Bimodal Bidding in Experimental All-Pay Auctions

Transcription

1 Bmodal Bddng n Expermental All-Pay Auctons Chrstane Ernst and Chrstan Thön August 2009 Dscusson Paper no Department of Economcs Unversty of St. Gallen

2 Edtor: Publsher: Electronc Publcaton: Martna Flockerz Unversty of St. Gallen Department of Economcs Varnbüelstrasse 19 CH-9000 St. Gallen Phone Fax Emal Department of Economcs Unversty of St. Gallen Varnbüelstrasse 19 CH-9000 St. Gallen Phone Fax

3 Bmodal Bddng n Expermental All-Pay Auctons 1 Chrstane Ernst and Chrstan Thön Author s address: Dr. Chrstan Thön FEW-HSG Varnbüelstrasse St. Gallen Phone Fax Emal chrstan.thoen@unsg.ch Webste Chrstane Ernst Emal c.a.ernst@lse.ac.uk 1 Acknowledgements: We thank the Brown-Bag Semnar Partcpants of the Unversty of St. Gallen, Eva Deuchert, Therese Fässler, Martn Huber, Henz Müller for helpful comments and suggestons.

4 Abstract We report results from expermental frst-prce, sealed-bd, all-pay auctons for a good wth a common and known value. We observe bddng strateges n groups of two and three bdders and under two extreme nformaton condtons. As predcted by the Nash equlbrum, subjects use mxed strateges. In contrast to the predcton under standard assumptons bds are drawn from a bmodal dstrbuton: very hgh and very low bds are much more frequent than ntermedate bds. Standard rsk preferences cannot account for our results. However, bddng behavor s consstent wth the predctons of a model wth reference dependent preferences as proposed by the prospect theory. Keywords All-pay Aucton; Prospect Theory, Experment JEL Classfcaton D44, D72, D80, C91.

5 1. Introducton Bddng behavor n all-pay auctons has so far only receved lmted attenton n emprcal aucton research. Ths mght be due to the fact that most of the applcatons of aucton theory do not nvolve the all-pay rule. However, occurrences of ths aucton format, where every bdder pays her bd are numerous: lobbyng battles, poltcal campagns, promoton tournaments n frms and applcatons for scence grants (Klemperer, 2004). In ths artcle we report expermental data on bddng behavor from the smplest possble all-pay aucton format. We conducted frst-prce, sealed-bd, common value auctons wth two or three bdders and no uncertanty wth regard to the value of the auctoned commodty. Every subject bds n an aucton for a prze of 100 monetary unts. Subjects choose ther bds smultaneously, the hghest bdder receves the prze, and all bdders pay ther bd. We analyze bddng behavor n ten subsequent auctons and under two dfferent nformaton condtons. In the NoRecall treatment subjects do not receve any nformaton about other subjects bds durng the ten rounds. In the Recall treatment subjects have full nformaton about the bddng hstory n ther group. Prevous evdence on bddng behavor n expermental all-pay auctons comes from Gneezy and Smorodnsky (2006), who conducted smlar all-pay auctons wth group szes of four to 12. They report persstent overbddng,.e., average bds were consderably hgher than predcted by the Nash equlbrum for all group szes. 1 Whle the exstng lterature focuses on aggregate outcomes, our am s to nvestgate ndvdual bddng strateges. Thus we study bddng behavor n the smplest case of auctons and n small groups. We fnd that subjects ndeed use mxed strateges, however, the observed dstrbuton of bds shows nterestng devatons from the predctons under standard assumptons. The mxed strategy Nash equlbrum under standard assumptons predcts unformly dstrbuted bds for groups of two players and a decreasng densty functon for larger group szes. We fnd that subjects bddng strateges dffer sharply from these predctons: on average, subjects apply bmodal bddng strateges whch gve most weght to both very low and very hgh bds, resultng n a bmodal bddng functon. Bmodal bddng occurs for both group szes and nformaton condtons. We show that bmodal bddng s consstent wth prospect theory. If players are rsk seekng n the doman of losses and rsk averse n the doman of gans then equlbrum bddng strateges are bmodal. 1 A related strand of lterature focuses on the rent-seekng game ntroduced by Tullock (1967), where hgher bds ncrease the bdder's share of the pe (or rent). See Mllner and Pratt (1991), Shogren and Bak (1991), and Davs and Relly (1998), Öncüler and Croson (2005) or Herrmann and Orzen (2008). 2

