3.1.1. Bayes’ rule

On seeing a given piece of evidence, the fact finder modifies her belief regarding guilt – just as one might change one’s beliefs regarding the speed of a horse on seeing it win a race. Beliefs, before and after evidence is observed, are represented as probabilities and are accordingly assumed to be subject to the mathematical properties thereof. One of those properties, Bayes’ rule, provides a formula for updating beliefs upon receiving new evidence.

3.1.2. Iterated application of Bayes’ rule

Conditional probabilities are themselves probabilities and Bayes’ rule may be applied to them as well, as though they were prior probabilities. The iterated application of Bayes’ rule will be useful in interpreting “trial selection bias,” as discussed in Section 3.2.2 below.

Thus, after seeing evidence E, the posterior odds of guilt – where “posterior” is defined relative to E – are, as already discussed, P(G|E)/P(I|E). If additional evidence F is then observed, the E-posterior odds of guilt may be further updated by treating P(G|E)/P(I|E) as the prior odds relative to F. This results in the formula:⁸ p. 212P(G|E∩F)P(I|E∩F)=P(F|G∩E)P(F|I∩E)×P(G|E)P(I|E).

The left-hand side of this equation represents the odds of guilt given that evidence E and F have both been observed. On the far right-hand side appear the “medial” odds of guilt, the odds of guilt given that evidence E has been observed; there are as yet no observations either way regarding event F. The other ratio on the right is the relevant likelihood ratio for the second round of updating, P(F|G∩E)/P(F|I∩E).

It is important to notice that the likelihood ratio for second-round updating takes into account that evidence E has already been observed. In general, P(F|G)/P(F|I), which does not account for E’s prior occurrence, is not the same as P(F|G∩E)/P(F|I∩E). For instance, a witness’ sighting of the defendant on the block where the murder took place (F) means something different if the defendant establishes that his elderly mother happens to reside on that block (E).

The updating process may then be repeated again for additional evidence G using P(G|E∩F)/P(I|E∩F) as prior odds, and so on.

3.1.3. The loss function

Having updated her beliefs on all the evidence, the fact finder’s second task is to decide on a verdict. Here she considers not just her posterior assessment of guilt, but also her view of the relative harm of wrongful conviction versus wrongful acquittal. Let W_C be the cost of wrongful conviction and let W_A be the cost of wrongful acquittal. Letting E now represent the conjunction of all evidence, the fact finder chooses to convict if P(I|E)WC<P(G|E)WA, that is, if the expected cost of wrongful conviction exceeds the expected cost of wrongful acquittal. (Here we are taking E to be the totality of the evidence presented in the case.) If, for instance, the cost of wrongful conviction exceeds the cost of p. 213wrongful acquittal, then conviction will not follow even if the fact finder assesses the probability of guilt to be greater than 50%. The elevated reasonable doubt standard of proof in criminal cases is sometimes justified in this way.

3.2.1. Base rate neglect

A large literature within evidence scholarship combines the prescriptive aspect of Bayes’ rule with experimental evidence⁹ to argue that fact finders are prone to misinterpret evidence. In particular, individuals are said to “neglect base rates” in drawing inferences from evidence. For instance, they know or are told that if the defendant were guilty, the evidence they are being shown would arise with 90% probability, and they incorrectly deduce from this that, having seen such evidence, they should regard the probability that the defendant is guilty as 90%. In making this improper deduction the fact finder, it is sometimes said, neglects the fact that the percentage of guilty individuals in the overall population – the “base rate of guilt” – is very low. Other times it is said that the fact finder is neglecting the base rate of the evidence itself. Still other times it is simply said that the finder is “neglecting base rates.”

Bayes’ rule helps uncover the precise nature of the mistake described in the example in the immediately preceding paragraph. It is also useful in describing how precisely the mistake deserves the name “base rate neglect” – that is, how the mistake is logically related to the base rate for guilt and/or evidence.

The fact finder’s mistake corresponds to confusing P(G|E) with P(E|G). As discussed in Section 3.1.1.1 above, the fact that P(E|G) is 90% does not imply that P(G|E) is also 90%. As is evident from the version of Bayes’ rule presented in equation (9.2), the reason that the probability of guilt conditional on the evidence P(G|E) does not necessary equal the probability of the evidence conditional on guilt P(E|G) is that the former depends on two other things besides P(E|G). It depends on the base rate of guilt, P(G) (as manifest in the prior odds of guilt). And it depends on P(E|I), the probability that the evidence would arise were the defendant innocent.

p. 214It is thus apparent that the mistake is not solely a matter of neglecting the base rate of guilt. It is also not solely a matter of neglecting the base rate of the evidence. The unconditional probability of the evidence may be written as a weighted average of the two conditional probabilities:¹⁰

If the fact finder, given P(E|G), also took into account P(E), but nothing else besides these two quantities, she would still not be able to determine P(G|E). In terms of the analysis in the previous paragraph, some algebraic manipulation shows that the fact finder would not be able to determine P(E|I) and P(G) from this limited information.¹¹ (However, it is easier to see that P(G|E) could not be so determined by examining equation (9.4) below.)

Yet, the simultaneous neglect of both base rates – that of guilt and that of the evidence – does adequately describe the fact finder’s mistake. This can be seen in two ways. First, from equation (9.3), it is clear that combining P(E|G) with both P(G) and P(E) makes it possible to recover P(E|I). This latter quantity, along with P(G) and P(E|G) can then be taken to Bayes’ rule to determine P(G|E).

Second, and more directly, we may use the definition of conditional probability to write

More precisely, as (9.4) makes clear, to reverse conditioning and conditioned events it is not necessary to know the separate values of the two base rates. It suffices to know their ratio. Thus, a more informative (if not more appealing) label for the fact finder’s mistake might be “ratio of base rates neglect.”

3.2.2. p. 215Trial selection bias and past crimes evidence

Another application of Bayes’ rule concerns the propriety of admitting evidence of the defendant’s past crimes. Lempert, Gross, and Liebman (2000)¹² argue against admitting past crimes evidence on the following basis: although past crimes evidence may be probative of guilt among the general population, it will not be as probative of guilt (and may even be probative of innocence) among the set of defendants whose cases actually make it to trial. Lempert, Gross, and Liebman (2000) offer several reasons why this phenomenon might arise. Most famously, they suggest that it is rooted in the tendency of the police to “round up the usual suspects.”

The trial selection bias argument can be formalized using an iterated application of Bayes’ rule, as described above.¹³ As we shall see, the exercise of formalizing the argument brings to the fore some potential weaknesses.

Fix an individual and an alleged crime. Let R be the event that the individual has a criminal record and let T be the event that he stands trial for the crime. The trial bias argument concerns the manner in which the individual’s past record affects the odds of guilt given that the individual now stands trial for the crime. We are thus interested in a statement of Bayes’ rule that shows how the existence of a past record converts the odds of guilt given just trial into the odds of guilt given both trial and a past record. This may be written as:

Thus, accounting for trial, the likelihood ratio of a past record of guilt is LTR=P(R|G∩​T)/P(R|I∩T). This likelihood ratio appears similar to the likelihood ratio of a past record, not accounting for trial, LR=P(R|G)/P(R|I). The difference however, is that this new likelihood ratio represents the information value of a past record for determining guilt among defendants who appear at trial, rather than among all potential defendants. As noted in the example presented in Section 3.1.2 above (involving the defendant and his elderly mother), these values may differ substantially.

The claim of the trial selection argument is that a past record, though generally informative of guilt, is not as informative of guilt – and may be informative of innocence – when such evidence is presented in the context of trial. Thus, the trial bias hypothesis is that while L^R may well be larger p. 216than one, LTR is less than LR, and possibly even less than one. What does this require? It is easy to confirm that the two likelihood ratios – describing the information value of a past record with and without trial – are related as follows:

The trial selection bias argument is thus equivalent to the assertion that the double ratio in brackets in the immediately preceding equation is less than one (and possibly even less than 1/L^R). This double ratio is less than one if and only if its denominator exceeds its numerator:

This inequality is perhaps more easily interpreted if rewritten in more compact notation:

Inequality (9.6) is a math-symbolic statement of the key condition for the existence of the kind of trial bias that has been proposed: having a past record must increase the individual’s chance of standing trial for the crime by a greater proportion if the individual is innocent, than if he is guilty.

