushng feature of a tournament s that workers are pad based on the rank of ther output relatve to others rather than the level output. Prze levels are set n advance and competton to wn generates the ncentve to exert effort to ncrease output. Lazear & Rosen (98) showed that rank-order tournaments can provde an effcent ncentve framework when workers are rsk neutral. A substantal lterature has compared the effcency of tournaments wth alternatve ncentve schemes, such as pece rates, wth regard to

3 such factors as the rsk-averson of workers and the flexblty of the ncentve framework to envronmental uncertanty (e.g. Nalebuff & Stgltz, 983). However, lttle work has explored ncentves n a tournament settng when workers choose not solely how much work effort to exert, but some other aspect of the work that s undertaken. In partcular, executves may have substantal scope for nfluencng the output varance, or rsk, of projects they undertake. Furthermore, they may be able to nfluence the mean output through choces other than work effort, such as by choce of producton process or regulatory complance. When montorng s mperfect t s lkely that workers ncentves wll not be perfectly algned wth those of ther employer because they wll have opportuntes to ncrease ther probablty of wnnng the tournament by engagng n actvtes whch do not serve the nterest of the frm. The most sgnfcant lne of research n ths area has been explorng nfluence actvtes : behavor that arses when workers can nfluence the choce of superors regardng who s promoted or otherwse rewarded n an organzaton through actons whch are nonproductve, rangng from ngrataton to brbery and sabotage of compettors (Mlgrom and Roberts, 988; Prendergast and Topel, 996; Km et al., 00; and Chen, 003). Bronars (987) was the frst to address rsk preference n tournament settngs, and showed that leaders n sequental tournaments have an ncentve to be rsk-averse whle followers are rsk-seekng. Hvde (00) frst modeled a tournament settng where workers smultaneously choose both effort and rsk, as we do here. Unlke our model, Hvde assumed that workers could costlessly alter the varance of ther output, whch yelds sgnfcantly dfferent mplcatons as we descrbe below. Nalebuff & Stgltz (983) term ths ssue the nfluence of prze on choce of technque, but do not model the problem. Stgltz & Wess (98) address a related problem of the nfluence of the nterest rate n bank lendng on the rsk nvolved n projects undertaken by borrowers. 3

4 Smlarly to Hvde (00) we develop a Lazear-Rosen tournament model n whch players can nfluence both the mean and varance of ther output dstrbuton. In our model players ncur a cost of ncreasng the varance of the output dstrbuton, whch reflects costs of searchng for hgh-rsk projects or a lower mean output (holdng effort constant) assocated wth very rsky projects. The prmary result from ths model s that multplayer (n>) tournaments whch award a sngle prze to the wnner (or sngle penalty to the loser) generate an ncentve for neffcent rsk seekng (rsk averson). Ths arses because, when workers are dentcal and effort levels are symmetrc n equlbrum, one worker s ncreasng the varance of hs output dstrbuton ncreases the probabltes of hs rankng both frst and last when facng multple opponents. Hvde examned -player tournaments where players never ncur a cost of ncreasng output varance and found that, f varance s unbounded the unque equlbrum nvolves both players choosng nfnte varance and zero effort. When varance s bounded players smply choose the maxmum varance and a suffcently hgh prze spread wll generate effcent ncentves; neffcency arses n ths case only f players are rsk averse. Conversely, our results llustrate the mportance of the number of players competng n the tournament on the rsk-ncentves, and show that neffcency arses from pursut of hgh (or low) rsk projects wthout relyng on rskaverse preferences among players or the noton of nfnte varance, whch, as Hvde acknowledged, s somewhat unclear. An mportant secondary result s that the prze structure of the tournament can be manpulated to elmnate the rsk ncentves generated by multplayer tournaments. Ths arses from the fact that ncreasng output varance symmetrcally ncreases the probabltes of rankng frst and last. Therefore f a penalty s mposed on the lowest rankng player exactly equal to 4

