Subgame-perfect punishment for repeat offenders.

Emons, Winand

I. INTRODUCTION

The literature on optimal law enforcement typically assumes that the government can commit to sanction schemes. (1) This means the government can use any set of threats to penalize wrongdoers. In particular, if a crime occurs, the government actually sanctions the wrongdoers even though, ex post, it may have no incentive to do so. Potential wrongdoers believe that the government will carry out the threat at any cost and, therefore, do not engage in the act in the first place.

In this article I give up the assumption that the government can commit to whatever sanction scheme. I consider the analysis of optimal sanctions without the possibility to commit important because judges often have a lot of discretion as to the size of the penalty; they may, for example, adjust sanctions to the financial possibilities, age, education, and so on of the wrongdoer. Accordingly, I allow only for sanctions that the government actually wishes to implement should a crime have occurred.

Ruling out full commitment changes the optimal enforcement schemes. Suppose, for example, the government does not care about the sanction, as is typically assumed in the literature. Then it will not enforce the penalty if a crime has happened given that there is, say, a small cost of doing so. The rational criminal will anticipate the ex post enforcement behavior of the government. Therefore, she will commit the crime because the threat of being sanctioned is not credible. Once one drops the commitment assumption, the typical deterrence equilibria of the law enforcement literature between potential wrongdoers and the government are based on empty threats. In the language of game theory, the equilibria are not subgame perfect or time consistent.

I study the problem of subgame perfect sanctions using the framework of Emons (2003). Agents may commit a crime twice. The act is inefficient; the agents are thus to be deterred. The agents are wealth-constrained so that increasing the fine for the first offense means a reduction in the possible sanction for the second offense and vice versa. The agents may follow history-dependent strategies, that is, commit the crime a second time if and only if they were (were not) apprehended the first time. The government seeks to minimize the probability of apprehension.

Ignoring the government's commitment problem, it is optimal to set the sanction for the first offense equal to the entire wealth of the agents while the sanction for the second offense equals zero. The intuition is as follows. A money penalty imposed for the second offense reduces the amount a person can pay for the first offense, because the wealth available to pay penalties is assumed to be fixed over the two periods. For that reason, a higher probability event--namely, a first offense that is detected--will be more effective use of the scarce money penalty resource than a lower probability event--namely, a second detected offense. Why is the probability of detection lower for the second rather than for the first crime? Simply because an agent faces the possibility of being sanctioned for the second crime if and only if he or she has already been sanctioned the first time. For further results it is important to note that the optimal probability of apprehension increases with the benefit from the crimes.

This decreasing sanction scheme raises of course the issue of time consistency. Will the government really charge the agent the entire wealth when she was apprehended for the first crime, knowing that then she will commit the second act for sure? Isn't it better for the government to renege and charge little for the first act so that the agent still has sufficient wealth to pay a sanction that deters the second crime? Given that the first act has been committed anyway, that way the government can at least deter the second act.

To study this problem I consider a rent-seeking government. The sanctions paid by the criminals enter the government's welfare function. Our government, therefore, has an ex post incentive to collect fines. The government can commit to a probability of apprehension but not to sanctions. Our basic result is that if the agent's benefit and/or the harm from the crime are not too large, then the scheme where the sanction for the first crime is the entire wealth and the sanction for the second crime is zero is indeed subgame perfect.

To see this, consider the government after the agent has been apprehended for the first crime. If it implements the decreasing sanction scheme, it appropriates the entire wealth yet incurs the harm of the second crime. Thus, the lower the harm of the second crime, the more attractive this option.

The alternative is to set the sanction for the second crime to a level that deters the act. With this option the government doesn't incur the harm of the second crime, yet forgoes the sanction for the second crime because it is deterred. If the benefit from the crime goes up, the optimal probability of apprehension increases, yet by more than the benefit; therefore, the actual sanction necessary to deter the second crime falls. Because a low sanction for the second crime means a high amount the government can charge for the first crime, a high benefit of the second crime makes this option attractive. Accordingly, only for low benefits the government sticks to the decreasing sanction scheme.