Given that (9.6) is its key condition, the trial selection bias argument against past crimes evidence is arguably more fragile and problematic than it may at first appear. Condition (9.6) concerns a difference in differences. The issue is not whether having a past p. 217record increases the individual’s chance of standing trial. The issue is whether having a past record increases the individual’s chance of standing trial more if the individual is innocent, than if he is guilty. That a past record makes a difference for appearance at trial seems reasonably clear. How this difference differs across innocent and guilty defendants is far from certain and would seem to vary greatly across circumstances.

At the very least, the explanations that have thus far been provided for trial bias seem inadequate as justifications for the real condition, (9.6). In particular, trial bias cannot be solely a matter of the tendency of the police to round up the usual suspects. Such a tendency implies that having a past record increases the individual’s chance of standing trial, period – both when the individual is innocent and when she is guilty. The usual suspects dynamic, by itself, is essentially silent on whether the increased chance of standing trial due to having a record is greater for the innocent than for the guilty.¹⁴

3.2.3. Across-person hindsight bias and its rational twin

Several authors have investigated the possibility that fact finders are subject to across-person hindsight bias: namely, “in hindsight, people consistently exaggerate what could have been anticipated in foresight . . . People believe that others should have been able to anticipate events much better than was actually the case.”¹⁵ Experimental evidence has been offered to establish that across-person hindsight bias exists.

Across-person hindsight bias has important legal ramifications when a litigant’s knowledge at the time of her alleged action or omission is at issue in the case. The bias may, for example, cause fact finders to misjudge whether due care was exercised under a negligence standard, whether warnings were adequate in product liability, or whether an accident was reasonably foreseeable under strict liability.

Yet, is it really a mistake to use knowledge that an event E has occurred in judging whether others should have or did in fact know that it was likely to occur ex ante? Bayes’ rule can be used to show that is not incorrect in the case that one believes that (1) these others may have been in a position to know about the chance of events like E; and (2) the fact finder does not have access to full information regarding what these others knew.¹⁶

Thus, assume that it is common knowledge that the defendant took a particular action. Let A be the event that an accident occurred. Let I be the ex ante information observed only by the defendant prior to her decision to take the action regarding whether an accident would occur as a result.

From the perspective of the fact finder, the value of the information I received by the defendant is an uncertain variable. Thus, the defendant’s various conditional probability assessments that are based on that information are, from the fact finder’s perspective, random variables.

The fact finder is interested in whether, having seen the information I, whatever its value may have been, the defendant “knew” that the accident p. 218would occur. More precisely, the fact finder must determine whether the defendant’s assessment of the conditional probability of an accident exceeded some legally determined state of mind threshold k:

The subscript “D” indicates that the preceding probability represents the defendant’s subjective beliefs. The value of the number PD(A|I) depends on I, which is uncertain from the fact finder’s point of view. Thus, from the fact finder’s perspective, PD(A|I) is simply a random variable and the question for the fact finder is whether this random variable exceeds k.

The above threshold condition may be restated in terms of the defendant’s I-posterior assessment of odds of an accident:

This odds ratio is, from the fact finder’s viewpoint, also a random variable.

The fact finder directly observes whether A or ¬A (“not A”) occurs. The fact finder then uses this information to update her beliefs regarding whether the random variable PD(A|I)/PD(¬A|I) exceeds threshold o.

The question whether outcome information rationally changes the fact finder’s assessment of defendant’s knowledge may be formalized as follows: is it true that

The subscript “F” indicates that the probability represents the fact finder’s subjective beliefs.

In words, the issue is whether the rational fact finder’s assessment of the probability that the defendant “knew an accident would occur” conditional on the defendant’s having observed I, whatever I might have been, is greater when the fact finder observes that an accident did occur than when the fact finder observes that an accident did not occur.

p. 219Substituting from the odds formulation of Bayes’ rule (shown above), (9.7) becomes

where o′≡o(PD(¬A))/(PD(A)). Therefore, the question is whether the defendant’s likelihood ratio for the information I – which information is probabilistic from the fact finder’s point of view – should be considered more likely to have been above the (transformed) threshold o′ when it is observed that an accident has occurred than when it is observed that an accident has not occurred.

Now, consider the special case in which I takes two values I¯ and I_<I¯.¹⁸ To make the problem non-trivial, assume that the defendant’s observation of I¯ raises the defendant’s assessed odds of an accident above the threshold, while the defendant’s observation of I_ does not. That is, in terms of the transformed threshold o′, assume

Therefore, from the fact finder’s perspective, the event that the defendant’s random odds ratio PD(I|A)/PD(I|¬A) was greater than or equal to o′ is equivalent to the event that the defendant saw I¯. Thus, condition (9.8) reduces to the condition

Condition (9.10) is the condition that the fact finder’s likelihood ratio for the defendant’s observation of I¯ exceeds 1. This is the condition that, were the fact finder to have learned – prior to learning of the occurrence of an accident – that the defendant had observed I¯, the fact finder would have raised her assessment of the odds that an accident would occur.

Condition (9.10) is satisfied if the fact finder has the same view as the defendant regarding what it means for the chance of an accident when the defendant sees I¯. More formally, if the fact finder’s likelihood ratios for the two values of the signal are ordered in the same manner as the defendant’s – that is, if

p. 220then condition (9.10) follows from condition (9.9):

Therefore, if the defendant received, prior to the accident, private information that determined her legal state of mind, and if the fact finder and defendant generally agree on the meaning of such information vis-à-vis the chance of accident, then, having observed that an accident did in fact occur, the fact finder should rationally increase her probability assessment of the event that the defendant had a culpable state of mind. In other words, when the fact finder cannot know all that the defendant knew, the fact finder’s hindsight that an accident did occur is rationally informative of the defendant’s foresight that an accident would occur.

Thus, something observationally similar to across-person hindsight bias is actually a rational response to outcome information. This does not, of course, imply that across-person hindsight bias is nonexistent. Nevertheless, many experimental designs that attempt to measure the unwarranted outcome adjustment of across-person hindsight bias do not appear to control adequately for the presence of the correlate rational adjustment. The result obtained from such experiments may thus overstate the magnitude of the irrational adjustment.

What if the fact finder also observes what the defendant observed, which is to say I? In this case, the occurrence of an accident is irrelevant for determining the defendant’s state of mind.

To see this, return to the more general case in which the information may take many values. (The following analysis will also serve to explicate this more general case.) The question is again whether the following inequality holds for any given value of I:

p. 221Let ℑ be the subset of information values I such that PD(I|A)/PD(I|¬A)>o′ Then the inequality above reduces to PF(ℑ|A∩I)>PF(ℑ|¬A∩I). Using the definition of conditional probability, this may be restated as

This condition is impossible. The left and right sides of this strict inequality are always equal. If I∉ℑ, then the numerators on both sides are zero, and so also the ratios. If I∈ℑ, then the appearance of ℑ in the numerators is redundant, and both ratios are equal to one.

In words, if the fact finder knows what the defendant knew at the time of the defendant’s decision, then the finder knows the defendant’s state of mind at the time of the defendant’s decision, and information regarding whether an accident later occurred is superfluous. Therefore, the rationality of using outcome information to determine state of mind relies on the (plausible) premise that the fact finder does not directly observe the information that was available to the defendant at the time of the defendant’s decision.

