From extropians-request@gnu.ai.mit.edu Wed Mar 17 16:24:21 1993 Return-Path: Received: from usc.edu by chaph.usc.edu (4.1/SMI-4.1+ucs-3.0) id AA11321; Wed, 17 Mar 93 16:24:12 PST Errors-To: Extropians-Request@gnu.ai.mit.edu Received: from churchy.gnu.ai.mit.edu by usc.edu (4.1/SMI-3.0DEV3-USC+3.1) id AA07034; Wed, 17 Mar 93 16:23:55 PST Errors-To: Extropians-Request@gnu.ai.mit.edu Received: by churchy.gnu.ai.mit.edu (5.65/4.0) id ; Wed, 17 Mar 93 18:16:48 -0500 Message-Id: <9303172316.AA00706@churchy.gnu.ai.mit.edu> To: ExI-Essay@gnu.ai.mit.edu Date: Wed, 17 Mar 93 15:12:29 PST From: Robin Hanson X-Original-Message-Id: <9303172312.AA16260@ptolemy.arc.nasa.gov> X-Original-To: ExI-Essay@gnu.ai.mit.edu Subject: Why Not Let Convicts Hire the Police? X-Extropian-Date: Remailed on March 17, 373 P.N.O. [23:16:47 UTC] Reply-To: Extropians@gnu.ai.mit.edu Errors-To: Extropians-Request@gnu.ai.mit.edu Status: OR Just over two months ago, I posted a then 9000 word paper with the above title. The paper is much revised, now 13,500 words, and I have just submitted it for publication to the Journal of Legal Studies. Here is a Latex version of the paper. Discussion welcome of course. ------------------------------------------ \newcommand{\set}{{\cal S}} \newcommand{\ave}[1]{\overline{#1}} \documentstyle[ijcai91,named]{article} \pagestyle{plain} \title{Should Privately Enforced Laws Fix Punishment or Restitution? \\ (or, Why Not Let Convicts Hire The Police?)} \author{Robin Hanson \\ {\normalsize 47164 Male Terrace, Fremont CA 94539} \\ {\normalsize hanson@ptolemy.arc.nasa.gov} \\ {\normalsize 510-651-7483} \\ {\normalsize March 12, 1993} } \date{March 12, 1993} \begin{document} \maketitle \begin{abstract} Privately enforced law might fix punishment, restitution, or some combination of the two. Such law might be uniform, depending only on the victim's complaint, or it might be contingent, depending on crime types determined after a conviction is obtained. After reviewing the general problem of designing private law and enforcement, I compare these three ways to fix law under both uniform and contingent law, comparing efficiency and ability to induce revelation of relevant information. Fixed punishment seems better at inducing victims to aid private law enforcers and at imposing high punishment levels. Fixed restitution, however, seems better at revealing criminal risk preferences, avoiding distortion from bribes, accommodating intangible preferences, and inducing criminals to aid enforcement. \end{abstract} \section{Introduction} Of the large and growing academic literature on law and economics, only a disappointingly small fraction is devoted to the task of imagining and evaluating alternatives to our current legal institutions. This is disappointing because, after our academic clouds of analysis settle (if ever), our main general (normative) choice will remain, in law as in most economic contexts, a choice between institutions. And we risk this being an empty choice if we have not bothered to imagine specific plausible alternatives to existing legal institutions. (And even those convinced of the superiority of existing institutions understand them better after comparing them to plausible alternatives.) Fortunately, there are exceptions to this trend, such as Gary Becker and George Stigler's proposal [1] that we privatize law enforcement, using competing private law enforcement agencies in the context of traditional state monopolies in law courts and legislation. David Friedman, elaborating on this suggestion, offers arguably the best current proposal [2]. He suggests that each privately enforced crime would have a complaining victim, and victims would initially own the right to collect fines from convicted criminals, rights which they would then sell to private law enforcers. Friedman's paper initially responds to a paper by Landes and Posner [3], who argue that private enforcers, competing to obtain a legally specified fine per crime, could not be economically efficient. In response, Friedman proposes that, rather than specifying a fine, laws could specify an expected punishment level for each type of crime. Using some unspecified method to assess criminal risk-aversion, each convicted criminal would be fined whatever amount is required to reach the law-specified expected punishment, given the enforcer's observed frequency of capturing and convicting groups of like criminals doing like crimes. With punishment fixed, victims own the residual after paying for enforcement costs, and so want to hire enforcers who minimize these costs. If the expected punishment is set at its optimum value, Friedman shows that enforcers have incentives to spend an optimum amount on enforcement. While Friedman's proposal is certainly an improvement over previous suggestions, it still seems less than ideal, ignoring criminal efforts to avoid enforcement, being susceptible to bribes, requiring large bundles of similar crimes, and requiring official knowledge of criminal risk aversion and preferences for intangibles like being treated with dignity. Therefore, this paper offers yet another (hopefully improved) alternative private law institution, where laws fix a specific level of restitution for each type of crime, rather than a punishment or fine. Here criminals would own the residual after enforcement costs, and so want to choose an enforcer to minimize these costs. A criminal's agent would estimate each criminal's risk preferences, and choose an enforcer offering the minimum expected punishment while paying the victim their due restitution. Specific institutions are described for validating case-specific probability estimates, without needing like crime bundles, and for validating an agent's case-specific estimates of criminal risk. This approach can induce criminals to cooperate with enforcers, can consider the intangible preferences of both parties, and better prevents bribes from distorting the parameter that the law fixes (preventing this completely under uniform law). However, it does give risk from uncertain enforcement costs to the party who can least afford it, requires more formal (and hence costly) rules to ensure that victims adequately aid enforcers, and does not allow punishment to be raised above the level where restitution begins to fall. A fourth and more general approach would have the law determine a social loss function of punishment and restitution which is to be minimized, and have the law directly hire the enforcer whose bid minimized this loss. Alternatively, both punishment and restitution could be fixed, with a victim's insurer owning the residual after paying enforcement costs. One might have doubts whether the process by which laws were set could adequately deal with such complexities, these approaches would less consider criminal intangibles, may make it difficult to estimate risk preferences, and may fail to entice criminals into aiding enforcement. But they might better allocate risk due to uncertain enforcement costs than either of the above alternatives. All of these approaches can be extended to allow the law to be contingent on later court-determined properties of convicted crimes. Such contingent law allows more carefully tuned incentives, and can directly deal with the case where the someone falsely claims to be the victim of a crime. Enforcement can also be contingent, allowing fines and probabilities of conviction to be contingent on later court-determined properties of the crime and criminal. However, under contingent law and enforcement, victims may suffer risk of uncertain restitution payment unless they can gain a reputation for being forthright in describing crimes. And bribes can raise both punishment and restitution above their legally fixed values. \section{Law As A Solution To Crime} But before going into detail on these different institutions, let start with the basics of law and enforcement, to provide a context. Let us call a ``crime'' between two people (or parties), an event which: \begin{enumerate} \item Changes the utility of both people, \item May occur even if one or both person objects, and \item Both can influence the chance of it occurring. \end{enumerate} (Such crime events often violate property rights of the victim.) If the crime happens a ``criminal'' will receive some benefit $B$, and a "victim'' will suffer some damages $D$. (We might choose the person labels so that $B+D$ is positive.) The probability $q[w,x]$ that the crime will occur can be influenced by ``watchfulness'' costs $w$ of the criminal, and ``defense" efforts $x$ of the victim. (These efforts $w,x$ are direct, and do not include "deterrence'' attempts to influence the effort of the other party by changing their payoff $B$ or $D$.) For example, a criminal might scratch a victim's car in a parking lot. This scratch might be accidental, or it might be an intentional act of vandalism (or artistic expression); it might even be an improvement to the car. Thus the net social benefit (something like $B-D$) might be negative or positive, as might the benefit $B$ and damage $D$. So efforts $w,x$ might be to avoid or to encourage the crime; it all depends on what consequences each party expects, and on what they expect the other party to do. Each party has only limited influence over the event; a scratch may happen even if both parties were trying hard to avoid it. Since one party can influence this event in the absence of consent from the other party, crimes can have externalities. The criminal may choose effort $w$ to tradeoff watchfulness costs with improved chance of gaining benefit $B$, but ignore the effect of increasing the victim's chance of suffering damage $D$. Similarly the victim's choice of defense effort $x$ may ignore the benefit $B$ to the criminal. A social optimum, however, (i.e., a solution all parties would prefer when bundled with initial wealth transfers) would adjust both $w$ and $x$ to instead minimize a social loss considering both $B$ and $D$. To deal with this problem, these two parties would prefer to contract before-hand to reduce these externalities. And if the efforts $w$ and $x$ are not cheaply observable (and so make a poor foundation for direct contracts), then a common second-best solution is to better divide up the net cost (or payoff) of the crime, $D-B$, among the parties. Ideally they would both get positive payoff when the net is positive, both negative when the net is negative, and the fraction each received would be in proportion to the relative influence they each had over the result. One way to do this is for these parties to agree to submit to a ``legal system'' (or a set