Truth-Revealing Mechanisms for Courts

Transcription

1 Truth-Reveaing Mechanisms for Courts by Robert Cooter * and Winand Emons ** January 2000 Abstract In trias itnesses often sant their testimony in order to advance their on interests. To obtain truthfu testimony the a reies on cross-examination under threat of prosecution for perjury. We sho that perjury a is an imperfect truth-reveaing mechanism. More importanty e deveop a perfect truth-reveaing mechanism. Under this mechanism the itness is sanctioned if a court eventuay finds that the testimony as incorrect; the court need not determine that testimony as dishonest. We expain ho truth-reveaing mechanisms coud combat distortions of observations by factua itnesses and exaggerations by experts incuding junk science. Journa of Economic iterature Cassification Numbers: D82 K4 K42. Keyords: itigation process truth-reveaing muti-dimensiona mechanism perjury rue. * Schoo of a Boat Ha University of Caifornia Berkeey CA USA rdc@a.berkeey.edu ** Department of Economics University of Bern Geseschaftsstrasse 49 CH-302 Bern Sitzerand inand.emons@vi.unibe.ch We thank Rafae Gey Hans Peter Grüner Martin Heig eis Kornhauser Benny Modovanu Eric Rasmusen Urs Scheizer and Matt Spitzer for hepfu comments.

2 . Introduction In deciding ega disputes courts must rey on observers to report facts and experts to provide opinions. Witnesses hoever often gain from providing fase testimony. Many itnesses have a materia interest in the case. For exampe the paintiff and defendant have a materia interest in the judgment and an expert has an interest in future empoyment as a itness. The interest of the itness in the case provides an incentive to sant testimony. To obtain the fu truth from sef-interested itnesses the ega process needs to provide incentives that offset the gain from santed testimony. Deterrence requires a rationay sef-interested itness to pay an expected price for santed testimony greater or equa to the expected gain. The a reies on cross-examination under the threat of prosecution for perjury to deter fase testimony. Cross-examination probes the quaity of testimony by the itness searching for interna inconsistencies or contradictions ith testimony by other itnesses. In a crimina tria for perjury the paintiff must prove that the defendant ied tod a fasehood knoingy or possiby that the defendant reckessy disregarded the truth. 2 If the a aos civi iabiity for fase testimony the paintiff in a civi case must prove the reevant form of faut by the defendant. In either case the paintiff must prove that the defendant vioated a ega standard hen testifying. Estabishing guit or iabiity often requires more information than anyone can prove in court so perjury trias or civi trias arising from perjury are rare. In practice a For the rapid groth of economists acting as expert itnesses see Posner 999 Thornton and Ward 999 and Mande 999. This form of consuting is no designated forensic economics. Severa associations such as e.g. the Nationa Association of Forensic Economics NAFE as e as a coupe of journas ike e.g. the Journa of Forensic Economics have emerged due to this boom in the demand for economists as experts. 2 According to the Mode Pena Code perjury requires testimony in court under oath that is miseading and materia. In addition perjury requires knoedge that the assertion as fase hen made or possiby reckess disregard for the truth. Fase testimony in an American court cannot support a civi suit for damages so a victim of sander or ibe in court has no ega remedy. For a more eaborate discussion of the ega institutions see Cooter and Emons 2000.

3 skifu itness can sant testimony ithout fear that the paintiff in a suit can prove that the itness vioated a ega standard. Moreover as e sho formay even hen a this information is avaiabe perjury rues are not perfect truth-reveaing. This paper appies the theory of mechanism design to the courtroom. We derive a system of sanctions hich is perfecty truth-reveaing. We repace the faut-based perjury rue ith a rue of strict iabiity. Under our rue of strict iabiity the itness pays a sanction for testimony that proves inaccurate regardess of hether inaccuracy as the itness s faut. Repacing a faut-based rue ith strict iabiity reduces the information needed by the paintiff to impose the sanction. In principe a rue of strict iabiity can deter distortions by factua itnesses and exaggerations by experts incuding junk science. 3 To further motivate our anaysis consider the fooing four exampes of testimony by itnesses to hich our mode appies: Exampe : A suit beteen to motorists over an automobie accident turns on ho as at faut. A pedestrian ho is neutra beteen the parties observed the accident. The pedestrian testifies on the question Was the stopight red? Exampe 2: A oman maintains a sexua iaison ith a handsome poor man and an ugy rich man. When a chid is born the mother needs to estabish the father s identity to in a paternity suit by testifying on the question Who is the chid s father? Exampe 3: In a suit for damages the defendant considers hether to argue that he is not iabe because he did nothing rong or to argue that he is iabe and the damages are sight. The second argument undermines the first argument so the defense attorney does not ant to make both arguments. The defense attorney s choice of an argument depends upon ho the defendant s expert ansers the question Did the defendant do rong? Exampe 4: A rich man and a poor man go hunting. The to hunters shoot at a bird and one of them injures a third person. In a suit for damages the judge has an interest in finding someone iabe preferaby the rich hunter. Fooing 3 Martha Nussbaum s testimony in Romer v. Evans seems to be an exampe of expert perjury. She seems to have miseadingy cited the ong superseded 897 edition of a Greek-Engish exicon isting no pejorative connotation of the Greek ord tomêma hereas in the ater 940 edition hich she normay cites in her academic ork shameess act is incuded as a possibe transation of tomêma. See ingua Franca Sept/Oct

