Relative Performance Evaluations in a Model of Financial Intermediation

Review of Economic Dynamics 3, 801᎐830 Ž2000. doi:10.1006rredy.1999.0091, available online at http:rrwww.idealibrary.com on

Relative Performance Evaluations in a Model of Financial Intermediation1 Satoshi Kawanishi Department of Economics, Sophia Uni¨ ersity, 7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan E-mail: [email protected] Received May 27, 1998; revised October 14, 1998

This paper focuses on the delegation by bank managers of lending decisions to their agents, typically subordinate employees of the bank. We assume that agents may base their decisions about lending to borrowers on decisions other banks have made about these same borrowers. Then we show that there exist some lazy or negligent agents who neither directly monitor the borrower nor imitate the other banks if managers use relative performance evaluations as incentive schemes. In addition, it is shown that the learning or adjustment process of agents exhibits cyclical dynamics. Journal of Economic Literature Classification Numbers: D82, D83. 䊚 2000 Academic Press Key Words: relative performance evaluation; imitation; negligence, random matching game; evolutionary game theory.

1. INTRODUCTION When financial intermediaries are unsure about a borrower’s ability to repay a loan, there arises the problem of ‘‘adverse selection.’’ As discussed by Broecker Ž1990., in order to mitigate this problem, banks often rely on active monitoring when they decide on the firm’s loan application. If active monitoring is too costly, banks may simply replicate the lending decisions of the other banks servicing the same firm. In this paper, the matter of bank decision making is examined in detail. In banks there is a section whose responsibility is to investigate the ability 1

I am grateful to Thomas Cooley and two anonymous referees for helpful comments, and I am especially grateful to Akihiko Matsui for his generous instruction. I also benefited from the comments of participants of the seminars held at University of Tokyo, the Japan Development Bank, and Sophia University. All remaining shortcomings are solely my responsibility. 801 1094-2025r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.

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of borrowers to repay their loans. Those in this section, referred to as monitoring agents, can obtain the information they need in two ways. One way is to monitor the borrowers themselves. We call this the ‘‘monitoring strategy.’’ Another is to observe the lending decisions of other banks servicing the same firm. We call this the ‘‘imitation strategy.’’ Bank managers Žprincipals. are willing to let their agents choose a strategy most favorable to the bank. Unfortunately, because the strategy chosen by the monitoring agent is not observable by the principals, a moral hazard arises. Without an appropriate incentive scheme, monitoring agents may actually not monitor the borrower nor imitate the other banks. We call this decision of the agent the ‘‘negligent strategy.’’ There are two general counter measures in the event of a moral hazard. The first is to impose part of the default risk onto the agent. In this, monitoring agents are granted high financial returns if the observable outcomes are good, and low returns if they are not. The second is the so-called ‘‘relative performance evaluation,’’ as discussed by Holmstrom Ž1982.. In this, rewards are set depending on how the agent in question fares compared to other agents. Though some of the literature on financial intermediation discusses the first counter measure, to the best of the author’s knowledge, there are few studies on the effects of relative performance evaluation on the behavior of agents. However, evaluations of agents seem to be based on relative performance in the real world, particularly in the Japanese corporations. The purpose of this paper is to examine the effects of relative performance evaluations on the behavior of banks in the equilibrium of the model. The main finding is that when we look at the whole economy, the supply of new loans from banks fluctuates cyclically. For the purpose of this analysis, a restrictive assumption that managers must evaluate their agent based only on the agent’s relative performance has been adopted. The effect of relative performance evaluations has been investigated by comparing how the restriction affects the equilibrium behavior of banks. Note that the purpose of this paper is not to show why relative performance evaluations are used in the real world.2 To make the analysis as simple as possible, this paper focuses on a ‘‘game’’ between two banks which get a request for financing from the same firm. Each bank is comprised of one manager and one monitoring 2

It is well known that the reason a principal uses relative performance evaluation is the concern with risk sharing. If agents are risk averse, imposing part of the default risk on the agents sacrifices the utility of monitoring agents. To avoid this problem, relative performance evaluation is useful. In addition, there may be reasons related to financial institutions. The profit of a bank depends on the performance of the other banks which compete with the bank for deposits. If deposits go to relatively good banks, bank managers will pay high wages to agents who perform better than others.

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agent. Each manager designs a wage contract in order to let that manager’s agent choose a favorable strategy for them. Then, we construct a random matching game which has the game above as the component game in the following way. We suppose that the rival of each bank is selected from a sufficiently large population. Each manager and each monitoring agent do not know the strategy of the rival in question but do know the strategy distribution within the population. Under this condition, the bank and the rival play the game above. The author takes note of two strategic interactions. Relative performance evaluations can be interpreted as an incentive scheme which makes use of the competition among agents. Managers let their respective agents compete with the agent of the rival bank. This competition characterizes the first strategic interaction. The second strategic interaction is characterized as free riding on the information produced by the competitor. Managers should make their agents imitate the rival’s lending decision if the rival monitors the common borrower and knows its ability to repay a loan. This is because the cost of imitation is generally lower than that of active monitoring. On the other hand, managers should make their agents actively monitor the borrower if the rival does not. The way in which these two strategic interactions affect the equilibrium behavior of the banks is of prime concern here. Two types of analysis are undertaken in this paper. The first is an analysis of the equilibrium of a random matching game. Equilibrium is defined as a state in which all managers have no incentive to change their contracts and all agents have no incentive to change their strategies given the equilibrium contracts. It becomes clear that, under the restrictive condition that managers must evaluate their agents based only on their relative performance, there exist some lazy negligent agents who neither monitor the borrower nor imitate the rival in equilibrium, though managers are not willing for their agents to be negligent. The two strategic interactions mentioned above are important in order to understand this result. The existence of imitation Žfree riding. weakens the power of competition as an incentive scheme. If there are no negligent agents in the population, an imitating agent can get the same wage as one who conscientiously monitors, while incurring lower costs. Under the restrictive condition, managers cannot prevent their agents from imitating the rival. This leads banks to a state where all agents in the population make unreliable decisions as a result of imitating each other. This state is the worst for managers, if the monitoring cost is not too high to pay. Relative performance evaluations lose power as an incentive scheme without the existence of negligent agents. Moreover, the state where all agents in the population are imitating cannot constitute an equilibrium. Suppose only