6 2. Theory and Expermental Desgn The game s played n groups of n=2 or 3 bdders. All players smultaneously choose ther bd b R +. The auctoned commodty has a value of unty for all bdders. A player s expected utlty u N : R R s defned as: + 1 max b f for all max b = b bj j u ( b, b ) = M ( b ). (1) b otherwse max M ( b ) counts the number of maxmal bdders. Followng, we ntutvely derve the Nash equlbra of ths game. A thorough theoretcal treatment of ths game s provded by Baye et al. (1996). Clearly the game cannot have an equlbrum n pure strateges snce the best reply to every bd n [0,1) s to overbd by the smallest amount possble. In every mxed strategy equlbrum t must hold that bdders are ndfferent between the mxed strategy and any pure strategy ncluded n ther mxed strategy. As long as the support of the mxed strategy ncludes zero, the expected utlty n the mxed strategy must be zero. For n=2 ths game has a unque equlbrum n mxed strateges where both players draw ther bd from a unform dstrbuton over the support [0,1]. The expected utlty then equals zero for both players. Could the two players mprove ther stuaton by restranng the support of ther mxed strategy from above,.e., both drawng ther bd from [0, b ] wth b < 1? No, because ths would offer the opportunty of earnng a strctly postve utlty by outbddng the other player wth a pure strategy of bddng slghtly more than b. Could they mprove by choosng ther bd from [ b,1] wth b> 0? Ths s also not possble as bddng b would then result n a certan loss and the player would prefer to bd 0. To conclude, both players choose ther bd from a unform dstrbuton wth support [0,1] and earn an expected payoff of zero. Expected bds are 0.5 and expected standard devaton s 1/ 12= Expected gross returns of the auctoneer are 1, whch equals the value of the auctoned commodty. For n=3 the theoretcal soluton becomes more complcated. There exsts a unque symmetrc equlbrum where all players draw ther bd from f b = b b. In addton, there s a 0.5 ( ) 0.5, [0,1] contnuum of equlbra of the followng knd: two players randomze on [0,1] whle the thrd player randomzes contnuously on an nterval [ b,1] and concentrates the remanng mass at zero, wth 0 b 1. The equlbra reach from b= 0, whch s the symmetrc case to b= 1, n whch player three does not take part n the aucton and the other two players choose ther bds accordng to the equlbrum strategy n the two player case. Expected bds for the two players who randomze wth full support range from one thrd to one half, expected bds from the thrd bdder range from zero to one thrd. All equlbra share the followng features: the expected bds are one thrd, expected utlty 3

7 of all bdders s zero and the revenue for the auctoneer s unty. Standard devatons of the bds depend on the equlbrum and range from for b= 0 to for b= 1. All equlbra also have n common that they are not lucratve for the bdders. It s straghtforward to show that colluson can substantally ncrease the expected utlty of the bdders. For example, f n=2 and both bdders bd zero, ther expected utlty s However, ncentves to devate from ths collusve strategy are obvously very strong. Our expermental subjects were frst year students from the Unversty of St. Gallen. The experment was programmed and conducted wth the software z-tree (Fschbacher, 2007). At the begnnng of the experment subjects were gven a show-up fee of CHF 20 (about $18). Losses n the experment were deducted from the show-up fee. The auctoned tem was 100 ECU (Expermental Currency Unt). We conducted two treatments, the Recall and the NoRecall treatment. In the Recall treatment subjects were allocated to groups of two or three subjects and played ten consecutve but ndependent all-pay auctons n unchanged groups. Full nformaton about bds of group members n all prevous rounds was provded. In the NoRecall treatment subjects also played ten consecutve all-pay auctons for a prce of 100 ECU. In each round they were randomly allocated to groups of ether two or three subjects. They were nformed about ther group sze but receved no nformaton at all about the outcome of the aucton and the other subjects bds. Pror to the experment subjects were gven detaled nstructons (see Appendx). Bds were restrcted n the nterval [0,125] and a resoluton up to three decmal places. 3 We report results from 52 subjects n two expermental sessons. We apply a wthn subject desgn where all subjects played both treatments, changng the order of the treatments between sessons. The experment lasted about an hour and the subjects earned on average CHF 19.4 (about $17.5), whch means that, on average, subjects made a small loss of CHF.6 n the 20 auctons they played. 3. Results We start by analyzng the data from the Recall treatment. In ths treatment subjects had access to the whole hstory of bds wthn ther group and played the game n stable groups. These are arguably the condtons most favorable for the establshment of a mxed strategy equlbrum. 4 2 In our experment the te rule was that all maxmum bdders pay ther bd and one randomly chosen bdder receves the prce. 3 The upper bound of 125 was ntroduced to prevent subjects from makng large losses due to erroneous entres. However, ths upper bound was not communcated to the subjects n the nstructons to prevent anchorng. 4 Once a bdder learns that other bdders use a pure bddng strategy she can smply outbd the others by the smallest amount and earn a secure payoff, whle the others make a loss. 4

8 Average bds over the ten rounds were 42.0 n groups of two and 36.9 n groups of three. Compared to the Nash predcton of 50 and 33.3 respectvely, we observe underbddng n groups of two and overbddng n groups of three. The dfferences are, however, not sgnfcant. 5 The underbddng n groups of two mght partally be due to attempts of colluson. If both players bd zero and let chance decde about the wnner, they have an expected proft of 50. The results by Huck et al. (2004) suggest that collusve strateges are most lkely to be observed n small groups. Average bds are, however, not too nformatve wth regard to the bddng strateges subjects played. If we calculate the standard devaton of the bds, we observe 40.0 n groups of two whch s hgher than predcted by the Nash equlbrum (28.9). In groups of three the observed standard devaton was 39.8 compared to the predcton whch les between 29.8 and However, the most strkng dfferences to the Nash predcton emerge f we take a closer look at the dstrbuton of bds. Fgure 1 shows hstograms of the bds separated by group sze. The dstrbuton of bds s clearly bmodal. Very low and very hgh bds (up to 100) are much more frequent than ntermedate bds. 5 A conservatve test based on the ndependent group averages does not allow to reject the null hypothess that average bds are equal to the Nash predcton (p=.140 for groups of two and p=.262 for groups of three, twosded Wlcoxon matched-pars sgned-ranks test). 6 In groups of two the dfference s sgnfcant at p=.036, n groups of three nsgnfcant wth p=.263 (two-sded Wlcoxon matched-pars sgned-ranks test). 5

