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arXiv:1006.5044v1 [q-fin.TR] 25 Jun 2010

Modelling savings behavior of agents in the kinetic exchange models of market

Anindya S. Chakrabarti

Indian Statistical Institute, 203 B. T. Road, Kolkata-700108

(Dated: June 28, 2010)

Kinetic exchange models have been successful in explaining the shape of the income/wealth dis-

tribution in the economies. However, such models usually make some ad-hoc assumptions when it

comes to determining the savings factor. Here, we examine a few models in and out of the domain

of standard neo-classical economics to explain the savings behavior of the agents. A number of

new results are derived and the rest conform with those obtained earlier. Connections are estab-

lished between the reinforcement choice and strategic choice models with the usual kinetic exchange

models.

PACS numbers: 89.65 Gh Economics, Econophysics- 05.20 Kinetic theory- 05.65.+b Self-organized systems

I.INTRODUCTION

The distributions of income and wealth have long been

found to possess some robust and stable features inde-

pendent of the specific economic, social and political con-

ditions of the economies. Traditionally, the economists

have preferred to model the left tail and the mode of the

distributions of the workers’ incomes with a log-normal

distribution and the heavier right tail with a Pareto dis-

tribution. For a detailed survey of the distributions used

to fit the income and wealth data see Ref. [1]. However,

there have been several studies recently that argue that

the left tail and the mode of the distribution fit well with

the gamma distribution and the right tail of the distribu-

tion follows a power law [2–4]. It has been argued that

this feature might be considered to be a natural law for

economics [2, 5].

The Chakraborti-Chakrabarti [6] model (CC model

henceforth) can explain the gamma-like distribution

very well whereas the Chatterjee-Chakrabarti-Manna [7]

model (CCM model henceforth) explains the origin of

the power law. Both models fall in the category of the ki-

netic exchange models. However, the first model assumes

a constant savings propensity and the second model as-

sumes an uniformly distributed savings propensity of the

agents. It may be noted that the distribution of the sav-

ings factor is exogenous in these models. It is not derived

from any optimization on the agent’s part or by some

other mechanism.

Ref.[8] considers an exchange economy populated

with agents having a particular type of utility function

and derives the CC model in the settings of a competitive

market. Here, we show that the same methodology could

be applied to derive the CCM model and we can explain

the savings factor accordingly. A further possibility is

also studied where the constancy of the savings propen-

sity over time is relaxed. More specifically, we examine

the cases where the savings propensity is dependent on

the current money holding of the corresponding agent.

[16] email-id : aschakrabarti@gmail.com

This model, as we shall see, shows self-organization and

in some cases, it gives rise to bimodality in the money

distribution.

It is well known the models of utility maximization has

been severely critisized on the grounds of limitations of

computational capability of human beings [9]. Hence,

to model the savings behavior of the agents, we study

some simple thumb-rules and derive the distributions of

savings therefrom and finally the income/wealth distri-

bution. In particular, we consider the savings propensity

as a strategy variable to the agents. Two cases are ex-

plored here. In the first case, the agents take their savings

decisions of their own by reinforcing their choices. In the

second case, the agents adopt the winning strategy. In

all cases, we study the final money distributions.

The plan of this paper is as follows. In section, II we

explain the savings behavior of the agents using argu-

ments from neoclassical economics. In the next section,

we study a self-organizing market where the savings is a

function of the current money holding of the correspond-

ing agent. In section V and VI, we study the savings

behavior of the agents where the savings decision follows

some very simple thumb-rules. Then follows a summary

and discussion.

II.KINETIC EXCHANGE MODELS IN A

COMPETITIVE MARKET

By competitive market we mean a market with atomistic

agents who trade with each other knowing that their in-

dividual actions can not possibly influence the market

outcome (that implies there is no strategic interaction be-

tween the agents). We assume that markets are always

cleared by equating supply and demand; that is the mar-

ket is completely free of any friction. Below, we elaborate

on the market structure and the behavior of the agents

more explicitly.

