Maintaining Auditor Independence AbstractIn this paper, we resolve the auditor independence problem that arises in the model studied in Antle (1982, 1984). We argue that the owner-manager-auditor relationship exhibits a "separability" that facilitates the use of a particularly simple mechanism that prevents collusion. The mechanism prevents collusion as long as the agents do not play strictly dominated strategies and assume the same to be true of the other agent; the owner's profits under this mechanism are arbitrarily close to second-best (the optimal Nash contract). The fact that a simple mechanism can be used to maintain auditor independence in a theoretical model strengthens our belief that, in practice, there exist institutions (e.g., the courts) that can maintain auditor independence with relative ease. This may be one reason why the institution of auditing is so enduring. 1. IntroductionAuditing is a vital institution. It seems unlikely that the modern firm, which is characterized by a separation of ownership and management, would be viable without auditor-verified financial statements. Preventing auditor-manager collusion (maintaining auditor independence) is critical to the survival of auditing, because when the auditing process "breaks down because of collusive behavior ... [it] can have severe economic repercussions and can erode public confidence in the profession" (Deloitte and Touche 1993, 13). Defining auditor independence precisely, however, has proved to be a daunting task for the standard-setting bodies. In a seminal paper, Antle (1984) articulated various definitions of auditor independence in the context of a formal model. Antle modeled both the manager and the auditor as economic agents and explored the extent to which the two parties may wish to cooperate while acting in their self interests. He showed that a naive extension of the owner-manager contract to include the auditor raises serious concerns about auditor independence. The solution to the naive program, the second-best solution, may call for the auditor and manager to play a Pareto-dominated (Bayes) Nash equilibrium. The question of interest is: can auditor independence be maintained and, if yes, at what cost?In this paper, we argue that the owner-manager-auditor relationship exhibits a "separability" that facilitates the use of a particularly simple mechanism in maintaining auditor independence. The mechanism prevents collusion as long as the agents do not play strictly dominated strategies and assume the same to be true of the other agent; as a corollary, obedient behavior is obtained as the unique Nash equilibrium in both pure and mixed strategies. As many texts on noncooperative game theory point out, a game with a unique strategy combination that survives a small number of rounds of iteratively removing strictly dominated strategies provides the players with an obvious way to play (see, e.g., Kreps 1990, 396). The problem is that most games cannot be solved in this way. Under the mechanism we construct, the owner's profits are arbitrarily close to second best. The ability to resolve the collusion problem almost costlessly using a simple mechanism provides us with some confidence that, in practice, observed institutions may be effective in maintaining auditor independence. In the last section of the paper, we attempt to relate our mechanism to the institution of the courts.The remainder of the paper is organized as follows. Section 2 formalizes definitions of auditor independence and motivates our study of simple mechanisms. Section 3 presents the model. Section 4 presents the result, and Section 5 concludes the paper. 2. Auditor Independence and Simple MechanismsThe owner-manager-auditor model studied in this paper was first presented in Antle (1982, 1984). As Antle (1984) points out, the degree of auditor independence can be related to the behavioral assumptions appropriate in predicting the auditor's actions. Although the following definitions are not identical to those presented in Antle, they are much in the same spirit. DEFINITION 1. An auditor is weakly independent if he will play the strategy prescribed by the owner only if it is the unique strategy that survives the removal of strictly dominated strategies (i.e., it is a strict dominant strategy) in the manager-auditor subgame. DEFINITION 2. An auditor is semi-weakly independent if he will play the strategy prescribed by the owner only if it is the unique strategy that