EFFECT OF UNCERTAINTY AND USEFULNESS OF PRIOR INFORMATION ON STRATEGIC RISK ASSESSMENT AND PLANNING Nahnhee Choi School of Professional Accountancy Long Island University - C.W. Post Brookville, NY 11548-1300 Romesh Saigal Department of Industrial & Operations Engineering The University of Michigan Ann Arbor, Michigan 48109-2117 Technical Report 95-4 April 1995

Effect of Uncertainty and Usefulness of Prior Information on Strategic Risk Assessment and Planning Nahnhee CHOI School of Professional Accountancy Long Island University - C. W. Post Brookville, New York 11548-1300, USA Romesh SAIGAL * Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2117, USA April 1, 1995 Abstract 'Ihis p)aper presents game theoretic lmodels to analyze the effect of uncertainty a l)bot1 tile auditee on the auditor's risk assessiment and planning in an internal audit iln wllich (l oth the auditor and the auditee mllake strategic moves. In addition, an itterinal audlit is conceptualized as a se(lleIItial. information-gathering activity during whlich tihe al(ditor obtains prior inforiiiatioin about the auditee to assess risk and to 1.plani strategically. In this context, this pal)er also examines the usefulness and the role of p)rior ilformation about the auditee iIn risk assessment and planning. TIhe analytic results indicate that, with incomplete information about the auditee tIpe, an internal auditor takes imore extreme actions, such as 100% testing or rl'llis lautllor's research has been partially supported by a grant from NSF.

no testing. The analytic results offer theoretical support for professional standards (Standards for the Professional Practice of Internal Auditing, SAS No. 53, and SAS No. 55): in certain settings, prior information about the auditee, if utilized properly, allows the internal auditor to plan the audit tests more effectively. Also, in some cases, prior information of different accuracy about the auditee type can be a substitute for actual testing. Support is also found for Statement No. 1 on Quality Control Standards: gathering sufficient prior information about management integrity before client acceptance is important. Key words: Strategic risk assessment and planning, Uncertainty, The usefulness and role of prior information, Internal audits. Abbreviated title: Uncertainty and strategic risk assessment i

1 Introduction In this paper, we analyze the effects of uncertainty and usefulness of prior information on strategic planning and risk assessment by an internal auditor. Towards that end we formulate three game models between the auditor and the auditee: a game with complete information; a game with incomplete information; and a game with incomplete information and information asymmetry. Existing audit game models in the literature (Fellingham and Newman, 1985; Shibano, 1990) typically assume complete information about the auditee and address the auditor's decision problems such as planning and risk assessment. In the real world, auditors do not have complete information about the auditee and gather such information by applying inquiries and observation. Professional standards (SAS No. 53, SAS No. 55, and Standards for the Professional Practice of Internal Auditing) suggest that the auditor assesses risk based upon background information about the auditee (hereafter, prior information about the auditee type). The auditor's assessment of prior information, in turn, determines the extent of audit tests. Prior information about the auditee, therefore, plays an important role in ain audit; however, the literature has never addressed the effects of incomplete information or the usefulness of prior information about the auditee on an audit. 1rlnclrtaitnty about the auditee seems to encourage product differentiation. The analytic resuIlts indicate that, with incomplete information about the auditee type, it is optimal for the i auitor to take more itree actions, such as 100% te assting or no testing. The analltic results on the usefulness of pr ior information offer theoretical support for professioiial st alldards (Standards for the Professional Practice of Internal Auditing, SAS No. 53, all(l SA\S No. 55); prior information aloutt he audlitee, if utilized properly, allows the interrnal auditor to plan the audit tests more effectively in certain settings. Prior information of differell accuracv about the auditee type call even be a substitute for actual testing if certain coiIditioiis are met. The analytic results also support Statement No. 1 on Quality Control Standllards. That is, more accurate information about the auditee does not necessarily mean highller ayoff. For example, the internal auditor could face lower expected payoff as the ilformnation about the high risk auditee gets more accurate. In the external-audit setting, 1

gathering sufficient prior information about management integrity before client acceptance, as required by Statement No. 1 on Quality Control Standards, would prevent this type of scenario. The paper is organized in 4 parts. Section 2 presents the game model with complete information. The next section describes the game with incomplete information and analyzes the effects of uncertainty on planning. The following section derives and discusses the analytic results on the usefulness of prior information in strategic internal audits. In Section 5, the paper concludes by discussing contributions of this research and broaching possible future research. 2 The Game Model with Complete Information Consider the situation where two types of internal auditors exist: experienced and less perced. An experienced auditor is characterized by higher testing cost and a higher detectic)on rate of fraud; a less experienced auditor is characterized by lower testing cost and a lower detection rate of fraud. Two different types of auditees (managers) represent two differentl levels of risks: a high risk of material fraud and a low risk of material fraud.1 The penalty for Ilaterial fraud when the auditee gets caught is a function of the fraud amount and is collstrailed to be more than the fraud amount. A high risk of fraud is attributed to the loNwer peiinalty for fraud, ani( a low risk of fraud is attributed to the higher pe alty for frall(l. ILac player knows the type of oppoIlenlt with certainty. Thus, only one type ot auditor andI oln tylpe of auditee exists il each gaMie'. I 'l( decision variable (strategy) of the auditor is to d(et ermini-e the probability of 100' testilng. andl the decision variable (strategy) of the atllitee is to determine the prolability of illaterial fraud.2'3 The strategy space of each player is colltinluolis from 0% to 100%. Exhilbit 2.1 i)resents notation. 1If t le iauditee commits fraud, he conImits frraud of the maximum possible amount, and, thus. the fraud is mlaterial. The maximum possible amount of fralud is the same for all auditees. '2T 1e auditor is assigned to the audit. unit after the auditee makes the move. 3According to SAS No. 22 (AU 311.0): planning the audit, te he auditor should consider the nature, extent, and timing of work to be performed...." 2