3.3.1. The conjunction problem

The first is the “conjunction problem,” itself the result of a conjunction of legal features. First, guilt or liability often turns on two or more findings of fact – as when a verdict requires findings regarding how the defendant acted, what harm she thereby caused, and whether she intended the consequences of her actions. Second, the law p. 222requires that each element of the charge, claim or defense be found to have obtained with a threshold probability, a “standard of proof” (commonly thought to be 50% in civil actions and something more than this, perhaps 90%, in criminal actions). The requirement that each element must individually be subject to a 50% threshold, say, implies a weaker requirement, possibly much weaker, for the probability of the conjunction of elements, which is to say the charge, claim or defense itself. In the extreme case in which the elements are statistically independent, the fact that each is more likely than not implies only that their conjunction is more likely than 0.5ⁿ < 0.5, where n is the number of components. Particularly troubling is the fact that the implied threshold probability for a charge, claim, or defense decreases (quite rapidly) in the number of elements it contains, a factor with uncertain relevance. With four independent elements, it is not 50%, but 6.25%.²²

3.3.2. The gatecrasher paradox

The second objection is implicated by the “gatecrasher paradox.” Suppose that the defendant was in attendance with 1000 others at an event at which it is known that 499 attendees purchased tickets and 501 crashed the gate. Assuming no other evidence is available, could (should) a fact finder conclude that the defendant more likely than not failed to pay for his seat, and could (should) the law hold the individual liable for the ticket price? While probabilistic analysis appears to imply that the defendant should be held liable, some see this as strongly contrary to intuition.²³

3.3.3. Accounting for the interests of the parties

Allen (1986) is drawn from a watershed symposium held at Boston University on the benefits and drawbacks of applying probabilistic deduction to evidence law. (Symposium (1986).) Many of the other papers in the corresponding issue of the Boston University Law Review are also well worth examining. Taken as a whole, the symposium papers provide a comprehensive picture of evidence scholarship at the time regarding the utility of applying formal theories of probability, conventional and unconventional, to the problem of legal evidence.

Yet the symposium (as well as contemporaneous evidence scholarship) largely ignores an arguably more serious drawback of the pure p. 223probabilistic approach.²⁴ The approach gives short shrift to a fundamental and distinctive fact about legal evidence. Unlike experimental evidence generated in a chemist’s laboratory or field evidence gathered in a macroeconomist’s survey of inventories, legal evidence is provided by conscious, animate individuals with strong interests in what the fact finder decides and a strong possibility of influencing that decision.²⁵

It is one thing to prescribe, as does the pure probabilistic approach, that the fact finder interpret evidence according to the relative likelihood of its production under alternative truths. It is another thing to explain how these relative likelihoods ought to account for the interests of the parties responsible for the production. And it is yet another thing to ask how fact finding should be structured to account for the necessity of this strategic accounting. If the defendant in a criminal case produces for us her oath-bound testimony that she is innocent, how do we evaluate the relative likelihood of observing such a performance if she really were innocent versus if she were in fact guilty? Wouldn’t she claim innocence either way? If so, what constitutes convincing evidence of innocence?

4.1.1. Single party model

Milgrom and Roberts show how the fact finder may costlessly determine the true state of affairs from a nonetheless interested party who is known to be informed regarding that state of affairs under the assumption that the party may not and will not provide information that is inconsistent with her knowledge. In this case, the fact finder need only announce ahead of time that if the informed party supplies her with ambiguous information, she will assume the worst outcome for the party that is consistent with the information supplied.²⁸ It follows that the worst deduction for the party that is consistent with the information that she provides will in fact be the truth. This is because the party would always have an incentive to clear up any ambiguity as between the truth and any less favorable deduction.

p. 225A simple numerical example will help to clarify the result and its logic. Suppose that the true state of the world is one of ten possibilities, each indexed with a number between 1 and 10. The fact finder (she) does not know the true state. The party (he) does know the true state, and the fact finder knows that the party knows. The fact finder would like to learn the true number (i.e., state of the world). The party would like the fact finder to think the number is as high as possible, and the fact finder knows this as well.

The party chooses what to report about the true number to the fact finder. (Other, “more evidentiary” interpretations of this game are presented in Section 4.2 below.) The fact finder makes a deduction about the true number based on this report.

The rules of the game are that the party may not lie. He may, however, omit to say all that he knows. That is, he may decline to report the exact number, providing instead a subset of numbers in which the true number lies. He might, for example, reveal that the number is odd (if in fact it is). Or that the number lies between 3 and 7 (if in fact it does).

Despite the party’s ability to omit information, the fact finder can be assured of learning the true number by announcing that upon hearing the party report a subset of numbers, she will assume that the true number is the lowest in that subset. The party would in that case never find it in his interest to report a subset containing a number lower than the truth. He can always do better – that is, inspire the fact finder to deduce a greater number – by removing any number from his report that is lower than the truth. Therefore, the lowest number in the party’s report, the number in fact chosen by the fact finder, will indeed be the truth. If, for example, the true state is 4, the party will never report the subset {2, 4, 6, 8, 10}. Doing so would cause the fact finder to deduce that the truth is 2. The party could do better by eliminating 2 from this report, whence the truth finder will deduce 4, which is the truth.

It is worth noting that the assume-the-worst rule used by the fact finder in Milgrom and Roberts is akin to an adverse inference from “spoliation,”²⁹ an ancient component of evidence doctrine dealing with a party’s refusal or purposeful inability to supply evidence alleged to be under her control. In terms of the numerical example in the immediately preceding paragraph, an additional document in the party’s possession p. 226might, for example, allow the fact finder to distinguish between state 2 and the subset of states {4, 6, 8, 10}. If the party refuses to provide the document – that is, if the party in effect reports {2, 4, 6, 8, 10} – the fact finder, under this rule, assumes that the truth is 2, the worst case for the party among the relevant possibilities.

Analysis of this kind of assume-the-worst rule also crops up in the legal scholarship on evidence preceding Milgrom and Roberts (1986). Indeed, the idea appears in attempts to resolve the gatecrasher paradox, as discussed in Section 3.3.2 above. Kaye (1979), for example, suggests that the plaintiff’s failure to present any information beyond the naked statistical evidence of ticket sales and attendance may be taken to indicate that the rest of the evidence that is likely available to him would hurt his case.

4.1.2. Multiple Informed Parties with Conflicting Interests

Milgrom and Roberts (1986) also put forward a related dual informed party model in which the parties’ interests are in conflict. In this model, the fact finder again learns the truth. Indeed, this occurs no matter what rule the fact finder uses to choose among the set of states that are consistent with both parties’ reports: the parties’ conflict of interest substitutes for the fact finder’s sophistication. If the parties have strictly opposing interests and each knows the truth, the fact finder will always end up learning the truth so long as she restricts her decision to the intersection of the parties’ reports. (Again, the parties may omit information, but may not lie.) This is because whatever the fact finder’s decision within that intersection, if it is not the truth, one party will prefer the truth and so will have an incentive to (and the ability to) refine her report.

Return to the subset reporting example presented above in which there were ten possible states. Imagine now that two parties know the truth and each reports a subset containing it. Suppose that one party wants the fact finder to think the number is as high as possible and the other as low as possible. Thus, the parties’ interests are in conflict. Suppose that the fact finder (unthinkingly) takes as the true state the average (rounded up or down) of the numbers in the intersection of the parties’ reports. Then, if the average of the points in the intersection of reports is not the truth, one party or the other will prefer the truth to this average and thus have an incentive to report the truth as a singleton. If, for example, the truth is 4, one party says “3 or greater” and the other “7 or less,” then the intersection of reports is 3 through 7, the average is 5 and the party that prefers lower numbers to high has an incentive to refine her report to “4.”

The logic of this result resonates with earlier less formal discussions within case law and evidence scholarship regarding the relative benefits of p. 227adversarial procedure.³⁰ According to the US Supreme Court, “the very premise of [the] adversarial system . . . is that partisan advocacy on both sides of a case will best promote the ultimate objective that the guilty be convicted and the innocent go free.”³¹

4.2.1. Subset reporting

The first interpretation is “subset reporting,” which was described above and is the main interpretation offered by Milgrom and Roberts (1986). The logic of truth revelation is clearest under this interpretation. Yet, as a description of actual evidence production, it might be regarded as too abstract.

