The Collusion Deterrence with Prisoner’s Dilemma

doi:10.4236/jssm.2010.34053

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J. Service Science & Management, 2010, 3, 470-478

doi:10.4236/jssm.2010.34053 Published Online December 2010 (http://www.SciRP.org/journal/jssm)

The Collusion Deterrence with Prisoner’s Dilemma

Aidong Liu, Jinfang Liu

Business School of Central South University, Changsha, China.

Email: Liujinfang79@yahoo.com.cn

Received September 29th, 2009; revised September 15th, 2010; accepted October 25th, 2010.

ABSTRACT

I tried to present a new method to prevent collusion through employing two auditors at the same time and inspiring

them provide true report by exploiting their prisoner’s dilemma. But I found this method cannot be put into practice

because of the high cost. So I analyzed whether sending the second auditor in a probability, a low cost method, can

deter the audit collusion. I find sending the second audito r in a probability, enfo rcing the rigidly la wful punishment and

perfecting the reward mechanism can prevent audit collusion. I also find the auditor’s ethical constraint do good to

prevent collusion and the charger of state assets management can play the same role as the real owner of the

state-owned enterprise in deterring collu sion. This finding pro vides theo ry support for the govern ment to implement th e

publicly audit bidding and random double auditing system on state-owned enterprise. The supervision of PCAOB on

auditors is also the operation of this theory.

Keywords: Audit Collusion, Prisoner’s Dilemma, Double Auditing, Ethical Constraint

1. Introduction

Who polices the police? This question has troubled

mechanism-designers ever since the early days of the

Roman Empire. The loose supervision on auditors will

induce them to collude with the auditee. Sunbeam,

Cendant, Waste Management, Enron and Worldcom all

involve the audit collusion [1]. Furthermore the failing

to deter audit collusion leads to American subprime

lending crisis to some extent. The auditors are sus-

pected of collusion because they do not obey the pro-

fessional morality to carry out the whole auditing

process and fail to disclose the serious risk of the

problematic company in subprime lending chain. This

outstanding economic crisis strongly reveals the im-

portance of collusion deterrence.

It’s urgent to find an effective way to deter collusion.

First of all, we need to analyze what caused the collu-

sion. An important reason is the divergence between

the goals of the principal and the auditor. Three kinds

of problems may arise from the divergence between the

goals of the principal and those of the auditor. 1) If the

auditor needs to spend (unwanted) effort to find out

compensation-relevant information about the agent, he

may shirk and report inaccurately. 2) If the auditor and

the agent can jointly manipulate compensation-relevant

information and can write self-enforcing side-contracts,

they may manipulate their information to play coopera-

tively against the principal. 3) If the auditor can ma-

nipulate by himself, compensation-relevant information

about the agent, he may frame and blackmail the agent.

Baiman et al. deal with the first problem in an auditing

model [2]. It seems that the third problem (framing)

has yet to be studied by economists. Our paper ana-

lyzes the problem of side-contracts.

Our concern is to find a way to prevent collusion in

hierarchies of self-interested agents and auditors. Bai-

man et al. have provided a excellent collusion-proof

contracts, they only use one auditor to supervise the

manager and make the auditor and manager supervise

each other to secure the pursued equilibrium [3]. Finkle

and Shin also give a optimal auditing policy according

to the accuracy and the frequency of audits [4]. But

here we would like to use a second auditor to monitor

the first one. If the principal decides to use two audi-

tors, the question arises as to who will monitor the

second auditor. Collusion deterrence depends on the

probability of detection, so if the second auditor is not

monitored he will collude and lose all his effectiveness

for the principal. This reasoning leads inevitably to an

infinite regress; we need a third auditor to monitor the

second, a fourth to monitor the third, and so on.

I will show that it is possible to design a system of

rewards and punishments so that the two auditors po-

lice each other. Even though double-checks are a good

The Collusion Deterrence with Prisoner’s Dilemma471

idea, they are costly. The cost of sending two auditors

may cause such an arrangement to be suboptimal. We

ask, then, whether it is possible to deter collusion by

sending the second auditor with probability less than

one.

