数理解析研究所講究録1108 巻 1999 年 89-105

89

Efficiency of Stochastic Transfers
in a Directed Graph
Akira Yamazaki
Graduate School of Economics
Hitotsubashi University
Kunitachi, Tokyo 186-8601, Japan
Abstract
A mathematical model of a directed graph with stochastic transfers is presented.
It will be used to analyze the optimality (or “competitiveness”) properties of a network of transactions involving risky transfers of assets in an economic system. These
properties are discussed in a model with some specific directed graph structures
which result in a decompositon of the graph into parts with “narrow” linkage.

1

Introduction

The purpose of this paper is to present a mathematical model of a directed graph withstochastic transfers, which will be used to analyze the optimality (or “competitiveness”)
properties of a network of transactions involving risky transfers of assets in an economic

system.
Amodel of stochastic transfers in a directed graph is presented in section
2. The concepts of optimality of stochastic transfers are discussed in terms of efficiency
and competitiveness in section 3. A specific model of risky transfers of assets, which will
be called thetriangle” plus a simple barter is presented in section 4. Optimality
properties of risky transfers in the specific model are briefly discussed in section 4.
$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\backslash$($‘ \mathrm{W}\mathrm{i}\mathrm{c}\mathrm{k}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{l}$2

A Model of Stochastic Transfers in a Directed Graph

2.1

A directed graph

An ordered pairconsisting of a nonempty finite set and a binary relation on
is a directed graph. The interpretation of the graphin this paper is the
:
$a\in
A$
a point
is a trader or an agent; an ordered pairshows a direction oftransfer, i.e., a transfer from to .
Two typical examples of a directed graphfrom commodity and financial assets
transfers are illustrated in.
andillustrates a simple barter tradeoftransfers commodities or assets. Agent sends commodities or assets to agent and$A$$(A,\mathcal{T})$$(A,\mathcal{T})$$\mathcal{T}$$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\dot{\mathrm{m}}_{\mathrm{o}}\sigma$$(t_{1}, t_{2})\in \mathcal{T}$$t_{1}$$t_{2}$$\mathrm{a}\mathrm{r}\mathrm{i}\sin_{\mathrm{o}}\sigma$$\mathrm{f}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{r}\mathrm{e}1$$\mathrm{f}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{r}\mathrm{e}2$$a_{1}$$A$$\mathrm{F}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{r}\mathrm{e}1$$a_{2}$

90

A$||\mathrm{S}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{r}^{\mathrm{I}\dagger}$Fig. 1

Fig. 2

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nice versa. Then, it is represented by a directed graphwith $A=\{a_{1}, a_{2}\}$ and. In figure 2, each agent sends commodities or assets counterclockwise to another agent, indicating the fact that there are no “double coincidence of wants.” This type of transfers is known as a ‘Wicksell triangle.” It is represented
bywith $A=\{a_{1},a_{2},a_{3}\}$ and.
gives an illustration of a general directed graph.
$(A, \mathcal{T})$$\mathcal{T}=\{(a_{1}, a_{2}), (a_{2},a_{1}), (a_{1}, a_{1}), (a_{2}, a_{2})\}$$(A, \mathcal{T})$$\mathcal{T}=\{(a_{1},a_{2}), (a_{2}, a_{3}), (a_{3}, a_{1}), (a_{1},a_{1}), (a_{2},a_{2}), (a_{3}, a_{3})\}$$\mathrm{F}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{r}\mathrm{e}3$

Probability structure

2.2

The stochastic transfer will be formalized in terms of a state spaceprobability measure spaceis given.

$\Omega$. We assume that a$(\Omega, Pr)$

Commodities, endowments and preferences

2.3

Let there be 1 kinds of commodities or assets. (Henceforth, assets and commodities are
not distinguished and they are simply referred to as commodities.)
represents the spaceof these commodities. The initial endowments of each agent $a\in A$ of these commodities
are specified by a
. Thus, each agent
mapping :endowed with thebundle of commodities.
In order to discuss state-contingent trade of these endowments, we adopt Arrow’s
convention that each type of commodity is a class ofingent commodities, one
for each state in . Thus the set of all commodities is. Each agent isendowed withunits of commodity. Formally, the commodity
for every
space isby 1$\mathbb{R}^{\ell}$$\ell$$e$$0\sigma \mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}$$Aarrow \mathbb{R}_{+}^{\ell}$$a$$\mathrm{i}.\mathrm{s}$$e(a)\in \mathbb{R}_{+}^{\ell}$$\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}- \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}$$\Omega$$\Omega\cross\{1, \ldots , l\}$$e^{j}(a)$

$L$

$a$

$\omega\in\Omega$$(\omega,j)$$0\sigma \mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}$$L\equiv(\mathbb{R}^{\ell})^{\Omega}$.

A commodity bundle $x\in L$ may be viewed as a random vector defined on
or a measurable mapping defined onvalues in .
We assume that each agent’s preference between commodity bundles conforms to
expected utility.
has a von-Neuman-Morgenstern utility function
:
.
$x\in
L$
Then, if
is a commodity bundle, the expected utility of for agent
is givenby$U_{a}(x)= \int_{\Omega}u_{a}(x(\omega))dPr$ .
$\mathbb{R}^{\ell}$$\mathbb{R}^{\ell}$$x$$\Omega \mathrm{t}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}_{\mathrm{o}}^{\sigma}$$\mathrm{A}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}$$\mathbb{R}^{t}arrow \mathbb{R}$$a$$u_{a}$

$x$

2.4

$a,$

$U_{a}(x)$Information partition

We shall consider two rounds of transfers. For each round of transfers, information ofis specified by an information partition
on . Denote by $P_{\tau\iota)}n=1,2$ , thepartitions corresponding to these rounds. It is assumed that information becomes finer asrounds proceed so that
. In a specific model presented in section 4, one assumes$P$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\Omega$$\mathcal{P}_{1}\subset \mathcal{P}_{2}$lIf one wants to conform to the Arrow-Debreu convention strictly) then one needs tocommodities by each state, the origin and the place of its consumption. Then, the commodity spaceshould be given by
where forshows theof the commodity
whereas shows its place of consumption. In this paper we do not adopt the convention because we liketo aaeume each agent derives utility from the consumption of commodities provided at his own location.
This assumption not only simplifiae our analysis but also helps us to focus on the effects of transfers on
the flnal consumption attained by each agent.
$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}_{\Leftrightarrow}o\mathrm{u}\mathrm{i}\mathrm{s}\mathrm{h}$$L$$L\equiv(\mathrm{R}^{p})^{\Omega \mathrm{x}\mathcal{T}}$$t_{2}$$t=(t_{1},t_{2})\in \mathcal{T},$$t_{1}$$\mathrm{o}\mathrm{r}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{n}$92Fig. 3Fig. 4

93

and $P_{2}=\{S,F\}$ . That is, there is no information available at round 1 and atround 2 information becomes available whether transfers have safely done, i.e.,orfailed, i.e.,.

