An Agent-Based Model of Strategic Adoption of Real-Time Payments.

An Agent-Based Model of Strategic Adoption of Real-Time
Payments
Katherine Mayo
University of Michigan
kamayo@umich.eduShaily Fozdar
University of Michigan
sfozdar@umich.eduMichael P. Wellman
University of Michigan
wellman@umich.edu
ABSTRACT
Real-time payments (RTPs) allow consumers to receive funds be-
fore the completion of payment clearing and settlement. This early,
irrevocable release of funds represents a credit risk to banks in the
event there are issues with the payment, such as the consumer’s
deposit holdings being insufficient to cover the payment they are
sending. We investigate the effects such risks may have on the
strategic adoption of RTPs by banks. We define a network game
in which consumer nodes with deposit holdings are assigned to
bank nodes responsible for routing consumer payments within the
network. Bank nodes make a strategic decision regarding which
consumers may send RTPs in the network by selecting from a set of
available strategies based on the initial deposits of the consumers.
Using agent-based modeling and empirical game-theoretic analysis,
we analyze this strategic decision in various game configurations.
Our results show that bank nodes tend to choose strategies that
allow many, but not all, consumer nodes to send RTPs. We find
that this outcome in strategic equilibrium reduces successful pay-
ments and increases the incidence of insufficient funds availability,
compared to a setting where RTPs are universally enabled. This
manifests in our model because RTP enables receivers of payments
to turn around those funds more quickly to make payments of their
own. As a result, banks are better off when all payments are real-
time, but a strategic bank node is inclined to avoid the liability of
allowing its own depositors to use RTPs when the risk is considered
high.
KEYWORDS
real time payments, financial network, game theory
ACM Reference Format:
Katherine Mayo, Shaily Fozdar, and Michael P. Wellman. 2021. An Agent-
Based Model of Strategic Adoption of Real-Time Payments. In ICAIF ’21:
2nd ACM International Conference on AI in Finance, November 2021. ACM,
New York, NY, USA, 9 pages. https://doi.org/10.1145/3490354.3494410
1 INTRODUCTION
The distinction between a standard payment and a real-time pay-
ment (RTP) lies in the speed and availability of the payment. Any
payment mechanism defines a series of steps that must occur for
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https://doi.org/10.1145/3490354.3494410the payment to be executed. To initiate a payment, the payer (entity
sending payment) issues a request to their payment service provider
(PSP), say a bank. The request entails removal of funds from the
payer’s account and routing the payment to the bank account of
thepayee (entity receiving payment). These steps encompass the
clearing andsettlement functions, which for a standard payment
must be accomplished for the payee’s bank to credit the payee’s
account. Clearing and settlement can delay completion of the pay-
ment, for example due to grouping of payments into batches for
processing, and communication required among banks to verify
sufficiency of funds. These delays can be particularly significant
for payments initiated outside of business hours.
A real-time payment allows the payee to receive the funds im-
mediately. We focus on the deferred settlement case, in which an
irrevocable credit of funds to the payee’s account occurs before
the clearing and settlement steps. Even if payment is initiated out-
side of regular business hours, the payee can still expect to receive
the funds immediately, though the processing of the remainder of
the payment steps is subject to business hours and batch process-
ing as a standard payment [Committee on Payments and Market
Infrastructures 2016].
Many RTP systems are in use today, including the Internet Bank-
ing System in China, the Real-Time Clearing system in South Africa,
and the Faster Payments Service of the United Kingdom [Commit-
tee on Payments and Market Infrastructures 2016]. The US Federal
Reserve is set to launch a real-time payment system, FedNow, in
2023 [Federal Reserve Board 2021]. The increasing provision of such
services is driven by consumer demand. In cases where merchants
insist on receiving funds before sending goods, RTPs enable the
payer to receive their goods sooner. The merchant also benefits
from more immediate opportunity to employ the funds in the busi-
ness. However, deferring settlement necessitates that banks take
on a credit risk. They face liability in the event problems arise, for
instance due to fraud or other errors. The expediency of RTPs can
also make it more difficult to catch potential problems before the
payment is sent.
We seek to understand how such issues might impact bank adop-
tion of RTP systems, particularly how they decide whom to offer
real-time payment options. We develop an agent-based model that
supports standard and real-time payments sent by consumer nodes
and routed through bank nodes in a financial network. We model
the risk of bank nodes using a scenario in which consumer nodes
sending RTPs may initiate payments that exceed their deposit hold-
ings. These insufficient payments are possible due to the deferred
settlement feature of RTPs, delaying the verification of the payer’s
deposits until after the payment process has already begun. When
such problematic payments occur, our model assigns liability to the
payer’s bank. We assume bank nodes in our model are willing to
ICAIF ’21, 3–5 November 2021, (virtual) Katherine Mayo, Shaily Fozdar, and Michael P. Wellman
extend short-term credit to consumer nodes sending RTPs. When
an insufficient payment occurs, the consumer node draws on this
credit and transfers short-term liability of the remainder of the
payment’s value to its bank.
