# Hybrid Quantum-Classical Algorithms for Loan Collection Optimization
with Loan Loss Provisions

Jirawat Tangpanitanon<EMAIL_ADDRESS>Quantum Technology Foundation
(Thailand), Bangkok, Thailand  Thailand Center of Excellence in Physics,
Ministry of Higher Education, Science, Research and Innovation, Bangkok,
Thailand  Jirawat Saiphet<EMAIL_ADDRESS>Optical and Quantum Physics
Laboratory, Department of Physics, Faculty of Science, Mahidol University,
Bangkok, Thailand  Pantita Palittapongarnpim  Chula Intelligent and Complex
Systems Lab, Department of Physics, Faculty of Science, Chulalongkorn
University, Bangkok, Thailand  Poompong Chaiwongkhot  Optical and Quantum
Physics Laboratory, Department of Physics, Faculty of Science, Mahidol
University, Bangkok, Thailand  Pinn Prugsanapan  Kasikorn Business-Technology
Group, Bangkok, Thailand  Nuntanut Raksasri  Kasikorn Business-Technology
Group, Bangkok, Thailand  Yarnvith Raksri  Kasikorn Business-Technology Group,
Bangkok, Thailand  Pairash Thajchayapong  Kasikorn Business-Technology Group,
Bangkok, Thailand  Thiparat Chotibut<EMAIL_ADDRESS>Chula Intelligent
and Complex Systems Lab, Department of Physics, Faculty of Science,
Chulalongkorn University, Bangkok, Thailand

###### Abstract

Banks are required to set aside funds in their income statement, known as a
loan loss provision (LLP), to account for potential loan defaults and
expenses. By treating the LLP as a global constraint, we propose a hybrid
quantum-classical algorithm to solve Quadratic Constrained Binary Optimization
(QCBO) models for loan collection optimization. The objective is to find a set
of optimal loan collection actions that maximizes the expected net profit
presented to the bank as well as the financial welfare in the financial
network of loanees, while keeping the LLP at its minimum. Our algorithm
consists of three parts: a classical divide-and-conquer algorithm to enable a
large-scale optimization, a quantum alternating operator ansatz (QAOA)
algorithm to maximize the objective function, and a classical sampling
algorithm to handle the LLP. We apply the algorithm to a real-world data set
with 600 loanees and 5 possible collection actions. The QAOA is performed
using up to 35 qubits on a classical computer. We show that the presence of
the QAOA can improve the expected net profit by approximately $70\%$, compared
to when the QAOA is absent from the hybrid algorithm. Our work illustrates the
use of near-term quantum devices to tackle real-world optimization problems.

##  I Introduction

Financial institutions often rely on the ability to solve computationally
intensive problems. Quantum computing holds the promise of highly efficient
algorithms that can provide speedup over some best-known classical algorithms
[1, 2, 3]. Applications of quantum computing on finance have been recently
explored both theoretically and experimentally on small quantum devices.
Examples include portfolio optimization [4, 5, 6, 7], option pricing [8, 9,
10], risk analysis [11], transaction settlement [12], and credit valuation
adjustment[13].

In this work, we introduce a new use-case of quantum computing in finance,
namely loan collection optimization. The goal is to find a set of optimal
actions to be taken on the loanees in order to maximize the expected net
profit presented to the lender. Examples of these actions include doing
nothing, creating a promise-to-pay agreement, debt restructuring and offering
various forms of discounted payoffs (DPOs) [14]. DPOs typically occur in
distressed loan scenarios, where loanees are experiencing financial or
operational distress, default, or are under bankruptcy. Usually, DPOs are a
last resort for lenders because they often involve taking a loss as the loan
is repaid for less than the outstanding balance.