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one agent becomes negligent and makes baseless decisions in the state. Then, the agent can get the same wage as the others without incurring any costs because the achievement is the same as that of their rivals as a result of imitation. Note that, under the condition above, negligent agents are never punished as long as the agent makes the same decision as their rival. This consideration implies that all agents have incentive to change their strategy from imitation to negligence in the state where all agents imitate. By definition, the state is not qualified as equilibrium. Consequently, we can conclude that the state without negligent agents cannot be equilibrium. We also show that the equilibrium distribution of each strategy is unique. The second analysis deals with the learning or adjustment process of monitoring agents outside the equilibrium, given the equilibrium contract found in the first analysis. It is shown that the adjustment process not only tends to be cyclical, but the equilibrium sometimes becomes a source of this process; i.e., the path diverges from the equilibrium. This result of behavioral cycles is the application of a well-known result in evolutionary game theory. What is interesting is that the structure of the game played by monitoring agents given the equilibrium contract bears a general resemblance to a ‘‘rock᎐scissors᎐paper’’ game. The result is derived from the fact that the learning or adjustment process of generalized rock᎐scissors᎐paper game tends to be cyclical and does not always converge to the Nash equilibrium. Along the cyclical path, the strategy with the highest increase rate changes in the following order: monitoring one, imitating one, negligent one, monitoring one, imitating one, etc. Because within each strategy there are different lending policies, the supply of new loans from banks also changes cyclically along the path. For example, where unconditional lending yields profit, the increase in the number of negligent agents expands the supply of new loans, though it also increases the default rate. The author mentions that these cyclical changes provide an explanation for financial instability. We often say that there seem to be two regimes. In one regime, banks lend to firms without having thoroughly investigated their capacity to repay. In the other, the banks’ investigation is so stringent that it is difficult for firms to borrow without the prospect of a very profitable project. Moreover, the two regimes are repeated one after the other. The result of this analysis suggests that the regime shifts may be an unexpected outcome of contract designs. The instability of the financial market is a curious puzzle. There are many explanations with different properties Žsee Chapter 6 of Freixas and Rochet, 1997, for a comprehensive survey.. Kiyotaki and Moore Ž1997. is one of the notable contributions in the literature, in this regard. They focus on two causal relationships between real assets and loan supply.

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First, the value of real assets as collateral constrains the loan supply. Second, if loan supply increases for some reason, it stimulates company investments and raises the value of real asset owned by the firms. These two relationships work as a propagation mechanism and lets the economy overshoot its stable state. Similar to the work of Kiyotaki and Moore, most other explanations point to the relationship between real activity and finance. On the contrary, the present explanation suggests that the cause may be in the financial market itself, as stated above. This property differentiates the current explanation from the preceding literature. The rest of this paper is organized as follows. In Section 2 a model of financial intermediaries using the concept of a random matching game is described. In Section 3 the equilibrium of the model without the restrictive assumption on relative performance evaluation is discussed. In Section 4, the restrictive assumption is introduced, and the effect on the equilibrium behavior is analyzed. In Section 5, learning or adjustment behavior of the monitoring agent is investigated, and it is shown that the dynamics of the learning process oscillate around the equilibrium. In Section 6, the macroeconomic implications of the cyclical dynamics are discussed. Section 7 is the conclusion.

2. MODEL As illustrated in Fig. 1, we consider two kinds of contracts: debt contracts between firms and banks, and labor contracts between bank managers and monitoring agents. We suppose that the number of banks is sufficiently large and exactly twice that of firms. Suppose also that there are one manager and one monitoring agent in each bank. Both monitoring agents and managers are assumed to be risk neutral. That is, we omit the problem of risk sharing. Since we want to concentrate on labor contracts, we assume the following simple debt contracts. If firms succeed in their project they must repay twice as much as they borrowed, but if their project goes wrong, they

FIG. 1. Two kinds of contracts in the model.

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cannot repay at all. That is, the interest rate is 100%. Results of projects are observable and verifiable for everyone without incurring any costs. However, it takes monitoring cost m for monitoring agents to observe the probability of success of the firms’ projects. For simplicity, we assume that there exist two kinds of firms Žtype G and type B. of the same number, and the probability of success for a firm of type G ŽB. is p Ž q .. We also assume that Assumption 1. 0 - q - 1r2 - p - 1. This means that it is optimal for a risk neutral lender to extend financing only to type G. Of course, all of the firms want to borrow money because they can reap private benefits from their projects. If monitoring agents incur monitoring cost m, they can be certain of the type of firm with which they are dealing. Because the shares of each type are the same, the unconditional expected profit of lenders is p q q y 1. We omit the case where p q q y 1 is zero as a special case. The process of financing. We assume the following simple financing procedure. Step 1. Each firm makes a request for financing to two banks selected from the population of banks at random. Because each bank can only finance one firm, all of the firms and banks can find trading partners after the matching procedure. Under this matching assumption, each monitoring agent in a bank has a rival who is delegated to monitor the same firm’s capacity to repay. Monitoring agents enter into a labor contract, which we characterize later, with their bank manager. Step 2. Monitoring agents must decide whether to tell their manager to finance, or to refrain from advising anything. They make this decision according to strategies that we shall discuss later. Step 3. Bank managers provide financing to the firm if and only if their monitoring agent tells them to finance. Step 4. After the result of the firm’s project is known, wages for agents are determined according to the labor contract signed at Step 1. Next, we provide the strategies of monitoring agents, specifying their decision in each case. Three strategies of monitoring agents. Monitoring agents delegated to monitor have three alternative strategies. Strategy M. Monitoring faithfully, incurring monitoring cost m, and telling their manager to finance the firm if and only if it is of type G.

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Strategy I. Imitating the decision of their rival, incurring imitation cost i.3 In other words, a monitoring agent adopting strategy I tells the manager to finance the firm if and only if the rival does so. Note that he cannot observe the strategy of his rival, but only his decision. If both the monitoring agent and his rival adopt strategy I, neither can report anything to the manager as they are unaware that they are observing each other. As a result, neither manager recommends the financing of the firm at all. Strategy N. If p q q y 1 is positive, the agent tells the manager to finance. If it is negative, the agent remains silent. This strategy incurs zero cost. As a natural assumption, we suppose the following: Assumption 2. m ) i ) 0. As a conventional assumption in random matching games, we also suppose that each agent cannot adopt a mixed strategy. Finally, we provide the assumptions concerning the labor contracts between bank managers and monitoring agents. Labor contracts. The labor contract prescribes the wage of a monitoring agent as a function of the outcome of his decision and his rival’s decision as wŽ i, j. ,

where i , j g  s Ž success . , f Ž failure . , n Ž no finance . 4 .

The first and the second arguments of this function are the outcome of the agent’s lending and that of the rival’s lending, respectively. Note that it is impossible that one bank succeeds and the other fails because the borrower is common. Payoff of monitoring agents. We suppose that the monitoring agents maximize the expected value of their private earnings, wage minus cost, which is needed to adopt each strategy. Since wage depends on the rival’s lending, the payoff of monitoring agents depends not only on their own strategy but also the rival’s strategy. Moreover, rivals of each monitoring agent are selected from the population of all agents. Thus, the payoff of an agent depends on the strategy profile of all monitoring agents except him. Let ⌰M and ⌰I be the percentage of agents pursuing strategy M and I in the population, respectively. The expected payoff of an agent adopting each pure strategy becomes a function of ⌰M and ⌰I . Because both the 3 Imitation cost includes the cost of determining who the rival is and the cost necessary to observe and imitate the ultimate decision, taking care not to be noticed.

