11institutetext:  IEOR, IIT Bombay, India
11email: {indrojit<EMAIL_ADDRESS>
# Financial replicator dynamics: emergence of systemic-risk-averting
strategies

Indrajit Saha  Veeraruna Kavitha

###### Abstract

We consider a random financial network with a large number of agents. The
agents connect through credit instruments borrowed from each other or through
direct lending, and these create the liabilities. The settlement of the debts
of various agents at the end of the contract period can be expressed as
solutions of random fixed point equations. Our first step is to derive these
solutions (asymptotically), using a recent result on random fixed point
equations. We consider a large population in which the agents adapt one of the
two available strategies, risky or risk-free investments, with an aim to
maximize their expected returns (or surplus). We aim to study the emerging
strategies when different types of replicator dynamics capture inter-agent
interactions. We theoretically reduced the analysis of the complex system to
that of an appropriate ordinary differential equation (ODE). We proved that
the equilibrium strategies converge almost surely to that of an attractor of
the ODE. We also derived the conditions under which a mixed evolutionary
stable strategy (ESS) emerges; in these scenarios the replicator dynamics
converges to an equilibrium at which the expected returns of both the
populations are equal. Further the average dynamics (choices based on large
observation sample) always averts systemic risk events (events with large
fraction of defaults). We verified through Monte Carlo simulations that the
equilibrium suggested by the ODE method indeed represents the limit of the
dynamics.

###### Keywords:

Evolutionary stable strategy (ESS), Replicator dynamics, Ordinary differential
equation, Random graph, Systemic risk, Financial network.

##  1 Introduction

We consider a financial network with large number of agents. These agents are
interconnected to each other through financial commitments (e.g., borrowing
-lending etc.). In addition they make investments in either risk-free (risk
neutral) or risky derivatives. In such a system the agents not only face
random economic shocks (received via significantly smaller returns of their
risky investments), they are also affected by the percolation of the shocks
faced by their neighbours (creditors), neighbours of their neighbours etc. In
the recent years from $2007-2008$ onwards, there is a surge of activity to
study the financial and systemic level risks caused by such a percolation of
shocks ([6, 5, 4, 1]). Systemic risk is the study of the risks related to
financial networks, when individual or entity level shocks can trigger severe
instability at system level that can collapse the entire economy (e.g., [6, 5,
4]). In this set of papers, the author study the kind of topology (or graph
structure) that is more stable towards the percolation of shocks in financial
network, where stability is measured in terms of the total number of defaults
in the network.

In contrast to many existing studies in literature related to systemic risk,
we consider heterogeneous agents and we consider evolutionary framework. In
our consideration, there are two groups of agents existing simultaneously in
the network; one group of agents invest in risk-free instruments, while the
other group considers risky investments. The second group borrows money from
the other members of the network to gather more funds towards the risky
investments (with much higher expected returns). These investments are
subjected to large (but rare) economic shocks, which can potentially percolate
throughout the network and can even affect the ‘risk-free’ agents; the extent
of percolation depends upon relative sizes of the two groups. We consider that
new agents join such a network after each round of investment; they chose
their investment type (risky or risk-free) based on their observations of the
returns (the surplus of the agents after paying back their liabilities) of a
random sample of agents that invested in previous round. The relative sizes of
the two groups changes, the network structure changes, which influences the
(economic shock-influenced) returns of the agents in the next round, which in
turn influences the decision of the new agents for the round after. Thus the
system evolves after each round. We study this evolution process using the
well known evolutionary game theoretic tools.

In a financial network perspective, this type of work is new to the best of
our knowledge. We found few papers that consider evolutionary approach in
other aspects related to finance; in [8], the authors study the financial
safety net (a series of the arrangement of the firms to maintain financial
stability), and analyze the evolution of the bank strategies (to take
insurance or not); recently in [7] authors consider an evolutionary game
theoretic model with three types of players, i) momentum traders ii)
contrarian traders iii) fundamentalists and studied the evolution of the
relative populations. As already mentioned, these papers relate to very
different aspects in comparison with our work.

Evolutionary stable strategies Traditionally evolutionary game models have
been studied in the literature to study animal behaviour. The key ingredients
of the evolutionary game models are a) a large number of players, b)the
dynamics and c) the pay-off function (e.g., see the pioneering work [13]).
Replicator dynamics deals with evolution of strategies, reward based learning
in dynamic evolutionary games. Typically it is shown that these dynamics
converge to a stable equilibrium point called Evolutionary Stable Strategy
(ESS), which can be seen as a refinement of a strict Nash Equilibrium ([13]);
a strategy prevailing in a large population is called evolutionary stable if
any small fraction of mutants playing a different strategy get wiped out
eventually. Formally, in a 2-player symmetric game, a pure strategy $\hat{s}$
is said to be evolutionary stable if

  1. 1.

$(\hat{s},\hat{s})$ is a Nash Equilibrium; i.e., $u(\hat{s},\hat{s})\geq
u(s^{{}^{\prime}},\hat{s})$ $\forall s^{{}^{\prime}}$ and

  2. 2.

If $(\hat{s},\hat{s})$ is not a strict NE (i.e., $\exists$ some
$s^{{}^{\prime}}\neq\hat{s}$ such that
$u(\hat{s},\hat{s})=u(s^{{}^{\prime}},\hat{s})$), then
$u(\hat{s},s^{{}^{\prime}})>u(s^{{}^{\prime}},s^{{}^{\prime}})$.

We study the possible emergence of evolutionary stable strategies, when people
chose either a risky or a risk-free strategy; the main difference being that
the returns of either group are influenced by the percolation of shocks. The
returns of the portfolios depend further upon the percolation of shocks due to
layered structure of financial connections, and not just on the returns of the
investments, i.e., not just on economic shocks. Our main conclusions are two
fold; a) when agents consider large sample of data for observation and
learning, the replicator dynamics can settle to a mixed ESS, at which the
expected returns of the two the groups are balanced; b) in many other
scenarios, through theoretical as well as simulation based study, we observed
that the replicator dynamics converges to one of the two strategies, i.e., to
a pure ESS (after completely wiping out the other group).

The analysis of these complex networks (in each round) necessitated the study
of random fixed point equations (defined sample path-wise in large dimensional
spaces), which represent the clearing vectors of all the agents ([1, 6, 5, 4]
etc). The study is made possible because of the recent result in [1], which
provided an asymptotically accurate one dimensional equivalent solution.

##  2 Large Population Finance Network

We consider random graphs, where the edges represent the financial connection
between the two nodes. Any two nodes are connected with probability $p_{ss}>0$
independent of the others, but the weights on the edges depend on (the number
of) neighbors. This graph represents a large financial network where borrowing
and lending are represented by the edges and the weights over them. The
modeller may not have access to the exact connections of the network, but
random graph model is a good approach to analyse such a complex system. In
particular we consider the graphs that satisfy the assumptions of [1].