5 the prze awarded to the wnner the ncentves precsely balance and players wll not have an ncentve to pursue rsker projects. More generally, the model llustrates the mportance of the use of penaltes for poor performance ( stcks ) as well as rewards for hgh performance ( carrots ) n rank-dependent compensaton, and how ther relatve magntudes nfluence the decson-makng of workers. Ths paper s organzed as follows: secton models the choce of varance n a tournament wth a sngle prze to the wnner. Sectons 3 shows that balancng the prze to the wnner of the tournament wth an equal penalty to the loser elmnates the workers ncentve to pursue rsky projects. Secton 4 addresses the analogous ssue of rsk-averson arsng n tournaments wth a penalty to the lowest-ranked player and no reward for fnshng frst. Secton 5 concludes wth a dscusson of general mplcatons regardng rank-dependant payoffs and rsk preferences. The Model We consder a tournament where each of n dentcal rsk-neutral players determnes the mean and varance of the dstrbuton of output he wll produce, ~ N ( µ, σ ) q, whch we assume to be normally dstrbuted for smplcty and concreteness. Work effort drectly determnes the mean wth cost of effort defned as c( µ ) and c ( µ ), c ( µ ) > 0. We assume there s exogenous varaton n ndvdual output, whch determnes the natural varance n outputσ. The ndvdual can then choose [ σ, ] σ wth z( σ ) 0 ; z, z > 0. The noton here s that there s a natural level of uncertanty regardng the output of the type of projects the ndvdual s employed to undertake, but he can acheve a mean-preservng ncrease n the spread of the dstrbuton by seekng out unusually rsky projects. Dong so, however, requres tme and effort 5

6 and s costly. Alternatvely, one could assume that ncreasng the varance of the output dstrbuton does not ental a drect cost, but that beyond some pont ncreasng the varance s possble only by choosng a project wth a reduced mean outcome an assumpton wth smlar mplcatons but whch ncreases the complexty of the model. Our assumpton that t s costly for ndvduals to ncrease the varance of ther output s central to the results we derve and merts some dscusson. In partcular, t may seem strangely at odds wth the common noton that nvestment decsons typcally nvolve a trade-off of greater expected returns n exchange for ncurrng ncreased rsk. Our vew s that workers have some dscreton n the projects they pursue whch ncludes a lmted ablty to vary the rsk ncurred, but that the nature of the frms actvtes establshes a natural baselne varance n output, represented by σ n our model. Workers can use ther dscreton to pursue projects that ental greater rsk, but dong so requres effort on ther part to push the usual bounds of the frms actvtes or ther role n the organzaton and may also nvolve choosng projects wth a lower expected return. We essentally assume that workers cannot transfer rsk costlessly: f they wsh to ncrease output varance from ts natural level ths must nvolve some cost. Smlarly, f an ndvdual wshed to reduce output varance below the natural level they could do so but only at a cost. An ncentve to engage n such rsk-averse behavor would arse n a tournament where no prze s awarded, only a penalty s meted out to the player who fnshes last. We address ths scenaro n greater depth n secton 4 below. We emphasze that the natural varance (σ) n workers output s exogenously determned and the total varance can be shfted only at some cost. Greater complexty arses because n such a model the cost of ncreasng output varance s ndrect (through the effect on the probablty of wnnng) rather than drect, but the results parallel those presented here. 6