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A.The Chakraborti-Chakrabarti (CC) Model

This model considers homogenous agents characterized

by a single savings propensity.

derivation of the model following Ref. [8]. The structure

of the economy is the following. It is an exchange econ-

omy populated with N agents each producing a single

perishable commodity. There is complete specialization

in production which means none of the agents produce

the commodity produced by another. Money is not pro-

duced in the economy. All agents are endowed with a

certain amount of money at the very begining of all trad-

ings. Money can be treated as a non-perishable com-

modity which facilitates transactions. All commodities

along with money can enter the utility function of any

agent as arguments. These agents care for their future

consumptions and hence they care about their savings

in the current period as well. It is natural that with

successive tradings their money-holding will change with

time. At each time step, two agents are chosen at ran-

dom to carry out transactions among themselves compet-

itively. We also assume that the preference structure of

the agents are time-dependent that is the parameters of

the utility functions vary over time (Ref. [10, 11]). For

a detailed discussion on the derivation of the resulting

money-transfer equations, see Ref. [8]. Below, we pro-

vide the formal structure and the solution to the model

only.

Let us assume that at time t, agent i and j have been

chosen. Also, assume that agent i produces Qiamount

of commodity i only and agent j produces Qj amount

of commodity j only and the amounts of money in their

possession at time t are mi(t) and mj(t) respectively (for

simplicity, mk(0) = 1 for k=1,2). Notice that the notion

of complete specialization in production process provides

the agents with a reason for trading with each other.

Naturally, at each time step there would be a net transfer

of money from one agent to the other due to trade. Our

focus is on how the amounts money held by the agents

change over time due to the repetition of such a trading

process. For notational convenience, we denote mk(t+1)

as mkand mk(t) as Mk(for k = 1,2).

Utility functions are defined as follows.

i,Ui(xi,xj,mi)=xαi

Uj(yi,yj,mj) = yαi

jwhere the arguments in both

of the utility functions are consumption of the first (i.e.,

xiand yi) and second good (i.e., xjand yj) and amount

of money in their possession respectively. For simplic-

ity, we assume that the sum of the powers is normal-

ized to 1 i.e., α1+ α2+ λ = 1.

ity prices to be determined in the market be denoted

by pi and pj. Now, we can define the budget con-

straints as follows. For agent i the budget constraint

is pixi+ pjxj+ mi≤ Mi+ piQiand similarly, for agent

j the constraint is piyi+pjyj+mj≤ Mj+pjQj. In this

set-up, we get the market clearing price vector (ˆ pi, ˆ pj) as

ˆ pk= (αk/λ)(Mi+ Mj)/Qkfor k = 1, 2.

By substituting the demand functions of xk, ykand pkfor

We briefly review the

For agent

for agent j,

ixαj

jmλ

1

and

iyαj

jmλ

Let the commod-

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5

m

2 2.5 3

P(m)

FIG. 1:

The usual CC model: money distribution among the agents.

Four cases are shown above, viz., λ = 0 (+), λ = 0.2 (×),

λ = 0.5 ( ∗), λ = 0.8 (?). All simulations are done for

O(106) time steps with 100 agents and averaged over O(103)

time steps.

k = 1, 2 in the money demand functions, we get the most

important equation of money exchange in this model. To

get the final result, we substitute αi/(αi+αj) by ǫ to get

the money evolution equations as

mi(t + 1) = λmi(t) + ǫ(1 − λ)(mi(t) + mj(t))

mj(t + 1) = λmj(t) + (1 − ǫ)(1 − λ)(mi(t) + mj(t))

(1)

where mk(t) ≡ Mk and mk(t + 1) ≡ mk (for k= i, j).

Note that for a fixed value of λ, if αiis a random variable

with uniform distribution over the domain [0,1−λ], then

ǫ is also uniformly distributed over the domain [0,1]. For

the limiting value of λ in the utility function (i.e., λ →

0), we get the money transfer equation describing the

random sharing of money without savings.

Interpretation of λ: Here, it is clearly shown that λ

in the CC model is nothing but the power of money in the

utility function of the agents and finally this turns out to

be the fraction of money holding that remains unaffected

by the trading action. However, in this form it can not

be directly interpreted as the propensity to save. Below,

we try to derive λ from an utlity maximization problem

while retaining the kinetic exchange structure and we

show that in this slightly alternative formulation, λ is

indeed the savings propensity as has been postulated.

