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Profit-Maximizing Pirates By:

Place: Date of defense: Opponent:


Marie Gaarden Gaardmark Department of Economics, University of Copenhagen April 29th, 2010

Katrine GrønbÌk Von F Ringsted Paul Sharp

Abstract The goal of this paper is to illustrate how economic theory can be applied to the institutions of pirates. In particular, I construct a model that shows how incentive-corrective measures can be implemented to avoid shirking, or staying behind in battle in this context. The tool used is game-theory, and both static and repeated game analysis is applied. The model builds on the description of contractual set-ups during the heyday of pirates by Leeson (2009). Even with the differences in time, setting and degree of crime, the analysis reflects considerations relevant to present day's corporate environment, as profit-maximization cannot be fully optimal without taking incentives to shirk and general alignment of interests into account.



According to Leeson (2009), life onboard a pirate ship during the 17 th and 18th century was surprisingly democratic due to economic incentives. To ensure full cooperation of the crew members, there could be no dictator captain. Why? The average pirate was not forced onboard and thus had to be compelled to sign up for the risky adventure. Before setting sail, the crew members signed up on a well-specified contract, securing that the terms were clear beforehand, all having consented (Leeson 2009, p. 60, 66). And the terms included the right to elect a captain and a relatively egalitarian pay scheme (Leeson 2009, pp. 68). At the same time, the pay was fully dependent on taking a ship over or not – no prey, no pay (Leeson 2009, p. 58). With more or less equal sharing of gains, it is natural to compare with profit-sharing in today's corporate environment: Weitzman and Kruse (1990) explain in economic terms some of the upsides and downsides of profit-sharing. On the upside, it has the potential to mitigate reduced effort on an agent's behalf, i.e. reducing the moral hazard problem stemming from disutility of effort, as the additional effort is fully or partially equalized by additional pay. However, with more than one agent, we have the incentive to free-ride on the effort of others, i.e. shirk, which in the context of pirates translates into keeping away from the center of battle (Leeson 2009, pp. 56). The more difficult it is to monitor the effort of the agent, the more incentive to shirk as the decline in profits is shared amongst all. With an all or nothing outcome as here, no decline is actually the case, given you are still victorious. A natural tool to apply when it comes to incentives is game theory as it is a tool that can be u sed to formally outline the economic motives entering the decisions of the agents, i.e. pirates. The tool is commonly used in labor economics, e.g. Shapiro and Stiglitz (1984) model how the imperfect knowledge of a worker's effort may result in shirking and how to align the incentives with the profit-maximizing entity – or principal, with a fixed pay. FitzRoy and Kraft (1995) set up two scenarios, comparing the equilibrium wages and profits of a system with a constant1 wage with risk of shirking to one of profit-sharing without moral hazard problems, both with risk-adverse agents. Huddart and Liang (2005) add the free-riding issue to a profitsharing set-up, including monitoring and signaling.

1 Supplemented by a bonus if observed effort is high.


The goal of this paper is not to question the descriptions of history and incentives in the book by Leeson (2009) but instead to take them as given. Based on this, I will in the following construct a model using the non-cooperative game-theoretic framework, examining the mechanism design required to prevent shirking, drawing upon inspiration from the aforementioned models. The analysis is thus relevant for even the non-criminal enterprises of today as much of the same incentives were in play during the era of pirates. Profit-sharing and alignment of incentives to avoid shirking are still highly relevant topics.


Setting the stage

What are your economic motivations for not taking full part in the battle? Well, you might get injured! The problem is that you then impose yourself negatively on the overall outcome of the battle – the risk of losing the battle is heightened. In economic terms: You are acting like a negative externality. In case of you still receiving full payoff and the outcome is positive, you are free-riding on the public good that a victory is 2. This gives you an incentive to shirk, when you are a rational, payoff-maximizing agent who behaves egoistically in the sense that you are only concerned with your own payoff and will free-ride if possible. The payoff as a shirker without any punishment or perceived influence on the outcome would be enough to have all pirates shirk, assuming they have the same preferences with the same disutility from being hurt. The following shows a simple one-stage game with perceived payoffs, signifying that the players are not internalizing the negative externality they impose: Pirate 2

Pirate 1









The arbitrary payoff of playing Don't, 3, is set to be slightly less than the full profit-share, 4, due to the higher risk of being injured when not slacking. Shirk is the optimal choice for both pirates as it is the strictly dominating strategy yielding strictly higher payoffs regardless of the action taken by the opponent. The unique Nash equilibrium is then (Shirk, Shirk). 2 Leeson (2009, p. 57) categorizes “full pirate ship effort� as a public good, as the benefits accrue to all. The opponents might not see a victory of their enemy as a public good!


