Applications of Game Theory in Finance and Managerial ... · Applications of Game Theory in Finance...

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.209-241 209

Applications of Game Theory in Finance and Managerial Accounting

Athanasios Migdalas Technical University of Crete, Dept. of Production Engineering and Management, Decision Support Systems Laboratory, University Campus, 73100 Chania, Greece

E–mail: [email protected]

Abstract Game theory has been applied during the last two decades to an ever increasing number of important practical problems in economics, industrial organization, business strategy, finance, accounting, market design and marketing; including antitrust analyses, monetary policy, and firm restructuring. In this paper we give insight into the growing role of game theory, and particularly of the principal-agent model, for the important fields of finance and managerial accounting. Keywords: Equilibria, Game Theory, Mathematical Programming, Portfolio Selection, Corporate Finance, Managerial Accounting 1. Introduction Game theory studies mathematical models of conflict and cooperation between utility optimizers whose decisions influence each other’s utility, and provides mathematical techniques for analyzing such situations. Thus, with respect to optimization, it parallels to operations research and particularly to mathematical programming which are usually concerned with a single optimizer [Shubik (2002)]. Game theory utilizes theoretical concepts, such as Karush-Kuhn-Tucker optimality conditions, as well as numerical techniques from mathematical programming in order to analyze and solve the mathematical models that it studies. In parallel with mathematical programming, such models are idealized representations of an assumed reality, since many aspects of real world cannot be formulated in terms of mathematical expressions. This kind of approximations have raised considerable philosophical issues [Gintis (2000)]. The concepts of strategies, Nash equilibrium and its descendants, such as Bayesian equilibrium, as well as the language for the study of information sets, provided by game theory, have made possible to formalize under the same umbrella situations of conflict and cooperation that arise in a variety of fields. Thus, game theory [Aumann (1985), Fudenberg and Tirole (1991)], which originally developed as an elegant mathematical discipline [von Neumann and Morgenstern (1944), Owen (1982), Forgó et al. (1999)], has been applied during the last two decades to an ever increasing

210 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.209-241

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number of important practical problems [Tirole (1986), Milgrom and Roberts (1992), Shubik (2002)] in economics [Friedman (1990), Gibbons (1992), Besanko et al. (1999)], management [McMillan (1992)], social and political sciences [Shubik (1982), Shiller (1996)], industrial organization and business strategy [Ghemawat (1998), Oster (1999)], finance [Elton and Gruber (1991), Thakor (1991), Allen and Morris (1998)], accounting [Baiman (1982), (1990)], market design and marketing [Chatterjee and Lilian (1986), Roth (1999)]; including antitrust analyses, monetary policy, industrial organization and firm restructuring.

In this paper we give an extensive, but not exhaustive, insight into the growing role of game theory in the important fields of finance and accounting with respect to applications of such concepts as expected utility maximization, asymmetric information, Nash and Bayesian equilibria, signaling games and agency models, emphasizing the mathematical programming problem faced by the optimizers of the games. Particular emphasis is given to the principal-agent problems [Myerson (1991), Gintis (2000)], due to personal bias but also because they are usually only briefly covered in non-specialized game theory books. The approach often followed by the researchers in these fields is to model the problem they are facing as a game, find the corresponding solutions, e.g., the Bayesian Nash equilibria, and then analyze the consequences and implications of these solutions.

Finance is concerned with the allocation of the budget available to investors among firms through financial markets and intermediaries which use it to support their activities. Thus, it is usually divided into two broad fields; asset pricing is concerned with the investors’ decisions, while corporate finance is concerned with the decisions of the firms. The field of finance developed due the necessity to explain many aspects of corporate finance and due to the uncertainty present in asset pricing. Early models in finance heavily relied on assumptions of perfect information and perfect markets. These early models were unable to explain certain empirical phenomena in finance; for instance, why firms pay dividends and how firms choose their levels of debt and equity. Thus, models that incorporate asymmetric information and signaling were introduced. Moreover, in order to analyze the connections between product markets and finance, models were introduced that take into consideration strategic interactions.

Managerial accounting, on the other hand, although linked to finance, has been mainly concerned with analyzing the statistical properties of accounting data and their relevance in decision making. With time, it was recognized that accounting data and strategic decisions within and across firms mutually affect each other. Hence, managerial accounting has shifted from ex post recording and ex post analysis of accounting data to strategic interactions among agents and the ex ante incentives for their behavior. Thus, contracting, incentives, and control of human behavior have come to become more important, and consequently game theory has increased its influence. In particular, principal-agent models have been developed in order to provide insight into, and design of mechanisms for performance and motivation via


incentive contracting, as well as firm activities coordination via budgeting and transfer pricing. Within a firm, for instance, a central decision maker could globally make optimal decisions and direct subordinates to implement centrally determined plans. Principal-agent modeling can provide understanding of incentives and information asymmetries in such a case and help in designing an optimal mechanism for an informationally decentralized organization. If a division manager is rewarded on the basis of the division’s profit, different division outputs concerning costs and revenues are aggregated into a single performance measure. Principal-agent modeling can provide insights in such cases. Through early or late accounting recognition policies, the manager can, at the margin, shift a portion of the reported performance measures, e.g., accruals, across periods. Under conditions of limited communication, performance management may turn out to be an equilibrium behavior that is encouraged by the principal. 2. Asset Pricing and Portfolio Selection A major contribution of game theory to finance was the notion of von Neumann’s expected utility [von Neumann and Morgenstern (1944)], on which important results on asset pricing and portfolio selection have been based.

The capital asset pricing model (CAPM) utilizes a special case of this utility notion, where investors are only concerned with the mean and variance of the payoffs, in order to provide theoretical basis for portfolio choice in the case when the investor’s utility of consumption is quadratic and/or the asset returns are multi-normally distributed. Thus, it was shown that, under such assumptions, in market equilibrium, the diversification of holdings is optimal, and that the benefit obtained depends on the covariances of the asset returns.