survives two rounds of iteratively removing strictly dominated strategies in the manager-auditor subgame. DEFINITION 3. An auditor is independent if he will play the strategy prescribed by the owner only if it is the auditor's strategy under the unique Nash equilibrium in the manager-auditor subgame. DEFINITION 4. An auditor is strongly independent if he will play the strategy prescribed by the owner only if it is the auditor's strategy under some Nash equilibrium in the manager-auditor subgame.Antle (1982, 1984) showed that it is costlier for the owner to prevent collusion when the auditor is weakly independent than when he is strongly independent. Following the approach taken in Ma, Moore, and Turnbull (1988), Sen (1994) and Yost (1995) showed that it is no more costly for the owner to prevent collusion when the auditor is independent than when he is strongly independent. In this paper, we show that the owner need incur only an arbitrarily small cost in order to prevent collusion when the auditor is semi-weakly independent rather than strongly independent.A mechanism that is effective in preventing collusion with semi-weakly independent auditors is appealing because it will (1) in some cases improve actual independence (if the auditor is semi-weakly independent but not independent or strongly independent) and (2) improve the appearance of independence even when the auditor is, in fact, independent or strongly independent. As Blough (1960, 58) points out, "[s]ince one's usefulness as an auditor is impaired by any feeling on the part of third parties that he is likely to lack independence, he has the responsibility of not only maintaining independence in fact but of avoiding any appearance of lacking independence."Potentially collusive strategy combinations would seem to be those under which both agents are at least as well off as they are under the owner's desired strategy combination. Even though the owner's desired strategy combination is achieved in two rounds of iteratively removing strictly dominated strategies under our mechanism, it has an appealing feature that is generally true of only strict dominant strategy mechanisms. Under our mechanism, all potentially collusive strategy combinations involve the play of strictly dominated strategies. Mechanisms typically used in the implementation literature, including Ma, Moore, and Turnbull-type mechanisms, have recently come under close scrutiny because of their complexity (Jackson 1992, Moore 1992). The criticisms include the use of infinite message spaces even for binary models and the use of tail-chasing constructions under which best replies are not well defined. These criticisms have led to the development of simple mechanisms, albeit in the context of structured settings. A contribution of the work on simple mechanisms is that it provides some assurances about the more general results obtained in the implementation literature, since the work demonstrates that, at least in some specific settings, less objectionable mechanisms can be effective. We believe that work on simple mechanisms may be of interest to accountants for two reasons. First, simple mechanisms are, presumably, easier to relate to observed institutional arrangements. When institutional arrangements are designed to coordinate the activities of managers in a firm, one would expect these institutional arrangements to include accounting practices. Second, work on simple mechanisms helps identify relationships (e.g., the auditor-manager relationship in this paper) and organizational structures in which control problems can be more easily dealt with. Although the literature on simple mechanisms is at a fairly early stage, one theme that has emerged from the work (including our paper) is that a certain amount of separability in the agents' environments often simplifies control. Roughly stated, separability helps the principal (mechanism designer) sort things out when an off-equilibrium-path event occurs. Antle's auditing model includes public information beyond the information produced by the accounting system. In this paper, we add the assumption that for each agent, there is at least one component of the nonaccounting information that provides direct information only about him. That is, given an agent's action and information, this component of nonaccounting information is uninformative of the other agent's action and information. The assumption of separability in the agents' environments is