B.The Chatterjee-Chakrabarti-Manna (CCM)

Model

As is clear from above, in the CC model the savings deci-

sion, the market clearence, the prices are all determined

at the same instant. But the savings decision is usually

made in separation. More specifically, we can model the

savings decision and the market clearence distinctly. The

CCM model takes into account the heterogeneity of the

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agents. In particular, it assumes that it is the savings

propensity of the agents which differs from each other.

To derive the same, we assume that the agents take de-

cisions in two steps. First, they decide how much to save

and in the second step, they go to the market with the

rest of the money and take the trading decisions.

Formally, we can analyze a typical agent’s behavior at

any time step t in the following two steps.

(i) Each agent’s problem is to make the decision re-

garding how much to save. For simplicity, we as-

sume that the utility function is of Cobb-Douglas

type. Briefly, at time t the i−th agent’s prob-

lem is to maximize U(ft,ct) = fλi

to ft/(1 + r) + ct = m(t) where f is the amount

of money kept for future consumption, c is the

amount of money to be used for current consump-

tion, m(t) is the amount of money holding at time

t and r is the interest rate prevailing in the market

which can be assumed to be zero in a conservative

framework. This is a standard utility maximization

problem and solving it by Lagrange multiplier, we

get the optimal allocation for the i−th agent as

f∗

t= λim(t) and c∗

decision is independent of what other agents are do-

ing. So now the agents will go to the market with

(1 − λi)m(t) only.

tc(1−λi)

t

subject

t= (1 − λi)m(t). Clearly, this

(ii) Now that each agent has made the savings deci-

sion, they can engage in competitive trade with

each other in the fashion descibed in subsection

IIA with λ → 0 (but λ ?= 0; it is a mathemati-

cal requirement). Note that the amount of money

used by the i-th agent is c∗

The resultant asset exchange equations are those

given by the CCM model [7].

t= (1 − λi)m(t) only.

mi(t + 1) = λim1(t) + ǫ[(1 − λi)mi(t) +

(1 − λj)mj(t)]

mj(t + 1) = λjm2(t) + (1 − ǫ)[(1 − λi)mi(t) +

(1 − λj)mj(t)]

(2)

Interpretation of λi: First we recall the solution of

the savings decision which is f∗

implies

t= λim(t). Note that it

λi=

f∗

t

m(t)

that is λi is nothing but the proportion of money kept

for future usage to the current money holding and this is

by definition the savings propensity.

III.

λ AS A FUNCTION OF MONEY: A

SELF-ORGANIZING ECONOMY

A distinct possibility is that the savings propensity

is a function of money-holding itself. To examine

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

10

100

1 10 100

P(m)

m

FIG. 2:

The usual CCM model: money distribution among the

agents. All simulations are done for O(106) time steps with

100 agents and averaged over O(103) time steps. The

straight line is a guide for a power law with slope -2.

that case, we need λ:[0,∞)→[0,1].

are two possibilities.

an increasing or decreasing function of money-holding.

The simplest forms that we may assume are the following.

However, there

The savings propensity can be

(i) λt = c1e−(c2m(t))with c1 <1: Savings propensity

is a decreasing function of money holding. The re-

striction on c1 does not allow any agent to have

savings propensity equals to 1. The system shows

self-organization and assumes a stable probability

density function in the steady state (See Fig. 3

where, for purpose of illustration, c1has been kept

constant at 0.95). It is seen numerically that as

c2increases the distribution converges to an expo-

nential density function. In the other extreme, it

tends to the CC model with λ = c1. For moder-

ate values of c2, the distribution resembles gamma

function. For other values of c1 also, the system

behaves similarly. Note that since c1 is the max-

imum possible savings propensity, for a very low

value of it, the system becomes indistinguishable

from an exponential distribution. While it seems

counter-intuitive that savings propensity falls with

the money holding, this might in fact be possible

since poorer people can not take any chance to gam-

ble whereas richer people can.