If the pirates share the same preferences, all will choose to shirk, and the public good of victory will most likely not be provided, yielding a non-positive payoff. The pirates' perception of payoffs above is thus misguided. The question is then how to keep the agents from shirking. The negative externality can be corrected by private property rights or regulation (Leeson 2009, pp. 53), or using Pigovian taxes (e.g. Varian 1992, pp. 433). As a shirking pirate has no incentive to inform others about his free-riding, we are in a situation with asymmetric information, which perhaps only to some extent can be rectified by monitoring, and thus potentially no visible misbehavior to tax or regulate. A natural departure for designing mechanisms that prevent shirking is the model by Shapiro and Stiglitz (1984). Here the wage of the employee is set so as to ensure good behavior given the outside option of (a lower) pay if fired, acting as a disciplining device. As in their model, I assume homogenous workers, or pirates, and therefore do not consider any problems concerning adverse selection problems3 but only the moral hazard aspect of not doing the job right. The incentives are somewhat dissimilar compared to their model: The outside option is different as you would not be fired as a pirate but excluded from pay (Leeson 2009, p. 57). As compensation we have the share of profit, and in the case of injury this is supplemented with a form of disability pay: “The effect of pirate social insurance was to encourage full effort from each individual pirate, or at least to reduce the private disincentive to shirk, which improved pirates' ability to profit through plunder.� (Leeson 2009, p. 72) For at least one crew this went on indefinitely (Leeson 2009, pp. 71)4. Another important difference is the addition of repeated interactions, although not mentioned by Leeson (2009): We have multiple battles and thus multiple private decisions about whether to shirk or not5. Additionally, profit-sharing means that monitoring from fellow workers may be added (Weitzman and Kruse 1990, p. 113). In contrast to the profit-sharing set-up in FitzRoy and Kraft (1995)

3 Blinder (1990): Selection bias with profit-sharing, as highly productive employees might be attracted to a workplace where the potential payoff is greater. 4 In addition, a bonus for exceptional courage was rewarded (Leeson 2009, p. 72). 5 E.g. Weitzman and Kruse (1990, p. 99) suggest modeling a long-term relationship with repeated games.


the mutual monitoring is assumed to be imperfect. I will in the following propose a model, showing how the behavior-corrective mechanisms can be used in the context of ensuring full ship effort, where no pirate shirks.



In this section I formulate a model by going through the key elements entering the decisions of a pirate in times of battle. First, we consider the payoffs, then the structure of the game.

3.1. Parameters and Payoffs Let us define some key parameters and variables and sketch out the main assumptions. First of all, I will assume that the pirates were risk-neutral agents. Potentially they could be seen as risk-adverse but then they could not have been overly so, presuming that a risk-adverse person would not be happy to hang from the gallows if apprehended by the authorities, thus not joining a pirate crew to begin with. On the other hand, being a risk-lover might actually be the case for at least some of the pirates but this is pure guess-work. To keep the model tractable, the middle ground of risk-neutrality is chosen. This means that maximizing your expected utility is equivalent to maximizing the utility of expected payoff (e.g. Varian 1992, p. 177), why we can focus on the problem of maximizing the latter. The profits from successfully capturing a ship are assumed positive: t0 . This would be changing from time to time, sometimes you strike gold, or come upon a golden goose, as Leeson (2009, pp. 12) puts it, other times you would encounter less precious booty. We therefore consider this variable stochastic with a mean of e 0 . With risk-neutrality, the variance is not of importance. In the case of overall defeat, the profits are assumed to be 0, and borne by all the pirates, regardless of shirking or not. The probability of overall defeat is 0≤ p n , N 1 . The number of pirates engaged in battle, N 0 , enters negatively as it would be expected that at least for a range of N it would be influential on the outcome, as 1) the more fierce you would appear as a pirate, the less likely that your would have to actually fight and not just take over a white4

flag waving enemy6, and 2) the more power combined, the more effective could you be. In addition, the probability is assumed to depend positively on the overall number, n, of pirates that shirk, thus modeling that the behavior of the shirking members act like a negative externality. This is an addition to the model of Shapiro and Stiglitz (1984). Entering battle is dangerous business even when the overall outcome is victory. This I model by adding a probability of getting a disabling injury rendering no possibility of working in the future combined with the additional pay besides the profit-share. The probability of injury is: 0q n , N ≤1 , again with