a demanding one, unless the setting being studied is one in which the agents have distinct roles. This is true of the auditor-manager relationship. The manager is responsible for stewardship and reporting, while the auditor's role is one of attestation. As we argue in Section 5, in the absence of separable information, implementation via the iterative removal of strictly dominated strategies would be impossible.Hereafter, when we refer to simple mechanisms, we are referring to mechanisms that are consistent with semi-weakly independent auditors. DEFINITION 5. A mechanism is simple if the strategy combination the owner most prefers is the unique strategy combination that survives two rounds of iteratively removing strictly dominated strategies in the agents' subgame. 3. ModelA principal (owner) contracts with two agents, a manager and an auditor. If the manager joins the firm, he takes an action a + {aL,aH} that affects the distribution of two variables, x + {xL,xH} and m + {mL,mH}. Denote the probability distributions byPr(x | a) and Pr(m | a). x is interpreted as the output (dividend) consumed by the owner, and m is interpreted as net earnings. Assume the two distributions satisfy first-order stochastic dominance (FOSD). That is, Pr(xH | aH) > Pr(xH | aL) and Pr(mH | aH) >Pr(mH | aL). If the auditor joins the firm, he takes an investigative act b + {bL,bH}. b generates a random variable y + {yL,yH}. Denote the probability distribution of y by Pr(y | a,b,m). The dependency of the distribution on a, b, and m allows for y to be interpreted as monitoring information about a, about m, or about both a and m.None of the variables a, b, m, y, or x are available for contracting. The reason a and b are assumed to be noncontractible is that each agent's action is observed neither by the owner nor by the other agent (a hidden action problem). The reason m and y are assumed to be noncontractible is that the manager privately observes m, while the auditor privately observes y (a hidden information problem). The reason x is assumed to be noncontractible is that the agents' contracts have to be settled before x is realized. The variables available for contracting are m^ , y^ , and z . m^ is the manager's report on m, y^ is the auditor's report on y, and z is publicly available information. z is a k-dimensional vector and has a probability distribution Pr(z | a,b,m,y). Eachcomponent of z can take on one of two values, zL or zH. By allowing z to be multi-dimensional, we allow for the possibility that various components of z provide different types of information. For example, one component could be a random variable generated by the agents' actions alone, while some other component could be a random variable generated by both the agents' actions and their private information. z can be thought of as a nonaccounting source of information. z can also include accounting information but only if the information cannot be manipulated by the manager or the auditor. The interpretations of the various components of z that Antle offers include a general economic indicator (such as GNP), a variable capturing local market conditions, and the product of litigation. We assume y is more informative of m if the auditor chooses bH than if he chooses bL. For expositional purposes, we assume Pr(y | a,bL,m) = 0.5 and Pr(yk | a,bH,mk) > Pr(yk | a,bH,mn), k _ n. These assumptions imply that if the auditor chooses bL, he receives an uninformative signal; if the auditor chooses bH, it is more likely m and y will match than not match.The owner's contracts with the manager and auditor are a function of m^ , y^ , and z and are denoted by s(.) and t(.), respectively. Denote (s(.),t(.)) by I(.). The sequence of events is as follows. First, the owner offers s(.) and t(.) to the agents. Second, assuming the agents accept their contracts, they choose their actions. Third, m and y are realized. Fourth, reports m^ and y^ are submitted. Fifth, z is realized and the contracts are settled. Finally, x is realized. In order to study the agents' subgame in normal form, Antle models the agents as choosing their reporting strategies m^(.) and y^(.) at the same time as they choose their actions. m^(.) is a function from {mL,mH} to {mL,mH}, and y^(.) is a function from {yL,yH} to {yL,yH}. The truth-telling strategies for the manager and the auditor are denoted by idm and idy, respectively. The manager and auditor's preferences are represented by (von Neumann-Morgenstern) utility