[This paper is a draft. It may be circulated to interested parties without charge, but it should be understood that it may not reflect the author's current opinion, nor is the author willing to stake that much of his reputation on any one claim made in it.] SHOULD PRIVATELY ENFORCED LAWS FIX PUNISHMENT OR RESTITUTION? (or, Why Not Let Convicts Hire the Police?) By Robin Hanson Draft 1/15/93 ABSTRACT Privately enforced law might fix punishment, restitution, or some combination of the two. Such law can be uniform, depending only on the victim's complaint, or it can be contingent, depending on crime types determined after a conviction is obtained. After reviewing the general problem of private law design, I compare these approaches under both uniform and contingent law, comparing efficiency, ability to reveal risk preferences, susceptibility to bribes, and ability to consider less tangible preferences, such as for privacy. 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 possible new legal institutions, alternatives to our current institutions. This is disappointing because, after our academic clouds of analysis settle (if ever), our main general choice will remain, in law as in most economic contexts, a choice between institutions. And we risk this choice being an empty one if we have not bothered to imagine specific plausible alternatives to existing legal institutions. Fortunately, there are exceptions to this trend, such as scattered proposals for alternative *private* legal institutions. For example, Gary Becker and George Stigler have proposed [1] we privatize law enforcement, a suggestion elaborated on by David Friedman, who offers arguably the best current proposal in [2]. Friedman considers the possibility of competing private law enforcement agencies in the context of traditional state monopolies in law courts and legislation. Each private law 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. If the expected punishment is set at its optimum value, Friedman shows that enforcers have incentives to spend the optimum amount on enforcement. While Friedman's proposal is certainly an improvement over previous suggestions, it still seems less than ideal, 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 even better) alternative private law institution, where laws fix a specific level of restitution for each type of crime, rather than a punishment or fine. A criminal's agent estimates that criminal's risk preferences, and chooses an enforcer who offers the minimum expected punishment while guaranteeing the victim the restitution due. Specific institutions are described for validating case-specific probability estimates, without needing like crime bundles, and for validating the agent's case-specific estimates of criminal risk when that agent has not been directly chosen by the criminal. This approach can consider the intangible preferences of both parties, and does not allow bribes to distort the parameter that the law fixes. A fourth and more general approach would have the law determine a social loss function of punishment and restitution to be minimized, and have the law directly hire the enforcer whose bid minimized this loss. One might have doubts whether the process by which laws were fixed could be trusted to deal with such complexity, and this approach would less consider criminal intangibles. But it would deal better with uncertainties in 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 court-determined properties of the crime and criminal. 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 the proper context. Let us call a "crime" between two people (or parties), an event which: 1) Changes the utility of both people 2) May occur even if one or both person objects, and 3) One or both of them can influence the chance of it occurring. If the crime happens a "criminal" will receive some benefit B, and a "victim" will suffer some damages D. (B and D are calibrated as the amounts of money that induce the same utility changes, and we choose the person labels so that B+D is positive.). The probability q that the crime will occur can be influenced by "wariness" costs w of the criminal, and "defense" efforts x of the victim, and in general q = q[B,D,w,x], a function of all these parameters. 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 sometimes even be an improvement to the car. Thus the net social benefit 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 have externalities. The criminal will choose wariness w to minimize loss L_c = w - qB, while the victim will choose defense x to minimize loss L_v = x + qD. 