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own agent and the rival have three strategies, there are nine matching cases in all. Before calculating the expected payoff, let us calculate payoffs of an agent in every case at first. The matrices in Table I summarize the payoffs. The first one is for the case where p q q y 1 is positive, and the second is for the case where p q q y 1 is negative. The parameters in Table I are defined as

w' wN ' aq' bq' ay' by'

p ⭈ w Ž s, s . q Ž 1 y p . ⭈ w Ž f , f . q w Ž n, n . 2

Ž p q q . ⭈ w Ž s, s . q Ž 2 y p y q . ⭈ w Ž f , f . 2 q ⭈ w Ž n, s . y w Ž n, n . q Ž 1 y q . ⭈ w Ž n, f . y w Ž n, n . 2

Ž 1.

q ⭈ w Ž s, s . y w Ž s, n . q Ž 1 y q . ⭈ w Ž f , f . y w Ž f , n . 2 p ⭈ w Ž s, n . y w Ž s, s . q Ž 1 y p . ⭈ w Ž f , n . y w Ž f , f . 2 p ⭈ w Ž n, n . y w Ž n, s . q Ž 1 y p . ⭈ w Ž n, n . y w Ž n, f . 2

TABLE I Payoffs of an Agent in Nine Matching Cases Rival’s strategy M

Own strategy

M I N

I

wym wym wyi w Ž n, n. y i wN y bq wN In the case where p q q y 1 ) 0

N w q aqy m wN y i wN

Rival’s strategy M

Own strategy

M I N

I

wym wym wyi w Ž n, n. y i w Ž n, n. y by w Ž n, n. In the case where p q q y 1 - 0

N w q ayy m w Ž n, n. y i w Ž n, n.

.

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In the rest of this paper, we need to analyze two possible cases depending on the sign of p q q y 1. We give the sign q to the variable in question in order to show the variable is of the case p q q y 1 ) 0 and give y in order to show the variable is of the case p q q y 1 - 0. Note that if both the agent and the rival are adopting the same strategy, or if one of the two agents is adopting strategy I, then their decisions become the same. Different decisions appear only when strategy M meets strategy N. Then, what we call relative performance evaluations display great power. It is summarized in parameters a and b. If wage does not depend on the rival’s performance, the parameters are zero, a can be interpreted as a reward given to a relatively good decision, and b can be interpreted as punishment given to a relatively bad decision. Next, we calculate the expected payoff of monitoring agents given the strategy profile of population Ž ⌰M , ⌰I .. Let EM, EI, and EN be the expected payoff from each strategy M, I, and N, respectively. Because the ex ante probabilities of matching with an agent adopting strategy M, I, and N for every agent are ⌰M , ⌰I , and 1 y ⌰M y ⌰I , respectively, we obtain that when p q q ) 1, EMqs w y m q Ž 1 y ⌰M y ⌰I . aq, EIqs ⌰M w q ⌰I w Ž n, n . q Ž 1 y ⌰M y ⌰I . wN y i ,

Ž 2.

ENqs ⌰M Ž wN y bq . q Ž 1 y ⌰M . wN , and when p q q - 1, EMys w y m q Ž 1 y ⌰M y ⌰I . ay, EIys ⌰M w q Ž 1 y ⌰M . w Ž n, n . y i ,

Ž 3.

ENys ⌰M w Ž n, n . y by q Ž 1 y ⌰M . w Ž n, n . . Next, we investigate the optimal contract and equilibrium of the model.

3. OPTIMAL INCENTIVE CONTRACT AND EQUILIBRIUM WITHOUT ANY RESTRICTION In this section, we first derive the optimal incentive contract when the strategy profile of population Ž ⌰M , ⌰I . is given. Then, we investigate the equilibrium where all managers make their own monitoring agent select the best response strategy toward the equilibrium strategy profile of population.

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Optimal incenti¨ e contract. As a conventional simplification, we allocate all bargaining power to managers of banks. Given the proportion of rivals Ž ⌰M , ⌰I ., and the reservation payoff of the monitoring agent u, the optimal incentive contract maximizes the managers’ expected profit Ž E⌸ .. When p q q y 1 is positive, the optimal incentive contract  w, w Ž n, n., wN , aq, bq 4 maximizes E⌸qs ␪ M p y 12 y w y Ž 1 y ⌰M y ⌰I . aq q ␪ I ⌰M Ž p y 12 y w . y ⌰I w Ž n, n . q Ž 1 y ⌰M y ⌰I . Ž p q q y 1 y w N . q Ž 1 y ␪ M y ␪ I . p q q y 1 y ⌰M Ž wN y bq . y Ž 1 y ⌰M . wN ,

Ž 4. subject to the following incentive constraints: the agent monitors Ž ⌰M ) 0., only if EMqG u, the agent imitates Ž ⌰I ) 0., only if EIqG u, the agent is negligent Ž ⌰M q ⌰I - 1., only if ENqG u. Similarly, when p q q y 1 is negative, the optimal incentive contract  w, w Ž n, n., ay, by 4 maximizes E⌸ys ␪ M p y 12 y w y Ž 1 y ⌰M y ⌰I . ay q ␪ I ⌰M Ž p y 12 y w . y Ž 1 y ⌰M . w Ž n, n . q Ž 1 y ␪ M y ␪ I . y⌰M Ž w Ž n, n . y by . y Ž 1 y ⌰M . w Ž n, n . ,

Ž 5. subject to the following incentive constraints: the agent monitors Ž ⌰M ) 0., only if EMyG u, the agent imitates Ž ⌰I ) 0., only if EIyG u, the agent is negligent Ž ⌰M q ⌰I - 1., only if ENyG u. If there is no restriction about the wage parameters, the manager can control the agent’s behavior perfectly. For example, the manager can make his agent monitor seriously by setting EM s u, EI - u, and EN - u. Because managers guarantee their agent only reservation payoff u, the managers’ expected profit becomes E⌸ s ␪ M ⭈ ␲ M q ␪ I ⭈ ␲ I q Ž 1 y ␪ M y ␪ I . ⭈ ␲ N ,

Ž 6.

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where

␲ M s p y 12 y u y m

¡⌰

␲I s

␲N s

~

M

Ž p y 21 . q Ž 1 y ⌰M y ⌰I . Ž p q q y 1. y u y i if p q q y 1 ) 0

⌰M Ž p y 12 . y u y i if p q q y 1 - 0

¢

½

Ž p q q y 1. y u yu

if p q q y 1 ) 0 . if p q q y 1 - 0

This means that it is not agents, but managers who incur the costs for monitoring or imitation in real terms. For example, managers need to pay a higher wage to a conscientious monitoring agent than a negligent agent by monitoring cost m. We concentrate on the case where the following natural assumption is satisfied. Assumption 3. p y 1r2 y m ) maxw p q q y 1, 0x. This assumption implies ␲ M ) ␲ N . Therefore, managers never allow their agent to become negligent Ž ⌰M q ⌰I s 1.. First, let us solve the optimal contract in case p q q ) 1. We obtain the following relationship from the definition of ␲ M and ␲ I ,

␲ M ) ␲ I m ⌰I )

1r2 y q pqqy1

⌰M q

q y 1r2 q m y 1 pqqy1

if p q q ) 1.