functions u1(s,a) = U1(s) - a and u2(t,b) = U2(t) - b, respectively. It is assumed that Ui' > 0 and Ui'' E[s+t; (a,m^(.) ), (b,y^(.) )] and E[x; a], respectively. For the time being, it is assumed that the owner ensures only that the desired behavior on the part of the agents is individually rational and constitutes a Nash equilibrium in their subgame. The former requirement is represented by the individual rationality constraints (IR-1) and (IR-2), while the latter requirement is represented by the incentive compatibility constraints (IC-1) and (IC-2). In order to facilitate her search for optimal contracts, the owner can invoke the Revelation Principle (Myerson 1986) and, without loss of generality, consider only contracts under which truth-telling is incentive compatible. The standard approach to solving the owner's contracting problem is to proceed in two steps. First, find payments that minimize the cost of motivating each (a,b) pair. Second, find the (a,b) pair that maximizes the expected net benefit to the owner. Step 1:C(a,b) = MinI E[s+t; (a,idm), (b,idy)] subject to:E[ui; (a,idm), (b,idy), I]|U- i i = 1,2(IR-i)E[u1; (a,idm), (b,idy), I]|E[u1; (a',m^(.) ), (b,idy), I] " (a',m^(.) )(IC-1)E[u2; (a,idm), (b,idy), I]|E[u2; (a,idm ), (b',y^(.) ), I] " (b',y^(.) )(IC-2)Step 2:Maxa b E[x; a] - C(a,b)As is standard in the agency literature, we will refer to the solution obtained at the end of the two steps, (a*, b*, I*), as the second-best solution. 4. ResultWe begin by studying a numerical example. Let U1(s) = +`s , U- 1 = 25, aL = 0,aH = 10, U2(t) = +`t , U- 2 = 5, bL = 0, bH = 0.5, xL = 0, xH = 2000. The probability distributions over m, y, and z , where z = (z1,z2), are given below. Let Pr(x | a) =Pr(m | a) and Pr(z2 | b,y) = Pr(z1 | a,m). Pr(m | a) Pr(y | a,b,m) = Pr(y | b,m) Pr(z1 | a,b,m,y) = Pr(z1 | a,m)ammLmH(b,m)yyLyH(a,m)zzLzHaL0.60.4(bL,mL)0.50.5(aL,mL)0.80.2aH0.20.8(bL,mH)0.50.5(aL,mH)0.70.3(bH,mL)0.80.2(aH,mL)0.30.7(bH,mH)0.20.8(aH,mH)0.20.8In the numerical example: (1) the distribution of y is unaffected by the manager's act, (2) the distribution of z1 is unaffected by the auditor's action and information, and (3) the distribution of z2 is unaffected by the manager's action and information. The second-best solution is (aH, bH, I*), where I* is as follows (the dollar payments are simply the square of the utile payments). (m^ ,y^ ,z1)U1(s*)(m^ ,y^ ,z2)U2(t*)(mL,yL,zL)17.2924(mL,yL,zL)5.4959(mL,yL,zH)37.2615(mL,yL,zH)5.6936(mL,yH,zL)17.2925(mL,yH,zL)3.7051(mL,yH,zH)37.2615(mL,yH,zH)5.5017(mH,yL,zL)13.9388(mH,yL,zL)4.5547(mH,yL,zH)36.9897(mH,yL,zH)5.5928(mH,yH,zL)30.7073(mH,yH,zL)5.0570(mH,yH,zH)38.3488(mH,yH,zH)5.6466Obedient behavior is a Nash equilibrium under the above contract. Assuming this equilibrium is played, the owner's payoff is 302.31. However, there is also a Nash equilibrium that has both agents choosing the low action and always setting m^ = mH andy^ = yH. Under the obedient equilibrium, the manager's payoff is 25 and the auditor's payoff is 5, while under the nonobedient equilibrium, the manager's payoff is 32.5412 and the auditor's payoff is 5.1985. This introduces the possibility that the agents may tacitly collude and play the nonobedient equilibrium; in this case, the owner's payoff is -296.671. That is, unless the auditor is strongly independent, he will be willing to collude with the manager under the second-best solution.We next present a condition under which it is possible to construct a simple mechanism, thereby preventing collusion, whether the auditor is semi-weakly independent, independent, or strongly independent. Assume the publicly available information, z , satisfies the following condition. Condition 1. (a)There exists components z1 and z2 of z such that Pr(z1 | a,b,m,y) = Pr(z1 | a,m) andPr(z2 | a,b,m,y) = Pr(z2 | b,y),(b)Pr(z1 = zH | aH) > Pr(z1 = zH | aL). (c)Pr(z1 | a,m) _ Pr(z1 | a',m') for all (a,m) _ (a',m') and Pr(z2 | b,y) _ Pr(z2 | b',y') forall (b,y) _ (b',y'). Condition 1(a), our separability assumption, says that there is at least one component z1 of z whose distribution depends only on the manager's action and private information and that a similar component z2 exists for the auditor. For example, if we allow for the possibility that the firm's output x is realized in time to be contracted on, z1 could be x. For the manager, other examples of information sources that may satisfy this (partial) separability assumption include stock price, liquidation