Exhibit 2.1: Notation Auditor's strategy "i" indexes type of auditor, i l {1, 2} 1 = experienced 2 = inexperienced "Xi" represents the strategy of type i auditor. Auditee's strategy "j" indexes the type of auditee, j C {1,2} 1 = low risk of material fraud 2 = high risk of material fraud "}1" represents the strategy of type j auditee. Auditor's cost elements "M" represents the loss to company due to undetected material fraud. "Si" represents the cost of 100% testing to type i auditor. "Ri" represents the detection rate of type i auditor given 100% testing. Auditee's payoff elements 'F" represents the maximum possible amount of fraud. "lj" represents the penalty for material fraud to type j auditee (P1 > P2). Trlhe cost function of each auditor has two elements: the expected cost of testing (XiSi) an(l tle exI)ectedl loss to the colampany due to undetected material fraud [(1 - R.X')~}M]. IFor loot h tvy)es of auditors, the expected( cost of testing is a linear function of the auditor's strategy. i.e., the probability of 100 testing. The loss to the company due to undetected ilater:ial fraud (Ali), i.e., the cost of a tlype 1 e ror, is a constant regardless of the type of auditor,, The probability that material firatud goes undetected, (1 - RiXi), given material fraudl. is all inverse function of (i) the detectionl rate, given the type of auditor and 100% testillg ani (ii) the auditor's strategy. If tie adllitee gets caught, he pays the penalty (PjF). Otherwise, the payoff to the auditee is the amount of the material fraud (F). The- game can be summarized as follows. 3

The expected cost to a type i auditor, given a type j auditee is: XS, + (1 - RX)YM (1) The expected payoff to a type j auditee, given a type i auditor, is: YjF - (RXR)PjYjF = {1 - RX,)Pj}YjF. (2) The decision problem of the type i auditor is to minimize equation (1) with respect to XI, given that the type j auditee does not change strategy Y;*; and the decision problem of the type j auditee is to maximize equation (2) with respect to Yj, given that the type i auditor does not change strategy X*. Theorem 1. A non-cooperative solution to the game with complete information exists. If RjPj > 1 and RiM > Si, then the unique equilibrium strategies are as follows. The equilibrium strategy for a type i auditor: X* = 1/(RiPj) The equilibrium strategy for a type j auditee: Yi* = Si/(RiM) Proof: See Appendix A. i The first hypothesis of the theorem ensures that, given 100% testing, the expected penallty for material fraud exceeds the fraud amount for any type of auditee. That is, it does not )pay for any type of auditee to defraud given 100% testing. The second hypothesis of the tleorem stipulates that, given material fraud, the cost of 100% testing is less than the bl)lenfit of 100% testing for any type of audlitor. The benefit of 100% testing is the reduction ill tie exl)ected loss to the compalny due to lundetected material fraud. The e(quilibrium strategy of a t)ype i atuditoi il(licates that the probability of 100% testing is al inverse function of the detection rate of fraud given 100% testing (Ri); accordingly, an experienced internal auditor works less thlall all inexperienced internal auditor. The penalty fol Ilatl-ial fraud (Pj) also affects, inversely, the probability of 100% testing; that is, any t ype of internal auditor audits less withl a low risk auditee. Each player chooses an equilibrium strlategy such that the cost of the opponent's action is the same as the benefit of the opponent's action. For example, the type j auditee chooses the equilibrium strategy (Y.*) such that the cost of 100% testing (Si) is the same as the benefit 4

of 100% testing (RiMI'7*) to a type i auditor. The type i auditor chooses the equilibrium strategy (X^*) such that the expected penalty for material fraud (cost) (X*RiPjF) is the same as the fraud amount (benefit) (F) of a type j auditee. 3 The Game Model with Incomplete Information 3.1 The Model Internal auditors of two different types - experienced or inexperienced - are randomly assigned to different audit units. Audit units represent either a low risk of material fraud or a high risk of material fraud. A player is assumed to have each type of opponent with probability 2; i.e., the opponent of a player is equally likely to be of either type. Players derivse tile weighted average of payoff or cost functions contingent on the type of the opponent, weighted by the probability distribution over opponent types. Each auditor minimizes the wveighted average of cost functions; and each auditee maximizes the weighted average of 1)a'off flunctions. Thrle cost function of a type i auditor, when the auditee is typej, is XiS+(1 -RiX )YjM. Si(ce l'ie aut(litee is equally likely to be type 1 or type 2, the expected cost to the type i 2 alditor is i {X1 S, + (1 - Ri1X1i)YM}. Thus, the game with incomplete information can.j = )e sullllrllarized( as follows.4 2 'Il1(e ('elpected cost to a type 1 auditor is: XiSli + {(1 - R1XI)M}( Z Y'}). j=1 2 '1I11( e'xp. te(d cost to a type 2 auditor is: X2.2 + {(1 - R2X2)i }( } ). j=1 2 I i e expected payoff to a type 1 auditee is: { 1 - P (1 y RXi)} Y F. IT1e exl)ecte(l payoff to a type 2 auditee is: {1 - P2( E RX i)}Y2F. i=1 'I'll (lecisioll l)roblem of each auditor, given the strategy of the auditee, is to minimize the;expect'ed cost with respect to IY or,A2: aiid the decision problem of each auditee is to 1in Sectiol 2. each player knows the type of opponent with certainty, and four different games with (collplete information are possible depending uponi the types of auditor and auditee. However, in Section 3, eacl opplonent is equally likely for each player, and only one game with incomplete information occurs. 5

maximize the expected payoff with respect to Y1 or Y2. 3.2 Equilibrium Concept When each type of opponent is equally likely, four different pairs of an auditor's expected cost and an auditee's expected payoff are possible; and, depending upon the values of the parameters, four different equilibria exist. Assume that S1/(R1M) > S2/(R2M). That is, when both types of auditors perform the same routine task, the cost to benefit ratio is higher for the experienced auditor than for the inexperienced auditor. This is due to diminishing marginal returns for such tasks. However, for complex tasks, the experienced auditor can attain increasing marginal returns, and therefore, may have lower cost to benefit ratio than the inexperienced auditor. Theorem 2. If S1/(RIM) > S2/(R2M[), a non-cooperative solution to the game with incomplete information exists for the following cases: (i) 0 < S2/(R2M) < 0.5, and 0 < 1/1P < 0.5R2 for j = 1,2. (ii) 0.5 < SI/(RM) < 1 for i = 1,2, and 0 < 1/P1 < 0.5R2. (iii) (0 < S,/(RlI) < 0.5 for i = 1,2, and 0.5R2 < 1/P2 < 0.5(R1 + R2) (iV) 0.; < S1/(RI1M) < 1. and 0.5R2 < 1/P, < 0.5(R1 + R2) forj = 1,2. /f 0I) 2( R2.11) < 0.5 and 0 < 1/P2 < 0.5J?2. then the unique equilibrium strategies are as follo 's. /h( (qtuilibrium strategy for a type 1 (mditor: A~ - 0 7'Th e quilibrium strategy for a type 2 auditor: X. = 2/(R2P2) Th( e(quilibrium strategy for a type I audtl'(: ) 0 = 0 7h(l quilibrium strategy for a typ( 2 udill(: )2 =I-2S2/(R2M) Proof: See Appendix A. U Although four different non-cooperatixe equilibria are possible depending upon the penalties for material fraud and the cost to benefit ratios, at all four equilibria, players exhibit the 6

