How To Play Reulation Games Between A Company And A Government

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1 Decision Analysis Vol. 9, No. 2, June 2012, pp ISSN (print) ISSN (online) INFORMS Reulation Games Between Government and Competin Companies: Oil Spills and Other Disasters May Cheun, Jun Zhuan Department of Industrial and Systems Enineerin, University at Buffalo, Buffalo, New York Oil spills are a characteristic risk of oil drillin and production. There are safety reulations set to reduce the risk of technoloical failures and human error. It is the oil company s decision to follow such laws and the overnment s decision to enforce them. Companies are balancin between safety efforts and production competition with other companies. To our knowlede, no previous research has considered the impact of competition in a overnment company reulatory ame. This paper fills the ap by modelin and comparin two ames: a one-company ame without competition and a two-company ame with competition, both with the overnment as a reulator. The objectives of all players are to maximize their expected revenue and minimize their losses. Our results indicate that competition increases a company s threshold for risk and therefore requires stricter overnment reulation. These results could be eneralized and applied to other industries includin airline, nuclear power, and coal minin. Key words: ame theory; pure and mixed strateies; reulation; competition between companies; oil spill; airline industry; nuclear power; coal minin; disaster; risk History: Received on May 31, Accepted on January 11, 2012, after 2 revisions. Published online in Articles in Advance April 6, Introduction The Deepwater Horizon ri explosion (also known as the British Petroleum (BP) oil spill) in April 2010 killed 11 workers and spilled 4.9 million barrels of oil in the Gulf of Mexico, by far the larest oil spill in history (BBC News 2010). Two key factors leadin to this disaster were the company prioritizin production over safety preparedness and a lack of reulation from the overnment (The Economist 2010). In such hih-risk, low-probability situations, underevaluation of the risk of disaster undermines the value of preparation (Becker 2010). A company s priority is to stay profitable, and there is very little incentive to take costly preventive measures toward a low-probability event (Posner 2010), includin both adaptive and nonadaptive threats. Game theory has been used to study the interaction between overnment and adaptive threats such as terrorists (Zhuan and Bier 2007, Hausken and Zhuan 2011). Althouh the occurrence of oil spills is a nonadaptive technoloical risk, it is still indirectly affected by the strictness of overnmental reulations and the cooperation of oil companies (Cohen 1987, Yeaer 1990). Reulation is an important factor, and therefore a hihly researched area. For example, Cohen (1987) analyzed the effectiveness of different types of reulation, includin incentive systems (Kambhu 1990, Lin 2004), cost observation (Laffont and Tirole 1986), surveillance (Oh 1995), liability (Shavell 1984), and fines (Polinsky and Shavell 1979, Kadambe and Seerson 1998). To more effectively reulate, possible factors that motivate an entity to take risk and disreard reulation must also be analyzed. Specifically, to strenthen safety preparedness in oil operations, factors such as competition need to be studied. Competition between companies could motivate each company to decrease their level of safety preparedness, focusin more on production in attempts to ain a competitive ede. To our knowlede, no previous paper has studied the impact of competition on a reulation ame, which we define as the strateic 156