9 Fgure 1 Hstogram of Bds n Groups of Two and Three Subjects n the Recall Treatment Group sze=2 Group sze=3 [0,10) [10,20) [20,30) [30,40) [40,50] Densty Densty (50,60] Bd (60,70] (70,80] (80,90] (90,100] (100,110] (110,125] [0,10) [10,20) [20,30) [30,40) [40,50] (50,60] Bd (60,70] (70,80] (80,90] (90,100] (100,110] (110,125] In the groups of three subjects very low bds are more frequent than very hgh bds. However, the overall adjustment of the bds to the larger group sze s much smaller than predcted. In order to compare the observed bd to predcted bds we plot the cumulatve dstrbuton of bds. Fgure 2 shows the cumulatve dstrbuton of the observed bds (bold lnes) and the cumulatve denstes predcted by the mxed strategy Nash equlbrum (thn lnes). In the rght panel we account for the fact that multple equlbra exst and depct the cumulatve denstes of two extreme cases: The knked curve corresponds to the equlbrum where one player abstans from the aucton (hence the ntercept at one thrd) and the other two draw ther bd from a unform dstrbuton; the smooth curve corresponds to the symmetrc mxed strategy equlbrum. For the smaller group sze, predcton and data are obvously very dfferent. In the auctons wth three bdders the large mass at very low bds s compatble wth an asymmetrc mxed strategy equlbrum. Stll, the mass of bds close to 100 s clearly not compatble wth the predcton. If we apply Kolmogorov-Smrnov tests for the null hypothess that the bds stem from the predcted denstes we can reject the null hypotheses for both group szes and all equlbra at p < Smple Kolmogorov-Smrnov tests yeld p values of vrtually zero. However, we have to take nto account that observatons wthn a group are not ndependent. We do ths by usng each group s Kolmogorov Smrnov test statstc as an observaton. We then run a smulaton (n = 1000) to calculate the test statstc for hypothetcal bds drawn from the denstes predcted by the symmetrc Nash equlbra. In case of the asymmetrc Nash equlbra we test the dstrbuton of the non-zero bds aganst the predcted dstrbuton most favorable to mass at very 6

10 Fgure 2 Nash Equlbrum (Thn Lnes) and Observed Cumulatve Dstrbuton Functons of Bds by Group Sze n the Recall Treatment (Bold Lnes) Group sze=2 Group sze=3 Cumulatve dstrbuton y Data Symmetrc NE Asymmetrc NE Bd Bd 3.1. Indvdual Bddng Behavor The data presented thus far dd not contan nformaton as to what extent the subjects actually played mxed strateges. In a next step we wll look at ndvdual patterns. Fgure 3 depcts small hstograms for ndvdual bddng behavor over the ten rounds of the Recall treatment, for both group szes. Each vertcal lne corresponds to one subject n the experment and shows the spread of the bds. Subjects are sorted accordng to average bd. The length of the small horzontal spkes corresponds to the frequency of the correspondng bd (bds are rounded to ntegers). The overwhelmng majorty of the subjects bd n the whole range from zero to (almost) 100. Three quarter of the subjects have a spread of 90 or more n ther ten bds. The majorty of the subjects changed ther bd frequently durng the ten rounds. If we calculate the number of dfferent bds a subject chose, we obtan an average of 7.96 dfferent bds. More than a quarter of the subjects chose dfferent bds n all ten rounds. Addtonally we can look at the number of changes n a subject s bd from one round to the next. In 90.4 percent of the cases subjects changed ther bd from t to t+1. 8 hgh bds, whch s the asymmetrc equlbrum wth one player abstanng from the aucton. The test statstcs for our data are always hgher than for the smulated data. A Wlcoxon rank-sum test gves p <.001 n all cases. 8 These numbers refer to all subjects n the Recall treatments, rrespectve of group sze. 7