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 L = L_v + L_c. Thus the criminal here 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. (Note ASCII math key: "_" starts a subscript, "^" a superscript, as in A_i^2.) To deal with this problem, these two parties would prefer to somehow 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 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 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 future legal obligations.) The criminal's "share" in the crime is now C = P-B, instead of -B, and the victim's share is now V = D-R, instead of D. The losses now become L_c = w + qC L_v = x + qV and the externalities 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, the victim's expectation of his or her loss, and w minimizes , the criminal's expectation of his or her loss. Note that what matters for minimizing social loss is not so much the actual values of transfers P and R, but what the criminal and victim estimate them to be when choosing efforts w and x. This analysis treats each possible crime independently, ignoring possible economies of scale of enforcement or deterrence. And this analysis ignores third party externalities, such as from convicting innocent persons, or from inabilities to discriminate between differing criminals or victims. RELATIVE INFLUENCE To understand "relative influence" better, let us neglect any dependence of enforcement costs P-R on efforts w,x, and take a quadratic Taylor expansion of the crime rate q[w,x] around the socially optimum w,x, q[w,x] ~= (q_ww)*w^2 + (q_xx)*x^2 + (q_wx)*w*x + k (redefining the zeros of w,x to allow this simple form). Here q_ww is shorthand for D_ww q[w,x], the second partial derivative of q with respect to w, and similarly for q_xx or q_wx. If we also neglect enforcement losses P-R, and assume q[w,x] is common knowledge, social loss becomes just the externality loss V^2 C^2 C V q_xx ----- + q_ww ----- - q_wx ------ ^2 ^2 L = ----------------------------------------- (C+V) (4q_xx*q_ww - q_xw^2) where and are each party's expectations of their own crime share at the time they choose efforts w,x. (To derive this, express quadratic social loss in terms of the first derivatives at each point, and substitute derivatives of q obtained from externalities found when assume each acts to minimize own loss.) This formula places a premium on shared responsibility, so that neither party is at risk of expecting a null share, and on predictable shares, since for uninformed parties expected social loss would have terms like = ^2 + s_v^2, where s_v is standard deviation of victim's share V, and similarly for C and s_c. If both parties share the same information about their expected shares, and , and are uninformed about this particular case, and if s_c (or s_v) is independent of (or ), then loss is minimized for ^2 (^2 + s_c^2) q_xx --------------------- = ---- ^2 (^2 + s_v^2) q_ww We see that the optimal crime share ratio depends weakly on how sharply the crime rate q peaks relative to each party's effort, and on the percentage variation in each party's shares. The party with a smaller percentage variation and with a wider peak should get a larger percentage share of the crime. So if the dependence of social loss on defense x were much more strongly peaked than wariness w, as it might be if the optimal victim defense effort were much smaller than the optimal criminal wariness, then the criminal should get more of the payoff, with R closer to D than P is to B. If one must choose which party gets how much of some residual uncertainty in the crime, such as due to uncertain enforcement costs, which party should suffer that uncertainty? If r_c and r_v are the shares of this uncertainty, with r_c + r_v = 1, then using the above crime share ratio formula and assuming q_wx = 0, ^2 + s_c^2 r_v ------------- = --- ^2 + s_v^2 r_c When the criminal has a larger share, the the victim should suffer more of the uncertainty, since, in a quadratic model like this, errors in larger quantities are more serious. This quadratic model is perhaps informative about "relative influence", but surely also misses a great deal. Reasoning more qualitatively, we might guess that if the victim chooses its defense levels well before the criminal chooses wariness, or is less able to guess the law's final best estimate of the criminal's benefit B (or influence) than the criminal is able to guess the law's final best estimate of the victim's damage D (or influence), then these lacks of information may further dilute the influence of the victim relative to the criminal. In the extreme case where the criminal could simply choose whether or not to act against a helpless victim, restitution should exactly compensate the victim for their damages. Thus we may guess that we want C >> V, with P,R nearer to D than to B, and r_v >> r_c. DESIGNING LAW The two main tasks of the legal system are to set the levels of punishment P and restitution R to minimize externality losses M[P,R], and to implement this transfer with minimal enforcement cost P-R. While the focus of this paper is on the second task, that of enforcement, the 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, and relative influence. The criminal wants to downplay B, the victim wants to exaggerate D, and both want to downplay their relative influence for bad crimes, and exaggerate it for good crimes. One hopes, however, that broader social negotiations over law could settle on reasonable values for P or R, indirectly forcing the parties to reveal expectations about D,B and relative influence. When the courts can later uncontroversially observe various characteristics of the crime or the parties, then the law might set punishment and restitution contingent on these characteristics, hopefully indirectly revealing expectations about D,B contingent on these characteristics. Of course the negotiations over law might do poorly