2 Decision Analysis 9(2), pp , 2012 INFORMS 157 interaction between the reulator and reulated entities throuh the medium of laws and rules. This paper fills the ap by modelin and comparin two ames: a one-company ame without competition and a two-company ame with competition. In particular, the overnment and oil companies are modeled as rational players who assess their risk and choose the best stratey to maximize their own utility. The safety reulations (i.e., the laws) are publically known: the overnment decides whether to spend resources to enforce the reulation, whereas the oil companies decide whether to spend resources to follow the reulation. If they are nelient in conformin with safety reulations, they risk ettin shut down if cauht. If the overnment is also nelient, not watchin over the companies to ensure safe oil operations, then all parties must risk the consequence of a spill. On the other hand, inorin safety reulations could save both the company and overnment money if no spill occurs. There is a trade-off between havin protection from future risks at a cost or savin expenses on safety at a risk. This trade-off is further complicated by the competition between companies, which will be studied in this paper. The next section introduces notation, assumptions, and problem formulation. Section 3 studies the onecompany ame, presentin the best response for the company and the optimal reulation stratey for the overnment. Section 4 adds a second company and restudies the ame, comparin results with those in 3. Section 5 summarizes the paper, eneralizes results with applications to other industries, and provides some future research directions. The appendix provides proofs on all theorems stated. 2. Notation, Assumptions, and Problem Formulation The parameters of our model are defined as follows: : the probability the overnment assins to checkin a company (companies); yz: the probability company Y Z assins to followin reulations; p: the probability of an oil spill; c 1 : the cost to follow the safety reulations (to satisfy requirements, companies must properly train employees, run routine inspections, and maintain equipment updates); c 2 : the cost to shut down operations (if an oil company is cauht not followin reulations, the overnment will not allow any more oil exploration or production); c 3 : the cost of an oil spill (which includes cleanup costs, liabilities, damaed public imae, and overnment fines; this cost affects both the oil companies and the overnment; the overnment may eventually et reimbursed by the oil company but in the meantime must cover the costs itself first); k: the cost to the overnment to check an oil company; r 0 : the revenue received by the company (when there is only one in the market); r 1 : the revenue received by both companies that follow safety reulations; r 2 : the revenue received by the company that does not follow (when the competitor follows); r 3 : the revenue received by the company that follows (when the competitor does not follow); r 4 : the revenue received by both companies that do not follow; L Y, L Z, and L G : the expected loss of company Y, company Z, and the overnment, respectively; R Y, R Z : the expected revenue for company Y and company Z, respectively; y, z : company s best response to the overnment s checkin probability ; : the overnment s optimal checkin probability; and 1, 2, and 3 : specific checkin probabilities for the overnment as defined later in this paper Assumptions For simplicity, we assume that if the safety laws are followed, there is no risk of spill. In reality, a risk of spill may be present because of natural causes, but we focus on the risk that can be controlled namely, human errors and technoloical failures. The probability of a spill p is exoenously iven and known by all players. Furthermore, in the event of a spill, all spills are of the same manitude. We study a sequential ame, one where the oil companies are the second mover. In practice, the oil companies may have the option of observin the overnment s strictness in enforcin laws