Loan collection actions are normally chosen based on the history of an
individual loanee and the experience of the collector. However, the network-
induced ‘domino’ effects of such actions, where financial distress can
potentially cascade through interconnected financial network of loanees, are
usually ignored due to the complexity of the underlying financial system [15].
For example, offering a DPO to one loanee usually signifies the cash flow
problem of such loanee which could cause disruptions in the supply chain [16].
Such scenarios would negatively impact cash flow of related parties. Some of
which may also be the customer of the lender. Therefore, it is important to
account for financial association among the loanees when determining the
optimal collection actions. The lender should aim to maximize the expected net
profit while minimizing the potential cascade of financial distress on the
financial network of the loanees.

Another important aspect of loan collection is Loan Loss Provision (LLP) [17].
LLP is regulated by the government to ensure the financial health of the bank.
The idea is to set aside funds as an expense in the financial statement to
account for potential loan defaults and expenses that occur as a result of
lending. The amount of LLPs depends on various factors such as types of
loanees (individuals, small businesses, large corporations), types of loans,
late payments, collection expenses, as well as the collection actions that
will be taken. LLPs typically constitute a significant portion of the
financial statement. For example, in the first half of 2020, seventy banks
worldwide report under Expected Credit Loss accounting provisions totaling in
$\$$161 billion [18]. Therefore, it is crucial to choose optimal collection
actions that keep LLPs at a minimum to promote financial liquidity.

Here, we model the above problem including the financial network and LLPs as
Quadratic Constrained Binary Optimization (QCBO) problems with both global and
local constraints. We solve this class of problems by devising a hybrid
quantum-classical algorithm which consists of three parts: a classical divide-
and-conquer algorithm to enable a large-scale optimization, a quantum
alternating operator ansatz (QAOA) algorithm to maximize the objective
function, and a classical sampling algorithm to handle the LLP. By
benchmarking with a real-world data set, we show that the presence of the QAOA
can improve the expected net profit by approximately $70\%$, compared to when
the QAOA is absent from the hybrid algorithm. Our work can be implemented on
near-term quantum devices, consisting of a few tens of qubits and sparse qubit
connectivity.

##  II problem formulation

Loan collection is a management problem that involves identifying which
collection actions to be taken to which loanees at what time. Due to the
complexity of the process, it is a common practice to follow a rule-based
guideline for collection activities. However, the actions and time can also be
personalized, as is the recent approach to loan collection [19, 20], in order
to respond to the loanee’s unique financial history. Moreover, as discussed
earlier, one collection action taken to a loanee can affect the financial
welfare of other loanees in the financial network which, in turn, affects the
expected net profit of the lender.

To capture both the personalized loan collection approach and the effect of
loanee’s financial network, we propose a heuristic QCBO model that captures
the impact of loan collection actions at a given time for a given loan
product. The objective (yield) function is defined as

$Y=(1-\epsilon)\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}h_{i,j}x_{i,j}+\epsilon\sum\limits_{\langle
i,i^{\prime}\rangle}^{N}A_{i,i^{\prime}}(1-x_{i,1})(1-x_{i^{\prime},1}),$ (1)

where $N$ is the number of loanees and $M$ is the number of possible actions.
Here, $\\{x_{i,j}\\}$ are binary decision variables: $x_{i,j}=1$ if action $j$
is taken to loanee $i$ and $x_{i,j}=0$ otherwise. By convention, we set $j=1$
to denote the DPO action. The coefficient $h_{i,j}\in\mathbb{R}$ is an
expected net profit that the bank would get if action $j$ is taken to loanee
$i$ 111The coefficient can also be seen as a way of ranking the bank’s
preferred action towards a loanee and can be personalized to the loanee’s loan
payment history. One approach for the personalized calculation of $h_{i,j}$ is
to model the loan collection problem as a Markov Decision Process (MDP) [19]..
$(A_{i,i^{\prime}})\in\mathbb{R}^{N\times N}$ is an association matrix,
defined as the averaged transaction between loanees $i$ and $i^{\prime}$.
According to the form of $Y$, association or ‘cash flow’ between loanees $i$
and $i^{\prime}$ vanishes if one of them is offered a DPO, i.e. $x_{i,1}=1$ or
$x_{i^{\prime},1}=1$. The hyper-parameter $\epsilon\in[0,1)$ tunes the
competition between the expected return to the bank and the financial welfare
in the network of loanees. Increasing $\epsilon$ will cause the optimal
solution to contain a fewer number of DPO actions being taken, which increases
the overall financial welfare of the loanees.