The agents are repeatedly investing in some financial projects. In each round
of investment, the agents borrow/lend to/from some random subset of the agents
of the network. Some of them may invest the remaining in a risk-free
investment (which has a constant rate of interest $r_{s}$). While the others
invest the rest of their money in risky investments which have random returns;
we consider a binomial model in which returns are high (rate $u$) with high
probability $\delta$ and can have large shocks (rate $d$), but with small
probability ($1-\delta$); it is clear that $d<r_{s}<u$. We thus have two types
of agents, we call the group that invests in risk-free projects as ‘risk-free’
group ($G_{1}$), the rest are being referred to as ‘risky’ group ($G_{2}$)

New agents join the network in each round of investment. They chose their
investment type, either risk-free or risky, for the first time based on the
previous experience of the network and continue the same choice for all future
rounds of investment. The new agents learn from network experience (returns of
agents of the previous round of investments) and chose a suitable investment
type, that can potentially give them good returns. The new agents either learn
from the experience of a random sample (returns of two random agents) of the
network or learn from a large number of agents. In the former case, their
choice of investment type depends upon the returns of the random sample in the
previous round. While in the latter case the decision can also depend on the
average utility of each group of the agents, obtained after observing large
number of samples.

Two strategies: As mentioned before, there are two strategies available in the
financial market. Risk-free agents of $G_{1}$ use strategy 1; these agents
lend some amount of their initial wealth to other agents (of $G_{2}$) that are
willing to borrow, while the rest is invested in a government security, for
example, bonds, government project etc. Risky agents of $G_{2}$ are adapting
strategy $2$, wherein they borrow funds from the other agents and invest in
risky security, for example, derivative markets, stocks, corporate loans etc..
These agents also lend to other agents of $G_{2}.$ Let $\epsilon_{t}$ be the
fraction of the agents in $G_{1}$ group and let $n(t)$ be the total number of
agents in round $t$. Thus the total number of agents (during round $t$) in
group 1 equals $n_{1}(t):=|G_{1}|=n(t)\epsilon_{t}$ and
$n_{2}(t):=|G_{2}|=n(t)(1-\epsilon_{t})$.

We consider that one new agent is added in each round 111this approach can
easily be generalized to several other types of dynamics and we briefly
discuss a few of them towards the end., and thus size of the graph/network is
increasing. The agents are homogeneous, i.e., they reserve the same wealth
$w>0$ for investments (at the initial investment period) of each round. Each
round is composed of two time periods, the agents invest during the initial
investment period and they obtain their returns after some given time gap. The
two time period model is borrowed from [5, 4, 1] etc. The new agents make
their choice for the next (and the future) round(s), based on their
observations of these returns of the previous round.

Initial investment phases: During the initial investment phases (of any round
$t$), any agent $i\in G_{1}$ lends to any agent $j\in G_{2}$ with probability
$p_{ss}$ and it lends (same) amount222This normalization, (after choosing the
required parameters, like $w$, appropriately) is done to derive simpler final
expressions.  $w/(n(t)p_{ss})$ to each of the approachers based on the number
that approached it for loan; let $I_{ij}$ be the indicator of this lending
event. Note that for large $n(t)$, the number of approachers of $G_{2}$
approximately equals $n(t)(1-\epsilon_{t})p_{ss}$, and, thus any agent of
$G_{1}$ lends approximately $w(1-\epsilon_{t})$ fraction to agents of $G_{2}$.
The agents of $G_{1}$ invest the rest $w\epsilon_{t}$ in risk-free investment
(returns with fixed rate of interest $r_{s}$).

Let $\tilde{w}$ be the accumulated wealth333These amounts could be random and
different from agent to agent, but with large networks (by law of large
numbers) one can approximate these to be constants. of any agent of $G_{2}$
out of which a positive fraction $\alpha$ is invested towards the other banks
of $G_{2}$ and $(1-\alpha)$ portion is invested in risky security. Thus the
accumulated wealth of a typical $G_{2}$ agent is governed by the following
equation,

$\tilde{w}=\underbrace{w+w\epsilon}_{\text{Initial wealth + Borrowed from
$G_{1}$}}+\underbrace{\tilde{w}\alpha}_{\text{Lend/borrow $G_{2}$}}\mbox{ and
thus }\tilde{w}=\frac{w(1+\epsilon)}{(1-\alpha)}.$ (1)

Thus the total investment towards the risky venture equals
$\tilde{w}(1-\alpha)=w(1+\epsilon)$. The $G_{2}$ agents have to settle their
liabilities at the end of the return/contract period (in each round) and this
would depend upon their returns from the risky investments. Thus the total
liability of any agent of $G_{2}$ is $y=(w\epsilon+\tilde{w}\alpha)(1+r_{b})$,
where $r_{b}$ is the borrowing rate444For simplicity of explanation, we are
considering constant terms to represent all these quantities, in reality they
would be i.i.d. quantities which are further independent of other rounds and
the asymptotic analysis would go through as in [1].; by simplifying

$y=\frac{w(\epsilon+\alpha)(1+r_{b})}{(1-\alpha)}.$

Similarly, any agent of $G_{2}$ lends the following amount to each of its
approachers (of $G_{2}$):

$\frac{\alpha{\tilde{w}}}{n(1-\epsilon_{t})p_{ss}}=\frac{\alpha
w(1+\epsilon)}{n(1-\epsilon_{t})p_{ss}(1-\alpha)}.$ (2)

Figure 1: Apportioning of $G_{1}$

Figure 2: Apportioning of $G_{2}$

Return and Settling phases, Clearing Vectors: We fix the round $t$ and avoid
notation $t$ for simpler notations. The agents of $G_{2}$ have to clear their
liabilities during this phase in every round. Recall the agents of $G_{2}$
invested $w(1+\epsilon)$ amount in risky-investments and the corresponding
random returns (after economic shocks) are:

$K_{i}=\begin{cases}w(1+\epsilon)(1+u)=:k_{u},&\text{w.p. (with probability)
}\ \delta\\\ w(1+\epsilon)(1+d)=:k_{d},&\text{otherwise}\end{cases}$ (3)

This is the well known binomial model, in which the upward moment occurs with
probability $\delta$ and downward moment with $(1-\delta).$ The agents have to
return $y$ (after the interest rate $r_{b}$) amount to their creditors,
however may not be able to manage the same because of the above economic
shocks. In case of default, the agents return the maximum possible; let
$X_{i}$ be the amount cleared by the $i^{th}$ agent of group $G_{2}$. Here we
consider a standard bankruptcy rule, limited liability and pro-rata basis
repayment of the debt contract (see [5, 4]), where the amounts returned are
proportional to their liability ratios. Thus node $j$ of $G_{2}$ pays back
$X_{i}L_{ji}(1+r_{b})/y$ towards node $i$, where $L_{ji}$ the amount borrowed
(liability) during initial investment phases equals (see the details of
previous subsection and equation (2)):

$L_{ji}=\begin{cases}I_{ji}\frac{w}{np_{ss}},&\text{if}\ i\in G_{1}\\\
I_{ji}\frac{\alpha w(1+\epsilon)}{np_{ss}(1-\alpha)(1-\epsilon)},&\text{if}\
i\in G_{2}.\end{cases}$ (4)

Thus the maximum amount cleared by any agent $j\in{\cal G}_{2}$, $X_{j}$, is
given by the following fixed point equation in terms of the clearing vector
$\\{X_{i}\\}_{i\in G_{2}}$ composed of clearing values of all the agents (see
[5, 4] etc):

$X_{i}=\min\left\\{\bigg{(}K_{i}+\sum_{j\in
G_{2}}X_{j}\frac{L_{ji}}{y}-v\bigg{)}^{+},\ y\right\\},$ (5)

with the following details: the term $K_{i}$ is the return of the risky
investment, the term $\sum_{j\in G_{2}}X_{j}\ L_{ji}/y$ equals the claims form
the other agents (those borrowed from agent $i$) and $v$ denotes the taxes to
pay. In other words, agent $i$ will pay back the (maximum possible) amount
$K_{i}+\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}-v$ in case of a default, and in
the other event, will exactly pay back the liability amount $y$.