3 158 Decision Analysis 9(2), pp , 2012 INFORMS (throuh observin historical data, actions with other companies, etc.). In contrast, it may not be as plausible to assume a situation where the overnment can observe an oil company s actions without actually checkin on it. The followin relationship between costs is assumed: c 2 > c 1 > pc 3 > 0. The larest cost is c 2, when an oil company s operations are shut down. The cost c 1 for each company to follow reulation is assumed to be reater than the expected cost of a spill, pc 3. This assumption is necessary for the oil company to have an incentive to take risk, to avoid dealin with trivial problems. The two oil companies are assumed to have the same company profile, meanin that they have identical resources, levels of oil production, and associated operation costs. They are competin in the same oil market, where the demand is assumed to be constant. If there is only one company, its revenue is constant at r 0 but is affected by its decision to follow reulations. If there is more than one company, they must compete for revenue; the followin relationship then applies: r 0 > r 2 > r 1 > r 4 > r 3. If both choose to follow reulations, each will have an equal share of r 1. If one chooses not to follow and the overnment checks, its operations will be shut down, ivin its competitor the entire market with a revenue of r 0. If one chooses not to follow and the overnment does not check, it will have a market advantae and therefore receive the hiher share r 2 while the other receives the lower share r 3. If both companies produce at their maximum level, the market will be oversaturated, and individual revenue will drop to r 4. Each player is assumed to prefer lower cost and hiher revenue. Furthermore, each player understands the setup of the ame and will choose whichever pure or mixed stratey will maximize its utility at equilibrium. When the players become indifferent between two courses of action (ivin equal utilities), they are assumed to choose the safer option (i.e., the overnment chooses to check and the company chooses to follow). When there are two companies, the overnment pays k to check each company and is liable for coverin the cost of each oil spill (i.e., if there are two spills, the overnment must cover the costs of both). Table 1 Payoff Table for Simultaneous Two-Company Game Company Z Company Y Follow Not follow Follow Not follow r 1 c 1 r 1 c 1 r 0 c r 3 c 1 c r 2 pc 3 c r 2 pc 3 r 0 c r 3 c 1 c r 4 pc 3 c r 4 pc 3 Notes. Companies Y and Z each have two possible actions: follow and not follow. For each payoff cell, the lower payoff is for Y and the upper payoff is for Z Problem Formulation The companies objectives are to maximize their profits, U i = R i L i i = Y Z, by choosin probabilities y and z, respectively, to follow reulations. The overnment s objective is to minimize its loss L G by choosin the probability to check companies to enforce reulations. These equations are derived from each ame tree in Fiure 1 throuh backward induction with respect to each player s expected payoffs: max U Y y = [ yr 0 c 1 +1 y c 2 ] +1 [ yr 0 c 1 +1 yr 0 pc 3 ] min L G y=k+1 1 ypc 3 If the company chooses to follow reulations (with probability y), it will receive revenue r 0 with cost c 1. If the company chooses to not follow reulations (with probability 1 y), it will receive cost c 2 with probability and a revenue r 0 and cost pc 3 with probability 1. If the overnment chooses to check (with probability ), it will cost k. If the overnment chooses to not check (with probability 1 ), it will receive a cost of pc 3 with probability 1 y. In the two-company ame, the companies play a simultaneous ame with each other first and then interact with the overnment. These outcomes are presented in Table 1. The objectives for the two-company ame are max U Y y z = [ yzr 1 c 1 + y1 zr 0 c yz c 2 ] +1 [ yzr 1 c 1 +y1 zr 3 c 1 +1 y zr 2 pc y1 zr 4 pc 3 ]