Therefore, the manager designs the optimal incentive contract to imply

␪ M s 1,

if ⌰I )

␪ I s 1,

if ⌰I -

1r2 y q pqqy1

⌰M q

q y 1r2 q m y i pqqy1

,

and 1r2 y q pqqy1

⌰M q

q y 1r2 q m y i pqqy1

.

On the boundary case, it is indifferent for the manager to make the agent monitor or make it imitate. The correspondence between the proportion of rivals’ strategy Ž ⌰M , ⌰I . and the own agent’s strategy implied by the optimal contract is shown in Fig. 2.1.

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FIG. 2.1. The own agent’s strategy implied by the optimal contract in case p q q ) 1.

Similarly. we can solve the optimal contract in case p q q - 1. If we note

␲ M ) ␲ I m ⌰M - 1 y

myi

if p q q - 1,

p y 1r2

then we obtain the optimal incentive contract as

␪ M s 1,

if ⌰M - 1 y

␪ I s 1,

if ⌰M ) 1 y

myi p y 1r2

,

and myi p y 1r2

.

On the boundary case, it is indifferent for the manager to make the agent monitor or make it imitate. The correspondence between the proportion of rivals’ strategy Ž ⌰M , ⌰I . and the own agent’s strategy implied by optimal contract is shown in Fig. 2.2. Equilibrium. We define an equilibrium as follows.

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FIG. 2.2. The own agent’s strategy implied by the optimal contract in case p q q - 1.

DEFINITION. conditions.

A state is an equilibrium if it satisfies the following two

Ži. No managers can increase their expected profit Ž E⌸ . by changing their contract. Žii. No agents can increase their payoff by changing their Žpure. strategy given the equilibrium contract. The first requires that all managers have no incentive to change their contract in equilibrium. The second requires that all agents have no incentive to change their strategy given the equilibrium contracts. As the most important element, we pay attention to the proportion of strategies in the population. Let Ž ⌰UM , ⌰UI . be the equilibrium values of the proportion of strategies M and I in the population. Figures 2.1 and 2.2 are useful to find equilibrium. In both figures, line S is the locus of ␲ M s ␲ I . The discussion above implies that all managers want to make their agent monitor when Ž ⌰M , ⌰I . is on the left-hand side of line S in Figs. 2.1 and 2.2. On the contrary, all managers want to make their agent imitate on the opposite side of S. First, any state with at least one negligent agent cannot be an equilibrium because the manager whose agent is negligent has incentive to change his contract in order not to allow his agent to become negligent. Thus, the equilibrium state Ž ⌰UM , ⌰UI . must be on the line ⌰M q ⌰I s 1. Next, any state on the line cannot be an equilibrium except

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point E, the intersection of the line and line S. This is because managers whose agent imitates will change the contract to make the agent monitor on the left-hand side of line S, and managers whose agent monitors will change the contract to make the agent imitate on the right-hand side of line S. Finally, point E satisfies the definition of the equilibrium because, as stated above, on line S, it is indifferent for managers to make the agent monitor or make him imitate. Consequently, we obtain the unique equilibrium as ⌰UM s 1 y

myi p y 1r2

,

⌰UI s

myi p y 1r2

.

Ž 7.

Because there is no negligent agent in the equilibrium, no bank lends to the firms of type B. Thus, the equilibrium is independent from the sign of p q q y 1. In the equilibrium, some banks monitor and the others try to imitate them. The number of monitoring agents increases with the gross profit of lending to type G firms Ž p y 1r2. and decreases as imitation become relatively easier Žas m y i becomes larger.. These results are consistent with intuition. We should note that the equilibrium is not Pareto efficient. This is simply because, though the information produced by a bank is useful for the rival, the bank makes decisions without considering the positive externality. As a result, the number of monitoring agents is smaller than ˜M , ⌰ ˜ I . be the the Pareto optimal level. We can confine this easily. Let Ž ⌰ Pareto optimal values of Ž ⌰M , ⌰I .. Because negligent agents are socially ˜M q ⌰ ˜ I s 1. In the Pareto effiuseless under assumption 3, we obtain ⌰ cient state, the average profit of bank managers under the incentive ˜M , ⌰ ˜ I . maximizes constraints must be maximized. That is, Ž ⌰

˜ M p y 12 y m y u ⌰ ˜I ⌰ ˜M Ž p y q⌰

1 2

. yiyu

˜M q ⌰ ˜ I s 1. As a result, we obtain subject to ⌰ ˜M s 1 y ⌰

myi 2py1

) ⌰UM .

Ž 8.

Contract design and the role of relati¨ e performance e¨ aluation. How does the manager design the optimal incentive contract to control the agent’s behavior? There must be two types of managers in equilibrium. One makes their agent monitor, and the other makes their agent imitate.

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In order to make the agent monitor, the manager should design the contract to satisfy the conditions EM s w y m s u EI s ⌰UM w q Ž 1 y ⌰UM . w Ž n, n . y i - u

¡⌰

~

U M

Ž wN y bq . q Ž 1 y ⌰UM . wN - u

if p q q ) 1 EN s U ⌰M Ž w Ž n, n . y by . q Ž 1 y ⌰UM . w Ž n, n . - u if p q q - 1.

Ž 9.

¢

Managers do not have to use the relative performance evaluation to satisfy the conditions above. Even if both a and b are zero, the inequalities are satisfied by setting w Ž n, n. and wN sufficiently small. On the other hand, the manager should design the contract to satisfy the following conditions in order to make the agent imitate, EM s w y m - u EI s ⌰UM w q Ž 1 y ⌰UM . w Ž n, n . y i s u

¡⌰

~

U M

Ž wN y bq . q Ž 1 y ⌰UM . wN - u

if p q q ) 1 EN s ⌰UM Ž w Ž n, n . y by . q Ž 1 y ⌰UM . w Ž n, n . - u if p q q - 1.

Ž 10 .

¢

When p q q ) 1, managers do not have to use relative performance evaluation to satisfy the conditions above. Even if both aq and bq are zero, the inequalities are satisfied by setting w and wN sufficiently small. However, when p q q - 1, managers must use relative performance evaluation to satisfy the conditions above. If by is zero, the expected payoff of strategy N is necessarily larger than that of strategy I. Consequently, the manager cannot make his agent imitate. To make his agent imitate, the manager needs to make the punishment by sufficiently large. This is possible by setting w Ž n, s . sufficiently small. In conclusion, relative performance evaluation does not play a significant role in the model without any restriction on the wage. In particular if a manager wants to make his agent monitor, relative performance evaluation is totally needless. Only when a manager wants to make his agent imitate and when p q q y 1 is negative does the manager use relative performance evaluation.