To study the problem of real-time payment adoption, we define
agame played by strategic bank nodes. The decision facing bank
nodes is which consumers, if any, should be allowed to send real-
time payments in the network. The banks select from strategies that
set varying thresholds on the amounts consumers must deposit in
the bank in order to be allowed to send RTPs. Banks must strategi-
cally balance benefits of offering RTP, including the ability to attract
consumers, with the cost of exposure to covering insufficient pay-
ments. We use a process known as empirical game-theoretic analysis
(EGTA) [Tuyls et al .2020; Wellman 2016] to identify Nash equilibria
of our game under a variety of configurations.
Our contributions can be summarized as follows:
(1)formulating the the real-time payments adoption question
as a strategic decision made by banks;
(2)defining an agent-based financial credit network model that
supports standard and real-time payments with deferred
settlement; and
(3)analyzing the effects of real-time payment adoption for a
particular payment scenario.
We find that banks in our model tend to select strategies that set
positive but low thresholds on the deposits required for sending
RTPs, resulting in outcomes where most but not all consumers have
access to the service. In aggregate banks are generally better off
when all consumers use RTP, as that increases the overall volume of
successful payments. Nevertheless, individual banks are generally
unwilling to assume a level of risk to grant all their own consumers
the use of RTP.
2 RELATED WORKS
The credit network model is represented by a directed graph with
weighted edges representing the capacity for agents in a network to
transact with one another. It has been used to study trust networks
for distributed payment in multi-unit auctions [Ghosh et al .2007],
informal borrowing [Karlan et al .2009], and liquidity [Dandekar
et al.2011, 2015]. Cheng et al .[2016] extended the model to the
financial credit model with the inclusion of interest rates on the
directed edges.
Game theory has previously been applied to the payments space
to study the real-time gross settlement (RTGS) system, which han-
dles the settlement of payments between banks. Bech [2008] study
the management of intraday liquidity by banks under different
credit policies of the central bank. Banks can manage their liquidity
by balancing the timing of settling payments. The authors find this
payment scenario leads to two well-known games: the prisoner’s
dilemma and the stag hunt. Additional work on RTGS systems by
Johnson et al .[2004] studies the effects of deferred settlement mech-
anisms on liquidity of banks. The authors use historical data from
the US Federal Reserve’s Fedwire Funds Service, a system used for
commercial payments.
As real-time payment systems are relatively recent additions to
the payment space, a number of works focus on a basic introduction.
Topics in these works include how the payments differ from more
Figure 1: An example of a consumer node with 20 deposits
currently in their bank account, but who is willing to hold
an additional 80 units in the account.
traditional payment methods, potential benefits and drawbacks
for both the consumer and PSPs, and discussions on existing RTP
systems throughout the world [Committee on Payments and Market
Infrastructures 2016; Hartmann et al .2019; Santamaría 2019]. Prior
literature in the RTP space also addresses RTP system design [Guo
et al.2015]. This includes proposals for additions to existing systems
[Kulk 2021], as well as the implementation of new systems such
as those incorporating the blockchain [Arshadi 2019; Zhong et al .
2019]. The design of RTP systems may be important when analyzing
how banks and consumers may use them in the real world.
Galbiati and Soramaki [2008] use an agent-based model to study
the liquidity demanded by banks with access to RTPs. Banks in
their model are able to choose their demanded liquidity reserves
for each day and payments are executed immediately as long as
the bank has the available liquidity. When there is no available
liquidity, payments suffer a delay, which can be costly. The authors
focus their analysis on how liquidity demand relates to network
efficiency, the number of banks in the system, and the volume of
payments.
To our knowledge, none of the existing literature specifically
addresses how banks might decide who can use RTP systems. Given
the potential risk to banks, it is worth exploring how they may
place a limitation on consumer use of RTPs and explore how such
a limitation may be chosen.
3 FINANCIAL PAYMENT MODEL
We model a financial network similar to that described by Cheng
et al.[2016]. The network consists of a set of nodes 𝐵={1,...,𝑏}
representing banks and a set of nodes 𝐶={1,...,𝑚}representing
consumers, with 𝑏≪𝑚. Nodes in the network are connected by a
set of weighted, directed edges 𝐸. Edge(𝑖,𝑗,type,𝑣)∈𝐸represents
a value𝑣owed by𝑖to𝑗iftype=debt or a value𝑣of credit extended
by𝑖to𝑗iftype=credit . A debt edge from a bank node to a consumer
node,𝑖∈𝐵and𝑗∈𝐶, can be interpreted as representing deposits
consumer𝑗holds in its account at bank 𝑖. A credit edge from a
consumer node to a bank node, 𝑖∈𝐶and𝑗∈𝐵, can be interpreted
as consumer 𝑖’s willingness to hold additional deposits in its account
at bank𝑗. For example, a consumer’s willingness to hold deposits
may be bounded for reasons such as account use, interest rates, or
FDIC insurance limits.