4 Decision Analysis 9(2), pp , 2012 INFORMS 159 max U Z y z = [ yzr 1 c 1 +1 yzr 0 c 1 +y1 z c 2 ] + 1 [ yzr 1 c yzr 3 c 1 + y1 zr 2 pc y1 zr 4 pc 3 ] min L G y z = 2k + 1 [ y1 zpc yzpc y1 z2pc 3 ] 3. One-Company Game We first look at a one-company sequential ame, where the overnment moves first and the company moves second, as depicted in Fiure 1(a). Company Y has a monopoly on the market and potential revenue is constant at r 0. Theorem 1. The best response of company Y to the overnment in the one-company ame is y = { 0 if < 1 1 if 1 where 1 c 1 pc 3 r 0 + c 2 pc 3 Remark 1. The boundary condition at which the company becomes fully deterred from takin any risk, switchin from followin to not followin, is at the overnment s checkin probability 1. In other words, if the overnment wants to motivate oil company Y to follow safety reulations, it must check with at least a probability of 1. Theorem 2. The overnment s optimal reulation probability for the one-company ame is { 0 if LG 00<L = G if L G 00 L G 1 1 where 1 c 1 pc 3 r 0 +c 2 pc 3 Remark 2. Theorem 2 states that the two possible reulation probabilities minimizin the overnment s loss are 0 and 1. The overnment should choose not to check at all or to check with a probability of 1 (the company will react, respectively, with not followin and followin). 4. Two-Company Game The addition of a second company, as depicted in Fiure 1(b), creates competition in the ame. With constant market demand, the two companies now must share the market, lowerin their potential revenue. There are four levels of potential revenues, r 1, r 2, r 3, and r 4, which depend on both companies decisions to follow or not follow safety reulations. In the sequential ame, the overnment moves first and then both companies move toether (i.e., both companies play a simultaneous-move ame). Fiure 1 Game Trees (a) One-company ame a (b) Two-company ame b Company Y Company Y Company Z z (2k, r 1 c 1, r 1 c 1 ) Government 1 y 1 y y 1 y (k, r 0 c 1 ) (k, c 2 ) (0, r 0 c 1 ) (pc 3, r 0 pc 3 ) Government a The two numbers in the parentheses are the utilities of the overnment and company Y, respectively. b The three numbers in the parentheses are the utilities of the overnment, company Y, and company Z, respectively. 1 y 1 y y 1 y 1 z z 1 z z 1 z z 1 z (2k, r 0 c 1, c 2 ) (2k, c 2, r 0 c 1 ) (2k, c 2, c 2 ) (0, r 1 c 1, r 1 c 1 ) (pc 3, r 3 c 1, r 2 pc 3 ) (pc 3, r 2 c 3, r 3 c 1 ) (2pc 3, r 4 pc 3, r 4 pc 3 )

5 160 Decision Analysis 9(2), pp , 2012 INFORMS Theorem 3. The company Y s and company Z s best response to each other and the overnment in the two-company sequential ame is y z 0 if < 2 r 4 + r 3 r 0 c 2 + pc 3 + r 4 r 3 + c 1 pc 3 = r 4 + r 3 + r 2 r 0 + r 4 r 3 r 2 + r 1 if 2 < 3 where 1 if 3 2 r 4 r 3 + c 1 pc 3 r 0 + r 4 r 3 + c 2 pc 3 3 r 2 r 1 + c 1 pc 3 r 2 + c 2 pc 3 and Theorem 4. The threshold checkin probability required to fully deter the oil companies from bein nelient in followin safety procedures increases from 1 in the one-company case to 3 in the two-company case, with an addition of threshold 2, where the companies start bein deterred; that is, 1 < 2 < 3. Remark 3. Theorem 3 shows that competition has introduced new boundary conditions 2 and 3. At boundary condition 2, the companies switch from usin pure strateies (not follow, not follow) to usin mixed strateies. At boundary condition 3, the companies switch from mixed strateies to pure strateies (follow, follow). Theorem 4 supports that competition increases an oil company s willinness to take risks, shiftin the boundary conditions up from the oneto the two-company ame. In particular, oin from panel (a) to panel (b) in Fiure 2, the reion where the companies play the pure stratey follow expands, whereas the reion where the companies play the pure stratey not follow shrinks. With the addition of competition, the decision to follow safety reulations becomes more complicated; there are more factors to weih in when an oil company is studyin the cost benefit analysis of takin such a risk. This means that two oil companies in competition have a hiher likelihood to disreard safety reulations than one company itself. The implication is that in the real-world oil industry, where there is more than just one rival company, competition is hiher, and so is the incentive to cut corners on safety. Theorem 5. The overnment s optimal reulation probability for the two-company ame is 0 if L G < L G 2 y 2 z 2 = 2 3 and L G < L G if L G 2 y 2 z 2 < L G and L G 2 y 2 z 2 < L G if L G < L G and L G < L G 2 y 2 z 2 Remark 4. Theorem 5 shows that competition amon the oil companies has introduced three other Fiure 2 Best Response Comparison (a) One-company ame (b) Two-company ame Pure (not follow, not follow) Pure (follow, follow) Pure (follow, follow) Pure (not follow, not follow) Mixed (y, z) pc 3 pc 3 c 1 r 4 r 3 + c 1 0 c 1 r 4 r 3 + c 1 r 2 r 1 + c 1 1 c 1 pc 3 r 4 r 3 + c 1 pc 3 r 2 r 1 + c 1 pc 3 2 r 0 + c 2 pc 3 r 0 + r 4 r 3 + c 2 pc 3 3 r 2 + c 2 pc 3