if there were too many such possible characteristics. Can damage D or benefit B be measured, even after the fact? The party who stands to benefit from the law's intervention, usually the victim (when R>0), is probably more easily examined than the party possibly hiding to escape conviction. (When the victim must pay the criminal, one may argue that the labels should be reversed.) If victim's defense efforts are done more in the open than criminal efforts, then victims may better reveal expected damages by whether they pay to increase or decrease the chance of crime at different levels of expected restitution. A more direct signal might come if victims could buy or sell changes in restitution level. Such open purchases by potential criminals, however, might focus unwanted attention on them in future investigations. 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. With just D=R, the victim loses incentives to adjust defense, and with P=B the criminal loses incentives to adjust wariness. 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 B is harder to measure and should be farther away. But if it 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 externality loss functions M_i[P,R], each contingent on some class i of crime and party characteristics. The law enforcement system would then be tasked with minimizing the sum of expected externality loss M plus enforcement loss P-R. Setting P=B is the same as making M sharply peaked around a particular value of P, but independent of R, and R=D makes M sharply peaked around a particular R, and independent of P. PURSUIT COSTS OF ENFORCEMENT The two main costs of law enforcement are the costs E to enforce a crime, i.e., to try to catch and convict a criminal, and costs from the risks imposed on criminals, who may suffer a small chance of having to pay a very large fine, risks they might find it expensive to insure against. Let us first consider the direct enforcement costs. Just because a criminal is legally required to pay a fine f, doesn't mean the enforcer will immediately receive a payment of this amount. Rather, the enforcer gets a certainty equivalent G[f] < f, with the difference f-G[f] due to delays, collection costs, and the possibility that the criminal simply does not have the means to pay. When the enforcer can, if needed, claim future criminal wages (without fear of bankruptcy), sell the criminal into slavery, or even sell the criminal's body to medicine, and if the criminal infinitely values his or her life, then there is clearly always some combination of minimal f and p to induce any possible desired punishment level P. If not, then punishments other than fines might sometimes be desired, and may incur net costs to punishers. In a private enforcement system, the private enforcer pays restitution to the victim, and pays for all court costs, though the court itself might not a private institution. The resulting total cost E[p,f,e] the enforcer incurs in trying to catch and convict a criminal varies strongly with both a (presumably hard to monitor) effort e, with the resulting probability p of conviction, and may vary weakly with fine f. Since the enforcer must pay restitution R to the victim, the enforcer's net loss (or negative profit) is R + E[p,f,e] - pG[f] (assuming the enforcer is large enough that risk in not an issue), which should average to zero. A net negative profit would bankrupt the enforcer, while we would treat a net positive profit as just a larger cost E; as an agent of the criminal and victim, the enforcers net gain is not a legitimate part of the social loss that those two parties would negotiate to minimize. Thus R[f,p] = p*G[f] - E[f,p,e] With private law enforcement, enforcers compete to gain the business of some party, such as the victim or the criminal, that benefits from lower enforcement costs. The point of private enforcement is to encourage enforcers to choose effort e to minimize the cost E[f,p,e]. This minimizes total social loss relative to e, because E is the only loss term e effects. With competition we expect no net enforcer profit and hence E is more literally the cost of enforcement. RISK COSTS OF ENFORCEMENT If, for a given crime, a criminal with a certain current wealth W has a probability p of being caught and required to pay a fine f, then that loss has a certainty equivalent wealth loss P[f,p,W] given by U[W,-P] = pU[W,-f] + (1-p)U[W], where U[] gives the criminal's utility for some combination of situation factors. Risk aversion makes P > p*f, and for all p, P_f[f=0,p]=p, i.e., P=p*f for low f. The wealth level W that matters here is the one the criminal used when choosing its wariness level w, i.e., the wealth W the criminal predicted it would have if the crime happened, namely the criminal's initial wealth, plus his or her crime benefit B, minus any costs he or she expects to incur while evading law enforcement. Thus the risk cost of punishment is about P[f,p,B] (where we now define the criminal's initial wealth level W to be zero). To insure against this risk, criminals would have to buy such insurance just after the crime was committed in order to avoid moral hazard problems regarding reduced incentives to adjust wariness effort w. Criminals would need to credibly communicate estimates