Surplus of any agent is defined as the amount obtained from various
investments, after clearing all the liabilities. This represents the utility
of the agent in the given round. The surplus of the agent $i\in G_{2}$:

$R^{2}_{i}=\left(K_{i}+\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}-v-y\right)^{+},$
(6)

while that of agent $i\in G_{1}$ is given by:

$R^{1}_{i}=\left(w\epsilon(1+r_{s})+\sum_{j\in
G_{2}}X_{j}\frac{L_{ji}}{y}-v\right)^{+}.$ (7)

In the above, the first term is the return from the risk free investment. The
second term equals the returns or claims form $G_{2}$ agents (whom they lent)
and $v$ denotes the amount of taxes.

##  3 Asymptotic approximation of the large networks

We thus have dynamic graphs whose size increases with each round. In this
section, we obtain appropriate asymptotic analysis of these graphs/systems,
with an aim to derive the pay-off of each group after each round. Towards
this, we derive the (approximate) closed form expression of the equations (6)
and (7), which are nothing but the per-agent returns after the settlement of
the liabilities.

The returns of the agents depend upon how other agents settle their
liabilities to their connections/creditors. Thus our first step is to derive
the solution of the clearing vector fixed point equations (5). Observe that
the clearing vector $\\{X_{j}\\}_{j\in G_{2}}$ is the solution of the vector-
valued random fixed point equations (5) in $n$-dimensional space (where $n$ is
the size of the network), defined sample-path wise.

Clearing vectors using results of [1]: Our financial framework can be analysed
using the results of [1], as the details of the model match555Observe that
$\alpha(1+\epsilon)/(\alpha+\epsilon)<1$. the assumptions of the paper. By [1,
Theorem 1], the aggregate claims converge almost surely to constant values (as
the network size increases to infinity):

$\displaystyle\mbox{(claims of agents of $G_{1}$)},$
$\displaystyle\displaystyle\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}\to$
$\displaystyle\frac{(1-\alpha)(1-\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}\mbox{
a.s.},\mbox{ and }$ $\displaystyle\mbox{(claims of agents of $G_{2}$)},$
$\displaystyle\displaystyle\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}\to$
$\displaystyle\frac{\alpha(1+\epsilon)}{(\alpha+\epsilon)}{\bar{x}}^{\infty}\mbox{
a.s.},$

where the common expected clearing value ${\bar{x}}^{\infty}$ satisfies the
following fixed point equation in one-dimension:

${\bar{x}}^{\infty}=E\bigg{(}\min\left\\{K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v,y\right\\}\bigg{)}^{+}.$
(8)

Further by the same Theorem, the clearing vectors converge almost surely to
(asymptotically independent) random vectors:

$X_{i}\to\bigg{(}\min\left\\{K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v,y\right\\}\bigg{)}^{+}\mbox{,
for each }i\in G_{2}.$ (9)

By virtue of the above results, the random returns given by equations (6) and
(7), converge almost surely:

$\displaystyle R^{1}_{i}$ $\displaystyle\to$
$\displaystyle\left(w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{(\alpha+\epsilon)}{\bar{x}}^{\infty}-v\right)^{+},\mbox{
for each }i\in G_{1}$ (10) $\displaystyle R^{2}_{i}$ $\displaystyle\to$
$\displaystyle\left(K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\right)^{+},\mbox{
for each }i\in G_{2}.$ (11)

Probability of Default is defined as the fraction of agents of $G_{2}$ that
failed to pay back their full liability, i.e., $P_{d}:=P({X_{i}}<y)$. For
large networks (when the initial network size $n_{0}$ itself is sufficiently
large), one can use the above approximate expressions and using the same we
obtain the default probabilities and the aggregate clearing vectors in the
following (proof in Appendix).

###### Lemma 1

The asymptotic average clearing vector and the default probability of $G_{2}$
is given by :

$({\bar{x}}^{\infty},P_{d})=\begin{cases}(y,\hskip 68.2866pt0)&\text{if }\ \
c_{\epsilon}>\frac{y-\underline{w}}{y}\\\ \left(\frac{\delta
y+(1-\delta)\underline{w}}{1-(1-\delta)c_{\epsilon}},\hskip
11.38109pt1-\delta\right)&\text{if}\ \
\frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}<c_{\epsilon}<\frac{y-\underline{w}}{y}\\\
\left(\frac{k_{d}(1-\delta)+k_{u}\delta}{1-c_{\epsilon}},\ \ \ \ \ \
1\right)&\text{if }\ \
c_{\epsilon}<\frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}\end{cases}$
(12)

where, $c_{\epsilon}=\frac{\alpha+\alpha\epsilon}{\alpha+\epsilon}$,
$E(W)=\delta k_{u}+(1-\delta)k_{d}$ , $\underline{w}=k_{d}-v$ and
$\overline{w}=k_{u}-v$. $\blacksquare$

Expected Surplus: By virtue of the Theorem developed in [1, Theorem 1] we have
a significantly simplified limit system, whose performance is derived in the
above Lemma. We observe that this approximation is sufficiently close
(numerical simulations illustrate good approximations), and assume the
following as the pay-offs of each group after each round of the investments:

$\displaystyle\phi_{1}(\epsilon):=E(R^{1}_{i})=\bigg{(}w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v\bigg{)}^{+},\mbox{
for any agent of }G_{1}$
$\displaystyle\phi_{2}(\epsilon):=E(R^{2}_{i})=E\bigg{(}K_{i}+\frac{(1+\epsilon)\alpha}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\bigg{)}^{+},$
(13)
$\displaystyle=\bigg{(}k_{u}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\bigg{)}^{+}\delta+\bigg{(}k_{d}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\bigg{)}^{+}(1-\delta),$

for any agent of $G_{2}.$ Observe here that the aggregate limits are almost
sure constants, hence the expected surplus of all the agents of the same group
are equal, while the random returns of the same group are i.i.d. (independent
and identically distributed).

##  4 Analysis of Replicator dynamics

In every round of investments, we have a new network that represents the
liability structure of all the agents of that round formed by the investment
choices of the agents, and, in the previous two sections we computed the
(asymptotically approximate) expected returns/utilities of each agent of the
network. As already mentioned in Section 2, new agents join the network in
each round, and chose their strategies depending upon their observations of
these expected returns of the previous round.

These kind of dynamics is well described in literature by name replicator
dynamics (e.g.,[12, 7, 2] etc). The main purpose of such a study is to derive
asymptotic analysis and answer some or all of the following questions: will
the dynamics converge, i.e., would the relative fractions of various
populations settle as the number of rounds increase? will some of the
strategies disappear eventually? if more than one population type survives
what would be the asymptotic fractions? etc. These kind of analysis are common
in other types of networks (e.g., wireless networks (e.g., [12]), biological
networks ([2])), but are relatively less studied in the context of financial
networks (e.g., [7]). We are interested in knowing the asymptotic outcome of
these kind of dynamics (if there exists one) and study the influence of
various network parameters on the outcome. We begin with precise description
of the two types of dynamics considered in this paper.

###  4.1 Average Dynamics

The new agent contacts two random (uniformly sampled) agents of the previous
round. If both the contacted agents belong to the same group, the new agent
adapts the strategy of that group. When it contacts agents from both the
groups it investigates more before making a choice; the new agent observes
significant portion of the network, in that, it obtains a good estimate of the
average utility of agents belonging to both the groups. It adapts the strategy
of the group with maximum (estimated) average utility.