$\mathcal{P}_{1}=\{\Omega\}$$\omega\in S$$\omega\in S$

A network of stochastic transfers

2.5

A given directed graphrepresents the inffastructure of a network of stochaetic
transfers. For the purpose of our present analysis, we consider two rounds of transfers.
Each agent $a\in A$ in the network decides amounts and constituents of his transfer to otherin the network at each round. We assume, however, that agents make an agreement
for transfers of commodities among them before the initial round. The agreement among
them is. The list of stochastic transfers among agents for each round of transfers
is denoted by:,$(A,\mathcal{T})$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\tau_{n}$$\Omega \mathrm{x}\mathcal{T}arrow \mathbb{R}_{+}^{\ell},n=1,2$

whereis -measurable,that agents can use information available at thetime of actual transfers.
represents a state-contingent transfer fromto in state at n-th round.
The stochastic nature of rounds of transfers will be described by a stochastic safety
rate function for each round, which indicates the safely arrived proportion of commodities
that are sent from one agent to another. Stochastic safety rate functions are given by$\tau_{n}$$P_{n}$$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\tau_{n}(\omega, (t_{1},t_{2}))$$\mathrm{a}_{\Leftrightarrow}\sigma \mathrm{e}\mathrm{n}\mathrm{t}$$t_{1}$$t_{2}$$\omega$$\sigma_{n}:\Omega\cross \mathcal{T}arrow[0,1]$$n=1,2$ ,,where both and are -measurable. Since
is the information available at initialround, -measurability ofmeans that agents do not know for initial round transfers
the actual proportion of commodities that reaches another, so that agents face risky
transfers. On the other hand, -measurability of the stochastic safety rate function
for second round transfers means that agents exactly know the proportion of commodities
that will reach anotherat the time transfers are to be made. Thus, agents can makesecond round transfer
contingent on the events in . When $0<\sigma_{2}<1$ , itmeans that second round transfers are costly. This cost which is sometimes called ascost,” can be viewed as a crude way of reflecting various intuitive considerations
time preference and exposure to business loss due to delayed availability oftransferred funds. We assume that the information$n=1,2$ , is publicly available atn-th round.
$\sigma_{1}$$P_{1}$$\mathcal{P}_{2}$$\sigma_{2}$$P_{2}$$\sigma_{1}$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}$$P_{2}$$\sigma_{2}$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}$$P_{2}$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{e}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}$(($\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{b}\mathrm{e}\mathrm{r}_{\mathrm{o}}^{\sigma}$$\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathcal{P}_{n},$2.6

A transfer network economy

An ordered pairtransfer. An ordered pair$\tau=(\tau_{1}, \tau_{2})$transfer technology.

of stochastic transfer functions will be called a stochasticof stochastic safety rate functions will be called a$\sigma=(\sigma_{1}, \sigma_{2})$94

A transfer network economy will be given by a quintuple consisting of a directed, a specification of a utility, an initial endowment function :, and an information, a transfer technology
function to each agent$\mathcal{E}$$Aarrow \mathbb{R}^{p}$$e$$\mathrm{o}^{\mathrm{T}\mathrm{a}\mathrm{p}\mathrm{h}}\mathrm{o}(A, \mathcal{T})$$\sigma=(\sigma_{1}, \sigma_{2})$$(u_{a})_{a\in A}$

structure$\mathcal{P}=\{P_{n}\}_{n=1,2}$, i.e.,$\mathcal{E}=\{(A,\mathcal{T}), e, (u_{a})_{a\in A}, \sigma,P\}$$A$.

Let us introduce some further notation for convenience. For any, we write$f$:$\mathcal{T}arrow R_{+}^{\ell},$$V,$ $W\subset$$f(V, W)= \sum_{t\in U}f(t)$a$f(a,W)=f(\{a\}, W)$ , and $f(V,a)=f(V, \{a\})$ . If we
where
$f(a,
W)$
commodities
is the total of
as a contingent transfer, thenmapping :$W$ , and $f(V, a)$ is the total of commodities
to
agent
from
to
that are sent
in V. (See . )fromthat are sent to, it is assumed
Given a transfer network economy and a stochastic transfer
that the consumptions take place after second round transfers are completed. Let us
the total of commodities in state sent from to where $t=(t_{1},t_{2})$ ,denote by$U=(V\mathrm{x}W)\cap \mathcal{T},$$f$$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{a}\mathrm{r}\mathrm{d}$$\mathcal{T}arrow \mathbb{R}^{\ell}$$a$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{n}\mathrm{g}$

$a$

$4.$

$\mathrm{f}\mathrm{i}_{\mathrm{o}}\sigma$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\tau=(\tau_{1}, \tau_{2})$$\mathcal{E}$$\tau^{s}(\omega,t)$i.e.,$\tau^{s}(\omega,t)\equiv\tau_{1}(\omega,t)+\tau_{2}(\omega,t)$

and byby , i.e.,$\tau^{\sigma}(\omega,t)$, the total of commodities in state$\omega$$t_{2}$$t_{1}$$\omega$,sent from$t_{1}$

to$t_{2}$

and actually received$t_{2}$$\tau^{\sigma}(\omega, t)\equiv\sigma_{1}(\omega,t)\tau_{1}(\omega, t)+\sigma_{2}(\omega, t)\tau_{2}(\omega,t)$.

represents the total of commodities
Thus, by our notational convention,$a\in
A$
represents the total of commodities
to other agents in andsent bybyinagentsreceived
We say that a stochastic transfer
byvarious
that
are
sent
, we haveis feasible for if for every $a\in A$ and$\tau^{s}(\omega, a, A)$$A$$7^{-\sigma}(\omega, A, a)$

$A$

$a$

.