4.2.2. Truth-consistent presentation

The “truth-consistent presentation” interpretation is somewhat more representational of actual legal evidence. Under this interpretation, rather than directly reporting to the fact finder a particular subset of states, the informed party makes an evidentiary presentation that is commonly known to correspond to a particular subset of states.

Thus, imagine that each state s is associated with a subset of evidentiary presentations or “messages,” E(s)⊂E. The subset E(s) represents the evidentiary presentations that are consistent with state s in some “natural” sense (which may not be fully specified). The association may be semantic. For instance, the state s may in part specify that the party paid a third person to supply him with a particular product. Several of the presentations in E(s) may involve providing the fact finder with a signed contract whose language is consistent with that fact.

A given presentation e∈E may be associated with more than one state: that is, there may exist two states s and s′ such that s≠s′ and e∈E(s)∩E(s′). Thus, a signed contract may also be consistent with the state in which the party failed to pay for the product. Indeed, the null presentation (i.e., the choice to make no presentation), which is presumably an available option for the party, is consistent with all states.

Two assumptions are imposed upon this structure. First, the correspondence E(.) is common knowledge between the party and fact finder. p. 228Second, given true state s, the party may make any presentation in E(s), but no presentation outside of E(s). Thus, the party’s presentation must be consistent with the true state, though it need not fully indicate the true state.

To understand the linkage between this structure and subset reporting, consider the following simple example: eight evidentiary presentations might be made, A, B, C, D, F, G, H, I, in each of three possible states, 1, 2, 3. Table 9.1, read row-by-row, shows for each state (row) the presentations (columns) that are naturally associated with that state. For example, presentations D, G, H and I are associated with state 3. The set E(s) corresponds to the set of shaded boxes in the row corresponding to state s. Thus E(3)={D,G,H,I}.

The key to linking this framework to the subset reporting model described above is to notice that the table can also be read column-by-column. Thus, given any evidentiary performance, the table tells us the subset of states with which such performance is naturally associated. For example, reading down column G, we see that performance G is associated with states 2 and 3. It follows that when the party makes a particular presentation to the fact finder, he is in effect truthfully reporting to the fact finder a subset of states, as in the subset reporting interpretation. When the party makes presentation G, for instance, he is in effect truthfully reporting to the fact finder that the true state lies in the subset {2, 3}. In general notation, on presenting evidence e, the fact finder is in effect reporting the subset {s|e∈E(s)} of states.

The truth revelation result discussed above may be restated in terms of this new interpretation. Indeed, this interpretation serves to highlight both an implicit requirement and a generalization of such result.

The additional requirement is that the correspondence E(s) be sufficiently “rich.” To take an extreme example, if every cell in Table 9.1 were shaded – signifying that every available evidentiary presentation was consistent with every true state – truth revelation would be impossible. (In subset reporting terms, this would correspond to the party’s somehow p. 229being unable to convey in language that the true state was in any subset other than the full set itself.)

Notice that quite contrary to this possibility, Table 9.1 has the following “extreme richness” property: every subset of states has at least one corresponding presentation in the sense that such subset precisely equals the subset of states consistent with such presentation. That is, for all subsets of states S there exists a presentation e such that {s|e∈E(s)}=S. For the singleton subset consisting solely of state 1, for instance, there are, in fact, two such presentations: A and B. For the subset of states {1, 2}, there is one, C. For the subset of states {1, 2, 3} there is also one, D, and so on. Thus, just as in the subset reporting interpretation, every subset of states may, in effect, be reported.

Extreme richness is sufficient (but not necessary) for truth revelation. Recall that in the subset reporting interpretation, the fact finder announces that she will assume that the truth is the party’s least favorite state among the states in the subset that the party reports to her. In the truth-consistent presentation interpretation, the fact finder announces that upon seeing a presentation A, B, C, . . ., or I, she will deduce that the true state is the party’s least favorite state among those states that are consistent with the presentation. Suppose, for example, that the truth is 2 and that the party prefers higher states to lower. Given that the true state is 2, the party is able to present C, D, F, or G. If the party makes presentation C or F, the fact finder, using an assume-the-worst rule, will deduce that the true state is 2. If the fact finder makes presentation D or G, the fact finder will deduce that the true state is 3, which is worse than 2 for the party. Therefore, the party will present C or F and the fact finder will decide that the state is 2, which is the truth.

Conversely, extreme richness is not necessary for truth revelation – this is the generalization referred to above. The truth revelation result does not require that every subset of states be effectively reportable via some presentation. All that is required is that for every state there be at least one presentation which allows the party to rule out in the fact finder’s mind those states that the party regards as worse than such state. Assuming the party prefers higher states to lower states, the requirement is that every state be the lowest state that is consistent with some presentation. Consider, for example, Table 9.2, which is a pared down version of Table 9.1. Notice that each state is the lowest that is consistent with some presentation. For example, state 2 is the lowest consistent with G. An assume-the-worst rule would also produce truth revelation under Table 9.2. Were state 3 the true state, for example, the party, who prefers higher states to lower states, would be able to report G, H, or I. But the party would never present G or H because, given the fact finder’s assume-the-worst rule, p. 230the party could do better presenting I, in which case the fact finder’s rule would lead it to the truth.

4.2.3. Feasible presentation

The third interpretation, “feasible presentation,” is similar in structure to truth-consistent presentation. The difference lies in the interpretation of the correspondence E(.). Under the feasible presentation approach, E(s) represents the presentations that are feasible for the party in any given state, rather than those that are naturally or semantically consistent with the true state. Interpreting Table 9.1 in this way, when the true state is state 3, presentations D, G, H, and I are the only presentations that the party is able to make. All other presentations are impossible.

The feasible presentations interpretation subsumes the truth-consistent presentation interpretation. Return for a moment to viewing Table 9.1 as a map of presentations that are naturally/semantically consistent with each state, as opposed to feasible under such state. If we layer on top of this the restriction that truthful presentations, and only truthful presentations are possible, Table 9.1 becomes also a representation of the set of feasible messages.

Conversely, the feasible presentations interpretation is conceptually broader than the truth-consistent presentations interpretation. The feasible presentations interpretation does not specify what the “technology” is that rules out certain presentations in certain states. An exogenous legal prohibition on untruthful presentation may well be doing some or all of the work. Physical impossibility or prohibitive cost may also be important factors.

4.2.4. Infinite or zero cost

A fourth interpretation, which is actually a gloss on the first three, merely replaces “feasible/permitted” with “of zero/negligible cost” and “infeasible/not permitted” with “infinitely/prohibitively costly.” In the subset reporting interpretation, with this gloss, the cost of reporting any subset containing the true state is zero, while the p. 231cost of reporting any subset not containing the true state is infinite. In the truth-consistent presentation interpretation, all evidentiary performances in E(s) have zero cost when the true state is s, while all performances outside E(s) have infinite cost. The source of infinite cost is presumably sanctions for truth-inconsistent reporting. The adjustment in the feasible presentation interpretation is similar: “feasible” becomes “of zero cost,” “infeasible” becomes infinitely costly. The source of costs is left unspecified and may be legal, technological, economic, or some combination thereof.

This third “binary cost” interpretation will be useful in comparing omission models with the endogenous cost signaling models discussed in Section 5.1 below.

4.3.1. Strategic search models and the possibility of pro-plaintiff or pro-defendant bias in the litigation system

The strategic search model of evidence production is a variant of the omission model that in effect combines the logic of the omission model with the logic of contest models of litigation, as discussed in Section 2.2. In strategic search models of litigation, parties sample from an exogenous distribution for “pieces of evidence,” deciding both when to stop sampling and what of their accumulated sample to show the court. Each time a party draws a sample she incurs a cost. Parties may decline to report elements of their accumulated sample to the court, but, as in Milgrom and Roberts (1986), they may not fabricate observations.

p. 232Froeb and Kobayashi (1996) use a strategic search model to show in effect that Milgrom and Robert’s (1986) two-party result, described above, extends to the case in which evidence is costly to acquire and the decision maker may be biased. The fact that evidence is costly to acquire does not prevent a correct decision because, all else the same, the party in the right, being favored by the distribution of evidence from which parties draw, ends up presenting more favorable evidence to the fact finder. Moreover, the litigation system automatically compensates for any decision maker bias because the party favored by such bias reacts, in Froeb and Kobayashi’s model, by slacking off on evidentiary effort.