Say it observes the average of each group with an error that is normally
distributed with mean equal to the expected return of the group and variance
proportional to the size of the group, i.e., it observes (here ${\cal
N}(0,\sigma^{2})$ is a zero mean Gaussian random variable with variance
$\sigma^{2}$)

${\hat{\phi}}_{i}(\epsilon)=\phi_{i}(\epsilon)+{\cal N}_{i}\mbox{ with }{\cal
N}_{1}\sim{\cal N}\left(0,\frac{1}{{\bar{c}}\epsilon}\right)\mbox{ and }{\cal
N}_{2}\sim{\cal N}\left(0,\frac{1}{{\bar{c}}(1-\epsilon)}\right),$

for some ${\bar{c}}$ large. Observe by this modeling that: the expected values
of the observations are given by $(\phi_{1}(\epsilon),\ \phi_{2}(\epsilon))$
and are determined by the relative proportions of the two populations, while
the variance of any group reduces as its proportion increases to 1 and
increases as the proportion reduces to zero. We also assume that the
estimation errors $\\{{\cal N}_{1},{\cal N}_{2}\\}$ (conditioned on the
relative fraction, $\epsilon$) corresponding to the two groups are
independent. Then the probability that the new agent chooses strategy 1 is
given by

$Prob({\hat{\phi}}_{1}(\epsilon)-{\hat{\phi}}_{2}(\epsilon)>0)=Prob({\cal
N}_{2}-{\cal N}_{1}\leq\phi_{1}(\epsilon)-\phi_{2}(\epsilon)),$

which by (conditional) independence of Gaussian random variables equals666
because
$\frac{1}{\epsilon}+\frac{1}{1-\epsilon}=\frac{1}{\epsilon(1-\epsilon)}$

$\displaystyle
g(\epsilon):=\int_{-\infty}^{\left(\phi_{1}(\epsilon)-\phi_{2}(\epsilon)\right)\sqrt{{\bar{c}}\epsilon(1-\epsilon)}}e^{-x^{2}/2}\frac{dx}{\sqrt{2\pi}}.$
(14)

Let $(n_{1}(t),n_{2}(t))$ respectively represent the sizes of $G_{1}$ and
$G_{2}$ population after round $t$ and note that
$\epsilon_{t}=\frac{n_{1}(t)}{n_{1}(t)+n_{2}(t)}$. Then the system dynamics is
given by the following ($g(\cdot)$ given by (14)):

$\displaystyle(n_{1}(t+1),\ n_{2}(t+1))$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{llll}\big{(}n_{1}(t)+1,&n_{2}(t)\big{)}&\text{w.p.}\
\ \epsilon_{t}^{2}+2\epsilon_{t}(1-\epsilon_{t})g(\epsilon_{t})\\\
\big{(}n_{1}(t),&n_{2}(t)+1\big{)}&\text{w.p.}\ \
(1-\epsilon_{t})^{2}+2\epsilon_{t}(1-\epsilon_{t})(1-g(\epsilon_{t})).\\\
\end{array}\right.$ (17) (18)

It is clear that (with $\epsilon_{0}$ and $n_{0}$ representing the initial
quantities),

$\displaystyle\epsilon_{t+1}$ $\displaystyle=$
$\displaystyle\frac{n_{1}(t+1)}{t+n_{0}+1}=\frac{(t+n_{0})\epsilon_{t}+Y_{t+1}}{t+n_{0}+1}=\epsilon_{t}+\frac{1}{t+n_{0}+1}\left(Y_{t+1}-\epsilon_{t}\right)\mbox{
where }$ $\displaystyle Y_{t+1}$ $\displaystyle=$
$\displaystyle\begin{cases}1&\text{wp}\ \
\epsilon_{t}^{2}+2\epsilon_{t}(1-\epsilon_{t})g(\epsilon_{t})\\\ 0&\text{wp}\
\ (1-\epsilon_{t})^{2}+2\epsilon_{t}(1-\epsilon_{t})(1-g(\epsilon_{t}))\mbox{,
for all }t\geq 1.\end{cases}$

One can rewrite the update equations as

$\displaystyle\epsilon_{t+1}$ $\displaystyle=$
$\displaystyle\epsilon_{t}+\frac{1}{t+n_{0}+1}\left(h(\epsilon_{t})+M_{t+1}\right),\mbox{
with, }\ \ M_{t+1}\ :=\ Y_{t+1}-\epsilon_{t}-h(\epsilon_{t}),\mbox{ where, }$
$\displaystyle h(\epsilon)$ $\displaystyle:=$ $\displaystyle
E\big{[}Y_{t+1}-\epsilon_{t}|\epsilon_{t}=\epsilon\big{]}=\epsilon(1-\epsilon)\left(2g(\epsilon)-1\right)\mbox{
for any }0\leq\epsilon\leq 1.$

and observe that (with ${\cal F}_{t}$ the natural filtration of the process
till $t$)

$\displaystyle E[M_{t+1}|{\cal F}_{t}]$ $\displaystyle=$ $\displaystyle
E[M_{t+1}|\epsilon_{t}]=0\mbox{ and }\ E[M_{t+1}^{2}|{\cal F}_{t}]\ \leq\
C\mbox{ for some constant }C<\infty.$

Further observe that

$\hskip 19.91692pt0\leq\epsilon_{t}\leq 1\mbox{ for all }t\mbox{ and all
sample paths}.$

Thus our algorithm satisfies assumptions777The assumptions require that the
process is defined for the entire real line. One can easily achieve this by
letting $h(\epsilon)=0$ for all $\epsilon\notin[0,1]$, which still ensures
required Lipschitz continuity and by extending $M_{t+1}=0$ for all
$\epsilon_{t}\notin[0,1]$. A.1 to A.4 of [11] and hence we have using [11,
Theorem 2] that

###### Theorem 1

The sequence $\\{\epsilon_{t}\\}$ generated by average dynamics (18) converges
almost surely (a.s.) to a (possibly sample path dependent) compact connected
internally chain transitive invariant set of ODE:

$\dot{\epsilon_{t}}=h(\epsilon_{t}).\hskip 56.9055pt\mbox{
{\hfill$\blacksquare$} }$ (19)

The dynamics start with initial condition $\epsilon_{0}\in(0,1)$ and clearly
would remain inside the interval $[0,1]$, i.e., $\epsilon_{t}\in[0,1]$ for all
$t$ (and almost surely). Thus we consider the invariant sets of ODE (19)
within this interval for some interesting case studies in the following (Proof
in Appendix).

###### Corollary 1

Define ${\bar{r}}_{r}:=u\delta+d(1-\delta)$. And assume $w(1+d)>v$ and observe
that $u>r_{b}\geq r_{s}>d.$ Assume $\epsilon_{0}\in(0,1).$ Given the rest of
the parameters of the problem, there exists a ${\bar{\delta}}<1$ (depends upon
the instance of the problem) such that the following statements are valid for
all $\delta\geq{\bar{\delta}}$:

  1. (a)

If ${\bar{r}}_{r}>r_{b}>r_{s}$ then $\phi_{2}(\epsilon)>\phi_{1}(\epsilon)$
for all $\epsilon$, and $\epsilon_{t}\to 0$ almost surely.

  2. (b)

If $\phi_{1}(\epsilon)>\phi_{2}(\epsilon)$ for all $\epsilon$ then
$\epsilon_{t}\to 1$ almost surely.

