a game theoretical approach to the international debt overhang

Vol. 58 (1993), No. 1, pp. 1-24 Journal of Economics Zeitschrift for National6konomie

�9 Springer-Verlag 1993 Printed in Austria

A Game Theoretical Approach to the International Debt Overhang

By

Mamoru Kaneko, Blacksburg, VA, USA, and Ibaraki, Japan, and Jacek Prokop, Evanston, 1L, USA*

(Received October 15, 1992; revised version received March 8, 1993)

We consider an international financial problem called debt overhang, by which we mean a situation where a sovereign country has borrowed money from foreign banks and has been unable to fulfill the scheduled repayments for some period. The problem is formulated as a noncooperative game with n lender banks as players where each decides either to sell its loan exposure to the debtor country at the present price of debt on the secondary market, or to wait and keep its exposure. This game has many pure and mixed strategy Nash equilibria. We show, however, that in any Nash equilibrium, the resulting secondary market price remains almost the same as the present price for a large number of banks. We also obtain the comparative statics result that in a mixed strategy equilibrium, a bank with a smaller loan exposure has a greater tendency to sell than one with a larger loan exposure. We discuss the implications of these results for the functioning of the secondary market and the resolution of debt overhang.

1. Introduction

This paper considers a game-theoretical aspect of the problem of debt repayments by a sovereign country indebted to many foreign banks. We model the problem as a noncooperative game of lender banks. The debtor country is treated as a part of environment, not as an active player in the game. The country, however, is assumed to be willing to buy back its debts at a discounted secondary market price. The main

* We thank J. Cr6mer, H. Hailer, S. Mendes, and the referees of this Journal for helpful comments on earlier drafts.

2 M. Kaneko and J. Prokop:

result of the paper states that the strategic decisions of lender banks keep the situation of the debtor country almost unchanged, in spite of the assumption of the country's willingness to buy back its debts.

The above problem is a part of the phenomenon called a debt overhang. The term "debt overhang" expresses a situation where a sovereign country has borrowed money from foreign banks and has not succeeded in fulfilling the scheduled repayments for some period. The existence of the debt overhang is a serious problem for the debtor country, which keeps the country in a bad economic situation. The debtor country should reduce its debt to have an access to the international market to finance its economic growth. Many proposals for resolution of the debt overhang have been discussed (cf. Versluysen, 1989). 1 For any kind of resolution, however, it is necessary to understand first the nature of the debt overhang. Before describing the details of our game theoretical investigation, we explain the economic background of the problem of debt overhang.

The debt overhang is closely related to the presence of the secondary market for debts. The secondary market has emerged as a result of the country's economic inability to make full repayments. On the market, loan exposures are traded by lender banks and other financial institutions, and each dollar of debt is priced much below one. For instance, in the case of Bolivia, the price of one dollar of debt was 5r in 1985 and 6r in 1986, and in the case of Peru, it was 19r in 1986 and 6r in 1988. Trade on the secondary market means that if a lender sells its loan exposure to some other financial institution at the price, say 5r then the lender obtains 5% of the lent money and gives up the remaining 95%, but the other institution will take over the right to the loan. Facing the bad economic situations of debtor countries, banks have given up some part of their rights to the loan exposures and have started trading them. Thus possible resolutions of debt overhang could not be considered without taking the secondary market into account.

It is important, particularly in this paper, to remark that currently

1 There are several attempts to investigate some problems related to the phenomenon of debt overhang. They focus mainly on the bargaining over debt reduction between the countur and the banks. For example, Fernandez and Kaaret (1988) considered a situation where one country borrowed money from two banks, one big and one small. Adopting the Nash bargaining solution, they investigate possible agreements on debt reduction. Fernandez and Rosenthal (1990), and Bulow and Rogoff (1989) provided different bargaining models with one country and one bank. Thus those authors considered bargaining over repayments between a borrowing country and a lending bank or banks, given the existing situation of debt overhang.

A Game Theoretical Approach to the International Debt Overhang 3

the debtor country could be a possible buyer of debts on the secondary market, while at the beginning of the secondary market, the debtor was excluded from trade. 2 In contrast to the trade of loans between lender banks and other financial institutions, the trade between the debtor country and a lender bank is expected to affect the country's situation. The trade between a debtor country and a lender bank (debt buyback) at the price, say 5r again, on the secondary market is regarded as 95% forgiveness, since the lender recoups 5% of its loan and the country is not indebted any more. In spite of this possibility of cheap buyback, most of the debtor countries have not succeeded in reducing their indebtedness significantly through the secondary market.

Another important fact is that, in many cases, a debtor country has borrowed money from many foreign banks at the same time. For example, in 1985 Mexico held loans from about 700 banks, Argentina from 370, and Venezuela from 460 banks (Bouchet, 1987, p. 75), though there has been a slight tendency for the number of lender banks to decline. This fact leads us to the consideration of the debt overhang problem from the viewpoint of the noncooperative behavior of lender banks, instead of the bargaining between the debtor country and the lender banks.

With the exclusion of the debtor country as a buyer, the secondary market might be approximated by a competitive model with financial institutions as buyers and sellers, where differences in their financial backgrounds and expectations about the debtor country might create possible trades. With the presence of the debtor country, however, the secondary market can no longer be regarded as a frictionless competitive market. Although we give a particular game theoretical model as a first-order approximation, one of our purposes is to examine and understand the functioning of the secondary market.