Daughety and Reinganum (2000) generalize Froeb and Kobayashi’s model and emphasize the dependence of Froeb and Kobayashi’s results on symmetries in sampling costs and sampling distributions. Daughety and Reinganum suggest that, all told, the adversarial system favors defendants.

Froeb and Kobayashi (2000), discussed in Section 4.3.2 immediately below, is another example of this type of model.

4.3.2. Adversarial versus inquisitorial procedure

Omission models have been employed to compare the relative merits of adversarial process (wherein litigating parties compete before a spectator/fact finder) and inquisitorial process (whereby the parties are restrained and the fact finder investigates and questions).³² Shin (1998) argues that inquisitorial process generates less information than adversarial process even when the inquisitor’s investigative ability is as good as that of each adversary (taken individually). In Shin’s model, as in Milgrom and Roberts (1986), parties can suppress evidence but cannot fabricate it. Shin shows, in essence, that the downside of adversarial process – the fact that evidence may be manipulated – can be significantly alleviated by the kind of assume-the-worst-of-omission deductions studied in Milgrom and Roberts (1986). With this downside mitigated, the upside of adversarial process – the fact that it offers multiple sources of evidence – becomes decisive in comparing systems.

Froeb and Kobayashi (2000) employ a strategic search model of evidentiary sampling similar to that in Froeb and Kobayashi (1996). Under adversarial process, each party decides first, how many times to draw from a given distribution of evidence and second, what of her sample to present in court. Once again, parties cannot fabricate any portion of their sample. The fact finder “averages” the evidence placed before her. This induces each party to present only the single piece of evidence in her sample that most favors her case. Under inquisitorial process, on the other hand, the inquisitor herself samples from the distribution and averages all the data. Which system does better? If the parties face the same sampling costs, both systems leave the fact finder with the same assessment on average, namely the true mean of the distribution. In other words, whether one takes the average of extreme draws in each case or the average of all draws in each case, one’s average assessment over all cases will equal the true mean of the sampled distribution: both estimating procedures are “unbiased.” However, the two estimators differ in their variance – a proxy for their degree of error.³³ Which system has less error is indeterminate. The p. 233outcome depends on the shape of the underlying distribution and the cost of sampling.

4.3.3. Link to primary activities

Bull and Watson (2004) adopt a variant of the feasible presentation interpretation discussed above. They link evidence production in a later phase of the model to action choice in an earlier phase: action choice determines the set of feasible evidentiary presentations.³⁴ In particular, Bull and Watson study how the prospect of transfers based on evidence production affects contracting and contractual performance. Their main result characterizes kinds of actions that are implementable under this endogenous evidentiary structure.

4.3.4. Partial provability in a multi-party context

Consider again the example from Section 4.1.2 in which each of two informed parties with conflicting interests reports to the uninformed fact finder something true, but perhaps not complete, about the actual state of the world, where the state of the world is indexed by the numbers between one and ten. In that example, the conflict of interest between the two parties implied that, whatever the fact finder’s rule for choosing among elements in the intersection of the parties’ reports, if the truth would differ from the fact finder’s determination, one of the parties would prefer to refine her report to precisely indicate truth. An implicit assumption in that analysis was that the party who preferred to refine her report was also able to do so.

What if the parties were not always able to precisely communicate the true state? What if, for example, both parties were able to report the true state precisely in all states except state 5, in which the only report that either side could make was the subset {2, 4, 5, 6}. How would this situation arise? Perhaps (to shift momentarily to the feasible or truth-consistent evidentiary presentation interpretation), the evidentiary presentations available to the parties in state 5 are simply “inconclusive:” everything the parties could show or do in front of the fact finder in state 5 might also be shown or done in states 2, 4 and 6.

Would the truth revelation result fall apart? This is the main question addressed by the sub-literature on partial or limited “provability.”

Section 4.2.2 discussed partial provability in the context of the single p. 234party model. This remainder of this section discusses partial provability in the context of the multi-party conflict of interest model.

4.4.1. The no-lying assumption in the subset-reporting and truth consistent presentation interpretations

Truth revelation results for omission models are precariously balanced on the assumption that agents cannot fabricate evidence. If, for instance, we remove the restriction on lying from the one-through-ten example used to explain Milgrom and Roberts’ (1986) single agent result in Section 4.1, the fact finder’s assume-the-worst rule merely induces the single agent to report her favorite outcome, “10,” regardless of the true state. Likewise, if we remove the no-lying restriction from the two agent conflict-of-interest example presented in Section 4.1.2, one agent will respond to the fact finder’s intersection rule by reporting “1,” and the other by reporting “10.” The fact finder’s decision rule will not lead to truth revelation. Indeed, the rule, which specifies a way of choosing from the intersection of the parties’ reports, will not even function: the agents’ reports will not intersect.

It is thus worth asking whether the no-lying assumption is justified, at least as an approximation of reality.

Many contributors to the omissions literature motivate the no-fabrication assumption by pointing out that lying in court is illegal by virtue of statutes criminalizing perjury, obstruction of justice and similar transgressions. This defense is problematic in several respects.

First, let us accept for purposes of argument the premise that behavior that has been rendered illegal is not an issue for the analysis of legal evidence. The problem for the omissions literature, then, is that much of the behavior that this literature would characterize as “omission,” rather than fabrication, is also illegal – by virtue of subpoena enforcement, compelled discovery, and statutes on obstruction of justice and contempt.⁴² Furthermore, instances of omission that are not now illegal might be made so. Thus, if illegality does (or can) rule out behavior, the omissions literature itself is left without a problem to solve. (Alternatively, it must explain why making omission illegal is not the best policy – see the fourth point below.)

Second, the implicit premise that illegal behavior is not an issue for evidence is troublesome. The illegality of fabrication does not rule it out, p. 240either in practice or in principle. As a matter of theory, the detection-probability-discounted penalty for lying may not outweigh the potential gains from doing so. As a matter of empirics, despite the fact that fabrication in court is often illegal, what limited data exist suggests that it is a regular occurrence.⁴³

Third, one is justified in asking why, if the no-lying assumption is legitimate in the context of judicial process, it is so rarely deployed by economists in other more frequently studied settings in which the government is also the principal. There is no corresponding no-fabrication assumption in the literature on optimal taxation, or in large swaths of the literature on optimal regulation. Perjury and related crimes, like lying to investigators and tax fraud, apply in these settings as well. If the existence of such sanctions justify the no–lying assumption in omission models, why does it not justify assuming that the optimal tax authority can observe wage rates as opposed to just labor earnings?

Fourth, and perhaps most importantly, by assuming that agents cannot lie, the omissions literature on evidence effectively assumes away one of its most fundamental challenges and, correspondingly, blocks off one of its chief sources of potential utility. Crucial and interesting questions central to the practical design of evidentiary procedure are shunted aside. How precisely do laws regarding perjury and obstruction function? Are such laws effective? Are they efficient? Are they effective and efficient in some settings and not in others? What are the alternatives to such laws for system design? More generally, what is the best way to structure litigation if we do not take as already solved the elemental problem that parties have an incentive to falsify their testimony and forge their tangible evidence?⁴⁴

4.4.2. Lingering problems in the feasible presentation interpretation

The assumption, in subset reporting, that a party may not lie, becomes, in truth-consistent presentation, the assumption that the party may not make a presentation that is inconsistent with the true state, where the p. 241correspondence E(.), as represented by Table 9.1, defines consistency. Presumably, in both of these interpretations, some external legal prohibition defines and enforces these reporting restrictions.