  3. (c)

When $r_{b}>{\bar{r}}_{r}>r_{s}$, and case (b) is negated there exists a
unique zero $\epsilon^{*}$ of the equation
$\phi_{1}(\epsilon)-\phi_{2}(\epsilon)=0$ and

$\hskip 28.45274pt\epsilon_{t}\to\epsilon^{*}\mbox{ almost surely; further for
$\delta\approx 1$,
}\epsilon^{*}\approx\frac{r_{b}-{\bar{r}}_{r}}{{\bar{r}}_{r}-r_{s}}.\hskip
42.67912pt\mbox{{\hfill$\blacksquare$} }$

From (13) and Lemma 1, it is easy to verify that all the limit points are
evolutionary stable strategies (ESS). Thus the replicator dynamics either
settles to a pure strategy ESS or mixed ESS (in part (c) of the corollary),
depending upon the parameters of the network; after a large number of rounds,
either the fraction of agents following one of the strategies converges to one
or zero or the system reaches a mixed ESS which balances the expected returns
of the two groups.

In many scenarios, the expected rate of return of the risky investments is
much higher than the rate of interest related to lending/borrowing, i.e.,
${\bar{r}}_{r}>r_{b}$. Further the assumptions of the corollary are satisfied
by more or less all the scenarios (due to standard no-arbitrage assumptions)
and because the shocks are usually rare (i.e., $\delta$ is close to 1). Hence
by the above corollary, in majority of scenarios, the average dynamics
converges to a pure strategy with all ‘risky’ agents (i.e., $\epsilon_{t}\to
0$). The group $G_{1}$ gets wiped out and almost all agents invest in risky
ventures, as the expected rate of returns is more even in spite of large
economic shocks. One can observe a converse or a mixed ESS when the magnitude
of the shocks is large ($d$ too small) or when the shocks are too often to
make ${\bar{r}}_{r}<r_{b}$.

###  4.2 Random dynamics

When the new agent contacts two random agents of different groups, its choice
depends directly upon the returns of the two contacted agents. The rest of the
details remain the same as in average dynamics. In other words, the new agents
observe less, their investment choice is solely based on the (previous round)
returns of the two contacted agents. In this case the dynamics are governed by
the following (see (10)-(11)):

$\displaystyle(n_{1}(t+1),n_{2}(t+1))$ $\displaystyle=$
$\displaystyle\begin{cases}\left(n_{1}(t)+1,\hskip
28.45274ptn_{2}(t)\right)&\text{wp}\ \ \epsilon_{t}^{2}\\\
\left(n_{1}(t),\hskip 45.5244ptn_{2}(t)+1\right)&\text{wp}\ \
(1-\epsilon_{t})^{2}\\\ \left(n_{1}(t)+G(\epsilon_{t}),\ \
n_{2}(t)+(1-G(\epsilon_{t}))\right)&\text{else, with}\\\ \end{cases}$
$\displaystyle G(\epsilon_{t})$ $\displaystyle=$ $\displaystyle
1_{\\{R^{1}\geq R^{2}\\}}$ $\displaystyle=$ $\displaystyle
1_{\left\\{\left(w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{(\alpha+\epsilon)}{\bar{x}}^{\infty}-v\right)^{+}\geq\left(K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\right)^{+}\right\\}},$

where ${\bar{x}}^{\infty}={\bar{x}}^{\infty}(\epsilon_{t})$ is given by Lemma
1. Here we assume people prefer risk-free strategy under equality, one can
easily consider the other variants. Once again this can be rewritten as

$\displaystyle\epsilon_{t+1}=\epsilon_{t}+\frac{Z_{t+1}-\epsilon_{t}}{t+n_{0}+1}\mbox{
with }\ \ Z_{t+1}=\begin{cases}1&\text{wp}\ \ \epsilon_{t}^{2}\\\ 0&\text{wp}\
\ (1-\epsilon_{t})^{2}\\\ G(\epsilon_{t})&\text{else}.\end{cases}$ (21)

As in previous section the above algorithm satisfies assumptions888In the
current paper, we consider scenarios in which $h_{R}(\cdot)$ is Lipschitz
continuous, basically under the conditions of Corollary 2.  A.1 to A.4 of [11]
and once again using [11, Theorem 2], we have:

###### Theorem 2

The sequence $\\{\epsilon_{t}\\}$ generated by average dynamics (4.2)
converges almost surely (a.s.) to a (possibly sample path dependent) compact
connected internally chain transitive invariant set of ODE:

$\dot{\epsilon}(t)=h_{R}(\epsilon(t)),\
h_{R}(\epsilon):=E_{\epsilon}\big{[}Z_{t+1}-\epsilon_{t}|\epsilon_{t}=\epsilon\big{]}=\epsilon(1-\epsilon)(2E[G(\epsilon)]-1).\mbox{
{\hfill$\blacksquare$} }$ (22)

One can derive the analysis of this dynamics in a similar way as in average
dynamics, however there is an important difference between the two dynamics;
we can never have random dynamics converges to an intermediate attractor, like
the attractor in part (c) of Corollary 1 (unique $\epsilon^{*}$ satisfying
$\phi_{1}=\phi_{2}$). This is because
$E_{\epsilon}[G]=P(R^{1}(\epsilon)>R^{2}(\epsilon))$ equals $0$, $1-\delta$ or
1 and never $1/2$ (unless $\delta=1/2$, which is not a realistic case).
Nevertheless, we consider the invariant sets (corresponding to pure ESS)
within $[0,1]$ for some cases (Proof in Appendix):

###### Corollary 2

Assume $\epsilon_{0}\in(0,1).$ Given the rest of the parameters of the
problem, there exists a $1/2<\delta<1$ (depends upon the instance of the
problem) such that the following statements are valid:

  1. (a)

If $E_{\epsilon}[G]=0$ for all $\epsilon$ or $1-\delta$ for all $\epsilon$,
then $\epsilon_{t}\to 0$ almost surely.

  2. (b)

If $E_{\epsilon}[G]=1$ for all $\epsilon$, then $\epsilon_{t}\to 1$ almost
surely.

  3. (c)

When $w(1+d)>v$ and $u>r_{b}\geq r_{s}>d$, there exists a ${\bar{\delta}}<1$
such that for all $\delta\geq{\bar{\delta}}$, the default probability
$P_{d}\leq(1-\delta)$ and $E[G]=1-\delta$ and this is true for all $\epsilon$.
Hence by part (a), $\epsilon_{t}\to 0$ almost surely. $\blacksquare$

Remarks: Thus from part (c), under the conditions of Corollary 1, the random
dynamics always converges to all ‘risky’ agents (pure ESS), while the average
dynamics, as given by Corollary 1, either converges to pure or mixed ESS
further based on system parameters (mainly various rates of return).

From this partial analysis (corollaries are for large enough $\delta$) it
appears that one can never have mixed ESS with random dynamics, and this is a
big contrast to the average dynamics; when agents observe sparsely the network
eventually settles to one of the two strategies, and if they observe more
samples there is a possibility of emergence of mixed ESS that balances the two
returns. We observe similar things, even for $\delta$ as small as $0.8$ in
numerical simulations (Table 4). We are keen to understand this aspect in more
details as a part of the future work.

To summarize we have a financial network which grows with new additions, in
which the new agents adapt one of the two available strategies based on the
returns of the agents that they observed/interacted with. Our asymptotic
analysis of [1] was instrumental in deriving these results. This is just an
initial study of the topic. One can think of other varieties of dynamics, some
of which could be a part of our future work. The existing agents may change
their strategies depending upon their returns and observations. The agents
might leave the network if they have reduced returns repeatedly. The network
may adjust itself without new additions etc.