Keeping in mind the background of the problem and the above em- pirical facts, we explain our model and main result. As was mentioned above, the problem is formulated as a one-period noncooperative game with lender banks as players and the debtor country is treated as a part of the environment. In our game, each bank decides either to sell its loan exposure to the country at the present secondary market price or to wait and keep its exposure. If a bank sells its exposure, then it obtains a payoff equal to the value of the exposure determined by the present secondary market price of debt. If a bank waits, its payoff

2 For historical and institutional description of the secondary markets for debts, see, for example, Sachs and Huizinga (1987), Huizinga (1989, pp. 5-8), and Hajivassiliou (1989).


is the present value of the loan exposure determined by the resulting secondary market price.

We assume that the secondary market for the country's debt is summarized as a function P[d] of the country's total outstanding debt d, which we call the price function. The price function is country- and time-specific, i.e., it depends on parameters other than the total debt, e.g., country's income, general economic performance, etc. These parameters are implicitly included in the price function. If a buyback takes place, the transaction is made at the secondary market price determined by the price function P [d].

It could be argued that a buyback increases the secondary market price of the country's debt, though the main result of this paper - - the country's situation remains unchanged - - does not need to assume this. When the country succeeds in buying back some of its debts at the discounted secondary market price, this will improve the access to the international capital market and help the country's international trade. Although money spent on a buyback could be used in a different way, e.g., to invest, this has a smaller effect on the improvement of the country's creditworthiness. A buyback is viewed as necessary to ameliorate the country's economy. 3

In this situation, each lender bank faces the decision problem where its resulting payoff highly depends upon other banks' decisions. On one hand, if many banks sell their loan exposures and the country's debt becomes smaller by paying the discounted amount of the loan exposures, then the secondary market price of the next period would become higher, and the banks that wait would receive a higher payoff by the increased price. This implies that there are some incentives for the banks to wait. On the other hand, if only a few banks sell, the total

3 The reader who makes an analogy between the country and a finn may wonder why a buyback of debt may change the secondary market price, since the sum of the country's disposable income and the debt evaluated at the secondary market price remains the same. To consider this question, we recall several aspects of the debt overhang problem. First, the country consists of people - - they are not simply labor supply with fixed wage. Second, foreign banks are not interested in the welfare of people in the country, but are interested in possible repayments and/or in increases in the secondary market price. If the country tries to return debts even by exploiting people's welfare, the country's creditworthiness would be improved. E.g., the secondary market price of Rumanian debt remained about 87 cents under the Ceaucescu government before the 1989 revolution, since the government returned almost all debts at the expense of people's welfare. For these reasons we view the secondary market price as having a tendency of a decreasing function of total debt.


outstanding debt would increase by the accrued interest, and the price of debt would decline, which implies that there are some incentives for the banks to sell. These two opposite tendencies balance in equilibrium.

We consider the concept of Nash equilibrium to represent the strate- gically stable behavior of banks. Our game has, in fact, many pure and mixed strategy Nash equilibria. Independently of the choice of a Nash equilibrium, however, we draw a definite conclusion on the behavior of the secondary market price. It states that the resulting secondary market price of debt remains almost the same as the present price for a large number of banks. Equivalently, the total outstanding debt of the country remains almost unchanged. When the distribution of loan exposures is relatively equal, this theorem gives the prediction that the proportion of banks selling is approximated by the interest rate. When the loan exposures are unequal, one of our results states that in a mixed strategy equilibrium, banks with smaller loan exposures have higher probabilities of selling than banks with larger loan exposures. Therefore the previous prediction is adjusted so that the proportion of banks selling may exceed the interest rate. Section 3 discusses these results. We also discuss the implications of these results for the functioning of the secondary market. We argue that the secondary market is quite different from a frictionless competitive market, mainly because of the presence of the debtor country as a buyer. We consider the structure of the set of mixed strategy equilibria in Sect. 4. Contrary to the almost unique determination of a new equilibrium price, the number of equilibria is large. Looking into the structure of the set of equilibria, we find that these seemingly contradictory aspects are, in fact, consistent.

2. Endurance Competition Game

In an endurance competition game G, we consider decision making of banks in one particular (representative) period. The economic situation of game G is described as a triple (N, {Di}, P). The symbol N denotes the set of banks 1, 2 . . . . . n, who have lent some amount of money to a foreign country, and Di denotes the present loan exposure of bank i. The symbol P denotes a real-valued continuous function on [0, +oo), which gives the secondary market price P [d] when the total outstanding debt is d. The present secondary market price is given as P[D], where D = Z i o N Di. One additional element is the market interest factor, denoted/~ (= 1 § the interest rate) > 1. W e assume that

P[D] > P[~D] and P[D] < P[O] . (2.1)


The first inequality of (2.1) states that if the country does not buy back any debt in the present period, then the total outstanding debt increases t o /3D by the accrued interest, and its market price declines to P[/3D]. The second inequality states that if all banks sell their loan exposures at the present price P [D], then in the next period the price of (an arbitrarily small) debt is higher than or equal to the present price P[D]. As was mentioned in Sect. 1, the price function P[d] is country- and time-specific.

Throughout the paper, we assume that the price function P is fixed, but the set N of banks and their loan exposures {Di} may vary. There- fore we denote our game G by (N, {DID. In this game, the banks are players and the country is a part of the environment.

Each bank i 6 N has two pure strategies, 0 - to sell its exposure at the present price P [D], and 1 - to wait and postpone the decision to the next period. 4 We denote the strategy space {0, 1} of player i by Si. Then S 1 x . . . x S n is the outcome space. When each bank i in N chooses its strategy si and the market transactions are completed, the price of the next period becomes

P[fld], where d = ~_s jDj . (2.2) j~N

If bank i keeps its loan exposure D i until the next period, it increases by accrued interest to flDi, and the total value becomes flDiP[fld] in the beginning of the next period. Since/3 is the interest factor, we discount this total value by 1//3 to have the present va lue) Thus the

4 In reality, loan exposures are regarded as partially divisible. The assumption that a loan exposure of each bank is indivisible is one extreme case, and the other extreme is that a loan exposure is completely divisible. We choose the first case, since exposures are often treated as indivisible packages - - the first extreme case seems to be better from this viewpoint. Nevertheless, we can obtain almost the same conclusion as our main result in the second case, without introducing mixed strategies.