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4. RESTRICTIONS ON ABSOLUTE PERFORMANCE EVALUATION The results in the previous section are theoretically consistent. However, they do not seem to be consistent with the practical world. We observe our rival’s achievements in many situations so as to stay ahead of our rival. This tendency seems particularly strong in financial institutions. It is said that, though traders have much secret information in order to perform well, they never discuss it even with their colleagues. These observations seem to imply that relative performance evaluation is functioning more effectively in the real economy than the model in the previous section predicts. To understand the effect of relative performance evaluation, we suppose that the inconsistency discussed above is derived from a restriction that managers cannot punish the agent when the rival makes the same mistakes. The restriction is formally given as follows. Assumption 4. w Ž s, s . s w Ž f, f . s w Ž n, n.. This implies w s w Ž n, n. s wN . Consequently, the expected payoff of each strategy becomes simple as follows. If we omit the signs q and y, we do not have to distinguish the two cases, EM s w y m q Ž 1 y ⌰M y ⌰I . a, EI s w y i ,

Ž 11 .

EN s w y ⌰M b. Under the restrictive assumption, managers need to use relative performance evaluation to control their agent’s behavior. If both a and b are zero, agents are always negligent because the expected payoff of strategy N is the largest. Moreover, managers cannot make their agent monitor if there are no negligent agents in the population because when 1 y ⌰M y ⌰I equals zero, EI is always larger than EM. Keeping this in mind, we investigate the equilibrium of the model under the restriction. Equilibrium under the restrictions. First of all, we need to discuss the upper bounds of a and b. To make agents monitor, or to make agents imitate, managers need to set a and b positive and large enough. However, because a and b are composed of the wages, it is impractical to assume that a and b can be infinitely large. Thus, we suppose there are upper bounds as follows. Assumption 5. ᭚A, B g Ž m, ⬁., a F A and b F B.

817

RELATIVE PERFORMANCE EVALUATIONS

Under Assumptions 4 and 5, managers cannot always control their own agent’s strategy. The following restrictions are important, 1 y ⌰M y ⌰I ⌰M -

i B

1 y ⌰M y ⌰I s ⌰M s

i B

myi A

« EM - EI

« EI - EN myi A

« EM F EI

« EI F EN.

When the number of negligent agents is strictly smaller than a critical value Ž m y i .rA, managers cannot make their agent monitor. Similarly, when the number of conscientious monitoring agents is strictly smaller than a critical value irB, managers cannot make their agent imitate. Under these cases, managers have to let their agent choose the better strategy from two possible choices. The boundary cases are more troublesome. In these cases, managers may succeed in letting their agent choose the most favorable strategy for them, or they may not. Restrictive Assumptions 4 and 5 change the equilibrium state drastically. The results are concluded in the following proposition. PROPOSITION. Under Assumptions 4 and 5, there always exists a unique UU . as equilibrium state Ž ⌰UU M , ⌰I

¡max ⌰UU M

s~

Ai q Ž A y m y i . Ž p q q y 1 . A Ž p y 1r2 . if p q q y 1 ) 0

¢B i

,

i B

Ž 12 .

if p q q y 1 - 0

⌰UU s 1 y ⌰UU I M y

myi A

.

Ž 13 .

Proof. See Fig. 3 for illustrations of state space Ž ⌰M , ⌰I .. We should note the restrictions. Managers cannot make their agent monitor when the state is in the strictly right-hand side of the down-sloping line 1 y ⌰M y ⌰I s Ž m y i .rA. Similarly, managers cannot make their agent imitate when the state is in the strictly left-hand side of the vertical line ⌰M s irB.

818

SATOSHI KAWANISHI

FIG. 3. The diagram used in the proof of the proposition.

We whittle the candidates of equilibrium states down by eliminating states which are not qualified as equilibrium. First, all states which satisfy 1 y ⌰M y ⌰I - Ž m y i .rA and ⌰M ) 0 Žall points in the strictly right-hand side of both the line 1 y ⌰M y ⌰I s Ž m y i .rA and the vertical line ⌰M s 0. cannot be an equilibrium state. This is because there exist some agents adopting strategy M Ž ⌰M ) 0., and they have incentive to change their strategy from M to I or N since EM is smaller than EI at least in the states. This contradicts the condition of equilibrium. Thus, the equilibrium state Žpoint E . in the previous section cannot be equilibrium under the restrictions. Because of a similar reason, all states which satisfy ⌰M - irB and ⌰I ) 0 cannot be an equilibrium state. Because EI is smaller than EN in the states, agents who imitate the rival Ž ⌰I ) 0. have incentive to switch their strategy from I to N or M. Second, all states which satisfy ⌰I ) 0, ⌰M - irB, and 1 y ⌰M y ⌰I ) Ž m y i .rA Žthe interiors of the right-angled triangle FGH . cannot be an equilibrium state. This is because there exist some managers who allow their agent to become negligent in the states, and they have incentive to change their contract so as not to allow their agent to become negligent since managers can control their agent’s strategy perfectly in the state Žremember that ␲ M ) ␲ N under Assumption 3.. This violates the equilib-

819

RELATIVE PERFORMANCE EVALUATIONS

rium condition that all managers must not have incentive to change their contract in equilibrium. Third, all states which satisfy ⌰I s 0 and ⌰M - 1 Žall points on the horizontal line except point Ž1, 0.. cannot be an equilibrium state. This is because there exist some managers who allow their agent to become negligent in the states, and they have incentive to change their contract since they can be better off at least by setting EM ) EN. Thus, the states are not qualified as an equilibrium state. Because of the same reason, we can easily eliminate open segment FG from the candidates. Despite the fact that managers can let their agent choose a better strategy M, there exist some managers who allow their agents to become negligent in the states. As a result, we succeeded to whittle the candidates down to open segment FH and point F. In order to finish whittling the candidates down to a unique state, we need to distinguish the two cases according to the sign of p q q y 1. Ži. p q q y 1 - 0. We start with the easier case. Note at first that ␲ N is strictly smaller than ␲ M and ␲ I in this case if ⌰M ) 0. In other words, strategy N is the least favorable strategy for managers. In any state on open segment FH, managers can prevent their agent from choosing strategy N by setting EI ) EN and EM ) EN. In spite of this, there exist some managers who let their agent choose strategy N on open segment FH. This implies that all states in open segment FH cannot be equilibrium. On the other hand, point F can be an equilibrium state. On point F, managers cannot always prevent their agent from choosing strategy N. Managers whose agents monitor or imitate do not have incentive to change their contract if they believe the change lets their agent choose the least favorable strategy N. ŽThis belief is consistent with the fact that strategy N gives agents the largest payoff.. Simultaneously, managers whose agent is lazy negligent do not have incentive to change their contract if they believe the change in contract would have no effect on the strategy of their agent. ŽThis belief is also consistent with the fact that strategy N gives agents the largest payoff.. Though it requires conditions on beliefs of managers, it has been shown that only point F can be an UU . equilibrium. Because F is on the boundaries of two restrictions, Ž ⌰UU M , ⌰I satisfies the conditions

UU 1 y ⌰UU s M y ⌰I

myi A

,

⌰UU M s

i B

.

This completes the proof for the latter part of proposition.

Ž 14 .