Markowitz’s work [Markowitz (1952), (1959), (2002), Farrar (1962), Philippatos (1973), Perold (1984), Elton and Gruber (1991), Zenios (1993), Rubinstein (2002)] on portfolio selection incorporates the notion of the interactive effects of expected returns from a group of investments by the use of covariances, and the notion of diversification and selection of investment combinations, i.e., portfolios, that minimize risk, in terms of variance, for specified levels of expected return. He proved that the rationale underlying the certainty equivalent method, i.e., a preference of expected returns and an aversion towards risk, can lead to the construction of diversified portfolios. The method of certainty equivalence is based on the behavioral assumption that the investor possesses an expected utility function of the form

2[ ( )]E U w µ ασ= − , where w is a random variable representing the investor’s wealth, [ ]E wµ = is the expected value of investor’s wealth, 2σ is the variance of w, and

0α > is the coefficient of risk aversion. In this setting, the investor no longer looks at each asset in isolation, rather the means and variances of the portfolios to which the


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asset would belong are compared. Hence, portfolios and not individual assets define the real set of available investment opportunities. Thus, the investor’s goal is to combine a group of assets such that for a given level of risk, the return is maximized, or, alternatively, for a given level of return, the risk is minimized. A group of assets or securities that achieves such a goal is termed an efficient portfolio. The investor bases his portfolio decisions on the principle of expected utility maximization, and his actions obey the rules of rational behavior. This results in the following mean-variance portfolio selection model (MVM):

Let a portfolio be denoted by a vector 1( )Tnx … x= , ,x assigning portions of

investments to the jth asset ( 1 )j … n= , , . A portfolio that contains every asset traded in the marketplace, i.e., 0jx j> ,∀ , is called a market portfolio. In mathematical programming terms, the portfolio selection problem can be stated [Farrar (1962), Schrage (1997)] as the right-hand side parametric problem

21 1

1

min ( )

s t

,

n nij i ji j

nj jj

x x

x

X

σ σ

µ ρ

= =

=

=

. .

= ,

∈

∑ ∑

∑

x

x

(1)

where }1

{ | 1 0 ( 1 )}nj j jj

X x x a j … n=

= = , ≤ ≤ = , ,∑x , and ρ is a parameter

representing the expected return of the portfolio. Short-sales are incorporated into the model by allowing xj to take negative values. A portfolio x* which achieves the minimum value of the objective in (2) for a given ρ is efficient. The efficient frontier is obtained by varying ρ . A self-financing portfolio is a set of long and short investments such that the net investment is zero. A mean variance analysis of such portfolios is given in [Korkie and Turtle (2002)], where the selection problem is modeled as (2) with

1{ | 0}n

jjX x

== =∑x .

It should be understood that the Markowitz portfolio selection model is a normative and not a descriptive approach. However, the MVM of Markowitz, popularized by Tobin’s work on liquidity preference [Tobin (1958)], spawned a considerable body of literature, and most notably literature on the CAPM [Sharpe (1964), Lintner (1965), Mossin (1966), Bawa and Chakrin (1979)], where it is used as the basis for an equilibrium theory; given investor demands for securities implied by MVM and assuming fixed supplies of assets, they solve for equilibrium security prices in a single-period environment without taxes, and demonstrate that in equilibrium, an individual security is priced to reflect its contribution to total risk, measured by its covariance of its return with the return of the market portfolio of all assets. More pricesly, the CAPM is a simple risk and return model that assumes that


there are no transaction costs, that all assets are traded, that investments are infinitely divisible, and that there is no private (incomplete) information – consequently, investors cannot find under- or over-valued assets in the market place. With these assumptions, diversification becomes attractive and implies holding every traded asset, in proportion to their market value, in one’s portfolio.

Although other asset pricing and portfolio selection models have been proposed [Ross (1977), Lucas (1978), Elton and Gruber (1979), Jarrow et al. (1995), Schrage (1997)], CAPM and MVM came to dominate due to a tractable theoretical basis, due to support by the contributions of other researchers [Tobin (1958), Elton and Gruber (1979), Zenios (1993), Jarrow et al. (1995)], but also because CAPM showed itself useful in a variety of topics, for instance, in deriving discount rates for capital budgeting, and in adjusting to risk. However, although the models do provide theoretical insights in the stock pricing and portfolio selection, they have received considerable theoretical criticism, due to their limited generality, a consequence of the assumptions on quadratic utility functions for the investors and/or normally distributed investment returns [Bawa and Chakrin (1979), Harvey and Siddique (2000)], as well as empirical criticism [Philippatos (1973), Bawa and Chakrin (1979), Jarrow et al. (1995), Bernstein and Damodaran (1998)] with respect to the validity of their predictions in practice, particularly in the presence of short sales [Michaud (1998)]. The quadratic utility implies increasing absolute risk-aversion, while the assumption of normally distributed investment returns carries the unrealistic implication of unlimited liability and rules out asymmetry or skewness [Konno and Suzuki (1995), Harvey and Siddique (2000)] in the probability distribution of returns. It is argued in Harvey and Siddique (2000) that if asset returns have systematic skewness, then expected returns should include rewards for accepting the risk. Thus, they propose an asset pricing model where skewness is priced.

The efficient market hypothesis (EMH) states that a market is efficient if it is possible to make economic profits by trading on available information. Thus, according to the EMH, in cases of competitive perfect markets, i.e., markets with symmetric information and no transaction costs or other frictions, differences in returns across assets are due to differences in risk. Hence, no investor can earn higher return without bearing higher risk. Although arguments exist that stock markets are at least partly predictable [Fama (1991), Thaler (1999), Harvey and Siddique (2000), Lewellen and Shanken (2002)], numerous empirical studies of stock prices indicate that it is difficult to earn above normal profits by trading on publicly available data, and, consequently, arguments exist that markets are efficient [Fama (1970), (1991)]. However, considerable number of anomalies have been documented [Jensen (1984), Jarrow et al. (1995)] for which the CAPM and other asset pricing models are unable to provide insight. Therefore, attempts have been made [Jarrow et al. (1995), Thaler (1999)] to provide psychological, behavioral explanations for such anomalies. This resulted in the field called behavioral finance, which, originally met by scepticism, replaces the assumption of the rational agents/investors who set the prices by adding a


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human element to the modeling, that is, by accepting that cognitive biases may influence asset prices. Thus, psychology motivates the behavior of the agents in these models. It is argued in Thaler (1999) that probably the most important contribution of this field is the careful investigation of the role of markets in aggregating a variety of behaviors, and progress is reported in explaining the “equity premium puzzle” with a full equilibrium model that incorporates consumption as well as returns. In Lewellen and Shanken (2002), it is argued that between rational behavior and irrational miss-pricing there exists a third potential source of predictability, namely parameter uncertainty. Thus, when investors have imperfect information about expected returns or cash flows, they must learn about the unknown process using whatever information is available using Bayesian analysis. They develop a simple equilibrium model to reflect this, i.e., investors are rational and use all available information when making decisions, however, their beliefs may diverge from the true distribution. They derive equilibrium with and without parameter uncertainty.

Asymmetric information in asset pricing is incorporated into the rational expectations equilibrium of Grossman and Stiglitz (1980). In their study, each market participant is assumed to learn from the market prices, however, believing that he cannot influence market prices. It was shown in Dubey et al. (1987) that an explicit strategic analysis was necessary in order to overcome a number of conceptual problems inherent to the rational expectations equilibrium. This direction was taken in the studies by Kyle (1985) and Glosten and Milgrom (1985). See Allen and Morris (1998) for a short historical overview of the area and O’Hara (1995) for an extensive treatment.