If the benefit and/or the harm of the second crime are large, the decreasing sanction scheme is no longer time consistent. The government prefers to deter the second crime should the first crime have occurred. Accordingly, only sanction schemes where each sanction by itself deters the corresponding crime are time consistent. In this case the optimal subgame-perfect sanction scheme entails equal sanctions in both periods. Enforcement costs are higher than with the decreasing sanction scheme.

The only article I am aware of that deals with the problem of time-consistent sanctions is Boadway and Keen (1998). They consider a government choosing a capital income tax rate and an enforcement policy. The government can commit to the enforcement policy but not to the tax rate. Ex ante the government wishes to announce a low tax rate to induce savings; ex post, when savings have been made, it will renege and apply a high tax rate. Boadway and Keen show that by committing to a lax enforcement policy the government can alleviate the welfare loss implied by its inability to commit to the tax rate.

In the next section I describe the model. In section III I derive the optimal sanctions for a government that can commit and in section IV for a government that cannot commit. Section V concludes.

II. THE MODEL

Consider a set of potential wrongdoers, which has measure 1. Individuals live for two periods. In each period the agents can engage in an illegal activity, such as illegal parking, illegally raising prices, polluting the environment, or evading taxes. If an agent commits the act in either period, she receives a monetary benefit b > 0. I consider crimes without social gains. Using the language of Polinsky and Rubinfeld (1991), b is the illicit gain and the crime creates no acceptable gain. (2) The act causes a monetary harm h to society, which is borne by the government. Because h > 0, the act is not socially desirable. The individuals are thus to be deterred from the activity.

To achieve deterrence, the government chooses sanctions and a probability of apprehension. The government cannot tell whether an agent is in the first or second period of life. The government only observes whether the crime is the first or the second one. Accordingly, the government uses fines [s.sub.1], [s.sub.2] [greater than or equal to] 0, where [s.sub.1] applies to first-time and [s.sub.2] to second-time observed offenders.

I assume that the government cannot commit to sanctions. This means that the government can choose a different sanction from the one announced at the outset once a crime occurred. Typically, a judge always finds good reasons to reduce or increase sanctions. In addition to sanctions, the government chooses a probability of apprehension p. This probability is the same for first- and second-time offenses. (3) It is irrevocably fixed before the agents take their actions. The government cannot easily change the amounts spent on, say, training the police. Accordingly, I assume that the government can commit to p while it cannot commit to sanctions. (4)

In the law enforcement literature, the optimal policy is derived by maximizing the sum of the offenders' benefits minus the harm caused by the offenses minus law enforcement expenditures. Sanctions do not enter the benevolent government's objective function because they are a mere transfer of money. (5) Within this framework the literature derives the results on optimal fines and optimal probabilities of apprehension. See, for example, Garoupa (1997) or Polinsky and Shavell (2000a).

Nevertheless, these results hold true if and only if the government can fully commit to the probability of apprehension and to the announced sanction. To see this, suppose the government incurs a small cost [epsilon] > 0 of collecting the fine. Suppose the agent has been apprehended for the crime and then the government strategically decides whether or not to impose the sanction. With such a sequencing, the rational government will not impose the fine: It does not care about the fine anyway and it can save the cost [epsilon]. If one anticipates this ex post behavior of the government, the threat of being sanctioned is not credible and the agent will commit the act in the first place. To put it in the language of game theory: The equilibrium in the game between the offender and the government is not subgame perfect.

If one wants to take the issue of subgame perfection (or time consistency) seriously, one must give the government an incentive to actually collect the fines. I do so by including the sanctions in the government's payoffs. (6) Our government maximizes revenues from sanctions minus the harms minus the enforcement expenditure and thus has an incentive to collect the fine should a crime have occurred. To save on notation, I take the probability of apprehension p as an indicator of the enforcement expenditure.

