Bank Runs and Interbank Markets: A Heuristic Example
Masuyuki Nishijima

Abstract
This paper offers an example of the game in which banks make lending and borrowing
decisions while depositors choose when to withdraw, to better understand how the interbank
market rescues a bank at risk of bank runs. Our example is based on Postlewaite & Vives(1987) and adds banks and the interbank market. We show that there is a Bayesian perfectequilibrium in which the interbank market collapses and a bank run occurs. There is also an
equilibrium where banks refuse to lend and a bank run happens when a situation of the bankrun (strategic withdrawal) is associated with a situation of default of the rescue loan. Therefore,
the interbank market alone does not necessarily save a bank at risk of bank runs.
Key Words: Bank runs, Interbank Markets, Coordination Failure, Bayesian Perfect Equilibrium,
Mismatch of Maturity between Lending and Borrowing

Ⅰ. Introduction
A bank run has been a central issue of financial crisis. It is not a long ago that economists
began to develop theoretical models of bank runs. Since Diamond & Dybvig (1983), a bank run
phenomenon has been understood as “sun-spot equilibrium” induced by coordination failure
among depositors. (See Freixas, X. & J-C. Rochet (2008, Ch.7) and Sakai & Maeda (2003), forexample.) At the same time, various policies that can prevent a bank run from occurring

have been discussed from reserve and equity capital ratio requirements to suspension of
convertibility.
A role of a central bank as the lender of last resort (a rescue loan to a bank at risk of bankruns) also has been debated. Goodfriend & King (1988), comparing private bank loans with acentral bankʼs discount window policy, conclude that there is no compelling reason why thecentral bank is superior to private banks in terms of monitoring and assessing a borrowing
bank, by analogy with a comparison between private and public firms in goods markets. To
my knowledge, there has been no theoretical model where private banks in interbank markets
can successfully rescue an illiquid but solvent bank. On the other hand, some economists,
suspecting a chain reaction of bank runs through financial markets, developed their models to
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support the central bank's role of the lender of last resort. Most of them (for example, Allen,
F. & D. Gale (2000), Freixas, X., Parigi, B. M. & J-C. Rochet (2000), Freixas, X., Martin, A. & D.
Skeie (2011)) are based on Diamond & Dybvig (1983), while Rochet & Vives (2004) is based on

Postlewaite & Vives (1987), with global game structure under incomplete information (Morris,
S. & H. S. Shin (1998)).

In these models, banksʼ lending and borrowing actions after risk of a bank run is perceived
are not analyzed because all banksʼ actions are treated in ex ante optimal deposit contract (as amaximization problem of depositorʼs expected utility subject to a bankʼs zero profit condition). This

structure makes it difficult to explicitly analyze how banks behave in the interbank market
when a bank faces a risk of a bank run (strategic withdrawals). As a result, these models do not
explicitly explain why the interbank market would not work to rescue a bank at risk of bankruns (in other words, what kind of market failure happens in the interbank market), but rather
focus on working of policy instruments such as reserve and equity capital ratio requirements
and the central bankʼs lending to the bank (discount window policy). Furthermore, these models
are complicated and not explicitly stated as a game form. This makes it hard for the reader
to understand these models.
In this paper, I attempt to offer a heuristic example to better understand how privatebanks would (fail to) provide liquidity to a troubled bank, at the same time taking into account
how its depositors withdraw strategically.
Participants in the interbank market are limited to a certain group of banks and otherfinancial institutions. It is easy and costless for them to participate in and retreat from themarket since what is traded is a short-term loan (or its rolling over). Thanks to new technology
of information delivery and processing, some of them are likely to have almost the same
information so that their actions are coordinated within their group, as in Morris, S. & H. S.
Shin (1998). This makes it appropriate to analyze the interbank market as a game theoreticalsituation rather than a competitive market, even if there appears many market participants.
In our example, we add banks as players to the game of strategic withdrawal of Postlewaite& Vives (1987). There banks receive signals and choose their loans demanded and offered, thenagents (depositors) at a troubled bank receive signals of their own types and choose when towithdraw. The novelty of our example is our formulation of the troubled bankʼs objective andthe interbank market. Following a separation of ownership from management, we assume
that the bank tries to avoid its bank run (whenever such a risk exists), i,e, its manager tries tominimize his expected penalty of a bank run. By doing so, we can avoid the deadlock in theexisting literature of being unable to explicitly treat banksʼ behavior in the interbank market
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Bank Runs and Interbank Markets: A Heuristic Example