I conclude that, under reasonable assumptions on the

size of rewards and punishments, the principal can

achieve truthful reporting only by ‘creating’ a new type

of auditor. When sending the two auditors sequentially

the principal cannot stop collusion if he tells them ei-

ther always or never whether they are the first or the

second auditor. However, by sometimes informing the

second auditor of his position and not telling the agent

whether the second auditor is informed, he can effec-

tively stop collusion. The intuition behind this result is

that the second auditor, when informed about his posi-

tion, will require a bribe unprofitable for the unin-

formed agent to pay, given that a bribe has already

been paid to the first uninformed auditor. In other

words, the second auditor will never collude when he

knows his position and when the first auditor does not

know his position.1 By introducing the imperfect in-

formation the principal ‘creates’ this new type of audi-

tor and is able to deter collusion. Let public Company

Accounting Oversight Board(PCAOB) supervise the

auditors is like designate another auditor to supervise

the first auditor. Although the PCAOB hold higher

authority and cannot inspect as careful as the auditor,

its role in detecting collusion is the same as a second

auditor.

The mechanism I propose is an example of Bayesian

Perfect implementation. That is to say, I trim the set of

equilibria of the game defined by the principal using

the Bayesian Perfect Nash criteria.

There is a variety of settings in the agency literature

wherein the principal gains from withholding informa-

tion from the agent [5]. Our analysis extends this idea

to a hierarchical setting where the principal garbles his

communication with the auditor as opposed to the

agent. In contrast to Maskin and Tirole, our principal

does not design a contract that maximizes ex post in-

formation asymmetry between him and the agent and

preserves his private information. In the framework,

the principal has no private information; instead, by

creating a hidden randomization-which does not affect

the exogenous parameters of the model-the principal

creates the source of his private in for mation.

Tirole was the first to study the phenomenon of

bribes in a hierarchical contract involving a principal, a

auditor and an agent [6]. However, Tirole rules out the

possibility of adding a second auditor.

In Kofman and Lawarrée, they make use of the audi-

tor’s prisoner’s dilemma to deter the collusion [7,8].

Khalil and Lawarrée [7] also use an external signal as a

credible auditor to prevent collusion [9]. But all their

paper assume that the manager will give in the outcome

produced with the principal’s assets. They do not con-

sider the real facts that the manager can earn by push-

ing the market value of the firm as high as possible, as

their compensation is proportional to firms’ market (or

share) value. Criminal investigations proved that on the

eve of the crisis, several large firm CEOs, resorted to

corrupt auditors to produce false, overly optimistic

financial statements, which helped inflate share values.

In what appeared to be outrageous conduct, some ex-

ecutives reaped million dollar gains by exercising

pending stock-options just before the collapse of the

company [10-12]. We will analyze this important ele-

ment in th is paper.

Laffont and Martimort also investigate the simulta-

neous use of two collusive auditors. They show that

information per se introduces increasing returns in the

benefits of side-contract. By duplicating auditors, the

principal can reduce their information and their discre-

tion, and, therefore, improve expected welfare [13].

Barry analyzes accountant’s ethical standards’ ef-

fects to their audit results [14]. We also consider this

element’s influence on the auditor’s behavior when

facing the temptation of collusion. The state owned

company dominates the economic development of

china. From the data of the Chinese stock market in

June 16, 2003, in all public companies almost 48% is

absolutely controlled by the government, while 81% of

all is relatively controlled by the government. This

means most of Chinese public company is dominated

by the state and the company’s major shareholder is the

state. While in china it is the state assets management

committee which exercises the rights of major share-

holder. When supervising the enterprise, deterring the

collusion, the charger of this committee gets the bene-

fits not only from confiscating the information rent

pocketed by the enterprise manager, but also from

gaining the good political achievement by deterring the

collusion. We will analyze the charger’s function in

preventing collusion in the following discuss.

The paper is organized as follows. In Section 2 we

model the problem as a game theoretical situation,

discuss reasonable assumptions about the magnitude of

penalties and rewards, and we present a simple, collu-

sion-proof mechanism. In Section 3 we explore the

possibility of sending the second auditor with a prob-

ability less than one and show that a simple model

yields counter-intuitive results. Section 4 presents a

resolution of this problem and restores our initial intui-

1If they both know their position, collusion cannot be prevented.