This approach can be motivated in several ways. Garoupa and Klerman (2002) take the public choice perspective of a self-interested, rent-seeking government that maximizes revenues minus the harm borne by the government minus expenditure on law enforcement. (7) Polinsky and Shavell (2000b) consider the standard benevolent welfare function and add a term reflecting individuals' fairness-related utility. If this fairness-related utility equals the actual sanction, their government maximizes the same welfare function as ours. (8)

Individuals are risk-neutral and maximize expected income. They have initial wealth W > 0. Think of W as the value of the privately owned house or assets with a long maturity. The agents hold on to their wealth over both periods unless the government interferes with sanctions. Any additional income they receive in both periods, be it through legal or illegal activities, is consumed immediately. Accordingly, all the government can confiscate is W. If the fine exceeds the agent's wealth, she goes bankrupt and the government seizes the remaining assets. This implies that the fines [s.sub.1] and [s.sub.2] have to satisfy the "budget constraint" [s.sub.1] + [s.sub.2] = W. (9)

To save on notation, let the interest rate be zero. An agent can choose between the following strategies:

* She can choose not to commit the act at all. I call this strategy (0, 0), which gives rise to utility U(0, 0) = W. This is the strategy I wish to implement.

* She can choose to commit the act in period 1 and not in period 2. Call this strategy (1, 0); here we have U(1, 0) = W + b - p[s.sub.1]. The act generates benefit b; with probability p the agent is apprehended and pays the sanction [s.sub.1].

* The agent can opt to commit the crime in period 2 but not in period 1. Call this strategy (0, 1) generating utility U(0, 1) = W + b - p[s.sub.1]. With strategy (0, 1) the agent has the same utility as with strategy (1, 0) because the government observes only one offense.

* Moreover, the agent can commit the act in both periods, which I denote by (1, 1) and U(1, 1) = W + b - p[s.sub.1] + b - p([1 - p][s.sub.1] + p[s.sub.2]). The second crime is detected with probability p. With probability p the agent has a criminal record in the second period and thus is fined [s.sub.2]; with probability (1 - p) she has no record and pays [s.sub.1] if apprehended.

* Finally, the agent can choose two history-dependent strategies. First, she commits the act in period 1. If she is not apprehended, she also commits the act in period 2; however, if she is apprehended in period 1, she does not commit the act in period 2. Call this strategy (1, (1 | no record;0 | otherwise)) with U(1,(1 | no record;0 | otherwise)) = W + b - p[s.sub.1] + (1 - p) (b - p[s.sub.1]). Because the agent stops her criminal activities if she is apprehended once, she is never sanctioned with [s.sub.2].

* Second, she commits the act in period 1. If she is not apprehended, she does not commit the act in period 2; however, if she is apprehended in period 1, she commits the act in period 2. Call this strategy (1,(0 | no record;1 | otherwise)) with U(1,(0 | record; 1 | otherwise)) = W + b - p[s.sub.1] + p(b - p[s.sub.2]). It turns out that this strategy defines the agents' binding incentive constraint for the optimal sanctions. (10)

Before I start deriving optimal sanctions, I have to ensure that the government indeed wants complete deterrence. I achieve this by assuming W - 2h < - 1. If the government completely deters, there is neither harm nor revenue, and the maximum possible expenditure for deterrence is 1 (recall that I take the probability of apprehension as a measure for enforcement cost). If the government doesn't deter at all, enforcement costs are zero, the government incurs the harm twice, and the maximal revenue it can obtain is the agents' wealth W. Accordingly, if the harm is sufficiently large, the rent-seeking government wants complete deterrence.

Let us now analyze sanctions that give the agents proper incentives not to engage in the activity in either period. I first derive the cost-minimizing sanction scheme that achieves perfect deterrence ignoring the government's commitment problem. This is the standard approach found in the literature. The literature does not further discuss why the government is able to commit. One argument coming to mind in favor of commitment is that the government plays repeated games with potential wrongdoers and, therefore, wants to build up a reputation of being tough. The analysis of the commitment scenario follows Emons (2003). I will then consider the government's incentives to actually implement this penalty scheme without commitment in section IV.