We now introduce local constraints that force only one action to be taken to
each loanee as

$\sum\limits_{j=1}^{M}x_{i,j}=1,\forall i\in\\{1,...,N\\}.$ (2)

LLP is treated as a global constraint as

$\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}l_{i,j}x_{i,j}\leq L,$ (3)

where $L\in\mathbb{R}^{+}$ is the upper bound for the total provision and
$l_{i,j}\in\mathbb{R}^{+}$ is the provision for taking action $j$ to loanee
$i$. Determining the values of $l_{i,j}$ depends on probability of default
estimates and expected loss. Exact details are bank-specific and depends on
government regulation, such as the Third Basel Accord, an international,
voluntary regulatory framework for banks developed by the Basel Committee on
Banking Supervision [17].

Although $L$ can be considered a fixed value and a hard constraint, in
practice when we are predicting the net income of a known collection of loans,
we might allow some leniency in terms of $L$ in favour of a solution that
yields high net income. In this scenario, we treat Eq. (3) as a soft
constraint with a secondary objective of minimizing the total provision during
optimization.

Figure 1: Schematic of the hybrid quantum-classical algorithm. Firstly, the
financial transaction network as indicated by $A_{i,i^{\prime}}$ is divided
into small groups using the Greedy Modularity algorithm and the Louvian
method. The edge nodes, labelled in red, are introduced to each group to
accommodate common loanees among the two groups. The optimal set of collection
actions for each group is obtained via the QAOA. Solutions from different
groups are then combined to reconstruct the optimal actions for the original
problem. The latter is fed to the GPR algorithm to minimize LLP. See the main
text for more details.

##  III Hybrid Quantum-Classical Algorithm

In this section, we present our algorithms, as depicted in Fig. 1, to find a
set of optimal actions $\\{x_{i,j}\\}$ that maximizes $Y$ subjected to the
constraints in Eq. (2) and Eq. (3). The idea is to use a classical divide-and-
conquer algorithm [22] to arrange the loanees into small groups based on the
association matrix. Loanees from the same group will have a higher association
compared to those outside the group. Next, we find the optimal collection
actions for each group using the Quantum Alternating Operator Ansatz (QAOA)
algorithm [23], without considering LLPs. Actions from different groups are
then combined to reconstruct the optimal actions for the original problem. The
latter process is referred to as state reconstruction. Lastly, we apply a
classical sampling method which we call Greedy Provision Reduction (GPR) to
adjust some actions to minimize LLPs while keeping the negative impacts of
those adjustments on $Y$ at a minimum. We lay out the details of each step
below. The pseudo-code is provided in Appendix B.

Division algorithm -. To divide the loanees into groups, we rely on the use of
two standard community detection algorithms, namely the Clauset-Newman-Moore
greedy modularity maximization [24] and the Louvain method [25]. After the
first iteration, each loanee will be assigned to exactly one group. For
example, in Fig. 1, the two groups $S_{1}$ and $S_{2}$ contain loanees
$i=1,2,3,4,5$ and $i=8,10,11$, respectively 222We note that, in general, the
combination of both community detection algorithms will iterate until
modularity maximization is convergence, which can result in multiple group
partition, as opposed to the example schematic in Fig. 1. As each group will
first be optimized separately, the sparse connection between $S_{1}$ and
$S_{2}$ through loanees $i=7,9$ will be ignored. To mitigate this algorithmic
artefact in this divide-and-conquer approach, we introduce ‘edge nodes’, which
are outsider loanees that have an association with $S_{1}$ and $S_{2}$. As
shown in Fig. 1, the groups $S_{1}$, $S_{2}$ have edge nodes $i=6,7,9$ and
$i=7,9$, respectively. By adding the edge nodes into each group, the new
groups now have loanees $i=7,9$ in common. The latter will play a crucial role
during state reconstruction. The Louvian method and the edge node introduction
are then applied recursively to $S_{1}$ and $S_{2}$ until the size of each
group is no greater than some fixed threshold $\nu\in\mathbb{Z}^{+}$. The
latter ensures that the size of each group will be small enough to be run on
near-term quantum devices.