of legal systems sharing treaties) which will, when a crime occurs, take $P$ from the criminal, through some expected punishment, and give restitution $R$ to the victim, with the difference $P - R$ being part of the social cost of enforcing a legal system. (Some initial or continuing wealth transfer might compensate those who expected to otherwise suffer net loss from the introduction of such a law.) More precisely, let harm $H_c$ to the criminal from loss $L$ be the negative of utility, with $H_c[L] = -U[-L]$ and $H_c[0] = 0$, and let additional harm of $l$ after suffering $L$ be $H_c[l,L] = H_c[l+L] - H_c[L]$, with all arguments to $H_c$ expressed in terms of equivalent amounts of money which induce the same utility changes. Let harm $H_v$ to the victim be defined similarly. If we define the initial wealth of each party to be zero, the criminal and victim expect to suffer losses \[ H_c[L_c] = H_c[w] + q[w,x] H_c[P-B,w] \] \[ H_v[L_v] = H_v[x] + q[w,x] H_v[D-R,x] \] \noindent where all arguments now defined as monetary equivalents. The criminal will pick $w$ to minimize $L_c$, while the victim picks defense $x$ to minimize $L_v$. If we assume efforts $w,x$ and loss $L$ are small (relative to each party's total wealth), and calibrate each $H$ so that $d_L H[L=0] = 1$ (where $d$ is the partial derivative operator), we get the approximations: \[ L_c = w + qC \] \[ L_v = x + qV \] \noindent where $C = H_c[P-B]$ and $V = H_v[D-R]$ are criminal's and victim's "shares'' in the crime. When $C$ and $V$ are also small, $C = P-B$ and $ V= D-R$. Without law ($P=R=0$), the criminal neglects the $w$ dependence of $qD$, ignoring damage to the victim, and the victim neglects the $x$ dependence of $-qB$, ignoring benefit to the criminal. An optimal prior legal deal, however, would choose $P$ and $R$ to minimize social loss $L = L_c + L_v$ and the externalities would become: the $w$ dependence of $qV$, and the $x$ dependence of $qC$. Social loss is thus minimized for $C$ and $V$ small in magnitude, i.e., for $R$ close to $D$, for $P$ close to $B$, and for low $|R|$, since larger wealth transfers incur larger absolute legal costs. (When $|B-D| > |P-R|$, this implies $B V$ and $\ave{(V-\ave{V})^2} > \ave{(C-\ave{C})^2}$, with $P,R$ nearer to $D$ than to $B$ and with $R$ varying more than $P$ does. \section{Contingent Crime} The above analysis treats each possible crime independently, ignoring possible economies of scale of enforcement or deterrence. And it ignores third party externalities, such as from convicting innocent persons, or from inabilities to discriminate between differing criminals or victims. Also neglected are economies of scope in encouraging or preventing different possible crimes. To correct one of these limitations, consider the possibility of multiple crime types $i$, with type-dependent impacts $B_i$ and $D_i$, and multiple criminal efforts $w = \sum_k w_k$ and victim efforts $x = \sum_l x_l$, all influencing the chance of each type of crime $q_i[\set_k w_k, \set_l x_l]$ (where $\set$ is a set collection operator). Then for equally type-ignorant parties, losses are \[ H_c[L_c] = H_c[w] + \sum_i q_i H_c[P_i-B_i,w] \] \[ H_v[L_v] = H_v[x] + \sum_i q_i H_v[D_i-R_i,x] \] \noindent and an ideal law would now set all $R_i$ and $P_i$ to minimize $L = L_c + L_v$, subject to the constraints that all $w_k$ are chosen to minimize $L_c$ and all $x_l$ chosen to minimize $L_v$. If we could redescribe the $w_k$ and $x_l$ so that each $q_i$ becomes a function of separate $w_ik$ and $x_il$, this analysis would largely reduce to that above, with $L = \sum_i L_i[P_i,R_i]$. But before considering more the complexities of contingent law, let us return to the simpler case. \section{Designing Law} As described above, the task of the law is to choose wealth transfers $P$ and R in order to set the crime shares $V = H_v[D-R]$ and $C = H_c[P-B]$ as well as possible, given the relative influence of the two parties. But while a law might be more efficient if expressed in terms of $V$ and $C$, laws are more easily made clear and predictable when expressed in terms of monetary equivalents of wealth transfer, the $P$ and $R$. So I assume law is expressed in terms of $P$ and $R$, or perhaps even more directly in terms of the fine f due from the criminal to the victim upon conviction. Given this assumption, the two main tasks of the legal system are to set the levels of punishment $P$ and restitution $R$ to minimize social loss $L[P,R]$, both by adjusting $P,R$ to proper levels, and by implementing this wealth transfer with minimal enforcement cost $P-R$. While the focus of this paper is on the second task, that of enforcement, this broader context helps us understand just what makes a good enforcement institution. A major problem with setting punishment/restitution levels is the difficulty of getting parties to credibly reveal the relevant information: damage $D$, benefit $B$, relative influence $d_{ww} q/d_{xx} q$, and relative risk $-d_P C/d_R V$. For small $P-B$ and $D-R$ relative to each party's total wealth, relative risk can be neglected. And even larger risk can be neglected if adverse selection and moral hazard do not overly restrict the use of insurance against the risk of crime. But other information is harder to obtain. The criminal wants to downplay $B$, the victim wants to exaggerate $D$, both want exaggerate their risk aversion, and both want to downplay their relative influence for bad crimes, and exaggerate it for good crimes. But there are many relevant clues which may reveal damage and benefit. A crime was probably a net bad ($D>B$) if, without prohibitive transaction costs, a similar event could have been induced, but wasn't, through a voluntary contract. If the criminal displayed ``intent'' to encourage the crime, benefit $B$ was probably expected to be positive. Similarly, victim expenditures on crime prevention indicate an expected positive damage $D$. Victims who purchase crime prevention services from a third party, and who use an incentive contract wherein the prevention service loses (though not necessarily to the victim) a certain amount when a crime happens, might thereby reveal their expected losses from a crime (after expected legal restitution). Victims might even thereby reveal risk preferences regarding large crime damages. A more direct signal might come if victims could buy or sell changes in restitution level somehow from potential criminals. (This suggestion was made by David Friedman to a private electronic mailing list.) Over all, we expect victims to better reveal their expectations through their prior efforts (at least when victims expect positive restitution), since potential criminals might fear that open expenditures would focus unwanted attention on them in future investigations. Unfortunately, such clues may in general be far from sufficient. One hopes, however, that broader social negotiations over law may settle on reasonable values for $P$ or $R$, indirectly forcing the parties to reveal expectations about $D,B$ and relative risk and influence. When such expectations are contingent on various features of the envisioned crime or parties, features that courts could later uncontroversially determine at conviction time, then the law might do better to set punishment and restitution contingent on these characteristics. Of course the costs of such negotiations, and the cognitive overhead for people to learn the law, may prohibit too many such possible contingencies from being considered. The simplest legal policy is $P=R=0$, i.e., no legal intervention, appropriate when costs of negotiating or enforcing a law would outweigh its externality reductions. The next simplest policies of $P=B$ or $D=R$ both have the advantage that only one parameter need be revealed, and for a crime of marginal social value, $L~=0$, they both work efficiently, eliminating externalities. But the more general case is more complex. If it must choose, the law might do better to focus on getting $P$ or $R$ to be the right distance from the damage $D$, since benefit $B$ is harder to measure and should be typically farther away. For a typical bad crime with mostly criminal influence, we want $P$ near $D$, or $R$ a fraction less than $D$. But if the law is going to fix one of punishment $P$ or restitution $R$, which should it fix? The answer is not obvious, depending on many subtle implications of making the law one way or the other, some of which will be described below. If legal complexity can be tolerated, a more general law might specify a set of official social loss functions $Ln_i[P,R]$, one for each type of crime $i$, or a single combined contingent crime loss function $Ln[\set_i (P_i,R_i)]$ when crime types are not determined until conviction. The law enforcement system would then be tasked with minimizing this social loss, by adjusting $P$ and $R$ and minimizing enforcement loss $P-R$. Setting $P=B$ is like making $L[P,R]$ sharply peaked around a particular value of $P$ and linear in $R$, and $R=D$ makes $L$ sharply peaked around a particular $R$, and linear in $P$. \section{Designing Enforcement} A good enforcement institution should induce the parties to reveal whatever information the law needs to fix the parameters it wants to, such as punishment $P$ or restitution $R$, and should offer incentives for parties to lower enforcement costs $P-R$ as much as possible. Enforcement costs consist primarily of two factors. There are the efforts that the parties, or their agents, expend to encourage or discourage a court conviction given that a crime has occurred, and the risk each party suffers from uncertainty about this conviction. The legal process may also impose ``intangible'' costs by not treating the two parties with dignity, or by violating their privacy. Finally, the parties may suffer reputation losses (or perhaps gains) if a conviction is publicized. Effort by a criminal to avoid being caught should be hard to measure, since such effort may be done in secret. This can make it difficult for law to fix punishment. Effort by a victim, however, can be much more visible if the victim hires an ``enforcer", at competitive prices, to do most of their investigation and court work. Even more victim effort becomes visible if the enforcer must the hire the victim, at the victim's standard wages, for time spend giving testimony, identifying line-ups, etc. (Of course when punishment is fixed, such formal methods to make victim effort visible are not needed.) Costs to run the court system itself are perhaps also reasonably paid by the enforcer requesting restitution (as might be the legal and other costs of those