interbank market, we simply assume that the interbank market determines the loan volume
at the short-side when excess demand (supply) happens at the lowest (highest) interest rate
and assume that the interest rate adjusts quickly in response to excess demand. Though theinterest rate may be indeterminate when demand and supply are balanced, it does not affectour results since we can assign either the highest or lowest rate in this case.
Our example shows two types of equilibrium. One is an equilibrium where the interbank
market collapses with no trade and a bank run occurs. The other is an equilibrium wherea bank run happens when a rescue loan is expected to default (due to mismatch of maturity
between lending and borrowing). What happens in the first type of equilibrium is a typical caseof coordination failure among banks. Both lenders and borrowers have pessimistic expectations
and shrink to avoid their transactions. The second type of equilibrium suggests that theultimate reason for default of the rescue loan (and thus the bank run) results from mismatch ofmaturity between lending and borrowing, not from insolvency of a bank (or a financial institution)or sudden devaluation of the deposits or liquid assets held at banks due to exogenous shocks.
This implies that we cannot avoid a bank run under the modern partial reserve banking
system.
This paper is organized as follows. In the next section we formulate an example of thegame among depositors and banks. In section 3, equilibrium of the game is analyzed and its
economic logic is discussed. We conclude by making some remarks on robustness and possible
extensions of our example.

Ⅱ. The Example
Our example is a modified version of the example in Postlewaite & Vives (1987). We add n+1
banks (managers of these banks) as players.
(1) Technology and Environments
There are four periods, 0, 1, 2, and 3. The production process exhibits constant returns toscale. For each unit planted at period 0, there will be α units available after one period, β units
available after two periods, and γ units available after three periods. For each unit left in period
1 of some amount planted in period 0 there will be β/α units in period 2 and γ/α units in period
3. For each unit left in period 2 of some amount planted in period 0 there will be γ/β units one
period later. We assume thatα < β < γ, 1/2 < α < 1, and γ > 1.

（1）
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If the production process is interrupted in period 1, one gets α, which is less than the initialinvestment 1. If the production process is not interrupted, one gets γ > 1. If it is interrupted
in period 2, one gets β, which is larger than α but less than γ.
In our example, we explicitly treat strategic behavior of withdrawal by agents at one of
the banks, denoted by bank 0, which is in a situation of a bank run.
At bank 0, there are two agents, each of whom has one unit of endowment and will live
for two or three periods. At the beginning of period 1, agent i (i = a or b) receives a signal si∈ Si, Si = { s2i , s3i }, which indicates his or her life span (type). We call an agent who lives fortwo periods short-lived, an agent for three periods long-lived. There is a joint distribution P
τ

on Sa×Sb. Each agent has his utility function: U(�i=1 xi ), where τ is the number of periods theagent will live and xi the consumption in period i.
(2) Banking Contract

Each agent deposits an amount in bank 0. During periods 1 and 2, an agent can withdraw his
deposit with no interest (penalty for early withdrawal). If the agent withdraws his deposit inperiod 3, he receives a share of what is left in the bank proportional to his deposit. If, during
any period, demand for withdrawal exceed assets, all assets will be distributed proportional towithdrawal demands.
(3) Banks and Interbank Market
There are n + 1 banks in the interbank market (n ≥ 2). They are only participants in it. Without
loss of generality, we assume that bank 0 is in a situation of bank runs without borrowing in
the interbank market. Bank i receives signal s i at the beginning of period 1 (i = 0, 1, 2, …, n).
The signals that banks receive indicate what pair of agent type at bank 0 happens. We assume
that the signals and the type pairs are perfectly correlated.1）(This means that all participants in
the interbank market know a situation of the bank at risk of bank runs.) s i ∈ S ≡{(s a2, s b2), (s a2, s b3),

(s a3, s b2), (s a3, s b3)} for i = 0, 1, 2, …, n. After receiving signals, banks decide how much to borrowor lend in period 1.