4 the practice in many civi a countries the judge chooses a baistic expert to testify on the question Which gun fired the buet that struck the paintiff? In Exampe the pedestrian ho itnesses the accident is neutra in the sense that the decision of the court does not affect her materia interests. In Exampe 2 the mother testifying about her chid s paternity has a direct materia interest in the case. In Exampe 3 a party to the case pays an expert itness as is the usua practice in the adversaria systems of the common a countries. By advancing the interests of the party retaining her an expert itness increases her prospects for empoyment in subsequent cases. In Exampe 4 the court seects an expert itness as is the usua practice in the inquisitoria systems of the civi a countries. The judge might have an interest in the case because he ants to resove it quicky ith east effort pease his superiors ho can promote him obtain a bribe or he is a friend of the paintiff. 4 By advancing the interests of the judge in Exampe 4 the expert itness increases her prospects of being hired by courts in subsequent cases. A of our itnesses observe a fact that is reativey good or reativey bad for the defendant or paintiff. The itness is either pretty certain or rather uncertain about the observation s accuracy. In terms of Exampe the pedestrian may have observed that the stopight as red but she may be rather uncertain because the sun binded her. In technica anguage a itness receives a signa that is better or orse ith high or o precision. When testifying in court a itness reports on the signa s content better/orse and precision high/o. An honest itness reports truthfuy about content and a dishonest itness reports fasey about content. A candid itness reports accuratey about precision and a miseading itness reports inaccuratey about precision. 4 Usman 999 endogenousy determines the eve of verifiabiity assuming a judge ho vaues correct ruings but disikes effort. 3

5 The court uses the avaiabe evidence incuding the testimony of itnesses to decide the case. 5 After a itness testifies subsequent events may prove that the testimony as right or rong. To iustrate by Exampe after the pedestrian testifies that the stopight as red someone may discover a photograph proving concusivey that the stopight as green. In Exampe 2 the mother may testify that the poor man as the father and after the tria subsequent deveopments in bioogy may resut in a proof that she as right. Our mode styizes these facts. We assume that after the itness testifies the court or simiar body subsequenty earns ith positive probabiity hether the itness as rong or right. In our mode a truth-reveaing mechanism uses the testimony of the itness and any subsequent findings by the court to possiby sanction the itness. A perfect truth-reveaing mechanism induces honest and candid testimony in a circumstances. 6 Our mechanism has a straightforard interpretation. A itness may gain from dishonest or miseading testimony. Against this gain the itness must baance the probabiity and magnitude of a sanction. A perfect truth-reveaing mechanism imposes an expected sanction equa to the gain from dishonesty or disingenuousness. To iustrate consider the mother in Exampe 2. Assume she is certain that the poor man is the father. She i hoever enjoy a arger court aard by asserting fasey that she beieves ith certainty that the rich man is the father. Perhaps officias i eventuay discover the truth through advances in bioogy. By definition the expected sanction equas the probabiity that such events i trigger a sanction mutipied by the sanction s magnitude. With our perfect truth-reveaing mechanism the expected sanction exacty equas the mother s gain from a fase report. 5 An episode in the 935 Hoard Haks movie Barbary Coast nicey fits our mode: As the defendant is being marched off to the gaos the tria occurs. The de facto judge asks to questions: Did you see the defendant shoot Bi? On this being ansered affirmativey the second question is: Are you sure. That too being affirmative the defendant is hanged. footnote needs to be doube checked 6 Our aim throughout the paper is to revea the truth. From a societa point of vie there may be cases here ies are better than the truth e.g. if somebody ies to protect a ady s honor or if a government officia ies about a foreign poicy event to protect ongoing deaings. 4

6 As another iustration consider the baistic expert in Exampe 4. Assume her tests indicate that the rich hunter fired the fata buet but she is uncertain about these tests. To promote her future business the expert can hep the judge by testifying fasey that she is certain that the rich hunter fired the fata buet. Doing so hoever the expert runs the risk that someone i subsequenty present irrefutabe proof that the poor hunter fired the fata buet. With our perfect truth-reveaing mechanism the expected sanction increases hen the expert asserts her concusion ith certainty rather than uncertainty and the increase exacty equas the gain to the expert from more future business. The sanctions in the perfect truth-reveaing mechanism can be interpreted as bond forfeited by the itness in the event that evidence disconfirms her testimony. 7 To iustrate by Exampe 3 assume the itness reports that the defendant did not do rong and that she is uncertain. The court and the paintiff s counse may suspect that the itness actuay observed ith uncertainty that the defendant did rong. The court or the paintiff s counse chaenges the itness to bond her testimony. To retain credibiity the itness has to post bond. What is the minimum bond inducing honest and candid testimony? The expected sanction must equa the itness s gain from dishonest and/or miseading testimony. In principe the court and paintiff s counse can compute the minima sanction from the itness s gain and the probabiity of disconfirming evidence. Thus the counse and court requires the itness ho testifies no rong and imprecise to post a truth-bond at east equa to the minima sanction in order to estabish credibiity. To iustrate by Exampe 2 assume the baistic expert testifies that the evidence indicates that the rich hunter fired the fata shot athough she cannot be certain. The defendant s attorney hoever suspects that the itness is dishonest. He may chaenge the itness to estabish her credibiity by bonding her assertion. Woud you bond the proposition that 3 out of 4 baistic experts ho examined the same evidence as you 7 See Cooter and Emons 2000 for an eaborate discussion on ho truth-bonding might ork in practice. 5