the court declares innocent). When the probability of conviction is small for any given crime, victim risk is dominated by uncertainty over whether a conviction is obtained or not. But victims can cheaply insure against most of this risk if such insurance is bought after it is clear a crime has occurred. (When bought from the enforcer, this insurance is equivalent to victims selling their right to restitution to the enforcer.) While victims might need some small stake in conviction to encourage them to aid the enforcer (though reputation effects and the usual drive for revenge may be sufficient), this risk cost should be small, particularly if the victim must offer a detailed complaint before an enforcer is hired. Thus we may typically neglect victim risk. Criminals avoiding capture, in contrast, cannot insure against conviction risk without significant moral hazard costs; they would be betting insurers that they would be caught and convicted. If the probability of conviction is near a socially optimal value, then criminal risks must remain a substantial fraction of total enforcement costs. If they were not, total enforcement costs could be substantially lowered by simply lowering the probability of conviction and raising punishment given a conviction. Thus to fix punishment, the court needs some way to estimate criminal risk. But with punishment fixed, criminals should want to exaggerate their risk-aversion to lower fines due. Reputation effects on the victim should not depend much on enforcement since the victim largely reveals their status when complaining of a crime. And even for criminals reputation effects should not depend much on the probability of conviction, if reputation consumers correct for that probability. Criminals may suffer risk costs from small changes of large reputation changes, but otherwise, it is not clear that reputation changes create net social losses, and in any case such costs should be hard for courts to estimate. Thus reputation costs will be largely ignored below. In summary, enforcement costs consist mainly of criminal risk to cash flow and (less predictably) reputation, visible enforcer effort, and hidden criminal avoidance. Because the relevant information is less available, punishment should be harder to fix than restitution. \section{Forms Of Punishment} When possible, a fine due directly from convicted criminal to vindicated victim seems preferable. Punishments such as prison or mutilation do not usually offer the victim a comparable restitution, and so raise enforcement costs $P-R$. When a criminal cannot pay a fine high enough to impose the intended punishment, it might seem that other types of punishments must be added on, even if such added punishments resulted in net reductions in restitutions. It should be understood, however, that how much fine a criminal can pay depends on other aspects of law, such as bankruptcy law and other limits on contracts. Even if one might ordinarily want to prevent people from selling themselves into indentured servitude or to a labor prison, it is hard to see why these shouldn't be acceptable alternatives to state prisons, if in fact the purpose of prison is to impose a given level of punishment. And even if isolation of the criminal were another purpose of prison, it seems likely that private prisons could still make net profits on most isolated prisoners. Thus such prisons should be willing to offer positive restitution for the rights to such prisoners, even if they must post a bond payable if one of their convicted criminals escapes. Thus it seems the criminal should be allowed to enter into such contracts for the purpose of paying a large fine due. And so if the criminal declines to enter such a contract voluntarily, the court should be able obtain the fine due by auctioning not only the criminal's personal possessions, but also the criminal's future labor, and any other assets. Limitations on the kinds of criminal assets a court could auction to obtain a fine due might serve to make fines sensitive to risk-aversion, on the reasoning that if the court must auction the shirt off of someone's back that person must be down to their last few dollars of assets. But more direct means of assessing risk-aversion, such as those described below, would be preferable. A particularly ruthless law might even auction off the criminal's life or bodily organs. If the law's customers judged punishments beyond a certain level as ``cruel and inhuman", then a maximum punishment could be set. Externalities due to such a maximum might be reduced by requiring everyone to obtain crime insurance which pays up to some standard maximum, but only after all other (humane) sources of fine payment are exhausted. In any case, it seems that there should almost always be some fine $f$ due (with certainty) which can induce any level of punishment that the law is willing and able to impose, with $d_f H_c[f] > 0$ (though perhaps death plus costly torture could impose a somewhat higher punishment than just death). Even so, a law might insist on imposing punishments through means other than fines, typically resulting in a net reduction in restitution for a given level of punishment, and perhaps even resulting in net negative restitution (and discouraging victims from complaining of such crimes). A law might also choose non-monetary restitution, such as sympathy, praise, or fame. \section{Optimal Enforcement} Let us introduce some notation to aid in modeling the enforcement process. When a court-determined fine $f$ is due from a convicted criminal to a victim, let $G[f] <= f$ be the expected amount that the victim's enforcer actually receives, correcting for delays, collection costs, and the possibility that the criminal simply does not have the means to pay. And let $p[a,e]$ be the probability of catching and convicting the criminal, estimated just after the crime is reported, a function of enforcer efforts $e$ and criminal avoidance efforts $a$ (and neglecting victim efforts to aid the enforcer). That said, the enforcer's net loss (or negative profit) is $R + e - p G[f]$; the enforcer pays the criminal a restitution $R$ up front, then exerts effort $e$, and in return has only a chance of being due a fine $f$. We assume the enforcer is wealthy enough that risk from this conviction is not an issue. A net negative profit would bankrupt the enforcer, while a net positive profit would be treated by the victim and criminal as just a larger effort cost $e$; as an agent of the criminal and victim, the enforcer's net gain is not a legitimate part of the social loss that those two parties would negotiate to minimize. Thus we can conclude \[ R[a,e,f] = p[a,e] G[f] - e \] \noindent With private law enforcement, enforcers compete to gain the business of the party that owns the residual after enforcement costs, and so benefits from lower enforcement costs. The point of private enforcement is to use competition to discourage waste and inefficiency, so that enforcers choose effort e appropriately; with competition we expect no net enforcer profit, making e more directly the enforcement effort. A criminal expending effort $a$ to influence a probability $p$ of being caught and required to pay a fine $f$ suffers a certainty equivalent net wealth loss $P[a,e,f]$ given by: Eqn. [2]: \[ H_c[P,-B] = H[a,-B] + p[a,e] H_c[f,a-B] \] \noindent Risk aversion makes $P > a + p f$ as $f$ gets large. Note that while punishment $P$ has been defined to include effort $a$, effort $e$ has been defined external to restitution $R$, since contracting with an enforcer allows most of this effort to be separated. Our goal is an enforcement institution with incentives for participants to set parameters a,e,f to minimize social loss $L[P,R]$. When all three of these parameters can be controlled, then ideally \[ \frac{d_a P}{d_a R} = \frac{d_e P}{d_e R} = \frac{d_f P}{d_f R} = \frac{-d_R L}{d_P L} = Q \] \noindent here $Q$ is defined by the last equation, and so the optimal $a,e,f$ satisfy \[ G d_e p = \frac{1}{1-F} > 0 \] \[ H d_a p = \frac{F}{1-F} (d_f H + (p-1) d_a H) > 0 \] \noindent where $G = G[f]$ and $H = H_c[f,a-B]$ and factor $F = H d_f G/G d_f H$, so that $0 < F < 1$ for risk-averse criminals. Since both $d_e p$ and $d_a p$ are positive, in this case both criminal and enforcer cooperate to increase the probability of conviction $p$ (which is presumed to be estimated similarly by criminal and victim). With cooperative criminals, restitution always rises with rising punishment (since $d_e R > 0$). It may be difficult, however, to induce the criminal to cooperate in setting avoidance $a$, since $a$ is hard to monitor. In that case we can at best minimize $L$ subject to $d_a P = 0$, which implies \[ H d_a p = - (d_f H + (p-1) d_a H) < 0 \] \noindent so that the criminal works to avoid conviction. With $a$ thereby determined, there remain two degrees of freedom, described as either $P,R$ or $f,p$ or $e,f$. At this optimum: Eqn. [3]: \[ \frac{d_{ea} P}{d_{fa} P} = \frac{d_e P - Q d_e R }{d_f P - Q d_f R} \] \noindent which implies \[ \frac{H - Q' (G - 1/d_e p)}{d_f H - Q' d_f G} = \frac{d_a H + H d_{ea} p/d_e p}{d_{ff} H + d_f H d_a p/p} \] \noindent with $Q' = Q d_P H[P-B]$. In this case there can be a maximum possible restitution, after which restitution falls with rising punishment. Note that in either case criminal avoidance $a$ depends on expected fine $f$, contrary to the simplifying assumption made in [4]. \section{Fixing Fines} The simplest legal approach is to have the law just set the fine $f$ due. This results in a clear and predictable law. However, since social loss is a function of at least three dimensions ($a,e,f$), just setting the one dimension fine $f$ (with no direct relation to social loss) is in general simply not enough degrees of freedom to induce the social optimum. Specifically, the probability $p$ that an unconstrained enforcer would choose would generally be higher than the social optimum. If one neglects criminal efforts $a$, so that probability $p$ is a function of only victim efforts $e$, the space reduces to two dimensions. In this case there exists some tax on enforcement, reducing $R$ relative to $f$, which can induce the social optimum, if those who set the tax could obtain the relevant information, and had incentives to set it well. However, such a tax would create an incentive for criminals to bribe enforcers not to turn them in, so that the criminal and enforcer could split the benefit of avoiding the tax. This problem with privately enforcing legally fixed fines, identified by Landes and Posner [3], is seen by some [6] as a justification for state-run law