(ii) λt= c1(1−e−(c2m(t))) with c1< 1: Savings propen-

sity is an increasing function of money holding. The

economy again organizes itself and the distribution

of money becomes stable over time. However, there

is something more. It is seen numerically that bi-

modality may apper spontaneously in the density

function of money. See Fig. 4. Ref. [12] discusses

such bimodal distribution of wealth (or money).

There a mixture of the agents was used where two

classes of agents were characterized by two different

and widely separated (but fixed!) savings propen-

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0

0.5

1

1.5

2

2.5

0 0.5 1

m

1.5 2

P(m)

FIG. 3:

Savings propensity is negatively related to the level of

money. All simulations are done for O(105) time steps with

100 agents and averaged over O(103) time steps. Here, c1

has been kept constant at 0.95 and c2 has been changed.

The plots include c2= 0.1 (+), 0.5 (×), 1 ( ∗), 2 (?), 4 (?).

It is seen that as c2 increases, the distribution becomes more

and more skewed finally converging to an exponential

density function.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2

m

2.5 3 3.5 4

P(m)

FIG. 4:

Savings propensity is positively related to the level of

money. All simulations are done for O(106) time steps with

100 agents and averaged over O(103) time steps. Here,

c1=0.95 and the curves are plotted for c2= 1 (+), 2 (×), 3

( ∗) and 4 (?). Bimodality is clearly seen in the distribution

of money.

sities. Such a segregated population gave rise to

bimodal distributions. However, such segregations

are exogenous since the λs are given from outside

the system. Here, however, we have a new model

in which savings decision is completely endogenous

and the economy organizes itself in such a way that

it gives rise to a class of bimodal distributions. It

may be noted that bimodality in the income/wealth

distribution has actually been observed in many

cases (see e.g., [13]).

It is seen numerically that bimodality appears for

c1 ≥ 0.92 and c2 ≥ 1. Another interesting fea-

ture of this model is that keeping c1 constant as

c2 increases, the monomodal distribution breaks

into a bimodal distribution which again becomes

a monomodal distribution for even larger values of

c2. For example, consider Fig.

maximum value of c2 considered is 4. But as c2

increases further, the distribution again becomes

monomodal. However, it should also be mentioned

that if c1 is too large (for example, if c1 ≥ 0.97),

then the system produces some strange-looking bi-

modal distributions.

4 in which the

IV. ‘IRRATIONAL’ DECISION MAKING

The standard economic paradigm of market clearence via

utility maximization has been criticised on the grounds of

limitations of computational capability of human beings

[9]. The main challenge is to derive the homogeneity in

savings behavior of the agents from a very simple thumb-

rule such that the final distribution of income/wealth

looks realistic. A few realistic components of decision-

making are noted in Ref. [14, 15]

(i) Players develop prefernces for choices associated

with better outcomes even though the association

may be coincident, causally spurious, or supersti-

tious.

(ii) Decisions are driven by the two simultaneous and

distinct mechanisms of reward and punishment,

which are known to operate ubiquitously in hu-

mans.

(iii) Satisficing or persisting in a strategy that yields a

positive but not optimal outcome, is common and

indicates a mechanism of reinforcement rather than

optimization.

Of particular interest is item (iii) which goes directly

against the derivations stated above (see Section II).

V.FROM ONE TO MANY ...

To model the savings behavior of the agents, we now

make the following assumptions.

(i) The agents do not perform static optimization.

(ii) There is reinforcement in their decision-making

process.

(iii) The agents look for better payoffs. But eventually

each of them converges to a single and simple strat-

egy or thumb rule.

To incorporate the three above-mentioned assumptions,

we model the agents’ saving behavior by Polya’s urn pro-

cess [14]. The model is as follows. Consider the i−th

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0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4

savings propensity

0.5 0.6 0.7 0.8 0.9 1

P(savings propensity)

B

C

A

D

FIG. 5:

The savings distribution by reinforcement. Four cases are

shown. Case A: a=1, b=1. Case B: a=2, b=2. Case C: a=4,

b=4. Case D: a=4, b=2. The solid and the dotted lines

show the theoretical results (the β(a,b) distributions).