5 Here we assume that the discount factor coincides with the interest factor. Since our subjects are financial institutions, this assumption seems to be plausible. If they were different, we should modify our result by multiplying appropriate variables by the ratio of these factors, and the main result of this paper would be modified into that the resulting new secondary market price becomes almost the present price multiplied by the discount factor. This mod- ification, however, could not be persistent, regarding our problem as a part of the entire economy, i.e., the discount factor would be changed into the interest factor by competition of financial institutions.

A Game Theoretical Approach to the Intemational Debt Overhang 7

present value of the exposure which will be sold in the next period becomes

1 -fi (ri Di P [rid]) --- Di P [rid] . (2.3)

We assume that this present value DiP[rid] is the payoff to bank i if it does not sell its loan exposure in the present period. We also assume that if bank i sells Di in the present period, then its payoff is simply Di P[D]. We do not take into account the possibilities of the bank's revenues by postponing selling its exposure after the next period. 6 Thus the payoff function of bank i is given as

[ DiP[fl~_.j~NSjDj] ifsi = 1 hi(s) = hi(s1 . . . . . Sn) = ' (2.4)

DiP[D] if si = 0 .

If all lender banks sell their exposures to the country, then the country must pay the total amount of DP[D]. To have G as a well defined normal form game, we assume that the country is able to afford these repayments, which implies that the current price P[D] is small enough for the repayments, or the total amount of debts is not large enough to prevent the total repayment. Practically, however, for our main result we need the assumption that the country is able to afford to pay a little more than the amount equal to the interests discounted at the secondary market price.

In addition to pure strategies, we allow the banks to play mixed strategies. Denote the set of mixed strategies of bank i b y / ) , i.e. Ti = {Pi : 0 < Pi < 1} for i E N, where Pi is the probability of waiting by bank i. Define the expected payoff to bank i by

-.(,. = ,.[ (1-I z + ScN\{i} jf~S jeS jeS

jr

+ (1 - pi)DiP[D]

for p = (Pl . . . . . Pn) in T1 x . . . x Tn �9 (2.5)

The first term of (2.5) is the expected payoff to bank i from waiting, and the second term (1 - Pi) Di P [D] is the expected payoff from selling.

6 This possibility is taken into account in the dynamic description of this problem with two banks in Prokop (1991, 1992). He shows, however, that this does not substantially change the structure of the equilibria.


We apply the Nash equilibrium concept to our game. A strategy n-tuple/3 = (/31 . . . . . /3n) is called a Nash equil ibrium if for all i E N,

Hi(~3) >- ni(/3-i, Pi) for all Pi E Ti ,

where /3-i = (/31 . . . . . /3i--1, /3i+1 . . . . . /3n). Before going to the results of the general case, we consider the two

bank case. In the case of two banks, under the condition

P[f ld] < P[D] < P[ f lDi] f o r / = 1 , 2 ,

game G has three equilibria:

(/31, t32) = (1, 0); (/31, /32) = (0, 1) ; and

P[ f iD j ] - P[D]

P[ f iD j ] - P[ f iD] for i, j = l , 2 (i T~ j ) .

Since fl is close to 1, the probability given by the third equilibrium is close to 1.

As a numerical example, consider the following specification of the basic structure:

1 D P[d] -- (1 + d ) D1 = D2 = 2 4.5, and fl 1.1 .

In this case, the present secondary market price is P[9] = 1/10, i.e., 10 cents. If both banks wait, then the price of the next period becomes P[9 x 1.1] = 1/10.9, i.e., about 9 cents. If one bank waits and the other sells its loan exposure, then the new price is P[4.5 x 1.1] = 1/5.95, i.e., about 17 cents. The first and second equilibria above state that this situation happens. Each bank hopes, however, that the other will give up and sell its exposure, and then it will enjoy the increased price, 17 cents. Here each bank has a strong incentive to wait. The mixed strategy equilibrium gives the probability to wait

/31 = /32 = 0 . 8 9 .

We would like to regard this equilibrium as better describing the incentive to wait for both banks than the other two pure strategy equilibria.

The above three equilibria are given by subgame perfect equilibria in Prokop's (1991) dynamic formulation with infinite horizon, i.e., the above equilibria occur in each period in the subgame perfect equilibria.


In Prokop (1991), the pure strategy equilibria are eliminated for the reason of a continuation of the situation. He finds some additional subgame perfect equilibria whose outcomes are almost the same as our third equilibrium. Thus we regard the third equilibrium as better representing the decision making of banks. This is the reason for our use of mixed strategies. Another implication is that our one-period (snap-shot) approach describes the problem without losing much of its dynamic nature.

Here we face the problem whether the banks choose and play such mixed strategies deliberately. The answer is, probably, negative unless the number of banks is very small, e.g., 2 or 3; otherwise we interpret mixed strategies as tendencies (or frequencies) of banks waiting or selling. For a large number of banks, only aggregation or average of such tendencies of banks are important for each bank's decision making and also for our analysis itself, which will be discussed in Sect. 4. We interpret mixed strategy equilibrium in this manner.

Let us return to the general problem. First we describe the set of pure strategy Nash equilibria of the endurance competition game.

Theorem 1: Assume condition (2.1). Let s be a pure strategy n-tuple, and S the set of banks choosing to sell, i.e., S = {i E N : si -=-- 0}. Then s is a Nash equilibrium (in mixed strategies) if and only if

P[f l (D- ~ Di)] >_ P[D] > P[fl(D - i~S

for all j 6 S .