A different direction is taken in the inter-temporal capital asset pricing model (ICAPM) of Merton (1969), (1973) which allows markets, even of few securities, to be made effectively complete by using the notion of continuous trading. Merton have used Ito calculus and stochastic control approach to find a solution to the problem in a Markovian model driven by Brownian motion process, for logarithmic and power utility functions. A comprehensive review of his work is presented in Merton (1990). These are continuous time models of greater realism that do not necessarily assume quadratic utility functions or normally distributed returns, and are particularly useful for the pricing of derivative securities, such as options [Ross (1992)]. For non-Markovian models, the partial differential equations approach of Merton is not adequate. Instead, a martingale approach using convex duality has shown remarkable success in solving portfolio optimization problems, particularly for incomplete markets. Under the no-arbitrage assumption, which is equivalent to the existence of a probability distribution π , i.e., the martingale measure, of the asset price X , and given the mean, [ ]E Xπ , the variance, [ ]V Xπ , of the underlying stock price, the problem of finding the best upper bound on the price of a European call option with strike price k and payoff max(0 )x k, − is considered within this framework and formulated as follows [Bertsimas and Popescu (2002)]:


0

max [max(0 )] max(0 ) ( )E X k x k x dxπ π∞

, − = , −∫ (2)

s.t.

{ }0

[ ] ( ) {0 }i iiE X x x dx q i … nπ π

∞= = ,∀ ∈ , ,∫ (3)

( ) 0xπ ≥ , (4)

where the expectation Eπ is taken over the unknown distribution π , and where the n first moments {qi} of the price of an asset are given, and q0=1.

The relations of the ICAPM to the Arrow-Debreu equilibrium have been studied in Harrison and Kreps (1979), Duffie and Huang (1985). The consumption CAP model [Breeden (1979)] and the term structure of interest rates [Cox (1985)] fall also in the category of ICAPM. In Uppal and Wang (2002), the difficulty in estimating the probability law for asset return is taken into consideration, and Merton’s model is extended to a model of inter-temporal portfolio selection where an investor accounts explicitly for the possibility of model misspecification. The model is calibrated to data on international equity returns, and it is observed that significantly under-diversified portfolio results for small differences in ambiguity for the marginal return distribution when the overall ambiguity about the joint distribution of returns is high.

In a recent special issue of “Mathematical Programming”, edited by Mulvey (2001), multi-period stochastic optimization models are developed [Consiglio and Zenios (2001), Mulvey (2001), Zhao and Ziemba (2001)] in order to address significant practical issues. Such problems are mainly solved by any of four primary approaches: (1) by solving a sequence of single period optimization problems, (2) by employing multi-stage stochastic optimization techniques, (3) by following the stochastic control approach introduced by Merton, and (4) by selected decision rules and Monte-Carlo simulation. Of these approaches, (4) are intuitively appealing to professional investors, however, although easy to implement may generate inferior results and require excessive computations. Stochastic programming provides a very general modeling framework, however, the solution methods are computationally very costly. The stochastic control approach provides another general framework but is practically feasible only as long as the state space can be kept manageable.

Konno and Yamazaki (1991), primely concerned with some computational difficulties inherent to the MVM in large-scale practical applications [Markowitz (1959), Philippatos (1973), Elton and Gruber (1979), Perold (1984), Schrage (1997)], especially in the presence of non-convex transaction costs [Konno et al. (1998), Konno and Wijayanayake (1999, 2001), (2002)], introduced a mean-absolute deviation (MAD) portfolio model. In this, Rj denotes the rate of return of the jth asset (j=1, …, n), and a portfolio is again denoted by a vector 1( )T

nx … x= , ,x assigning portions of investments to each asset. The rate of return of portfolio x is given by

1( ) n

j jjR R x

== ∑x . If rj denotes the expected value of Rj, the absolute deviation of the


216

rate of return of portfolio x is given by ( ) [ ( [ ( )] ]W E R E R= | − |x x x . Assuming that

1( )TnR … R= , ,R is distributed over a finite set of points { }1{ ( ) 1 }T

t t ntr … r t … Q= , , , = , ,r ,

and that the probability of occurrence of rt is pt, t=1, …, Q, one has 1

Qj t jtt

r p r=

= ∑

and 1 1

( ) ( )Q nt jt j jt j

W p r r x= =

= | − |∑ ∑x . The MAD optimization model is then defined

as follows:

1 1

1

min ( ) ( )

s t

Q nt jt j jt j

nj jj

W p r r x

r x

X

ρ

= =

=

= | − |

. .

= ,

∈ ,

∑ ∑

∑

x

x

(5)

where }1

{ | 1 0 ( 1 )}nj j jj

X x x a j … n=

= = , ≤ ≤ = , ,∑x }, and ρ is a given constant

representing the expected rate of return of the portfolio. Konno and Wijayanayake (2002) consider the transaction cost c(x) associated with purchasing a portfolio x as consisting of two components; a transaction fee, which is determined by the fee table of each agent, and an illiquidity/market impact cost, which is incurred when xj reaches some bound. The transaction fee, cj(xj) is an increasing piecewise linear concave function, since the unit transaction cost is larger for a small amount of purchase and increases as the amount increases. On the other hand, since the purchase is associated with a sale of some other investors, the unit purchasing price will increase if there is not sufficient sale. It is shown in Konno and Wijayanayake (1999), that problem (2) with the transaction costs incorporated in the objective function can be stated as a linearly constrained non-convex optimization problem. Ogryczak and Ruszczynski (1999, 2001) demonstrated that absolute deviation is an authentic measure of risk, and the resulting MAD model compatible with the von Neumann’s principle of expected utility maximization [von Neumann and Morgenstern (1944)], however, with certain undesirable characteristics. Moreover, as shown in Konno and Wijayanayake (1999), the MAD model (2) can be casted into a linear programming problem, and consequently solved more efficiently for large-scale applications that the corresponding MVM. In [Konno and Wijayanayake (2001), (2002)], a branch-and-bound approach for the non-convex problem, with and without minimal transaction unit constraints, is proposed and results are reported using monthly data of 200 stocks chosen from Nikkei 225 Index. In Goldfarb and Iyengar (1991), robust formulations of the portfolio selection problem are derived in order to overcome the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. They introduce uncertainty structures and show their correspondence to confidence regions associated with the statistical procedures used


in estimating the market parameters. The resulting optimization problems are so-called second-order code programs whose computational difficulty is comparable to that of convex quadratic problems. They present computational results for 43 assets among the top ten Dow Jones industry categories. 3. Corporate Finance Under the EMH, the market value of a firm reflects the present value of the firm’s expected future net cash flows, including cash flows from future investment opportunities [Jensen (1984)]. Consequently, there is no ambiguity about the firm’s objective of maximizing the firm’s current market value, there is no benefit in manipulating earnings per share, security returns are meaningful measure of firm performance, and if new securities are issued at market prices, then concern about the sharing of positive net present value projects with new security holders is eliminated [Jensen (1984)].