7 oud agree ith your concusion? The optima bond equas the increase in the payoff to the baistic expert from testifying rich hunter rather than poor hunter divided by the probabiity that at east 2 out of 4 baistic experts oud disagree ith her. et us no turn to a imitation of our truth-reveaing mechanism. In a of our exampes the itness benefits more from testifying that she observed better rather than orse given the precision. 8 She aso benefits more from testifying that she is certain rather than uncertain given the content. Accordingy payoffs increase henever the report s content and certainty improve. This is anaogous to a portfoio of stocks becoming more vauabe henever the mean increases and/or the variance decreases. To iustrate in Exampe 2 the mother gains most from testifying ith certainty that the rich man is the father. Next she gains from testifying ith certainty that the poor man is the father. The orst possibiities occur hen she cannot testify ith certainty about the father s identity in hich case she may not in in court. Exampe 3 aso fits this pattern of reasoning. The defendant s expert benefits the defendant most and hence benefits hersef most by asserting that the defendant certainy did nothing rong. Next the expert benefits the defendant by asserting that the defendant did something rong and the damages are certainy sma. The expert benefits the defendant east by asserting that she is uncertain hether the defendant did right or rong. This pattern of benefits hich fits the conception of the defense in Texaco v. Pennzoi 9 appies henever the defendant prefers to take a stand on ony one issue. Our mechanism no onger orks in scenarios here the itness benefits from being uncertain rather than certain. To iustrate by an exampe: If an aeged monopoist s expert finds out that the company has indeed a high market share the expert and the monopoist benefits from testifying to this fact ith uncertainty rather than certainty. Such a pattern of payoffs vioates a monotonicity requirement necessary 8 For purposes of our anaysis the opposite is aso acceptabe because better and orse are arbitrary. 9 Texaco s ayer refused to offer expert testimony on damages on the theory that doing so oud undermine the confidence of the court that Texaco as not a faut. This proved a disastrous mistake because Pennzoi caimed damages equa to the sae vaue of the oi fied rather than the expected profits that the oi fied oud yied. See e.g. Petzinger

8 for the existence of a perfect truth-reveaing mechanism sanctioning ony rong testimony. More possibiities to sanction are then necessary to induce truth-teing. We sketch such a mechanism hich aso sanctions right testimony. 0 Finay e anayze the faut-based perjury rue in our set-up. Under the perjury rue if the testimony ere rong the court must use this information to compute the probabiity that the itness as dishonest. If this probabiity exceeds the ega standard the court imposes the sanction for perjury. As e sho formay these Bayesian inferences are fairy difficut to make and require a ot of information. This fact provides one reason hy perjury prosecutions are so rare. Next e sho that under perjury a a neutra itness i never report a high precision signa. Since a court is more ikey to find perjury hen testimony as given ith certainty rather than uncertainty a neutra itness minimizes the probabiity of being sanctioned for perjury by understating her certainty. Finay since a simpe perjury rue does not adjust the sanction to the probabiity of detecting and prosecuting the perjury it acks the sophistication hich is necessary to induce an interest itness to te the truth. Because of these imitations a perjury rue is therefore an imperfect truthreveaing mechanism. The remainder of the paper is organized as foos. In the next section e describe our basic frameork. In section 3 e anayze the itness s incentives. In section 4 e derive the perfect truth-reveaing mechanism. In section 5 e discuss the perjury rue. The ast section concudes. Proofs are reegated to the Appendix. 2. The Mode A court s decision in a case depends on the outcome of a random event. This random variabe ~ X can take the to reaizations X=A and X=B. To iustrate a drug may have 0 In a subsequent paper e hope to expore this probem further and sho that ithin the cass of mechanisms sanctioning ony rong testimony our mechanism is second-best optima hen no perfect truth-reveaing mechanism exists. 7