value, a competitor's performance, and industry-wide performance. For the auditor, examples include the number of hours devoted to an audit and the auditor's working papers. Of course, Condition 1(a) allows for the possibility that there are other components of z that provide information about both the auditor and the manager (e.g., bankruptcy not preceded by a going concern qualification). Condition 1(b) says that the distribution over z1 satisfies FOSD in a. Condition 1(c) says that the distribution of z1 is distinct for every (a,m) pair and the distribution of z2 is distinct for every (b,y) pair; this distinctness assumption is similar to that of Ma (1988). Note that the numerical example, under which a multiple equilibria problem arises, is consistent with Condition 1. The collusion problem is uninteresting if the second-best solution has one or both agents choosing the low action, since the second-best payments are constant for any agent who is induced to choose the low action. Hence, we assume the parameters of the problem are so chosen that the second-best solution prescribes high actions for both agents. The following proposition presents our result. The proof is presented in the Appendix. Proposition. Suppose Condition 1 holds. Then auditor-manager collusion can be prevented by a simple mechanism under which the equilibrium payments are arbitrarily close to second best.

Under the mechanism, each agent is asked to submit a second report on his private information and a report on the act he has chosen. That is, each agent now submits a total of three reports. The role of these additional reports is to fix the collusion problem; of course, from the Revelation Principle, we know the additional reports cannot be used to improve the second-best solution itself. The manager's first report on m is denoted by m^ 1 (this was referred to as m^ earlier); his second report on m is denoted by m^ 2; the manager's report on a is denoted by a^ . Similarly, the auditor's reports are denoted by y^ 1 (referred to as y^ earlier), y^ 2, and b^ . m^ 2, a^ , y^ 2, and b^ are submitted after m^ 1 and y^ 1 are submitted but before z is realized. (The construction will also work if all reports are submitted simultaneously.) The manager's and auditor's payments are as follows, where Vi denotes the inverse function of Ui:s = V1[a1(m^ 1,b^ ,y^ 2,z ) + b1(a^ ,m^ 2,z1) + d1b1(a^ ,m^ 1,z1 )]t = V2[a2(a^ ,m^ 2,y^ 1,z ) + b2(b^ ,y^ 2,z2)]Closed form expressions for each of these components are provided in the proof. The basic idea of the mechanism is to use each agent's two reports on his private information in the following fashion. In each agent's compensation, only a small weight is placed on his own second message and it is used in a manner that provides him with incentives to report his second message truthfully. For example, m^ 2 affects the manager only through its use in the b1 component, and b1 is a truth-inducing function. The existence of such bs relies on the separability and distinctness assumptions (Condition 1). Each agent's compensation depends primarily on his own first message of his private information and on the other agent's (truthful) second message. For example, in the case of the manager, m^ 1 is used in the a1 component along with the auditor's second message y^ 2. Because of the moral hazard problem, a report on effort is also introduced. We now construct the mechanism in the context of our numerical example and highlight key arguments. If we were to present the agents' normal form subgame in a bimatrix, it would be of size 64 + 64, since each agent's strategy would be a 6-tuple with each entry taking on one of two values. For example, the manager's strategy would be (a,a^ ,m^ 1|mL,m^ 1|mH,m^ 2|mL,m^ 2|mH), where the first entry is the manager's act, the second entry is his report on the act, the third entry is the first report on m the manager submits if m = mL, etc. This would be cumbersome, although possible, to present. In contrast, under the Ma, Moore, and Turnbull/Yost approach, at least one agent has an infinite number of strategies. This infinite bimatrix game yields a unique Nash equilibrium; our finite bimatrix game yields not only a unique Nash equilibrium but also a unique strategy combination that survives two rounds of iteratively removing strictly dominated strategies. Instead of presenting the agents' bimatrix subgame, we will present the construction and demonstrate the outcome of each