6 Decision Analysis 9(2), pp , 2012 INFORMS 161 possible optimal checkin probabilities: 0, 2, and 3. As Theorems 3 and 4 indicate, the companies risk threshold has increased. Therefore, to maintain the same level of safety from the one-company ame to the two-company ame, the overnment must implement a hiher checkin probability. The implication here is that in the real-world oil industry, where there are more than just two oil companies to oversee, a strinent level of reulation is even more necessary. Understandin that competition is a persistent factor and how it affects a company s decision in reard to safety reulations is a key step in preventin another disaster such as the BP oil spill. 5. Conclusion and Extensions 5.1. Summary In the one-company ame, we solved for one company s best response, which included only pure strateies with one boundary condition, and for the overnment s optimal checkin probabilities, which included two. In the two-company ame, the companies used both pure and mixed strateies in their best response with three boundary conditions, and the overnment had three optimal checkin probabilities. Our results indicated that the competition introduced by addin a second company shifted the boundary conditions and, therefore, the optimal checkin probabilities upward. We conclude that competition effectively increases a company s motivation for risk takin and in turn requires stricter reulation from the overnment. Understandin that competition affects the willinness of oil companies to comply with safety standards could help the overnment more effectively reulate and prevent oil disasters, especially when in reality there are more than just two oil companies. It is interestin to note the counterintuitive observation that competition can have such neative impacts. Competition is well known to be able to help reduce prices for consumers, motivate innovation in companies, and produce lean systems of production (Rosenau 2003). However, our result indicates that, in contrast, competition makes it more difficult for the overnment to reulate companies, because they become more willin to take on risk (i.e., havin an increased risk threshold) to ain a competitive ede in the market Application to Other Industries In this subsection, we discuss potential applications of our results to other industries includin U.S. airlines, U.S. nuclear enery, and coal minin in China. In all the examples provided, increased competition has motivated companies in various industries to find alternative methods to stay profitable. These alternatives have reduced safety levels and increased the risk of disaster. Since the U.S. airline dereulation in 1979, competitiveness has proven difficult because of the volatile nature of airline profits, which are heavily affected by fuel prices and the attacks on September 11, 2001 (Gowrisankaran 2002). In the two instances discussed below, U.S. airlines seem to be prioritizin profitability over safety in attempts to stay competitive. One instance is that the U.S. Airport Transportation Association (ATA) sent a letter requestin the U.S. Federal Aviation Administration to reconsider its pilot fatiue reulation (the reulation requires loner rest periods for pilots) (Crawley 2011). In that letter, the ATA concluded that the reulation measure was too costly to comply with, estimatin a loss of $2 billion dollars and 27,000 jobs. Another instance is the 2011 court trial where the U.S. Airline Pilots Association sued US Airways for forcin pilots to fly under unsafe conditions (to improve company performance and maintain competitiveness) (Portillo 