Bank 0 has initially no liquid asset. Its strategy is how much per agent to borrow from theinterbank market for each signal, denoted by d(s0)∈D≡[0, ∞). Other banks are assumed tohave liquid assets and be immune to any bank run even if they lend whatever amounts of their
1） In other words, banks know types of both agents at bank 0 at the beginning of period 1.
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liquid assets to bank 0. (We strategically assume away a chain reaction of bank runs through theinterbank markets and instead focus on whether other banks (or financial institutions) participate intransactions in the interbank market.) Bank i (i = 1, 2, …, n) decides how much to lend to bank

0 for each signal, denoted by li (si) ∈ Li≡[0, mi] where mi is the amount of liquid asset bank iholds at the beginning of period 1. All banks make their decisions on borrowing and lending
simultaneously.
The loan in the interbank market is borrowed in period 1 and repaid with its interest
R－1 in period 3. The loan in question is of a rescue loan so that there is no opportunity forbank 0 to access to any financial market in period 2 and that the lenders will have to wait until
period 3 when bank 0 will get a higher yield of its (long-run) production. The interbank market

is short-sided constrained (though the interest rate responds to excess demand), “anonymous”(except that bank 0 is at risk of bank runs and the only borrower) in the sense that the aggregate

supply and demand, not identity of participants, affect the interest rate) and “centralized” inthe sense that all demands and supplies are matched at one place. (We assume away a favorite
relationship between bank 0 and some other banks.) However, the number of banks potentiallyparticipating in the interbank market is limited so that the interaction among the banks
matters. Note that since we assume constant-returns-to-scale technology, demand for loans
may be infinity, which is eventually limited by a fixed amount of liquid assets possessed bybanks at the maximum price. When excess supply happens at the minimum price, the demandside determines the amount of loan traded (and the supply side is rationed in proportion to theinitial offers). The interest rate is adjusted so as to equate demand with supply (2d =�ni=1 li for 1<R< γ/α, 2d ≤�ni=1 li for R=1 and 2d ≥�ni=1 li for R = γ/α) in the interbank market. We assume

that 1 ≤ R ≤ max{(β/α), (γ/α), (γ/β)} = (γ/α) (maximum willingness to pay for the loan), though wewill not explicitly analyze the equilibrium interest rate in some cases.
The banking contract, implicitly made in a competitive deposit market, is supposed toleave zero profit to a manager of a bank in the literature. This makes it difficult to formulatea bankʼs objective function in a context of strategic withdrawal. We assume that a manager
of bank 0 tries to avoid a bank run, i.e., to maximize expected payoff of (－1)×P0+(0)×(1－P0),
where P0 is the probability of bank 0ʼs bank run, which depends on the pair of strategy ofagents at bank 0 specified below. As for other banks, however, we assume that bank iʼs payoff(or its managerʼs payoff) is its expected yield of lending in the interbank market. For i = 1, 2, …,

n, (R－1) li if both agents repay their loans in period 3 (both live until period 3) or at least one ofagents (who is long-lived) repays the loans of the bank, (1/2){(R－1) li + (－li)} if one of the agents
fails to repay (he lives until period 2), and －li if both fail to repay the loans (they are short-lived).

This asymmetric treatment between bank 0 and other banks may be justified by interpretinga managerʼs preference as a lexicographic one: he first tries to avoid a bank run whenever sucha risk exists, but otherwise tries to maximize the bank's expected yield of loan.
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An agent at bank 0 decides when to withdraw his deposits after receiving his signal (type) andobserving how much banks have borrowed and lent. A strategy for an agent is a function
σ i, which indicates when agent i will withdraw his money from bank 0 for each possible
signal he receives and each possible combination of lending and borrowing amounts of banks.
nσ i：S i×D×� i=1 Li → A, where A = {w1, w2, w3} and wi represents withdrawal in period i. Agenti ʼs belief of agent jʼs type is denoted by μ i (s kj ︱sk'i , d, l), where l = (l1, l2, …, ln).

Table l shows payoffs to agents at bank 0 when there is neither borrowing nor lending(d(s0) = 0, li (si ) = 0 for i = 1, 2, …, n).2） Note that a strategy of withdrawal in period 3 is notavailable to s 2i type (short-lived) player.