11 Fgure 3 Indvdual Hstograms of the Bds n the Ten Rounds of the Recall Treatment Group sze = 2 Group sze = 3 Bd Bd Note. Vertcal lnes show the spread of the bds of each ndvdual, small horzontal lnes depct the frequency of the correspondng bd (bds are rounded to ntegers). In order to test for tme effects we ran OLS and Tobt estmates for the subjects bds. To take nto account statstcal dependence of observatons wthn groups we estmate standard errors usng the group as cluster. Table 1 reports the results of the OLS estmates (Tobt results are almost dentcal). Explanatory varables n Model (1) are the Round number, a dummy for the group sze, a dummy for the order of the two treatments, Recall and NoRecall. 9 None of these varables explans the bddng behavor and the model as a whole s nsgnfcant. Ths s surprsng snce the theoretcally predcted expected bds dffer consderably wth group sze (33.3 vs. 50). As shown above, subjects n groups of three overbd and subjects n groups of two underbd. Thus, n our data the nfluence of the group sze on bds n absolute terms s much smaller than predcted. Ths s n lne wth Gneezy and Smorodnsky (2006), who fnd that subjects bddng behavor does not suffcently react to the number of compettors at least n early rounds. They also report that bds decrease over tme, reducng the amount of overbddng In our data we also have nformaton about soco-economc characterstcs of the partcpants, such as gender, age, wealth of the famly of orgn and urban background. In all estmates these controls proved to be nsgnfcant (alone and jontly). We dd not nclude them n the estmates reported n Table Gneezy and Smorodnsky (2006) dscuss the logt equlbrum proposed by Anderson et al. (1998) concept as an explanaton for the nsuffcent adjustment of bds to group sze. In such an equlbrum players do not play 8

12 In order to check whether there s pressure towards the equlbrum n our experment we have to keep n mnd that the drecton s not clear. In groups of two, bds should ncrease, n groups of three they should decrease. We allow for dfferent tme trends n Model (2) wth the nteracton term. However, the model does not beneft from ths at all. Fnally, n Model (3) we add lagged varables. Ths gves us a model wth some explanatory power. A subject s bd n the prevous round s postvely related to her bd n the actual round. The maxmal bd n the prevous round has no effect. The most nterestng result s the (weakly) sgnfcant coeffcent for the dummy ndcatng whether the subject won the prevous aucton. Wnners of the prevous aucton tend to reduce ther bd n the actual round (controlled for the bd n the prevous round). Choosng a hgh but losng bd seems to ncrease bds, whle a wn nduces subjects to bd more conservatvely n the next round. Table 1 OLS Estmates for Bds n the Recall and NoRecall Treatment Dependent varable: Bd n t Recall NoRecall Model (1) Model (2) Model (3) Model (4) Model (5) Group sze=3 (D) *** ** (6.728) (9.548) (9.089) (5.083) (9.248) Round (0.506) (0.745) (0.990) (0.545) (0.711) Round x Group sze= (0.998) (1.065) (1.154) Recall NoRecall (D) ** ** (7.366) (7.373) (5.111) (7.857) (5.000) Bd n t ** 0.438*** (0.094) (0.083) Maxmum bd n t (0.071) Won aucton n t-1 (D) * (6.007) Constant *** *** ** *** *** (5.537) (6.629) (9.804) (5.260) (7.207) F-test Prob > F R² N Notes. OLS estmates for bd n the Recall and n the NoRecall treatment. In parentheses we report robust standard errors usng the group as cluster. * p<0.1, ** p<0.05, *** p<0.01. strct best response but play strateges wth hgher expected value wth hgher probablty. In the extreme case (very large error term) strateges are drawn from the unform dstrbuton, thus trvally generatng a postve relaton between the sum of bds and the number of bdders. Bmodal dstrbutons of bds are, however, ncompatble wth ths soluton concept. Our results suggest that devatons from Nash bddng behavor under standard assumptons s not drven by errors as formalzed n the logt equlbrum. 9

13 3.2. Reducng Informaton Thus far we have only analyzed data from the Recall treatments, where subjects learn other subjects bds and can adapt ther bds accordngly n the next round. In the NoRecall treatments subjects receve no feedback at all from the game and cannot fnd out the success of ther strateges. In such a game t s even more dffcult to nfer a subject s bddng strategy from the observed bds. Even f subjects play a mxed strategy n the sense that they use a random draw from a probablty dstrbuton to determne ther bd, t s unclear whether they draw every round or only once n the experment. It was n fact the case that subjects changed ther bds less frequently n the NoRecall treatment compared to the Recall treatment, despte the fact that n the NoRecall treatment group sze changed over tme. Durng the ten rounds, subjects chose on average 6.60 dfferent bds (as opposed to 7.96 n the Recall treatment). Stll, the overall dstrbuton of bds was clearly bmodal for both groups of two and three subjects. Fgure 4 shows hstograms for the bds n the NoRecall treatments. Fgure 4 Hstogram of Bds: Groups of Two and Three Subjects n the NoRecall Treatment Group sze = 2 Group sze = 3 [0,10) [10,20) Densty [20,30) [30,40) [40,50] (50,60] Bd (60,70] Densty (70,80] (80,90] (90,100] (100,110] (110,125] [0,10) [10,20) [20,30) [30,40) [40,50] (50,60] Bd (60,70] (70,80] (80,90] (90,100] (100,110] (110,125] Models (4) and (5) n Table 1 apply the estmaton of Models (1) and (2) to the NoRecall data. Interestngly we observe effects of the group sze and the order of the treatments. Bds are lower by 16.8 unts (standard error: 5.08) n groups of three compared to those of groups of two. The lack of adjustment of bds to group sze observed n the Recall treatment (and also reported by Gneezy and Smorodnsky, 2006) dsappears f we observe wthn subject varaton of the group sze. In fact, the 10