enforcement, at least when the optimal probability of conviction is low (supposedly not the case for most torts). State enforced law also offers the supposed benefit of allowing political interest groups to influence which laws are actually enforced in what contexts (and on whom). \section{Fixing Expected Punishment} David Friedman responds to this problem with fines in the following proposal. Friedman proposes [2] that ``offenses belong to the victims and must be purchased [by private enforcers] before or immediately after they occur", and that ``the state ... imposes an expected punishment'' $P$. He explains in an example (that ignores criminal risk): \begin{quote} ``Suppose, for example, that the expected punishment is set at \$1000. A particular firm has purchased 100 occurrences from the victims. If it succeeds in catching all 100 perpetrators, it can find them \$1000 each ... If it catches and convicts only one criminal, it can fine him \$100,000 - again an expected punishment of \$1000.'' \end{quote} The victim must pay for the cost of enforcement, and ``criminals who are unable to pay the fine ... must be punished in other ways ... such as flogging or execution ... [or] imprisonment". The result is that ``the firm must weigh the cost of catching more criminals against the advantage of being able to collect a larger fraction of the fines [the criminals should] pay.'' This approach also has a problem with criminal bribes to enforcers. Here bribes raise the actual punishment $P$ above the fixed $P_{\mbox{law}}$, by making the actual rate at which criminals must pay fines or bribes larger than the $p$ figure suggests. To combat this, Friedman suggests \begin{quote} ``The court system need only observe the rate at which crimes occur against the customers of each firm. If the rate is consistently 'too' low then the firm should be instructed to lower its expected punishment; if 'too' high, to raise it." \end{quote} In summary, the victim sells an enforcer the right to collect a fine $f$, and $f$ is set by solving $Pn[f,p] = P_{\mbox{law}}$, where p is the frequency of convictions in some enforcer-chosen bundle of crimes, and $Pn[f,p]$ is presumably some official estimate of $P$ taking into account risk-aversion and avoidance costs. Restitution $R$ is whatever the victim can get, and may be negative, if the victim so consents. We expect competition between enforcers to find the highest possible restitution $R$ given the fixed punishment $P$. If we assume that criminals cannot be induced to cooperate in setting their avoidance effort $a$, and if $d_R L < 0$ near the subspace of $e,f$ which satisfies $P[f,p] = P_{\mbox{law}}$ (and which an enforcer could feasibly implement), then an enforcer maximizing $R$ under these constraints would minimize social loss $L$. We expect $L$ to fall as $R$ rises, for fixed $P$, as long as the potential harm to the criminal from the victim wanting less to prevent the crime is not too severe. Though Friedman is not explicit about this, it seems that the enforcer crime bundles must be chosen before the enforcer starts to incur enforcement costs, and after a punishment level has been set for the crime. If bundles could be chosen after enforcement, then bundles with different frequencies and the same punishment would just not be as profitable as merging them into one bundle, at least for risk-averse criminals. If punishment were chosen after the bundles, then one couldn't create bundles of all the same punishment, giving enforcers incentives to have higher than advertized probabilities of conviction for cases for which they expected higher punishments, at the expense of lower punishment crimes in the same bundle. (This makes such bundles bad for contingent law, discussed later.) Advantages of this proposal include intuitive elegance and historical precedence. With victim contributions, punishments can be set higher than is possible with fines alone. Victims have clear incentives to gain a reputation for aiding enforcers, since that raises their expected restitution. The party best able to afford it suffers the risk of uncertain enforcement costs. And enforcers, hired by victims, have incentives to attend to less tangible preferences of victims, like being polite and discreet, though unfortunately a legalistic $Pn[f,p]$ function offers no such incentives to respect the dignity or privacy of criminals. The law would have to know explicitly how to take them into account, or they would be ignored. There are other problems with this proposal. Victims must suffer the uncertainty due to varying expected enforcement costs, even though the quadratic model above suggests this risk would be better assigned to a criminal with more influence. Crimes may not occur in the neat sets of near-identical crimes reported at the same time, needed here to validate $p$ estimates. No institution is described for discovering an explicit punishment function $Pn[f,p]$, capturing both avoidance and risk-aversion costs, required to implement the above policy. Friedman's solution to bribes seems less than satisfactory, and his approach has, at least in its simplest form, given up on encouraging the criminal to aid enforcement. \section{Case Specific Probabilities} Before going on to describe a restitution alternative to fixing punishment, let me describe some improvements to the above approach, improvements which were invented in the context of the fixed restitution alternative, but which might also be applied to fixed punishment. Friedman's proposal to bundle similar crimes and estimate probability from the frequency of conviction in the bundle has the advantage of simplicity, but the disadvantage of discouraging small enforcement firms, by creating significant economies of scale in bundling crimes. For a given crime, it might take a large pool of customers to find several more crimes reported at the same time, with the same criminal risk preferences, optimal fine, and probability of conviction. An alternative is to use bets to validate probability estimates. When an enforcer takes on a case, they declare a probability estimate p of conviction, and for a short time offer to bet anyone that the probability of conviction is less than this number (i.e., enforcer sells ``\$$X$ if convict'' for \$$pX$). People who take them up on this offer are essentially buying into the enforcement project, gaining assets contingent on a conviction. With more bets, the enforcer has a smaller potential payoff contingent on a conviction, and hence loses interest in pursuing a conviction, creating a natural limit to the amount of bets taken. Competition between betting speculators should ensure the validated probability p is not an underestimate; any speculator who thinks that the enforcer is trying to get away with too low a probability $p$ can bet on that, both making money if right and actually lowering the probability by lowering the enforcers interest in enforcement. A criminal who secretly tries to bet on being convicted, and then confesses, is like one who signed up to be their own enforcer; they just pay a lower net fine by raising their probability of conviction toward one. This betting option also offers incentives for anonymous informants to contribute information, after they have bet for a conviction (though one might worry about incentives to falsely testify). Such speculators could not prevent an overestimate of $p$, which could allow actual punishment to be less than intended. Such incentives to under-enforce, however, only happen when $d_e R$ is negative, which only happens when increases in punishment result in decreases in restitution, and this does not happen when $a$ is socially optimized. But it is possible if the criminal works to avoid conviction. Such high punishments are legally desired only when $d_P L$ is negative, when great harm to the victim from lack of criminal care justifies higher punishment even without higher restitution. In such cases one would want to allow betting against enforcement, but restrict this ability to parties clearly unable to help prevent conviction. Since the probability would be declared earlier, the criminal might better know what fine they face if convicted. \section{Estimating Criminal Risk} To fix or minimize a punishment level $P$, using (observable) control parameters fine $f$ and probability of conviction $p$, one needs to know $P[f,p]$, how expected punishment for a particular criminal (estimated after a particular crime is reported) varies with $f$ and $p$. And if one needs to impose a punishment higher than any allowed fine could give, one would need to know $P$ as a function of prison time, degree of torture, etc. While these problems can always be punted to be dealt with by the expensive social negotiations that choose laws, we prefer a more direct and case-specific approach. Contributions to punishment include risk-adjusted cost of owing the fine, cost of effort to avoid/encourage conviction, loss to reputation, and other intangibles. Most of these costs are rather difficult for a court to measure, and hence difficult to account for in fixing or minimizing punishment. However, risk-aversion can be measured, at least if one is willing to offer reduced fines to some fraction of convicted criminals. A criminal who has just received the benefit $B$ of the crime should be indifferent between having to pay an amount $P$ for sure, or having to risk a probability $p$ of paying fine $f$, after paying effort $a$ for sure. (This is actually the definition of punishment $P$.) But a criminal who has just been convicted has already received $B$ and paid $a$. So they should be indifferent between paying $P-a$ or suffering risk $p$ of paying $f$. Thus if we can see how much they are willing to pay to avoid risking a probability $p$ of owing fine $f$, we can see $P-a$. If in addition the law knew $a$, that would reveal $P[f,p]$ for this one combination $f,p$. To help induce honest revelation of willingness to pay, we can introduce some competition. Place each convicted criminal into a group of $2n$ criminals ($n=1$ may be reasonable) with similar fines, probabilities, declared functions $P$, estimates of $a$, and any other characteristics that indicates risk preferences (like a history of gambling). Do not introduce these people to each other (or allow insurance against the following risk). Instead, introduce a small random probability $z$ of freeing this entire group from their obligation to pay fine $f$. Usually each person in the group would owe their court-determined fines, but $z$ percent of the time they would instead be ``freed'' to suffer only a $p$ probability