of both p and f to the insurer, and even then an insured criminal would have moral hazard problem of reduced incentives to avoid being caught and convicted. Thus it seems that criminal risk can not be effectively insured away. Victim risk, on the other hand, can be effectively insured, at least in the simple case now considered. Once the victim has complained of a crime, there is no reason why an enforcer can't pay the victim a lump sum equal to their expected amount of restitution, without remaining moral hazard problems. Thus total enforcement costs are P - R = P[f,p,B] - p*G[f] + E[f,p,e], a function of parameters p,f and e. (P[f,p,B] is defined in the next section.) Our goal is an enforcement institution with incentives to set these parameters to minimize M[P,R] + P - R. FIXING EXPECTED PUNISHMENT Since the parameters p and f effect terms outside the enforcer's loss, a large externality would result if the enforcer could choose them at will. For example, having the law set a fine f and letting the enforcers compete to raise p or lower E has a worse problem than the difficulty of determining the right fine f. There would be no reason why enforcers would consider the added harm to the criminal from a larger probability p of conviction. This is the problem, posed by Landes and Posner, that David Friedman responds to in the following proposal. Freidman proposes 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): "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." 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 introduces a problem of criminal bribes to enforcers raising the actual punishment P above the P_law, by making the actual rate at which criminals pay enforcers larger than the p figure suggests. To combat this, Friedman suggests "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." In summary, the victim sells an enforcer the right to collect a fine f, and f is set by solving P[f,p] = P_law, where p is the frequency of convictions in some enforcer-chosen bundle of crimes. Restitution R is whatever the victim can get, and may be negative sometimes, if the victim so consents, in order to pay for non-fine punishments. Assuming an optimum punishment P for a homogeneous set of crimes (and no bribes, etc.), the result is a tradeoff between fine f and probability p that minimizes enforcement costs P - R. The reason this works is that in general one can minimize the sum of two terms, here P[f,p] and -R[f,p], by first finding the value of one of the terms, here P, at the global minimum of the sum, and then minimizing the other term subject to this constraint, here minimizing -R[f,p] subject to the constraint P[f,p] = P_min. Actually, we really want to pick f and p to minimize M[P,R] + P[f,p] - R[f,p], instead of just P-R, but the same approach works for this as well, as long as M is monotonic in R at the optimal P. 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. Advantages of this proposal include intuitive elegance and historical precedence. With victim contributions, punishments can be set higher than is possible with just fines. Victims are placed in charge of enforcement, and must suffer all of the uncertainty in enforcement costs, as the quadratic model above suggests they should. Enforcers, hired by victims, have incentives to attend to less tangible preferences of victims, like being polite and discrete, though unfortunately a legalistic P[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. 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 P[f,p] required to implement the above policy. In addition to bribes, for which Friedman's solution seems less than satisfactory, there is a similar problem that victim claims about crimes that never happened may also distort p. And a general problem with laws fixing punishments P is that to set the optimal P one would really have to know the enforcement cost, which this enforcement institution is intended to discover. 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 restitution alternative, but which also can be applied to fixing 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. It might take some hunting to find another crime at the same time, with the same criminal risk preferences, optimal fine, and probability of conviction as yours. An alternative is to use bets to validate probability estimates. When an enforcer takes on a case, they declare a probability estimate 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 betting activity. 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. Such speculators could not prevent an overestimate, which would allow actual punishment to be less than intended. But since this reduces expected fines, and hence restitution, competition between enforcers for victims should suppress such overestimates when punishment is fixed. 