14 sze of the effect s very close to the predcted effect of 16.7 (= ). The dummy for the order of the two treatments s sgnfcant as well. When subjects have experenced the Recall treatment before playng the NoRecall treatment, bds are lower by 20.3 unts (standard error: 7.86) compared to when they start wth the NoRecall treatment. On the other hand, tme effects reman nsgnfcant. Model (5) presents the OLS estmate ncludng the bd from the prevous round as explanatory varable, whch agan has a sgnfcant postve nfluence. We do not add the other lagged varables to the equaton because subjects n the NoRecall treatment had no nformaton about the outcome of the aucton durng the ten rounds. Comparng the data from the NoRecall treatment to the Recall treatment also provdes us wth nformaton about the subjects' attempts to collude. Collusve strateges can only be sgnaled by the subjects n the Recall treatment, because only there, the other bdder learns about the bds. In the groups wth two subjects a bd of exactly zero was chosen n 5.4 percent of the cases n the Recall treatment but only n one percent of the cases n the NoRecall treatment. Ths dfference s hghly sgnfcant. For groups of three we fnd that the percentage of zero bds drops from 9.2 to 6.7 percent when we compare Recall and NoRecall, and the dfference s not sgnfcant at all. We conclude from these observatons that some of the bds n the two bdder auctons are motvated by collusve attempts, however already n groups of three colluson s suffcently dffcult to acheve that t does not nfluence bddng behavor Why Bmodal Bddng? Why should one adopt a bmodal bddng strategy? Gven that the subjects n our experment dd not play accordng to Nash strateges, a proft maxmzng player would not be ndfferent between all possble bds when playng aganst ths populaton. We can use the observed dstrbuton of bds to calculate the bd that maxmzes expected profts, gven the bmodal bddng behavor of the other subjects. The optmal bd les somewhat above the lower mode of the bmodal bddng strategy n the regon between 5 and 15. The worst thng to do s to bd slghtly below the upper mode of the bddng strategy used by our subjects. There s already a great amount of emprcal evdence whch demonstrates that many people do not behave lke expected proft maxmzers n rsky stuatons (see e.g. Dohmen et al., 2005). It s thus natural to explore the game theoretc predctons under dfferent assumptons wth regard to the utlty functon. The prme concern s rsk preferences. The predctons derved n Secton 2 hold only for players whose utlty s lnear n the monetary payoff of the game. Let us consder an arbtrary utlty 11 We use probt estmates for b =0 wth the treatment dummy, round effects and the treatment order dummy as explanatory varables. The treatment dummy s hghly sgnfcant for groups of two bdders (p =.003) and nsgnfcant for groups of three bdders (p =.340). 11

15 functon u( x ) where x s the monetary payoff and u ' > 0. For every mxed strategy equlbrum wth n homogeneous players t must hold that: n ( ) EU b = F b u b + F b u b = u. (2) n 1 1 ( ) ( ) (1 ) 1 ( ) ( ) (0) The expected utlty of every bd b used n the strategy equals the probablty of wnnng the aucton tmes the utlty n case of a wn, plus the probablty of losng the aucton tmes the respectve utlty. Both terms depend on F( ), whch s the cumulatve densty of the bddng strategy of the other bdders. For a mxed strategy wth full support to be a best reply ths expresson must be constant and equal to the utlty of bddng zero. Followng we assume u (0) = 0 wthout loss of generalty. Thus the utlty of wnnng the aucton wll be postve and losng the aucton wll yeld zero or negatve utlty. From ths expresson we can easly derve the cumulatve densty of the equlbrum bddng strategy wth arbtrary (but symmetrc) utlty functons as: 1 n 1 u( b ) F( b ) = wth 0 b 1. (3) u( b ) u(1 b ) What denstes f ( b ) can we generate wth standard rsk preferences? If we ntroduce dfferent rsk preferences nto the utlty functon usng a functon wth hyperbolc absolute rsk averson, we can only predct unmodal bddng strateges n groups of two players. Relatve to the rsk neutral case, rsk averson shfts mass towards low bds whle rsk seekng preferences shft mass towards hgh bds. 12 However, n lght of the arguments by Rabn (2000) t s not very plausble that behavor n small stake games s explanable by the standard expected utlty maxmzng approach. Explanng behavor whch devates from expected payoff maxmzaton n small stake stuatons calls for a dfferent explanaton an explanaton that does not rely on a sngle concave or convex utlty functon n overall wealth. It turns out that the bddng strateges observed n our experment can be explaned by the curvature of the utlty functon f we allow for concave and convex regons. Ths leads us to a very well known theoretcal alternatve to expected utlty theory, namely Kahneman and Tversky s (1979) prospect theory. A core element of prospect theory s that players evaluate ther outcome relatve to a reference pont. If they earn more than ther reference pont, they are n the doman of gans, otherwse n the doman of losses. Kahneman and Tversky propose a value functon that s concave n the doman of gans and convex n the doman of losses. In addton to that, Kahneman and Tversky ntroduce a loss averson parameter, whch ncorporates the noton that most people suffer more 12 For example, f we assume a utlty functon wth constant relatve rsk averson (CRRA), such as u(x) = x (1-γ) /(1-γ) we cannot produce a bmodal bddng functon for groups of two bdders. For groups of three t s possble to generate bmodal bddng functons but t requres strong rsk lovng preferences. 12