of having to owe their common median fine $f$ (to the enforcer). In this case, they each would each offer a sealed bid saying the amount they are willing to pay to avoid this chance of having to owe $f$. Half, or $n$, of these offers are taken and all pay the lowest bid price $P'$ of that half group. This price should be near a median estimate of $P[p,f]-a$ for that group. If secrecy can be maintained so that freed criminals do not suffer reputation losses, this approach also measures punishment through reputation loss. At best, however, this approach only measures punishment $P$, even though the total criminal share $C = H_c[P-B]$ is of more direct interest to the law. Of course this test is not yet enough to allow the law to set an official function $Pn[f,p]$ to use in setting the fine $f$ from the probability $p$ set by the enforcer. Avoidance and other costs are not included, and the test can only be conducted for one combination $f,p$, and then only when the criminal is freed from paying the full fine. In principle estimates of $P'[f,p]$ could come from a full set of contingent markets valuing payments $P'$ by a criminal, conditional on particular possible $f,p$ combinations being randomly chosen for testing on that criminal. But in practice such markets might well be too thin to be useful. Worse, criminals might have incentives to bid too high when the law fixes punishment, in order to exaggerate their risk aversion. While this might cost them on the current test, it might induce future markets to raise estimates of criminal risk, and hence induce lower fines, for ``similar" future crimes. The effectiveness of this strategic bidding signal would depend of course on how easily markets could correlate current crimes with previously tested criminals, which would depend on exactly when the markets were asked. Signaling could be direct if markets were asked to estimate $Pn[f,p]$ after criminals have been identified and convicted, but would be harder if markets were asked just after a crime was reported. But such signaling may still be possible, particularly if criminals could join into larger cooperative groups with distinguishable modes of operation or geographic areas. Such signaling might become harder if the information available to agents about a crime were limited, but this would reduce the quality of the agent's estimates of criminal risk aversion, avoidance costs, etc. In the worse case, a broad cultural preference might form to always demand a $P' >> P$ when tested. While I do not know a general way to escape these problems if laws fix punishment, I do know of plausible approaches if laws fix restitution. So let us turn to that case. \section{Fixing Restitution} In Friedman's proposal, social loss $L[P,R]$ is minimized by legally fixing the optimal punishment $P$, and then letting the enforcer trade $p$ vs. $f$ to maximize restitution $R$. However, we could similarly have the law fix an optimal restitution $R$ and have the enforcer trade $p$ vs. $f$ to minimize punishment $P$. Instead of having the victim, who seeks maximum $R$, hire the enforcer, we could instead have the criminal, who seeks minimum $P$, hire the enforcer. And a criminal placed in charge of all three parameters ($a,e,f$) might be induced to choose all three for a simultaneous social optimum, instead of just the $e,f$ optimized when the victim is in charge. Minimizing $P$ fails to minimize $L$, however, if $d_P L < 0$, so that great harm to the victim from lack of criminal care justifies higher punishment even without higher restitution. It also fails if lack of criminal cooperation with enforcement makes restitution fall with rising punishment. And this approach may assign risk from uncertain enforcement costs to the party least able to afford it. Restitution R can be fixed by fixing the amount that the enforcer must pay the victim after reporting the crime. As mentioned before, if the victim must give a standard detailed crime report, and if the enforcer must pay a standard wage for victim testimony, etc. then victim effort costs might reasonably be neglected. And victim risk can be reasonably neglected, even if enforcers are allowed to give a small percentage of restitution via payment contingent upon conviction. Of course criminals in hiding may not want to reveal themselves to explicitly choose $f$ and $p$. However, this problem can be dealt with by introducing a criminal's agent, who has clear incentives to do whatever the criminal would want. If a criminal could somehow indicate clearly ``at the scene of the crime" who they wanted for an agent, there is no reason why that agent couldn't have complete discretion about who is hired as an enforcer, what fines they are promised, how those fines are constrained relative to probabilities of conviction, etc. All it would need to do was pick an enforcer who guaranteed the legally set amount of restitution $R$ to the victim. The criminal would have an incentive to pick an agent who would set terms to minimize the expected punishment to the criminal. If the agent could not find an enforcer willing to pay the full restitution under the constraints the agent usually preferred, then the agent would have to relax those constraints until some enforcer was so willing. And if no enforcer was willing under any constraints to pay the full restitution, then the agent would have to accept whatever enforcer offered the victim the most restitution, regardless of the promised fine, etc. Here, with restitution fixed, enforcers could have direct incentives to attend to the intangible desires of criminals for respect and privacy in the enforcement process (assuming agency problems are managed), but not directly to similar desires of victims. Victims could, however, directly pay enforcers for such consideration, or make their degree of cooperation in obtaining a conviction visibly contingent upon it, an option not easily available to criminals in hiding when punishment is fixed. When restitution is fixed instead of punishment, problems such as criminal bribes to enforcers to avoid conviction are internalized to the criminal/agent/enforcer group, and so do not directly distort the law's handle on crime in the way such bribes do if punishment is fixed. The criminal would want to choose avoidance $a$ to reach a social optimum, if it could convince the enforcer to bid based on a compatible estimate of $p[e]$, the probability of conviction given that enforcer's effort. An enforcer who expects a cooperative criminal will expect that a lower fine is sufficient to achieve a given fixed restitution to the victim. While the criminal's choice of $a$ for this crime is unlikely to influence this enforcer's estimates, a distinguishable cooperative community of criminals, with similar victims, modes of operation, etc., might create and be rewarded for a reputation for cooperation with law enforcement. \section{Appointed Agents} When the criminal can not directly designate an agent, one would need to be appointed, and such appointed agents would need to have clear incentives to help minimize the criminal's expected punishment, and to not be representing other interests, such as that of the enforcer. One way to help align agent interests with that of criminals would be to give agents a stake in the punishment, by testing criminals a fraction $z$ of the time, as described above, and making an appointed agent responsible for paying (some multiple of) the same amount $P'$ (also to the enforcer) that (half of) the criminal's test group pays (to avoid the probability p of again owing fine $f$). This requires some validation of a probability $p$, such as the betting suggestion above. To give the agents as strong an interest in the criminal's punishment as the criminal has, and to (mostly) correct for the fact that the probability of conviction affects the probability of a test, appointed agents should have to pay $P'/pz$ the enforcer instead of just $P'$. To avoid bias through how the agents are appointed, the job could just be given to whoever offers the lowest bid $Y (~= P'_{\mbox{min}})$ to take on the job, (though bidders should be restricted to those passing tests of clear independence from other interested parties). $Y$ would be paid by the enforcer once chosen. To avoid bribes from enforcers to induce agents to choose them, appointed agents might be required to choose the enforcer based on an auction with explicit untainted criteria. For example, the agent could publish a function $Pn[f,p]$, a best estimate of criminal risk preferences, and agree to accept whichever enforcer offered the combination $f,p$ with the lowest estimated $P$. In this case, enforcers who overestimated $p$ should lose the competition, so that competition again suppresses both over and under estimation, as long as restitution increases with increasing punishment. The agent should want to set $Pn[f,p]$, or whatever criteria are used to select an enforcer, to induce the lowest possible $P'$ if the criminal is tested. And it turns out that if the criminal is not going to be cooperative in setting $a$, it does not matter for enforcer incentives that $P'$ is not the same as $P$, since all the terms in Eqn. [3] describing this optimum are the same if $P$ is replaced by $P-a$. (If one wants to fix $P$ though, instead of minimizing it, not knowing $a$ is still a problem.) If, however, the criminal might be cooperative, then we want an appointed agent to pay an estimate $a'$ of avoidance $a$ when a criminal is tested in addition to paying $P'$ (all to the enforcer). While the court convicting the criminal might be able to make some crude estimate of $a$, we might do better to just accept the criminal's estimate $a'$, as long as it were within a broad range deemed plausible by the court. This ability of the criminal to arbitrarily reward or punish the appointed agent would give an added incentive for the agent to want to please the criminal. In estimating $a'$, criminals might also take into account other costs such as reputation risk losses and general mistreatment from the enforcement process. An agent who has good reason to believe that the criminal would have wanted to appoint them, if only they could have so signaled, could expect the criminal would reward them with a low $a'$ estimate, and so could bid lower for the job because of its expected lower cost $Y ~= P'+a'$. In this way the law could let criminals often get the agents they want, without explicit legal communication on the subject. With restitution fixed, criminals should no longer have an interest in exaggerating their risk aversion; they should not want to game this institution