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 to 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). And since the probability would be declared earlier, the criminal could better know what fine they face. ESTIMATING CRIMINAL RISK To control a punishment level, one needs to know P[f,p,B], i.e., how expected punishment for a particular criminal, after a particular crime, varies with fines and probability of conviction. And if one needs to impose a punishment higher than any fine could give, one would need to know P as a function of prison time, degree of torture, etc. While such problems can always be punted to be dealt with by the expensive social negotiations that choose laws, we prefer a more direct approach. If one is willing to offer reduced fines to some fraction of convicted criminals, one might actually measure P[f,p,B] for one particular set of f,p values for each such criminal. Since the crime will have already happened, each criminal will already have their benefit B. If one then fines that criminal a reduced amount P'' < f (where P'' is a hopefully good estimate of P), and then asks the criminal how much they would have to be paid to again accept a chance p of having to pay a fine f, they should answer with P' ~= P[f,p,B+P'-P''] ~= P[f,p,B]. At least if they have incentives to answer honestly, which they might in a competitive market situation, and are not using their bid to send some sort of signal. Such competition could come by placing each convicted criminal into a group of 2n criminals (n=1 may be reasonable) with similar fines, probabilities, declared functions P, and any other characteristics that indicates risk preferences (like a history of gambling). Do not introduce these people to each other. Instead, introduce a random probability z of freeing this entire group from their obligation to pay fine f. Instead they must each pay a reduced fine P'' (to the enforcer), and must each offer a sealed bid saying the amount they would have to be paid to be willing to again suffer a random chance p of having to pay the median fine f. Half, or n, of these offers are taken and all paid the highest bid price, P' (hopefully ~= P''), of that half group. This price P' should be near a median P[p,f,B] for the members of the group. Again, this ignores desires to be treated with dignity, etc. Also, any insurance the criminal has to help pay fines must apply equally to test fines as to ordinary fines. Of course what one wants, when punishment is fixed, is to know P[f,p] well enough to know the f value which, with the validated p, gives the legally set P, and one wants to know this in the usual case where the criminal is not freed from paying fine f. And the enforcer would like an estimate of this to help it to choose probability p. In principle such estimates could come from a full set of contingent markets valuing obligations to pay a criminal their bid P', conditional on particular random f,p combinations being tested for that criminal. But in practice such markets might be too thin to be useful. In addition, criminals might have incentives to bid too high when the law fixes punishment, in order to exaggerate their risk aversion. While this would cost them on the current test, it might help them induce future markets to raise their estimates of the criminal's risk, and hence induce lower future fines. To work, there would have to be signals associated with a new crime (where no particular criminal is yet suspected) that would indicate that the criminal is likely to engage in such strategic bidding. This might require criminals to join into cooperative groups with distinguishable modes of operation, but this could be possible. While I do not know how 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. FIXING RESTITUTION In Friedman's proposal, social loss M[P,R] + P[f,p] - R[f,p] is minimized by legally fixing the optimal punishment P, and then letting the enforcer trade p vs. f to maximize restitution R. However, a symmetric alternative is to 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 can instead have the criminal, who seeks minimum P, hire the enforcer. And while criminals may not want to reveal themselves to do this hiring, this problem can be solved 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 guarantee 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, 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, but not directly to similar desires of victims. Victims could however directly pay enforcers for such consideration, 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 would be internalized to the criminal/agent/enforcer group, and so would not directly distort the law's handle on crime in the way such bribes do if punishment is fixed. 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 fractional stake in the punishment, by testing criminals a fraction z of the time, as described above, and make an appointed agent responsible for paying the amount P' the criminal requires to again accept a probability p of paying a fine f. This requires some validation of a probability p, such as the betting suggestion above. To (mostly) correct for the fact that the probability of conviction affects the probability of a test, appointed agents should have to pay P'/p instead of just P', paying P' to the criminal and the rest to the enforcer. To provide an estimate P'' just before the test, all the agents of a test group of criminals could auction off their obligation to the lowest bidding new agent, with this bid price being nP'', and then have the new agent actually responsible for paying nP' to the tested criminals. To avoid bias through how the agents are appointed, the job could just be given to whoever offers the lowest bid Y (~= zP/2) to take on the job. Y would be paid by the victim, who might then gain the right to receive this much more restitution. 