2011). When the U.S. electricity utility industry started dereulation durin the early 1990s, Edison Electric Institute studied nuclear power s competitive potential aainst other retail competition (Riccio 1996). This study concluded nuclear power plant operations and maintenance to be expensive. To stay competitive, plants have increased productivity beyond their established capabilities, violatin many safety reulations, and in response, the Nuclear Reulatory Commission has weakened safety standards so that ain nuclear reactors can stay in compliance (Donn 2011). These practices have certainly increased the risks of a nuclear meltdown. In the mid-1990s, coal minin conditions in China s state-owned enterprises (SOEs) stopped improvin, keepin fatality rates hih (Wriht 2004). This is because the rise of township and villae enterprises (TVEs) created a competitive environment that depressed waes and conditions. TVEs use cheap

7 162 Decision Analysis 9(2), pp , 2012 INFORMS labor from China s rural areas, forcin the decline of profits for SOEs. Accordin to Wriht (2004), this competitive environment pressures SOEs to cut costs, with safety bein larely impacted. Outdated technoloy continues to be used by inexperience workers, who are often asked to work extra shifts under hihdemand production requirements. In all three aforementioned industries, the difficulties of stayin competitive have incentivized companies to cut corners on safety preparedness. The U.S. airline industry is resistin improvements to standards and orderin planes to fly under unsafe conditions, the U.S. nuclear enery industry is pushin ain nuclear plants to operate beyond their limitations, and the Chinese coal minin industry is subjectin inexperience workers to stressful and danerous environments. In such cases, stricter reulation, in terms of both settin and enforcin laws, is crucial to maintainin a hih safety standard and preventin disasters. The absence or corruption of safety reulation will and has led to formidable consequences Future Research Directions In this paper, we have seen a sinificant difference in the company s best response and overnment s optimal stratey when there is an addition of just one other company. An extension to an N -player ame would help eneralize the conclusions made. There may be further interestin results to be found, such as the effect of competition approachin a limit as more companies are added. We have studied a sequential ame where the oil companies had perfect information about the overnment s checkin probability. Various forms of reulation ames with competition could also be studied, includin those with imperfect/incomplete information and simultaneous-move ames between the overnment and oil companies. The assumption of the oil companies havin identical resources and utility functions simplifies the players interactions with each other and the overnment. In the future, we could study heteroeneous company profiles. In addition, the players were assumed to be risk neutral. To build a more robust model, we would like to allow players to have some risk preferences (i.e., risk seekin, risk averse; see Laffont and Rochet 1996). Finally, addin a revenue component to the overnment will ive more dynamic interactions between the players. The overnment also receives royalties from the same companies that they reulate. This conflict of interest was another crucial component causin the BP oil spill (Walsh 2010). We can then study the direct impact of competition on the overnment. In addition to competition, vertical interation and control issues between the oil companies are other major factors that could be studied. Acknowledments The authors thank the special issue editors and two anonymous referees of Decision Analysis for useful comments and suestions. This research was partially supported by the U.S. Department of Homeland Security throuh the National Center for Risk and Economic Analysis of Terrorism Events (CREATE) [Award 2010-ST-061-RE0001]. However, any opinions, findins, and conclusions or