An example of the relationship between a consumer and a bank
node in our model can be seen in Figure 1. The dashed debt edge
(𝐵1,𝐶1,debt,20)represents consumer 𝐶1holding 20 units in its
account at bank 𝐵1. The solid credit edge (𝐶1,𝐵1,credit,80)repre-
sents𝐶1’s willingness to hold up to 80 more units in its account at
𝐵1. Thus, at any given time, consumer 𝐶1is willing to hold at most
100 total units in its account.
An Agent-Based Model of Strategic Adoption of Real-Time Payments ICAIF ’21, 3–5 November 2021, (virtual)
There may exist multiple edges of each type between any pair of
nodes𝑖and𝑗. We refer to the total debt between the pair of nodes
as𝑑𝑖𝑗=Í
(𝑖,𝑗,debt,𝑣)∈𝐸𝑣. Similarly, the total credit extended from
𝑖to𝑗is𝑐𝑖𝑗=Í
(𝑖,𝑗,credit,𝑣)∈𝐸𝑣. We constrict edges in our network
to never be between two consumers such that if 𝑖∈𝐶and𝑗∈𝐶,
𝑑𝑖𝑗=0and𝑐𝑖𝑗=0.
We model a payment as a series of new edges added to the net-
work. To initiate a standard payment, the payer must hold enough
deposits in its account to cover the value of the payment. For con-
sumer𝐶1with an account at 𝐵1to send a payment of value 𝑣, it
must be that 𝑣≤𝑑𝐵1𝐶1. We will later relax this constraint to allow
for insufficient payments. By using 𝑣of its deposits for a payment,
𝐶1is now owed 𝑣fewer deposits by 𝐵1, which we represent with
the creation of edge (𝐵1,𝐶1,debt,−𝑣). With𝑣fewer deposits in its
account,𝐶1’s willingness to hold more deposits must increase by 𝑣,
since they were willing to hold that many deposits previously. We
capture this by adding the edge (𝐶1,𝐵1,credit,𝑣)to the network.
Suppose the payee is consumer 𝐶2with an account at a different
bank,𝐵2. Consumer 𝐶1’s bank will use the interbank network,
which connects all banks in the network allowing payments to flow
between them, to route the payment to 𝐶2’s bank. We model the
interbank network as a set of debt and credit edges, which connect
any given bank node to every other bank node in the network. Our
model assumes that banks have an infinite willingness to route
payments on behalf of their consumers and thus models the credit
edges between banks with infinite value. For simplification, we
omit these credit edges from our figures. Routing the payment from
𝐵1to𝐵2is reflected in the creation of the debt edge (𝐵1,𝐵2,debt,𝑣).
Upon receiving the funds from 𝐵1,𝐵2is able to credit those funds
to the payee’s account. In doing so, 𝐵2now owes these additional de-
posits to𝐶2, creating edge(𝐵2,𝐶2,debt,𝑣). The increase in deposits
in𝐶2’s account decreases their willingness to hold more deposits
by the value of the payment, adding edge (𝐶2,𝐵2,credit,−𝑣).
We model batch processing with a queue (𝑄). When the edges of
a new payment are created, such as those above, they are stored
in this queue. At regularly scheduled intervals, the clearing period ,
the queue is cleared by removing the edges and adding them to the
network.
An example of a payment from consumer 𝐶1of 10 units to con-
sumer𝐶2is shown in Figure 2. At time 𝑡=0when the payment is
initiated, the network remains unaffected. The queue however, now
holds the edges created by the payment: 𝑄={(𝐵1,𝐶1,debt,−10),
(𝐶1,𝐵1,credit,10),(𝐵1,𝐵2,debt,10),(𝐵2,𝐶2,debt,10),
(𝐶2,𝐵2,credit,−10)}. It is not until the next clearing period, 𝑡=𝑋,
that the the edges are removed from the queue and added to the
network. The edges in the figure display the total debt ( 𝑑𝑖𝑗) and
total credit ( 𝑐𝑖𝑗) values instead of showing each individual edge.
3.1 Real-time Payments
We can implement real-time payments with deferred settlement
in our network simply by changing which edges are added to the
queue. Instead of placing all the edges in the queue, a RTP adds
both the credit and debt edges between the payee and its bank to
the network immediately. This models the irrevocable credit of
funds to the payee’s account. The deferred settlement is handled
Figure 2: An example of processing a standard payment of
10 units from consumer 𝐶1to consumer 𝐶2in our model.
Figure 3: An example of processing a payment of 10 units
from consumer 𝐶1to consumer 𝐶2as a real-time payment
in our model.
by placing the remaining edges in the queue until the next clearing
period, similar to a standard payment.