round separately. For expositional purposes, we concentrate on the manager. a1 = 20.5556 if b^ = bL and z1 = zLa1 = 39.1741if b^ = bL and z1 = zHa1 = U1[s*(m^ 1,y^ 2,zL)] if b^ = bH and z1 = zLa1 = U1[s*(m^ 1,y^ 2,zH)] + .1if b^ = bH and z1 = zHb1(aH,mH,zH) = 3b1(aH,mH,zL) = -7b1(aH,mL,zH) = .7b1(aH,mL,zL) = -.3b1(aL,mH,zH) = .32b1(aL,mH,zL) = -.08b1(aL,mL,zH) = .036b1(aL,mL,zL) = -.004d1 = .0004To see that the only strategies that survive the first round for the manager have him reporting a^ and m^ 2 truthfully, note that a^ and m^ 2 affect the manager's compensation only through their use in the b1 and d1b1 components. Hence, it suffices to see what the reporting incentives are for a^ and m^ 2 under these two components. When these two reports are submitted, the manager has chosen his act, seen m, and reported m^ 1 (which may be true or false). The First Round: Manager sets (a^ ,m^ 2) = (a,m)(a^ ,m^ 2)(a,m,m^ 1)(aL,mL)(aL,mH)(aH,mL)(aH,mH)(aL,mL,mL).004003*0.000003-.100070-5.000070(aL,mL,mH).004000*0-.103500-5.003500(aL,mH,mL).008006.040006*0-4.000000(aL,mH,mH).008028.040028*-.160112-4.160110(aH,mL,mL).024961.200961.416011* .016011(aH,mL,mH).032006.208006.400000*0(aH,mH,mL).029121.241121.5200141.020010*(aH,mH,mH).037607.249607.5400281.040030*Each entry denotes the manager's payoff from b1(a^ ,m^ 2,z1) + d1b1(a^ ,m^ 1,z1 ). * in each row denotes the manager's preferred (a^ ,m^ 2) report. As the above table shows, the manager has strict incentives to set (a^ ,m^ 2) = (a,m). Similarly, the auditor has strict incentives to set (b^ ,y^ 2) = (b,y). In the game that remains after the first round of iteratively removing strictly dominated strategies, we demonstrate that the manager (auditor) has strict dominant strategy incentives to choose aH (bH) and report m^ 1 (y^ 1) truthfully. We again focus on the payoffs to the manager. Given (a^ ,m^ 2) is truthful (from the first round), we restrict attention to the remainder of the manager's strategy, denoted by (.,.,.), where the first entry denotes his act, the second entry denotes m^ 1 when m = mL, and the third entry denotes m^ 1 when m = mH. Since y^ 1 does not affect the manager's payoff and (b^ ,y^ 2) is truthful (from the first round), the only uncertainty the manager faces because of the auditor is due to the auditor's choice of b. The Second Round: Manager sets (a,m^ 1) = (aH,m)(a,m^ 1|mL,m^ 1|mH)b(aL,mL,mL)(aL,mL,mH)(aL,mH,mL)(aL,mH,mH)(aH,mL,mL)(aH,mL,mH)(aH,mH,mL)(aH,mH,mH)bL25.04240225.04240725.04240125.042406 25.97720825.993214*25.974006 25.990013bH22.12742125.04239722.12743525.04241123.84553525.993219*23.84233325.990018Each entry denotes the manager's payoff. *in each row denotes the manager's preferred (a,m^ 1|mL,m^ 1|mH) strategy. As the above table shows, the manager has strict incentives to set a = aH and m^ 1 = m. Similarly, the auditor has strict incentives to set b = bH and y^ 1 = y. 5. Concluding RemarksWe conclude with some comments on the role of e, the role of z, an interpretation of the mechanism, and possible extensions. In the absence of e, we could have achieved implementation in two rounds of iteratively removing weakly dominated strategies. Hence, the use of e in our setting is in much the same spirit as the use of e to convert a weak dominant strategy equilibrium into a strict dominant strategy equilibrium (e.g., Demski and Sappington 1984). The use of z , contractible information that cannot be manipulated by the manager or the auditor, was critical in facilitating the construction of our simple mechanism. A mechanism is a message space and an outcome function that maps messages to allocations. In the presence of z , the set of allocations consists of z -contingent payments to the agents. In the absence of z , the set of allocations consists of payments to the agents. In the latter case, the agents' preferences over allocations are unaffected by their private information. Hence, implementation via the iterative removal of strictly dominated strategies would be impossible. We now turn to an interpretation of this paper's mechanism. The additional reports (beyond those required by the second-best solution) submitted by the manager and auditor, might be viewed as court testimony of the agent's own guilt or innocence. Such testimony is consistent with the common law doctrine known as the Opinion Rule. Here, we might view z1 as the jury's conclusion about the manager's act and information and z2 as the jury's conclusion about the auditor's act and