recommendations in this document are those of the authors and do not necessarily reflect views of the U.S. Department of Homeland Security or CREATE. Jun Zhuan is the correspondin author. Appendix Proof to Theorem 1 The overnment checks with probability ; company Y reacts to the overnment with probability y. We solve for the optimal checkin probability that makes company Y indifferent between its pure strateies follow and not follow. By settin equal their expected payoffs, Expected payoff of followin = r 0 c r 0 c 1 = r 0 c 1 Expected payoff of not followin = c r 0 pc 3 we et 1 = c 1 pc 3 /r 0 + c 2 pc 3. If < 1, the expected cost of followin is reater than not followin, and thus y = 0. Likewise, if > 1, we have that the expected cost of followin is less than that of not followin, and thus y = 1. Proof to Theorem 2 First, any probability 0 1 is dominated by = 0 because increasin the checkin probability in this reion will not chane the company s decision accordin to Theorem 1; it will only increase the cost for the overnment. The company will always choose to not follow. Second, any probability 1 1 is dominated by = 1 because aain increasin the checkin probability in this reion will not chane the company s decision accordin to Theorem 1; it will only increase the overnment s cost. The company will always choose to follow. Therefore, there are only two possible optimal checkin probabilities: 0 and 1. The overnment s objective function for the one-company ame is min L G = k ypc 3

8 Decision Analysis 9(2), pp , 2012 INFORMS 163 The overnment will compare the two expected loss outcomes L G 0 0 = pc 3 and L G 1 1 = c 1 pc 3 /r 0 + c 2 pc 3 k, and choose the that will minimize its loss. Proof to Theorem 3 We solve for the simultaneous ame between the companies first, for which then we can define the boundary condition for. Table 1 is a payoff table for the two companies for their two pure strateies. It is worthy to note that because the company profiles are identical, the payoff table is also symmetric, ivin identical equilibriums for companies Y and Z. For (not follow, not follow) to be a Nash equilibrium, the expected payoff of playin not follow must be reater than the expected payoff of playin follow : c r 4 pc 3 > r 0 c r 3 c 1 Solvin for, we et < 2 r 4 r 3 + c 1 pc 3 r 0 + r 4 r 3 + c 2 pc 3 For (follow, follow) to be a Nash equilibrium, we must have Solvin for, we et r 1 c 1 c r 2 pc 3 3 r 2 r 1 + c 1 pc 3 r 2 + c 2 pc 3 For (follow, not follow) and (not follow, follow) to be Nash equilibria, we must have r 1 c 1 < c r 2 pc 3 and c r 4 pc 3 < r 0 c r 3 c 1 Solvin for, we et, respectively, r 4 r 3 + c 1 pc 3 r 0 + r 4 r 3 + c 2 pc 3 < r 2 r 1 + c 1 pc 3 r 2 + c 2 pc 3 and Simplifyin this, we et 2 < 3. Because there is no dominant pure stratey for this condition, we assume that both companies use mixed strateies. To find the mixed stratey equilibrium, we look for the z and y that will make companies Y and Z, respectively, indifferent to their pure strateies: Expected payoff (Y follow) = zr 1 c z [ r 0 c r 3 c 1 ] Expected payoff (Y not follow) = z [ c r 2 pc 3 ] + 1 z [ c r 4 pc 3 ] Expected payoff (Z follow) = yr 1 c y [ r 0 c r 3 c 1 ] Expected payoff (Z not follow) = y [ c r 2 pc 3 ] + 1 y [ c r 4 pc 3 ] By settin these two payoffs equal and solvin, we et y =z = r 4 r 3 +r 0 +c 2 pc 3 r 4 +r 3 c 1 +pc 3 r 4 r 3 r 2 +r 0 r 4 +r 3 +r 2 r 1 Proof to Theorem 4 The boundary conditions where the companies et deterred from bein nelient are 1 and 2 for the one- and twocompany ames, respectively. We prove below that 1 < 2 in the followin two cases for pc 3 (refer to Table 1): (1) 0 pc 3 < c 1. We set a c 1 pc 3 > 0, b c 2 c 1 + r 0 > 0, c = r 4 r 3 > 0. After substitutin these values into we et r 4 r 3 + c 1 pc 3 < r 2 r 1 + c 1 pc 3 r 0 + r 4 r 3 + c 2 pc 3 r 2 + c 2 pc 3 a a + b < a + c a + b + c which simplifies to 0 < bc. This inequality always holds, so 1 < 2. (2) pc 3 c 1. When pc 3 = c 1, 1 = 0 while 2 > 0, so 1 < 2. We prove next that 2 < 3 in the followin two cases for pc 3 (refer to Table 1): (1) 0 pc 3 < r 4 