III. OPTIMAL SANCTIONS IF THE GOVERNMENT CAN COMMIT

I assume that agents have enough wealth so that deterrence is always possible, that is, 2b < W. The agent does not follow strategy (1, 0), if U(1, 0) [less than or equal to] U(0, 0), she does not follow strategy (0, 1), if U(0, 1) [less than or equal to] U(0, 0), and so on. Straightforward computations confirm that the agent does not engage in strategies (1, 0), (0,1), and (1,(1 | no record;0 | otherwise)), if

(1) [s.sub.1] [greater than or equal to] b/p;

she does not pick strategy (1, 1), if

(2) [s.sub.2] [greater than or equal to] (2b/[p.sup.2]) - [s.sub.1]([2/p] - 1);

and she does not pick strategy (1,(0 | no record; 1 | otherwise)), if

(3) [s.sub.2] [greater than or equal to] (b[1 +p]/[p.sup.2]) - [s.sub.1]/p.

Accordingly, with all sanction schemes ([s.sub.1], [s.sub.2]) to the right of the bold line in Figures 1 and 2, the agent has proper incentives and commits no crime. For example, the scheme [s.sub.1] = [s.sub.2] = b/p induces no crimes.

[FIGURES 1-2 OMITTED]

Next I minimize the enforcement costs, as given by p while providing incentives not to commit any crime. (11) I will minimize p taking the incentive constraint (3) into account. Then I show that the optimal [??] also satisfies the incentive constraints (1) and (2).

Obviously, Becker's (1968) maximum fine result applies here, meaning that to minimize p the government will use the agent's entire wealth for sanctions. (12) Accordingly, plugging the budget constraint [s.sub.1] + [s.sub.2] = W into (3) and differentiating the equality yields

dp / d[s.sub.1] = (p - [p.sup.2])/(b - [s.sub.1] - 2p[W - [s.sub.1]]) < 0

for b < [s.sub.1] [less than or equal to] W. Consequently,

[[??].sub.1] = W, [[??].sub.2] = 0, and [??] = b / (W - b).

Because b/p<2b / p(1 - p) < b(1 + p)/p [for all] p [member of] (0, 1), the incentive constraints (1) and (2) are also satisfied.

I thus find that the optimal sanction scheme sets [[??].sub.1] = W and [[??].sub.2] = 0. First-time offenders are punished with the maximal possible sanction, and second-time offenders are not punished at all. The sanction [s.sub.1] is high enough that it not only deters first-time offenses but also second-time offenses even though they come for free.

The intuition for this result follows immediately from the incentive constraint (3). The agent pays the sanction [s.sub.1] with probability p and the sanction [s.sub.2] only with probability [p.sup.2]. To put it differently: The agent is charged [s.sub.2] with probability p if and only if he has paid already [s.sub.1]. Because paying the fine [s.sub.1] is more likely than paying [s.sub.2], shifting resources from [s.sub.2] to [s.sub.1] increases deterrence for given p. Consequently, p is minimized by putting all the scarce resources into [s.sub.1].

It is perhaps somewhat surprising that strategy (1,(0 | no record; 1 | otherwise)) and not strategy (1,(1 | no record; 0 | otherwise)) defines the binding incentive constraint in the optimal penalty structure. Given that the optimal penalties are declining, an agent who was not apprehended for the first crime has a strong incentive not to commit the act a second time: If she is apprehended, she pays the high sanction [s.sub.1]. If the agent was, however, apprehended for the first crime, the second crime comes for free. The sanction [s.sub.1] has to be high enough so that she doesn't commit the first crime in the first place.

IV. OPTIMAL SANCTIONS IF THE GOVERNMENT CANNOT COMMIT

I now check under which conditions the sanction scheme [[??].sub.1] = W, [[??].sub.2] = 0 together with the minimal enforcement probability [??] = b/(W - b) is subgame perfect. This means: Does the government really implement these sanctions once the agent has committed a crime? To do so, consider the subgame starting when the agent has been apprehended for the first crime.

If the government sticks to the penalty scheme [[??].sub.1] = W, [[??].sub.2] = 0, the agent will commit the second offense for sure because it comes for free. The government's payoff then is W - 2h - [??]. It incurs the harm twice and seizes the agent's entire wealth with [s.sub.1].