same behavior ( strategies for the four different cases of Theorem 2 appear in Appendix B ). That is, players go more extreme when the opponent is equally likely to be of either type. At each equilibrium, the benefit of 100% testing (or material fraud) is the same as the cost of 100% testing (or material fraud) only for one type of auditor (or auditee). For the other type, either the benefit exceeds the cost or the cost exceeds the benefit. Therefore, the other type of player takes the extreme strategy such as no testing (or no material fraud) or 100% testing ( or material fraud). When the benefit of action is the same as the cost of action for a player, since the player can influence the behavior of the opponent with probability -2 the player exaggerates the move, that is, the player doubles the probability of 100%testing or doubles the probability of material fraud over the strategy of the complete information game studied in Section 2. 4 The Game Model with Information Asymmetry Although two different types of auditors and auditees are equally likely to be matched, the auditors; have more information about the auditee than the auditees have about the auditor, and thus there is information asymmetry in this game model. In this section we propose a mlolcel to investigate this asymnletry. 4.1 The Model 11erIe wle coinceptualize an internal audit as a sequential, information-gathering activity in \whlicl tte auditor gathers background infoirmation about the auditee to assess auditee's ilcentive for fraud; and to generate a strategic I)lan based upon this assessment. This sectiol f urther hypothesizes that only tihe e.xperienced auditor can utilize this background illnforlatl ion. An auditee is, hovWever. reulliredl to make a move before an auditor is assigned to an1 audit unit; an auditee does not have any specific information about the auditor type. InIfornaltion asymmetry is modeled as follows. Type 1, the experienced auditor, is modeled( to o(>serve a signal about the t ype of opponent. There are two signals: 01 and 02; and the accuracy of the signals, q > 0.50. If the signal is 0i, the probability that the opponent is ty-pe 1 is 6; and if the signal is 02, the probability that the opponent is type 2 is A. Exhibit 7

4.1 explains the necessary changes in notation to incorporate this information asymmetry. Exhibit 4.1: Notation q represents signal accuracy and ) > 0.5. X191 represents the strategy of the type 1 auditor contingent on 01. X1l2 represents the strategy of the type 1 auditor contingent on 02. X2 represents the strategy of the type 2 auditor. The type 1 auditor, observes a signal which indicates the type of auditee with probability 6, and comrnits to a strategy contingent upon the signal observed. For example, given the signal is 01 and the auditee is type 1, the cost to the type 1 auditor is: X0l Sl + {(1 - RiX0l )YM}. When the type 1 auditor observes the signal 01, the probability of a type 1 auditee as an opponent is 0, and the probability of a type 2 auditee is (1 - I ). When type 1 auditor observes 02, the probability of a type 1 auditee as an opponent is (1 - <), and the probability of a tNyIp)e 2 auditee is Q. The opponent of a type 2 auditor is equally likely to be type 1 or type 2 auditee. Tlhuls. the expected cost to a type 1 auditor when the signal observed is 01 is: o[il,.qs'1 + {(1 - RX.1lo0 ))i;M11}] + (1 - 0)[X10eS1 + {(1 - RllXlol)YAi2l,}]. (3) The (exI)ecte(l cost to a type 1 auditor whlelI the signal observed is 02 is: (1 - o)[ X,,25 + {(1 - 1 - t),X A )] 112} + o[X S {(1 - R1.2 2)YE }]. (4) 'li'(e cxl)ct( ctl ost to a type 2 ad(litor is: 12 XA2(2 + {( - 22 ) ( ). (5) j=1 Ih1e op)p)onent of each type of a(uditee is equ(ally likely to be a type 1 auditor or a type 2 audlitor. IlHowevsser. when a type 1 audtlitor is matched with a type 1 auditee, the probability, that tile type 1 auditor observes 01 is o anld the probability that the type 1 auditor observes 02 is (1 - o). Thus, the probability distribution of the type 1 auditee is as follows: the prolability that the type 1 auditee is matched with a type 1 auditor and the type 1 auditor 8

uses the strategy contingent on 01 is 0.5q$, the probability that the type 1 auditee is matched with a type 1 auditor and the type 1 auditor uses strategy contingent on 02 is 0.5(1 - 0). Also the prolbability that the opponent is a type 2 auditor is 0.5. The probability distribution of matching a type 2 auditee can be derived similarly. Thus, the expected payoff for the type 1 auditee is: 0.5q${i - P1(R1Xl9 )}Y1F ~ 0.5(1 - 5){1I - P1(RlXl62)}Y1F + 0.5{1I - Pi (R2X2)}Y1F. (6) The expe-cted payoff for the type 2 auditee is: 0.5(1 5){1I - P2(R1Xlol)}Y2F +0.54{1I - P2(RlXl02)}Y2F +0.5{1I - P2(R2X2)}Y2F. (7) The game with information asymmetry can be described by Equations (3)-(7). The decision problem of the type 1 auditor is to minimize Equation (3) with respect to XI01 and Equation (4) with respect to X602. The decision of the type 2 auditor is to minimize Equtationi (5) with respect to N2. The decision problem of the type 1 and type 2 auditee is to miaximilze Equation (6) and (7) with respect to Y1 and Y2 respectively. 4.2 Equilibrium Concept 'Ilie av-ailab1lity of a, signal abouit the type of auditee to a, type 1 auditor changes the game wi-th i11ic'(uij)lete inforination in tw\o different, ways. The type 1 auditor commits to a strategyr 'onitinigent uipon the signal observed and inicurls a different expected cost contingent upon l ie signali. rfThus, the game wvith inforniat ion asymmnetry consists of five objective functions inlsteall Of four. Theorern 3. 14 nofl-cooperatir(' so/u/iou to Il/( yame with i'nformation asymmetry exi~sts. For f/u( followingI'l' sqi cases. (i) (1<@ ~ F1(RIA1) < 1, and 0.5){(1 0)I?1 +R2} ~_ 1/Pj < 0.5(RI +R2) forj 1,2, (i)0 Sil~(RZ Al) ~ (1-P foi- I 1.2. anid O.,5(O5R + R2) ~< 1/1P2 ~ 0.5(lRi + R2), (ii) S/R I Al) < 1, anrd 05RO 2 ~< 1/1 PI 0 I5( )RI + R2} (i) S-/(R-M) < 0 for- i 1. 2, aind 0.5RhK1P.(51+R) 9