r 3 + c 1. We set a r 4 r 3 pc 3 + c 1 > 0 b c 2 c 1 +r 0 > 0 c r 4 +r 3 +r 2 r 1 > 0. After substitutin these values in we et r 4 r 3 + c 1 pc 3 < r 2 r 1 + c 1 pc 3 r 0 + r 4 r 3 + c 2 pc 3 r 2 + c 2 pc 3 a a + b < a + c a + b + c which simplifies to 0 < bc. This inequality always holds, so 2 < 3. (2) pc 3 r 4 r 3 + c 1. When pc 3 = r 4 r 3 + c 1, 2 = 0 while 3 > 0, so 2 < 3. Combinin these two results, we have 1 < 2 < 3. Proof to Theorem 5 Usin the same reasonin from the proof to Theorem 2, any 0 2 is dominated by = 0. The companies will choose (not follow, not follow) in this reion. Similarly any 2 3 is dominated by = 2, where the companies use a mixed stratey. Last, any 3 1 is dominated by = 3, and the companies will choose (follow, follow) in

9 164 Decision Analysis 9(2), pp , 2012 INFORMS this reion. The overnment objective for the two-company ame is represented by min L G yz=2k+1 [y1 zpc 3 +1 yzpc 3 +1 y1 z2pc 3 ] and because y =z, the equations is simplified to min L G yz=2k+1 1 y2pc 3 Comparin the three expected loss outcomes, L G 000=2pc 3 L G ( 2 y 2 z 2 ) = 2 2k+1 2 ( 1 r 4 r 3 +r 0 +c 2 pc 3 r 4 +r 3 c 1 +pc 3 r 4 r 3 r 2 +r 0 r 4 +r 3 +r 2 r 1 L G 3 11= 3 2k+1 2pc 3 ) 2pc 3 the overnment will choose the that minimizes its loss. References BBC News (2010) Timeline: BP oil spill. (September 19), Becker G (2010) The ulf oil leak and optimal safety precautions. Beckner-Posner Blo (July 4), -becker.html. Cohen MA (1987) Optimal enforcement stratey to prevent oil spills: An application of a principal-aent model with moral hazard. J. Law Econom. 30(1): Crawley J (2011) US airlines say pilot fatiue rule would cost jobs. Thomson Reuters (September 15), article/2011/09/15/airlines-fatiue-iduss1e78e1wt Donn J (2011) U.S. nuclear reulators weaken safety rules, fail to enforce them: AP investiation. Huffinton Post (June 20), Economist, The (2010) The oil well and the damae done. (June 17), Gowrisankaran G (2002) Competition and reulation in the airline industry. Federal Reserve Bank of San Francisco Economic Letter , January 18. Accessed January 5, 2012, el html. Hausken K, Zhuan J (2011) Governments and terrorists defense and attack in a T -period ame. Decision Anal. 8(1): Kadambe S, Seerson K (1998) On the role of fines as an environmental enforcement. J. Environ. Plannin Manaement 41(2): Kambhu J (1990) Direct controls and incentives systems of reulation. J. Environ. Econom. Manaement 18(2): Laffont J-J, Tirole J (1986) Usin cost observation to reulate firms. J. Political Econom. 94(3): Laffont J-J, Rochest J-C (1996) Reulation of a risk averse firm. Games Econom. Behav. 25(2): Lin S-W (2004) Desinin incentive systems for risk-informed reulation. Ph.D. dissertation, Department of Industrial and Systems Enineerin, University of Wisconsin Madison, Madison. Oh Y (1995) Surveillance or punishment? A second-best theory of pollution reulation. Internat. Econom. J. 9(3): Polinsky AM, Shavell S (1979) The optimal tradeoff between the probability and manitude of fines. Amer. Econom. Rev. 69(5): Portillo E (2011) US airways, pilots union spar in court. Charlotte Observer (Auust 19), 08/19/ /us-airways-pilots-union-spar-in.html. Posner R (2010) The ulf oil leak. Becker-Posner Blo (July 4), Riccio J (1996) Competition and nuclear power: A danerous combination. Synthesis/Reeneration Maazine Online 11 (Fall), Rosenau PV (2003) The Competition Paradim: America s Romance with Conflict, Contest, and Commerce (Rowman and Littlefield, Lanham, MD). Shavell S (1984) A model of the optimal use of liability and safety reulation. RAND J. Econom. 15(2): Walsh B (2010) The Gulf disaster: Whose asses need kickin? Time Maazine (June 10), article/0,9171, ,00.html. Wriht T (2004) The political economy of coal mine disasters in China: Your rice bowl or your life. China Quart. (179): Yeaer PC (1990) The Limits of Law: The Public Reulation of Private Pollution (Cambride University Press, Cambride, UK). Zhuan J, Bier VM (2007) Balancin terrorism and natural disasters Defensive stratey with endoenous attack effort. Oper. Res. 55(5):