The alternative is to lower [s.sub.1] and at the same time increase [s.sub.2] such that the agent doesn't commit the second act. Obviously, the rent-seeking government will set [s.sub.2] = b/[??], the minimal sanction-achieving deterrence. The government goes for the minimal sanction guaranteeing deterrence because, by its very nature, the government will not get this money; that way, [s.sub.1] is as large as possible. Using [??] = b/(W - b), I find [s.sub.2] = W - b and [s.sub.1] = b. If the government follows this strategy, its payoffs are -h + b - [??]. It incurs the harm from the first crime, collects [s.sub.1] = b, and there is no more crime.

Comparing the two payoffs, obviously the government prefers to stick to [[??].sub.1] = W, [[??].sub.2] = 0 if W - h [greater than or equal to] b. The government gets the entire wealth less the harm by sticking to the optimal incentive scheme, whereas it gets [s.sub.1] = b if it chooses to deter the second offense. One may, therefore, conclude that [s.sup.*.sub.1] = W, [s.sup.*.sub.2] = 0 is subgame perfect if the agent's benefit b and/ or the harm are not too large (see Figure 1).

I now determine the optimal subgame-perfect sanction scheme together with the probability of detection p if W - h < b. Consider again the government deciding on sanctions after the wrongdoer has been apprehended for the first act. If the government wants to deter the second act, it will set [s.sub.2] = b/p. It chooses the minimal sanction ensuring deterrence because it will not get the money. This way it can collect the maximum amount [s.sub.1] = W - b/p for the first act from the agent.

In contrast, the government may wish to induce the second crime. It does so by setting [s.sub.2] < b/p. The government collects [s.sub.2] only with probability p; it collects [s.sub.1] for sure because we are in the node where the government has just apprehended the agent for the first crime. Because W = [s.sub.1] + [s.sub.2], the revenue maximizing government sets [s.sub.1] = W and [s.sub.2] = 0 if it wants to induce the second crime. This generates a payoff of W - 2h - p for the government.

The government prefers the strategy of inducing the second crime to optimally deterring the second crime if W - 2h - p > W - h - b/p - p [??] b/p > h. Deterring the second crime has the cost of the forgone revenue [s.sub.2] = b/p; encouraging the second crime has the cost of the harm h.

The left-hand side of the inequality b/p > h is decreasing in p. Therefore, if it is not satisfied for the minimal probability of apprehension inducing no crimes [??] = b/(W - b), it does not hold for any p deterring both crimes. Thus, if b/[??] < h [??] W - h < b, the government prefers to deter the second crime and does so optimally by setting [s.sup.*.sub.1] = [s.sup.*.sub.2] - W / 2 and [p.sup.*] = 2b/W (see Figure 2).

A low probability of apprehension increases b/p, the sanction that is necessary to deter the second crime. Deterring a second crime thus becomes unattractive. By choosing a low p, the government commits not to raise [s.sub.2] to a level that deters. This result is similar to Boadway and Keen (1998) where the government commits to a lax enforcement not to raise tax rates after savings decisions have been made.

I summarize the preceding observations with the following proposition.

PROPOSITION 1. If W - h [greater than or equal to] b, the optimal subgame-perfect sanction scheme is given by [s.sup.*.sub.1] = W, [s.sup.*.sub.2] = 0 and [p.sup.*] = b / (W - b).

If W - h < b, the optimal subgame-perfect sanction scheme is given by [s.sup.*.sub.1] = [s.sup.*.sub.2] = W / 2 and [p.sup.*] = 2b / W.

Obviously, the government is better off in the first case, where it uses the decreasing sanction scheme. In both cases crime is completely deterred. With the decreasing sanction scheme the probability of apprehension and hence enforcement cost is lower than in the second case of constant sanctions.

V. CONCLUSIONS

The purpose of this article is to analyze subgame-perfect sanction schemes, that is, sanctions the government indeed wants to implement should a crime have occurred. I consider the problem of time consistency important because judges tend to have a lot of discretion as to the size of the penalty. They anticipate that a high penalty now may reduce the potential for future sanctions. Rational criminals will anticipate this and thus not be deterred by empty threats.