∂q ∂N

0 and

∂q ∂n

0 . The negative externality of fellow pirates

shirking on the risk of injury is not mentioned by Leeson (2009). q n , N  is simply assumed to be zero when you are shirking yourself. I denote the disability pay D and assume D≥0 . The expected profits available for profit-sharing for the ship in its entirety can be written as: S n , N ≡ [ 1− p n , N  ] [e −{ N −n } q n , N  D ] , where the profits as victorious has the

combined disability pay subtracted: N −n reflects the number at risk of injury as shirkers are not exposed. The expected utility for the individual pirate if no pirate is shirking is: S 0, N / N [ 1− p 0, N  ] q 0, N  D−q 0, N  e , where the disability pay only is rewarded if the battle is won and e reflects the disutility from being disabled. The equal shares simplifies the more or less egalitarian pay scheme described. In the case where a pirate shirks and is caught at it, he is punished by exclusion from the pro fits. With 0≤m1 denoting the probability of getting caught, the expected utility for a nonshirker becomes S / N −nm[ 1− p ] qD−qe , where functional dependence has been suppressed. Note that the profits are divided by N −nm , as it is shared amongst those that either do not shirk or not caught at it. The expected utility of a shirker is 1−mS / N −nm0m , seeing that he may still receive a profit-share. With no risk of injury, no disutility or disability pay enters. You cannot pretend to be injured in this game, perhaps in contrast to football. The pay for the scenarios with and without shirking is thus dependent on the outcome of the battle, which is a generalization of the fixed wage modeled in Shapiro and Stiglitz (1984), and the disutility is assumed to only arise in the of case injury, whereas in their model working it 6 Leeson (2009, e.g. pp. 89) explains how signaling and branding could ensure peaceful overtaking.


self requires effort that creates disutility. Their outside option of being fired is here the exclusion of pay to act as a disciplining device.

3.2. One-period game As in Shapiro and Stiglitz (1984), our goal is to ensure that no shirking will occur, which means that the expected payoff as a non-shirker must be greater than or equal to that of a shirker, corresponding to the incentive constraint in the game theoretic literature (e.g. Varian 1992, p. 442). The method used below follows that described by e.g. Gibbons (1997): Solve dynamic games by backwards induction, seeing that the first part of this meta game is setting the disability pay, and the second part is where the pirate decides to assert full effort or not. That is, we solve the last part first, finding the equilibrium strategy given the D, then find the optimal disability pay. To ensure non-shirking behavior, it must hold that:

S S 1 m S 1− pqD−qe≥1−m ⇔ D≥ e− N −nm N −nm 1− p q N −nm


where the pirate takes the profit-share as given. The higher the level of disutility, e, the greater the pay required. Also, if the expected profit-share is great enough, no pay is needed. This effect is dampened for a high risk of injury, q, but enhanced for a higher m, as the payoff of a shirker decreases with the greater probability of being caught, ceteris paribus. For the first part of the meta game, we need to find the overall optimal D. With this in effect, no shirking will occur, i.e. n=0 . Substituting for S and seeing that optimality requires choosing the minimal level of D, the solution becomes7:


D=max 0,

1 1 m e e− 1−m 1− p q N


For D0 we have that the higher the mean value of profits, e , the lower the required D. A higher risk of injury, q, and disutility, e, naturally increases the pay. N, the number of pirates, does the same. This latter effect can be understood by the increased number of pirates sharing the profits. At the same time, recall that N influences q and p negatively, working in the direction of a lower D: A lower p affects the pay negatively due to the expected higher profits from 7 See appendix, section 6.1, for calculations as well as the resulting profits available for sharing.


winning. A higher chance of catching a shirker, m, can both entail a lower and greater D: e e ∂ D 1−m − /qN  e /1− p−m /qN  e /1− p −e /qN = = ∂m 1−m2 1−m2

The sign of the derivative will be positive for relatively high levels of disutility and low chan ce of victory, reflecting that the payoff is negative for a non-shirker 8. In fact, in this case we would be violating the participation constraint, i.e. the pirate would not sign up for the adventure to begin with. Therefore, for the payoffs implicitly considered, the sign of the derivative is negative, meaning a higher m would naturally result in a lower requirement for D.