820

SATOSHI KAWANISHI

Žii. p q q y 1 ) 0. Unlike the previous case, when p q q y 1 ) 0, strategy N is not always the least favorable for managers. Note the following relationship obtained from the definition of ␲ I and ␲ N ,

␲ N ) ␲ I m ⌰I )

1r2 y q pqqy1

⌰M y

i

if p q q ) 1.

pqqy1

Line T in Fig. 3 is composed by the points that satisfy the equation ⌰I s

1r2 y q pqqy1

⌰M y

i pqqy1

.

Ž 15 .

This line is parallel to line S and located in the left-hand side of it. In the state on the left-hand side of line T, strategy N is more favorable for managers than strategy I. We should note this fact. If point F is located in the right-hand side of line T, this fact is not important because strategy N is the least favorable for managers in the candidate states Žopen segment FH and point F .. Therefore, only point F can satisfy the requirements for equilibrium when F is located in the right-hand side of line T because of the same reasons with case Ži.. On the contrary, if point F is located in the right-hand side of line T, this fact is important. First, consider a state on open segment FH and in the right-hand side of line T. These states cannot satisfy the requirements for equilibrium because of the same reason why states on open segment FH cannot satisfy the requirements for equilibrium in case Ži.. That is, managers whose agent chooses strategy N have incentive to change their contract in this state. Next, consider a state on open segment FH and in the left-hand side of line T. The state cannot be an equilibrium because managers whose agent chooses strategy I have incentive to change their contract in this state. By setting EN ) EI ) EM, the managers can be better off Žremember ␲ N ) ␲ I in these states .. Because of the same reason, point F that is located in the left-hand side of line T cannot be an equilibrium. Finally, consider a state on the intersection of open segment FH and line T. In this state, strategy M is the most favorable, and strategies I and N are indifferent for managers. In this state, managers whose agent chooses strategy M do not have incentive to change their contract. On the other hand, managers whose agent chooses strategy I or N do not have incentive to change their contract if they believe that the change in contract does not improve their payoff. ŽThis belief is consistent with the facts EM F EI and ␲ I s ␲ N in this state.. Thus, though it requires conditions on beliefs of managers, only the state on the intersection of open segment FH and line T can be an equilibrium state.

821

RELATIVE PERFORMANCE EVALUATIONS UU . satisfies the conditions Therefore, Ž ⌰UU M , ⌰I

UU 1 y ⌰UU s M y ⌰I

myi A

⌰UU s I

,

1r2 y q pqqy1

i

⌰UU M y

pqqy1

.

Ž 16 . We obtain

⌰UU M s

Ai q Ž A y m y i . Ž p q q y 1 . A Ž p y 1r2 .

,

⌰UU s 1 y ⌰UU I M y

myi A

.

Ž 17 . To complete the proof, we need to consider the special case where point F is just on line T. Also in this case, all points on open segment FH cannot be an equilibrium state because they are located in the right-hand side of line T. That is, the managers whose agent chooses strategy N have incentive to change their contract. Only point F can be an equilibrium state because of the same reason why the state on the intersection of open segment FH and line T can be an equilibrium when point F is located in the left-hand side of line T. Summing up the result in the case where p q q y 1 ) 0, if F is located in the left-hand side of line T, the unique equilibrium state is on the UU . satisfies intersection of line FH and line T, and Ž ⌰UU M , ⌰I ⌰UU M s

Ai q Ž A y m y i . Ž p q q y 1 . A Ž p y 1r2 .

⌰UU s 1 y ⌰UU I M y

myi A

)

i B

,

Ž 18 .

.

UU . satisOtherwise the unique equilibrium state is point F, and Ž ⌰UU M , ⌰I fies

⌰UU M s

i B

G

Ai q Ž A y m y i . Ž p q q y 1 .

⌰UU s 1 y ⌰UU I M y

A Ž p y 1r2 . myi A

.

,

Ž 19 .

822

SATOSHI KAWANISHI

Putting them together, when p q q y 1 ) 0, we obtain ⌰UU M s max ⌰UU I

s1y

Ai q Ž A y m y i . Ž p q q y 1 . A Ž p y 1r2 . ⌰UU M

y

myi A

,

i B

,

Ž 20 .

.

This completes the proof for the proposition.

Q.E.D.

Notable properties of the equilibrium under Assumptions 4 and 5 are twofold. The first property is variety of equilibrium strategy. Note that there coexist all of the three strategies including strategy N in the equilibrium. This fact implies that relative performance evaluations cannot substitute the role of absolute performance evaluations in the situation of this model. Some may think this is because relative performance evaluations take effect only when there are relatively bad agents. Though this sounds plausible, relatively bad agents do not have to actually exist, but they have only to exist potentially. The true reason why the negligent agents actually exist in equilibrium is the existence of imitation strategy. To understand this fact, consider a world where agents do not have strategy I. In the world, strategy M is always the most favorable for managers. The state where all the managers achieve their goal of making their agents conscientious can be equilibrium if they set b ) m. ŽIn addition, the equilibrium state where all agents monitor conscientiously is Pareto efficient because there is no externality in the world.. This consideration tells us that relative performance evaluations can substitute the role of absolute performance evaluations if strategy I does not exist. Strategy I changes the structure of the model drastically. The following two facts are especially important. Agents prefer strategy I to M if there is no negligent agent in the state in question. Agents prefer strategy N to I if there is no conscientious monitoring agent in the state in question. On the other hand, managers always prefer strategy M to N under Assumption 3. This conflict of interest generates a complicated structure of the model. The second property is fragility of equilibrium. Unlike the equilibrium of the previous section, the equilibrium under the restrictions requires some presumptions on the belief of agents whose strategy is not favorable for their manager. For example, when F in the right-hand side of line S is the unique equilibrium, all managers with a conscientious monitoring agent must believe that, even if they set EI s EN ) EM, their agent will necessarily choose strategy N. In fact, this belief sounds too pessimistic. If we require managers to let their agent choose the most favorable strategy,

RELATIVE PERFORMANCE EVALUATIONS

823

even the unique equilibrium is denied existence in many cases. Only when point F in the left-hand side of line S is the unique equilibrium and when all managers set w s u q i, a s A, and b s B in the state, the equilibrium satisfies the requirement. Though all managers try to let their agent monitor by setting EM s EI s EN in the equilibrium state, some managers fail to do it unfortunately. In addition to these two static properties, the equilibrium has an interesting structure, which may cause cyclical changes in the behavior of monitoring agents. In the following section, we show this by focusing on the equilibrium where all managers set w s u q i, a s A, and b s B as a comparatively robust equilibrium.