Figure 2: Qubit representation of loanees. (Left) A group of loanees and their
interactions as indicated by $A_{i,i^{\prime}}$.(Right) The qubit topology to
run the QAOA to find optimal collection actions for this set of loanees. Each
loanee is represented by $M$ qubits ($M=3$ in this figure). The interactions
among qubits as labelled by solid grey lines and dashed red lines are captured
by $\hat{H}_{A}$ and $\hat{H}_{B}$, respectively.

QAOA algorithm -. To apply the QAOA, we represent a set of actions applied to
the loanees as a basis state of a quantum system consisting of up to
$\nu\times M$ qubits. Action $j$ is taken to loanee $i$ ($x_{i,j}=1$) if qubit
$(i,j)$ is in the excited state $|1\rangle$; otherwise, the qubit is in the
ground state $|0\rangle$. The QAOA is executed by running the following
evolution on a quantum hardware,

$|\psi_{T}(\underline{\theta})\rangle=\prod_{t=1}^{T}e^{-i\hat{H}_{A}\gamma_{t}}e^{-i\hat{H}_{B}\beta_{t}}|\psi_{0}\rangle,$
(4)

where $T$ is the number of driving cycles, $\gamma_{t},\beta_{t}\in\mathbb{R}$
are variational parameters and
$\underline{\theta}=\\{\gamma_{1},\beta_{1},..,\gamma_{T},\beta_{T}\\}$. The
Hamiltonian $\hat{H}_{A}$ involves interactions among qubits with $j=1$ as
labeled by solid grey lines in Fig. 2. It is defined as

$\hat{H}_{A}=-(1-\epsilon)\sum\limits_{i=1}^{N^{\prime}}\sum\limits_{j=1}^{M}h_{i,j}\hat{n}_{i,j}-\epsilon\sum\limits_{\langle
i,i^{\prime}\rangle}^{N^{\prime}}A_{i,i^{\prime}}(1-\hat{n}_{i,1})(1-\hat{n}_{i^{\prime},1}).$
(5)

Here, $N^{\prime}\leq\nu$ is the number of loanees in the group and
$\hat{n}_{i,j}$ is the number operator acting on qubit $(i,j)$. The mixing
Hamiltonian $\hat{H}_{B}$ involves ring-type interactions among qubits with
the same $i$ as labeled by red dotted lines in Fig. 2. It is defined as

$\hat{H}_{B}=-\frac{J}{2}\sum_{i=1}^{N^{\prime}}\left[\sum_{j=1}^{M}\left(\hat{X}_{i,j}\hat{X}_{i,j+1}+\hat{Y}_{i,j+1}\hat{Y}_{i,j}\right)\right],$
(6)

where $\hat{X}_{i,j}$ and $\hat{Y}_{i,j}$ are Pauli’s operators acting on
qubit $(i,j)$ and $J=1$ is the coupling strength. The periodic boundary
condition is applied.

The initial state $|\psi_{0}\rangle$ is prepared as a product state such that
action $j=1$ is taken to every loanee, i.e., $x_{i,j}=\delta_{j,1}$ where
$\delta_{j,1}$ is the Kronecker delta. Note that since both $\hat{H}_{A}$ and
$\hat{H}_{B}$ preserve the number of qubit excitations per loanee, it follows
that all basis states involved in $|\psi_{T}(\underline{\theta})\rangle$
always satisfies the constraints in Eq. (2).