The Collusion Deterrence with Prisoner’s Dilemma

472

tion. Finally, Section 5 gathers our conclusions.

2. Description of the Model

2.1. The Game Sequence

We consider a vertical structure represented by a three-

layer hierarchy: principal-auditor-agent. The principal

owns a productive technology, but lacks the skills or

the time necessary to operate it and must hire an agent

for that purpose.2 The agent is the productive unit. The

principal also lacks the knowledge to supervise the

agent. He can hire auditors whose only role is to audit

the agent.3 We assume the audit cost is and the

principal would like to give reservation wage to

the auditors, which is equal to the cost the principal

spend when he do this work by himself.4 All players

are risk neutral.

The agent’s ability to perform depends on a charac-

teristic unobservable to the principal. This characteris-

tic (or type) is the agent’s private information and de-

termines his productivity. We assume that the agent

can only be of two types: high productivity or low

productivity. To simplify the analysis and gain the in-

tuitional results, we assume the agent’s operation cost

is the same no matter what his type is. The prin-

cipal pay the agent according to their report, he gives

the wage to those agents who present high

productive report and to those agents who

present low productive report. Here i







denote the

report presented by the agents, , h



,ilh

represent

high productive report, represent low productive

report, obviously hl

.In the second-best

contract without auditors if the high productive agent

report low output he can obtains



w











, while the low

productive agent can obtains



if he report high

output.5 (For an underlying structure yielding this re-

sult, see [15,16]. Now I denote i



as the information

rent the agent can earn in the end, . For the

high productive agent,



,il













lh lh

ws ws



 . Because









hl lh



h l



 

swsand in

most cases











,













lhhl

sswsws





, then we

get hl





.6 I assume that the agent has limited liabil-

ity and the contract must award him a non-negative

payoff in any state of the world.

To reduce the informational rents of the agent, the

principal can employ a self-interested auditor at cost

. We assume that the auditor cannot buy the right to

audit. He can be subject, though, to negative transfers

if he is caught lying. The auditor learns the agent’s

private information without mistakes and obtains veri-

fiable evidence. His report to the principal, however,

can differ from his observation. If he can gain by ma-

nipulating the report and the agent agrees, he will do so.

We require that the agent collaborates in manipulating

the infor mation to avoid cases in which the auditor can

‘frame’ the agent. The auditor can gain by manipulat-

ing his reports because the agent can give him a condi-

tional side-transfer (a bribe, ). The agent may share

his rent i



to get the auditor to present false informa-

tion to th e princ ipal.

To prevent collusion, the principal could match the

agent’s bribe with a reward . This solution does not

improve the principal’s payoff. Since the agent will

lose i



if reported to the principal, he will be willing

to pay up to i



to the auditor to present a false report.

To discourage the auditor from doing that, the principal

will have to match the bribe and pay i



to the auditor

when his report extracts the agent’s rents. This is a

‘bounty-hunter’ scheme where the auditor obtains all

the informational rents from the agent.

Another strategy to prevent collusion is to hire a

second auditor at cost . This auditor is similar to

the first. He is self-interested, learns perfectly the

agent’s private information and obtains verifiable evi-

dence. He can manipulate this evidence with the help

of the agent to give the principal a false report. Being

the first auditor, he cannot pay for the right to audit but

can be subject to negative transfers if caught lying. The

way in which an auditor can be caught lying is that the



 





hlhl

ws ws



 ,

for the low productive agent,

2I do not allow the principal to sell the firm. While in china, it is the

state assets management committee supervising the state-owned enter-

rise. It sends the agent to manage the enterprise. The leader of the state

assets management committee plays the role as the principal.

3The use of auditor(s) may not be the only way for the principal to

achieve better control of his agent. In some cases, it is not even feasible

as in the case of relationships between doctors and patients, lawyers

and clients, advisors and Ph.D. students. The principal may also dupli-

cate the agents to get more information. This method, however, may be

very inefficient. A typical example is the regulation of a private firm

(agent) characterized by increasing returns to scale. The existence o

competition can reduce the incentive problem, but some of the benefits

of scale economies will be wasted. Our model studies the efficiency o

using a third party (auditor) to lessen the information asymmetry prob-

lem. It is then assumed that other institutional arrangements are not

feasible.