In feasible presentation, the no-lying assumption is transmuted into the bare assumption that the party’s choice of presentation is constrained by the table – for reasons that are not specified. Thus, assume that the true state is 1. In subset reporting, the party is prohibited from lying and reporting that the true state is in the subset {2, 3}. In truth-consistent presentation, the party is prohibited from making presentation G, which is consistent with 2 and 3, but not 1. In feasible presentation, G is simply designated as impossible. G may be impossible because it is semantically inconsistent with the truth and truth-inconsistent presentations are legally prohibited. Or it may be somehow technologically or economically impossible, without the aid of legal prohibition. That is, G is the kind of presentation that, for whatever reason, cannot be faked when the truth is neither 2 nor 3.

Arguably, eschewing the explicitly truth-semantic subset reporting and truth-consistent presentations interpretations in favor of the feasible presentation interpretation does little to solve the problem identified in the preceding subsection. Rather, the feasible presentations interpretation appears merely to relocate the problem. In lieu of wondering why the defendant/surgeon could not lie about her performance during the seven-hour operation, the reader is left to wonder why there should exist an evidentiary performance that would be possible (at zero cost) for the surgeon if she had mindfully followed best practices, and impossible if she had not. If the response to the reader’s inquiry relies on the assertion that the surgeon is legally prohibited from presenting evidence that semantically contradicts the truth, then we are back to the problem with the explicit no-lying assumption. If the response relies on the assertion that there exists a kind of evidentiary presentation that is physically/technologically/economically impossible when the surgeon has not followed best practices, the question then becomes: where can such evidence be found? The time, expense, and endemic, lingering ambiguity of actual litigation suggest that, even if this unicorn exists, it is rarely discovered and harnessed by litigants.

Put another way, in the feasible presentation interpretation, the richness condition takes on added importance and correspondingly requires greater scrutiny. By contrast, with truth-consistent presentation, it is not difficult to imagine that there exists a presentation that is truth consistent with each given subset of states. As noted, simply reporting the subset of states is, after all, a presentation. What is perhaps difficult to imagine in truth-consistent presentation is the absolute prohibition on lying. In the feasible interpretation, we are not specifically asked to imagine an absolute p. 242prohibition on lying. Rather, we are asked to imagine that for (not all, but) a key collection of subsets of true states, it is possible to find an evidentiary presentation that would be effectively impossible in any state outside that state. This may be likely true for some subsets of states. But is it true for enough subsets of states to make possible the level of discernment that the fact finder requires? Is the evidentiary space rich enough?

For example, in one possible presentation, the party may appear before the fact finder missing a leg. This presentation is effectively impossible in a number of possible states. For example, the fact finder may be effectively certain that the true state is one of those in which the party has no leg. Moreover, the fact finder may be effectively certain that the true state is one of those in which party’s leg was involuntarily severed. Yet, this presentation tells the fact finder little about how the leg was severed. Was it caused by an accident? If so, who was the injurer? Was the injurer negligent? Was the party herself contributorily negligent? How will missing a leg impact the party economically? Mentally and emotionally? The fact finder may be interested in these issues as well, and it seems unlikely that the evidentiary space is rich enough to contain unfakeable presentations for all, or even most of what will be at issue in the case.

4.4.3. The illusory solution of adopting the infinite or zero cost interpretation

Recall that a fourth interpretation replaces “feasible/permitted” with “of zero/negligible cost” and “infeasible/not permitted” with “infinitely/prohibitively costly.” Arguably, this cost-based interpretation adds little to the omission model besides, perhaps, rhetorical appeal. After all, one could argue that economics as a field is partly defined in opposition to the mistake of regarding as predetermined variables that are actually subject to choice by self-interested agents. Could it be that the binary cost interpretation absolves omissions models of this mistake with respect to fabrication and lying? If so, then presumably any model that exogenously prohibits an action might improve itself by specifying that actors, with finite budgets, may indeed choose the action, so long as they pay an infinite price.

Indeed, in some respects, the binary cost interpretation makes the omission model look less appealing. Recasting the model in terms of binary costs places in stark relief the questionable binary nature of permissibility/feasibility. Under the cost interpretation, all evidentiary performances inside E(s) are equally (un)costly. For instance, an evidentiary performance e that pinpoints the state – i.e., E-1(e)={s} – costs as much to present, namely zero, as an evidentiary performance e′ that communicates nothing about the state E-1(e′)={1,....,10}. Furthermore, all evidentiary performances outside E(s), regardless of whether they merely spin the truth or turn it fully on its head, are equally (prohibitively) costly. p. 243Accordingly, the cost difference between any performance in E(s) and any performance outside E(s′) is always the same: it is always infinite.

4.4.4. The partial solution of partial provability

Allowing for only partial provability, as do Lipman and Seppi (1995), does weaken the no-lying/infeasibility assumption and so restores some degree of plausibility to the omission model. But, relative to the substantial distance separating omissions models from the basic outlines of real litigation, the improvement seems negligible. Lipman and Seppi’s truth revelation result requires, inter alia, that it be impossible for the second mover to falsely refute the true state. If the first mover/defendant/surgeon has actually taken adequate care and makes a presentation that is contingently so interpreted, and if a given presentation by the second mover/plaintiff/patient would be interpreted as a refutation of the surgeon’s assertion, such presentation must be, in the state of adequate care, infinitely costly for the plaintiff.

Moreover, for truth revelation to work, any allowance for paucity in the evidence available to the second mover must be compensated for by additional richness in the evidence available to the first mover. Whenever there is no means for the second mover to distinguish state s from state s′ (that is, whenever every feasible second mover report that contains s also contains s′), there must be a first mover report that is of zero cost when the true state is s and of infinite cost when the true state is s′. Thus, if the patient cannot distinguish best practice and negligence, there must (again) be a report that is feasible for the surgeon if she followed best practice and impossible for the surgeon if she was negligent.

4.4.5. Other problems

There are other drawbacks to omission models in addition to the severity of the no-lying assumption. Several important issues are masked by the extreme binary nature of evidence costs, as discussed in Section 4.4.3 above. For instance, the manner in which the structure of evidence costs determines the range of supportable rewards and punishments is hidden from view. In an omission model, the stakes of the case are never so large as to inspire the production of false evidence; the cost of false evidence is infinite. Furthermore, all evidence that is actually presented in an omissions model is of zero cost (or at least constant cost), and so the question of how to design efficiently evidentiary process – taking account of the potential tradeoff between detail and certainty, on the one hand, and the deadweight cost of investigation and presentation, on the other – is simply not encountered in this literature.

These points will resurface in Section 5.2.1 when the omission model is compared to the costly signaling model.

5.1.1. In general

The idea of exogenous cost signaling is often attributed to Spence (1974), who models educational attainment as a signal of natural ability. It is, in fact, also the basis of Mirrlees’ (1971) optimal income tax model.⁴⁶ Almost 40 years after publication of these papers, costly signaling remains a commonly deployed mechanic in economic modeling generally. Yet it is oddly uncommon in the economics literature on legal evidence, for which it is arguably perfectly suited.

Spence’s approach was roughly this: Suppose an employer would be willing to pay higher wages to individuals of higher ability if only she knew p. 245who these individuals were. Merely asking job candidates whether they are of high ability won’t do: given the prospect of a higher wage, many candidates would answer “yes” whatever the truth. A candidate’s possession of a college degree might, however, act as a more reliable signal. Even though individuals of all abilities could conceivably raise their wage by earning a degree, this might be cost justified only for individuals of high ability, for whom earning the degree is less arduous.

One may agree or disagree with this account of education. But Spence’s (1974) general point regarding credible information transmission has resonance: an individual’s action reliably “signals” that she is of a particular “type” – i.e., has particular characteristics or knows particular information – when it would not have been in her interest to take that action were she some other type. In other words, talk is cheap and actions speak louder than words.

5.1.2. Applied to legal evidence

In the case of legal evidence, parties’ “types” are what they know about the event or condition in question, or how they acted in the primary activity. The relevant actions/signals are parties’ “performances” before the fact finder. This includes whether they present documents and things that are difficult to forge, as well as whether the witnesses they offer give consistent, detailed, robust and coordinated testimony. Focusing on the role of cognitive limitations, Sanchirico (2004a) explains why these “performances” might be more costly for some “types” than for others.