L)i)] i~S \{ j } (2.6)

The proof of Theorem 1 is sketched as follows. Consider the case where sj = 1, i.e., bank j waits. If j changes its strategy to 0, i.e., selling, then its new payoff is DjP[D] instead of the present payoff DjP[fl(D - Y~ iES Di)]. In the case where sj = 0, by changing its strategy to sj = 1, bank j would obtain the new payoff Dj P[~(D - Zi~s\{j} Di)] instead of DjP[D]. In either case, new payoff should not exceed the present payoff for a strategy n-tuple s to be a Nash equilibrium. This is summarized as (2.6).

Consider implications of Theorem 1 under the assumption that

P [d] is a decreasing function . (2.7)


Under this assumption, condition (2.6) is equivalent to

fi(D - E Di) < D <_ fl(D - E Di) for a l l j E S . (2.6*) i~S i~S\{j}

This states that in a pure strategy equilibrium, the remaining debt fl ( D - ~iEs Di) with accrued interest differs from the original total debt D by at most flDj for any j in S. Therefore when flDj is small relative to the total debt D, the resulting total debt fl(D - Y~i~s Di) is almost the same as D, which implies that the resulting secondary market price P[fl(D-Y~i~s Oi)] remains almost the same as the present price P[D]. This will be discussed more generally below.

The existence of a pure strategy equilibrium also follows from (2.6*). Indeed, let S be a minimal subset of N with the property fl(D - E i 6 s Di) ~ D. Then s = (Sl . . . . . sn), defined by: si = 0 if i ~ S and si = 1 otherwise, gives a Nash equilibrium. Actually, there are many pure strategy Nash equilibria. Under the additional condition

> fl E Di for all j 6 N , (2.8) D iCj

- - this condition could be true for small n - - there are exactly n pure strategy Nash equilibria. Each of them is represented as s i = (1 . . . . ,

i 1, 0 , 1 . . . . . 1) for some i, since the set S in Theorem 1 is the single- element set {i}.

We return to the general case, i.e., without condition (2.7). Since D = E i E s Oi is represented as ~'~i~U siDi, condition (2.6) is represented as

P[fl E s i D i ] >__ P[D] >_ P[ f l (Es iD i + Dj)] i~N i~N

for all j with sj = 0 . (2.6**)

This means that in a pure strategy equilibrium, the resulting price is not smaller than the present price, but the difference between them is bounded by P[fl ~-'~icU siDi] - P[I~(Y~.i~N siOi -}- Oj)]. I f Oj is small, then the new price does not differ much from the present price. When the number of banks is large and their loan exposures do not vary a lot among them, this difference between the present and new prices, a fortiori, can be regarded as small. This will be the main result of this paper, but it is not clear yet that a similar result holds for


mixed strategy equilibria, since we have already seen that pure and mixed strategy equilibria differ considerably for two-person games. To show such a possibility, we formulate our result in a more explicit manner.

The above observation for a large number of banks will be formulated as follows - - the same formulation will be used to state a more general result. We introduce a sequence {G ~} = {(N ~, {D~})} of the endurance competition games with

IN~I ~ ~ as 1) ~ oo ; 7 (2.9)

K for s o m e K , D ~ < - - for a l l i r N ~ a n d v > 1 . (2.10)

Denote D ~ = Z i o N v D~, and note that D ~ is bounded by K. Then the above observation is formulated as follows.

Theorem 2: Assume condition (2.1) for each G ~ For any sequence {s ~ } where each s ~ is a pure strategy Nash equilibrium in G ~, we have

Ie[ 0 1 ] - : 0 i~N

(2.11)

3. Behavior of Nash Equilibria for a Large Number of Banks

In pure strategy equilibria, the resulting secondary market price of debt remains almost the same as the present price when the number of banks is large. In this section, we show that the same holds for mixed strategy Nash equilibria. We also show a comparative statics result that in a mixed strategy equilibrium, a bank with a larger loan exposure has a greater tendency to wait than a bank with a smaller exposure. With these results, we discuss the functioning of the secondary market.

To state the first result, we translate the outcomes of mixed strategies into random variables. For a strategy n-tuple p = (p~ . . . . . Pn) we define X = (X1 . . . . . Xn) by

[ Di

Xi = | I 0

if the realization of bank i 's mixed strategy Pi is to wait ;

o therwise.

(3.1)

7 IS[ stands for the cardinality of S.


That is, bank i keeps or sells its exposure Di with probability Pr(Xi = Di) = Pi or Pr(Xi = 0) = 1 - P i , respectively. These random variables are independent because of the basic assumption that the randomiza- tions of P l , . . . , Pn are independent. 8 The resulting secondary market price of debt in the game G is given as P[fl ~-,icU Xi] and is also a random variable. Assertion (2.11) becomes stochastic in the case of mixed strategy equilibria. Now we can state the main result of the paper, which is proved in the Appendix.

Theorem 3." Let {G ~ } = {(N v, {Dy})} be a sequence of endurance games with conditions (2.1), (2.9), and (2.10), and {pV} a sequence of Nash equilibria for G v's. Let {X v } be the sequence of random variables where each X v is defined with pV by (3.1). Then for any E > 0,

ioN v

In fact, this convergence is uniform on the choice of a Nash equilibrium pV for each GV. 9 If for each v there is some player i such that 0 < py < 1, condition (2.1) becomes unnecessary.