5. ADJUSTMENT DYNAMICS OUTSIDE THE EQUILIBRIUM Do managers and agents learn to play the equilibrium? The answer for the equilibrium of Section 2 is obviously affirmative. Without restrictions, managers will learn to let their agent choose strategy M in the left-hand side of line S in Figs. 2.1 and 2.2, or to let their agent choose strategy I in the right-hand side of line S. The adjustment converges to equilibrium point E as illustrated by arrows in the figures. On the other hand, the answer for the equilibrium under restrictive Assumptions 4 and 5 is not obvious. Ideally, we should consider not only the adjustment of strategy by agents, but also the adjustment of contract by managers. However, it is difficult to describe the adjustment of contract by managers because we cannot always say what is the best contract for managers under Assumptions 4 and 5 as discussed in the previous section. Thus, though it is unwilling, we concentrate on the adjustment of strategy by agents given the equilibrium contract by managers. As stated above, we focus on the comparatively robust equilibrium where all managers set w s u q i, a s A, and b s B. The equilibrium state is UU Ž ⌰UU M , ⌰I . s

ž

i B

,1y

i B

y

myi A

/

.

Ž 21 .

If we consider how monitoring agents pursue learning or adjustment outside the equilibrium state, we find that adjustment behavior not only tends to be cyclical, but does not always converge to equilibrium. This occurs mainly because the structure of the strategic situation among agents resembles a rock᎐scissors᎐paper game given the equilibrium contract. First, if all the rival agents are monitoring faithfully, the strategy of ‘‘imitating the judgment of other banks’’ is the best response

824

SATOSHI KAWANISHI

because imitating agents can obtain almost the same evaluation as faithful agents without incurring monitoring costs. Second, if all the rivals are imitating the judgment of other banks, the strategy of ‘‘neither monitoring nor imitating’’ is the best response because agents who make a baseless judgment can get almost the same evaluation as agents who imitate, without incurring the associated imitation costs. Third, if all the rivals are neither monitoring nor imitating, the ‘‘monitoring faithfully’’ strategy is the best response because relative performance evaluations functioning as incentive schemes give faithful agents higher expected wages than negligent agents. As is shown in texts on evolutionary game theory, such as Weibull Ž1995., Vega-Redondo Ž1996., and Fudenberg and Levine Ž1998., the corresponding dynamic system for replicator dynamics oscillates in a rock᎐scissors᎐paper game. Let us demonstrate this result briefly. Replicator dynamics is the prototype of evolutionary dynamics. Under replicator dynamics, the growth rate of the frequencies of pure strategies is linearly related to their payoffs. Adapting this to our model, the dynamics are given as

˙ M s ⌰M Ž EM y EU . ⌰ ˙ I s ⌰I Ž EI y EU . , ⌰

Ž 22 .

where EU ' ⌰M EM q ⌰I EI q Ž 1 y ⌰M y ⌰I . EN. The frequencies of pure strategies increase Ždecrease. if and only if it brings the players higher Žlower. payoffs than average. ŽThe original idea behind this adjustment process is the natural selection of species.. Note that the equilibrium state is a steady state, because EM s EI s EN s EU is realized there. We can easily investigate the behavior of the dynamics in the neighborhood of the equilibrium state. Substituting Ž11. into Ž22. and linearizing Ž22. at the equilibrium state, we obtain

˙M ⌰

ž / ž ˙I ⌰

sJ⭈

⌰M y ⌰UU M , ⌰I y ⌰UU I

/

Ž 23 .

where J'

UU UU ␪M  m y Ž1 y 2⌰UU M y ⌰I . Ž A y B . y A 4

UU ⌰UU M  i q ⌰M Ž A y B . y A 4

UU ␪ IUU  m y Ž 1 y 2⌰UU M y ⌰I . Ž A y B . 4

UU ⌰UU I  i q ⌰M Ž A y B . 4

.

Let ␭1 and ␭ 2 be the two eigenvalues of the Jacobian matrix J. As is well known, the properties of the dynamic system are sum up to ␭1 and ␭2 .

RELATIVE PERFORMANCE EVALUATIONS

825

Substituting Ž21., we obtain the the relationship

␭1 , ␭2 s

tr J "

'Ž tr J .

2

y 4 det J

2

,

Ž 24 .

where tr j ' ␭1 q ␭2 s y

Ž A y B. AB

iŽ m y i. ,

det J ' ␭1 ⭈ ␭ 2 s ⌰UU I i Ž m y i . ) 0, where tr J and det J are the trace and the determinant of matrix J, respectively. To understand the general properties of the dynamic system, consider first the critical case where A y B s 0. The eigenvalues are complex numbers and the real part is zero because tr J s 0 and det J ) 0. This implies that the adjustment behavior not only tends to be cyclical, but that it does not converge to equilibrium. Next, consider the case where A y B / 0. According to Ž24., the eigenvalues remain complex numbers if the absolute value of A y B is not too large. This implies that cyclical behavior can be observed fairly generally. On the other hand, the real part of the eigenvalues becomes negative Žpositive. when A y B becomes positive Žnegative.. In other words, in the case where A Žthe advantage of adopting strategy M . is larger than B Žthe disadvantage of adopting strategy N ., the equilibrium state is at least a local attractor of the dynamics. In fact, the cyclical adjustment paths converge to equilibrium in this case as illustrated in Fig. 4a. By contrast, in the case where A Žthe advantage of adopting strategy M . is smaller than B Žthe disadvantage of adopting strategy N ., the equilibrium state is repelling. In fact, the cyclical adjustment path diverges to the boundary of the state space as illustrated in Fig. 4b. Some readers may think that this result depends solely on the assumptions of replicator dynamics. However, this is not true. According to Gaunersdorfer and Hofbauer Ž1995., two popular learning processes, fictitious play and best response dynamics, have similar properties to replicator dynamics concerning the stability of the Nash equilibrium in the rock᎐scissors᎐paper game.4 As Gaunersdorfer and Hofbauer stated, fictitious play is the oldest and still most fundamental learning process, and best response dynamics is the prototype of modeling rational behavior. Their 4

Unfortunately, because fictitious play and best response dynamics are not smooth in a Nash equilibrium, we cannot investigate their stability by checking the eigenvalues of the Jacobian matrix.

826

SATOSHI KAWANISHI

FIG. 4. The cyclical adjustment paths under replicator dynamics. Ža. Parameters A s 5, B s 3, m s 2.5, i s 1; prior state, ⌰M s 0.1, ⌰I s 0.1. Žb. Parameters A s 4, B s 5, m s 3, i s 1.5; prior state, ⌰M s 0.28, ⌰I s 0.38.

results imply that it is a fairly general result that adjustment behavior not only tends to be cyclical, but does not always converge to equilibrium. Why are there cases where the adjustment behavior does not converge to the equilibrium, even though all the learning processes infer that players change their strategies in order to obtain higher payoffs? Some may think that this is because agents are indifferent as to which of the three strategies to adopt. However, we cannot state that the fact is critical because the equilibrium is not always unstable. The true cause of instability is the disharmony of adjustment behavior, which in turn generates oscillation. The mechanism of oscillation is simple and intuitive. In a state where many monitoring agents adopt strategy M, agents who imitate can obtain the highest expected payoff of the three strategies. As a result of natural adjustment, the percentage of agents adopting strategy I gradually grows. This change creates a comfortable environment for negligent agents because imitation obscures their negligence. In such a situation, agents are wiser to adopt strategy N, which will not incur any cost. As a result of natural adjustment, the percentage of agents adopting strategy N gradually increases. As stated above, faithful agents are justly evaluated only when there are many negligent agents. Relative performance evaluation succeeds in temporarily excluding negligent agents to some degree and in breeding faithful agents. One cycle of adjustment is then complete, the next one starting simultaneously. If the degree of disharmony is small, the cyclical adjustment paths converge to equilibrium; otherwise they diverge. What is important is that the adjustment paths generally exhibit oscillation. Because there are always various stochastic shocks in the real world,