An example RTP is detailed in Figure 3. Similar to the previous
example, consumer 𝐶1sends a payment of 10 units to consumer 𝐶2,
however this time it is sent in real-time. Again, the edges depicted
show the total debt ( 𝑑𝑖𝑗) and total credit ( 𝑐𝑖𝑗) values. The payment
is initiated at time step 𝑡=0and the immediate change is demon-
strated by the state of the network at the next time step, 𝑡=1. The
edges between consumer 𝐶1and bank𝐵1and between banks 𝐵1
and𝐵2are not changed until the next clearing period, 𝑡=𝑋. It can
be seen that after clearing, the standard and real-time payments
have the same effect on the network.
We also support the insufficient payment case, which we con-
sider to be a byproduct of the expediency of the real-time payments
system. Consumer 𝐶1’s payment is considered insufficient if its
value𝑣exceeds𝐶1’s current deposits at bank 𝐵1:𝑣>𝑑𝐵1𝐶1. Insuffi-
cient payments can occur in our model only when the payment is
sent in real-time.
An example of an insufficient payment sent as a RTP is out-
lined in Figure 4. Consumer 𝐶1sends a payment of 25 units to
consumer𝐶2. Similar to the RTP case, the edges between the payee
and its bank are updated immediately, while the remaining edges
are added to the queue. The queue will look similar to in the pre-
vious two examples: 𝑄={(𝐵1,𝐶1,debt,−25),(𝐶1,𝐵1,credit,25),
ICAIF ’21, 3–5 November 2021, (virtual) Katherine Mayo, Shaily Fozdar, and Michael P. Wellman
Figure 4: An example of processing an insufficient payment
of 25 units from consumer 𝐶1to consumer 𝐶2in real-time.
(𝐵1,𝐵2,debt,25),(𝐵2,𝐶2,debt,25),(𝐶2,𝐵2,credit,−25)}. The spe-
cial case of an insufficient payment will be handled during the
clearing of the queue. When the edges between the payer and the
payer’s bank are processed, the model will detect the insufficiency
of the payment. The payer’s bank will force the payer to cover
as much of the payment as possible by supplying all of its cur-
rent deposit holdings. This is reflected in a change in the edge
(𝐵1,𝐶1,debt,−25)to(𝐵1,𝐶1,debt,−20), which is then added to the
network. Note the consumer’s total deposits are now equal to 0.
As a result, the accompanying credit edge is also changed from
(𝐶1,𝐵1,credit,25)to(𝐶1,𝐵1,credit,20), to reflect the amount that
was actually taken from the account, and that the consumer is will-
ing to replace. The payer’s bank, must supply the remaining value
of the payment ( 𝑣−𝑑𝐵1𝐶1) which enables the bank to pass on the
full amount to the payee’s bank, leaving the remaining edges of the
payment unchanged.
4 PAYMENT GAME
We define a real-time payments game played by bank nodes in the
financial payment network described. Banks in our network must
decide which consumer nodes, if any, are allowed to send RTPs by
selecting from one of 6 available strategies. Each strategy sets a
different threshold on the amount consumer nodes must initially
deposit in their accounts to be allowed to send RTPs. The thresholds
range from allowing every consumer node to send RTPs to allowing
no consumer nodes to send RTPs. In our model consumers may
make insufficient payments, for which the payer’s bank will be held
liable in the short-term. As described in Section 3, the difference
between the insufficient payment’s value and the payer’s deposit
holdings, the insufficient coverage , must be covered by the payer’s
bank. Thus, the strategic real-time payments decision for bank
nodes is to choose a strategy that balances the benefits of RTPs,
including attracting consumer nodes, with a desire to avoid a large
amount of insufficient coverage.
The game begins by initializing consumer nodes with a random
amount of initial deposits drawn from an exponential distribution.
This creates a set of consumers with realistic variation in levels
of wealth, including a larger group with average deposit holdings
and a small group of very wealthy individuals. Consumer nodes
are also randomly assigned to one of two preferences for receiving
payments with equal likelihood. Some consumer nodes are willingto accept any kind of payment, while others accept only RTPs.
We also initialize a set of bank nodes and the edges forming the
interbank network. Bank nodes then select a strategy for adopting
RTP use for their consumers.
All consumer nodes are assigned to a bank under the assump-
tion that consumers prefer banks that allow them to send RTPs.
Consumer nodes do not further differentiate between the banks in
any way. The full assignment procedure is as follows:
•If no bank allows the consumer to send RTPs, the consumer
is randomly assigned to any bank.
•If only one bank allows the consumer to send RTPs, the
consumer is assigned to that bank.
•If multiple banks allow the consumer to send RTPs, the con-
sumer is randomly assigned to one such bank.