16 from the loss of a certan amount of money than they enjoy the wn of the same amount. For llustraton purposes we use the parametrc specfcaton proposed n Tversky and Kahneman (1992): α x f x 0 v( x) = α λ( x) else. (4) We denote the amount of money a player earns n an aucton by x; α s a parameter for the curvature of the value functon. Rsk averson n gans and rsk seekng n losses requres 0< α< 1. λ s a shftng parameter n the doman of losses, whch s larger than one for loss averson. We assume that n every aucton the reference pont s the actual wealth when enterng the aucton. Thus, wnnng the aucton wth a bd below 100 puts a player n the doman of gans whle losng the aucton wth a postve bd puts a player n the doman of losses. 13 A second ntegral part of prospect theory s the probablty weghtng functon whch maps objectve probabltes nto subjectve probabltes. For smplcty we do not consder the probablty weghtng functon n our context because, unlke n the typcal applcaton of prospect theory, subjects n our game do not know the probablty of wnnng and losng. 14 Our expermental data offers an unusual approach to estmatng preference parameters of prospect theory. Levy and Levy (2002) argue that emprcal evdence on prospect theory focused largely on lotteres nvolvng ether postve or negatve payoffs, whereas real decsons under rsk usually nvolve postve and negatve outcomes. They face ther subjects wth a set of mxed prospects and conclude form ther data that behavour s better explaned by an reversed S-shaped value functon rather than the S-shaped value functon proposed by Kahneman and Tversky. Thus, unlke n prospect 15 theory, Levy and Levy suggest a value functon wth α > 1. We use a least squares method to ft a cumulatve densty for the bds to our data for both group szes. We pool all data from the Recall and NoRecall treatments. The estmates for the preference 13 The reference pont s usually defned as the status quo. Köszeg and Rabn (2006) dscuss the role of reference pont determnaton and present a model where the reference pont can dffer from the status quo. In our context we thnk that an outcome of zero s a natural reference pont. 14 In our context the probablty weghtng functon as usually assumed n cumulatve prospect theory does not offer addtonal predctve power. If subjectve probabltes devate from objectve probabltes, then equaton (3) provdes a condton for the subjectve denstes n equlbrum. In prospect theory t s usually assumed that changes n probabltes close to zero and one have more weght than changes n ntermedate probabltes. Thus, n the extreme case t could be that two prospect theory agents draw ther bds from the unform dstrbuton and perceve the wnnng probabltes as depcted by the bold lne n the rght panel of Fgure 5,.e., we could have an equlbrum where probablty weghtng offsets the effects of the prospect theory specfc shape of the utlty functon. 15 Baucells and Heukamp (2004) however show that the expermental results reported by Levy and Levy (2002) are n lne wth an s-shaped value functon when probablty weghtng s taken nto account. 13

17 parameters are α =.48, and λ= The left panel n Fgure 5 depcts the value functon. The rght panel depcts the ftted cumulatve denstes and the observed bds. Thus, unlke Levy and Levy our data s well organzed by an S-shaped value functon. A reverse S-shaped value functon would predct a mode at ntermedate bds,.e., exactly the opposte of what we observe. Fgure 5 Left Panel: Prospect Theory Value Functon wth Estmated Parameters. Rght Panel: Observed (Bold Lnes) and Predcted (Thn Lnes) Cumulatve Denstes for Both Group Szes u(x) Alpha:.475 Lamda: Group sze=3 Group sze= x Bd Fgure 5 shows that the combnaton of rsk averson n the doman of gans and rsk seekng behavor n the doman of losses as proposed by Kahneman and Tversky (1979) can account for bmodal bddng behavor. Intutvely the subjects use a make-or-break strategy,.e., they ether submt a very low bd and hope for the lucky punch or they submt a very hgh bd n order to ncrease the wnnng probablty. The potentally hgh loss connected wth ths strategy s acceptable due to the rsk seekng preferences n the doman of losses. Surprsngly, the estmate for the loss averson parameter λ s smaller than one, whch means the opposte of loss averson,.e., some sort of loss tolerance. Ths s at odds wth many observatons from experments on loss averson n rsky choce stuatons (see e.g. Gächter et al., 2007 or Abdellaou et al. 2007). We thnk that the man dfference between ths lterature s that we observe rsky choces n a strategc stuaton wth compettve characterstcs. We speculate that there s a 16 The lterature provdes a relatvely wde range of parameter estmates. Tversky and Kahneman (1992) report α =.88 and λ = 2.25; Camerer and Ho (1994) fnd α =.32; Wu and Gonzalez (1996) fnd α =.50, the latter two do not nclude the loss averson parameter. Booj et al. (2007) estmate the parameters n a representatve sample allowng for dfferent powers n gans and losses. They report no sgnfcant dfferences for the power n gans and losses and fnd α =.86 and λ =