$\tau$$\omega\in\Omega$$\mathcal{E}$$0\leq\tau^{s}(\omega, a, A)\leq e(a)$.

Consumptions resulting from a stochastic transfer

2.7

be a feasible stochastic transfer for . We now provide an explicit definition ofis endowed with
consumptions resulting from a stochastic transfer. Eacha bundle $e(a)$ of various commodities to start out with. Out of what has, transfers
and receives aa bundleof various commodities to agents in. In general one hascommodity bundleLet$\mathcal{E}$$\tau$’$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$

$a$

$a$

$a$

$A$

$\tau^{s}(\omega, a, A)\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}\mathrm{i}\sin_{\mathrm{o}}\sigma$$\tau^{\sigma}(\omega, A, a)$$\sum_{a\in A}\tau^{\sigma}(\omega, A,a)<\sum_{a\in A}\tau^{s}(\omega, a, A)$(that is, the total of commodities received by various agents are strictly
for .
and
as first round transfers areless than that of commodities sent out by thesecond round transfers are costly.) Thus, after the stochastic transfer is completed, theconsumption, isamount of commodities at hand for agent , which is available for
given by.
$\mathrm{a}.\mathrm{e}$$\omega\in\Omega$$\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}\gamma$$\mathrm{a}_{\mathrm{o}}^{\sigma}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\tau$$a$$e(a)-\tau^{s}(\omega, a, A)+\tau^{\sigma}(\omega, A, a)$$a’ \mathrm{s}$95

3Efficiency and Competition

3.1

Efficiency

Given a stochaetic transfer for and a nonempty subset, we say is C-inefficientif there is another stochastic transfer feasible for such that
$C\subset A$$\mathcal{E}$$\tau$$\tau$$\mathcal{E}$

$\mu$

(1)

$(\forall a\in C)E[u_{a}(e(a)-\mu^{s}(\omega, a, A)+\mu^{\sigma}(\omega, A, a))]$$\geq E[u_{a}(e(a)-\tau^{s}(\omega, a, A)+\tau^{\sigma}(\omega, A, a))]$(2),$(\exists a\in C)E[u_{a}-\mu^{s}(\omega, a, A)+\mu^{\sigma}(\omega, A, a)]$$>E[u_{a}(e(a)-\tau^{\mathrm{s}}(\omega, a, A)+\tau^{\sigma}(\omega_{f}A, a))]$(3)$(\forall\omega\in\Omega)(\forall t=(t_{1}, t_{2})\in \mathcal{T}$and$\mu_{1}(t)=\tau_{1}(t)$

with

or

$t_{1}\not\in C$$\mu_{2}(\omega, t)=\tau_{2}(\omega,t)$

(3.1)

,
$t_{2}\not\in C$).

is -efficient if it is not -inefficient. A feasible stochastic transfer is Pareto efficient
if it is -efficient. A feasible stochastic transfer is totally efficient if for any nonempty, it is C-efficient.
$C$$\tau$$C$$\tau$$A$$\tau$$C\subset A$

Competition

3.2

Since we arewith efficiency and competitiveness properties of a network transfer
economy without explicit consideration of “market prices”, we appeal to the game theoretic notion of core. For a nonempty, a stochastic transfer is C-subcompetitive(or -dominated) if there is another feasible stochastic transfer satisfying (1) and (2)above and$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{d}$$\cdot$$C\subset A$$\tau$

$C$

$\mu$

(3)

$(\forall\omega\in\Omega)(\forall t=(t_{1}, t_{2})\in \mathcal{T}$and$\mu_{1}(\omega,t)=0$

with$t_{1}\not\in C$$\mu_{2}(\omega,t)=0$or$t_{2}\not\in C$

)
(3.2)

.

A feasible stochastic transfer is said to be -competitive if it is not C-subcompetitive.
Finally, a stochastic transfer is called totally competitive if it is -competitive for all. In particular, is called individually rational if it isnonempty-competitive
$a\in
A$
.
for all$C$$\tau$$C$$\tau$$C\subset A$

3.3$\{a\}$$\tau$

Facts

One can check the following lemma and propositions without difficulty.
Lemma 1 Assume$C_{1}\subset C_{2}$. If is$\tau$$C_{2^{arrow}}efficient_{J}$then it is$C_{1}$-efficient.

Proof Let
. Assume is -efficient but not efficient. Then, there is anotherfeasible stochastic transfer satisfying. the properties (1),(2) and (3) with $C=C_{1}$ . Since, it also implies that is-inefficient contradicting to the assumption. I$C_{1}\subset C_{2}$$\tau$$C_{2}$$C_{1}$$\mu$$C_{1}\subset C_{2}$Proposition 1$\tau$$\tau$$C_{2}$

is totally efficient if and only if it is Pareto efficient.

96

Proof Assume is totally efficient. Then, it is trivially A-efficient.
One the other hand, assume is Pareto-efficient or -efficient. Then, by Lemma 1,
is -efficient. Thus, it is totally efficient. I
for any$\tau$$A$$\tau$$C$$C_{1}\subset A,\tau$

Proposition 2 If$\tau$

is totally competitive, then it is totally efficient.

Proof Let be totally competitive. Assume is not -efficient. Then, there is anotherthe properties (1),(2) and (3) with $C=A$ . Since
feasible stochastic transfer
the complement of is empty, must be -subcompetitive contradicting the fact that it
is totally competitive. Thus, it must be -efficient. It follows from Proposition 1 thatis totally efficient. 1$A$$\tau$$\tau$$\mu \mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{y}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$

$A$

$A$

$\tau$$A$$\tau$A Specific Result

3.4

At this point we would like to introduce a network transfer economy with a very specificdirected graph which has relevance to some economic problems. (See, for example, [2]and [3].)The structure of a specific directed graph that we consider here is what we call a“Wicksell triangle” plus a simple barter which is given by the following (see fig. 5) :$\bullet A=\{a_{1}, a_{2}, a_{3)}a_{4}\}$$\bullet \mathcal{T}=\{(a_{1}, a_{2}), (a_{2}, a_{3}))(a_{3}, a_{1}), (a_{3},a_{4}), (a_{4}, a_{3}), (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4})\}$For a transfer network economy with a directed graph which is given by a Wicksell
plus a simple barter, we have the following characterization of its total competitiveness.