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 P[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 with a betting approach to probability validation, again competition suppresses both over and under estimation. The agent should want the lowest possible punishment cost, so that it will pay the least should the criminal be tested. And criminals as a group should not want to game this institution because they want it to work, encouraging agents to help minimize punishment of criminals. Thus criminals as a community should be free to choose the parameters z and n. Unfortunately, such enforcer auctions cannot directly give enforcers incentives to attend to the intangible preferences of either victims or criminals, though direct payments could again be made by victims. 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 actually distort very little. TRADING PUNISHMENT VS. RESTITUTION When enforcement costs are uncertain, the optimum is to give both parties a piece of that risk, as opposed to giving it entirely to the victim, when fixing punishment, or to the criminal, when fixing restitution. If the law specifies an externality loss function M[P,R], and a criminal's agent sets P[f,p] (constrained to satisfy D_f P[f=0,p]=p), then the above approach to selecting enforcers by auction can be extended to pick the enforcer whose bid {p,f,R} minimizes L[P,R] = M[P[f,p],R] + P[f,p] - R Instead of fixing fines, punishments, or restitution, this approach can be said to more directly minimize loss. However the agent's incentives might become distorted, as he or she might reduce the net punishment P he or she will have to pay by clever choice of the function P[f,p] declared. And there may be incentive problems for the criminal similar to the case where one is trying to measure criminal risk aversion when punishment is fixed; by acting more risk adverse than one really is, one might reduce actual fines imposed. I don't know of a good way to validate risk aversion estimates in this case. 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 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. A set of M_i[P,R] would be a general set of contingent laws, with each type i describing another possible legally anticipated contingency. Contingent law increases uncertainty for parties who do not know the contingent factors, but allows a better tuned law for parties who do [4]. 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], while others may be more difficult to track down and convict. 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. An important special contingency, let us call it i=0, is when a "victim" has falsely claimed that a crime occurred. Contingent law must specify a zero fine and punishment here (f_0 = 0), since there is no criminal to be fined in this case, and thus enforcement costs must be paid for by the victim through a negative restitution R. Since the punishment a criminal expects depends on how much they know about the crime's type, the type i should 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 when a court declares a type when a conviction is obtained, one can't give the victim a type-dependent restitution R_i regardless of whether a conviction is obtained. Instead the law in general can offer the victim a take t_i only if there is a conviction declared of type i. Thus the victim may now suffer 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. The enforcer's profit equation now has R_i replaced by p_i*t_i, where p_i is the joint probability of getting a conviction and having it declared to be of contingency type i (with p = Sum_i p_i). CONTINGENT RISK The extra risk contingent law imposes on victims can create a new cost of enforcement, 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, these problems are worse here, because a victim openly complaining of a crime can be much more easily identified to be the same victim as in a previous crime. This makes much more direct the reward from faking a high risk aversion when tested. Victims could attempt to insure against this risk, for example by being paid by an insurer an amount = Sum_i p_i*t_i, where p_i is the joint probability of getting a conviction and having it declared to be of contingency type i (with p = Sum_i p_i). (The enforcer, having chosen the p_i, might be a natural group to sell this insurance.) However, if the victim may have superior information about the crime type, not reflected in the p_i, and if the victim choose whether or not to buy this insurance at the last minute, then adverse selection would push insurers to offer lower restitution amounts. Otherwise the victim might buy the insurance only when it expected the actual R_i to be less than the estimated . A victim who committed ahead of time to be paid would still suffer from incentive problems, wanting to only reveal information relevant for estimating p_i when that would raise the estimated . Others estimating p_i might then presume, often erroneously, that lack of detail from the victim indicates a less serious crime. This would then induce too much effort to collect