information. Agent i's compensation under our mechanism can be written as a function of the agents' first messages (and z ) and additional terms that enable the courts to (1) provide each agent with strict dominant strategy incentives to testify truthfully and (2) adjust agent i's compensation when agent j, j _ i, admits in court to having lied with his first message and/or having shirked. The role of the courts is to ensure that agent i, in considering the possibility of colluding with agent j, will not view any (implicit) promise by agent j to collude as credible, since he realizes that agent j will have strict dominant strategy incentives to confess in court to his own guilt. If agent j were to confess, agent i knows that any benefit he receives by colluding will be undone by the courts. This deters the agents from colluding. The ability of the courts to provide the agents with strict dominant strategy incentives to testify truthfully does not rely on the courts being perfect. It merely requires that the courts be able to obtain some imperfect (and separable) information. Since the courts are a costly mechanism, it is undesirable to use them on the equilibrium path with probability one. Our mechanism will continue to work if either (1) the courts are used only with a small positive probability (and payments to one of the agents are not bounded) or (2) the use of the courts can be made contingent on the agents' reports. In the latter case, z1 and z2 would be available only off the equilibrium path and, hence, would not be used under the second-best solution. Of course, other interpretations of z and the implementing mechanism are possible. In practice, both institutions (explicit mechanisms) and reputation considerations (implicit mechanisms) are, presumably, used to maintain auditor independence. In future work, it might be useful to study the role of reputation (and its interaction with explicit mechanisms) by modeling repeated auditor-manager interactions. Another issue that we have not dealt with is sidepayments. A conclusion that one might reach from the existence of a simple mechanism that avoids the collusion problem in our paper is that auditor independence may be more related to the issue of side payments than to multiple equilibria. It has been suggested that awarding lucrative management advisory service (MAS) contracts to a firm's auditor is a way of engaging in sidepayments and, hence, may impair auditor independence. A natural question would seem to be: to what extent can sidepayment agreements be undone by liability rules or other institutional arrangements? AppendixProof of the Proposition. The manager and auditor's utile payments are as follows:U1(s) = a1(m^ 1,b^ ,y^ 2,z ) + b1(a^ ,m^ 2,z1) + d1b1(a^ ,m^ 1,z1)U2(t) = a2(a^ ,m^ 2,y^ 1,z ) + b2(b^ ,y^ 2,z2) We first provide a characterization of the components that make up the agents' utile payments.Constructing a1 and a2The idea behind the construction of a1 is to ensure that if a1 were the only component of the manager's compensation, it would be incentive compatible for him to choose aH and report m^ 1 truthfully, assuming the auditor is reporting b^ and y^ 2 truthfully. The idea behind a2 is similar. Denote the cost minimization program that is solved in Step 1 by P(a,b). The second-best payments, (s*,t*), are the solution to the program P(aH,bH). Denote the manager's payments under P(aH,bL) by s' and the auditor's payments under P(aL,bH) by t'. a1 = U1(s') if b^ = bL and z1 = zLa1 = U1(s') + g, g > 0if b^ = bL and z1 = zHa1 = U1(s*) if b^ = bH and z1 = zLa1 = U1(s*) + gif b^ = bH and z1 = zHa2 = U2(t') if a^ = aL and (y^ 1,m^ 2) + {(yL,mH), (yH,mL)}a2 = U2(t')+ gif a^ = aL and (y^ 1,m^ 2) + {(yL,mL), (yH,mH)}a2 = U2(t*) if a^ = aH and (y^ 1,m^ 2) + {(yL,mH), (yH,mL)}a2 = U2(t*)+ gif a^ = aH and (y^ 1,m^ 2) + {(yL,mL), (yH,mH)}The construction of a2 is different from the construction of a1. This is because at the time the manager is choosing his act, the distribution of z1 does not depend on his beliefs about the auditor's act. In contrast, when the auditor is choosing his act, the distribution of z2 does depend on his beliefs about the manager's act, since the manager's act influences y, both directly and indirectly (through m). Constructing b1 and b2b1 is used to ensure that if it were the only component of the manager's compensation, he would have strict incentives to report a and m truthfully. List the distribution over z1 in order of