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10

m

100

P(m)

A

B

FIG. 6:

The steady state money distributions for two cases:- (A)

a=1, b=1 and (B) a=4, b=2. Case (A) produces the same

distribution of money as the CCM model (slope -2 in log-log

plot) whereas in case (B) we get a distribution with slope -3

in the log-log plot. The lines drawn have slopes -2 and -3

respectively. All simulations are done for O(106) time steps

with 100 agents and averaged over O(103) time steps.

agent. The choice is binary, he can take any of the two

decisions, to consume (c) or to save (s). His strategies

(c and s) such that c,s = 0,1 and c + s = 1. At each

instant, he chooses the values for c and s. Define Ct(St)

as the number of times c (s) has been assigned a value of

unity in t time periods. The savings propensity at time t

(that is λt) is defined as the ratio of Stto (St+Ct). We

can assume that initially S0= a and C0= b. The rein-

forcement mechanism is incorporated by assuming that

the probability of choosing s = 1 at any time t + 1, is

simply λt. Basically, it is the Polya’s urn model and the

famous result that follows from it is the following (Ref.

[14]). The random variable λtconverge almost surely to

a limit λ. The distribution of λ is β(a,b). (See Fig. 5 ).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

m

2.5 3 3.5 4

P(m)

A

B

FIG. 7:

The steady state money distributions for two cases:- (A)

a = b; a, b→ ∞ and (B) a=4, b=4. Note that case (A) is

identical to CC model with λ = 0.5. All simulations are

done for O(105) time steps with 100 agents and averaged

over O(102) time steps.

A.Two limits

(i) (a=b=1): From the above result, λtconverges to λ

where λ ∼ uni[0,1]. The resultant distribution of

money follows a power law. This is the basic CCM

model. See Fig. 6.

(ii) (a=b; a,b→ ∞): λt converges to λ where λ is a

delta function at 0.5. This corresponds to a special

case of the CC model where λ = 0.5. See Fig. 7.

B. For moderate values of a and b

Clearly, (1<a, b<∞): This model gives the gamma-

like part as well as the Pareto tail of the income/money

distribution for different values of a and b. For example,

we show the results of two cases.

(i) (a=4, b=2): The resulting distribution of savings

propensity is clearly β(4,2). The distribution of

money in the steady state follows a power law with

a slope -3 in the log-log plot. See Fig. 6.

(ii) (a=4, b=4): The resulting distribution of savings

propensity is β(4,4). The distribution of money in

the steady state is gamma function-like. See Fig.

7.

VI. FROM MANY TO ONE ...

In Section V, we have discussed how one can derive a

set of distributed savings propensities starting from a

unique value. Here, we discuss the reverse side of the

same coin. We shall show that the agents with different

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savings propensities, can converge to a single value over

time.

To model this situation, we assume that the agents treat

savings propensity as a strategy which evolves over time.

A very simple rule of evolution is the following. The

winner in any trade retains his strategy whereas the loser

adopts the winner’s strategy. Note that by winning in a

trading action, we simply imply that the agent who gets

the lion’s share in that particular trading is the winner.

Since it is a relative term, (by refering to Eqn.2) winning

is determined by the value of the stochastic term ǫ. If

ǫ ≥ 0.5, then the i-th agent wins (in Eqn. 2) else the

j-th agent wins. Note that the most important support

of this type of strategy evolution comes from the third

observation by Flache and Macy [15] noted in Section IV.

Let us assume that the possible saving propensities are

finite and denoted by λ1, λ2, ..., λketc. Also, let us de-

note the number of agents with λisavings propensity at

time t by ni(t) (for i= 1, ..., k). At each time-period two

of the agents are randomly selected and they trade ac-

cording to Eqn. 2 and then the loser adopts the winner’s

savings propensity. This process is repeated untill the

sytem reaches a steady state in terms of savings propen-

sities. After the whole system becomes steady with the

agents with a fixed saving propensity, the system is al-

lowed to evolve further to reach a steady state in terms

of money.