##  5 Numerical observations

We performed numerical simulations to validate our theory. We included Monte-
Carlo (MC) simulation based dynamics in which the clearing vectors are also
computed by directly solving the fixed point equations, for any given sample
path of shocks. Our theoretical observation well matches the MC based limits.

In Tables 4, 4 and 4 we tabulated the limits of the average dynamics for
various scenarios, and the observations match the results of Corollary 1. The
configuration used for Table 4 is:
$n_{0}=2000,\epsilon_{0}=0.75,r_{s}=0.18,r_{b}=0.19,w=100,v=46,\alpha=0.1$,
while that used for Table 4 is:
$n_{0}=2000,\epsilon_{0}=0.5,r_{s}=0.17,r_{b}=0.19,w=100,v=40,\alpha=0.1$. For
both these tables risky expected rate of returns ${\bar{r}}_{r}$ is smaller
than $r_{b}$ and the dynamics converges either to ‘all risky’ agents
configuration or to a mixed ESS. In Table 4, the risky expected rate of
returns ${\bar{r}}_{r}=.1250$ which is greater than $r_{b}$ and $r_{s}$, thus
the dynamics converges to all risky-agents, as indicated by Corollary 1.

$u$ | $d$ | $\delta$ | $\phi_{1}$ | $\phi_{2}$ | $\epsilon^{*}$
---|---|---|---|---|---
0.2 | -0.05 | 0.8 | 72 | 0 | 1
0.2 | -0.1 | 0.8 | 72 | 0 | 1
0.2 | -0.15 | 0.8 | 72 | 0 | 1
0.2 | -0.2 | 0.8 | 72 | 0 | 1
0.2 | -0.25 | 0.8 | 72 | 0 | 1
Table 1: When the shocks are too large along with larger taxes ($v=46$), the
average dynamics converges to a configuration with all ‘risk-free agents’!

$u$ | $d$ | $\delta$ | $\phi_{1}$ | $\phi_{2}$ | $\epsilon^{*}$
---|---|---|---|---|---
0.2 | -0.1 | 0.95 | 78.33 | 78.27 | 0.3326
0.2 | -0.11 | 0.95 | 78.24 | 78.31 | 0.3791
0.2 | -0.12 | 0.95 | 78.14 | 78.14 | 0.4288
0.2 | -0.13 | 0.95 | 78.04 | 78.04 | 0.4820
0.2 | -0.14 | 0.95 | 77.92 | 77.92 | 0.5385
Table 2:  Average dynamics converges to mixed ESS, at which both populations
survive with $\phi_{1}=\phi_{2}$,

$u$ | $d$ | $\delta$ | $\phi_{1}$ | $\phi_{2}$ | $\epsilon^{*}$
---|---|---|---|---|---
0.15 | -0.1 | 0.9 | 0 | 82.12 | 0
0.16 | -0.1 | 0.9 | 0 | 83.24 | 0
0.17 | -0.1 | 0.9 | 0 | 84.29 | 0
0.18 | -0.1 | 0.9 | 0 | 85.19 | 0
Table 3:  Average Dynamics converges to all ‘risky-agents’; Configuration:
$n_{0}=2000,\epsilon_{0}=.5,r_{s}=0.10,r_{b}=0.12,w=100,v=30,\alpha=0.5$

Config | $\epsilon^{*}$(Theory) |  $\epsilon^{\star}$(Monte Carlo)
---|---|---
($d,\delta,v$) | Avg | Rndm | Avg | Rndm
0.10, 0.95, 40 | 0 | 0 | .0016 | 0.0011
-0.10, 0.95, 40 | 0.33 | 0 | .3214 | 0.0004
-0.15, 0.95, 40 | 0.6 | 0 | .5988 | 0.0014
0.10, 0.80, 46 | 1 | 0 | .9896 | 0.0065
Table 4:  Average and Random dynamics, Comparison of MC results with theory
Configuration: $n_{0}=2000,u=0.2,r_{s}=0.17,r_{b}=0.19,w=100,\alpha=0.1$

In Table 4 we considered random dynamics as well as average dynamics. In
addition, we provided the Monte-Carlo based estimates. There is a good match
between the MC estimates and the theory. Further we have the following
observations: a) random dynamics always converge to a configuration with all
‘risky’ agents, as given by Corollary 2; b) when ${\bar{r}}_{r}>r_{b}$, the
average dynamics also converges to $\epsilon^{*}=0$ as suggested by Corollary
1; and c) when ${\bar{r}}_{r}<r_{b}$, the average dynamics converges to mixed
ESS or to a configuration with all ‘risk-free’ agents, again as given by
Corollary 1.

As the ‘risk increases’, i.e., as the amount of taxes increase and or as the
expected rate of return of risky investments ${\bar{r}}_{r}$ decreases, one
can observe that the average dynamics converges to all ‘risk-free’ agents
(last row of Table 4) thus averting systemic risk event (when there are large
number of defaults, $P_{d}$). While the random dynamics fails to do the same.
As predicted by theory (the configurations satisfying part (b) of Corollary
2), random dynamics might also succeed in averting the systemic risk event,
when the expected number of defaults is one for all $\epsilon>0.$ It is
trivial to verify that the configuration with $w(1+u)<v$, is one such example.
Thus, average dynamics is much more robust towards averting systemic risk
events.

##  6 Conclusions

We consider a financial network with a large number of agents. The agents are
interconnected via liability graphs. There are two types of agents, one group
lends to others and invests the rest in risk-free projects, while the second
group borrows/lends and invests the rest in risky ventures. Our study is
focused on analysing the emergence of these groups, when the new agents adapt
their strategies for the next investment round based on the returns of the
previous round. We considered two types of dynamics; in average dynamics the
new agents observe large sample of data before deciding their strategy, and in
random dynamics the decision is based on a small random sample.

We have the following important observations: a) when the expected rate of
return of the risky investments is higher (either when the shocks are rare or
when the shocks are not too large) than the risk-free rate, then ‘risk-free’
group wipes out eventually, almost all agents go for risky ventures; this is
true for both types of dynamics; b) when the expected rate of risky
investments is smaller, a mixed ESS can emerge with average dynamics while the
random dynamics always converges to all risky agents; at mixed ESS the
expected returns of both the groups are equal; more interestingly, when the
risky-expected rate is too small, the average dynamics converges to a
configuration with all risk-free agents.

In other words, in scenarios with possibility of a systemic risk event, i.e.,
when there is a possibility of the complete-system collapse (all agents
default), the average dynamics manages to wipe out completely the risky
agents; the random dynamics can fail to do the same. Thus when agents make
their choices rationally and after observing sufficient sample of the returns
of the previous round of investments, there is a possibility to avoid systemic
risk events. These are some initial results and we would like to investigate
further in future to make more affirmative statements in this direction.