(v,) 0.5 < S$/(R-M) < 1 forj 1, 2, and 1/P1 < 0.51R2, arid (Vi) 0 < S2/(R2M) ~ 0.5, and 1/Ps < 0.5R2 Jor = A1,2 the equili1Zbri'um strategies also sati'sfy 0- 5)(Y2* - Yl*) < { S1/(RM) - S2/(R2M)} (8) and -0.5)(X*6 X- 1 < { 1/RP2) - 1/(R1P1)}. (9) Iff0 < S —/(R-M) < 0 for i=1,2, and 0.51R2 ~< 1/P2 < 0.5(OR, + R2), then the uniqu equilibrium strategies are as follows. Th, eq(uilibriumn strategy for a type 1 auditor with 01:X*0 0 The ~U eqilibrium, strategy for a type 1 auditor with 02 = X 2/(q5R2P2) - R O, Thec equi'librium strategy for a type 2-9 auditor: X* 1 The, equili'11b rium -strategy for a type 1 audilee: 1? 0 Th (qpulibruirni.trategy for a type 2_ auditeec: Y* S,/(~RjM) Proof: Scc Appendix A.U L~jnlihinmstrategies for all six ca-ses of Theorem- 3 appear in Appendix B. Cond~itioni (S8) on equilibrium strategies of Theorem 3 imposes a limit on the accuracy of' prior Hittorination. The right-hand sidle of t le inequality represents incremental cost per (lollar rc turni for an exp erienced auditor asstiliingi material fraud and 100% testing. The left-han(I side of the inequality is the 1Cincrase ii the probability of material fraud due to prior information of accuracy 0) The six caseIs considered in this theorem limit the accuracy of' prior information such that i1crIneIital cost for an experienced auditor is higher than exI)ecte(l incremental return due to prior- Hiformnatiorn. In other words, it is more costeffective to use inexperienced auditors f'or routine tasks. Cond —ition (9) on equilibrium strategies of Theorem 3 also limits the accuracy of prior in-form-ation. The right-hand side of the inequality represents the difference between the 10

probability of 100% testing for a type 1 auditor with a type 2 auditee and such probability of 100% testing with a type 1 auditee. The left-hand side of the inequality is the difference between the probability of 100% testing contingent upon 02 and such probability of 100% testing contingent upon 01. 01 is related to a type 1 auditee with accuracy X, and 02 is related to a type 2 auditee with the same accuracy. Condition (9) on equilibrium strategies limits the value of X such that, the difference between the probability of 100% testing contingent upon 02 of accuracy 4$ and such probability of 100% testing contingent upon 01, is less than the differeence between the probability of 100% testing for a type 1 auditor with a type 2 auditee and such probability of 100% testing with a type 1 auditee. When we compare the strategy of a type 2 auditee obtained in this model with the strategy obtained in Theorem 2, we note that prior information about the auditee type decreases the probability of material fraud by this auditee ( which represents a high risk of material fraud ). The probability of 100% testing by a type 1 auditor is also decreased when the signal observed is 02. 4.3 Usefulness of Prior Information about Auditee Type A.XSB StIatemtent of Financial Accounting Concepts No. 2 defines that the decision usefulIless of accolllnt ig illformiation depends upon relevance and reliability. Relevant accounting ilforlmal ion is (lefiiled to the information that can make a difference in a decision by helpinlg misers to foIrm I)redictions ablout the outcomes of past, present, and future events or to c(onfilrml (r correct prior expectations. Tleoreml 4. Prior information aboutt Ih/ auditff type is useful in the first four cases of 'It or( J.. In' tfl last two cases of '1 h(or(.3, prior information about the auditee type /I(s,! iino rol(. s in the strategy of a tJyp/ I auditor. Proof: See Ap.\lpendix A. IiI tlie first four cases of Theorenm:3, t lie (lequilibrium strategy of an experienced auditor is coltilt iiit lupoin the signal observed. In the last two cases, the penalties for material fraud are tlie highest, and the probability of material fraud is the lowest. In these two cases, the cost of test ing is always greater than the benefit of testing for an experienced auditor, and, thlus. tlie eq(uilibrium strategy is no testing regardless of the signal observed. 11

The risk analysis model for internal and external auditors suggests that auditors assess risk based upon prior information about the auditees. Based upon the assessment of risk, auditors determine the nature, the timing, and the extent of audit tests. The role of prior information is to enable auditors to plan more effectively. The higher the risk, the larger the extent of the tests should be. In the game models of this study, the auditee's strategy (Y,*) is equivalent to risk. The extent of testing is, on the other hand, the strategy of a type 1 auditor contingent upon prior information (X 0,). Since 02 represents a higher risk of material fraud than 01, an effective audit would mean X2 > X*. In the first four cases of Theorem 3, X'2 > X'1; that is, prior information about the auditee type allows an experienced auditor to plan more effectivelv. 4.4 Prior Information of Different Accuracy about the Auditee Type This subsection assumes that any internal auditor gathers the same amount of prior information as requested by professional standards. Thus, the cost of gathering prior information is irrelev\ant or negligible as prior information should be obtained in any circumstances. This subsection also hypothesizes that the accuracy of prior information is solely a function of exerience. lThat is, the accuracy of prior inforn-ation depends upon an auditor's ability to recognize red flags. If prior inforimation of different accuracy! ai)out the auditee type is a substitute for actual testillg. prior information has two possible roles: (i) to reduce the extent of actual testing or (ii) to (liscourage material fraud. Trllls. as tlle accuracy of prior information about the auditee t1yxpe increases, the probability of 100(/i test ing and the probability of material fraud will ldecrease provided they change. Theoremi 5. Prior information of diffre Ot accIralcy about the auditee type is not a substitlut for actuall testing in the first thre( ca((I. of Theoremn 3. Prior information of different accuriacy about the auditee type is a sulb.sitlF for actual testing in the fourth case of Theore7n 3. In oth1erl words, in the fourth ca.se of T7'heorem71 3, OY1*/9 < 0 and &X* /9ld < 0. 12