$\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}_{\mathrm{o}}\sigma 1\mathrm{e}$be a transfer network economy with a
Proposition 3 Let$a\in A$ ,as specified above. Assume each $agent^{f}s$ utility fun tion
directed graphbe locdly nonsatiated in his own endowment goods. Let be a stochastic transfer feasible
for . Then, is totally competitive if and only if the following conditions hold: is-competitive.
andindinidually rational, totally efficient,$\mathcal{E}=\{(A, \mathcal{T}), e, (u_{a})_{a\in A}, \sigma, P\}$$(A, \mathcal{T})$$\mathrm{c}$$u_{a},$$\tau$$\mathcal{E}$$\tau$$\tau$$\{a_{3}, a_{4}\}$$\{a_{1}, a_{2}, a_{3}\}-$

Proof Assume is totally competitive. Then, it must satisfy the conditions by definition and proposition 2.
Conversely, suppose that a stochastic transfer is individually rational, totally effi-competitive. The only nonemty subset for which could-andcient,
be -subcompetitive without explicitly violating one of the three conditions are then:$\tau$$\tau$$C$$\{a_{3}, a_{4}\}$$\{a_{1}, a_{2}, a_{3}\}$$\tau$$C$$\bullet$$\bullet$

A. The subsets consisting of$a_{4}$

together with either

B. The subsets to which exactly two members of$a_{1}$

or$a_{2}$$\{a_{1}, a_{2}, a_{3}\}$; andbelong.

Call these type A and type subsets respectively. We now show that any individually
rational and totally efficient stochastic transfer cannot be -subcompetitive for any
subset of either type A or type B.
Assume that a stochastic transfer satisfied the conditions (1), (2), and (3’) for asubset . Suppose $C$ is of type A. Since is totally efficient, it is -efficient and thus
the only possible satisfying the conditions (1) and (3) is autarky, i.e.,$\mathrm{B}$$\tau$

$C$

$\mu$

$C$

$\tau$$\mu$$C$$\mu^{s}(\omega, a, \mathrm{A})=0$

97

Fig. 5A Wicksell Triangle pius a Simple Barter

98. But since is individually rational, cannot
orfor $a\in A$ when issatisfy condition (2) for either of those two subsets.
Now suppose that is of type B. That is, either $C=\{a, b\}\subset\{a_{1}, a_{2}, a_{3}\})$ or else
$C=D\cup\{a_{4}\}$ and $D=\{a,b\}\subset\{a_{1},a_{2}, a_{3}\}$ . In the.former case, without loss of generality,$a=a_{j},$ $b=a_{i}$ with $j=i+1$ (mod 3). The only transfer that can occur in any state ofendowment goods. Thisis for to receive some ofnature between these twomust happen with positive probability, in order for (2) to be satisfied for . In that case,applies in the latter
though, neither (1) nor (2) can be satisfied for . The same
(and
agent
goods
the
some
of
endowment
case, unless is agent
who receivesby). If so, define a stochastic transfer
is$C$$\{a_{2}, a_{4}\}$$\{a_{1}, a_{4}\}$$\mu$$\tau$$C$$b’ \mathrm{s}$$a$$\mathrm{a}_{\mathrm{o}}^{\sigma}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$

$a$

$b$

$b$

$\mathrm{a}\Gamma_{\mathrm{o}}^{\sigma}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}$$a_{3}$$a$$a_{4}$$0.\mathrm{f}$$\theta=(\theta_{1}, \theta_{2})$$\mathrm{a}_{\Leftrightarrow}\sigma \mathrm{e}\mathrm{n}\mathrm{t}a_{1}$$0$

if $t=(a_{3}, a_{1})$$\mu_{\tau\iota}(\omega, t)$otherwise$\theta_{n}(\omega,t)=\{$

strictly prefers to , and agent is indifferent between and .
for $n=1,2$ .
-subcompetitive for
Therefore, if were -subcompetitive for , then would be, contrary to hypothesis. I$\tau$$\theta$$\theta$$\mathrm{A}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}a_{3}$

$C$

$\mu$

$a_{4}$$\mu$$\tau$$\mu$$\{a_{3)}a_{4}\}$$\theta$

Decomposition of a graph

3.5

Before proceeding to obtain further results on efficiency and competition, we would liketo discuss concepts concerning the decomposition of a graph.
if.
is a decomposition of
be a directedLet,. We write
is a partition of the set and$i\neqj$. A decompositionforand put$i\neqj$ .
perfect
if
for
all
graph
is calledof a directed
be a transfer network economy. A feasible stochasticLettransfer is said to be -totally competitive if it is -competitive for all nonempty $C’\subset C$ .
a very easy result.
We now$(A, \mathcal{T})$$A$$\{A_{\mathfrak{i}}\}_{i=1,\cdots,n}$$(A, \mathcal{T})$$\{(A_{i}, T_{i})\}_{i=1,\cdots,n}$$0\sigma \mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}$$\mathcal{T}_{ij}=T\cap(A_{\dot{\mathrm{t}}}\mathrm{x}A_{j})$$\mathcal{T}_{i}=\mathcal{T}\cap(\mathrm{A}_{i}\mathrm{x}A_{i})$$\{(A_{i}, \mathcal{T}_{i})\}_{i=1,\cdots,n}$$\mathcal{T}_{ij}^{-1}\equiv\{(a, b)\in A_{i}\cross A_{j}|(b,a)\in \mathcal{T}_{ji}\}$$\mathcal{T}_{ij}=\emptyset$$\mathcal{E}=\{(A, \mathcal{T}), e, (u_{a})_{a\in A}, \sigma, \mathcal{P}\}$