Assuming that such cost differences exist – at least probabilistically – their role is illustrated by the following example. Suppose that some piece of evidence cost $10 to produce when it is real and $100 when it is fake. If it is understood that production of this piece of evidence increases the party’s payoffs at litigation by some amount strictly between $10 and $100, then the evidence would be worth producing only when it is real. That is, the fact finder may reliably deduce that the evidence is real from the fact that it is produced, because it would not have been in the party’s interest to produce it otherwise.

One could question whether evidence is reliably more expensive to fake than to truthfully present. One could also question whether litigation payoffs can be reliably calibrated to separate the truthful from the untruthful (though we shall see that separation is not necessary). Yet, arguably, something like the dynamic in this (albeit stark) example must be at work if evidence production is to have value in a world of interested parties and hidden information.⁴⁷

5.1.3. Comparison to omission models

Before moving on to endogenous cost signaling, it is worth comparing exogenous cost signaling with the zero/infinite cost interpretation of omission models (see Section 4.2.4). It could be said that exogenous cost signaling smoothes the cost structure relative to this interpretation of omission models. Whereas the cost of an evidentiary performance must be either zero or infinite in omissions models, it may have some middling value in exogenous cost evidence models. Alternatively, one can say that omissions models rule out the choice to lie and fabricate (what is possible at infinite cost is, in fact, not possible), whereas exogenous cost signaling models incorporate such choices into the analysis. Moreover, exogenous cost signaling models recognize even truth-consistent evidentiary presentations are expensive, which raises several interesting issues of system design.

Tables 9.3 and 9.4 show the omission model version and the costly signaling version, respectively, of the numerical example in the previous section. These tables help to clarify the difference between the two approaches.

Under the omissions model, if the state is 1, no matter how small the (positive) “reward” for presenting the evidence, the party always presents it. Moreover, when she presents it, the social cost is zero. On the other hand, if the state is 2, then no matter how large the reward for presenting the evidence, the party never presents it.

p. 247Under the “smoothed” costly signaling version, if the true state is 1, the agent presents the evidence if and only if the reward is greater than $10, and this cost (which includes the cost of investigation and presentation) is added to the social cost of litigation. Notice that even though the evidence is true, the party will not present it, if the reward for doing so is insufficient. On the other hand, if the state is 2, then the agent presents the evidence if and only if the reward exceeds $100, and this larger amount is added to the social cost of litigation. Even though the evidence is false, the party will still present it, if the stakes are high enough.

Two issues arise naturally in the costly signaling model that are absent from the omission model.

First, notice that, in the costly signaling version, the agent’s payoff in state 1 cannot be more than $90 greater than her payoff in state 2. In state 2, the agent always has the option of paying an additional $90 in evidence costs to achieve the reward (or lesser punishment) that is available to her in state 1. Thus, the structure of evidence costs imposes limits on implementable payoff differences across states. This phenomenon does not arise in omission models.

Second, suppose there was another evidentiary performance whose costs were $5, $50 rather than $10, $100. This new evidentiary performance imposes lower costs on society, but is only capable of producing a payoff difference of $45, as opposed to $90. Thus, a potential tradeoff arises between the size of the payoff difference that is implemented and the deadweight cost of evidence production. This phenomenon also does not arise in omission models.

The second phenomenon suggests another drawback of omission models. Section 4.4 criticized the assumption in omission models that there exists evidence that is of infinite (or prohibitive) cost when the state that it is taken to represent is not true. The tradeoff discussed in the last paragraph points to the fact that existence is not the only issue. Even if effectively-impossible-to-fake evidence did exist, it might be woefully inefficient. Suppose it were true that evidence that was extremely expensive when false was also quite expensive, though less so, when true. Perhaps the many witnesses, documents, and things that make the performance so difficult to fake, also make it difficult to present truthfully. Thus, imagine that there is another piece of evidence in the example above that costs $1,000,000 (which we will regard as effectively infinite) to present when false, and $10,000 to present when true. If system design requirements were such that the payoff difference need only be $85, it would be much more efficient to use the $10, $100 evidence. The latter is sufficiently difficult to forge, and much cheaper when truthfully presented. These kinds of cost considerations are very real; they are frequently cited in evidence p. 248case law and in the history of evidentiary rule-making. (See, for example, Federal Rule of Evidence 403.)⁴⁸

5.2.2.1. Truth Revelation Versus Primary Activity Incentive Setting

In most scholarship on legal evidence, finding out what really happened between the parties is taken to be the objective of trial. Consider the widespread use of the phrase “fact finding” to describe the central activity of trial. And recall that the chief issue in the literature on omission models is whether there is truth revelation. In an endogenous cost signaling model, by contrast, the reason to have trials is to create primary activity incentives. Importantly, truth finding is neither sufficient nor even necessary for this task. What is important for truth finding is some separation in evidentiary actions. What is important for primary activity incentive setting is sufficient separation in evidentiary payoffs. In an endogenous cost signaling model, either of these may occur without the other.

It is fairly clear that truth finding is insufficient for incentive setting – that some separation in evidentiary actions does not imply sufficient separation in litigation payoffs. In the example that was provided above to illustrate the basic mechanic of endogenous cost signaling, imagine dividing all the dollar figures by 1000 – except compliance costs, which remain at $100,000. The firm still presents the evidence (now costing either $140 or $20) if and only if it has complied. Therefore, the regulator still learns the truth about compliance. However, compliance increases the firm’s prospective hearing payoff by only $110, which is insufficient to the additional $100,000 primary activity cost of compliance.

p. 250To see that truth finding is unnecessary – that sufficient separation in evidentiary payoffs is possible without any separation in evidentiary actions – consider again the example from above with all numbers restored to their original magnitudes. Recall that the regulator announced that it would fine the firm $130,000 unless it presented the evidence costing either $140,000 or $20,000. With this reward structure, only the compliant firm presents the evidence. Thus, the regulator learns whether the firm has been compliant. But the fact that the regulator learned the truth from the evidence was purely collateral. What mattered was the fact that the hearing payoffs were sufficiently higher for the compliant firm than for the noncompliant firm. And since evidence production costs differ, this does not require that compliant and noncompliant firms present different evidence. Suppose, for example, that the regulator instead announced that the firm would be fined $150,000 if it did not present the evidence. Then both compliant and noncompliant firms would find it worthwhile to present the evidence, and the regulator would never learn whether a given firm had been compliant or not. On the other hand, the hearing payoffs for both firms would now consist solely of presentation costs, and the difference in these costs (between $140,000 and $20,000) would still be enough to overcome the additional $100,000 cost of compliance in the primary activity.⁴⁹

5.2.2.2. p. 251Questioning the Emphasis on “Verifiability” in Contracts Scholarship

Contract theorists sometimes suggest that it is optimal for contracts to condition obligations only on contingencies that can be “verified” to a court (see, e.g., Hart and Moore 1988; Schwartz 1992). One implication of this principle, it would seem, is that an optimal contract should not specifically induce parties to fabricate evidence should a legal dispute arise.

Yet, to most contracting parties, verifiability is an intermediate goal. Adjudication creates value primarily through its ex ante effect on performance incentives. Anticipating the judicial resolution of future disputes, contracting parties are likely to be interested in the likelihood or cost of judicial truth finding only to the extent that the court’s ability to discern the truth efficiently improves contract incentives and the gains from trade.

Indeed, Sanchirico and Triantis (2008) argue that the parties themselves might prefer to permit evidence fabrication as part of a conscious contracting strategy that emphasizes efficient performance incentives over accuracy or fairness in the resolution of disputes.

The authors make their point using a probabilistic endogenous cost p. 252signaling model of evidence. A buyer and a seller design a sales contract in which they seek to motivate the seller to “perform” (e.g., deliver a good or service of a particular quality within a particular time frame). In particular, they wish to induce the seller to perform while incurring the lowest possible prospective litigation costs.