18 preference for competng, whch makes wnnng n an aucton more attractve than earnng the same amount of money n a smple lottery. Smlar effects are also found n market entry games wth a compettve structure (Fschbacher and Thön, 2008). Prospect theory can account for bmodal bddng and for the fact that mass s shfted from the hgher mode to the lower mode of the dstrbuton when the number of contestants n the aucton s ncreased. Ths corresponds to the observatons n our experment and also to the data reported by Gneezy and Smorodnsky (2006). Furthermore, by allowng a λ lower than one we can account for the fact that n larger groups bds are typcally hgher than predcted by the Nash equlbrum under standard assumptons. 5. Concluson We nvestgated bddng strateges n very smple common value all-pay auctons wth no pure strategy equlbra. Bdders n our experment use mxed strateges that are remarkably dfferent from the mxed strateges predcted by the Nash equlbrum under standard assumptons. Bdders n our experment drew ther bds not from unform or unmodal denstes but from bmodal denstes. They seemed to apply an all-or-nothng strategy, where they ether chose a very low bd as a low rsk strategy or a very hgh bd wth hgh wnnng probabltes but large potental losses. Bmodal bddng strateges are observed under two very dstnct nformaton condtons: They occur when bdders are n stable groups wth full nformaton about the bddng hstory and also when bdders do not receve any nformaton about other bdders' strateges. The bmodalty n the dstrbuton of bds cannot be explaned by standard rsk preferences but fts very well to the S-shaped value functon proposed by Kahneman and Tversky (1979) n ther prospect theory. We use our data to estmate preference parameters. For the curvature of the value functon we fnd values that are comparable wth the values reported n the lterature. For the second ngredent of prospect theory's value functon loss averson we fnd strkngly dfferent results. The observed bddng strateges are best explaned when assumng the contrary of loss averson. The reason for ths s presumably not because our subjects lke losses, but because the compettve structure of the game offers them addtonal utlty when they wn the aucton. Ths hypotheszed addtonal utlty for wnnng an aucton can addtonally explan why bds tend to become excessve n larger groups, leadng to systematc losses for the bdders n such all-pay auctons. Anderson et al. (1998) propose the logt equlbrum as a soluton concept to account for excessve bds. In ths framework players are boundedly ratonal n the sense that they make random errors when choosng ther bd. The probablty of choosng a strategy that s not best reply s negatvely related to the expected payoff of that strategy. The dstrbuton of bds we observe n our data makes ths explanaton hghly unlkely, because a bmodal dstrbuton of bds s not compatble wth ths knd of erroneous bds. Errors smply shft the denstes predcted by the Nash equlbrum 15

19 under standard assumptons towards the unform dstrbuton, but cannot produce a second mode at hgh bds. What are the consequences of our bdders devatng from strateges derved under standard assumptons? On the one hand bmodal bddng denstes ncrease the varance of the bds. On the other hand, the fact that bdders appear to gan utlty from wnnng the aucton that goes beyond the fnancal benefts reduces ther expected monetary payoff. Many compettve stuatons n the real world nvolve aspects of all-pay auctons, lke lobbyng battles or competng for research money. Whle our experment certanly represents a very stylzed stuaton we stll thnk t s nformatve for real lfe stuatons. People are aware of the fact that mxed strateges are optmal but they seem to be especally attracted to pure strateges at the boundares of the strategy space and seem to be consderably more tolerant to the rsk of losng money than n noncompettve stuatons. 6. References Abdellaou, M, Blechrodt, H., Paraschv, C Loss Averson Under Prospect Theory: A Parameter-Free Measurement. Management Sc., 53(10), Anderson, S. P., J. K. Goeree, C. A. Holt Rent seekng wth bounded ratonalty: an analyss of the all-pay aucton. J. Poltcal Econom. 106(4) Baucells, M., F. Heukamp Reevaluaton of the Results by Levy and Levy (2002a). Organzatonal Behavor and Human Decson Processes, 94(1), Baye, M. U., D. Kovenock, C. G. de Vres The all-pay aucton wth complete nformaton. Econom. Theory. 8(2) Booj, A. S., B. M. S. van Praag, G. van de Kulen A Parametrc Analyss of Prospect Theory s Functonals for the General Populaton. Mmeo. Camerer, C. F., T.-H. Ho Volatons of the Betweenness Axom and Nonlnearty n Probabltes. J. Rsk Uncertanty. 8(2) Davs, D. D., R. J. Relly Do too many cooks always spol the stew? An expermental analyss of rent-seekng and the role of a strategc buyer. Publc Choce. 95(1-2) Dohmen, T., A. Falk, D. Huffman, U. Sunde, J. Schupp, G. Wagner Indvdual Rsk Atttudes: New Evdence from a Large, Representatve, Expermentally-Valdated Survey. IZA Dscusson Paper Seres. 16