A rent-seeking government will stick to the optimal decreasing sanction scheme if it gets more money by allowing the second crime and cashing in the agent's entire wealth with the first sanction than by deterring the second crime. In the opposite case the government prefers to deter the second crime. It does so with equal sanctions for both crimes.

Accordingly, the constraint of time consistency has bite. If the government can commit, decreasing sanctions are always optimal; if the government cannot commit, decreasing sanctions may still be optimal but so may be equal sanctions. I haven't explained escalating sanctions based on offense history, which are embedded in many penal codes and sentencing guidelines. Explaining escalating sanctions seems to be fairly difficult for the law enforcement literature. (13) Nevertheless, in my set-up the commitment issue ruled out decreasing sanction schemes in some cases. Perhaps the problem of time consistency is a fruitful track for future research to explain escalating sanction schemes.

(1.) See Garoupa (1997) or Polinsky and Shavell (2000a) for surveys.

(2.) See also Chu et al. (2000) for an analysis of crimes without social gains. They argue that the gains to the offender are not considered because the crime is not socially acceptable or because the gains of offenders (such as theft or other zero-sum crimes) offset with the victims' losses.

(3). I thus rule out the case where agents with a criminal record are more closely monitored than agents without a record. See Landsberger and Meilijson (1982) for an analysis of optimal detection probabilities.

(4.) Boadway and Keen (1998) use the same commitment structure when studying the time consistency problem in the taxation of capital income.

(5.) In the explicit formulation, welfare is the criminal's utility (benefit minus expected sanction) plus the government's utility (expected sanction minus harm) minus enforcement costs.

(6.) In terms of the explicit welfare function given in the preceding note, I simply exclude the criminal's utility (benefit minus expected sanction).

(7.) Dittmann (2001) uses a similar approach.

(8.) In Rubinstein (1979) the government's payoffs also depend on whether or not it punishes the offender. Unlike the other studies, Rubinstein's government is worse of f if it punishes the offender, independently of whether the act was committed intentionally or not.

(9.) This assumption distinguishes our approach from Polinsky and Shavell (1998) who work with a maximum per period sanction [s.sub.m]. Accordingly, they may set [s.sub.1] = [s.sub.2] = [s.sub.m], which is typically the optimal enforcement scheme. In their framework [s.sub.m] is like a per period income that cannot be transferred into the next period. Burnovski and Safra (1994) use the same budget constraint as we do.

(10.) These history-dependent strategies distinguish my work from Burnovski and Safra (1994), where individuals decide ex ante simply on the number of crimes.

(11.) Because in my setup the harm of the crime exceeds its acceptable benefit, maximizing social welfare boils down to minimizing enforcement costs.

(12.) If [s.sub.1] + [s.sub.2] < W, sanctions can be raised and p lowered so as to keep deterrence constant.

(13.) See Emons (2003) for a discussion of the problems the law enforcement has in explaining escalating sanctions.

REFERENCES

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--. "The Economic Theory of Public Enforcement of Law." Journal of Economic Literature, 38(1), 2000a, 45-76.

--. "The Fairness of Sanctions: Some Implications for Optimal Enforcement Policy." American Law and Economics Review, 2(2), 2000b, 223-37.

Rubinstein, A. "An Optimal Conviction Policy for Offenses that May Have Been Committed by Accident," in Applied Game Theory, edited by S. Brains, A. Schotter, and G. Schwoodiauer. Wuurzburg: Physica-Verlag, 1979, 406-13.

WINAND EMONS, I thank Nuno Garoupa, Manfred Holler, Thomas Liebi, Francois Salanie, and an anonymous referee for helpful comments.

Emons: Professor, University of Bern, Department of Economics, Gesellschaftsstrasse 49, CH-3012 Bern, Switzerland. Phone 41-31-631 3922, Fax 41-31-631 3992, E-mail winand.emons@vwi.unibe.ch