9 to side-effects one B somehat orse than the other A. Or B might mean the shot as fired by the poor hunter and A the rich hunter fired the fata shot. The court has some information about the ikeihood of the to events hich e denote by ProbX=B:=ProbB and ProbX=A:=ProbA =-ProbB. The defendant can base his defense on either A or B. His case is somehat stronger for A than for B. A good defense for A is hoever a pretty bad one for B and vice versa. The defendant is therefore interested in obtaining as much information as possibe about hich state of the ord i materiaize. If the to states are equay ikey he prefers of course A. Consider e.g. the side effects caused by the drug. If both possibiities are equay ikey the drug company has to argue that it took appropriate measures against both side-effects. If the defendant can convince the court that one of the to effects occurred rather than the other he can focus his defense rather than fighting on to fronts ike Germany in Word Wars One and To. Formay e define the indicator variabe U A = and U B = 0. The defendant s utiity function is then given as ~ V = EU X ~ αvaru X = Prob A αprob A Prob B here E stands for the expected vaue Var for the variance of the indicator variabe and α measures the dis- taste of the associated risk. We assume α to be sufficienty arge hich e i make precise beo. A itness observes a fact ith an attached probabiity hich e ca a signa that is reevant to the court s decision. 2 To iustrate concretey the itness observes a medica fact and infers a definite probabiity about the occurrence of the side-effects. Or the itness runs some tests and infers a definite probabiity about hich gun fired the fata shot. In notation the itness has observed a signa YP Y {AB} P {H} If the court has no further information by the principe of insufficient reason ProbB =ProbA =/2. 2 As in Shin 999 e treat the information coection process as exogenous in order to focus on the incentives to discose the evidence. In Deatripont and Tiroe 999 information gathering is costy; their focus is on the incentive to gather information. 8

10 meaning that the state of the ord Y i occur ith probabiity P. The signa BH for exampe means that the state B i occur ith probabiity H hich of course impies that the state A occurs ith probabiity -H. Simiary the signa A means that the state A occurs ith probabiity etc. First of a e assume that a four signas reduce uncertainty; furthermore it is convenient to define H and so that one is more precise than the other in the sense of conveying more information or formay > H > > max[ Prob B; Prob A] / 2. By this condition a signa B or A is o precision and a signa BH or AH is high precision. We can no make precise the defendant s distaste of risk i.e. e assume α > H / H H. If α satisfies this condition the defendant prefers A H to B H to A to B to no signa at a. 3 As an exampe consider tossing to dice. The state A corresponds to the sum of the to dice exceeding 7 hie B occurs hen the sum is 7 or ess. Accordingy ProbA = 5/2 and ProbB =7/2. After the to dice are roed the itness observes something about the number on one of them. The itness observes either one of the to high precision signas {56} and {234}. The information that the first die is either 5 or 6 transates into the signa that the good state occurs ith probabiity 3/4 i.e. A3/4 and the information that the first die is ess than 5 corresponds to the signa B3/4. Aternativey the itness observes one of the to o precision signas {456} corresponding to A2/3 and {2345} corresponding to B2/3. The itness testifies in court on her private information. She announces a state of the ord and the probabiity ith hich this state i occur. 4 To avoid confusion e use sma etters for her reported vaues. Formay the itness announces yp 3 Our anaysis easiy carries over to the case here the defendant s preferences are A H f A f B H f B. In Proposition 4 e sho that truth-reveation is impossibe if e sanction ony rong testimony hen montonicity in the precision is vioated. 4 We thus confine our attention to direct reveation mechanisms. By the reveation principe this impies no oss of generaity. See e.g. Myerson

11 y {ab} and p {h}. We i use the fooing semantics: If y=y testimony is honest; otherise testimony is dishonest. If p=p testimony is candid; otherise testimony is miseading. Our aim is quite naturay to get an honest and candid testimony. 5 Depending on her reported vaues the itness receives a remuneration age yp 0 from a third party. Taking future consequences into account remuneration is higher hen the testimony is more favorabe to the party for hom the testimony is given. In a tria testimony that favors one party disfavors the other party. Consequenty e i vie testimony from the viepoint of one party e i take the defendant and scae vaues accordingy. In our setup the defendant prefers high to o precision signas and better a to orse b nes. Consequenty a a. 6 The age depends upon the ega and contractua status of the itness. An interested itness receives a age for testimony that increases ith the strength of her testimony. Formay for an interested itness a three of the above age inequaities are strict. Typicay an interested itness is a party to the suit or an expert paid by a party to the suit. Under US rues expert itnesses are interested. A neutra itness receives a constant age for testifying meaning that the equaity hods in a of the above eak inequaities. If this constant age is zero e i ca the itness disinterested. Under European rues expert itnesses are more often neutra. Under European and American rues itnesses to the facts are typicay unpaid for testimony so they are neutra uness connected to the paintiff or defendant. After the itness has testified further deveopments in the tria may refect upon the accuracy of the itness s testimony. We styize this fact by assuming that the court 5 In our set-up the itness can ie i.e. report fase information. There is a reated iterature comparing the adversaria partisan procedure of the Ango-Saxon a in hich partisan advocates present their cases to an impartia jury ith the inquisitoria procedures of Roman-Germanic countries in hich judges take an active roe in investigating a case Deatripont and Tiroe 999 and Shin 999. In these papers a party can concea information but cannot report fase information. Shin 999 justifies the assumption of no fase evidence a reported information is verifiabe ith the effectiveness of perjury rues. Our resuts on perjury rues tend to quaify this assumption. 0