3.3. Infinite number of interactions We now turn to the set-up of repeated games since more than one battle would be assumed to take place while aboard the ship and the actions taken today as well as the potential for dis abling injury would be determinant of the payoffs of the future. The motivation for assuming an infinite horizon, is that the discount factor in an infinite, dynamic game might be interpreted as both including the rate of time preference but also the probability of the game ending (e.g. Gibbons 1997), as the number of battles could not be thought to be known exactly at the time of agreement of the terms. As for punishment in the periods following being caught shirking, the payoff could be set to 0 for all periods. A severe penalty like receiving no future income would lower the incentive to shirk in the first place, and can be compared to the tacit collusion set-up in Tirole (1988, pp. 245), where two players set prices in a repeated Prisoners' Dilemma game, and punishment for undercutting is future profits of zero9. In the current set-up this harsh punishment may not be a credible threat seeing that the shirker may have a chance of boarding another ship and earning wages there, as staying onboard would entail a certain payoff of 0. This would howe-

8 The appendix, section 6.1.3, briefly shows this. 9 It can be sustainable that both set the price to the monopoly price and thus receive higher (>0) profits than under perfect, Bertrand, competition. The punishment for undercutting the price slightly and obtaining almost full monopoly profits is for the other player in the next period to set the price to the competitive one, fol lowed by competitive competition for all eternity by both players even though the overall profit-maximizing choice is to collude, i.e. set the higher price.


ver depend on if his reputation would suffer adequately. Alternatively, he could find a (lowerpaid) job in the marine or on shore. For simplicity, I therefore assume an outside option of w≥0 , thus extending the notion of simple pay-exclusion in Leeson (2009)10. Another aspect is what the disabled should receive in future battles. Here, 1− p  D is chosen to reflect the continuity of disability pay while on the vessel and thus in the horizon considered, though without a profit-share, nor disutility. Of course, a lower payoff chosen would enhance the incentive to shirk for a given D.  is defined as the discount factor, 0≤1 , including the probability of the game ending, assumed to be constant reflecting the same length between all periods. With this, here is the game tree as a non-shirker, where the payoffs are for the period at hand in present value: 1q




 / N −nm [ 1− p ] D−e

 / N −nm



 [ 1− p ] D


 [ 1− p ] D

  / N −nm


 [ 1− p ] D




1 2

 [ 1− p ] D



 [ 1− p ] D



  / N − nm

Given you become disabled the first round, the total payoff is: p D≡



 [ 1− p ] D−e[ 1− p] D2 [ 1− p ] D...=  −e [1− p] D N −nm N −nm 1−

If disabled in period two, the payoff is  p D for that period and onwards. Total expected payoff thus becomes, with p N ≡S / N −nm defined for convenience, and 10 Alternatively, a shirker could be allowed back after a given number of battles, if it was maximizing the overall outcome. This depends on the payoff's dependence on N. See Tirole (1988, pp. 262) for a related collusion-set-up. More on the dependence of N in section 4.


assuming independent realizations: D 2 pay non≡qp 1−q p N 1−q q  p D 1−q  p N 1−q q  2 p D1−q2 p N ... ⇒    1st stage

pay non=q

2nd stage given not yet disabled



p p 1−q ⇒ pay non = 1−1−q  1−1−q


3rd stage given not yet disabled S

[1− p] D  −e  1− N −nm 1−1−q 

Note that this assumes constancy of the variables and parameters, where the actual outcome would depend on N 11. As a potential shirker, we are looking at the period from where you choose to deviate from non-shirking. If caught slacking, you receive a payoff of 0 now and w the following periods: C



p ≡0 w w...=w  1 ...=

w 1−

The payoff when not caught is for that period equal to the profit-share, p N . The probability of getting exposed is still m, and the total payoff is found to be12: pay shirk =m

pC pN m w /1−1−m S / N −nm 1−m = 1−1−m 1−1−m  1−1−m

The incentive constraint results in the following requirement for D13:


1−1−q q 1−m−m mw   q [ 1− p ] 1−1−m 1−1−m


1− S 1−  e q [1− p ] N −nm 1− p

 

Seeing that D is unambiguously greater for higher w, this is for ease of exposition set to 0. Optimally, D should now be set to the maximum of 0 or14:

11 As fellow pirates are disabled and shirkers leave, the profit-share will be affected as q and p are dependent on the number of agents ready for battle. More on this in section 4. 12 Calculations available in the appendix, section 6.2.1. 13 For derivation, see section 6.2.2. 14 Calculations, including for the relative size of the first coefficient, can be found in section 6.2.3.