$C’$

$C$

$\tau$$\cdot \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}$be a transfer network economy. Let
Proposition 4 Let. Then, a feasible stochastic transfer
be a perfect decomposition of, .
is totally competitive if and only if it is -totally competitive for each $i=1,$$\mathcal{E}=\{(A, \mathcal{T}), e, (u_{a})_{a\in A}, \sigma, P\}$$\{(A_{i},T_{i})\}_{i=1,\cdots,n}$$(A, \mathcal{T})$$n$$\cdots$$\mathrm{A}_{i}$$\tau$. If
Proof Letbe a perfect decomposition of the directed graph-totally
triviallyitiscompetitive,
totallythenwhichstochastictransferisafeasibleis, .
competitive for all $i=1,$. Assume
Conversely, assume that is -totally competitive for each $i=1,$$C\subsetA$
suchtransferstochastic
nonempty
and a feasible
subsetthat there existed athat is not -competitive with respect to . Then, there exists $j\in\{1, \cdots , n\}$ such thatstrictly prefer to . Define a feasible stochastic transfer
some of the agents in. (which is possible as the decomposition is
to be identical with for agents in-subcompetitive under
perfect), and to be outside. Then, becomes, contradicting the fact that is -totally competitive. Therefore, must be totallycompetitive. I$(A, \mathcal{T})$$\{(A_{i}, \mathcal{T}_{i})\}_{i=1,\cdots,n}$$A_{i}$$\tau$$n$$\cdots$$\cdots,$ $n$$A_{i}$$\tau$$\mu$$\tau$$C$$\mu$$C\cap A_{j}$$C\cap A_{j}$$\theta$$\mu$$\theta$$\theta$$\tau$$\mu$$0$$C\cap A_{j}$$\tau$$A_{j}$$\tau$$C\cap A_{j}$$\tau$

99

The proposition 4 shows that when a transfer network economy is decomposed into
independent parts, the competitiveness of the entire part simply reduces to the competitiveness of each independent part. This simple result motivates us to investigate
circumstances under which the competitiveness of each part is equivalent to that of theentire economy even when the entire graph cannot be decomposed in a perfect manner.
Intuitively, one would expect this to happen when the linkage between the decomposedparts is ‘narrow’ in an appropriate sense.
Given a directed graph, a subset is a path from to if $C=\{a_{1},a_{2}, \cdots , a_{n}\}$with, $n-1$ . If all
for $i=1,$,
are distinct, it is called a strict
path. A pathfromtois a cycle if $a_{1}=a_{n}$ and if
is a strict
path. In this case we also say that is a cycle into or . A path from to or acycle into $a_{1}=a_{n}$ is called unilateml if
for all $i=1,$$n-1$ . It is calledbilateral or a barter if
for all $i=1,$, $n-1$ .
Let
be a decomposition of a directed graph. Define two subsets
andoffor each $i=1,$.
$C$$(A,\mathcal{T})$$(a_{i}, a_{i+1})\in \mathcal{T}$$\cdots$$C$$a_{1}$$a_{n}$$a_{1}$$a_{1},$$\cdots$$a_{n}$$\{a_{1}, a_{2}, \cdots , a_{n-1}\}$$a_{n}$$C$$a_{1}$$C$$\mathit{0}_{n}$$(a_{i+1}, a_{i})\not\in \mathcal{T}$$(a_{i+1}, a_{i})\in \mathcal{T}$$\cdots,$$(A, \mathcal{T})$$A_{i}$$L_{ij}$$a_{n}$$\cdots$$\{(A_{i}, \mathcal{T}_{i})\}_{i=1,\cdots.n}$$B_{ij}$$a_{1}$$\cdots,$ $n$$L_{ij}\equiv\{a\in A_{i}|(\exists b\in A_{j}) (a, b)\in \mathcal{T}_{ij}\}$,.

$B_{ij}\equiv\{a\in A_{i}|(\exists b\in A_{j}) (a, b)\in \mathcal{T}_{ij}\cap \mathcal{T}_{ij}^{-1}\}$

An element a inan element inwill be called a transfer linkage ofwith respect to . Similarly,
will be called alinkage ofwith respect to . If, for any, $j,$ $a\in A_{i},$ $b\in A_{j}$ , there always exists a path from to , then thegiven $i,j=1,$
directed graph is said to be.
We would like to extend proposition 4 to the case of a transfer network with anirreducible directed graph. This will be done in a manner corresponding to proposition 3
where the directed graph is decomposed into two parts that has a narrow barter linkage.
We now state a proposition which extends both proposition 3 and proposition 4 when the
directed graph is decomposed into two parts.
$a$$L_{ij}$$A_{i}$$ba\hslash er$$B_{ij}$$A_{j}$$A_{i}$$A_{j}$$\cdots$$a$$b$$i7\mathrm{v}’ educible$

Proposition 5 Let
be a transfer network economy andbe a decomposition of. Assume each agent’s utility function $a\in$
be locally nonsatiated. Let be a feasible stochastic transfer. Then, is totally competitive if and only if the followin9 conditions hold: is totally efficient, -totally competitive-competitive, provided that each
$i=1,2$ , is a unilateralfor each $i=1,2$ andcyde and each barter linkage is singleton.
$\mathcal{E}=\{(A, \mathcal{T}), e, (u_{a})_{a\in A\sigma},, P\}$$\{(A_{i}, \mathcal{T}_{i})\}_{i=1,2}$$(A, \mathcal{T})$$A_{j}$$u_{a},$$\tau$$\tau$$A_{i}$$\tau$$B_{12}\cup B_{21}$$A_{i},$

Proof Assume is totally competitive. Then, it must satisfy the conditions by definition and proposition 2.
Conversely, suppose that the decompositionis such that each
is aunilateral cycle and each barter linkage is singleton. Let $B_{12}=\{a^{*}\}$ and $B_{21}=\{b^{*}\}$ .
Assume is totally efficient, -totally competitive for each $i=1,2$ andcompetitive. Suppose there existed another feasible stochastic transfer for which were-subcompetitive for some nonempty $C\subset A$ .
Since is -and -totally competitive, by proposition 4 one must haveandfor each .
$\tau$$\{(A_{i}, \mathcal{T}_{i})\}_{i=1,2}$$\tau$$A_{i}$$A_{\mathrm{i}}$$B_{12}\cup B_{21^{-}}$$\mu$$\tau$$C$$\tau$$A_{1}$$C\cap A_{i}\neq\emptyset$$\bullet$$A_{2}$$\{a^{*}, b^{*}\}\subset C$$i$