because they want it to work, encouraging agents to pick enforcers who help minimize criminal punishment. Thus criminals as a community should be free to choose the parameters $z,n$. Testing more often allows smaller organizations to become agents, because of the reduced risk, but raises average estimated punishment levels by making conviction harder to obtain, $p[a,e] \rightarrow (1-z)p[a,e]$. Unfortunately, such enforcer auctions cannot directly give enforcers incentives to attend to the intangible preferences of either victims or criminals, though victims could plausibility negotiate for such consideration. Bribes, however, would only distort validations of probabilities $p$ bid by each enforcer. This might allow one enforcer to unfairly win over another, to the detriment of the criminal, but should not distort the restitution fixed by law. When punishments are small, and each enforcer has the same percentage of bribed crimes, then bribes may actually distort very little. \section{Trading Punishment vs. Restitution} In general, the enforcer's bid determines $P$ and $R$ when the enforcer is chosen, and so risk due to uncertainties in enforcement costs after this point is taken by the enforcer (for whom it is presumed relatively costless). However, the criminal and victim can still suffer risk, at the time they choose their efforts $w,x$, about what the enforcer will bid. To deal with this, a general solution might be to give both parties a piece of that risk by specifying an official legal social loss function $Ln[P,R]$, having an agent set $Pn[f,p]$ (constrained to satisfy $d_f Pn[f=0,p]=p$), and then extending the above approach to selecting enforcers by auction to pick the enforcer whose bid $p,f,R$ minimizes $Ln[Pn[f,p],R]$. Instead of fixing fines, punishments, or restitution, this approach can be said to more directly minimize loss. In order to avoid distorting the incentives of the agent who estimates a case-specific $Pn[f,p]$, that agent would have to be more of a social agent than a criminal's agent, paying $Ln[P'+a',R]/pz$ when tested instead of just $(P'+a')/pz$ as before. Even so there could remain incentive problems for the criminal similar to the case where one is trying to measure criminal risk aversion when punishment is fixed. A cooperative group which can signal its presence through information available about a reported crime could act more risk adverse than they really are when tested, and thereby reduce future fines imposed. I don't know of a robust general way to avoid this problem, though there remains the possibility that it might not actually be much of a problem in practice. This approach would deal with both conditions $d_R L < 0$ and $d_P L > 0$, where the fixed approaches fail, though these cases could also be dealt with by fixing punishment for some crimes and restitution for others. And beyond this, it's not clear that this more general solution really buys much, especially considering how the added complexity might increase costs to negotiate a law. The simple model behind Eqn. [1] suggests that all risk should be given to the victim, if possible. And which ever party suffers this risk might in principle insure against this variation, even if general insurance against crime penalties were disallowed. If a neutral agent, representing neither criminal nor victim, is to be in charge of hiring the enforcer, then both restitution $R$ and punishment $P$ could be fixed at their prior expected values. The enforcer offering to pay the highest amount while fixing both $R$ and $P$ would be hired. This amount would be sometimes positive, sometimes negative, but supposedly average to zero. This fee might be paid to a victim's insurer, who paid or was paid by the victim for this right. When the average amount paid across the legal community for a particular type of crime were non-zero, that would be a signal for the law to adjust their official estimates of $R$ and $P$. This approach of course still has a problem with inducing criminals to reveal risk preferences while fixing punishment, or to cooperate in encouraging a conviction. \section{Contingent Law} To minimize the externality costs of crime, it is crucial that the court set punishment $P$ or restitution $R$ with the best possible estimates of the benefit $B$ to the criminal, the damage $D$ to the victim, and their relative risk-aversion and influence. The court ideally wants to know as much as either party could plausibly have known when choosing efforts $w$ or $x$. The best time to make these assessments is after a conviction, but the law needs to declare its policy before efforts are made. A solution is for the law to be contingent, declaring different punishments or restitutions contingent on factors which the criminal and victim might have known before choosing efforts, and which the court could later find out as a conviction is obtained. A function $Ln[\set_i P_i,R_i ]$ would be a general contingent law, with each type i describing another possible legally anticipated contingency. Contingent law increases legal uncertainty for parties who do not know the contingent factors, but allows a better tuned law for parties who do [5]. For example, law might be contingent on whether the criminal showed intent, on victim expenditures on crime prevention, on whether a similar effect could have been obtained voluntarily, on the state of knowledge of either party, etc. In addition to contingent types for setting punishment and restitution, enforcers may want to break these law types into further enforcement subtypes, to distinguish different classes with different enforcement costs. Some types may be able to afford higher fines, i.e., have a higher $G_i[f]$, some might be more likely to have a high risk aversion, and others may be more difficult to track down and convict. For example, the asset mix of a person, might be informative about difficulty of collection and risk-aversion. And criminals with prior convictions or who cross state lines to avoid being caught may be more difficult to catch. When restitution is fixed, enforcers should want to pick types to distinguish criminals likely to be cooperative in picking avoidance $a$. For example, criminals who seemed to immediately turn themselves in should have a high probability of conviction with low enforcer effort. Like ordinary insurers trying to discriminate customer types with different levels of risk, enforcers should be free to use any subtypes they find useful, as long as these types can be determined by a court if the criminal is convicted (and if the cost for this determination is borne by the enforcer). A serious problem with the non-contingent approach to law described above is that a ``victim'' may falsely claim a crime has occurred. Contingent law can deal with this by introducing special false crime contingencies, here labeled them $i0$. Contingent law must specify a zero fine and punishment ($f_{i0} = 0$) in this case, since there is no criminal to be fined, and thus enforcement costs must be paid for by the victim through a negative restitution $R_{i0}$. Contingent law might also allow law to account for the costs of enforcing crimes, such as embezzlement, where significant regular enforcement costs must be expended to even discover that a crime has occurred. One might regularly hire an enforcer to try and uncover such crimes, expecting to usually have to pay negative restitution $R_{i0}$, but recovering crime detection costs if a crime occurred. This should of course only pay for detection costs, and not for efforts to prevent crime via threats of punishment. Since each party's ability to avoid or encourage crime can depend on how much they know about the crime's type, the type i might also encode the court's final judgement on how much the criminal or the victim knew about the crime type. Since crime types are only officially revealed by a court declaring a type when a conviction is obtained, the law can't easily give the victim a type-dependent restitution $R_i$ regardless of whether a conviction is obtained. Instead the law can in general only offer the victim a take $t_i$ when there is a conviction declared to be of type $i$. Thus the victim may now suffer conviction risk as well as the criminal, having a small chance of winning a possibly large take. Thus enforcement costs may now include a term for victim risks imposed, and estimates of victim risk would have to be validated, such as with a similar random testing approach (more about this below). With probabilistic restitution, bribes can now distort fixed restitution, as well as fixed punishment, by raising restitution and punishment above their legal values. This is in the interest of the victim, but not the criminal, suggesting that allowing the victim discretion to hire the enforcer, as in fixed punishment, might lead to more bribe problems than having the criminal hire the enforcer, as in fixed restitution. With proper competition, so that each enforcer offers their best deal in all possible contingencies, we expect zero enforcer profit in each possible contingency. So the enforcer's profit equation for each contingency now implies \[ g_i t_i = R_i = g_i G[f_i] - e_i \] \noindent where $g_i$ is the enforcer's estimated probability of a conviction conditional on the type $i$, and where $e_i$ is the effort the enforcer expects to have made by the end of their attempted pursuit of a criminal of type $i$ (meaningful even if the criminal is never caught or the crime type never revealed). Summing across the contingencies, the enforcer expects to pay the victim $\sum_i p_i t_i$, and to gain $\sum_i p_i G[f_i]$ from the victim. Here $p_i$ is the enforcer's estimated joint probability of getting a conviction and having it declared to be of contingency type $i$ (with total conviction probability $p = \sum_i p_i$). The ratio $p_i/g_i$ gives the probability of each type $i$. \section{Contingent Risk} Eqn. [2] can be directly generalized to give contingent punishment as: \[ H_c[P_i,-B_i] = H_c[a_i,-B_i] + g_i H_c[f_i,a_i-B_i]. \] \noindent This is not the same as the punishment the criminal expects after the crime occurs, since the criminal may not know the crime type, but this contingent punishment is what the law may want to fix to optimize incentives in choosing initial efforts $w_k$ and $x_l$. Testing of criminal risk-aversion would go as before, now using $f_i$ and $g_i$ in place of $f$ and $p$. At first glance, it seems that contingent law also imposes significant new costs of victim risk, requiring a similar random measuring approach to validate estimates of victim risk aversion. And if the law