We make the following assumptions on parameters in Table 1 to assure that bank 0 is ina situation of bank runs without borrowing and lending.
Table 1: Payoff Matrix of Agents at Bank 0 without borrowing and lending
Agent a/Agent b
w1
w2
w3

			

w1
α, α
(2α－1)(β/α), 1(2α－1)(γ/α), 1w2
1, (2α－1)(β/α)β, β(2β－1)(γ/β), 1

β < 1, (2α－1)(β/α) < α < (2α－1)(γ/α), 1 < (2β－1)(γ/β).

w3
1, (2α－1)(γ/α)
1, (2β － 1)( γ /β)
γ, γ

（2）

Under the above conditions, both short-lived agents have a dominant strategy of withdrawal
in period 1, while both long-lived agents have a dominant strategy of withdrawal in period 3 inthe payoff matrix of Table l. This implies that a bank run occurs (where both agents withdraw
their deposits in period 1) whenever both agents are short-lived, as demonstrated in Postlewaite& Vives (1987).
When bank 0 borrows d (per depositor) from the interbank market, payoffs to agents atbank 0 must be classified into the four possible cases of types. Though claims and debts arein the agentʼs hand and carried over to period 3 when bank 0 is resolved, short-lived agents
can never pay their debts in period 3 because they will no longer live in period 3. The debt
the short-lived agent carries over to period 3 will be default. This makes payoffs different,
depending on what pair of types occurs.

2） This table is the same as Table 1 in Postlewaite & Vives (1987).
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Let A(x, y, d)≡(2x－1+2d)(y/x), B(x, y, d)≡(2x－1+2d)(y/x)－2Rd and C(x, d)≡x+(1－R)d.
We assume that d < 1/2. Otherwise a bank will be able to meet unilateral withdrawal withoutliquidating part of its long run investment. Each cell of Table 2 is divided into the four possible
states of type, in each of which payoffs to agents are shown. The payoffs in case of bothplayers being short-lived is in the northwest, those for short-lived agent a and long-lived agent
b in the northeast, those for long-lived agent a and short-lived agent b in the southwest, andthose in case of both being long-lived in the southeast. The dash (-) indicates that there is no
possible strategy pair since at least one agent is short-lived in the cell. When d=0, Table 2
coincides with Table 1.
Table 2: Payoff Matrix of Agents at Bank 0 with borrowing and lending
a/b
w1
w2
w3

d + α, d + αC(α, d ), d + α
A(α, β, d ), 1
B(α, β, d ), 1
B(α, γ, d ), 1

w1

d + α, C(α, d )C(α, d ), C(α, d )A(α, β, d ), 1
B(α, β, d ), 1
B(α, γ, d ), 1

1, A(α, β, d )1, A(α, β, d )d + β, d + βC(β, d ), d + β
B(β, γ, d ), 1

w2

1, B(α, β, d )1, B(α, β, d )d + β, C(β, d )C(β, d ), C(β, d )B(β, γ, d ), 1-

w3

1, B(α, γ, d )1, B(α, γ, d )1, B(β, γ, d )1, B(β, γ, d )C(γ, d ), C(γ, d )(5) A Summary of the Game
The players are agent a and agent b at bank 0, managers of the banks numbered from zero ton. The timing of the game is as follows. Each player receives his signal (on his type for agentsand on a pair of type for banks) at the beginning of period 1. Then banks borrows or lends inthe interbank market. After observing these transactions, agents at bank 0 decides when towithdraw.
Agent i ʼs strategy is when to withdraw, given his type and transactions in the interbanknmarket: σ i : S i × B × � i=1 Li → A, where A = {w1, w2, w3} and wi represents withdrawal inperiod i. Agent iʼs belief of agent jʼs type, μi (skj︱sk'i , d, l) is updated if possible, where k, k' = 2,

3 and l = (l1, l2, …, ln). A manager of bank 0 has a strategy of how much to borrow, given his

received signal on a pair of type: d(s0): S → D ≡ [0,∞). A manager of bank i (i = 1, 2, …, n) has astrategy of how much to lend, given her received signal: l (s i ): S → Li ≡ [0, mi] where mi is theamount of liquid asset bank i holds at the beginning of period 1. Banksʼ beliefs of a pair of typeare omitted because signals are perfectly correlated with actual pairs of type.

Payoff to agent i (i = a, b): Let πi (σ i (s i, d, l), σ j (s j, d, l): ski , sk'j ) be agent iʼs payoff function when

he is type k and the other agent is type k' where k, k' = 2, 3, which appears in Table 2. For (ski,d, l ), his expected payoff is		�πi (σ i (s i, d, l), σ j (s j, d, l): ski, sk'j ) μ i (sk'j ︱ski, d, l).

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Payoff to bank 0: Let P0 (σ i (s i, d, l), σ j (s j, d, l)) be the probability that a bank run occurs atbank 0. For (s i, s j), bank 0's (its managerʼs) expected payoff is		(－1)×P0 (σ i (s i, d, l), σ j (s j, d, l)) + (0)×(1－P0 (σ i (s i, d, l), σ j (s j, d, l))).