RELATIVE PERFORMANCE EVALUATIONS

827

FIG. 5. The sample paths of replicator dynamics with stochastic shocks in the case of Fig. 4a. Ža. The changes in the share of strategies M and I. Žb. The changes in the supply of new loans Žthe capacity is 1..

the realized adjustment path exhibits oscillation even when the cyclical adjustment paths converge to the equilibrium. For example, there may be stochastic changes in exogenous parameters, or trembling in strategy choice, or experimental strategy change. If there are such shocks when the equilibrium is a global attractor, the sample paths become as illustrated in Fig. 5a Žsee the Appendix for the method of simulation.. The paths cannot stop oscillating because the stochastic shocks impede the convergence to the equilibrium. As a result, the path infinitely repeats this irregular circulation.

828

SATOSHI KAWANISHI

6. MACROECONOMIC IMPLICATIONS OF CYCLES As each strategy has exactly different lending policies, the aggregate amount of new loans change cyclically during the cyclical changes in the population of each strategy. The change in the aggregate loan supply corresponding to the sample path of Fig. 5a is illustrated in Fig. 5b. Note that the lending policy is different depending on the sign of p q q y 1. When unconditional lending yields negative profit Ž p q q - 1., only strategy M and strategy I matching with M extend loans to type G, and other strategies never extend loans to any firms. On the other hand, when unconditional lending yields positive profit Ž p q q ) 1., strategy N and strategy I matching with N extend loans to any firm, strategy M and strategy I matching with M extend loans only to type G, and strategy I matching with I never extend loans to any firms. Thus, loan supply in case p q q - 1 is generally lower than that in case p q q ) 1, and the paths of both cases do not synchronize. We think that the mechanism of oscillation can provide a practical and persuasive explanation for financial instability. For example, the changes in case p q q ) 1 capture some symbolic phenomena of the behavior of Japanese banks in the decade from 1985 through 1995. At the beginning of the eighties, it is known that many Japanese banks apparently based their decisions on other banks’ judgment in the so-called Japanese main bank system. In other words, bank managers seemed to let their agents choose strategy I in that period. However, the tendency has changed in several years. During the Heisei boom Žin particular 1986᎐1990., it was said that Japanese banks lent to firms without having thoroughly investigated their repayment capacity. This lending policy is exactly the same with strategy N in case p q q ) 1. In that period, bank loans kept on growing at the rate of 20% per year. Soon these judgments turned out to be bad. This has been a serious problem in the current Heisei recession in Japan. Recently, banks are said to be so stringent in their investigations that it is difficult for firms to borrow without the almost certain prospect of a profitable project. In the decade of 1985᎐1995, we have observed three different strategies, i.e., imitation, negligence, and conscientious monitoring, one after another. The instability of the financial market is a curious puzzle. There are many explanations for it. There are two features that differentiate our explanation from others. First, our explanation captures two peculiar strategies, i.e., imitation and negligence. Though the existence of these strategies is symbolic as discussed above, other explanations do not seem to have paid sufficient attention to it. Of course, our explanation does not capture all of the symbolic phenomena. For example, we cannot explain the changes in the stock prices, which play the most important role in the

RELATIVE PERFORMANCE EVALUATIONS

829

explanation by Kiyotaki and Moore Ž1997.. Second, though most of others point to the relationship between real activity and finance ŽHyman Minsky, for example, emphasizes the impact of an unsustainable economic expansion on the financial structure ., our explanation suggests that the cause may be in the financial market itself, i.e., the way of evaluation. These features may provide helpful leads to understand the true mechanism of financial instability.

7. CONCLUSION This paper has two main results. First, relative performance evaluation cannot substitute for the role of absolute performance evaluations when monitoring agents can imitate the decision of the other banks. Second, the use of relative performance evaluation as an incentive scheme may cause the cyclical changes in the behavior of monitoring agents. In order to demonstrate the mechanism as simply as possible, we give restrictive assumptions in this paper. The author is interested in whether the results will be affected if we relax the assumptions. The author hopes that these findings and future research can promote greater stability in the economy by helping to better understand business cycles.

APPENDIX To simulate the sample path of replicator dynamics with stochastic shocks, we approximate the differential equations Ž22. by the difference equation with a small adjustment parameter ␥ ) 0, and add random variables ␧ M , ␧ I , and ␧ N , ⌰M tqI y ⌰M t s ␥ ⭈ ⌰M t Ž EMt y EUt . q ␧ M , t y 0.5 Ž ␧ I t q ␧ N t . ⌰I tqI y ⌰I t s ␥ ⭈ ⌰I t Ž EIt y EUt . q ␧ I t y 0.5 Ž ␧ N t q ␧ M t . .

Ž 25 .

 ␧ M t , ␧ I t , ␧ N t 4 is a sequence of i.i.d. random variable ; N Ž0, ␴ 2 .. We interpret these random variables as follows. There will be some shocks that change the absolute numbers of each strategy. ␧ M , ␧ I , and ␧ N represent the effects of the change in the absolute number of agents adopting strategy M, I, and N respectively. Figure 5 is derived from the parameters in Table II.

830

SATOSHI KAWANISHI

TABLE II The Simulation Pararneters Used to Derive Fig. 5 Basic parameters

Prior state

Simulation parameters

A

B

m

i

⌰M

⌰I

␥

␴2

5

3

2.5

1

0.4

0.3

0.08

0.004

REFERENCES Broecker, T. Ž1990.. ‘‘Credit-Worthiness Tests and Interbank Competition,’’ Econometrica 58, 429᎐452. Freixas, X., and Rochet, J. Ž1997.. Microeconomics of Banking. Cambridge, MA: MIT Press. Fudenberg, D., and Levine, D. Ž1998.. Theory of Learning in Games. Cambridge, MA: MIT Press. Gaunersdorfer, A., and Hotbauer, J. Ž1995.. ‘‘Fictitious Play, Shapley Polygons, and the Replicator Equation,’’ Games and Economic Beha¨ ior 11, 279᎐303. Holmstrom, B. Ž1982.. ‘‘Moral Hazard in Teams,’’ Bell Journal of Economics 13, 324᎐340. Kiyotaki, N., and Moore, J. Ž1997.. ‘‘Credit Cycles,’’ Journal of Political Economy 105, 211᎐248. Vega-Redondo, F. Ž1996.. E¨ olution, Games, and Economic Beha¨ ior. Oxford: Oxford Univ. Press. Weibull, J. W. Ž1995.. E¨ olutionary Game Theory. Cambridge, MA: MIT Press.