12 observes the true state of the ord after the tria s end. 7 We i say the testimony is right if X=y; otherise the testimony is rong. Conditiona on the reationship beteen the testimony and the court s observation the itness can be rearded or sanctioned. Formay e denote a sanction/reard by SXyp here S > 0 is a sanction and S < 0 a reard. We ant to derive mechanisms orking ith minima sanctions and reards. Therefore e set the sanction equa to zero henever the testimony is right i.e. SB = SAa = SB = SAa = 0. The itness s expected payoff equas her age minus the expected sanction. ~ ~ Formay the payoff is given as y p E S X y p Y P here E S X y p Y P stands for the expected sanction given her reported testimony yp and the true information Y P. She chooses her reported testimony yp so as to maximize her expected payoff. 3. The Incentive Constraints We ant to derive a system of sanctions that induces the itness to be honest and candid. We ca such a mechanism perfect truth-reveaing. This means that reporting the true signa must generate at east as much payoff as announcing any other signa. Formay this requirement means ~ ~ Y P E S X Y P Y P y p E S X y p Y P yp {a} {h} YP {BA} {H}. Consider for exampe the case in hich the true signa is YP=B. Here one of our tasks is to guarantee that announcing yp= is at east as good as reporting 6 Note that e do not further anayze the reationship beteen the defendant and the itness. We have specified the defendant s preferences ony to motivate the itness s age schedue. 7 Since everybody in our mode is risk neutra it is straightforard to extend the anaysis to the case in hich the court observes the true state ony ith a probabiity γ<. Then a the sanctions in the perfect truth-reveaing mechanism have to be mutipied by /γ. Note that e coud ork ith any imperfect signa of the itness s observation YP. To save on notation e have chosen the true state of the ord.

13 a. Formay this means --SA a-sba. 8 If the itness tes the truth she receives the age. With probabiity - the state A materiaizes and the itness has to pay the sanction SA. If in contrast she reports a she receives the higher age a. No the sanction is SBa triggered by the state B hich occurs ith the hig probabiity if B is the true signa. Simiary if the true state is YP = B e must guarantee that the message yp= is at east as good as the reports and a. Anaogous incentive constraints hod for the other 3 signas so that overa e end up ith 2 incentive constraints. After some agebraic manipuation and rearranging e have the fooing system of 6 eak inequaities. -SBa--SBa a-a -HSBa--HSBa 2 -SA--SA - -HSA--HSA 3 SBa--SA a- -SBa-SA 4 HSBa--HSA a- -HSBa-HSA 5 SBa--SA a- -HSBa-HSA 6 SBa--SA a- -HSBa-HSA; ca the first inequaity in such a chain a and the second one b. Before deriving the perfect truth-reveaing mechanism in detai e can aready state a preiminary resut namey that truth-reveaing mechanisms differ for interested and neutra itnesses. For a neutra itness sanctions must be constant hereas for an interested itness the sanctions increase ith the strength of the testimony. Proposition : Perfect truth-reveaing sanctions for interested itnesses satisfy SA < SA and SBa < SBa. If the itness is neutra SA = SA = SBa = SBa. The intuition for this resut is straightforard. An interested itness s age increases ith the strength of the testimony being maxima for the reported vaue a. 8 Reca that e set the sanction to zero henever the testimony is right so SBb =SAa=0. 2

14 If the sanctions ere say constant an interested itness oud aays report a. To compensate for the increasing age schedue sanctions must increase ith the strength of the testimony. Conversey if the itness is neutra the age schedue provides no incentives not to te the truth. In order not to distort the age schedue s proper incentives the sanctions must be neutra too. 4. The perfect truth-reveaing Mechanism et us no determine the truth-reveaing mechanism. We focus on mechanisms empoying minima sanctions/reards. This means first that e set as many sanctions as possibe to zero; second e set those sanctions hich need to be positive to the minima vaues sti providing proper incentives. We make the sanctions as o as possibe in order to minimize the monetary strain on the itness. See aso the fooing discussion on individua rationaity in Proposition 3. The reason hy e do not ork ith reards is tofod. First e ant to keep the cost of the judicia system o and reards are costy. A second probem arises if reards become so high that before having observed the signa the itness knos she i receive an expected reard. Then agents ithout any knoedge of the case may try to be caed upon as itness simpy to cash in on the expected reard. 9 Proposition 2: The perfect truth-reveaing mechanism using minima sanctions/reards is given by a a / S * / Xyp= a / 0 + a / if X = B y = a p = h; if X = A y = p = h; if X = B y = a p = ; otherise. The truth-reveaing mechanism obviousy refects Proposition. If the itness is neutra a sanctions are zero. If the itness is interested sanctions increase ith the strength of the testimony. 9 See Emons and Sobe 99 for a more eaborate discussion of the probems generated by expected reards. 3

15 The sanctions are constructed as foos. When the itness orks out for instance hether to report the true eak or a fase strong signa she compares the increase in the age ith the increase in the expected sanction. Accordingy a e have to do is to ensure that the increase in the expected sanction is at east as great as the increase in the age. This task is somehat tedious due to the stochastic nature of our probem; sanctions appear in severa incentive constraints at the same time. This generates severa oer bounds for certain sanctions and of these e have to take the maximum. With this type of construction for a certain deviation the itness is just indifferent hie for other deviations the incentives are strict. Finay e have to check that e did not overdo it i.e. set the sanctions so high that they distort the itness incentives esehere. After a this technica parance it seems a good idea to iustrate the truth-reveaing mechanism using the dice exampe. Reca that =2/3 and H=3/4. et =0 a=6 =8 and a=0. Then SA=0 SA=24 SBa=9 and SBa=2. If the itness has observed for exampe A expected sanctions for the possibe reports are given by Figure. Given the true report a the margina expected sanctions are greater or equa the margina ages making any deviation from reporting the true signa unattractive. Figure Perfect Truth-Reveaing Mechanism for the signa A report a a Wage to itness Expected sanction given the signa A Net payoff