detail about an accused crime. The resulting deviation of paid from R_i specified by law would also reduce the incentives of victims to adjust their defense x. Still, in the absence of better ways to estimate victim risk, these may be reasonable prices to pay for allowing contingent law. And since the victim suffers the cost of this problem, it seems conceivable that the victim could solve at least part of it through a reputation for being forthright. If the criminal knew the crime type, then contingent punishment would be given by U_c[-P_i] = g_i*U_c[-f_i], where g_i is the probability of a conviction conditional on the type i, U_c is the criminal's utility, and where we calibrate utility U so that U[W=0]=0, and D_W U[W=0]=1. Similarly, if victim risk is a factor then contingent restitution for a victim who knew the crime type would be given by U_v[R_i] = g_i*U_v[t_i]. If, however, either party did not know the crime type, then their expected wealth transfer would depend on their beliefs p_i about the chance for each type of conviction. Punishment would be given by U_c[-P_i] = Sum_i p_i*U_c[-f_i], and similarly for restitution. To validate estimates of punishment and restitution, then, the court would have to estimate how much each party knew about the crime type, with simple choices being that they probably knew the type, or that they were about as ignorant as the validated estimates for p_i (validation is discussed below). To test a criminal who knew the type, the test would go as before, using f_i and g_i in place of f and p. To test an ignorant criminal, criminals would bid to be subjected to the set of risks of probability p_i of owing fine f_i. CONTINGENT PUNISHMENT OR RESTITUTION? A general law might appoint the enforcer whose bid {p_i,g_i,f_i,t_i} for all i minimized expected social loss = Sum_i (p_i/g_i) (M_i[P_i,R_i] + P_i - R_i) where p_i/g_i is the probability of the crime being of type i, and M_i[P,R] is set by law. Note that this requires a set of functions P_i[f,g] and R_i[t,g], perhaps declared by a criminal's agent and a victim's agent. For small punishments and restitutions, not significantly changing the wealth of either party, this social loss becomes approximately ~= Sum_i p_i*( M_i[ g_i*-U_c[-f_i] , g_i*U_v[t_i] ]/g_i + -U_c[-f_i] - U_v[t_i]), where g_i has dropped out of the enforcement terms corresponding to P and R, but still remains in the externality term M. 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. Enforcers would be free to offer victims either contingent restitution, or some average value up-front. Criminals would be fined the f_i that solves P_i[f_i,g_i] = P_i,law, though its not clear how to validate a set of P_i[f,g] functions in this case. Note that if we allow that P_i for ignorant criminals should depend on f_j, g_j for other contingencies j, then solving the P_i equations may be complex, perhaps even impossible. If the law were to fix contingent restitution, we might ignore victim risk because of the presumption of insurability described above. In this case, appointed criminal agents would publish a set of P_i[f_i,g_i], and accept that enforcer whose bid {p_i,g_i,f_i} promised the lowest expected punishment Sum_i (p_i/g_i) P_i[f_i,g_i], and who promised to pay the victim a take t_i = R_i/g_i. Actually, enforcer bids would also have to include a (negative) R_0, so the winning bid would actually minimize: - R_0 (p_0/g_0) + Sum_i (p_i/g_i) P_i[f_i,g_i] Note that for contingent enforcement, as opposed to contingent law, fines and probabilities may be contingent, but restitution need not be. Thus there is no problem with a fixed up-front restitution paid to victims in this case. CONTINGENT PROBABILITIES Contingent law needs validation of both p_i and g_i estimates. Joint probabilities p_i are somewhat more difficult to validate than p. Validation by bundle frequency requires even more similar cases to form bundles, not to mention that bundles need to be chosen after punishment or restitution is declared, yet contingent types are not revealed until conviction. If, however, enforcers validate p_i estimates by offering to bet 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. A solution is to only allow such bets 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 p_i such that had some subset J of those been {p'_i, i in J} instead of {p_i, i in J}, that bid would have lost to another bid in the auction. Then 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 in J} p'_i that a criminal would be convicted and the contingency type would fall in J. Again this offer would only be open for a short time, and would be naturally limited in amount. 