 1−1−m   m−q 1−m e 1 D= e− 1−m1q  m/1−  1− p q N .   lessthan

1 1−m

less than

1 1− p

less than

m q

For =0 , D reduces to the one-period result. The last coefficient, m−q 1−m , may be negative, depending on the parameter values, especially for small values of m. Hence, even with the other coefficients being lower, D may possibly be higher (or less negative) than in the one-period game. With varying  and three different values of m, the above expression can be illustrated as: e=3600, q=0.2, N =130,1− p=0.7, e=95


m=0 m=0,8 m→1

1,0 0

0,9 0

0,8 0

0,7 0

0,6 0

0,5 0

0,4 0

0,3 0

0,2 0

δ 0,1 0

0,0 0

160 140 120 100 80 60 40 20 0

The more you value the future and/or the higher the probability of the game continuing, the less becomes the required pay in the sketched scenario. In terms of the minimal participation constraint, the required mean value of profits in case of victory is smaller, the greater  is:

e ≥1− Nq 1−e p 15. A lower payoff is thus required for the pirate to join the crew.


Potential extensions and comments

Other types of gains could be included in the above model, e.g. utility from winning a battle and there could be disutility from being disabled besides the pure future, monetary loss. Also, if the pirates were risk-adverse, the incentive to shirk would remain and the uncertainty of profits and disability would create an increased payoff-requirement to sign up in the first place. Similarly, Weitzman and Kruse (1990) say that fluctuations in profits and risk averse agents most likely calls for combining a reduced profit-share with a fixed wage.

15 Section 6.2.4 shows the derivation, and section 6.1.3 does the same for the one-period game. Both calculated for a reservation payoff of 0.


I have in the model assumed that the agents are already onboard the ship and thus not focused on the decision about joining the crew to being with, apart from requiring a positive payoff. This could be extended to include the utility from life aboard ship without a cruel merchant captain, the disutility of risk of death and unknown income streams in the participation constraint. Also, the very negative outcome of being caught by a military opponent and receiving the capital punishment afterwards is ignored. In other words, I have assumed that the overall participation constraint of the pirate is fulfilled, meaning the pirates receive utility higher than their reservation utility (e.g. Varian 1992, pp. 441). What about the optimal number of pirates onboard? To reuse the term of meta games, we could proceed by adding a part to the game before the ones analyzed and solve for the number of pirates that should join the crew, i.e. N, given the the payoff required by the pirates to behave. The N would have to be high enough to ensure a good chance of being victorious and a low risk of injury at the same time as keeping the profit-share relatively high. Perhaps it might not even by profitable to set sail in the first place. Another point is the assumed constancy of N in the repeated game set-up. It would be affected effectively by the number of pirates being disabled along the course of the game, even while N is not reduced by slackers leaving the ship as the equilibrium disability pay prevents this. A lower effective N would reduce the chance of victory and further heighten q. The required D would be correspondingly higher. The pool of pirates available for battle could however be replenished: Leeson (2009, pp. 134) describes how, as ships are taken over, conscription of new pirates takes place. The probability of catching a shirker has been assumed exogenous but would in reality depend on the number of fellow pirates and their incentives. A non-shirker would want to expose a shirker due to the heightened risk of injury. Also, all fellow pirates would dislike the negative externality imposed on the chance of victory and would want to cut a shirker off from his profit-share, which for the last part relates to the assertion by Weitzman and Kruse (1990). Additionally, having caught a shirker would result in a higher share to the others, given a victorious outcome. The monitoring by fellow pirates is why no monitoring cost was included in the profit-maximizing problem. Another, quite different, perspective, is the moral hazard problem of ratting on an innocent that would arise from him being excluded from sharing the profits, resulting in a higher share of S available to the rest.