When a consumer node is assigned to a bank, a credit edge is
created from the consumer to the bank and a debt edge is created
from the bank to the consumer. The value on the debt edge is set
to the consumer node’s initial deposits, representing the consumer
placing all of its deposits into its account. We assume each consumer
node is willing to hold an infinite amount of deposits in its bank
account and set the value on the credit edges to infinity. A consumer
node’s bank assignment lasts for the duration of the game.
The game proceeds in discrete time steps, 𝑡={0,...,𝑇}. The
game starts each time step by checking if it is in a designated
clearing period. In every clearing period, the queue is cleared by
removing the edges from 𝑄and adding them to the network. When
an insufficient payment is found, the edges between the payer and
its bank node are handled as described in Section 3.1.
After performing any necessary clearing, the game attempts to
create𝐿new payments in the network. For each payment a payer,
payee, and value are randomly selected. The value of the payment is
drawn uniformly from a fixed interval. Consumer nodes are limited
to at most one payment per time step. Each payment is processed
in the following manner:
•If the receiver accepts only RTPs and the sender cannot send
RTPs, they are deemed incompatible and the payment fails.
•If the sender can send RTPs, the payment is processed as a
RTP.
•If the sender cannot send RTPs and the receiver is willing
to accept all payment types, the payment is processed as a
regular payment.
Drawing each payment from a fixed interval may result in an
insufficient payment. The first chance to catch a possibly insufficient
payment is when it is initiated. We define a consumer 𝐶1’savailable
funds ,𝐴𝐶1as its current total deposits minus the value of any
pending payments in the queue for which 𝐶1is the payer, or 𝐴𝐶1=
𝑑𝐵1𝐶1+Í
(𝐵1,𝐶1,debt,𝑣)|𝑣<0𝑣. A payment of value 𝑣initiated by𝐶1is
marked as potentially insufficient if 𝑣>𝐴𝐶1. Note that we use the
term potentially insufficient here, as it is possible for the consumer
node to receive funds between the payment’s initiation and the
next clearing period, such that at clearing, 𝑣≤𝑑𝐵1𝐶1. A standard
payment deemed potentially insufficient will terminate the payment
process immediately. While the expediency of a real-time payment
does not allow the bank nodes to catch an insufficient payment
before it is sent, our game provides an opportunity for the payer
to potentially do so. Consumer nodes check the validity of their
An Agent-Based Model of Strategic Adoption of Real-Time Payments ICAIF ’21, 3–5 November 2021, (virtual)
payments before sending them with probability 𝑝∈ [0,1], the
consumer check probability . We define this probability as a game
configuration variable that applies to all consumer nodes in the
network. If a consumer node checks their payment and finds it
may be insufficient, they adjust the payment’s value to be valid
so that𝑣=𝑚𝑎𝑥(0,𝐴). A payment with a value equal to 0 triggers
the termination of the payment process. Anytime the payment
process is terminated early, the payment attempt is still counted
and subsequently decrements the total number of payments left to
attempt in the time step. If the payment process does not terminate
for any reason, the appropriate processing steps continue.
The final time step, 𝑡=𝑇, only clears the queue and no new
payments are created. At the end of the game, bank nodes are
awarded a payoff for selecting their chosen strategy that relies on
three values: the total value of initial deposits attracted ( 𝐷), the total
value of RTPs routed ( 𝑅), and the amount of insufficient payments
coverage (𝐼). The deposits a bank holds for its consumers may be
viewed as representing a consumer’s level of business with the bank,
however the deposits themselves are a liability that must be paid
upon demand. Therefore, we model the utility a bank node receives
from deposits as a fraction of their value. Similarly, the utility a
bank derives from routing payments in real-time for its consumers
is a fraction of its value representing consumer satisfaction and
continued business. We model our payoff assuming a consumer
derives slightly more satisfaction from being able to send a 100
unit payment in real-time than from being able to send a 5 unit
payment in real-time. Conversely, the loss to bank nodes in our
game is equal to the value of insufficient coverage required. Banks
hold the short-term liability for insufficient payments, even if they
are able to push the liability onto consumers in the long-run.
The equation for the payoff to bank 𝐵is:
𝑝𝑎𝑦𝑜𝑓𝑓𝐵=0.5∗𝐷+0.02∗𝑅−𝐼
We model games with 𝑚=225consumer nodes, 𝑏=3bank
nodes and𝑇=720time steps, with 𝐿=45payments attempted
per time step with values 𝑣∼U{1,..., 100}. We test clearing period
lengths𝑋∈ {4,6,12,24}and consumer check probability 𝑝∈
{0,0.25,0.5,0.75}, for a total of 16 different game configurations for
analysis.