20 Fschbacher, U z-tree: Zurch Toolbox for Ready-made Economc Experments. Exper. Econom. 10(2) Fschbacher, U., C. Thön Excess entry n an expermental wnner-take-all market. J. Econom. Behavor Organ. 67(1) Gächter, S., E. J. Johnson, A. Herrmann Indvdual-Level Loss Averson n Rskless and Rsky Choces. CeDEx Dscusson Paper No Gneezy, U., R. Smorodnsky All-pay auctons an expermental study. J. Econom. Behavor Organ. 61(2) Herrmann, B., H. Orzen The Appearance of Homo Rvals: Socal Preferences and the Nature of Rent Seekng. CeDEx Dscusson Paper No Huck, S., H.-T. Normann, J. Oechssler Two are few and four are many: number effects n expermental olgopoles. J. Econom. Behavor Organ. 53(4) Kahneman, D., A. Tversky Prospect Theory: An Analyss of Decson Under Rsk. Econometrca. 47(2) Klemperer, P Auctons: theory and practce. Prnceton Unversty Press, Prnceton, NJ. Köszeg, B., M. Rabn A Model of Reference-Dependent Preferences. Quart. J. Econom. 121(4) Levy, H., M. Levy Prospect theory: Much Ado about nothng? Management Sc Mllner, E. L., M. D. Pratt Rsk averson and rent-seekng: An extenson and some expermental evdence. Publc Choce. 69(1) Öncüler, A., R. Croson Rent-seekng for a rsky rent A model and expermental nvestgaton. J. Theoret. Polt. 17(4) Rabn, M Rsk Averson and Expected-utlty Theory: A Calbraton Theorem. Econometrca. 68(5) Shogren, J. F., K. H. Bak Reexamnng effcent rent-seekng n laboratory markets. Publc Choce. 69(1) Tullock, G The welfare costs of tarffs, monopoles and theft. Western Econom. J. 5(3) Tversky, A., D. Kahneman Advances n prospect theory: cumulatve representatons of uncertanty. J. Rsk Uncertanty. 5(4) Wu, G., R. Gonzalez Curvature of the Probablty Weghtng Functon. Management Sc. 42(12)

21 6. Appendx: Expermental Instructons General explanaton for the partcpants You are now takng part n an economc experment that s fnanced by dfferent research-promotng facltes. If you are readng the followng explanaton carefully, you can dependng on your decsons make a consderable amount of money. Hence, t s mportant that you read ths explanaton carefully. The nstructons you wll receve are for your prvate nformaton only. Durng the experment absolute slence s requred. Communcaton s prohbted. If you have any questons please drect them towards us. Non-observance of these rules wll lead to excluson from the experment and any payments. Durng the experment we do not talk about Swss Francs. Your ncome wll be calculated n ponts. At the end of the experment, the attaned ponts wll be transferred nto Swss Frances, where 1 pont = 1 centme. At the end of the experment you wll receve your earned ponts (n CHF) plus CHF 20 for showng up, n cash. If you make a loss, t wll be deducted from the CHF 20. You cannot make an overall loss. The next pages descrbe the detaled procedure of the experment. Experment Instructons: Recall Treatment You are takng part n an aucton. In total there are 10 rounds and n each of these rounds a prze of 100 ponts s auctoned. You wll be put n a group of ether two or three partcpants. Hence there wll be one other or two other bdders n your group besdes yourself. The group composton wll be constant durng the 10 rounds,.e. n each round you are n the same group wth ether one or two other partcpants. You wll not know who else n ths room s n your group; your dentty wll be kept secret. When the aucton begns you wll have to place your bd. All partcpants do so at the same tme. You can place a bd up to three decmal places. In each group the partcpant placng the hghest bd wns. If more than one partcpant bd the same hghest value, the computer randomly assgns the wnner. Dfferent to usual auctons you mght know, not only the wnner but all bdders have to pay ther bd. As soon as the experment starts you wll see the followng screen: 18

22 On the screen, you can see whether you are placed n a group wth 2 or 3 partcpants. In the rght hand upper corner you can note the remanng tme you have to place your bd. Type your bd nto the feld. After submttng the bd [pressng OK] you wll not be able to change t agan. You only bd once n each round. You can place bds from and wth zero and up to three decmal places. As soon as all the partcpants have submtted ther bds, the computer wll calculate the hghest bd and determne the earnngs n ponts. It wll then appear a screen showng the outcome of the aucton: You wll be notfed f you won the aucton and of your revenue n ths round. Addtonally you wll see the bds of the other partcpant(s) as well as your own. Subsequently press the contnue button to proceed to the next round. 19

23 An example to clarfy the rules: Assumng that n a group of three partcpants the followng bds are submtted: Anton: 10 ponts Berta: 50 ponts Claus: 80 ponts Claus wns the aucton and has a revenue of 20 ponts (=100-80) n ths round. Anton and Berta do not wn, but have to pay ther bd nevertheless. The revenue of Anton s therefore -10 and that of Berta s -50 n ths round. Do you have any questons? Experment Instructons: NoRecall Treatment Explanaton to the second experment The second experment also conssts of 10 rounds, n each of whch 100 ponts are to be auctoned. There are two mportant modfcatons: You wll not receve any nformaton whether or not you won the aucton n ths round. You wll also not receve any nformaton about the bds of the other partcpants. The constellaton of the group changes n each round. The group sze vares between 2 and 3. The computer wll randomly assgn you n a group of 2 or of 3. You wll be able to see the sze on the screen. Do you have any questons? 20