fixes restitution, there are incentive problems in getting victims to reveal their true risk aversion, similar to the problems of getting criminals to reveal risk aversion when punishment is fixed. Actually, such problems are much worse here, because a victim openly complaining of a crime can be much more easily identified to be the same victim as in a previously tested crime. This makes much more direct the reward from faking a high risk aversion when tested. Under contingent law, an uninsured victim suffering conviction risk would have restitution given by $H_v[-R_i,D_i] = g_i H_v[-t_i,D_i]$. However, victims could insure against this conviction risk by instead accepting from the enforcer a restitution $\ave{R} = \sum_i p_i t_i$. Adverse selection could be limited if the victim commits ahead of time to accept this restitution $\ave{R}$, and a law might even insist on this. (Though a small contingent percentage to encourage victims to aid enforcement might be allowed.) If a victim told potential enforcers most everything they knew about the crime, and if the enforcers believed that victim, then the enforcer bids $p_i$ should be no worse estimates than the victim's original estimates, and therefore victim incentives to choose defense $x_l$ carefully would suffer little from this insurance. Risk due to the variation of the enforcers estimate $\ave{R}$ from the victim's original estimated $R$, should typically be much smaller than the risk from just owning the right to be paid $t_i$ if a conviction is obtained. A victim who committed ahead of time to be paid $\ave{R}$ might want to only reveal information relevant for estimating $p_i$ when that would raise the estimated $\ave{R}$. Others estimating $p_i$ might then presume, often erroneously, that lack of detail from the victim indicates a less serious crime. This might then induce too much effort to collect detail about an accused crime. Still, in the absence of better ways to estimate victim risk, these may be reasonable prices to pay for allowing contingent law, and so we may just assume $R_i = g_i t_i$. And since the victim suffers the cost of these problems, it seems conceivable that the victim could solve at least part of through through a reputation for being forthright. For a victim believed to be forthright, the enforcer estimate $\ave{R}$ would embody everything the victim knew. Incentives to gain a reputation as a forthright victim may also result in incentives to aid the enforcer in obtaining a conviction, even without a conviction-contingent percentage of payment. \section{Contingent Punishment Or Restitution?} When a general social loss $L$ can be split into separate contingent losses $L_i$, a general law might appoint the enforcer whose bid $\set_i (f_i,t_i,p_i,g_i)$ minimized expected social loss \[ \ave{L} = \sum_i (p_i/g_i) Ln_i[ Pn_i[f_i,g_i] , Rn_i[t_i,g_i] ] \] \noindent where $p_i/g_i$ is the probability of the crime being of type $i$, $Ln_i[P,R]$ is set by law, and a set of functions $Pn_i$ and $Rn_i$ might be declared by a criminal's agent and a victim's agent respectively. And as discussed above, it might be reasonable to assume that all $Rn = g_i t_i$. Strong competition between enforcers should force zero profit in each contingency. But again such a general law could have problems inducing criminals (and victims if victim risk is included) to reveal risk preferences. If the law were to fix contingent punishment, then victims would shop for the enforcers they think offer the highest expected restitution, using victim's beliefs about crime types, if contingent restitution were allowed. If accepting some average $\ave{R}$ were required, they would just shop for the highest offered average. It is not clear how to validate a set of official punishment functions $Pn_i$ in this case. If the law were to fix contingent restitution $R_i$, it might ignore victim risk because of the presumption of insurability described above. In this case, an appointed criminal agent (lowest bidder in an action to become such) would publish a set of punishment functions $Pn_i$. The agent would then accept that enforcer whose bid $\set_i (f_i,p_i,g_i),\set_{i0} t_{i0}$ promised the lowest expected punishment \[ \sum_i (p_i/g_i) Pn_i[f_i,g_i] - \sum_{i0} p_{i0} t_{i0} \] \noindent while promising to pay the victim either a contingent take $t_i = R_i/g_i$ or a certain restitution $\sum_i p_i t_i$, with the $R_i$ (except the $R_{i0}$) fixed by law. Note that if $P,B,a$ are all small compared to the criminal's wealth, punishment can be approximated as $a_i + g_i H_c[f_i]$, and so the winning enforcer minimizes $\sum_i (p_i H_c[f_i] + a_i/g_i)$. When $a$ is roughly independent of $i$, such as when the criminal doesn't know the type $i$ any better than the market, then punishment depends only on $p_i$, and no longer directly on $g_i$. Note that it is possible for only enforcement to be contingent, and not the law itself. In this case fines and probabilities would be contingent, but with fixed punishment the criminal would always suffer the same net punishment, and with fixed restitution the victim would always receive their legal payment soon after reporting a crime. The law could avoid the complexities of contingency, while still allowing the enforcement process to take advantage of it. \section{Contingent Probabilities} For either restitution or punishment fixed, what the law fixes depends only on the $g_i$ estimates and not on the $p_i$ estimates. However, the $p_i$ are perhaps more the focus of the competition between enforcers. Joint probabilities $p_i$ are somewhat more difficult to validate than the total conviction probability $p$ was. Validation by bundle frequency requires even more available cases to form similar bundles, and bundles need to be chosen after punishment or restitution is declared, even though contingent types are not revealed until conviction. If, however, enforcers validate $p_i$ estimates by offering to bet anyone at those odds, they run the risk of betting against an informed criminal or victim. Such bets could allow a criminal to effectively pay a much lower fine if convicted. One solution is to only allow anyone to bet on the total $p$, but only allow bets on $p_i$ by other serious bidders in the auction to become the enforcer, a set unlikely to contain the criminal or victim in disguise. Imagine that the winning bid for enforcement used a set of probability estimates $\set_i p_i$ such that had some subset $J$ of those been $\set_{i\mbox{ in }J} p'_i$ instead of $\set_{i\mbox{ in }J} p_i$, that winning bid would have lost to another bid in the auction. In this case we could allow that losing bidder to challenge the winner, claiming that those $p'_i$ were a better estimate than the $p_i$, by betting at probability $p'_J = \sum_{i\mbox{ in }J} p'_i$ that a criminal would be convicted and the contingency type would fall within $J$. Again this offer would only be open for a short time, and would be naturally limited in amount. If restitution falls as punishment rises, so that the enforcer would rather overestimate $p_i$, then such losing bidders could similarly be allowed to bet against a conviction. Estimating $g_i$ is more difficult still. The complication is that while bets can be clearly settled about the types of convicted crimes (since the court is supposed to label crimes with a conviction), allowing validation of $p_i$ estimates, the conditional probability $g_i$ also includes information about the types of unconvicted crimes. To validate $g_i$, one wants a way to find out the type of a reported but unconvicted crime, and do so with near certainty. One approach would introduce a very small chance $y$ that a reported crime will be tested, and then offer a very large prize $Z$ to any person or group, including the criminal or victim, who can demonstrate the type of this tested crime to the court's satisfaction before a declared (and distant) deadline. If one auctioned (shares in) obligations to pay off the prize for each type (before choosing the enforcer), then each total obligation should sell for about $y(1-b)Z(p_i/g_i)$, where $b$ is the probability that no prize will be claimed. Combined with estimates of $p_i$ these should give estimates of $g_i$. Underestimates of $p_i/g_i$, via overestimates of obligation prices, should be directly discouraged by competition in the auction for each prize obligation. These estimates should sum to one (since $\sum_i p_i/g_i = 1$), so overestimates are also discouraged if anyone could create more bundles of co-prize shares (paid to those who issue them) for all the contingencies, and sell them in the same auctions. These bundles would have to also include a special non-prize awarded if no other prize is awarded. These non-prizes are actually useful in letting the law specify a given accuracy of measurement $b$. The auctions could simultaneously set $Z$, raising it however high is needed so that the relative price of non-prize shares falls to $b$ times that of a bundle of prize shares. Alternatively, a separate prior auction could set $Z$ by offering to give a co-prize of $Z$, paid whenever someone else wins a basic prize, to the bidding $Z$-validator that offers to pay a non-prize $Z(1-b)/b$ should no one collect a basic prize, and who is willing to do so for the lowest prize amount $Z$. The law might limit such bidders to those unlikely to include the criminal or victim in disguise, such as to the serious enforcement bidders, but it is not clear this would be needed or desired. Type-wise criminals or victims could earn up to $\simeq yZ$ by buying obligations to pay on types they know are not the true type (or more if extra prize bundles are sold), and in the process would give enforcers information which might aid in catching criminals. So as long as $y$ is small enough, this need not distort the legal process greatly, though with $y$ too small $p_i/g_i$ may be estimated poorly. In this approach, then, the $p_i/g_i$ estimates are not part of enforcer bids, but are made prior to the auction for enforcer. As before, payments which must be made before an enforcer is hired can be made by the complaining victim, who would then gain the right to collect that much more later from the chosen enforcer. Like the parameters $z,n$ describing how criminal's are tested, free parameters in this approach such as $y,b$ should be chosen by criminals if restitution is fixed, by victims if punishment is fixed, and by a more expensive combined social negotiation if social loss functions are minimized. \section{Comparing Approaches} Three plausible approaches to private law have been described here. The law