increasing Pr(z1 = zH | a,m). Since the probabilities are distinct for each (a,m) pair, there are no ties. Without loss of generality, assume the sequencing of the distributions is as in the example, i.e., Pr(z1 = zH | aL,mL) Pr(z1 = zH | aL,mH) e1 = [p2 - p1]e2, where e > 0. b1(aH,mH,zH) = [1 - p3]e4 and b1(aH,mH,zL) = -p3e4b1(aH,mL,zH) = [1 - p2]e3 and b1(aH,mL,zL) = -p2e3b1(aL,mH,zH) = [1 - p1]e2 and b1(aL,mH,zL) = -p1e2b1(aL,mL,zH) = [1 - p0]e1 and b1(aL,mL,zL) = -p0e1, where 0 To see that b1 provides incentives for truthful reporting of a and m, note that if the truth is (aL,mL) the expected value of the lottery is positive if and only if (aL,mL) is reported; in this case, the expected value is p1(1 - p0)e1 + (1 - p1)(-p0e1) = (p1 - p0)e1. If the truth is (aL,mH), the expected value of the lottery is positive if (aL,mH) or (aL,mL) is reported, but the choice of e2 relative to e1 ensures that the former lottery is preferred. Similarly, the choices of e3 and e4 ensure that if the truth is (aH,mL) the manager's preferred report is (aH,mL) and that if the truth is (aH,mH) the manager's preferred report is (aH,mH). b2 is constructed as above, except that (a,m) is replaced with (b,y). Constructing d1The idea behind the construction of d1 is to ensure that incentives provided by b1(a^ ,m^ 2,z1) to report a^ truthfully are not upset by d1b1(a^ ,m^ 1,z1) if m^ 1 is not truthful. This is accomplished by setting d1 > 0 small enough. Consider the b1(a^ ,m^ 2,z1) component. Find the minimum benefit obtained by reporting the truth instead of a false (a^ ,m^ 2). Denote this minimum benefit by B. Note that B > 0. Next, consider the d1b1(a^ ,m^ 1,z1) component. Find the maximum cost of reporting a^ truthfully given that m^ 1 is false. Denote this maximum cost by C. Since all values of b1(a^ ,m^ 1,z1) are between -p3e4 and [1 - p3]e4, C In order to ensure that incentives provided by b1(a^ ,m^ 2,z1) to report a^ truthfully are not upset by d1b1(a^ ,m^ 1,z1), we choose d1 so that B > C. Any d1 such that 0 Finally, we explain how under the above construction, obedient behavior is obtained as the unique strategy combination that survives two rounds of iteratively removing strictly dominated strategies. Round OneThe only strategies that survive the first round have the manager reporting a^ and m^ 2 truthfully and the auditor reporting b^ and y^ 2 truthfully. a^ and m^ 2 affect the manager's compensation only through their use in the b1 and d1b1 components. As argued earlier, b1(a^ ,m^ 2,z1), if it were the only component of the manager's compensation, would provide the manager with strict incentives to report a^ and m^ 2 truthfully. The only problem is that a^ is also used in the d1b1(a^ ,m^ 1,z1) component and, since m^ 1 is also used in the a1 component, there is no reason (at this point) to assume that m^ 1 is truthful. However, the choice of d1 given above ensures that the incentives provided by the b1(a^ ,m^ 2,z1) component dominate the incentives provided by the d1b1(a^ ,m^ 1,z1) component. In the case of the auditor, the argument is even more transparent, since b^ and y^ 2 are used only in the b2 component (there is no d2b2 component). Round TwoThe only strategy combination that survives the second round has the manager choosing aH and reporting m^ 1, m^ 2, and a^ truthfully and the auditor choosing bH and reporting y^ 1, y^ 2, and b^ truthfully. First, consider the incentives provided by the a1 component. Given the first round, a1 provides the manager with strict incentives to choose aH and at least weak incentives to report m^ 1 truthfully. To see this, note that the manager's payments are found by solving the program P(aH,bL) if b^ = bL and P(aH,bH) if b^ = bH; in either case, given that the auditor is reporting b^ and y^ 2 truthfully, it is a best response for the manager to choose aH and report m^ 1 truthfully. The weak incentives to choose aH are made strict by the e-bonus that is contingent on z1. The weak incentives to report m^ 1 truthfully are made strict by the d1b1 component. This follows from the fact that d1b1 is the only place, other than a1, where m^ 1 is used, a^ is truthful (from the first round), and the b-lotteries are constructed so as to provide strict incentives for truthful reporting. Consider the incentives provided by the a2 component. Given the first round, a2 provides the auditor with strict incentives to choose aH and report m^ 1 truthfully. 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