The buyer can sue under the contract to collect damages and, in litigation, can present either true or fabricated evidence.⁵⁰ An “evidentiary state” is defined to be the quantum of truly existing evidence of nonperformance, and the probability of a given evidentiary state is endogenous to whether the seller in fact performed. True evidence of nonperformance is costly for the buyer to present, and the buyer can choose to present all or part of the existing quantity of such evidence. Importantly, the buyer may also choose to present a quantum of evidence that exceeds the quantity of existing evidence. That is, the buyer may also fabricate evidence of nonperformance. The marginal cost of fabricated evidence is higher than that of true evidence.

Because the parties care about litigation costs, and because fabricated evidence is more expensive, the parties would prefer – all else the same – to induce seller performance without inducing the buyer to go beyond the quantity of truly existing evidence of nonperformance. Were this factor the only consideration, the parties would indeed wish to avoid the circumstance in which the reward that the buyer receives for evidence of nonperformance is so great that it induces the buyer to fabricate. Avoiding this circumstance would effectively bound the seller’s liability (which equals the buyer’s reward for evidence) in each state.

However, all else is not the same. The effect on the seller’s ex ante performance incentive of increasing the seller’s liability in any given evidentiary state depends in part on the difference between the probability that the state will occur if the seller performs (p) and the probability that the state will occur if the seller does not perform (q). Moreover, the effect on ex ante litigation costs of the buyer’s presentation of a given unit of evidence of nonperformance depends not only on the ex post cost of such unit of evidence, but also on the ex ante likelihood that such state will occur (r). Thus, ignoring for a moment the difference in cost between truthful and fabricated evidence, the parties can reduce the cost of achieving any given p. 253performance incentive by increasing the seller’s sanction in states in which the ratio r/(q – p) is low and reducing it where such ratio is high.

The consideration described in the immediately preceding paragraph constitutes a second factor in optimal contract design, and it is independent of the difference in cost between truthful and fabricated evidence. Importantly, this second factor may well be decisive. Specifically, where the ratio r/(q – p) ranges widely across different evidentiary states – and so is much greater in some states than in others – the parties may prefer to increase the seller’s liability in states with low ratios even if that requires paying the greater evidentiary costs of the fabrication by the buyer that is thereby induced in those states.

5.2.2.3. “Presumptions” and Litigation-Primary Activity Feedback

Bernardo, Talley, and Welch [BTW] (2000) apply the idea of endogenous evidence costs (Sanchirico 1995, 2000, 2001b) to study the effect of legal “presumptions.” The definition and real impact of legal presumptions is a complex and unresolved issue in evidence scholarship. In BTW (2000), the effect of legal presumptions is interpreted as a parameter in the kind of exogenous probability function studied in Posner (1973), as discussed in Section 2.2.⁵¹

The theoretical analysis in BTW (2000) concerns the positive effect of presumptions on litigation and primary activity behavior. Normative issues regarding the optimal design of litigation are also analyzed, but the analysis is confined to numerical simulations. BTW apply their findings to three substantive law areas: private securities litigation, the business judgment rule in Corporations law, and fiduciary duties to lenders in financially distressed firms.

The remainder of this discussion concerns BTW’s positive findings and the structure of the BTW model.

BTW’s positive findings center on the counterintuitive consequences of feedback effects from litigation design to primary activity behavior. The authors highlight the possibility that shifting “presumptions” in favor of defendants may increase, rather than decrease, plaintiff filings and may even result in a larger, rather than a smaller, frequency of plaintiff victory among filed cases. These effects can be explained by imagining that the changes induced by modifying the presumption occur in sequence.

The shift in the presumption initially increases the (potential) defendant’s p. 254litigation payoffs. This is true regardless of the defendant’s marginal cost of evidence (which may be “high” or “low”; evidence costs are linear). Given BTW’s functional form assumptions,⁵² however, the increase in litigation payoffs is greater for high marginal evidence cost defendants. The defendant has high marginal evidence costs if and only if she has “shirked” in the primary activity. Thus, the favorable shift in the “presumption” increases the defendant’s incentive to shirk. As a result, the defendant more often shirks. Therefore, given a “bad outcome” in the primary activity, the likelihood that this outcome is a result of the defendant’s shirking – and not just bad luck – is now greater than it was before the shift in the presumption.⁵³

When (and only when) there is a bad outcome, the (potential) plaintiff may choose to file suit against the (potential) defendant. The plaintiff chooses whether to file based on his expected litigation payoffs. These expected payoffs are determined in part on his assessment of the chance that the bad outcome has been caused by the defendant’s shirking, and that the defendant thereby has high marginal evidence costs. (The plaintiff does not directly observe the defendant’s evidence costs or primary activity choices.)

What impact does shifting the presumption in favor of defendants have on the number/frequency of filings? There are three effects.

The first effect is the most direct: conditional on a bad outcome, and conditional on the defendant’s marginal evidence cost, the change in the presumption in favor of defendants lowers the plaintiff’s expected litigation payoffs, and thereby acts to reduce filings.

Second, however, conditional on a bad outcome, the defendant is more likely to have shirked, and so more likely to have high marginal evidence costs. Hence, the defendant is more likely to present less evidence. This acts to increase the plaintiff’s expected litigation payoff. Notice that this second effect is a result of litigation-primary activity feedback: litigation design influences primary activity behavior, which in turn influences litigation outcomes. If the second effect dominates the first, then there will be more filed cases per bad outcome. This is the possibility that BTW emphasize.

Third, because the defendant more often shirks, there will be more bad outcomes. This will also act to increase the number of filed cases. This third effect, another kind of litigation-primary activity feedback effect, p. 255is not emphasized in BTW and has deep roots in the law and economics literature on litigation.

What is the impact of the shift in the presumption on the frequency of plaintiff victory at trial? Consider a filed case after the change in the presumption. Per the first effect identified above, the presumption has shifted against the plaintiff, which lowers the plaintiff’s chance of victory. However, per the second effect, the defendant is more likely to have high marginal cost of evidence, and so more likely to present less evidence. This increases the chance of plaintiff victory given any level of plaintiff evidence. One possible result, therefore, is that the plaintiff more often wins filed cases – despite the fact that the litigation playing field has been tilted against him.

5.2.2.4. Allocation of Proof Burdens and Strategic Complementarities in Evidentiary Choice

Sanchirico (2008) uses an endogenous cost signaling model of evidence to investigate the question of which party in litigation should be assigned the burden of proof. He finds some justification for the regularity in actual law that the burden of proof is assigned to the opponent of the party whose primary activity incentives are being set by the law in question – as when the plaintiff has the burden of proving the defendant’s negligence and the defendant the burden of proving the plaintiff’s contributory negligence.

Sanchirico (2005) generalizes these findings by considering the question of how optimal litigation design structures strategic complementarities between parties’ evidentiary choices. The payoff structure of the adversarial litigation game is such that one party strategically complements (i.e., mimics her opponent’s advances and retreats), while the other strategically substitutes (i.e., does the opposite of her opponent). Which party plays which role depends on how the litigation transfer function – the mapping from evidence onto liability – is structured. Since the litigation transfer function is a policy choice, the question arises: should litigation be designed to induce the plaintiff to strategically complement and the defendant to substitute, or vice versa? On the basis of an asymmetry derived from the envelope theorem, Sanchirico (2005) argues that the answer depends on whether and whose primary activity incentives are being set by the particular evidentiary contest in question. Within each subsidiary evidentiary contest, the “incentive target” should be induced to complement and her adversary to substitute. In some cases, the defendant will be the incentive target, as when the issue is the defendant’s negligence or contractual breach. In other cases, the plaintiff will be the target, as when the defendant defends by claiming that the plaintiff has been ”contributorily negligent.”

p. 256The analysis of the optimal allocation of proof burdens in Sanchirico (2008), cited above, is a special case of the general phenomenon described in Sanchirico (2005). As discussed in the former paper, the party who is not assigned the proof burden tends to be the one who strategically complements. Thus, to induce the incentive target to complement, per Sanchirico (2005), is to assign the proof burden to the opponent of the incentive target.⁵⁴