Case 1: $C\cap A_{2}=\{b^{*}\}$

100. Since is
Put $C_{1}=C\cap A_{1}$ . As is totally efficient, we must have
who cannot receive any
is not a cycle and hence, there isa unilateral cycle,
Thus,
under , no agents incommodities from other agents under the transfer .
can transfer endowment goods to other agents. This means that only agents
just consume
who would strictly prefer over are or . Since agents in$\{a^{*},b^{*}\}=B_{12}\cup
B_{21}$-subcompetitive
their own endowment goods, it follows that is
hypothesis.
with respect to , contradicting to the$C_{1}\neq A_{1}$$\tau$$A_{1}$$a”$$C_{1}$

$\mu$

$\mu$

$C_{1}\backslash \{a^{*}\}$$\mu$$a^{*}$$\tau$$b^{*}$$C_{1}\backslash \{a^{*}\}$$\tau$$\mu$$\bullet$Case 2: $\#(C\cap A_{2})\geq 2$
or
Put $C_{i}=C\cap A_{i}$ for $i=1,2$ . As is totally efficient, we must have,
, then the agents in. Since bothare unilateral cycles,$a_{1}=a^{*}$, must consume only their endowments goods under the
and
where
competitive. Without loss of generality, assume-totallytransfer since is. Then, since agents other than $a_{2}=b^{*}$ consume only their endowment
goods, arguments essentially reduces to the one in Case 1.
$C_{1}\neq A_{1}$$\tau$$C_{2}\neq A_{2}$$C_{i}\backslash \{a_{i}\}$$\mathrm{i}\mathrm{f}C_{i}\neq A_{i}$$A_{i}’ \mathrm{s}$$a_{2}=b^{*}$$\tau$$\mu$$A_{i}$$C_{2}\neq A_{2}$We thus proved that$\tau$

must be totally competitive. 1

The above proposition 5 essentially shows that even when the entire graph cannot be
decomposed in a perfect manner, when a transfer network economy is decomposed into
parts, the competitiveness of the entire part simply reduces to the competitiveness ofeach ‘almost’ independent part provided that the linkage between the decomposed parts
is ‘narrow’ in an appropriate sense.

4

Competitive Stochastic bansfers in the Wicksell
biangle plus a Simple Barter

4.1

A further specification, we are interested in
Given a transfer network economy
transfers. For
totally
competitive
stochastic
giving a more specific characterization ofplus athis purpose, we shall go back to a very specific model of the Wicksell
simple barter which was presented in the subsection 3.4. Thus, throughout this section:
to be given by the following
we take the directed$\mathcal{E}=\{(A, \mathcal{T}), e, (u_{a})_{a\in A}, \sigma,P\}$$\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}_{\mathrm{o}}\sigma 1\mathrm{e}$$(A,\mathcal{T})$$0\sigma \mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}$$\bullet A=\{a_{1},a_{2}, a_{3}, a_{4}\}$$\bullet$$\mathcal{T}=\{(a_{1}, a_{2}), (a_{2}, a_{3}), (a_{3}, a_{1}), (a_{3}, a_{4}), (a_{4}, a_{3}), (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4})\}$The transfer network economy
ting:$\bullet a_{i}=i$,$\mathcal{E}=\{(A, T))e, (u_{a})_{a\in A}, \sigma,P\}$$i=1,$ $\ldots)4$$\bullet L=R^{5}$$\bullet e(i)=(0, \ldots, 1i, \ldots, 0)$$\bullet$$P_{1}=\{\Omega\},$$P_{2}=\{S, F\},$$Pr(S)=\overline{\sigma}$

is further specified by let-101$\bullet$bansfer technology-For $t=(2,3)$ ,-For all other-For all$\sigma=(\sigma_{1}, \sigma_{2})$$(\forall\omega\in\Omega)\sigma_{1}(\omega,t)=\chi_{S}(\omega)$$t\in \mathrm{T},$$t\in \mathrm{T},$is given by:$(\forall\omega\in\Omega)\sigma_{1}(\omega,t)=1$$(\forall\omega\in\Omega)\sigma_{2}(\omega,t)=\rho$

.

.

.

$-1> \overline{\sigma}>\rho\geq\frac{1}{2}$$\bullet$We$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}6^{r}$

the agents’ utilities as follows.
$u_{1}(x)$$=$$\ln(x_{1}+\beta x_{3})$$u_{2}(x)$$=$$\ln(x_{2}+\beta x_{1})$$u_{3}(x)$$=$$\ln(x_{3}+\beta x_{2}+\psi x_{4})$$u_{4}(x)$$=$$\ln(x_{4}+\varphi x_{5})$with$\beta>\max\{\overline{\sigma},\rho^{-1}\}\iota,$(4.3)$0<\varphi\psi<1$.

Here, goods received in trade are “better” substitutes for endowment goods for essential
participants 1,2,3. Agent 4 considers agent
good to be a “worse” substitute for his
own endowment good, and agent 3 c.onsiders good to be a “worse” substitute for agent$3’ \mathrm{s}$$4’ \mathrm{s}$$2’ \mathrm{s}$good or even for his own endowment good.

4.2

Interpretation

Motivation ofa model described in the previous subsection 4.1 is discussed
in [2] or [3]. Here, let usa brief description of an interpretation of the model in an
economic context.
It is a model of a network or an arrangement of transactions that involve a risky
transfer of assets. Transactions are generated endogenously. There is a risk in asset
transfers and one is concerned with the question of optimal risk management in sucha network. Assets may well be usual commodities and not limited to financial assets.
Some specific questions
risk management in a network of transactions are thefollowing. If there is some risk of failure in a transfer from one
to another, should thetransfer be done through that arrangement? If so, then what considerations are relevant
to determining whether third parties ought to share that risk? Are there conditions underwhich thepublic or the government (in the case of a private arrangement) ought
to bear some risk and, if so, what level of compensation would it be appropriate for themto receive? One can address these questions by analyzing a schematic, formal} model of astochastic transfer introduced above. The particular model of the Wicksell triangle plus
a simple barter has been motivated as a simplest model to analyze the questions posed.
Suppose that our task is to formulate a model of a transaction that involves a risky
asset transfer. The model should be richto describe such a transaction
ably, but simple enough to be analytically tractable. Consider whatmodel could
satisfy both the requirements of richness and simplicity. A transaction is a related setof asset transfers between agents. An asset transfer involves two, the donor andthe recipient, but a transaction can general.ly involve more than two. Therefore,
at the very least, a model of a transaction
a risky transfer should include three$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$0\sigma \mathrm{i}\mathrm{v}\mathrm{e}$$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{p}\mathrm{a}\mathrm{I}\mathrm{t}\mathrm{y}$$0\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}$$\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{u}_{\mathrm{o}}\sigma \mathrm{h}$$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}_{\mathrm{o}}^{\sigma}\mathrm{n}\mathrm{i}\mathrm{z}-$$\mathrm{s}\mathrm{o}\iota*\mathrm{o}\mathrm{f}$$\mathrm{a}_{\mathrm{b}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$