Payoff to bank i (i = 1, 2, …, n): For (s a, s b), bank iʼs expected payoff is 3）－li(1/2){(R－1) li + (－li )}(1/2){(R－1) li + (－li )}(R－1) liif (s a, s b) = (s a2, s b2) and σ i (s i, d, l) = w1 or w2 for i = a, b,

if (s a, s b) = (s a2, s b3) and σ b (sb, d, l) = w2 ,if (s a, s b) = (s a3, s b2) and σ a (sa, d, l) = w2 ,otherwise.

The equilibrium concept we employ is Bayesian Perfect Equilibrium.

Ⅲ. Equilibrium and Its Economic Logic
In this section, we show that there is an equilibrium in which a bank run happens with positive
probability even if all the other banks in the interbank market have no risk of their own bankruns. In other words, our example demonstrates that the interbank market alone does notnecessarily work to avoid a bank run.
n

Let M ≡ (1/2)�ni=1 mi , m ≡ (1/2) max{m1, m2, …, mn} and t ≡ min{d, (1/2)� i=1 li}, as the

maximum amount of loan (per depositor) made by all banks, the maximum amount of loan(per depositor) made by an individual bank. and the realized amount of loan (per depositor)

determined by the short side between demand and supply, respectively. We define
t1 ≡ 1－β, t2 ≡ {α2－(2α－1)β}/(2β－α),t3 ≡ min{(γ－1)/(R－1), (β－(2β－1)γ)/(γ－Rβ)} for R≠1 if R > γ/β.
Otherwise t3 ≡ (γ－1)/(R－1).

Proposition 1
Suppose that a joint probability of (s a2, s b2) is positive. In addition to Assumptions (1) and (2),we make the following assumptions.

3） For ease of understanding, we separately describe payoffs depending on the signals received since weassume that signals and the actual pair of type are perfectly correlated.
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Bank Runs and Interbank Markets: A Heuristic Example(a) For all t ≤ m and for all R (1 ≤ R ≤ γ/α),
		(i) t + α > A (α, β, t), (ii) 1 > t + β, (iii) C (α, t) < B (α, γ, t),		(iv) max {C (β, t), 1} < B (β, γ, t), (v) 1 < C (γ, t).

(b) (vi) t1 < t2 < t3 , (vii) t2 < M < t3 .
Then there is an equilibrium in which a bank run happens with the joint probability of (s a2, s b2),and equilibrium strategies are as follows.
For i = a, b,

σ i (s2i, d, l ) = w1, with any belief, σ i (s3i, d, l) = w3, with any belief, for (d, l) such that t < t1,

σ i (s2i, d, l ) = w1, with μi (sk'j ︱ski, d, l) = 0 (k'≠k and j≠i) for (d, l) such that t1 ≤ t ≤ t2,			σ i (s3i, d, l) = w3, with any belief, for (d, l) such that t1 ≤ t ≤ t2,

σ i (s2i, d, l ) = w2, with any belief, σ i (s3i, d, l) = w3, with any belief, for (d, l) such that t2 < t ≤ M.
			d (s0) = 0 for all s0, li (s i) = 0 for all s i, (i = 1, 2, …, n).

Proof: Note that the assumptions in Proposition 1 are satisfied for some combinations ofparameters if we take γ large enough and β close to one, and mi (i = 1, 2, …, n) appropriately.
For agent i (i = a, b), they face the payoff matrix of Table 2 where d is replaced witht. When t < t1, the above conditions (i) to (v) hold. So a short-lived agent has his dominant

strategy of w1, while a long-lived agent has her dominant strategy of w3. In this case, we canallow any belief of type. When t1 < t < t2, inequality of condition (ii) is reversed. Since any
arbitrarily specified belief of type is allowed in Bayesian perfect equilibrium, we specify the

belief that the other agent is of the same type. Under this belief, a short-lived agent thinks thathe plays the two-by-two northwest payoff matrix, in which there are two equilibria (w1, w1) and(w2, w2). We choose the former as the equilibrium path. A long-lived agent has his dominant

strategy of w3 (with any belief) when t1 < t < t2. (We employ weak dominance so that we can include
the cases of equality in the second range of t without changing weakly dominant strategies.) Whent2 < t ≤ M < t3, inequalities of conditions (ii) and (i) are reversed. A short-lived agent has his

dominant strategy of w2, while a long-lived agent has her dominant strategy of w3.