This theorem states that the probability of the distance between the prices of the next and present periods to be less than ~ converges to 1 as the number of banks becomes large. When every pV is a pure strategy Nash equilibrium, the assertion of Theorem 3 is equivalent to Theorem 2. Indeed, Theorem 2 implies (3.2), and conversely, when every p~ is a pure strategy equilibrium, (3.2) means that for any E > 0 there is a v0 such that ]P[fi Z i E N v X v] -- p[DV][ < E for all v > v0, which is (2.11).

8 Each random variable X i is a function from S = Sa x . . . x Sn to {0, Di} with Xi (s) = Di if si = 1 and Xi (s) = 0 if si = 0. Also, the space S has the probability measure/z determined by/z({s}) = (I]si=l Pi)(I~si=0(1 - Pi)) for all s 6 S. Since it is not necessary to return to this basic probability space in this paper, we work on the random variables without referring to the probability space.

9 From the mathematical point of view, Theorem 3 is a variant of the (weak) law of large numbers - - for a large number of independent random variables, the mean of these variables is close almost surely to the true average. The proof of Theorem 3 itself, given in the Appendix, is based on Chebyshev's inequality, which is central also in proving the law of large numbers.


Consider implications of Theorem 3. In Nash equilibria, the price of debt remains almost unchanged when the number of banks is large. On the one hand, the total debt may increase by accrued interest, but, on the other hand, it may decrease as a result of the country's buyback of some debt. Theorem 3 implies that these two effects balance and the total outstanding debt does not change much. The proportion of debt bought back is approximately

D I ) - - - -

( D ~ /3 ) / D~ /~ l f i (3.3)

Since the interest factor is close to 1, this proportion is also close to the interest rate fl - 1. The country succeeds in buying back a part of debt tantamount to the interest on the total outstanding debt discounted at the present secondary market price. This also implies that it is sufficient to assume that the country affords to pay a little more than the interests discounted at the secondary market price.

When the distribution of loan exposures is relatively equal, the proportion of banks selling is also approximately (fl - D/f t . Therefore the number of banks involved is decreasing by about this ratio as time passes. For example, if the number of creditor banks is 460 (Venezuela 1985), and if the annual interest rate is 8% formula (3.3) states that the number of creditor banks will decrease to 213 in 10 years. If, however, the loan exposures are unequal, then we could expect a larger number of banks to sell their exposures. This follows from the next theorem which is proved under condition (2.7). It states that in a mixed strategy equilibrium, a bank with a smaller loan exposure has a greater tendency to sell than one with a larger loan exposure.

Theorem 4: Suppose condition (2.7). Let (/31 . . . . . /3n) be a Nash equilibrium. Then it holds that for any i, j with 0 </3i, /3j < 1,

Di > Dj if a n d only if/3i >/3j . (3.4)

A proof of Theorem 4 is given in the Appendix and is sketched as follows: when banks play mixed strategies in equilibrium, each of them is indifferent between selling and keeping its loan exposure, given the others' behavior. This implies that each bank expects the same payoff from waiting as from selling now. Suppose that Di > D j, but their probabilities Pi, Pj a re identical. If bank j decides to wait, then it may expect the future prices to be affected by i 's selling Di with probability


1 - Pi, and if i decides to wait, it may expect the future prices to be affected by j ' s selling Dj with probability 1 - p j . Since Di > Dj, the effect of i 's selling incurs a higher price than j ' s selling, i.e., with the same probability, bank j can expect a higher price by waiting. This cannot happen in equilibrium. In equilibrium, these effects must be the same, which implies 1 - Pi < 1 - p j , i.e., Pi > Pj.

The phenomenon described by Theorem 4 could be observed in the real world; for example, in the case of Bolivia, all American banks with larger debt exposures have kept their loans but some banks with small exposures have sold theirs, cf., Sachs (1989).

Recall that the proportion of debt bought back is almost constant. Hence, since banks with smaller exposures have higher probabilities of selling, the number of banks selling has an increasing tendency when the loan exposures are more unequally distributed. The previous prediction from (3.3) for the number of creditor banks to become 213 after 10 years must be adjusted downward. It remains open to evaluate how large this adjustment should be.

Now, consider the implication of the almost unchanged price of debt for the functioning of the secondary market. For the secondary market to function, there must be (at least potentially) some trade of loan exposures. Each creditor bank who wants to sell its loan exposure is indifferent between selling it to the debtor country and to some other bank. Therefore the equilibrium strategy for a creditor bank is applied to the trade with other banks as well as to the trade with the country, though the market consequences of those trades differ substantially. If the trade among banks is extensive (many banks are potential buyers), then we need to modify our theory to take into account the influence of the presence of many buyers. Since a loan is sold, with the same probability, to the debtor country and to other banks, the equilibrium probability of selling should be increased - - the trade between banks does not affect the situation substantially - - to have the almost unchanged price.

We do not, however, expect an extensive trade of loan exposures among banks. Assume that banks trade their loans extensively. Then the debtor country could hire agents to buy back loans from banks, and succeed in reducing its total debt, leading to a price increase. This behavior of the debtor country, however, would make creditors more cautious about their decisions, and consequently, they would rather wait longer. Thus the assumption of an extensive trade itself tends to obstruct trade among banks. Besides this economic reasoning, another reason for small trade among banks is that potential buyers could be only banks with some interests specific to the debtor country. For these reasons we do not expect an extensive trade among banks. This together


with Theorem 3 implies that each bank has a limited tendency to sell its exposure to other banks as well as to the debtor country. Thus the secondary market is regarded as quite different from a frictionless competitive market. 1~

The prediction that the secondary market prices are almost constant is, however, not confirmed, looking at the data of prices in the 1980s. For some countries the secondary market prices seemed to be constant, for some they looked increasing, and for some others decreasing. In totality the tendency of decreasing prices has been regarded as slightly dominant (cf. Hajivassiliou, 1989). In our game, we regard one period as short, i.e., one or two months; the corresponding decreasing tendency could also be small. Nevertheless, for the long run there remains some small discrepancy between our prediction and the evidence.