7 We follow the Lazear and Rosen (98) tournament model n whch ndvduals sell ther labor n a compettve market where the value of ther output s V per unt. Intally we assume that there are only two prze levels, a payoff of W to the hghest ranked player and W to all others. 3 Applyng the usual Cournot-type assumpton that players optmze over the choce varables gven ther belefs regardng the choces made by other players, we frst solve for the workers best response functons and then mpose symmetry to fnd ther choces of effort and varance n Nash equlbrum as a functon of the przes. We then fnd the prze par that maxmzes workers expected utlty subject to the constrant that frms earn zero profts (whch arses from the assumpton of a compettve labor market). Let P represent the probablty of wnnng the larger payoff W, then workers maxmze max π PW + µ σ, P ( P) W c( µ ) z( σ ) ( W W ) + W c( ) z( σ µ ) () The probablty of wnnng s a functon of the probablty densty functon chosen by agent, whch we wll denote f(x), and the cumulatve dstrbuton functon G(x) representng player 's expected probablty of wnnng for any realzed output, whch s a functon of hs belefs regardng the choces of the other players: P f n ( x) ( G( x) ) dx. () Expresson () ndcates that the probablty of player wnnng the tournament s found by ntegratng over the player s own densty functon (whch s determned by hs choce of effort 3 Lazear and Rosen found that, where players choose only effort, t s suffcent to have only two payoffs to acheve effcency from the tournament and that results from -player tournaments generalze to n-players. They also note that n ther -player tournament model when ndvduals are rsk-neutral the soluton s not affected by the ablty of ndvdual s to choose the varance. However, although these statements are separately correct, they do not hold n combnaton. In other words, we show that when there are more than two players the soluton s affected by the ablty of ndvduals to choose rsk, and tournament ncentves dffer when more than two possble payoffs are consdered. 7

8 and varance) multpled by the probablty of wnnng assocated wth each possble output that he may realze, represented by the expresson ( G ( x) ) n. Notce that as the number of opponents ncreases, the shape of ths functon changes: wth two players (a sngle opponent) the functon s symmetrc around the opponent s mean output. But as the number of players ncreases the functon becomes ncreasngly skewed to the rght of the opponents mean output, reflectng the fact that the probablty of wnnng when facng multple opponents remans very small unless one acheves an output substantally greater than the opponents mean. Utlzng the assumpton of normalty on f, we can wrte P σ e π ( x µ ) σ n ( G( x) ) dx. (3) Of course the probablty of wnnng s strctly ncreasng n effort, as shown by the followng expresson: P µ x µ σ n ( x) ( G( x) ) dx 0 f >. (4) The effect of σ on the probablty of wnnng s gven by: P σ x µ f σ σ ( ) ( ) n x ( G( x) ) dx. (5) PROPOSITION : For n>, when players effort levels are symmetrc (µ µ - for all ), each P player s probablty of wnnng s ncreasng n hs choce of output varance: > 0. (Proof n σ the appendx.) 8

9 To see ths, frst note that the expresson n (5) equals zero for n: for any two symmetrc dstrbutons wth the same mean the probablty of recevng a hgher draw on one dstrbuton s not affected by ncreasng the varance. However, for n> the ( ( x) ) n G term s no longer symmetrc around µ -, ts mass s shfted to the rght as dscussed earler. Of course ncreasng the varance of f reduces mass n the center of the dstrbuton and places more of the mass n the tals. Increasng the mass n the left tal has only a small negatve mpact on the probablty of wnnng snce outcomes below the symmetrc mean are extremely unlkely to wn n any event, but ncreasng the mass n the rght tal substantally ncreases the probablty of wnnng because n a multplayer competton a rght-tal outcome s usually necessary to wn. Movng probablty mass from the center of the dstrbuton to the tals ncreases the probablty of fnshng both frst and last, but t s only the former that s relevant n ths settng. We can now solve for the symmetrc Nash equlbrum of the game as a functon of the payoffs. Nash best response functons are defned by the frst-order condtons: P µ (6) ( W W ) c ( µ ) 0 : µ P σ. (7) ( W W ) z ( σ ) 0 : σ Imposng symmetry we fnd the Nash equlbrum s descrbed by x µ n c ( µ ) ( W W ) f ( x) ( F( x) ) dx (8) σ ( ) ( ) ( ) ( ) x µ n z W W f x ( F( x) ) dx σ. (9) σ σ 9