Given li (s i) = 0 for all s i (i = 1, 2, …, n), we have t = 0 whatever amount of loan bank 0

demands. Thus a bank run happens only when the type pair is (s a2, s b2). Bank 0 cannot change
the probability of bank runs so that d (s0) = 0 for all s0 is its best response. Given d (s0) = 0 for

all s0, any other bank cannot change t = 0 whatever amount of loan it offers. Thus li (s i) = 0
for all s i (i = 1, 2, …, n) are their best responses. Q.E.D.

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What happens in equilibrium of Proposition 1 is a typical case of coordination failure
among the participants in the interbank market, even if bank 0 faces risk of a bank run. Bank
0 expects other banks to lend no loan so that it demands no loan. Other banks expect bank 0 to
demand no loan so that they offer no loan. These expectations are realized in equilibrium. As
a result, there occurs no transaction in the interbank market, which eventually collapses. This
happens not only when a bank run occurs (both agents are short-lived) but also in all the othercases. Whether this kind of coordination failure takes place is independent of whether bank0 has risk of bank runs. Rather this kind of coordination failure always happens whenever
the number of market participants is limited and their expectations are likely to converge (asperfectly correlated signals in our example).

In equilibrium of Proposition 1, the interbank market collapses so that the interest rate is
not determined.
The reader might wonder that the equilibrium in Proposition 1 may be pathological in
the sense that bank 0 demands no loan in the interbank market. The next proposition shows
another equilibrium in which bank 0 demands a positive amount of loan.
Proposition 2
Suppose that all the assumptions in Proposition 1 are satisfied. Suppose further that		(viii) t2 < M － m < t3 .

Then there is an equilibrium in which a bank run happens with the joint probability of (s a2,s ), and equilibrium strategies are the same as those in Proposition 1 for agents at bank 0 andb2

those of banks are		d (s0) = d > M for all s0.

For i = 1, 2, …, n, li (s i) = 0 if s i = (s a2, s b2), li (s i) = mi otherwise.
Proof: Note that the additional condition (viii) is satisfied if we take mi and n appropriately.
The proof of equilibrium strategies for agents at bank 0 is the same as that in Proposition
1, because what is relevant to agentsʼ strategies is the realized amount of loan in the interbank
market.
Given li (s i) (i = 1, 2, …, n), whatever demand for the loan is bank 0ʼs best response fors = (s a2, s b2) since t = 0. In this case, a bank run happens. In other cases of s0, whatever amount0bank 0 chooses, no bank run occurs because at least one agent is a long-lived type (having her
dominant strategy of w3) so that there is no case of both agents choosing withdrawal in period
110Bank Runs and Interbank Markets: A Heuristic Example

1. Thus d > M is bank 0ʼs best response for all s0, taking into account equilibrium behavior ofagents at bank 0.
Given d (s0) = d > M for all s0, there is always excess demand whatever total supply of loan
is provided. This means that R = γ/α. For s i = (s a2, s b2), full default of loan happens in period 3 sothat bank i will not lend money to bank 0, i.e., li (s i)=0. In this case, even if bank i deviates fromthis strategy (to a positive amount of loan), agents at bank 0 will not change their strategies dueto the conditions in (a) of Proposition 1. This deviation lowers bank iʼs payoff since its payoff
function is －li in this case. For s i = (s a2, s b3) or s i = (s a3, s b2), the short-lived agent fails to pay his

debt of the loan but the long-lived agent repays all debt the bank owes so that bank i lends its

maximum amount of liquid asset, mi. Even if bank i deviates (reduces a mount of loan) from thisstrategy, agents at bank 0 will keep their strategies intact due to condition (viii). This deviation
does not increase bank iʼs payoff. For s i = (s a3, s b3), all the loans are surly paid in period 3. It is

optimal for bank i to lend mi to bank 0. Any deviation from this strategy to a smaller amountof loan decreases bank iʼs payoff since its payoff function is (R－1) li . Q.E.D.