5 EMPIRICAL GAME-THEORETIC ANALYSIS
To analyze the real-time payments game we use extensive simula-
tion of strategy combinations in a process referred to as EGTA. The
process selects a strategy profile , a list of strategies and the number
of players employing each strategy, and uses the profile in repeated
simulations of the payment game. In each game, consumer and
bank nodes are randomly initialized, banks are randomly assigned
to a strategy in the profile such that the specified number of banks
employ each strategy, and random payments are generated and
cleared over 720 time steps. Profiles are selected for simulation in
an iterative procedure with the aim of finding symmetric mixed-
strategy Nash equilibria similar to the manner used in previous
EGTA studies [Cassell and Wellman 2013; Wellman et al .2013]. The
calculated payoff to a bank node for employing a strategy is the
sample average of payoffs observed over the many simulation runs.The equilibria identified by EGTA for the different game config-
urations classified by clear period and consumer check probability
can be seen in Table 1. We sort the 6 strategy thresholds into three
categories low,medium , and high based on the minimum amount
of initial consumer deposits each requires for real-time payments
use. Most of the Nash equilibria identified were mixed-strategy
equilibria, except for the case where the consumer check proba-
bility is equal to 0.75which has a pure-strategy Nash equilibria
for all clear periods. We report the total probability a bank node
will adopt a strategy with a threshold belonging to each category,
calculated from the probability assigned to playing the strategies
in equilibrium.
The results show an increase in the clear period and consumer
check probability leads to an increase in the probability a bank
node adopts strategy with a low threshold on deposits. Intuitively,
as consumer nodes are more likely to check the validity of their
payments and correct errors, the number of insufficient payments
decreases. Thus, bank nodes may be more willing to allow broader
use of RTPs. A lower clearing period, on the other hand, allows
payee’s of standard payments to receive funds in a more timely
manner than when the clearing period is very high. The difference
between a RTP and a standard payment becomes smaller. In this
case, it may not be as worthwhile for bank nodes to provide as
many consumer nodes the use of RTPs.
6 EFFECTS OF EQUILIBRIA
We analyze the effect bank nodes adopting the Nash equilibrium
strategies has on the network, in particular on consumer nodes’
access to RTPs, the success of payments attempted, and insufficient
payments made. Bank nodes play the payment game 1,000 times
as described: randomly initialize bank and consumer nodes, bank
nodes adopt a strategy, consumer nodes are assigned to banks, and
random payments are created and cleared in the network over 720
time steps. However for this analysis, bank nodes are required to
adopt the equilibrium strategy identified by EGTA for the game
configuration being analyzed. When the equilibrium is mixed, the
bank nodes will be assigned to one of the pure strategies that make
up the equilibrium (the support) according to a weighted draw with
weights equal to the probability assigned to playing each strategy
in equilibrium.
We also compare the success of payments and insufficient pay-
ments under the equilibria to the all-RTPs and no-RTPs cases. For
the first case, bank nodes allow all consumer nodes the use of RTPs.
This represents a situation in which banks might be required, for
instance by federal regulation, to allow everyone access to RTPs. In
the second case, bank nodes do not allow any consumer nodes to
send RTPs, representing the status quo before RTP systems were
introduced. A consumer’s preference to accept only RTPs in this
case can be viewed as unwillingness to accept payments through
the banking system without a RTP system in place.
The proportion of a bank node’s consumers that are allowed to
send RTPs when the bank adopts the equilibrium strategies is shown
in Figure 5. In all game configurations, the majority of consumer
nodes are allowed to send RTPs, although in many cases not all.
The proportion tends to decrease as the clearing period length
decreases and increase as the consumer check probability increases.
ICAIF ’21, 3–5 November 2021, (virtual) Katherine Mayo, Shaily Fozdar, and Michael P. Wellman
Table 1: The probability of a bank node selecting a strategy that sets a low threshold in equilibrium increases as the consumer
check probability and clear period increase.
Consumer Check 𝑝 0 0.25 0.5 0.75
Clear Period 𝑋 Strategy Threshold low med high low med high low med high low med high
24 0.58 0.42 0 0.61 0.39 0 1 0 0 1 0 0
12 0.54 0.46 0 0.46 0.54 0 1 0 0 1 0 0
6 0.51 0.49 0 0.37 0.63 0 0.92 0.08 0 1 0 0
4 0.48 0.52 0 0.39 0.61 0 0.75 0.25 0 1 0 0
Figure 5: The proportion of consumer nodes allowed to send
real-time payments under the equilibria increases as the
clear period and consumer check probability increases.
This aligns with the trends seen in the equilibria in Table 1. As the
clear period decreases, the probability of selecting a strategy with a
lower threshold decreases. As a result, the proportion of consumer
nodes allowed to send RTPs also decreases. On the other hand,
when the consumer check probability increases, the probability
of choosing a strategy with a lower threshold increases and the
proportion of consumer nodes allowed to send RTPs increases as
well. The exception occurs when the consumer check probability is
0.75and the pure strategy Nash equilibrium is to allow everyone to
send RTPs. This results in a proportion of 1 for all clearing periods.