10 By proposton above the rght hand sde of (9) must be postve for n>, whch allows us to state the followng: PROPOSITION : For n> n the symmetrc Nash equlbrum ndvduals chooseσ > σ, whch s neffcent snce t entals costly effort wthout rasng expected output. Furthermore, players equlbrum choce of varance (and the magntude of neffcency) s ncreasng n the prze spread (W -W ). We now solve for equlbrum przes and choces of effort and rsk. Let µ and σ represent the symmetrc choces of effort and rsk of all players n Nash equlbrum. Then a frm s expected revenue s the product of the number of workers t employs, ther mean output, and the value of the output: nµv. Competton n the labor market ensures expected revenues equal total przes (or equvalently, the average prze equals average value of output Vµ) ( ) nvµ W n or (0) + W ( n ) W + W V µ ( W W ) + W. () n n Notng that at symmetrc equlbrum P /n for all, then workers expected payoffs are π n ( W W ) + W c( µ ) z σ ( µ ) z( σ ) ( ), substtutng from () above () π Vµ c. (3) Frms competng for workers choose the prze spread to maxmze ths expected payoff, and ths maxmum s found where σ µ j. (4) W [ V c ( )]( µ / W ) z ( σ ), j, j 0

11 PROPOSITION 3: For n>, snce workers chose σ > σ and z ( σ ) > 0, expresson (4) mples that V c ( µ ) > and therefore ndcates that workers exert suboptmal effort. In multplayer tournaments ncreasng the prze spread ncreases the ncentve to wastefully pursue rsk whle also encouragng greater work effort. Therefore, n compettve equlbrum the prze spread wll not be suffcent to nduce effcent effort. 3 Stcks and Carrots The fact that the problem of wasteful rsk-seekng arses n wnner-take-all multplayer tournaments suggests that the rsk-seekng ncentve may be elmnated by structurng the tournament wth more than prze levels. In partcular, an obvous canddate s to penalze the player who ranks last as well as rewardng the player who ranks frst and thereby establsh symmetry n the game s payoffs. We now consder a multplayer tournament wth three payoffs: W >W >W 3. Let P represent an ndvdual s probablty of rankng frst and earnng W as before, and R be the probablty of rankng last and recevng W 3. An ndvdual s expected utlty s now ( W ) R( W W ) c( µ ) z( σ ) π P W 3 (5) wth the probablty of recevng W just as before P ( x µ ) σ e σ π n ( G( x) ) dx (6) and the probablty of recevng W 3

12 R σ e π ( x µ ) σ n ( G( x) ) dx. (7) The probablty of fnshng last s of course decreasng n effort R µ x µ σ n ( x) ( G( x) ) dx 0 f <. (8) Just as the probablty of fnshng frst s ncreasng n σ at symmetrc effort levels, the probablty of fnshng last s also ncreasng n σ R σ x µ f σ ( ) ( ) n x ( G( x) ) dx > 0 σ for µ µ -. (9) Furthermore, by symmetry of the G dstrbuton around the common mean output µ, n ( ( )) n G x ( G( x) ) at µ. Therefore R P σ σ at the symmetrc Nash equlbrum. The frst order condtons are µ P R ( ) ( ) C W W W W ( σ ) 0 (0) : 3 µ µ P R σ ( ) ( ) z W W W W ( σ ) 0. () : 3 σ σ PROPOSITION 4: In a multplayer tournament when wnnng and losng prze spreads are equal, (W -W ) (W -W 3 ), expresson () ndcates that workers choose not to expend effort pursung σ hgh rsk projects z ( ) 0 c ( µ ) V at the compettve equlbrum.. The tournament wll therefore generate effcent ncentves wth