Many factors could be thought of as causing the discrepancy between our prediction and the evidence. Disregarding the political dis- turbances, some part of discrepancy between our prediction and the evidence could be explained by considering some unformulated costs of decision making. In reality, the decision making involves transaction costs, specifically, monetary and informational costs, and it requires also time. In our theory, the decision making by banks is instantaneous and costless. Such costs could be regarded as effectively increasing the probability of waiting. Incorporating this fact, our theoretical prediction of the almost constant price needs to be adjusted to a slightly decreasing tendency.l 1

4. The Structure of the Set of Nash Equilibria

To state and prove the limit theorem of Sect. 3, we did not need to know the structure of the set of all Nash equilibria. The limit theorem holds independently of the choice of a Nash equilibrium. From this result one might expect that the number of mixed strategy Nash equilibria could

10 If the secondary market was a frictionless competitive market, we could obtain immediately the result of the constant secondary market price, since the total demand and supply balance only when the price of the next period is the same as the present price. In this case, as was discussed above, the country can buy back loan exposures extensively by, for example, hiring agents, but consequently, this behavior itself would destroy the secondary market.

11 One might argue that the equilibrium probability should be determined after taking this consideration into account, and thus it should be the same as before. This is, however, in the framework of an equilibrium theory of instantaneous and costless decision making of higher order.


be small. The fact, however, is quite different from this expectation: many mixed strategy equilibria may exist in addition to the pure strategy Nash equilibria described by Theorem 1. For a better understanding of the debt overhang and the future continuation of the research on this problem, we investigate the structure of the set of equilibria. Here we will present results without giving proofs. The proofs are available from the authors on request.

First we have the following theorem, which is proved using Kaku- tani's fixed point.

Theorem 5: Suppose conditions (2.7) and (2.8). Let T be a subset of N with ]TI _> 2. Then there exists a Nash equilibrium/3 such that 0 < Pi < 1 for all i c T; and Pi = 1 for all i 6 N \ T.

Theorem 5 states that each choice T with [TI ~ 2 gives a mixed strategy equilibrium. Thus there are at least 2 n - n - 1 mixed strategy equilibria, since condition IT] _> 2 excludes the possibility of T being a singleton or empty. As we mentioned after Theorem 1, there are n pure strategy Nash equilibria under conditions (2.7) and (2.8). Thus the total number of equilibria is at least 2 n - 1.

Condition (2.8) holds for a relatively small n. Thus Theorem 5 gives the structure of mixed strategy Nash equilibria in the case of a relatively small number of banks. The next theorem gives the structure of mixed strategy equilibria for any number of banks under the assumption of identical loan exposures, i.e.,

D1 = D2 . . . . . D . > 0 . (4.1)

Theorem 6: Suppose conditions (2.7) and (4.1). Let T be a subset of N with D > 3 ( D - ITI-1 D) Then there exists a unique Nash equilibrium P = (Pl . . . . . Pn) such that 0 < Pi < 1 for all i 6 T, and Pi = 1 for a l l / ~ N \ T .

Under conditions (2.8), D > f l ( D - ITI-1 D) implies IT[ > 2. In n

this case, Theorem 6 implies that the total number of Nash equilibria becomes exactly 2 n - 1.

Consider the case without (2.8). Let t = ]TI be the smallest integer

with D > f l ( D - ]T]-I D). Theorem 6 gives exactly Y~=t (~) as the n


number of mixed strategy equilibria. Every subset T of N with IT[ = t - 1 satisfies condition (2.6"), which implies that T gives one pure strategy Nash equilibrium. Thus the total number of Nash equilibria is given as Y~=t-1 (~). This number is approximately 2 n for large n and fl < 2. More formally,

Theorem 7." Under the assumption of Theorem 6 and/3 < 2,

I the set of all Nash equilibria I lim = 1 .

n~ec 2 n

From these theorems, we can regard the structure of the set of Nash equilibria essentially as described by Theorem 5. In each equilibrium, the banks are partitioned into the groups of banks waiting surely and the banks having a positive tendency (probability) to sell their exposures. The aggregated tendency faced by every bank in the first group is calculated as the sum of tendencies of the banks of the second group. This aggregated tendency gives each bank in the first group a higher expected value from waiting surely than from selling. A bank in the second group is in a slightly different decision situation. It also com- pares the expected value from waiting with that from selling, but the aggregated tendency it faces is smaller than that a bank in the first group faces, since the bank's tendency to sell itself is not counted. This difference makes banks in the first and the second group behave differently.

According to the above theorems, the partition of such two groups is arbitrary. In each partition, however, the aggregated tendency to sell in the second group makes the expected values from waiting and selling almost equal; Theorem 3 implies that the aggregated tendency to sell makes the new expected price almost the same as the present price. Thus although the number of equilibria is large, they have quite similar structure. In equilibria with different number of banks in each group, banks may have different tendencies to sell, but the aggregated tendencies are almost the same.

5. Conclusions

In this paper, we have considered the problem of international debt overhang. We formulated the problem as a one-period noncooperative game with lender banks as players. The main result is that the secondary market price remains almost unchanged when the number of banks is


large. It has several implications: 1) the total debt also remains almost the same, equivalently, the debtor country buys back about the accrued interest evaluated at the present secondary market price; and 2) the number of banks who sell their exposures is approximated by the interest rate when the loans are equally distributed. The second implication is adjusted upward by the result of Theorem 4 when the distribution is unequal. On the contrary to the almost unique determination of the secondary market price, there are many Nash equilibria. Nevertheless, they are regarded as symmetric and create similar market outcomes. With these results, we reflected on the functioning of the secondary market. The secondary market is quite different from a perfectly competitive market, mainly because of the presence of the debtor country as a possible buyer, and is not expected to have an extensive trade among banks themselves.