Asset pricing defines the opportunity cost of capital for a firm’s capital budgeting decisions and helps determining the expected returns linking present day asset price with expected future returns. Black and Scholes (1973) study the problem of valuation of assets, like call options, which have returns that are contingent on the value of other assets. Their analysis results in an option valuation model for the firm’s equity and debt. In particular, they found that an increase in the value of the firm’s assets increases the expected return and the coverage of the debt, increasing the current value of both. Also an increase in the face value of the debt, reduces the current value of the equity.

In the classical contribution of Modigliani and Miller (1958, 1961), where the implications of market equilibrium for optimal debt policy on financial structure are studied, it is shown that, in the case of perfect markets with no taxes or contracting costs, the current total value of a firm is independent of the firm’s choice of financing policy, i.e., of debt/equity ratio and of the levels of dividends. In Modigliani and Miller (1958), they formulate the capital structure irrelevance proposition (IP) according to which the firm’s financing policy with respect to capital structure cannot affect the value of the firm as far as it does not affect the probability distribution of the total cash flow to the firm. In Modigliani and Miller (1961), the IP is extended to include the firm’s dividend policy. This stresses the importance of taxes and capital market imperfections in determining corporate financial policies, that is, corporate financial decisions do not create value except through tax effects and reduction in transaction costs and other frictions. Unfortunately, their work provides no explanations for the corporate, financial policies observed in practice. Neither the trade-off theory of capital structure, where tax deductibility of interest but neither dividends nor bankruptcy costs are incorporated, nor the tax-augmentation theory of dividends, where the fact that capital gains are taxed less at the personal level than


218

dividends is incorporated, provide insightful explanations of what a firm does in practice. For instance, “equity market timing”, which refers to the practice of issuing shares at high prices and repurchasing at low prices in order to exploit temporary fluctuations in the cost of equity relative to the cost of other forms of capital, remains unexplained by the Modigliani and Miller theory, since in an efficient and integrated capital market, the cost of different forms of capital do not vary independently, and consequently there is no gain from opportunistically switching between equity and debt. In practice, however, equity market timing appears to be an important aspect of real corporate financial policy [Baker and Wurgler (2002)]. Black (1976) addresses the questions regarding why firms pay dividends and the effects of alternative dividend policies when the firm’s cash flow distributions are allowed to vary with dividend policy and coins the expression “the dividend puzzle”. The assumptions of perfect information and perfect markets are indeed strong. On the other hand, game theory provides a methodology to approach unexplained financial phenomena by incorporating asymmetric information and strategic interaction into their analysis.

Pioneering in this respect is the work of Bhattacharya (1979) on signaling and commitment models, where firm managers, having superior information on the profitability of their firm’s investment, signal this to the capital market by committing to a sufficiently high level of dividends. This commitment was subsequently removed in models suggested in John and Williams (1985), Miller and Rock (1985). A signaling game involves two players, a sender and a receiver, where some aspect of the problem being solved by the sender (the state of nature, or the sender’s type) is not known to the sender. The sender selects a message after observing the state of the nature, while the receiver, who cannot observe the true state of the nature, must try to infer it from this message. Having observed the sender’s message, the receiver selects an action, and the game ends as each player receives a payoff which depends on the sender’s message, the receiver’s action, and the state of the nature. The appropriate definition of equilibrium in such games is generally considered as an unresolved question, since the Nash equilibrium concept does not seem powerful enough to provide a full answer. In particular, such models of asymmetric information posses often multiple Nash equilibria, and some kind of refinement mechanism is required in order to select an equilibrium and justify such a selection. In Cadsby et al. (1998), experimental evidence is provided for the predictive power of one particular refinement, namely equilibrium dominance. Moreover, they observed that the mechanism of equilibrium dominance often failed to predict well when a Pareto superior equilibrium is also available. In Kumar (1988), a signaling theory consistent with the fact that firms smooth dividends is developed. None of these models explain, however, why firms use dividends rather than share repurchases. In Brennan and Thakor (1990) it is suggested that repurchases have a disadvantage in that informed investors are able to bid for undervalued stocks and avoid overvalued ones. A survey of game theoretic signaling approaches to the dividend puzzle is given in Allen and Morris (1998).


Concerning the trade-off theory of capital structure, the first game theoretic approach, based on signaling, was proposed in Ross (1977). In this, managers signal the prospects of the firm to the capital markets by choosing an appropriate level of debt, instead of dividends. In Leland and Pyle (1977), a situation is considered where entrepreneurs use their retained share of ownership in a firm in order to signal its value in the presence of asymmetric information. The “capital structure puzzle” under asymmetric information is studied in Myers (1984). In this, and in Myers and Majluf (1984), it is reasoned that instead of using equity to finance investment projects, it would be better to use less information sensitive sources of funds. The preferred order is retained earnings, followed by debt, and equity. This and other results consistent with stylized facts are, however, derived under strong assumptions, such as bankruptcy aversion of managers. Moreover, it is shown in Dybvig and Zender (1991) that sub-optimal managerial incentive schemes are assumed, and that under optimal schemes, the IP can still hold both for capital structure and dividends despite asymmetric information. Surveys of game theoretic approaches to the capital structure puzzle are found in Harrison and Raviv (1991), Allen and Morris (1998). See also Mahrt-Smith (2000).

Asymmetric information is important in initial public offerings (IPO). Numerous studies of IPO show that the initial short-run return on these stocks are significantly positive (See for instance Lowry and Schwert 2002). This phenomenon of “hot IPO markets” has been studied extensively. In [Rock (1986)], it is explained that the under-pricing occurs because of adverse selection. Rock’s model assumes two groups of buyers of which only one is informed about the true value of the stock. Of these, the informed group will buy only when the price is at or below the true value, while the uninformed will receive a high allocation of overpriced stocks. In order for the uninformed group to participate, Rock suggests that they should be compensated for the overpriced stock they buy. Under-pricing on average is one way of doing this. Modeling under-pricing as signals is the approach taken in Allen and Faulhader (1989), Grinblatt and Hwang (1989), Welch (1989). Significant long-term under-performance of newly issued stocks is documented by Ritter (1991). Although game theoretic approaches have been used in explaining underpricing, overpricing is usually explained by diverting from the assumption on rational investors. See Allen and Morris (1998) and Mahrt-Smith (2000) for an overview of the area.

Strategic interaction and asymmetric information constitute the heart of takeover contests. Consequently, game theoretic models have a very important role in such studies [Allen and Morris (1998)].

Intermediation and banking constitute another area of research that has been heavily affected by game theory. A detailed survey is given in Bhattacharya and Thakor (1993), and a short overview of the main results is given in Allen and Morris (1998). In Allen and Santomero (1996), the theory of financial intermediation is revised to fit the significant changes that have occurred during the last decades, and important contributions to the literature are reviewed. In particular, they stress the


220

role of risk trading and participation costs in the new context as opposed to the role of transaction costs and asymmetric information in traditional theories of intermediation.