16 Here the surprising feature is that the highest sanction is imposed hen the itness has reported and A materiaizes. This resut foos immediatey from 2a. Reporting rather than increases the age by a steep 8. This increase has to be compensated by SA hich is imposed ony ith the o probabiity -=/3. We have chosen the exampe deiberatey such that the monotonicity of the age scheme is not entirey refected in the sanctions. 20 Wages are monotone increasing in the strength of the testimony but sanctions are not. The highest sanction is imposed for the second highest testimony. Note that such non-monotone incentive schemes are the rue rather than the exception. In the principa-agent probem for exampe the agent s remuneration is typicay not monotone in the outcome. See e.g. Grossman and Hart 983. Another unpeasant feature of truth-reveaing mechanisms is that agents often do orse if they participate in the mechanism than if the do not participate. In mechanism design jargon participation in an incentive scheme ith this feature is not individuay rationa. Agents do better if they stay out of the incentive scheme than if they take part: they must be forced to participate. 2 Transated into our probem interim individua rationaity requires that the itness s expected payoff is non-negative hatever signa she received. 22 To put it differenty expected sanctions may not exceed the age or formay ~ Y P E S X Y P Y P 0 Y P { B A} { H}. Fortunatey our mechanism is e behaved. Proposition 3: The mechanism S * Xyp defined in Proposition 2 is individuay rationa. This resut hich foos more or ess immediatey from our construction has the impication that the itness hatever signa she receives need not be forced to testify in 20 Note that e are taking no about actua rather than expected sanctions. 2 See e.g. Emons 994 for a detaied discussion of individua rationaity. 22 Reca that under our rue the itness reports truthfuy a signas. See Homström and Myerson 983 for a definition of the ex ante interim and ex post concepts. 5

17 court. 23 She i do so vountariy because her expected payoff from doing so is nonnegative. Ex post hoever hen the testimony has actuay turned out to be rong and the itness is sanctioned she may end up ith a negative payoff as can be seen by our exampe. We may therefore concude that from an interim point of vie it is individuay rationa to testify in court athough ex post the itness may regret to have done so. et us concude this section by shoing that truth-reveation is impossibe hen the age is decreasing in the precision such that e.g. > meaning the defendant prefers bad nes ith o rather than ith high precision. Proposition 4: If > and/or a > a no perfect truth-reveaing mechanism ith SXyp 0 if X y and SXyp=0 if X = y exists. If the age increases hen the reported precision decreases the probabiity that triggers the sanction if the itness is miseading decreases ith the age. But then it is impossibe that the expected sanction increases ith the age as is necessary for truthteing. The non-existence probem arises because e sanction the itness if and ony if testimony is rong. The probabiity of a rong testimony is oer for the high than for the o precision signa. To provide proper incentives hoever the expected sanction for rong testimony must be higher for the high rather than the o precision signa. The non-existence phenomenon disappears if e aso ao the itness to be sanctioned hen the testimony is right i.e. if e et SXyp>0 for X = y. This can be seen by the fooing extension of our mechanism. Consider the case here a a >. We introduce sanctions σ y p hich are evied in addition to S X y p. We charge the itness σ y p simpy for reporting y p independenty of hether this report turns out to be right or rong. σ a = σ a = σ = 0 and σ =. Then the itness s age et 23 For a neutra disinterested itness a perfect truth-reveaing sanctions are zero so that individua rationaity is triviay satisfied. 6

18 net of σ W : = σ satisfies W a W a W = W. Construct S * Xyp as given in Proposition 2 using W rather than. Obviousy the extended mechanism σ ; S* is perfect truth-reveaing. 24 If e sanction the itness ony for rong testimony e do not have enough everage to induce the report b rather than b. etting the itness pay σ > 0 gives us this everage aternativey e can aso reard the itness ith σ < We pan to expore this case further in a subsequent paper. 5. The Perjury Rue et us no compare our truth-reveaing mechanism S * ith the perjury rue. As in our mechanism under a perjury rue the sanction is zero henever the testimony is right. If hoever the testimony is rong the court uses this information to compute the probabiity φ that the itness did not te the truth. If this probabiity exceeds a ega standard φ the court imposes a sanction s>0; if the probabiity φ is beo the ega standard the sanction is zero. 26 Formay S p Xyp= s if φ Xyp φ; 0 otherise. Computing the probabiity φ of not having reported the truth turns out to be tricky. First the court has to kno the probabiities ProbB and ProbA ith hich the to states of nature occur. Note that e did not use this piece of information for our mechanism S *. Moreover the court needs to kno the probabiity distribution over the signas YP hich e denote by ProbYP. 27 To have some structure assume that the 24 Even more to the point is the mechanism σ y p = y p y p and no further sanctions for rong testimony. Under this mechanism the itness s payoff equas zero for a reports. 25 It is of course harder to satisfy individua rationaity ith mechanisms sanctioning correct testimony. 26 For φ 0 the perjury rue essentiay orks ike a negigence rue ith a due care standard φ. If φ =0 the perjury rue functions ike a rue of strict iabiity and if φ = ike a rue of no iabiity. 27 This probabiity distribution depends of course on the stochastic process generating the outcomes A and B. For an appication see the fooing discussion of our dice exampe. 7