## References

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## Appendix

Proof of Lemma 1 : We consider the following:
Case 1: First consider the case when downward shock can be absorbed , in this
case the clearing vector ${\bar{x}}^{\infty}=y\delta+y(1-\delta)=y$, default
probability is $P_{d}=0$. The region is true if the following condition is
meet i.e., if

$K_{d}-v+yc_{\epsilon}>y\implies c_{\epsilon}>\frac{y-\underline{w}}{y}$

Case 2: Consider the case with banks receive shock will default and the
corresponding average clearing vector
${\bar{x}}^{\infty}=y\delta+(\underline{w}+c_{\epsilon}{\bar{x}}^{\infty})(1-\delta)$
which simplifies to :

${\bar{x}}^{\infty}=\frac{y\delta+\underline{w}(1-\delta)}{1-c_{\epsilon}(1-\delta).}$

This region lasts if the following conditions hold to be true

$K_{d}-v+c_{\epsilon}{\bar{x}}^{\infty}<y,\mbox{ and
}K_{u}-v+c_{\epsilon}{\bar{x}}^{2\infty}>y.$

Substituting
${\bar{x}}^{\infty}=\frac{y\delta+\underline{w}(1-\delta)}{1-c_{\epsilon}(1-\delta)}$
we have,

$\hskip
159.33542pt\frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}<c_{\epsilon}<\frac{y-\underline{w}}{y}.$

Case 3: In this we first calculate ${\bar{x}}^{\infty}$ which is obtained by
solving following fixed point equation:

$\displaystyle{\bar{x}}^{\infty}$ $\displaystyle=$
$\displaystyle(K_{d}-v+c_{\epsilon}{\bar{x}}^{\infty})(1-\delta)+(K_{u}-v+c_{\epsilon}{\bar{x}}^{\infty})\delta\
\ \implies{\bar{x}}^{\infty}\ =\ \frac{EW}{1-c_{\epsilon}}.$

In this case the default probability is $P_{d}=1$. The regime satisfies if the
following hold

$\displaystyle K_{u}-v+c_{\epsilon}\frac{EW}{1-c_{\epsilon}}<y\ \ \implies
c_{\epsilon}<\frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}.\hskip
14.22636pt\mbox{ {\hfill$\blacksquare$} }$

Proof of Corollary 1: First consider the system with $\delta=1$, i.e., system
without shocks. From Lemma 1, $P_{D}\leq(1-\delta)$ for all $\epsilon$ because
(with $\delta=1$)

$y\left(c_{\epsilon}-\frac{y-{\bar{w}}}{y}\right)=w(1+\epsilon)(1+u)-v-w\epsilon(1+r_{b})=w(1+u)-v+w\epsilon(u-r_{b})\geq
w(1+u)-v,$

for all $\epsilon$ (the lower bound independent of $\epsilon$). Under these
assumptions, there exists ${\bar{\delta}}<1$ by continuity of the involved
functions such that

$y\left(c_{\epsilon}-\frac{y-{\bar{w}}}{y-(1-\delta)({\bar{w}}-{\underline{w}})}\right)>0\mbox{
for all }\delta\geq{\bar{\delta}}\mbox{ and for all }\epsilon.$

Thus from Lemma 1 ${\bar{x}}^{\infty}=y$ or ${\bar{x}}^{\infty}=\frac{\delta
y+(1-\delta)\underline{w}}{1-(1-\delta)c_{\epsilon}}$ for all such
$\delta\geq{\bar{\delta}}$. We would repeat a similar trick again, so assume
initially ${\bar{x}}^{\infty}=y$ for all $\epsilon$ and consider
$\delta\geq{\bar{\delta}}$. With this assumption we will have:

$\displaystyle R^{1}(\epsilon)$ $\displaystyle=$
$\displaystyle\left(w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{(\alpha+\epsilon)}y-v\right)^{+}$
$\displaystyle=$
$\displaystyle\left(w\epsilon(1+r_{s})+w(1-\epsilon)(1+r_{b})-v\right)^{+}$
$\displaystyle=$
$\displaystyle\left(w(1+r_{b})-v+w\epsilon(r_{s}-r_{b})\right)\mbox{, under
the given hypothesis, and }$ $\displaystyle R^{2}(\epsilon)$ $\displaystyle=$
$\displaystyle\left(K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}y-v-y\right)^{+}=\left(K_{i}-w\epsilon(1+r_{b})-v\right)^{+}$
(24) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{llll}R^{2}_{u}&\mbox{ w.p.
}\delta&\mbox{ where }R^{2}_{u}:=w(1+u)-v+w\epsilon(u-r_{b})\\\
\left(R^{2}_{d}\right)^{+}&\mbox{ w.p. }1-\delta&\mbox{ where
}R^{2}_{d}:=w(1+d)-v+w\epsilon(d-r_{b}).\end{array}\right.$ (27)

Note that $R^{2}_{u}\geq w(1+u)-v>0$ (for any $\epsilon$) under the given
hypothesis.
Proof of part (a): When ${\bar{r}}_{r}>r_{b}$, from (13), it is clear that
(inequality only when $R^{2}_{d}$ is negative)

$\displaystyle\phi_{2}(\epsilon)-\phi_{1}(\epsilon)\geq
R^{2}_{u}\delta+R^{2}_{d}(1-\delta)-\phi_{1}(\epsilon)=w({\bar{r}}_{r}-r_{b})+w\epsilon({\bar{r}}_{r}-r_{s})>0.$

Thus in this case $\phi_{2}>\phi_{1}$ for all $\epsilon$ and hence

$g(\epsilon)<1/2\mbox{ and }2g(\epsilon)-1<0\mbox{ for all $0<\epsilon<1$. }$

Therefore with Lyaponuv function $V_{0}(\epsilon)=\epsilon/(1-\epsilon)$ on
defined on neighbourhood $[0,1)$ of $0$ (in relative topology on $[0,1]$) we
observe that

$\frac{dV_{0}}{d\epsilon}h(\epsilon)=\frac{\epsilon}{1-\epsilon}(2g(\epsilon)-1)<0\mbox{
for all }0<\epsilon<1\mbox{ and equals }0\mbox{ for }\epsilon=0.$

Further $V_{0}(\epsilon)\to\infty$ as $\epsilon\to 1$, the boundary point of
$[0,1)$. Thus $\epsilon^{*}=0$ is the asymptotically stable attractor of ODE
(19) (see [11, Appendix, pp.148]) and hence the result follows by Theorem 2.

For all $\delta\geq{\bar{\delta}}$, from Lemma 1, we have the following

$\displaystyle\sup_{\epsilon}|y-{\bar{x}}^{\infty}|=\sup_{\epsilon}(1-\delta)\bigg{|}\frac{y-c_{\epsilon}-\underline{w}}{1-(1-\delta)c_{\epsilon}}\bigg{|}<\frac{1-\delta}{\delta}\eta$
(28)

for some $\eta>0$ , which decreases to 0 as $\delta\to 1$. ( The last
inequality is due to $c_{\epsilon}<1$ and then taking supremum over
$\epsilon$). By continuity of the above upper bound with respect to $\delta$
and the subsequent functions considered in the above parts of the proof, there
exists a ${\bar{\delta}}<1$ (further big if required) such that all the above
arguments are true for all $\delta>{\bar{\delta}}$.