102, so that a distinction can be drawn between a participant in the broad transaction
and a participant (that is, the donor or the recipient) in the specific transfer where therisk occurs. In order for the third-party participant in the transaction –that is, theparticipant who is neither the donor nor the recipient of the risky transfer–to be essena mutually beneficial transaction, there should be no “double coincidence
tial toof wants” between the donor and the receiver. This consideration suggests modeling thethree participants as a “Wicksell triangle.”
There is a distinction between two types of third party (or potential third party) that a
party to risky transfer in a Wicksell triangle mightgood model ought to capture. A
be intrinsically necessary in the sense that the donor and recipient of the risky transfer
would have no double coincidence of wants, even if the transfer did not involve risk. For
characterizing the differences between the roles of these two types of third parties, a fourtrader model (including both an intrinsic third party and a trader whose only involvement
would be to share risk) can be useful. On the basis of these considerations, one is leadto the four-agents model of the Wicksell triangle plus a simple barter, whereis aesumed to be essential to a mutually beneficial transaction but that agent 2 is theis the recipient of the risky transfer. The attributes of agent 4, whodonor and
constitutes to be the potential partner of the essential participants, is specified in such a
way that agent 4 can only participate in a risk-sharing capacity. This should be clearlyseen from the specification of the agents’ utilities in (4.3).

$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\mathrm{m}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{t}\prime \mathrm{h}\mathrm{i}\mathrm{r}\mathrm{d}$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}1$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}3$

4.3

Some notation

Given a stochastic transfer , we shall write for convenience$\tau$$\tau_{i}^{1},$$\tau^{1}(\omega, t_{1}, t_{2})$

$=$

$\{$

for $t_{1}=1,2,4$
for $t_{1}=3,$ $t_{2}=1$
for $t_{1}=3,$ $t_{2}=4$$i=i_{1}$$\tau_{3}^{1}$$\tau_{5}^{1}$for all$\tau_{i}^{C},$$=$$\tau^{2}(\omega.

’t_{1}, t_{2})$$\{$$\tau_{5}^{C}\tau_{3}^{C}$$\omega\in\Omega$$i=t_{1}$,for $t_{1}=1,2,4$
for $t_{1}=3,$ $t_{2}=1$
for $t_{1}=3,$ $t_{2}=4$.
for
is clear from the context, we may write for simplicity$\omega\in C\in P_{2}$When a stochastic transfer$\tau$$C_{i}^{S}=$$1-\tau_{i}^{1}-\tau_{i}^{S}+a_{i-1}(\tau_{i-1}^{1}+\rho\tau_{i1}^{S})$$=$$1-\tau_{\mathfrak{i}}^{1}-\tau_{i}^{F}+a_{i-1}(\tau_{i-1}^{1}+\rho\tau_{i-1}^{F})$$C_{i}^{F}$

,

(4.4)

for $i=1,2,4$ where $i-1=3$ for $i=1$ and $i-1=5$ for $i=4$ , and. For $i=3$ , we have
, and$a_{i}=\beta$

for $i=1,2,3$ ,$a_{5}=\varphi$$a_{4}=\gamma$$C_{3}^{S}=$$C_{3}^{F}$$=$$1-\tau_{3}^{1}-\tau_{3}^{S}+\beta(\tau_{2}^{1}+\rho\tau_{2}^{S})-\tau_{\overline{\mathrm{Q}}}^{1}-\tau_{5}^{S}+\gamma(\tau_{4}^{1}+\rho\tau_{4}^{S})$$1-\tau_{3}^{1}-\tau_{3}^{F}+\beta\rho\tau_{2}^{F}-\tau_{5}^{1}-\tau_{5}^{F}+\gamma(\tau_{4}^{1}+\rho\tau_{4}^{F})$

.

,
(4.5)

103

is interpreted ae “real” consumption level of agent in event $E$ in the sense that it
directly determines
utility level in event $E\in P_{2}$ .
Given a stochaetic transfer , a net transfer gap of agent is defined by:$C_{i}^{E}$$i$$i’ \mathrm{s}$

$i$

$\mu$

$g_{i}( \mu)=1-\mu_{i}^{1}-\chi\langle 3\}\mu_{5}^{1}-\max\{\mu_{i}^{S}+\chi\{3\}\mu_{5}^{S},\mu_{i}^{F}+\chi_{\{3\}}\mu_{5}^{F}\}$.

It represents the maximal amount that agent can further transfer to other agents.
We may view the Wi&sell triangle as a transfer system, in which case we may refer tothe agents belonging to the Wicksell triangle as essential participants to the system andas the inessential participant. Then, the amountis the maximal amountthat agent can further transfer to other agents including the inessential participant.
Hence, if we define transfer gap
by$i$$g_{i}(\mu)$$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}4$$i$$\overline{g}_{i}(\mu)$$\bullet$$\overline{g}_{i}(\mu)=g_{i}(\mu)$$\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$

for $i=1,2$ , and$\bullet\overline{g}_{3}(\mu)=1-\mu_{3}^{1}-\max\{\mu_{3}^{S},\mu_{3}^{F}\}$,

represent the maximal amount that agent can further transfer to other agentsthenamong essential participants.
$i$$\overline{g}_{i}’ \mathrm{s}$4.4Assumption on parameter values

We shall require conditions which guarantee to generate nontrivial stochastic transfer
in the given transfer network economy. We assume thefollowing conditions on parameter values:$\mathcal{E}=\{(A, \mathcal{T}),e, (u_{a})_{a\in A}, \sigma,P\}$$\bullet$, and$\varphi\gamma>(\frac{1-\overline{\sigma}}{\rho})\frac{1}{1-\rho}$$\bullet\beta\rho>\sqrt[3]{2},$$\rho>\sqrt[3]{2}/2$.