Jensen and Meckling (1976) use the framework of agency models (Section 2) in order to provide analysis of the conflicts of interest that exist among stockholders, bondholders, and managers, since corporate decisions that increase the welfare of one of these groups may reduce the welfare of the others. They argue that viewing the financial structure problem as one of determining the optimal quantities of debt versus equity is too narrow. Analysis of the monitoring and bonding technology for control of such conflicts are discussed in detail by Myers (1977), Smith and Warner (1979), Mayers and Smith (1982), Smith and Watts (1982), while Smith and Warner (1979), Kalay(1982) analyze the restrictions on dividends specified in corporate contracts. Herd behavior of agents in cases of sequential investment are studied in Scharfstein and Stein (1990), Ottaviani and Sørensen (2000). These studies are concerned with an agent who acts after observing the behavior of another ex ante identical agent. It is argued in Scharfstein and Stein (1990) that reputational herding requires that better agents have more correlated signals (private information) conditionally on the state of the world. They claim that without correlation the second agent would not have the incentive to attempt manipulation of the market inference about ability by imitating the behavior of the first agent. In Ottaviani and Sørensen (2000), it is shown that correlation is not necessary for herding, other than in degenerate cases. The agency game theoretic framework is used in Hart and Moore (1988), Hart and Moore (1994) in order to gain insights into the role of incomplete contracting in determining financial contracts and particularly debt. They consider an entrepreneur who wishes to raise funds to undertake a project. The focus is in designing an incentive for the entrepreneur to pay the borrowed funds. Hart (1995) surveys this particular area. In Sherman and Titman (2000), an agency model with costly information is developed for the problem of building IPO order book with moral hazard. In it, the underwriter selects a group of investors along with pricing and allocation mechanism in a way which maximizes the information generated during the process of going public at a minimum cost. The investors must purchase their information, paying for a signal that may be good, bad or uninformative. Two incentive constraints are present in order investors to both buy the information and to report it accurately. The Nash equilibrium, in which each investor selected by the underwriter optimally purchases information and truthfully reveals the information to the underwriter, requires that the underwriter use optimal pricing and allocation rules. The initial allocation of financial securities in an IPO is also analyzed by Mahrt-Smith (2000) within the framework of a contractual game. A firm owned by debt and equity investors with differential abilities/costs of interfering with management choices is considered. Since the managers have a preference as to which of these investors should have the right to determine the course of the firm, they can be motivated in their work by a contractual setup which leaves the managers’ preferred investors in charge only if managers perform well. A bi-level optimization model [Migdalas et al.


(1998)], with constrained second level, is developed in which the interaction of the capital structure and the ownership structure of the manager-run firm is analyzed. Empirical results are derived for the interaction of equity ownership dispersion, debt ownership structures, bank debt, and dispersed public debt. It is demonstrated that the capital and ownership structures are useful for providing incentives for both managers and investors. Moreover, the usefulness of the model in the analysis of the initial allocation of financial securities in an IPO is also discussed. The next section is devoted to agency models and their applications in managerial accounting and incentive contracting.

4. Managerial Accounting 4.1 Principal-Agent Models and Incentive Contracting The role of compensation contracts in determining risk taking decisions in the financial industry is very important. Empirical evidence [Orphanidis (1996)] suggests that incentive compensation has substantial influence on risk decisions by money managers. Similar considerations have led to the introduction of agency models in order to analyze and design incentive contracting mechanisms.

Since the introduction of the principal-agent model by Mirlees (1976) and Holmström [Holmström (1979), (1982)], it has been used extensively in order to provide insight into incentive contracting and design of incentives and performance measures that alleviate moral hazard [Baiman Demski (1980), Baiman (1982), (1990), Baiman and Evans (1983), Murphy (1984), Kanodia (1985), Jewitt (1988), Jensen and Murphy (1990), Garen (1994), Baiman and Rajan (1995)], but also in order to provide insight into market failures that associate relationship specific assets and franchise contracts [Williamson (1971), (1979), (1983), Grossman and Hart (1986), Hart and Moore (1988), Minkler and Park (1994), Klein (1995), Wimmer and Garen (1997)]. It should be noted that these models differ from the evolutionary agent models used in agent-based computational economics, where economies are modeled as evolving decentralized systems of autonomous interacting agents in order to study the apparently spontaneous formation of global regularities in economic processes [LeBaron (1998), Tesfatsion (1998), (2001)].

The very basic principal-agent model is formulated for a principal and an agent involved in a contractual game. The agent faces a choice of accepting the principal’s offer or declining it and seek employment elsewhere. The agent exerts an effort x that results in an output y observable by the risk neutral principal, who offers the agent a wage w(y). The technology, which is common knowledge, is represented by the distribution function of output dependent on effort, i.e., F(y | x). The risk-neutral principal observes output but not the agent’s effort. It is assumed that F(y | x) is


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absolutely continuous with respect to the same nonnegative measure for each x, thus, ( )F ⋅ has a density f and

( | ) ( | ) ( )y

F y x f t x d tσ−∞

= ∫ (6)

It is assumed that the agent has a separable vol Neumann-Morgenstern utility ( ( )) ( )u w y v x− , where v( )⋅ is a private cost of effort for the agent. Since the agent is

strictly risk and effort averse, ( )u ⋅ is an increasing concave and ( )v ⋅ is an increasing convex function. An optimal contract between the principal and the agent trades off risk sharing and incentives. In this setting, the agent is maximizing his income. The agent’s reservation amount ρ denotes the expected utility he could receive in a different line of business, so the principal’s choice of ( )w ⋅ must be such that the agent’s maximized utility must not be less than ρ . Thus, the principal’s problem is then:

max [ ( )] ( ) ( )w x y w y f y x d yσ, − |∫ (7)

s.t. ( ( )) ( ) ( ) ( )u w y f y x d y v xσ ρ| − ≥∫ (8)

argmax ( ( )) ( ) ( ) ( )x u w y f y t d y v tσ= | −∫ (9)

The individual rationality or participation constraint (8) requires that the expected utility of the agent will be at least as much as its reservation price ρ . The incentive compatibility constraint (9) provides the agent with the motivation he needs to choose the effort level that the principal prefers, given the contract it is offered. Note that this is essentially a bi-level optimization problem [Migdalas et al. (1998)], where the second-level problem, the agent’s utility maximization problem, is unconstrained. If the effort levels x are restricted to a set X, problem (1)-(1) turns explicitly into a constrained bi-level programming problem. In the absence of such constraints, the first-order approach [Mirrlees (1976), Holmström (1979), Milgrom (1981), Grossman and Hart (1983), Jewitt (1988)] replaces the incentive compatibility constraint with the condition that the agent’s expected utility be stationary in effort, i.e.,