Proof of part (b): The proof follows in similar way, now using Lyaponuv
function $V_{1}(\epsilon)=(1-\epsilon)/\epsilon$ on neighbourhood $(0,1]$ of
1, and by observing that $g(\epsilon)>1/2$ for all $\epsilon<1$ and hence

$\frac{dV_{1}}{d\epsilon}h(\epsilon)=-\frac{1-\epsilon}{\epsilon}(2g(\epsilon)-1)<0\mbox{
for all }0<\epsilon<1\mbox{ and equals }0\mbox{ for }\epsilon=1.$

Proof of part (c): It is clear that $\phi_{1}(\epsilon)=R^{1}(\epsilon)$
decreases linearly as $\epsilon$ increases:

$\phi_{1}(\epsilon)=w(1+r_{b})-v+w\epsilon(r_{s}-r_{b}).$

For $\epsilon$ in the neighbourhood of $0$, $\phi_{2}(\epsilon)>0$ and is
decreasing linearly with slope ${\bar{r}}_{r}-r_{b}$, because
$R^{2}_{d}(0)=w(1+d)-v>0$ and thus for such $\epsilon$

$\phi_{2}(\epsilon)=w(1+{\bar{r}}_{r})-v+w\epsilon({\bar{r}}_{r}-r_{b}).$

From (24), $R^{2}_{d}({\epsilon})$ is decreasing with increase in $\epsilon.$
There is a possibility of an ${\bar{\epsilon}}$ that satisfies
$R^{2}_{d}({\bar{\epsilon}})=0$, in which case $\phi_{2}$ increases linearly
with slope $\delta w(u-r_{b})$, i.e.,

$\phi_{2}(\epsilon)=\delta\left[w(1+u)-v+w\epsilon(u-r_{b})\right]\mbox{ for
all }\epsilon\geq{\bar{\epsilon}}.$

When ${\bar{r}}_{r}<r_{b}$ we have,

$\hskip 68.2866pt\phi_{1}(0)=w(1+r_{b})-v>w(1+{\bar{r}}_{r})-v=\phi_{2}(0).$

By hypothesis $\phi_{1}(\epsilon)<\phi_{2}(\epsilon)$ for some $\epsilon$,
hence by intermediate value theorem there exists at least one $\epsilon^{*}$
that satisfies $\phi_{1}(\epsilon^{*})=\phi_{2}(\epsilon^{*}).$ Further the
zero is unique because $\phi_{2}$ is either linear or piece-wise linear (with
different slops), while $\phi_{1}$ is linear.

Consider Lyaponuv function
$V_{*}(\epsilon):=(\epsilon-\epsilon^{*})^{2}/(\epsilon(1-\epsilon))$ on
neighbourhood $(0,1)$ of $\epsilon^{*}$, note $V_{*}(\epsilon)\to\infty$ as
$\epsilon\to 0$ or $\epsilon\to 1$ and observe by (piecewise) linearity of the
functions we will have

$\displaystyle\phi_{1}(\epsilon)$ $\displaystyle>$
$\displaystyle\phi_{2}(\epsilon)\mbox{ and thus }(2g(\epsilon)-1)>0\mbox{ for
all }0<\epsilon<\epsilon^{*}\mbox{ and }$ $\displaystyle\phi_{2}(\epsilon)$
$\displaystyle>$ $\displaystyle\phi_{1}(\epsilon)\mbox{ and thus
}(2g(\epsilon)-1)<0\mbox{ for all }1>\epsilon>\epsilon^{*}.$

Thus we have999When $\epsilon<1/2$ and $\epsilon<\epsilon^{*}$ then clearly
$\frac{(\epsilon-\epsilon^{*})(2\epsilon-1)}{\epsilon(1-\epsilon)}>0$. When
$\epsilon>1/2$ we have $(2\epsilon-1)/\epsilon<1/2$ and with
$\epsilon<\epsilon^{*}$ we have $\epsilon^{*}-\epsilon<1-\epsilon$ and thus
$2+\frac{(\epsilon-\epsilon^{*})(2\epsilon-1)}{\epsilon(1-\epsilon)}\geq
3/2>0\mbox{ for all }\epsilon<\epsilon^{*}.$ In a similar way
$\epsilon>\epsilon^{*}$, then we will have that the above term is again
positive. ,

$\frac{dV_{*}}{d\epsilon}=2\frac{\epsilon-\epsilon^{*}}{\epsilon(1-\epsilon)}+\frac{(\epsilon-\epsilon^{*})^{2}(2\epsilon-1)}{\epsilon^{2}(1-\epsilon)^{2}}\mbox{
and hence }$
$\frac{dV_{*}}{d\epsilon}h(\epsilon)=(\epsilon-\epsilon^{*})\left(2+\frac{(\epsilon-\epsilon^{*})(2\epsilon-1)}{\epsilon(1-\epsilon)}\right)(2g(\epsilon)-1)<0\mbox{
for all }\epsilon\notin\\{0,1,\epsilon^{*}\\}.$

Thus $\epsilon^{*}$ is the asymptotically stable attractor of ODE (19) and
hence the result follows by Theorem 2. The result can be extended for
$\delta<1$ as in case (a) and the rest of the details follow by direct
verification (at $\delta=1$), i.e., by showing that
$\phi_{1}(\epsilon^{*})=\phi_{2}(\epsilon^{*})$ at $\delta=1$ and the equality
is satisfied approximately in the neighbourhood of $\delta=1$. $\blacksquare$

Proof of Corollary 2: For part (a),
$h_{R}(\epsilon)=-c_{G}\epsilon(1-\epsilon)$, where the constant $c_{G}=1$ (or
respectively $c_{G}=2\delta-1$). Using Lyanponuv function of part (a) of
Corollary 1, the proof follows in exactly the same lines.

For part (b), $h_{R}(\epsilon)=\epsilon(1-\epsilon)$, and proof follows as in
part (b) of Corollary 1. For part (c), first observe (using equations
(Appendix)-(24) of proof of Corollary 1)

$\displaystyle R^{2}_{u}(\epsilon)-R^{1}(\epsilon)$ $\displaystyle\geq$
$\displaystyle
w(1+u)+w\epsilon(u-r_{s})-y+{\bar{x}}^{\infty}\bigg{(}\frac{2\alpha+\epsilon-1}{\alpha+\epsilon}\bigg{)}$
$\displaystyle=$ $\displaystyle
w(1+u)+w\epsilon(u-r_{s})+({\bar{x}}^{\infty}-y)-{\bar{x}}^{\infty}\bigg{(}\frac{1-\alpha}{\alpha+\epsilon}\bigg{)}$
$\displaystyle=$ $\displaystyle
w(u-r_{b})+w\epsilon(u-r_{s})+({\bar{x}}^{\infty}-y)\bigg{(}1-\frac{1-\alpha}{\alpha+\epsilon}\bigg{)}>0.$

The last inequality is trivially true for $\delta=1$ (and so
${\bar{x}}^{\infty}=y$) for the given hypothesis, and then by continuity as in
proof of Corollary 1, one can consider $\bar{\delta}<1$ such that for all
$\delta\geq\bar{\delta}$, the term
$({\bar{x}}^{\infty}-y)\big{(}1-\frac{1-\alpha}{\alpha+\epsilon}\big{)}$
(uniformly over $\epsilon$) can be made arbitrarily small. When $P_{d}=0$,
i.e., ${\bar{x}}^{\infty}=y$ for some $\epsilon$, then
$R^{2}_{d}(\epsilon)-R^{1}(\epsilon)=w(d-r_{b})+w\epsilon(d-r_{s})<0$ for all
such $\epsilon$. When $P_{d}\neq 0$, then $R^{2}_{d}=0\leq R^{1}$. Thus in
either case $R^{2}_{d}(\epsilon)\leq R^{1}(\epsilon)$ for all $\epsilon$.

By virtue of the above arguments we have $P_{D}\leq(1-\delta)$ and
$E[G]=1-\delta$ and this is true for all $\epsilon$, for all
$\delta\geq{\bar{\delta}}$. The rest of the proof follows from part(a).
$\blacksquare$