A.Convergence in savings propensity

Let

k

?

i=1

ni(0) = N,(3)

as the total number of agents remains fixed over time

(recall that ni(t) has been defined above as the number

of agents with a particular savings propensity λi). The

agents only shift from one savings propoensity to another

over time. Note that at each (trading) time point, the

number of agents with a particular savings propensity

rises or falls by unity with equal probability (i.e. depend-

ing on whether ǫ ≥ 0.5 or not) or it remains unchanged

if its agents are not selected to trade. To put it formally,

let us assume that the two agents selected two trade have

savings propensities λiand λjrespectively. Then

ni(t + 1) = ni(t) ± 1with equal probability(4)

ni(t + 1) + nj(t + 1) = ni(t) + nj(t) (5)

nk(t + 1) = nk(t) for all k?= i,j. (6)

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 2000 4000 6000 8000 10000 12000 14000 16000

time

average saving propensity

FIG. 8:

The convergence in savings propensity. As examples, three

cases are shown. To study the phenomena of convergence in

the savings propensities we focus on the fluctuation of the

average savings propensities. Initially all agents are assigned

with uniformly distributed savings propensities. Hence the

average savings propensity is initially very close to half. But

over time it evolves to reach a steady state where it

maintains a fixed value indicating that all the agents have

the same savings propensity. Thereafter the system behaves

like the usual CC model.

Hence, the number of agents with a particular savings

propensity performs a random walk of unit step and also

note that the walk is bounded below since ni≥ 0 for all

i and also above by Eqn. 3. In fact, the formulation is

akin to the n-players ruin problem [16] where the random

walk occurs on a simplex given by Eqn. 3. The general n-

players ruin problem is very difficult to solve analytically

for k ≥ 4. Ref. [16] presents a matrix-theoretic approach

to the problem which reduces the complexity of the com-

putation. However, we are not interested in finding the

exact solutions to the problem. We simply note that the

given enough time the system will ultimately evolve to a

state where there is only one savings propensity and this

is not unique. It can be any of the initial λis (i = 1, 2,,,

k). See Fig. 8.

B. Steady state money distribution

Once all the agents have a single savings propensity,

the system then behaves like the CC model (see Fig. 1).

VII. SUMMARY AND DISCUSSION

The kinetic exchange models have been very successful

in explaining the origin of the gamma function-like dis-

tribution and the power law in the income/wealth distri-

bution. However, these models use the notion of savings

extensively on an ad-hoc basis without offering much the-

oretical understanding of it. The aim of the present paper

is to provide support to the kinetic exchange models by

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deriving and explaining them from standard neo-classical

economics paradigm and the not-so-standard models of

reinforcement learning and strategic selection.

Ref. [8] provides a microeconomic basis for the kinetic

exchange models with homogenous agents. Here, we ex-

tend that model to explain the heterogenous exchange

models where the agents have different savings propensi-

ties (Sec. II). A further possibility is investigated in Sec.

III where the savings propensity of an agent is dependent

on the money holding of that particular agent and hence

it changes over time (as the money holding changes). It

is shown that even in that case, the economy organizes

itself in such a way that the distribution of money be-

comes stable over time. In some cases, the distribution

produces bimodality. Bimodal income/wealth distribu-

tions have indeed been seen in many countries (see e.g.,

Ref. [13]).

However, it is also noted in Sec. IV that the market

clearing, competitive models used extensively in the eco-

nomics literature has been criticised on the grounds of

limitations of computational capability of human beings

(see e.g., [9]). So we try to explain the kinetic exchange

models assuming that the agents follow some simple rules

of thumb. It is shown that the mechanism of reinforcing

one’s own choice leads to the CCM model [7] (Sec. V).

The basic result regarding the distribution of the fixed

points follows from the famous Polya’s Urn problem (Ref.

[14]). Next, we show in Sec. VI that the agents following

a simple rule of thumb of selecting the best strategy leads

to the CC model (Ref. [6]). The game of strategy selec-

tion reduces to the generalized Gambler’s Ruin problem

or the N-player Ruin problem (Ref. [16]).

Acknowledgments

The author is grateful to Bikas K. Chakrabarti and

Arnab Chatterjee for some useful suggestions.

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