The first inequality is satisfied, if, for example,
It is also satisfied wheneverwhere$\varphi\gamma>0.63$

when$\overline{\sigma}=0.9$

and $\rho=0.8$ .

$\varphi\gamma>r^{\overline{\sigma}\rho}$$r^{\overline{\sigma}\rho} \equiv(\frac{1-\overline{\sigma}}{\overline{\sigma}})(\frac{\rho}{1-\rho})$

The second and the third inequalities are to ensure that second round transfers are
not too costly to make such transfers.

4.5

Specific results

Finally, let us present two specific results concerning the efficiency and optimality properties of stochastic transfers in the model of the present section which have been proved
elsewhere (see, e.g., [2]). These properties are stated in terms of total competitiveness oftransfers.

104

Proposition 6 A totally competitive stochastic tmnsfer always specifies
gent tmnsfers. A typical totally competitive stochastic transfer satisfies:$\mu$$state\sim$contin-$\mu$

for $i=1,2,3,5,$$\mu_{i}^{1}>0$$\mu_{4}^{1}=0$,$\mu_{1}^{F}>0,\mu_{2}^{F}>0,\mu_{3}^{F}>0,\mu_{4}^{F}>0,\mu_{5}^{F}=0$$\mu_{1}^{S}=\mu_{3}^{S}=\mu_{4}^{S}=\mu_{5}^{S}=0,\mu_{2}^{S}>0$,,

in which case we have:$\frac{C_{2}^{F}}{C_{2}^{S}}$$=$$\frac{C_{1}^{F}}{C_{1}^{S}}$$\geq$$\frac{C_{3}^{F}}{C_{3}^{S}}=r^{\overline{\sigma}\rho}$$( \frac{1-\overline{\sigma}}{\overline{\sigma}})\frac{1}{1-\rho}>r^{\overline{\sigma}\rho}$

vrith equality holding when $g_{1}(\mu)>0$$=$$\frac{C_{4}^{F}}{C_{4}^{S}}$$\frac{1-\overline{\sigma}}{\overline{\sigma}}(\frac{1}{\varphi\gamma\rho(1-\rho)}-1)>(\frac{1-\overline{\sigma}}{\overline{\sigma}})\frac{1}{1-\rho}$’
consumptions that a typicaJ totally competOne may note the extent to which
itive stochastic transfer induces are state contingent. Given a typical totally competitiveof the statement of the proposition above, forstochastic transfer as in the,and agent 3 the consumption level in event relative to that in event is.
maybewhich is less than 1 but approaches 1 as the value of becomes closer to This
interpreted to say that the failure of receipt by agent 3 is compensated by other agentsis as responsible as agent 3 for the loss as hisby the factor of $(\rho/(1-\rho))-1$ .
relative consumption level in event is reduced to the level of agent 3. Agent 1 in turncompensates agent 2 but extent to which he joins in the compensation is less than that
. It is very instructiveof agent 2 so that his relative consumption in event exceeds
to note that agent 4 also participates in this compensation scheme but extent to which
he does compensate agent 3 is much less than those of other agents in the sense that his
relative consumption level in event is higher than those of all the essential participants.
$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\mathrm{b}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$

$S$

$F$

$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}2$$r^{\overline{\sigma}\rho}$$\overline{\sigma}$$\rho$$\mathrm{A}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}2$

$F$

$F$

$r^{\overline{\sigma}\rho}$$F$

Proposition 7 Let$\mu$be a totally competitive stochastic tmnsfer. Then:

1. At least one essential participant must be sending all his endowment to other agentsin some event. That.
$is_{j}$$(\exists i\in\{1,2,3\})g_{\mathfrak{i}}(\mu)=0$

2. Suppose that agent 3 is not sending all of his endowment to other essential panici. Then, is a totally competitive stochastic transfer if andpants so that
only if agent is making a transfer to agent 4 either by the amount of his transfer
gap or by the amount of ltfeasibility bound given by$\overline{g}_{3}(\mu)>0$

$\mu$

$S$

$v( \varphi\gamma,\overline{\sigma}, \rho,\beta)=\frac{(1-\overline{\sigma})(1-\varphi\gamma\rho(1-\rho))}{\beta[\varphi\gamma\rho(1-\rho)-(1-\overline{\sigma})]}$

,

105

whichever is smdler, $i.e.$ ,,$\mu_{5}^{1}=\min\{\overline{g}_{3}(\mu))v(\varphi\gamma,\overline{\sigma},\rho,\beta)\}$and agent 4 in $tum$ is mahng a state contingent transfer in event
amount given by$\tau_{4}^{F}=(\frac{\varphi\gamma\rho(1-\rho)-(1-\overline{\sigma})}{\overline{\sigma}\varphi\gamma\rho(1-\rho)})(1+\beta\tau_{5}^{1})$$F$

at most the

.

(4.6)

The first part of the proposition 7 is due to our specification of preferences of essential
participants that they prefer the endowment of another agent to his own. The second part
results from two factors. One is that a totally competitive stochastic transfer in general
specifies positive second round transfers in both eventsand as well as a positive
first round state non-contingent transfer from agent 2 to agent 3. This ensures agent
consumption in event relative to that in event $S,$
, to be given by. Secondis that under this circumstance, the expected utility of both of the agents 3 and 4 canbe increased whenever first round state non-contingent transfer from 3 to 4 and second
round state contingent transfer in event from 4 to 3 can be increased. For a first roundstate non-contingent transfer from 3 to 4, , the maximal amount that agent 4 would bejust willing to send to 3 is given by the amount shown in (4.6). For details, one is referredto [2] and [3].
$F$

$S$

$3’ \mathrm{s}$$F$$r^{\overline{\sigma}\rho}$$C_{3}^{F}/C_{3}^{S}$$F$$\tau_{5}^{1}$

References[1] Allen, Beth, “Supermechanisms,” University of Minnesota, undated.

[2] Fujiki, Hiroshi, Edward J. Green, and Akira Yamazaki, “Sharing the Risk of Settlement Failure,” October 1998.
[3] Yamazaki, Akira, (‘Risk Sharing in a Network of)’$\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{S}},$

August 1998.