( ( )) ( ) ( ) ( ) 0xu w y f y x d y v xσ′ ′| − =∫ (10)

However, the two problems (7)-(9) and (7)-(8), (10) are not generally equivalent since not all stationary points are global maxima. On the other hand, if the agent’s expected utility is concave in effort, the two problems have the same solution. Holmström (1979) showed that the necessary optimality conditions for problem (7)-(8), (10), the optimal compensation w(y) satisfies:


1 ( | )( ( )) ( | )

xf y xu w y f y x

λ µ′

= + ,′

(11)

where it has been established [Holmström (1979), Jewitt (1988)] that 0µ > . Thus, the optimal compensation is such that the principal’s expected payoff is strictly increasing in the agent’s action. Equation (1) is related to the monotone likelihood ratio property (MLRP) which states that the compensation w(y) depends on the properties of the likelihood ratio /xf f′ and implies, for instance, that w(y) is monotone increasing in y. This assumption is quite standard in the literature, and has fairly natural interpretations. It implies for instance that more effort means more output, and that the agent’s payment increases with observed output [125]. However, the derivation of (1) requires the additional assumption that the distribution function of output in the agent’s effort be convex at each level of output. This is disputed both theoretically and empirically [Milgrom (1981), Rogerson (1985), Jewitt (1988)]. Jewitt (1988), in particular, derives conditions under which equation (1) and the first order approach are valid without assuming the MLRP, even in the more general case where the output is ( )y xφ ε= , , with ε being a random disturbance. The case where the principal is risk averse, or in case the effort of the agent is constrained in some set, the first order approach cannot be justified and the problem should be analyzed and solved within a bi-level programming framework [Migdalas et al. (1998)].

These models easily extend to the case of multiple outputs (signals), i.e., 1[ ]n

i iy ==y , with joint density ( )f x|y , particularly when the outputs are

independently distributed, i.e., 1

( ) ( )ni ii

f x f y x=

| = |∏y . This is known as the the conditional independence (CI) assumption. For instance, for 2n = , problem (1)-(1) takes the form [Holmström (1979), Jewitt (1988)]

1 1 2 1 1 2 2max [ ( )] ( ) ( )y w y y dF y x dF y x− , | |∫ ∫ (12)

s.t. 1 2 1 1 2 2( ( )) ( ) ( ) ( )u w y y dF y x dF y x v x ρ, | | − ≥∫ ∫ (13)

1 2 1 1 2 2argmax ( ( )) ( ) ( ) ( )x u w y y dF y t dF y t v t= , | | −∫ ∫ (14)

and equation (11) becomes

1 21 1 2 2

1 2 1 1 2 2

( | ) ( | )1( ( , )) ( | ) ( | )

y yf y x f y xu w y y f y x f y x

λ µ′ ′

= + + ′ (15)

It was shown in Holmström (1979) that, under IC, if either of (y1, y2) is sufficiently statistic for both, then the optimal contract based on both outputs strictly dominates, in the Pareto sense, the optimal contract written on either of them.


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When the principal’s objective function depends on both private information (hidden information) and the unobservable actions (hidden actions) of the agent, such situations are usually modeled in terms of a generalized principal-agent problem. In cases of asymmetric and incomplete information in the contractual game, it is typically the agent who may have more information than the principal. For instance, the principal may not know the utility of the agent or may lack information concerning the environment (Bayesian Nash game). In such cases the revelation principle (RP) implies that by providing a monetary incentive, the principal can ask for additional information from the agent, who would reply honestly, since for every contract that leads to lying, there is a contract with the same outcome for the agent but without inducement for the agent to lie. Thus, RP implies that it is enough for the principal to consider only contracts where it is in the agent’s own interest to react honestly. RP requires strong pre-commitment on the part of the principal. Moreover, RP often leads to multiple equilibria, and even worse, an undesirable equilibrium may dominate the desirable one in the Pareto sense. These problems are further elaborated in Arya et al. (1998a, 1998b).

In Kraus (1993, 1994), incentive contracting for agents in non-collaborative environments is overviewed. In multi-agent environments, where agents do not have a common goal, an agent may try to contract some of the tasks that it cannot perform by itself, or that may be performed more efficiently by other agents. Kraus studies strategies which are in equilibrium, since, if these are adopted by the agents, their interaction will be more stable. Depending on the situation; perfect versus asymmetric information, single versus multi-stage interaction, Kraus considers the concepts of Nash, Bayesian-Nash, and perfect equilibrium.

In Cabrales and Charness (2000), it is examined whether agents are acting as pure material wealth maximizers or whether they are also motivated by social preferences. In Bhattacharya (1979), the question “Who sets CEO pay” is investigated under two views; the contacting view represented by the principal-agent theory, and the skimming view which argues that CEOs set their own pay by manipulating the compensation committee. In Shiller (1996), the potential of improving individual risk management through new risk management contracts and associated index-settled derivatives is investigated. In particular, the idiosyncratic individual risks that can be hedged only at some resource cost due to moral hazard are taken into consideration. 4.2 Monitoring, Control and Aggregation Monitoring in a contractual game is concerned with the stochastic augmentation of an initially designed performance measure by additional information collected at a cost, for instance if the principal undertakes a costly investigation.

Baiman and Demski (1980) addressed, under CI between y1 and y2, the question whether the principal would be willing to pay a cost in order to collect the additional


output y2 when the signal y1 has already been observed. Such investigations on behalf of the principal could be used as a carrot, when y1 is high, or stick, when y2 is low, in order to motivate the agent, given that the cost is not so large that it never pays to monitor. It was shown by Dye (1986) that, under MLRP and IC, the risk aversion of the agent decides whether monitoring is used as a carrot or a stick.

The case where the principal must pay a fixed fee c in order to observe a signal y2 in addition to y1 is modeled by extending the model (1)-(1). A principal can observe y1 before deciding whether to pay for an investigation. In general, he may choose to randomly investigate conditionally on the observed output. He has to choose w(y1, y2) and w(y1), and probability p(y1) conditional on observed output y1. Thus, his problem becomes:

[ ] [ ][ ]{ }1 1 2 1 1 1 1 1 1 2 2max ( , ) ( ) ( ) 1 ( ) ( | ) ( | )y w y y c p y y w y p y dF y x dF y x− − + − −∫ ∫ (16)

s.t. [ ]{ }1 2 1 1 1 1 1 2 2( ( , )) ( ) 1 ( ) ( ( )) ( | ) ( | ) ( )u w y y p y p y u w y dF y x dF y x v x ρ+ − − ≥∫ ∫ (17)

[ ]{ }1 2 1 1 1 1 1 2 2arg max ( ( , )) ( ) 1 ( ) ( ( )) ( | ) ( | ) ( )x u w y y p y p y u w y dF y t dF y t v t= + − −∫ ∫ (18)

This model is linear in p(y1) implying that the optimal monitoring is of “all-or-nothing” nature. At each y1, the principal either monitors with probability 1 or does not monitor.