As described in Section 4, issues with consumer compatibility
and deposit values may lead to early termination of payments. We
examine the frequency of such instances by studying the proportion
of a bank node’s payments that are deemed successful because they
did not terminate early. To calculate, we use the expected number of
payments attempted per bank node. We also compare the equilibria
case to the case where all consumer nodes may send RTPs. The
results are in Figure 6.
Most payments attempted in the network are successful, regard-
less of game configuration and in both the equilibria and all-RTPs
cases. Under the equilibria, the success tends to slightly decrease as
the clear period decreases. At lower clear periods the bank nodes
tend to allow fewer consumer nodes to send RTPs. With fewer
consumers sending RTPs, there may exist more cases where the
payer’s allowed payment type and payee’s desired payment type are
incompatible and therefore, increases the number of unsuccessful
payments. When compared to the equilibria, we see that allowingeveryone to send RTPs leads to a higher proportion of successful
payments in all game configurations. The only exception is when
consumer check probability is 0.75, where allowing everyone to
send RTPs is the equilibrium. Instead of a decrease in the proportion
of successful payments, the all-RTPs case shows a slight increase in
the proportion of successful payments as the clear period decreases.
This may be attributed to the larger number of payments made
between clearing as a result of longer clearing periods. Thus, there
are more cases of consumer nodes validating payments and finding
they have no available funds.
We also compare these results to the no-RTPs case, where the
average proportion of successful payments across all game config-
urations is only 0.32. This highlights the importance of real-time
payments for promoting consumer liquidity. Allowing RTPs for
at least some participants in the network promotes increased con-
sumer participation in the banking system. Furthermore without
a delay in receiving funds for which a consumer is the payee, the
funds are available for the consumer’s immediate use as a payer.
The disincentive for bank nodes to offer real-time payments ser-
vices to consumers is the risk of insufficient payments. We measure
the effect insufficient payments have on bank nodes by the percent-
age of a bank’s payments that are insufficient, as well as the average
total insufficient coverage required by a bank. The results when
bank nodes adopt the equilibrium strategies are shown in Table 2
and when bank nodes allow all consumers to send RTPs in Table 3.
If banks do not allow anyone to send real-time payments, insuf-
ficient payments in our model are caught and terminated before
processing.
Few payments made by consumer nodes in our network turn
out to be insufficient, however when they occur the bank nodes
tend to cover the majority of the payment. This trend is evident
regardless of whether bank nodes adopt the equilibrium strategies
or allow all consumer nodes to send RTPs. As the consumer check
probability increases, both the number of insufficient payments and
the insufficient coverage required tend to decrease. With consumer
nodes more likely to check their payments when the consumer
check probability is high, the likelihood they catch and correct po-
tentially insufficient payments is also high. Conversely, as the clear
period decreases, the number of insufficient payments tends to in-
crease and the insufficient coverage required tends to decrease. The
exception occurs when the consumer check probability is 0.75. In
this case both the number of insufficient payments and insufficient
coverage required increase as the clear period decreases.
To explain this phenomenon, we must consider consumer nodes
as both payers and payees. When the clear period decreases, the
An Agent-Based Model of Strategic Adoption of Real-Time Payments ICAIF ’21, 3–5 November 2021, (virtual)
Figure 6: When bank nodes adopt the equilibrium strategies,
the proportion of successful payments decreases as the clear
period decreases. Furthermore, its value is always smaller
than when all consumers send real-time payments.
number of time steps between clearing becomes smaller. At a
smaller clear period, there are fewer opportunities for a given con-
sumer to receive funds as a payee. Now when a clear period occurs,
this consumer has accumulated fewer deposits to use when settling
payments for which they are the payer than they might have if the
clear period were longer. Thus, we see an increase in the number
of instances in which a payment’s value is strictly larger than the
payer’s deposit holdings at the time of settlement. On the other
hand, a smaller clearing period also limits the number of payments
for which a given consumer node may be the payer. Note that this
limitation has a stronger impact on a consumer node’s ability as a
payer than as a payee, as consumers make at most one payment,
but may receive any number of payments per time step. With fewer
opportunities to make payments between clearing, the total amount
owed in payments for any given consumer node at clearing time
will be smaller with a clearing period of 4 than 24. So while we may
see an increase in the number of payments for which consumer
nodes do not have enough deposits, the amount of deposits a con-
sumer node has tends to be closer to the total value of the payments
made.
The insufficient payment coverage trend at consumer check
probability 0.75may be understood by analyzing the effect of con-
sumer check probability on payment value. When the probability aconsumer node checks their payments is high, it is likely that an
insufficient payment will be caught and corrected. By definition,
a corrected insufficient payment’s value will be smaller than the
original value. Thus, the value of payments on average is less when
consumer nodes correct insufficient payments more often. Smaller
payment values means fewer deposits. The increase in the number
of insufficient payments along with the decrease in deposits causes
the insufficient coverage requirement for bank nodes to slightly
increase.