Proof: See Appendix A. 1 In Case (iv) of Theorem 3, the penalties for material fraud are such that a type 1 auditee does not have any incentive to commit material fraud; however, a type 2 auditee clearly has the incentive to commit material fraud. The cost to benefit ratio of testing for an experienced auditor is such that the optimal strategy for an experienced auditor, contingent upon 01, is no testing; however contingent upon 02, an experienced auditor has the incentive to do testing. Thus, as the accuracy of prior information about the auditee type improves, a type.2 auditee decreases the probability of material fraud, and an experienced auditor decreases the extent of testing contingent upon 02. Thus, in Case 4 of Theorem 3, prior information of different accuracy about the auditee a type is a substitute for actual testing. In the first two cases of Theorem 3, the penalties for material fraud are the lowest for both types of auditees, and the probability of material fraud is the highest. The equilibrium strategy of an experienced auditor, contingent upon 02, is 100% testing regardless of the accuracy of prior information about the auditee type. Thus, as an experienced auditor gets more accurate information that the auditee is a high risk of material fraud, the experieliced auditor faces higher expected loss due to undetected material fraud and, thus, higher exp)ected cost. In (Case (iii) of Theorem 3. testing is costly for an experienced auditor, and, thus, the )probablility of 100(% testing byN an experienced auditor, contingent upon 02, is a function of tlc a(ccutracy of prior information about the auditee type. As 02 is more closely related to a hligllhe( risk of material fraud. and the accuracy of the signals improves, an experienced alluitor incrieases the probability of 100( testilg. contingent upon 02. 5 Conclusion and Future Research 5.1 Conclusion Thlis stludy presents an audit game mlodel which conceptualizes an internal audit as a se(lluential, information-gathering activity. In the game model, the auditor gathers prior information about the auditee, assesses the risk of material fraud, and plans the audit test strategically based upon the assessment. Using the audit game model, this research also 13

analyzes the effects of uncertainty on the strategic relationship between the auditor and the auditee and, further, addresses the role of prior information about the auditee in strategic audits. The analytic results indicate that both the auditor and the auditee take such more extreme actions as 100% testing or no testing (material fraud or no material fraud) with uncertainty about the type of opponent. The scenario is similar to that where two policemen try to get the truth out of a suspect. It is more effective (or strategic) for them to exaggerate the difference between their moves in a given situation. That is, one of them plays a nice guy, and the other a bad guy. Even without formal coordination of strategies between two types of auditors or auditees, the same phenomenon seems to occur in the game with incomplete information.5 One of them increases the extent of testing (or the probability of material fraud), and the other decreases the extent of testing (or the probability of material fraud) due to uncertainty about the type of opponent. The usefulness of costless prior information seems to depend upon the control environment. If the penalty for material fraud for any type of auditee is so high that the probability of nmaterial fraud is not high enough to warrant any, testing, then the optimal strategy is to do no o t(esting regardless of the accuracy of prior information. In this case, prior information al)out aly typ)e of auditee plays no role in planning the audit test. Allotllher analytic result is that more accurate prior information, even if costless, does tiot guilalaIltee' ligller payoff or lower expected cost. If the penalty for material fr'aud is so low t1o warraiit 100%o testing regardlless of prior information, then more acculiat-e prior infoirinat ioi about the auditee whliclih re[)reseilts a high risk of material fraud merely increases tlie exl)eclted loss due to undetecteld iilterial flraud. This increase in the expected loss in iievitiable due to imperfect audit tecliiiologv. In the external-audit setting, gathering sufficieni inlformnation about manaIgemenlt integirity before client acceptance would prevent this typel) of scenario. Thle role of prior information of different accuracy about the auditee type deperids upon thle Ilpenalties for mlaterial fraud for differenet types of auditees and the cost to benefit ratio 5oAccording to the Folk Theorem, outcomes usually associated with cooperation can be supported by non-cooperative equilibrium strategies (Friedmllan (1986), p. 103). 14

of testing for the auditor who can properly utilize the information. For example, prior information of different accuracy can be a substitute for actual testing if the following two conditions are met: first, one group of auditees have no incentive to commit material fraud; however, the other group of auditees have incentive to commit material fraud; second, the equilibrium strategy for experiences auditors is no testing if the opponent is more likely to be a low risk; and, the equilibrium strategy is to do testing if the opponent is more likely to be a high risk. In this scenario, as the accuracy of prior information about the auditee type improves, a high risk type auditee decreases the probability of material fraud, and, thus, an experienced auditor can also decrease the extent of testing, contingent upon 02. 5.2 Future Research The strength of game theory as a research tool is its ability to address conflicts of interests among different interest groups. Accordingly, game theoretic results should be most useful to the policy makers. Despite its usefulness, no attempt was made to evaluate the efficacy of anyr auditing policy with game theory until Newman and Noel (1989) investigated the equilil)ritum impact of policies (e.g., the Foreign Corrupt Practices Act) on the auditor's risk assessment and planning. 'FThle welfarie implications of some accounting and/or auditing issues on auditors and alI(titees can be derived by means of the game model. For example, the implications of diffeHrent dlegirees of auditing on different interest groups can be analyzed in the game theoretic colt ext. Because of excessive litigation against auditors, questions have been raised about tlie ade luiac\v of audit procedures or thie iquality of audits. A need exists to understand the welfacre ilml)lications of different (legrees of all(lit ilIg on different interest groups. Ihlie (audit game model can be extended(l to nlult iple period scenarios. Since internal audits arce t pl)icallyv lone once every three or four years, an internal audit is likely to be a singlesllot gameln( ill response to changes in p)ersolInel or some other environmental characteristics. I-loswex'er, an internal audit can also be a finitely repeated game. The literature on reputation (IKreps alnd Wilson, 1982; Kreps et al. 1982: Milgrom and Roberts, 1982a; Milgrom and Rolerts, 1982b) predicts bluffing in a nmllltiple-period game. That is, the weaker player tries to influence the probability assessment of the opponent over its type by mimicking the 15