can fix either criminal punishment, victim restitution, or it can set a more general social loss function of the two. All these options seem preferable to the familiar forth alternative of fixing criminal fines. This paper has compared these approaches, and in the process hopefully illuminated some basic issues in designing institutions of private law enforcement. Assuming each possible crime is independent, and ignoring economies of scale and third-party externalities, we find that a criminal and a victim should, to reduce externalities between them, want to agree on a law. Before they each choose how much they want to work to encourage or discourage the possibility of a ``crime", they want to agree on some wealth transfer plus a legal system which forces the criminal to pay the victim a certain fine if the crime happens and the criminal is convicted. This fine should be such that the resulting shares in the crime, the wealth change for each party if the crime happens, are both non-zero, the same sign, and somewhat larger for the party with more influence and better information. If this is the criminal, then the punishment and restitution should be closer to the level of the victim's damage than to the criminal's benefit, and the damage level should be easier to measure. Risk due to uncertain enforcement costs would ideally allocated via a general social loss function $Ln[P,R]$. But if just one of the criminal or victim is to suffer this risk, a simple model suggests that it should be victim, and that giving the victim all this risk may be ideal. This argues for fixed restitution. However, if the criminal is not given the enforcement risk, i.e., the residual after paying actual enforcement costs, then it is hard to induce them to cooperate in estimating their risk preferences or in encouraging a conviction. This argues for preferring fixed restitution. The mostly subcontracted efforts of victims to promote conviction, in contrast, can be more easily encouraged, and victim risk can be more easily insured against. Both fixed restitution and punishment suffer the possibility of regions of $L[P,R]$ where they fail to actually minimize L, though fixed restitution seems more likely to actually suffer this problem. Moving beyond having the law simply fix fines requires validation of probability estimates, and this paper suggests betting auctions as an improvement over watching frequencies in bundles of like crimes. Moving beyond fixed fines also requires validation of risk aversion estimates, of the criminal in general and perhaps also of the victim when there are different difficult to insure contingent restitutions. This paper also suggests a method for validating risk estimates. However, if punishments are fixed then it may be difficult to measure criminal risk, if restitutions are fixed then it is surely difficult to measure victim risk, and both may be difficult to measure when a general social loss function is fixed. However, the feasibility of insurance suggest we can neglect victim risk, and argues for fixed restitution. The possibility of bribes from criminals to enforcers can make it difficult to legally fix punishment, but only affects contingent, not uniform, restitution. Even then, it seems better to place criminals in (indirect) charge of managing enforcement, through fixed restitution. With criminals in charge we can expect better attention to their intangible desires, while with victims in charge their preferences should be attended to. However, since victims should in any case be able to buy or negotiate for such attention, this consideration also prefers fixed restitution. The option to set some general social loss functions of punishment and restitution might allow better allocation of enforcement risk, but would require some social agent be placed in charge of enforcement, instead of either the victim or criminal. Thus intangible criminal preferences might again be neglected, as might criminal aid to enforcement. And criminals here again can have incentives to distort tests of risk aversion. Details of the auction and testing processes would require broader social negotiation, and social negotiations to estimate official $Ln[P,R]$ functions might be considerably more complex. So which approach is better? If we presume against the more complex social loss function (on admittedly shallow grounds), we face a choice between fixing punishment or restitution. Since only fixed restitution currently has a plausible way to estimate criminal risk or to induce criminal cooperation with enforcement, then for desired punishment not too high, fixed restitution seems tentatively the best approach. \section{Getting There From Here} So far I have described some institutions within which law might be privately enforced, and argued the plausibility that these institutions may be more efficient than either previous proposals or existing institutions. However, while there may be a place for evaluating concrete radical proposals, it is also reasonable to ask whether there are incremental paths toward such changes. Private contracts could of course specify the use of alternative dispute resolution, such as arbitration or private courts, which use such procedures (as long as their procedures were legal). However, victims might be reluctant to use the above procedures at the time they become aware of a crime if they did not expect that the unknown criminal had probably signed a related contract. This might be less of a problem within the scope of a private community. Criminals guilty of many ordinary crimes can also be sued for related torts, though jurors are unfortunately usually reluctant to award damages without a criminal conviction. More fundamentally, however, the above approaches rely heavily on the victim being able to insure against the risk of failure to convict, usually by selling their award rights to an enforcer. And selling such rights is currently forbidden (at least in personal injury cases - I don't know for sure about wider prohibitions). The betting approach to validating probabilities unfortunately may requires the relaxation of pervasive anti-gambling laws. However, the above descriptions in terms of buying shares of enforcement or prize obligations may help deflect such an attack. Frequency bundle methods would not suffer such a limitation, but the need for a minimum number of similar cases would make difficult the spontaneous birth of such an enforcement industry. Once allowed, however, we might hope that for torts with a low probability of convicting the offender, courts would come to treat enforcers like the victim's lawyer. That is, in addition to awarding compensation for lawyer fees, compensation could also be awarded for enforcer fees, and for insurance against the possibility of a failure to convict. And what better way to demonstrate the reasonableness of the fees than an explicit auction intended to obtain the direct damages appropriate to the crime with a minimum harm to the criminal? Initially, such validation might substantially increase the chance that the court would actually award compensation for such costs. Eventually, there might arise a presumption against compensating such costs in the absence of such validation. Uncertainty about the actual direct damage awarded could be handled in part by contingent enforcement, with different probability of conviction estimates for different possible ranges of damages awarded. \section{References} [1] Gary Becker, George Stigler, ``Law Enforcement, Malfeasance, and Compensation of Enforcers", 3 J. Legal Studies, 1 (1974). [2] David Friedman, 'Efficient Institutions for the Private Enforcement of Law', J. Legal Studies, June, 379-397 (1984) [3] William Landes, Richard Posner, ``The Private Enforcement of Law", 4 Legal Studies 1 (1975) [4] Mitchell Polinsky, Steven Shavell, ``Enforcement Costs and the Optimal Magnitude and Probability of Fines", J. Law \& Econ v35, 133-148. (1992) [5] David Friedman, ``Reflections on Optimal Punishment, Or: Should the Rich Pay Higher Fines?'' 3 Research Law \& Econ. 185 (1981) [6] Richard Posner, Economic Analysis of Law, 4th Ed., 1992 [7] Bruce Benson, The Enterprise of Law, 1990. [8] Robert Cooter, Thomas Ulen, Law and Economics, 1988. [9] Mitchell Polinsky, An Introduction to Law and Economics, 1983. [10] David Friedman, The Machinery of Freedom, 2nd Ed. 1989. \section{Acknowledgements} The following people offered useful comments on portions of previous drafts of this paper: Tom Bell, Rebecca Crowley, Tim Freeman, David Friedman. \section{Glossary} (of terms appearing in multiple contexts) \begin{tabular}{lll} $d_u$ & $=$ & partial derivative operator, done relative to subscripted parameter \\ $\ave{u}$ & $=$ & the expected value of some variable parameter \\ $H$ & $=$ & harm to criminal, inverse of utility \\ $B$ & $=$ & benefit to criminal \\ $D$ & $=$ & damage to victim \\ $q$ & $=$ & probability of crime happening \\ $w$ & $=$ & watchfulness, effort by criminal to encourage or prevent crime \\ $x$ & $=$ & defense, effort by victim to prevent or encourage crime \\ $P$ & $=$ & punishment of criminal, certainty-equivalent cash \\ $R$ & $=$ & restitution to victim, certainty-equivalent cash \\ $C$ & $=$ & criminal's loss from crime given law \\ $V$ & $=$ & victim's loss from crime given law \\ $L_v$ & $=$ & victim's total loss \\ $L_c$ & $=$ & criminal's total loss \\ $L = L_v + L_c$ & $=$ & total loss victim and criminal negotiate to minimize \\ $Q$ & $=$ & ratio of restitution vs. punishment rates of loss increase \\ $W$ & $=$ & wealth level of criminal \\ $f$ & $=$ & fine criminal owes victim given conviction, cash due \\ $t$ & $=$ & take victim is paid given conviction \\ $G$ & $=$ & what enforcer actually gets from victim, certainty equivalent \\ $p$ & $=$ & probability of catching and convicting the criminal \\ $a$ & $=$ & effort by criminal to avoid being convicted \\ $e$ & $=$ & effort by victim's enforcer to catch and convict criminal \\ $Pn$ & $=$ & official function estimating $P$, given by law or criminal's agent \\ $Rn$ & $=$ & official function estimating $R$, given by law or victim's agent \\ $Ln$ & $=$ & official function estimating $L$, given by law \\ $z$ & $=$ & probability of testing criminal's risk aversion \\ $n$ & $=$ & number of criminals pairs tested as a single group \\ $Y$ & $=$ & amount appointed criminal's agent paid for services \\ $i$ & $=$ & type of crime \\ $g$ & $=$ & probability of convicting criminal, given crime type \\ $Z$ & $=$ & prize amount to one who informs about crime type \\ $b$ & $=$ & probability that no prize will be claimed \\ \end{tabular}