Finally, consider some implications of our results for possible resolutions of debt overhang. A direct conclusion is that the pure secondary market transactions will not resolve the debt overhang. One possible resolution, often considered, is to organize collective bargaining over debt reduction between the debtor country and creditor banks. For such collective bargaining, the number of involved creditor banks must be reasonably small. As was discussed in Sect. 3, it would take about 10 years for the number of creditor banks to decline from 460 to about 200 when the annual interest rate is 8% - - for more acurate prediction we need a quantitative evaluation of the difference in equilibrium mixed strategies of banks with different loan exposures. This decrease in the number of banks would only slightly ease the coordination of banks' positions in the bargaining process. Thus it would still be difficult to expect collective bargaining to take place voluntarily. Therefore we need some institutional arrangement of a central authority to resolve the debt overhang problem. For the consideration of an institutional arrangement, we would need to know more about the functioning of the secondary market, especially, its dynamic aspects.

Appendix

Proof of Theorem 3

We show the claim of Theorem 3 under the assumption that every pV has at least one player who plays a completely mixed strategy. If this is done, then this together with Theorem 2 implies Theorem 3. Indeed, suppose that the claim is proved under this assumption. If only a finite number of pV's in {pU} have mixed strategies, Theorem 2 is applied.


If only a finite number of pV's in {pV} are pure strategy equilibria, the above claim is applicable. In the case where the sequence {pV} is divided into two subsequences with pure strategies and mixed strategies, respectively, then we apply Theorem 2 and the above claim to these subsequences, respectively. Then we have the general assertion of the theorem. Thus it suffices to consider the case where every p~ has at least one player playing a completely mixed strategy. Note that (2.1) will not be used in the following proof.

Let us prepare several notions and some lemmas. Since Pr(X~ = D~) = p~ and Pr(X~ = 0) = 1 - p~, the mean and variance of X~ are given as

E ( X y ) = ~ Pi Di , and (A.1)

v(x;) e((xy -= -- Pi D i ) ) = (DV)2piV(1 - P~) < �9 IZVN

Denote ZioN v Xy by S v. Then

E p 1) E ( S v ) = Pi Di , and i E N v

v(sv) = } 2 v(x ) < 2 - K 2

i E N v I 1 ' 1 IN~I

(A.2)

L e m m a 3.1" lim Pr(IS ~ - E(S~)I ~ E) = 1 for any e > 0. 1)--->oo

Proof." Applying Chebyshev's inequality (Feller, 1957, p. 219) to S ~, we have

/ Pr {,IS '~ - E(S ' ) ]

It follows from (A.2) that

Pr ( ]S v - E(SV)[ \

> t < ~ for any t > 0 .

> t < ~ for a n y t > 0 .


Putting t = [NVJ 1/4, we have

( K) 1 Pr IS ~ - E(S~)I > iN~-[1/4 _ IV~-~-----~

K Choose a v0 so that ~ < E for all v > v0. Then we have

Pr(IS ~ - E(S~)I > E) _ - - for all v > v0 .

Therefore we have

P r ( I S ~ - E ( s ~ ) I < ~) > 1 _ _ - + 1 as v ---> ~ .

Q.E.D.

Lemma 3.2: For any E > 0, l im Pr ( [ P [ i Sv] - P[iE(SV)][ < E) = 1. 1) ---~ O ~

Proof: Since the price funct ion P is continuous on [0, + o e ) , P is uniformly continuous on its relevant domain [0, ELK]. Therefore there is a 8 > 0 such that IS ~ - E(S~)I < 8 = , I P [ i S ~] - P[flE(S")]I < ~. Thus it fol lows f rom L e m m a 3.1 that

P r ( I P [ i S ~] - P[flE(S")]I < E) >_

> Pr (IS ~ - E ( S ' ) I _ 8) ~ 1 as v --+ oe . Q.E.D.

Lemma 3.3: 1) l irn lE (P t iSV] ) - PtflE(SV)]I = 0 ;

2) lim I E ( P [ t S q ) - P [ D q I = 0 ;

3) l i m J e [ I E ( S ~ ) ] - e [ o q I ---- 0 .

Proof: 1) Let E be an arbitrary positive number. Since P is a uni formly continuous funct ion on the relevant domain [0, i K ] , there is a 8 > 0

E F rom such that IS ~ - E(S~)I < 8 =~ IP[pS v] - e [ I E ( S V ) ] l < ~.

A Game Theoretical Approach to the Intemational Debt Overhang 21

Lemma 3.1, there is a vl such that for all v > u1,

E Pr(IS ~ - E ( S ~ ) I > 8 ) < 2 M ' w h e r e M = max P[d].