13 Ths result suggest that n multplayer tournament settngs where workers have scope for rskseekng whch cannot be perfectly montored by the employer t s mportant that the tournament desgn ncorporate both a carrot and a stck. The ncluson of a penalty for rankng last exposes the worker to both rght and left tal effects from ncreasng output dsperson, thereby elmnatng the ncentve to pursue rsky projects. 4 Rsk Averson n Tournaments wth No Prze, Only a Penalty Although t s conventonal to model tournaments as settngs n whch workers compete for a hgh payoff or prze, t s easy to conceve of settngs n whch lttle reward s pad to workers wth top-rankng output, but sgnfcant penaltes are appled to those who fnsh last. For example, ths may apply to organzatons whch offer lttle opportunty for promoton or advancement, but where relatvely poorly performng workers are lkely to be dsmssed. Of course n two-player tournaments t s meanngless whether the hgher payoff s desgnated a prze to the wnner or the lower payoff a penalty to the loser. But multplayer tournaments wth a sngle penalty to the player wth the lowest output generate very dfferent rsk-takng ncentves than those n whch a sngle prze s awarded to the player who fnshes frst. The model dscussed above can easly be altered to show that, when the tournament mposes only a penalty on the lowest-rankng player and workers can reduce the varance of ther output but ncur a cost of dong so, workers wll engage n neffcent rsk-averson. All the results are parallel: for any symmetrc effort level workers reduce ther probablty of fnshng last by reducng the varance of output. Ths also reduces the probablty of fnsh frst, but snce there s no prze for dong so workers gnore ths effect. Workers therefore wastefully seek to reduce the varance of output, and the ncentve to do so ncreases wth the magntude of the 3

14 penalty assocated wth fnshng last. Just as n the wnner-take-all multplayer tournament descrbed earler, worker effort wll be below the frst-best n the symmetrc equlbrum. 5 Concluson In many labor market settngs performance ncentves are lkely to be a complex mx of monetary and non-monetary payoffs, some of whch have the rank-dependant character of a tournament, whle others are closer to pece-rates or playng aganst a standard (such as meetng a quota) n character. We beleve the model presented here provdes mportant nsght nto the manner by whch the balance of payoffs carrots versus stcks nfluences the decson-makng of workers. When rank-dependent payoffs nvolve prmarly postve rewards such as bonuses and promoton for the few most productve workers, wth comparatvely lttle lkelhood of dsmssal or other penaltes for those who rank at the bottom, workers wll be encouraged to pursue hgh-rsk strateges. Conversely, f rewards are few but unusually poor performers consstently face dsmssal or other rebukes, workers wll be dscouraged from takng rsks n ther actvtes. These results suggest that frms are lkely to use the balance of payoffs to nfluence the rsk preferences of ther workers. In partcular, large rewards for top-ranked performance may be a means for frms n ndustres where a wllngness to take large rsks s crtcal to success to balance the nherent rsk-averson of most employees. Frms n other ndustres where rsk-takng by workers s undesrable may use prmarly negatve ncentves to motvate employees wthout encouragng the rsk-takng that bonuses and other postve ncentves generate. In general, the nature of compensaton and dfferences among frms and ndustres wll reflect frms efforts not 4

15 just to motvate employee s work effort, but to nfluence the decsons they make and the rsks they ncur on behalf of the frm. 5

16 Appendx A: Proofs PROOF OF PROPOSITION : We frst show that for n P / for any σ. Ths follows from the symmetry of both the densty f and dstrbuton G around a common mean µ: P n ( x) ( G( x) ) dx [ f ( x) G( x) + f ( x) ( G( x) )] dx f ( x) f µ µ dx. () P Snce for n P / at any symmetrc effort equlbrum wth common µ then 0 for n. σ From equaton (5) we know that generally P σ x µ f σ σ ( ) ( ) n x ( G( x) ) dx Notce that the ntegrand of ths expresson contans three terms, each of whch s symmetrc around µ for n, and the ntegrand takes on negatve values for µ -σ >x> µ +σ and postve values outsde ths range. The negatve and postve values precsely balance, renderng the expresson equal to zero. For n> the last term becomes asymmetrcally skewed to the rght, reducng the value of the ntegrand compared to the n case for all x, but reducng t progressvely less for hgher values of x. Ths necessarly renders the expresson postve by placng greater weght on the postve values of the ntegrand found where x> µ +σ relatve to the negatve values that occur around µ.. 6

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18 Rothschld, M. and Stgltz, J., 970, Increasng Rsk I: A Defnton, Journal of Economc Theory, Vol., pp Smth, V.L., 98, Mcroeconomc Systems as an Expermental Scence, Amercan Economc Revew, 7, pp Stgltz, J., and Wess, A. Credt Ratonng n Markets wth Imperfect Informaton, Part I. Amercan Economc Revew, Vol. 7, No. 3 (June 98), pp