Assuming that c is neither too high making monitoring uneconomical nor to low making monitoring desirable for all output values y1, an investigation policy is said to be lower tailed if it takes place only for output values below some preassigned level, and upper tailed when it takes place only for output values above such a level. It can be shown [Baiman and Demski (1980), Jewitt (1988)] that the investigation will be lower tailed if

1 2 1 1 1 1( ( , )) ( | ) ( ( )),u w y y dF y x u w y y≤ ∀∫ (19)

and upper tailed if the opposite inequality holds. Thus, the monitoring will be lower tailed if inspection is bad, in terms of expected utility, for the agent, and upper tailed otherwise. Hence, there are criteria for whether it is optimal to use a carrot or a stick to motivate the agent [Baiman and Demski (1980)]. However, Kanodia (1985) has found that optimal monitoring policies are stochastic rather than deterministic when the incentive problem concerns the revelation of hidden information in addition to moral hazard. This seems to be consistent with casual empiricism since most audits and investigations are stochastic in order to incorporate the element of surprise. Moreover, the model assumes that the principal can commit ex ante to a monitoring policy p(y1).On the other hand, it seems to often be ex ante optimal to threaten with an investigation on some observed output but ex post irrational to execute such a threat. Thus, sequentially rational monitoring policies can be shown to be always lower


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tailed. It was shown in Melumad and Mookherjee (1989) that delegation of audits to an independent party could serve as a commitment device.

In Trester (1993), a multi-period framework is developed, in which optimal contract choice is driven by considerations of monitoring difficulty in an environment of potentially asymmetric information with moral hazard. It addresses the role asymmetric information plays in determining why venture capitalists use equity, and preferred equity, rather than debt, to finance entrepreneurial projects. Empirical findings supporting the theoretical results are also provided. In Zhao (2001), two agents are involved in multiperiod, possibly infinite-horizon, contractual relationships. In every period, the agents simultaneously take hidden actions (moral hazard), each of which independently affects the distribution of a separate random public signal. The realizations of the public signals jointly determine the output of a perishable final good, which the agents consume. Zhao examines the nature of Pareto optimal contracts in this environment with respect to both technological and informational constraints.

While much of the literature has treated the hidden information and hidden action separately, in Myerson (1982), a generalized principal-agent model is introduced in order to study the case where the principal’s payoff function depends on both the private information and the unobservable actions of the agent. In Laffont and Tirole (1986), a generalized agency model in a regulation context, where both the regulator and the firm are risk-neutral is studied. They show that the second-best solution is implementable by a partial cost sharing contract. That is, a contract based on the mix between the efficient private information and moral hazard contracts. Closely related is the work in McAfee and McMillan (1986), Baron and Besanko (1987), (1988). It is shown in Faynzilberg (1997) that the MLRP is no longer sufficient for the optimal contract to be monotone in the sharing rule. They therefore introduce the concept of separability of technology, and demonstrate that separability and MLRP are jointly sufficient for the monotonicity.

Since the seminal contribution of Tirole (1986), a large number of research papers in agency theory focus on modeling collusive behavior within multiagent organizations. According to Tirole (1992), incentive compatible schemes designed in accordance with the revelation principle may not be optimal in agency problems that involve two or more agents who may side-contract with each other. In Andrianova (1999), a three-layer model of delegated monitoring is developed, in which the principal is the residual claimant of a hierarchical relationship with an agent who exerts some effort in a production process and with a supervisor who is in charge of a monitoring technology. Neither production nor monitoring is observable or verifiable by the principal, thus making it a problem of double moral hazard. Moreover, the agent is unable to detect if and when he is being monitored. Low cost of collusion, i.e., negligible communication costs and efficient bribing technology, result under certain conditions in a trade-off between ex post and ex ante collusion difference and in an optimal contract without monitoring. On the other hand, if both employee


collusion is sufficiently costly and monitoring secures verifiable evidence of the agent’s effort, the principal optimally saves on supervision costs by randomly hiring the supervisor. In Faure-Grimaud et al. (1999), a three-tier model of a firm’s bureaucracy is considered, where a principal delegates contracting with a productive agent to a supervisor. The model provides a theory of supervision with soft information which allows tracing out the origin of the agency costs of delegated authority in hierarchy. Comparisons to Tirole’s model of collusion are also provided.

In Antle and Demski (1998) and Arya et al. (1998c), the principal-agent model is used in order to make the different control notions in responsibility accounting more precise. A basic premise of responsibility accounting is that managers should be held accountable for variables they control. However, there is ambiguity in this. According to Arya et al. (1998c), a casual notion of control, which they define as controllability, is that a manager’s pay should depend on variables whose marginal distribution he can affect by his supply of inputs. In Antle and Demski (1998), a manager controls a variable if the manager’s input, conditioned on whatever other information is present, influences the distribution of the variable. This notion of control is defined as conditional controllability or informativeness. Controllability does not imply informativeness nor is it implied by conditional controllability. Informativeness explains why certain measures are included in a manager’s performance evaluation and reward system, although the manager may not have direct control over the measures. Conditional controllability is a necessary but not a sufficient condition for a variable to be valuable in contracting. In Arya et al. (1998c), the focus is on the value of information. Two settings are investigated; one in which agents are involved in both team and individual production, and another with focus on owner intervention. In the latter case, the basic model is of one-sided moral hazard, while modeling of two-sided hazard, where the agent’s pay may depend on variables that are conditionally controlled by the principal, is also considered.

Performance measures are often aggregates of different outputs. For instance, double-entry bookkeeping is a linear process of aggregating numerous transactions into few account balance, and so are the quarterly financial statements of future daily reports. There are several benefits in aggregation [Arya et al. (1998d)]: computational costs and information overload may make detailed information less valuable than aggregate, aggregation may lead to cancellation of measure errors, the aggregation process may itself add information, and in cases where the principal shows limited ability to commit, aggregated information may show optimal. Banker and Datar [20] investigate how desirable it is to reward managers on the basis of divisional income and also to aggregate overhead costs into cost pools for allocation purposes. For an n-output model (1)-(1), they investigate the conditions for the existence of a linear aggregate

1

ni iiyλ

=∑ , where the weights λi are independent of y but may depend on the agent’s effort x. They demonstrated that the weights are dependent on the precision and sensitivity of the corresponding outputs with respect to the agent’s action. Thus,


228

the relative information brought by each signal is a decisive factor. The role of aggregation in easing the provision of incentives is studied in Arya et al. (1998d) using a two-period principal-agent model in which either period-by-period output or aggregate output over the two periods is measured. The cost of aggregation is the loss of information available to the principal for use in compensating the agent. The benefit of aggregation is that it provides the agent with less pre-decision information about the first period’s output at the time the agent chooses his second period action. They also study the value of additional information, taking disaggregated or aggregated output as given. The cancellation of errors in aggregate performance measurements is studied in Arya and Glover (1999).