There would be an incentive to keep captured booty to yourself as a pirate. To ensure all the gains were shared, Leeson (2009, p. 59-65) describes how this was countered by requiring taking an oath on the bible combined with searches for hidden booty, where the punishment would be expulsion. Apparently, eternal condemnation was not a sufficient threat. A final point relates to the assumption of shirking being a binary choice and homogeneity of the pirates: Heterogeneity allows for some agents shirking, while others do not. Continuity of effort allows for a fraction of full effort being exerted, where n could be redefined to reflect this fraction times the number of agents. It may be profit-maximizing to have some shirking occurring in this set-up, as enough effort may be asserted to ensure victory. This depends on the probabilities p and q. If these are dominated by the lucrative effects of a high N compared to n, then for a given m the disability payout would be lower, thus raising overall profits, S .



Why did the era not persist? Well, the payoffs changed in that especially the associated risk of capture increased (Leeson 2009, p. 145). Greif (2006, pp. 182) explains how an institutional equilibrium may change endogenously when parameters shift. Here, the participation constraint became violated, making it unprofitable to continue pirating in the changed environment. The goal with this paper was to derive a formalized argument for the description of the incentive-corrective measures in Leeson (2009) in relation to the shirking choice. It has been shown that the disability pay and pay-exclusion can ensure that no free-riding will take place in a pirate environment, when treating the crew as economically-thinking agents. I have compared the incentives with those in both the fixed-wage scenario in Shapiro and Stiglitz (1984), the profit-sharing environment in Weitzman and Kruse (1990) and the monopolistic competition game (Tirole 1988). Hence, the motivation of agents in today's corporate world can be compared to that of the sea rovers of bygone times – perhaps even more than for the pirates of today16.

16 Leeson (2007, pp. 1088) describes how the pirating of today seems to be organized very differently.


6. Appendix 6.1. One-stage game 6.1.1. Incentive constraint S N −nm



1− p qD−qe≥1−m N −nm ⇔ 1− p qD≥−m N −nm qe ⇔ D≥ 1−1 p  e− mq

S N −nm

6.1.2. Optimal D and resulting profits D≥ 1−1 p  e− mq D≥

1 1− p


1− p e −NqD N m e q N

⇔ D≥ 1−1 p e− mq

m D ⇔ D 1−m≥


D=max 0,

1 1−m

1 1− p


1 1− p

m e q N

 e −NqD N m e q N


⇔ ⇒


The resulting overall profits for D≥0 :


1 S =[1− p]  e −N q 1−m  1−1 p e− mq

e N


⇔ e

1 m   S =[1− p] e −[1− p ] N q 1−1 m 1−1 p e[1− p ] N q 1−m ⇔ q N 1 1 1 S e S e  =[1− p]  1−m −N q 1−m e ⇔  = 1−m  [1− p] −N q e 

Higher Nqe means a lower profit available for sharing due to the cost of disability payouts (direct effect) but indirectly a higher N lowers q and increases 1− p , thus raising S . The more effective the monitoring, m, the higher the profits available to share due to the reduced incentive to shirk and its effect on the disability payments required.

6.1.3. Sign of

∂D ∂m

and participation constraint

The payoff to a non-shirker with optimal D set is positive when

[ 1− p ] [  −NN qD ] [ 1− p ] qD−qe = [ 1− p ] N −qe≥0 for e


e 1− p


 ≤ qN .

This is the minimal requirement for joining the adventure to begin with. With this, the sign of the derivative is negative, so the greater the risk of being caught, the less disability pay is required.


6.2. Repeated game set-up 6.2.1. Payoff to non-shirker and shirker D 2 pay non≡qp 1−q p N 1−q q  p D 1−q  p N 1−q q  2 p D1−q2 p N ...    1st stage

2nd stage given not yet disabled

3rd stage given not yet disabled

⇒ pay non =qp D 11−q 1−q 2 2 ...1−q p N 11−q 1−q2  2... ⇒ p p pay non =q 1−1−q 1−q 1−1−q   D