stronger player as the game is repeatedly played and the assessment of the opponent's type is dependent on the prior moves. Bibliography Anderson, U. and Young, R.A., 1988, Internal Audit Planning in an Interactive Environment, Auditing, A Journal of Practice & Theory, Fall: 23-42. Antle. R., 1982, The Auditor as an Economic Agent, Journal of Accounting Research, A utumn, Pt. II: 505-527. Antle, R., 1984, Auditor Independence, Journal of Accounting Research, Spring, 1-20. Antle, R. and Lambert, R.A., 1985, Accountant's Loss Functions and Induced Preferences for Conservatism, Working Paper, September. Ameriican Institute of Certified Public Accountants, 1988, Statement on Auditing Standards N;o. 53, The Auditor's Responsibility to Detect and Report Errors and Irregularities, New York, NY: AICPA. American Institute of Certified Public Accountants, 1988, Statement of Auditing Standards.\o. 55, Consideration of the Infternal Control Structure in a Financial Statement.1 dit. d 1BaiTallnl. S., Evans III, J.H., andl Noel.... 1987, Optinmal Contracts with a Utility-.M[axiilizing Auditor, Jourinal of A.ccoulntin Research, Autumn: 217-244. (C'oi., N. and Blocher, E., 1991, The Efffcls of' Incomplete Information, the Due Care Level, (ld thle Mlateriality Level Inmposed by the Standard-Setting Boards, Working Paper. Clloi. N. and Saigal, R., 1991, ThI.lMa ager-Auditor Came Revisited: The Effects of lncomplete Information and nlformationl Asymmetry, Working Paper. lThe (1Committee of Sponsoring Organization of the Treadway Commission, 1991, Internal Co ntro'l-Integrated Framevwo rk. 16

Dresher, M. and Moglewear, S., 1986, Statistical Acceptance Sampling in a Competitive Environment, Operations Research, May-June: 503-511. Fellingham, J.C. and Newman, D.P., 1985, Strategic Considerations in Auditing, The Accounting Review, October: 634-650. Fellingham, J., Kwon, Y., and Newman, P., 1988, Auditing for Fraud: An Application of Statistical Game Theory, Working Paper, February. Fellingham, J., Newman, D.P., and Patterson, E.R., 1988, Sampling Information in Strategic Audit Settings, Auditing: A Journal of Practice & Theory, Spring: 1-21. Friednman, James, 1986, Game Theory with Applications to Economics, Oxford Press, Inc. Ilarasanvi. J.C., 1967a, Games with Incomplete Information Played by "Bayesian" Players, I: The Basic Model, Management Science, March: 159-182. Ilarasanri, J.C., 1968a, Games with Incomplete Information Played by "Bayesian" Players, II: 3Bayesian Equilibrium Points, Management Science, March: 320-334. Ilarasal i..J.C.. 1968b, Games with Incomplete Information Played by "Bayesian" Play(T s. III: tlie Basic Probability Distribution of the Game, Management Science, March: I - ()502. Inst it lute of Internal Auditors, 1978, Standards for Professional Practice of Internal Audit-.Jaolson. A.. 1990, How to Detect (a Fraud Thrllough Auditing, The Institute of Internal.\ i (litors Research Foundation. Iantlllerl. I... \lcEnroe, J.E., and lKes. I.(C. 1990, Developing and Installing an Audit I1 isk Mo(del, Inteernal Auditor. Decellmber: 51-55. IKimiev. \V.RI. Jr., 1975a, A Decision Thleoretic Approach to the Sampling Problem in.\1luditinIg. Journal of Accounting Research, Spring: 117-132. 17

Kinney, W.R. Jr., 1975b, Decision Theory Aspects of Internal Control System/Design Compliance and Substantive Tests, Studies on Statistical Methodology in Auditing, Supplement to the Journal of Accounting Research: 14-29. Kreps, D.M., Milgrom, P., Roberts, J., and Wilson, R., 1982, Rational Cooperation in the Finitely Repeated Prisoner's Dilemma, Journal of Economic Theory 27: 245-252. Kreps, D.M., Milgrom, P., Roberts, J., and Wilson, R., 1982, Reputation and Imperfect Information, Journal of Economic Theory 27: 253-279. Magee, R.P. and Tseng, M., 1990, Auditing Pricing and Independence, The Accounting Review, April: 315-336. Milgronm, P., and Roberts, J., 1982a, Prediction, Reputation, and Entry Deterrence, Journal of Economic Theory 27: 280-312. Milgrom, P., and Roberts, J., 1982b, Limit Pricing and Entry under Incomplete Information: An Equilibrium Analysis, Econometrica, March: 443-459. The National Commission on Fraudulent Financial Reporting, 1987, Report of the National (Colm tission of Fraudulent Financial Reporting. Newxl.an. P. and Noel, J.. 1989, Error Rates, Detection Rates, and Payoff Functions in Allditilg. Auditing: A Joulrnal of Practice &' Theory, Supplement: 50-66. Ng. S.N. and Stoeckenius. J., 1979a. Audt(iting: Ilncentive and Truthful Reporting, Journal of c.-ccounting Research, Suppl)lemen: 1-16(i. Ng. S.N. and Stoeckenius, J., 19791). I)iscllssion of Auditing: Incentives and Truthful Repl)orting, Journal of Accounltig tl( r(I arh. Supplement, 32-34. Slil)ano, T., 1990, Assessing Audit Risk fromI Errors and Irregularities, Journ'al of Accoulting Research, Supplement: 110- 110. Sittenfelcl I.. 1991, Audit Plainning with the Grid Model, Internal Auditor, February: 32-37. 18

Wallace, W., 1980, The Economic Role of the Audit in Free and Regulated Markets, New York, Touche Ross. Walz, A.P., 1991, An Integrated Risk Model, Internal Auditor, April: 60-65. Watts, J.S., 1990, Discussion of Assessing Audit Risk from Errors and Irregularities, Journal of Accounting Research, Supplement: 141-147. Appendix A Theorem 1. A non-cooperative solution to the game with complete information exists. If RiPj > 1 and RiM > Si, then the unique equilibrium strategies are as follows. The equilibrium strategy for a type i auditor: X* = I/(RiPj) The equilibrium strategy for a type j auditee: Yj = Si/(RiM) Proof: Let XI and Y* be the equilibrium strategies. IThen. (1 - X;RiPj)(Y7* - })) > 0 for every 0 < Y} < 1 and (,'i) - RiY;A'Il)(X, - X?) > 0 for every 0 < Xi < 1. If (1 - Xi RiPj) = 0 and (.Si - RIYA'l) = 0, the above two conditions are satisfied. T'lIs. since RPj > 1 and RiMI > Si, X* = 1/(RiPj) and Y* = Si/(RiM). To see the uniqueness, if 1 - X'RiP> > 0, then Y* = 1, aidc.' - RYi;.l < 0, so.' = 1 and 1 > RZPj a contradiction. Also. if I - XIRPj < 0, then 1 =- 0 adii( ' = 0 a contradiction. Theorem 2. If S1/(R1M) > S2/(R2 11). a non-cooperative solution to the game with ir 'offlplrfe info rmzation exists for the followinrg cases: (i) 0 < S2/(R2A1) < 0.5, and 0 < 1/P, < 0.5/2 forj = 1,2. (ii) 0.5 <,S;/(R;M) < 1 for i = 1.2. and 0 < 1/P1 < 0.5R2. (iii) 0 <C Si/(Ri MI) < 0.5 for i = 1, 2. aind 0.5R2 < 1/P2 < 0.5(R1 + R2). (1i) 0. 5 < '1/(R1M) < 1, anzd 0.5R2 < l/Pj < 0.5(R1 + R2) for j = 1,2. 19