According to Arya et al. (1998e), one role of accounting is in disciplining more manipulable sources of information. For instance, earnings forecasts are kept in check by subsequent audited earning numbers. Their analysis, based on a principal-agent model of hidden action and hidden information, shows that it may be necessary to take a long term perspective to uncover the disciplining role.

In Jensen and Meckling (1976), it is argued that agency problems emanating from conflicts of interest are virtually all cooperative activity among self-interested individuals whether or not it occurs in the hierarchical fashion suggested by the principal-agent model. They define agency costs as the sum of the costs of structuring contracts, i.e., monitoring expenditures by the principal, bonding expenditures by the agent, and the residual loss, which is the opportunity cost associated with the change in real activities that can occur because it is not profitable to enforce all contracts perfectly.

4.3 Transfer Pricing, Budgeting and Audits The key components of organizational architecture are the assignment of decision rights, the measurement of performance, and the reward system. Where decisions rights are placed in a firm’s hierarchy determines the extend to which the firm is centralized or decentralized [Arya et al. (1998f)]. In decentralized organizations transfer of goods and services takes place between the different divisions. In such a process, the producing division records an intrafirm revenue, while the purchasing division registers an intrafirm cost. Such activities necessiate the need for planning and coordination. Accounting practices such as allocations, transfer pricing and budgeting are used to coordinate. Cost allocation coordinate activities by assigning the cost of one activity to others in proportion to some measure of use [Rajan (1992)]. Budgeting attempts to coordinate activities by assigning targets for costs, revenues, production, etc, to managers. Most large corporations pay considerable attention to design elaborate capital budgeting systems in order to provide for decentralized decision making that provides incentives for agents at various level of the organization to make optimal choices [Harris and Raviv (1998)]. Budgeting is an ex


ante process with communication and negotiation between divisional managers and a central firm manager. The term transfer price refers to the amount of monetary units of the interdivisional exchange. This phenomenon of pricing intrafirm transactions is called transfer pricing. Divisional profits reflect transfer prices. Thus, when managers are evaluated on the basis of accounting incomes in their divisions, transfer pricing affects managerial decisions [Vaysman (1998)]. Tranfer-pricing methods can be either administered or negotiated. In the first case, the firm’s top management specifies a set of rules, while in the second case, divisional managers are free to negotiate whether intrafirm transfers take place, the quantity to be transferred, and the transfer payment. The main difference between transfer pricing and cost allocation is that the former is based on ex ante calculations of marginal cost, while the latter is based on ex post average observed costs.

Kanodia (1993) examined coordination and budgeting within the framework of the principal-agent model under the assumption that the managers’ participation constraints must be satisfied state by state, rather than in an ex ante sense. He found that the firm is best off if it splitts its operations so that the performance measure of any manager is unaffected by the performance of other managers, and that the optimal coordination mechanism is a budget based mechanism. In Melumad et al. (1992), for stochastic production costs and revenues, under the RP, it is examined whether deviations from budget are entirely controllable by divisional managers.

Vaysman (1986) demonstrated how coordination mechanisms can be framed as transfer price mechanism. Under RP, coordination mechanisms result in budgeting rather than transfer pricing. However, when communication is limited, transfer pricing is superior to budget coordination mechanisms. In Melumad et al. (1995) it is shown that with limits on communication between divisional managers and the central manager, delegation of decisions has the advantage of allowing decisions to be made based on richer information processed by divisional managers and thus of greater flexibility gains.

Kanodia and Mukherji (1994) analyze a two- and a three-period model in order to obtain insights into the dynamics of audit pricing. The models assume that there is a pool of auditors, with identical technologies, who compete for the audit business of a client firm, that there is a start up cost when the auditor performs a first time audit, that there is a cost when the client switches auditors, and that there is an operating cost of an audit per period. For the two-period model, where the client expects to be in business for two periods and its financial statements are required to be audited in each of the two periods, the equilibrium is found by backward induction. The result is that given the informational advantage of the incumbent auditor, the client must make a “take-it-or-leave-it” price offer to him. Kanodia and Mukherji found out that for the three-period model, given the constraint that clients can write contracts only one period at a time, the optimal audit mechanism cannot be characterized using the RP, i.e., the incentive constraints [Laffont and Tirole (1986)] are not satisfied. The mechanism investigated by Kanodia and Mukherji involves determination of audit


230

price through Bertrand competition among auditors. Morgan and Stocken (1998) investigate the effect of the litigation risk following

an audit report (audit risk) on audit pricing and auditor turnover. The informed incumbent and the uninformed competitors simulatneously submit sealed bids for the audit. The client accepts the lowest bid. For a two-period model, a perfect Bayesian equilibrium involving mixed strategies is derived for the bidding game by backward induction. Their results indicate that auditor turnover will be higher for high-risk clients than for low-risk ones, that the expected litigation costs of high-risk firms are subsidized by low-risk firms, and that, on average, auditors make losses on high-risk audits and compensate for this from excess profits on low-risk audits.

Agents in a hierrachy are commonly delegated authority to communicate and contract with agents at lower levels in order to reduce the burden of communication and information processing on the principal, introducing, however, additional incentive problems [Melumad et al. (1995)]. Harris and Raviv (1998) study the case when a level of a managerial hierarchy will delegate the allocation of capital across projects and time to the level below it. The found that an optimal scheme involves an initial capital spending limit that can be negotiated upward. Depending on the cost of auditing projects, requests by managers for additional capital can be ignored, partially granted without investigating the manager’s projects, or granted fully after a careful audit. When requests are ignored, the allocation of the given capital amount across projects is partially delegated to the manager. The model predicts a trade-off between sensitivity of the aggregate allocation to project characteristics and the extent of delegation of the allocation across projects.

In Fagart and Sinclair-Desgagné (2002), the information systems induced by auditing policies in a principal-agent model with moral hazard is studied. They conclude that the design of optimal auditing policies involves not only the trade-off between risk sharing and incentives, but also an examination of the location of risk.

5. Conclusions Game theory has significantly contributed to the normative rules for the selection of portfolios as well as to the design of measures in and analysis of incentive contracting phenomena. We have also seen that game theoretic models have helped in gaining insights into and explain many phenomena, previously considered as paradoxes or anomalies, in finance. However, quite a few phenomena remain unexplained and further effort is required. There seems to be two directions of research developing; behavioral modeling that moves away from the assumption of rational behavior [Thaler (1999)], and richer game theoretic models that employ information cascades, higher order beliefs and heterogeneous prior beliefs [Allen and Morris (1998)]. Similarly, while the game theory approach to managerial accounting has been successful in taking care of incentive problems that arise from hidden information and


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