Our analysis also illustrates an important relationship between
banks and consumer use of RTPs. From our analysis, it can be
seen that all parties involved tend to be better off when all con-
sumer nodes send their payments in real-time. This is evident in
the proportion of successful payments being larger and insufficient
payments, both the number and resulting required coverage, being
smaller when all consumer nodes send RTPs compared to the other
cases for all game configurations. An individual bank node playing
our payment game, however, is more likely to play a strategy that
limits the number of its consumers who send RTPs in the event
insufficient payments are likely. We can conclude that, while bank
nodes might be better off when all consumers in the network are
allowed to send RTPs, a strategic bank node would prefer not to
assume the risk required to allow all of its consumers the use of
RTP.
7 CONCLUSION
In this study, we analyze how the adoption of real-time payments by
banks may be affected by potential payments risk, specifically the
possibility of insufficient payments. We introduce a payment model
that supports consumer nodes sending both standard and real-time
payments with deferred settlement through the interbank network.
Within this framework, we model insufficient payments as the
case where the value of a RTP is greater than the current deposit
holdings of the payer and for which the payer’s bank becomes
liable. Such a scenario captures both the risks of the expediency of
RTP and the credit risk borne by banks in the deferred settlement
case. We ask which consumers banks should allow to send RTPs in
this scenario by modeling the decision as a strategic game played
by bank nodes. Bank nodes select a strategy based on the initial
deposits of consumer nodes. The strategic decision for bank nodes
requires balancing the benefits of real-time payments and a desire
to attract consumers, with a desire to limit their liability.
Our results show that while bank nodes never choose strategies
with high thresholds, the likelihood of allowing all consumers to
send RTPs is dependent on different game configuration variables.
When consumer nodes are less likely to send insufficient payments,
bank nodes are willing to allow all, or nearly all consumers to
send real-time payments. However, if consumer nodes may send
many insufficient payments, bank nodes become more likely to
select a strategy with a medium level threshold. Bank nodes also
tend to prefer strategies with medium level thresholds to strategies
with lower thresholds when the clearing period is lower. This is
likely because a shorter clearing period makes real-time payments
more similar to standard payments. In this case, it becomes less
worthwhile for bank nodes to allow as many consumers the use of
RTPs.
ICAIF ’21, 3–5 November 2021, (virtual) Katherine Mayo, Shaily Fozdar, and Michael P. Wellman
Table 2: When bank nodes adopt the equilibrium strategies, few insufficient payments are made, but the insufficient coverage
required by banks is large.
Consumer Check 𝑝 0 0.25 0.5 0.75
Clear Period 𝑋 % insufficient % coverage % insufficient % coverage % insufficient % coverage % insufficient % coverage
24 4.6 79 3.3 75 1.8 70 0.75 65
12 4.9 74 3.9 72 2.3 68 1.1 65
6 5.1 72 4.4 72 2.8 69 1.3 67
4 5.3 72 4.5 71 3.0 69 1.4 68
Table 3: If all payments are sent as real-time payments, the number of insufficient payments made is small, but bank nodes
cover the majority of the payment.
Consumer Check 𝑝 0 0.25 0.5 0.75
Clear Period 𝑋 % insufficient % coverage % insufficient % coverage % insufficient % coverage % insufficient % coverage
24 4.2 78 2.9 74 1.7 70 0.76 65
12 4.6 73 3.4 70 2.2 68 1.1 65
6 4.8 70 3.7 69 2.6 68 1.3 67
4 4.9 70 3.8 69 2.7 68 1.4 68
We compare the effects of bank nodes adopting the equilibrium
strategies to the cases where bank nodes allow everyone or allow
no one to send RTPs. The results of this analysis demonstrate that
bank nodes are better off when all consumer nodes are sending
payments in real-time than when bank nodes adopt the equilibria or
don’t allow RTPs. This is evident in the proportion of payments that
are successfully made in the network, the number of insufficient
payments made in the network, and the insufficient coverage re-
quired by bank nodes as a result of insufficient payments. However,
the Nash equilibrium in many situations for banks in our game
is to select a strategy that places a limitation on the number of
consumers who are allowed to send RTPs. From these results, we
can infer that although outcomes for the banks are more favorable
when all consumers send RTPs, strategic banks are unwilling to as-
sume the liability risk required to provide all their own consumers
the use of RTPs.
While our work offers some insight into the strategic decision
faced by banks in the real-time payments space and the effects
the decision may have on the network, it is important to note this
model is rather simplistic. Future work may explore how other
factors may affect the real-time payments decision. For example,
consumers may be endowed with additional features that would
impact their payments, the network may be made of more node
types than just consumers and banks, and issues with payments
could stem from other scenarios such as fraud. Addressing these, as
well as other potential additions, may help paint a more complete
picture of the real-time payments question.
ACKNOWLEDGMENTS
This work was supported in part by a grant from JP Morgan Chase
AI Center.
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Katherine Mayo
Shaily Fozdar
Michael P
Wellman