If 0 < S2/(R2M) < 0.5, and 0 < 1/Pj < 0.5R2, then the equilibrium strategies are as follows. The equilibrium strategy for a type 1 auditor: X' = 0 The equilibrium strategy for a type 2 auditor: X* = 2/(R2P2) The equilibrium strategy for a type 1 auditee: Y1 = 0 The equilibrium strategy for a type 2 auditee: Y2* = 2S2/(R2M) Proof: Let X,, X, Y* and Y2* be equilibrium strategies of the four players. The equilibrium strategies of the four players must meet the following conditions. {S1 - RllM(* + 2)/2}{XA' - X1} > 0 for all 0 < X1 < 1 {, 2- R,2Ai("-1* + Y2)/2}{'X2 - X2} > 0 for all 0 < X2 < 1 {1 -:1(X~R1 + X2*R2)/2}{"1 - Y/1 > 0 for all 0 < Y1 < 1 {1 - [2(-R1l + X2R2)/2}{12 - Y2} > 0 for all 1 < Y2 < 1 1=- (i1 + ):27)/2 and. = (A<Rl + X2R2)/2:lsing t ie result of Theorem 1, it canl be shown that equilibrium strategies, for each case, satisfy: (ase N V _i _ / /).^P/( R2 11) _i 1_ / l/.2 /( t?2 M) iii l1/2,I/(RIM), iv 1/Is S,/(RIM) For case (i) ' = ( ')* + Y' e)/2 = S '/(J 1 ). l'Ierefore, {S2 - R2 n(Y1* + 27)/2} = 0. I'lhenl, {S,' - IM11('7 + AY*)/2} > 0 since,s'/(RlMA) > S2/(R2M). h1'lus..V is arbitrary and X1 = 0. N' = (.X;1, + X2;R2)/2 = 1/P2. Therefore. {1 - P2(XRR + X2R2)/2} = 0. Then. {1 - PI(Xl Ri1 + X2R2)/2} < 0 sinlce P > P2 20

Thus, EY = 0 and Y2* is arbitrary. Since {1 - P2(X1R1 + X2R2)/2} = 0 and X* = 0, X* = 2/(P2R2). Since {S2 - R2M(Y1* + Y2*)/2} = 0 and Y* = 0, Y2* = 2S2/(R2M). Since 0 < S2/(R2M) < 0.5, and 2 < R2P2, 0 < X2 < 1 and 0 < Y* < 1. The equilibrium strategies for the other cases can be derived similarly. Theorem 3. A non-cooperative solution to the game with information asymmetry exists. For the following six cases, (i) (1 ->) < Si /(RM) < 1, and 0.5{(1 - O)R + R2} < 1/Pj < 0.5(Ri +R2) forj = 1,2, (ii) 0 < S-/(R-M) < (1 - ~) for i = 1,2, and 0.5(OR1 + R2) < 1/P2 < 0.5(R1 + R2), (iii) ~ < Sl/(R1M)) < 1, and 0.5R2 < 1/P1 < 0.5{(1 - )R1 + R2}, (it) 0 < Si/(R MI) < f for i 1,2, and 0.5R2 < 1/P2 < 0.5(qR1 + R2), (v) 0.5 < Si/(RM) < 1 for i = 1,2, and 1/P1 < 0.5R2, and (vi) 0 < S2/(R2MI) < 0.5, and 1/Pj < 0.5R2 for j =1,2, tl( equilibrium rn strategies also satisfy (0- 0.5)(7 - Y ) < {Sl/(RlM)- S2/(R2MI)} ( (I (~ - 0.5)(X-~o2 - ~o ) < {l/(R1P2) - 1/(RiP1)}. If 0., /(RllM) < c for i = 1,. (ad 0..5)I?2 < I/P2 < 0.5(RI + R2), then the unique (qu iliblr'i ni strategies are as follows. 7l( (qtilibriunm strategy for a tylpe aIl(uitd or w'ih 01:'X1 = 0 7/' ((iqilibriun strategy for a type I audi/or wlit 02: ' 02 = 2/((3R1P2)- R2/(-R1) 7The quilibriurn strategy for a type 2 audiolr: X2 = 1 7Ih( equilibrium strategy for a type 1 auditee: Y' = 0 Th( equilibriu7u strategy for a type 2 auditee: Y2 = 51/(R1iM) 21