Under the assumptions of Proposition 2, when a bank run occurs without borrowing, both
agents at bank 0 are short-lived. So other banks expect full default of their loans in period 3
when the agents will no longer exist. This makes other banks lend no money to bank 0. In
other cases when a bank run does not occur without borrowing, at least one agent is long-lived
so that no default will take place, because the long-lived agent has her dominant strategy of w3
and owes the loans (when the bank is resolved in period 3). This implies that other banks expect

high enough expected yields of their loans since the production process without interruption
generates a high yield. They lend their liquid assets as much as possible. Banks do not offerloans to a bank at risk of a bank run because a situation where a bank run will happen isassociated with a situation where the loans will default.
This linkage prima facie appears to correspond to the free market schoolʼs view that asolvent bank cannot be illiquid in the well-functioned interbank market. In this view, however,
it is supposed that (possibility of) insolvency of a bank induces its depositors to make strategic
withdrawals. In our example, it is supposed that even if a lending bank unilaterally makes
a loan to a bank in a situation where strategic withdrawals will take place, the lending bankexpects a default of its loan due to the linkage to a situation where agents cannot repay theloan they inherit. The causality directs from strategic withdrawals to agentsʼ default of the
rescue loan in our example, while in the free market schoolʼs view the causality directs from
insolvency of a bank to strategic withdrawals (bank runs).
The linkage between the two situations in our example results from the state where both
agents are short-lived. However, it is impossible to avoid this state (or possibility) because
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新しい金融経済学

mismatch of maturity between lending and borrowing is inherent in the modern partial reservebanking system. The interbank market does not necessarily rescue a bank at risk of bankruns by providing liquid assets when the risk of bank runs is linked to a risk of loan default.
The bank run is unavoidable since this linkage comes from mismatch of maturity between
borrowing and lending in the modern partial reserve banking system.

Ⅳ. Concluding Remarks
This paper offers an example of the game in which banks decide how much to lend or borrowand depositors at a bank at risk of bank runs choose when to withdraw strategically, in order tobetter understand how the interbank market would (fail to) work to rescue the troubled bank.
We modify Postlewaite & Vives (1987)ʼs example, by adding banks as players and a simple
structure of the interbank market. We show two types of equilibrium in our example. Thereis a Bayesian perfect equilibrium in which the interbank market collapses (with no transaction)
and a bank run occurs. There is also an equilibrium where banks refuse to lend and a bank run
happens when a situation of a bank run (strategic withdrawal) is linked to a situation of defaultof the rescue loan. This linkage comes from the fundamental fact of mismatch of maturity
between lending and borrowing under the partial reserve banking system. As a result, theinterbank market alone does not necessarily save a bank at risk of bank runs.
Several remarks are in order. We have assumed that the borrowing bank (or its manager)
tries to minimize a chance of a bank run. This objective of the bank may be justified as akind of the strategic assumption used when we try to show our result — the interbank market
fails to rescue a bank at risk of runs — under the condition — a borrowing bank directlytries to avoid its bank run — that seems disadvantageous to our intention. We can escapefrom asymmetric treatment between borrowing and lending banks and keep our propositions
intact, by assuming that every bank tries to minimize probability of its own bank run. Thisis a technical trick since the lending banks have no risk of bank runs so that their payoffs areconstant, which does not affect equilibrium.
A small amount of liquid assets held by lending banks seems restrictive for our propositions
to hold. However, this does not necessarily mean that our results are irrelevant to the realworld. If a bank at risk of runs is relatively large to lending banks, the conditions on liquid
assets of the lending banks can be satisfied.
A few extensions are worthwhile to examine. We have assumed that the signals received
by banks and the pairs of actual type are perfectly correlated. If signals received by lendingbanks are not informative (not correlated to actual pairs of type) and prior probability of both112

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agents being short-lived is small enough, the lending banks will always lend money to bank0 in our example (because a bank run occurs only when both agents are short-lived even on offequilibirum paths and expected payoff of these banks may become positive). On the other hand,

if signals received by these banks are not informative and prior probability of both agents
being short-lived is not small enough, these banks will not make any loans to bank 0. These
cases could be a clue for how a bank's ability to collect information would affect whether theinterbank market would rescue a bank at risk of runs.
In the interbank market, banks usually have their specific relations with one another anddevelop networks among themselves. A bank can receive a rescue loan easily from its closely
related banks. We have not taken account of this network structure in our example. As Allen& Gale (2000) demonstrates, the structure of network in the interbank market can be crucial
to how the interbank market would function. It is an interesting issue to examine how thestructure of network in the interbank market would affect a rescue lending. When lendingbanks face risk of bank runs, how lending and borrowing through the interbank market are
carried out is also a future research topic.
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