19 o precision signas are at east as ikey as the high precision signas i.e. ProbBH ProbB and ProbAH ProbA. Note once again that for our mechanism S * e did not use the probabiity distribution ProbYP. If the itness has reported say a and nature has chosen B the probabiity of not having tod the truth is 28 φba = - Prob A H B = - - Prob B A H Prob Prob B B A H Prob A H Prob B = - = H Prob Prob B A H The probabiity that a as not the true signa given B equas the sum of the probabiities that the itness has observed a and given B hich in turn equas minus the probabiity that a as the true signa given B. Anaogousy e compute φba= - φa= - Prob Prob B Prob Prob A A B φa= -. H Prob Prob A B H. and et us iustrate these probabiities by means of our dice exampe. Before the dice are tossed nature chooses ith equa probabiity /2 hether the itness i observe high or o precision signas. If she is to observe a o precision signa and the outcome of the first toss is 4 or 5 she receives the signa A and B ith equa probabiity /2. With this signa generating process e have ProbAH = ProbA = /6 and ProbBH = ProbB = /3. We then compute φba = 3/4 φba = 9/2 φa = 4/5 and φa = /5. Given these four probabiities of not having tod the truth e may no state the resut that a neutra itness s testimony is miseading under the perjury rue hen she has observed a high precision signa. 28 A more precise yet more cumbersome notation for φ B a oud be φ a B. 8

20 Proposition 5: Under the perjury rue for a neutra itness the o precision signa eaky dominates the corresponding high precision signa. This resut is easiy expained. Suppose the itness has observed the signa AH and she compares the honest and candid message a ith the honest but miseading message a. The probabiity that the testimony is rong the outcome hich may trigger the perjury rue s sanction is the same for both messages. Nevertheess the probabiity of not having tod the truth is higher for a than for a i.e. φba> φba. Consequenty the expected sanction for a is at east as great as for a and reporting a eaky dominates a: If φba< φ [φ<φba] so that the itness is never [aays] sanctioned miseading does not hurt. If in contrast φba< φ<φba being disingenuous is stricty better than being candid. 29 Given that a neutra itness is miseading for high precision signas the next natura question to ask is: under hat conditions is her testimony under a perjury rue at east honest? Here e have the fooing straightforard resut. Proposition 6: If max[ φ A φ B a ] <φ or min[ φ A φ B a ] φ the neutra itness testimony is honest. From Proposition 5 e kno that the itness aays reports a o precision signa. If the probabiities of untruthfu testimony are beo the ega standard for both reevant messages and a the itness is never sanctioned and accordingy indifferent beteen the to messages. Therefore her report is honest though not necessariy candid. If both probabiities of being untruthfu are above the ega standard for both messages the itness is sanctioned by the amount s henever the testimony is rong. If the victim reports honesty the probabiity of being rong -H resp. - is oer than for an dishonest report H resp.. et us finay anayze the incentives a perjury rue gives an interested itness. As can be expected the resut is negative. 9

21 Proposition 7: If the itness is interested a perjury rue is never truth-reveaing. Incentive compatibiity requires for an interested itness that the sanctions increase ith the strength of the testimony. To compensate for the increasing age the increase in the sanctions has to take on more vaues than the perjury rue does here the sanction is either 0 or s. To put it differenty: the binary perjury rue acks the sophistication to give an interested itness proper incentives. et us concude this section ith a discussion of the informationa requirements of our sanction system S * and the perjury rue S p. Our mechanism focuses essentiay on the itness s age schedue and uses this information to derive the incentive compatibe sanctions. The perjury rue in contrast focuses on the stochastic processes generating the signas and the fina outcomes to determine the probabiity that the itness did not te the truth. Accordingy both mechanisms use different pieces of information. 6. Concusions Economists have devoted much effort to deveoping truth-reveaing mechanisms but these deveopments have not been appied to courts. 30 We have shon for a simpe frameork that existing ega practices create incentives for itnesses to give santed or fase testimony. More importanty e have deveoped a mechanism that prevents santed or fase testimony. A fe quaifications are in order. First e have ooked at the incentive probem of preventing santed and fase testimony. There are of course other incentive probems paying an important roe such as e.g. ithhoding unfavorabe information. Second e did not orry about the itness s effort to gather the information. The more effort a itness provides the more precise her signa say. If effort ere observabe the court coud use this information to infer the quaity of the testimony. Third the process generating the evidence confirming or disconfirming the testimony is exogenous in our 29 See e.g. Kreps 990 pp for a discussion of the pro and cons of the concept of eak dominance. 30 Exceptions are Bernardo Taey and Wech 999 and Sanchirico

22 set-up. Endogenizing this process as e the other to quaifications seem interesting tasks for future research. Many obstaces impede institutionaizing our mechanism but a move to strict iabiity has the promise of significanty improving the quaity of testimony in court. Moreover it shoud economize significanty on transaction costs. 2