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 them), allowing a similar approach to validating estimates of p_i, the conditional probability g_i also includes information about the types of unconvicted crimes. To validate g_i, one needs 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 (paid by the enforcer) 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 the obligations to pay off the prize for each type (before choosing the enforcer), then each obligation should sell for about y(1-e)Z(p_i/g_i), where e is the probability that no prize will be claimed. Combined with estimates of p_i this should give estimates of g_i. To reach a given law-specified accuracy of measurement e, one could first set the prize amount 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 Z(1-e)/e should no one collect a basic prize, and who is willing to do so for the lowest prize amount Z. Criminals or victims could earn up to ~yZ by buying obligations to pay on types they know are not the true type, and in the process would give enforcers information which might aid in catching criminals. So as long as y is small enough, this should not distort the process greatly. If serious enforcement bidders (delimited again so as to be unlikely to include the criminal or victim in disguise) could sell more prize obligations in the same prize obligation auction, then this would block enforcers from bidding too low in order to inflate g_i estimates. To suppress underestimates of Z, Z-validators could be required to also offer to pay other serious enforcement bidders a proportional fee if no basic prize is collected, if those bidders will pay the Z-validator a proportional co-prize otherwise. These payments would be made by the victim, which would then gain the right to collect that much more from the enforcer, once the enforcer was chosen using these estimates of p_i and g_i. Like the parameters z,n describing how criminal's are tested, free parameters in this approach such as y,e should be chosen by criminals if restitution is fixed, by victims if punishment is fixed, and by a more expensive combined social negotiation in the law if social loss functions are minimized. 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 clearly preferable to a forth alternative of fixing criminal fines. This paper has compared these approaches, and in the process hopefully illuminated some basic issues in designing systems of private law enforcement. Assuming each possible crime is independent, 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, before they choose how much they want to work to encourage or discourage the possibility of a "crime", on some wealth transfer plus a legal system which forces the criminal to pay the victim a certain fine if the crime happens (assuming this is uncontroversial) 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. Moving beyond having the law 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 contingent restitutions. This paper suggests a method for validating risk estimates. But if punishments are fixed then it is difficult to measure criminal risk, if restitutions are fixed then it is difficult to measure victim risk, and both may be difficult to measure when a general social loss function is fixed. The possibility of bribes from criminals to enforcers can make it difficult to legally fix punishment, but only affects restitution when restitution is not certain, as with contingent restitution. Even then, it seems that fixing restitution discourages bribes more than fixing punishment. Fixing punishment allows for consideration of the intangible desires of victims, while fixing restitution allows for such consideration of criminal desires only when agents are appointed directly. However victims should be able to directly buy such consideration from enforcers more easily than criminals. The fact that a simple quadratic model suggests residual enforcement cost variations should go to the victim argues for fixing punishment, as does the fact that fixing punishment allows punishments greater than fines, if such things are possible. The option to set a general social loss functions of punishment and restitution would allow better reductions in crime externalities when enforcement costs are uncertain at the time the law is fixed. But it would not consider intangible desires of either party, and would require more expensive social negotiation about the details of the enforcement process, details which would otherwise be the responsibility of just one party. So which approach is better? Should we find better ways to validate estimates of risk, none of the above approaches is obviously out of the running. But at present, the only known approach which includes validation of risk is to fix expected restitution, and to assume insurance allows risk-neutral victims when restitution is contingent. 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] David Friedman, "Reflections on Optimal Punishment, Or: Should the Rich Pay Higher Fines?" 3 Research Law & Econ. 185 (1981) GLOSSARY B = benefit to criminal D = damage to victim q = probability of crime happening w = wariness, 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 = P - B = criminal's share = criminal's loss from crime given law V = D - R = victim's share = victim's loss from crime given law L_v = victim's total loss = x + q V L_c = criminal's total loss = w + q C L = L_v + L_c = M + P - R = total loss for society M = externality losses W = wealth level of criminal p = probability of catching and convicting the criminal f = fine criminal owes victim given conviction, cash due t = take victim is paid given conviction, cash due G = what enforcer actually gets from victim, certainty equivalent E = cost of trying to catch and convict criminal e = effort by enforcer to enforce 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