Proof: Let X*6, Xh2, Xl, Y1* and Y2* represent the equilibrium strategies of the four players. The equilibrium strategies of the four players must meet the following conditions. [S1 - tlM{I Y1* + (1 - )Y2*}][Xlo, - Xll] > 0 for all 0 < X1el < 1 [S1 - RM{(1 - )Y1* + Y2*}][Xlo2 - X12] > 0 for all 0 < X1o2 < 1 [S2 - t2M(Y* + Y2*)/2][X2 - X*l > 0 for all 0 < X2 < 1 [1 - PI{RX01 - +- (1 - q)R1Xl2 + R2X2}/2][Y* - Y1] > 0 for all 0 < 17 < 1 [1P{(1 - P )RiX*o + <R1X02 + R2X*}/2][Y2* - Y2] 0 for all 0 < Y2 < 1 ~i = qjy; + (1 - q)~Y, Y2 = (1 - )YW +, 2* and Y3 = 0.5(Y1* + 2*) Xi' = 0.5{q5RXlA + (1 - 0Q)R1X012 + R2X }. X2 = 0.5{( -- )R1X01 + qR1X/ 2 + R2X'}. Using the result of Theorem 1, it can be shown that the equilibrium strategies, for each case, satisfy: Case X1 or X2 1?, Y or 13 i - 1 / /P,1 - 1 = /( R1 M) ii X2 =- /P2 ~' = S,/(R1M) ii Xi. l 1 /- I=,2 1/(RiM) iv -'X2 = /2 Y= S1/(R1,[M) v X= 1= ~ 3 - S2/(R2M) vi -\2 1/1)2, - S /(R2M) Foi case (iv ) Y = (1 -)'1+ = S1/(.1 \/). Therefore, [S, -RlM{(I - 0)* + 0"2}] = 0. Tlien. [SI' - RiM{/Y/)* + (1 - -)2'}] > 0 since 2'* > Y'.* )* = Y'* or 12* < ) * leads to a conltIradiction. [-52 -- R21iM( + 1'/2 < O since S2/(R2I) < l3 < S1/(RaM) due to condition (8) on the e(quilibrium strategies. Thus. X2 = 1 and X*0= 0. X=,2 is arbitrary. 22

X2 = 0.5{(1 - )R1Xl1 + qRlX* + R2X2 } = 1/P2. Therefore, [1 - P2{(1 - q)RXl1X + OR1AX*2 + R2X*}/2] = 0. Then, [1 - Pi+-R1,X*1 + (1 - o)R,X*2 + R2X2}/2] < 0 since P2AX2 < P1X due to condition (9) on the equilibrium strategies. [1 - Pi{qORiX61 + (1 - O)R1XiR 2 + R2X*}/2] > 0 cannot happen. Thus, Y* = 0 and Y* is arbitrary. Since [1 - P2{(1 - )RIlX*i + qR- X2 + R2X*}] = 0 and X = 0 and X' 1,X= 0 = 2/(ORi2) - R2/(OR1). Since Y2 = (1 - )Y+* + -Y2* = S1/(R1M) and Y* = 0, Y* = S1/ORiM Since 0 < S-/(RiM) < < for i = 1,2, and 0.5R2 < 1/P2 < 0.5(OR1 + R2), 0 < X'1*2 < 1 and 0 ~2 2 *1. The equilibrium strategies for the other cases can be derived similarly. Theorem 4. Prior information about the auditee type is useful in the first four cases of Th eorf)m 3. In the last two cases of Theorem 3, prior information about the auditee type play.s no roles in the strategy of a type 1 auditor. Proof: The equilibrium strategies of a. type 1 auditor in each case are as follows. cased' Xi-S, X*12 i *I 2/(oR1 P) - - ( -)- R2/(R1) 1 ii 2/{(1P -O )? P }-/(l - ) - R2/{(1 - O)R} iii 0 2/{(1 - O)R,Pl} - R2/{(1 - -)R1} iv.0 o 2/(/R1P2)- R2/(OR1) V 0 0 \vi 0 0 As it ca(1l Ie seen fronm the above table, for t lie first four cases, X1*1 X1 02 and X01 < AX-1. Fol r li last two cases, X101 = A-2 Thleorerm 5. Priorl informatiovp of diff(ir / (i ccuirlcy about the auditee type is not a substilt/f for/ actual testing in the first thle-r cal.(i(s of Theorem 3. Prior information of different (Iccurl / (lboft tl auditee type is a subs/i/ifi for actual testing in the fourth case of Theorem.-. iIl o/l(Ir l'ords.. in the fourth casi of Th7orem 3, <Y/9*l4 0 and aXl./9 I < 0. Proof: I lHe equilibriumlnl strategies of a typle 1 auditor in each case are given in the proof foI 'llicorlin -1, and the equilibrium strategies of the auditees in each case are as follows. 23

caseY1*2 i Si/($R1,M)-(1-q )/& 1 ii 0 S0/{(l - )RiM} iii S1/{(1 - M.)R1M} - 4(1 - 4>) 1 iv 0 Si/(qRiM) The signs of the follows. available derivatives of X*0 or X*2 and Y* or Y* with respect to b are as case signs of derivatives i Qa01/04> a /0 aY*/ > 0 -- ii 0X'196 _<0 Y/9 > 0 iii X'/ > 0 ay*/a0 < 0 iv X /0 < 0 05Y//0a < 0 In Case (iv), ".7 =:S1/((R iM) a9Y:,/ - = {S1/(RlM)}{ —1/2} < 0 9XAo2/ = {2/(R1P2)}{-1/2} - {R2/R1}{-1/I2} 1/02{R2j/R -'2/(RIP2)} = l/02('2/R1){R2/2 - 1/P2} < 0 because 0.5R2 < 1/P2 < 0.5(qR1 + R2). I'he signls of tlie derivatives in the other cases can be derived similarly. Appendix B For t lie four cases of Theorem '2 thle equilibluilll st rategies of the players are: (i) A —1 = 0, X2 = 2/(R2P2) Y; =. 2' = -'2./(1?21M) (ii) -X = 0.' X = 2/R2P1 Y*' = 2,2/(1?2.1) - 1,; = 1 (iii) X.' = /IR1[1/P2 - R2],}X I = 1;' = 0. 12' = 2S,/(RiM) (iv).-i = 1/R[2/P1 - R2]X = 1 = 2,/(RIM) - 1, 1 For tlie cases of Theorem 3, the equilibrium strategies of the players are: 24

(i) X;,9 = 2/(OR1Pi) - (1 )- R)/ (O),X, -, = 1, = 1 Y;* = S1/(RlM) - (1 - )/, Y2 = 1 (ii) X*1 = 2/{(1 - )RP2 - /(1 - ) - 2/(1- )Rl2 = l, = 1 Y = o, Y2 = Sl/{(l - )R1M} (iii) Ax1 = - 0 = 2/{(1 - -)PR1} - R2/{R1(1 - )}, X2 = 1 = S2/{R1M(l - 6)}- /(1 - ), Y2* = 1 (iv) -I = O'0,X = 2/(P2R1)- R2/(=R1), X = 1 }1; =0, 2=' - S1/(R1M) (v) X1 = 0,X2 = - 02 = 2/(P1R2) 7 = 2,-,2/(R2M) - 1,Y2 = 1 (vi) \Xj01 = z0,X'2 = ~0,A = 2/(P2R2),- = 0o = 2)2/(R2Al ) 25