O<d<flK

E i f IS~'-E(S~) I < 3 The difference IP[flSv]-P[flE(S~)] I is less than ~ and is less than M if IS ~ - E(S~)I > 3. Therefore we have

E(P[fS')]- = E ( P [ f l S ~] - P[fiE(S )]) <_

_< ( 2 ) P r ( I S " - E(S ' ) I _%< 8 ) + MPr(IS " - E(S ' ) I > (~) <

< - + M x - - = e f o r a l l v > v l . - 2 2M

2) Recall that in each equilibrium p~, at least one player i plays a completely mixed strategy p~ (0 < p~ < 1). We rewrite E(P[fiS~']) as

E(Pt Sm = (VI p;) I-I( 1 - E D;] = RcN jeR j~R jeR

(A.3) jr

+ (1 -- Pi ~ (17 P ; ) ( 1-I (1 - p;) )P[ f Z D;] "RGN\{i} jaR j•R jER

j#i

Notice that there is the one-one correspondence between the terms of the first and second brackets, and that each term of the first bracket differs from the corresponding term of the second in that the first has the additional/~D~ in P[.]. Since 0 < p~ < 1, the player i is indifferent between his choices of waiting and selling. Since the expected payoff from waiting is given as the first bracket and the expected payoff from selling is P[DV], the first bracket of (A.3) is equal to P[DV]. Since maxj~u~ Dj v < K _ ~ --+ 0 as v --+ e~, the difference between the first and second brackets of (A.3) converges to 0 as v --+ co, because of uniform continuity of P on the relevant domain. As we mentioned above, the first bracket is always the same as P[DV]. This means limv~oo IE ( P[fiS~]) - P[ D~']I = O.


3) It follows from 1) and 2) that for any e > 0, there is a v0 such that for all v > vo,

E

E E ( P [ f l S V ] ) - p[Dv][ < ~ .

and

Therefore we have

P[ f lE(SV)] - p [ D v] <

<_ P [ f l E ( S V ) ] - E(P[ f lSV]) + E ( e [ f l S V ] ) - p [ D v] <_

E E _ < z + z = E for a l l v > _ v o . Q.E.D.

Z Z

Finally we prove the assertion of the theorem. From 3), for any E > 0, there is v0 such that for all v > vo, IP[~E(S~)] - P[D~]I < ~.

Thus, using Lemma 3.2, we have,

Pr(IP[r v] - P[DV]I _< E) _>

> Pr([P[flS ~] - P[flE(S~)][ + [P[flE(SV)] - p[Dv][ < ~) >

E > Pr( lP[ f lS v] -- P t f l E ( S ' ) ] I < 7) ---> 1 as v ---> o e .

Q.E.D.

Proo f o f Theorem 4

Since 0 < /3i , /~j < 1, banks i and j are indifferent between waiting and selling. Therefore the expected payoffs from waiting and selling are the same. This is expressed as

( 1-I RC_N\{k} m e N mER

mf~RU{k}

= P [ D ] for k = i, j .

m c R (A.4)


The left hand side for k = i is written as

m~i,j m~R m~R

+

+(1-/~j)[ Z ( H /}m)(H(1-/~m)) • t-RCNX{i,j} m~NXR m~R

m~i,j

X P[f l (O-- Z Om)]] " meRU{j}

Each term in the second bracket corresponds to one in the first bracket with the same R, and each is bigger than the corresponding one by (2.7). Therefore the whole sum in the second bracket is bigger than that in the first one. When Di -= D j, the left hand side of (A.4) for bank j is obtained from the above formula by replacing/3j by/~i- In this case, if/~i ~ /3j, then the value of the obtained formula for bank i is also different from the value of the formula for j . But these two must be the same as P[D]. Hence O i = Oj implies/3 i = /3j. NOW it is sufficient to prove that Di > Dj implies/3i >/3j . When Di > D j, the left hand side of (A.4) for bank j is obtained from the above formula by replacing/3j by /3 i and Dj by Di in P[D - Y~m~eu{j} Dm] in the second bracket. In this case, the value of the new second bracket is greater than that of the original one. Hence to keep the left hand side of (A.4) equal to P[D] for i, the probability coefficient/3 i for the first bracket must be higher than/3j. Q.E.D.

References

Bouchet, M.H. (1987): The Political Economy of International Debt. New York: Quorum Books.

Bulow, J., and Rogoff, K. (1988): "The Buyback Boondoggle." Brookings Papers on Economic Activity 2: 675-698.

- - (1989): "A Constant Recontracting Model of Sovereign Debt." Journal of Political Economy 97: 155-178.

Feller, W. (1957): An Introduction to Probability Theory and Its Applications, Vol. I. New York: Wiley. [2nd ed.]

Fernandez, R., and Kaaret, D. (1992): "Bank Heterogeneity, Reputation, and Debt Renegotiation." International Economic Review 33: 61-78.

Fernandez, R., and Rosenthal, R. (1990): "Strategic Models of Sovereign-Debt Renegotiations." Review of Economic Studies 57:331-349.

24 Kaneko and Prokop: International Debt Overhang

Hajivassiliou, V. (1989): "Do the Secondary Markets Believe in Life after Debt?" Cowles Foundation Discussion Paper No. 911, Yale University, New Haven.

Huizinga, H. (1989): "How Does the Debt Crisis Affect Commercial Banks?" Working Paper, International Economics Department, The World Bank, Washington, D.C.

Prokop, J. (1991): "Dynamics of International Debt Overhang with Two Lender Banks." Mimeo.

- - (1992): "Duration of Debt Overhang with Two Lender Banks." Eco- nomics Letters 38: 473-478.

Sachs, J. (1989): "Efficient Debt Reduction." Working Paper, International Economics Department, The World Bank, Washington, D.C.

Sachs, J., and Huizinga, H. (1987): "U.S. Commercial Banks and the Devel- oping-Country Debt Crisis." Brookings Papers on Economic Activity 2: 555-601.

Versluysen, E. L. (1989): "A Review of Alternative Debt Strategies." Working Paper, International Economics Department, The World Bank, Washing- ton, D.C.

Addresses of authors: Professor Mamoru Kaneko, Department of Eco- nomics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA, and Institute of Socio-Economic Planning, University of Tsukuba, Ibaraki 305, Japan; and Professor Jacek Prokop, Department of Management and Strategy, J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208-2001, USA.

a game theoretical approach to the international debt overhang

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