23 22 Appendix Proof of Proposition : The resut foos immediatey from - 6. Q.E.D. Proof of Proposition 2: We use the first inequaities a-6a to determine the smaest incentive compatibe sanctions. Proposition impies that e can set S * A = 0. 3a then impies S * Ba = a-/. 2a then defines S A = -/- hie 6a defines S 2 A = + b a a h b. Here e have S A S 2 A ] [ ] [ b a b h b hich hods because 2 / >. Hence S * A = S A. Given this a then defines S Ba = a h a + b a 5a impies S 2 Ba = b h a and 4a defines S 3 Ba = H h b h a + H H b h b. Whie it is straightforard to see that S Ba S 2 Ba proving the second inequaity is more tricky. Here e have S Ba S 3 Ba a H + H a H H + H H = a H + H + H a b hich hods given H > and our assumptions on. Consequenty SBa= a h g + b a.

24 It remains to be shon that b --- 6b aso hod. b 2b and 3b are obvious. Substracting 3b from 2b yieds -a -HSA--SBa -H SA-HSBa impying 6b. Adding b to 3b generates a- -H SBa+H-SBa > -HSBa-HSBa meaning 5b is satisfied. ast but not east subtracting 2b from 5b yieds a- -HSBa--HSA -HSBa- HSA hich is 4b. Proof of Proposition 3: Q.E.D. If Y=B --SA = 0. If Y=BH --HSA = - H + H 0 since H>. If Y = A a--sba = a + 0 as /2. Finay if Y=AH a-- HSBa = a H +a H H + H 0 because H> /2. Q.E.D. Proof of Proposition 4: If the true state is e.g. B reporting b must be better than b meaning S A S A. If the true state is B H must be better than b meaning H S A H S A. Rearranging gives us H[ S A S A ] > > [ S A S A ] hich cannot hod since [ S A S A ] has to be positive and H <. The same argument appies to the signa A. Q.E.D. 23

25 Proof of Proposition 5: If the itness has observed the high precision signa BH the expected sanction equas -HS p A if she is honest and candid and -HS p A if she is honest but not candid. Since φa>φa e have S p A S p A. The same argument appies to the signa AH. Q.E.D. Proof of Proposition 6: Proposition 4 impies that the itness either reports or a. If min[ φ A φ B a ] φ the sanction is s for both messages henever the testimony turns out to be rong. Suppose the itness has observed BH. If she reports honesty but not candidy the expected sanction is -Hs < Hs hich is the expected sanction hen she reports a. The same reasoning appies to the other three signas. If max[ φ A ; φ B a ] <φ the expected sanction is zero for both a and. Accordingy the itness i provide an honest testimony. Q.E.D. Proof of Proposition 7: 3a impies that SBa and SA cannot both be zero for an interested itness. This observation together ith Proposition means that for an interested itness incentive compatibe sanctions have to take on at east three different vaues. The perjury rue S p takes on at most to vaues 0 and s. Q.E.D. 24

26 References BERNARDO A. E. TAEY AND I. WECH: A Theory of ega Presumptions University of Southern Caifornia a Schoo Working Paper No COOTER R. AND W. EMONS: Truth-Bonding and Other Truth-Reveaing Mechanisms for Courts University of Bern Dept. of Econ. and U.C. Berkeey Schoo of a Working Paper 2000 forthcoming. DEWATRIPONT M. AND J. TIROE: Advocates Journa of Poitica Economy EMONS W. AND J. SOBE: On the Effectiveness of iabiity Rues hen Agents are not Identica Revie of Economic Studies EMONS W.: The Provision of Environmenta Protection Measures under Incompete Information: An Introduction to the Theory of Mechanism Design Internationa Revie of a and Economics GROSSMAN S. AND O. HART: An Anaysis of the Principa-Agent Probem Econometrica HOMSTRÖM B. AND R. B. MYERSON Efficient and Durabe Decision Rues ith Incompete Information Econometrica KREPS D.: A Course in Microeconomic Theory Harvester Wheatsheaf 990. MANDE M. J.: Going for the God: Economists as Expert Witnesses Journa of Economic Perspectives MYERSON R. B Bayesian Equiibrium and Incentive-Compatibiity: An Introduction in Socia Goas and Socia Organization Essays in Memory of Eisha A. Pazner. Huricz D. Schmeider and H. Sonnenschein Eds. pp Cambridge: Cambridge University Press 985. PETZINGER T. Oi & Honor : The Texaco-Pennzoi Wars Beard Books 999. POSNER R. A.: The a and Economics of the Economic Expert Witness Journa of Economic Perspectives SANCHIRICO C. W.: Enforcement by Hearing: An Integrated Mode of Evidence Production mimeo University of Virginia Schoo of a SHIN H. S.: Adversaria and Inquisitoria Procedures in Arbitration Rand Journa of Economics THORNTON R. AND J. WARD: The Economist in Tort itigation Journa of Economic Perspectives USMAN M.: The Endogenous Determination of Verifiabiity mimeo Koç University Coege of Administrative Sciences and Economics Istanbu

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