C 2 pay shirk ≡mp 1−m p N 1−mm  pC 1−m p N 1−m m 2 pC 1−m2 p N     1st stage

2nd stage given not yet caught pC shirk 1−1−m

⇒ pay




3rd stage given not yet caught pN 1− 1−m 

6.2.2. Incentive constraint S S [1− p ] D  mw  1−m  −e   1− N −nm 1− N − nm non shirk pay ≥ pay ⇔ ≥ ⇔ 1−1−q  1−1− m S S [1− p ]D mw  1− m  q  1− 1− N −nm N −nm qe ≥ −  ⇔ 1−1−q  1−1−m 1−1−q 1−1−q  S mw  1− m 1−1−q   1− N −nm [1− p ] D S q ≥ − qe ⇔ 1− 1−1−m N −nm S S 1−1− q 1−1−q 1−m  1− mw   1−  1− D≥  −  e ⇔ q [1− p ] 1−1−m  1−1−m  q [1− p ] N −nm q [1− p] N − nm 1− p S 1−1−q  q  1− m− m mw  1−  1− D≥   e q [1− p ] 1−1− m 1−1−m  q [1− p ] N −nm 1− p


 

 

    

6.2.3. Optimal D, with w=0  1−m− m 1− [1− p ][ − N qD ] D= q1−1−  1− e ⇔ m    q[ 1− p ]  N 1− p  e

 1−m −m −m 1−  D= q1− − q1−1−m  1−  D 1− e ⇔ 1−m    q  N 1−m   1− p  e

1−m −m  1−m−m 1−  D  1 q1−  1−  = q1−1−m  1− e ⇔ 1−m     q  N 1− p  e


1− 1−m  q  1−m−m   1−  1− 1−m


q 1−m −m 1− 1−m 

  1− q

e N

1−  1− e ⇔ p

The nominator in the parenthesis on the left-hand side becomes: L

nom ≡1−1−m q 1−m−m1−⇒ nom L =1−m[1−1q ] m0 After dividing on both sides with the left-hand side parenthesis and rearranging, optimal D becomes: 14

1− 1−m  1− m−q  1−m 1−  D=  1−m [1−q e−  1−m [1−q ⇔  1−] m  1− p 1−]m  q N e

D= 1−m [ 1−1− 1q ] m D=


1− 1−m   1− p

1  1−m 1q m / 1−

1 1− m

The first coefficient is less than

e−  m−q q 1−m  N


 1−1−m   1− p


 m−q 1−m  e q


, since the following holds:

1 1  1−m 1q  m/ 1− 1−m m  1−  1−m

⇔1−m1−m1q  m/1−⇔ m m  ⇔−q  1− ⇔−q 1− 1−m 1−m 

11q 

6.2.4. Minimal participation constraint 1 1−1−q 


[1− p] D 1−


p] D −e   N −nm ≥0 ⇒ q [1−1− −qe  1− p  N −NqD  ≥0 ⇔ S


1  e  qD  1− −1 − 1−qep  N ≥0 ⇔ D  1− − 1−e p  qN ≥0 ⇔ D≥ 1−   1− p − qN  e



Inserting for optimal D>0:

≥ 

 1−1−m    m −q 1−m  e 1 1− e e  1−m 1q m / 1− 1− p q N  1− p qN 1−1− m  m−q  1−m e e 1− e 1− e  1− p  qN 1−m  1q  m /1− 1− p  1−m 1q m / 1− qN 1− m−q  1−m e 1− 1−1−m  e  1−m1q m / 1− qN   1−m 1q m / 1− 1− p


1− 


[1−m  1q  m / 1− ]−mq  1−m    1−m 1q m / 1− qN m 1−[ 1−m 1q  1− ]−m q 2 1−m  e  qN 2 e qN e 2  qN [1−q ] e 1− qN e

⇔ ⇔

1− 

[1−m  1q  m / 1−]−11−m  e 1−m  1q  m /1− 1− p m  1−[ 1−m 1q  1− ]−1−m  2 e  1− p

≥ ⇔ [1−m1q 1−q  1−m] ≥[1−m1q 1−−1−m 1−m 2 ] 1−e p ⇔ [ 1q 1−q  ] ≥[1q 1−−1− ] 1−e p ⇔ ≥[1q −] 1−e p

This yields the minimal participation constraint: e e ≥[1−] qN 1− p

For D :=0 , the constraint reduces to the one-period result: 1 1− 1− q 

[ q −e 

S N −nm



⇒ − 1−qe p  N ≥0 ⇔

e qN

≥ 1−e p




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Profit-Maximizing Pirates  

Economic paper

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