Following the standard QAOA procedure, we then readout the observable
$\langle\hat{H}_{A}\rangle_{\underline{\theta}}$ from the quantum hardware.
The procedure is repeated with a new value of $\underline{\theta}$ as
suggested by a classical optimizer until
$\langle\hat{H}_{A}\rangle_{\underline{\theta}}$ is minimized with
$\underline{\theta}=\underline{\theta}_{\rm opt}$. The optimal quantum state
$|\psi_{T}(\underline{\theta}_{\rm opt})\rangle$ now concentrates around bit-
strings with high objective function values. The probability of sampling a bit
string $\underline{x}=\left[x_{1,1},x_{1,2},..,x_{N,M}\right]$ from
$|\psi(\underline{\theta}_{\rm opt})\rangle$ is given by
$p(\underline{x})=|\langle\underline{x}|\psi_{T}(\underline{\theta}_{\rm
opt})\rangle|^{2}$, where $|\underline{x}\rangle$ is a basis state
representing a set of actions $\underline{x}$.

Reconstruction algorithm -. To combine optimal actions from two groups, for
example $S_{1}$ and $S_{2}$, we simply sample bit-strings from
$p(\underline{x})$ of each group. If the bit-strings from $S_{1}$ and $S_{2}$
have the same values at the common edge nodes, we append the two bit-strings.
For example, as depicted in Fig. 1, $S_{1}$ has optimal actions
$j=2,4,3,1,5,1,1,2$ for loanees $i=1,2,3,4,5,6,7,9$, respectively and $S_{2}$
has optimal actions $j=1,3,2,3,4$ for loanees $i=7,8,9,10,11$, respectively.
Since the common loanees $i=7,9$ have the same actions $j=1,2$ in both $S_{1}$
and $S_{2}$. The optimal actions for the combined group $S_{1}\cup S_{2}$ is
$j=2,4,3,1,5,1,1,3,2,3,4$ for loanees $i=1,2,...,11$, respectively. If the
actions to the common loanees from the two groups are incompatible, then the
bit-strings are resampled from $p(\underline{x})$ until the compatible one is
found. This procedure is repeated for every group until the optimal actions
for the original problem with $N$ loanees is obtained.

Greedy Provision Reduction (GPR) algorithm -. Lastly, we apply a classical
sampling algorithm to handle the LLP constraint in Eq. (3) by minimizing the
total provision. The idea is to quantify the impact of changing action $j$ of
loanee $i$ to $j^{\prime}$ using a finesse score $f_{i,j\to
j^{\prime}}\in\mathbb{R}$. For a given set of actions $\\{i,j\\}$, there are
$N\times M$ values of $f_{i,j\to j^{\prime}}$’s. We then choose $f_{i,j\to
j^{\prime}}$ with the highest value and update the action accordingly. To
define $f_{i,j\to j^{\prime}}$, we first define a reward $a_{i,j\to
j^{\prime}}$ as a reduction in the total provision and a penalty $b_{i,j\to
j^{\prime}}$ as a reduction in $Y$, when changing action $j$ of loanee $i$ to
action $j^{\prime}$. By definition, the finesse score increases when switching
actions result in a higher reward and a lower penalty. Specifically, if the
total provision does not increase ($a_{i,j\to j^{\prime}}>=0$) and $Y$ does
not decrease ($b_{i,j\to j^{\prime}}<=0$), we will choose the action with the
highest reward, i.e., $f_{i,j\to j^{\prime}}=a_{i,j\to j^{\prime}}$. If both
the provision and $Y$ decreases ($a_{i,j\to j^{\prime}}>0$ and $b_{i,j\to
j^{\prime}}>0$), we set $f_{i,j\to j^{\prime}}=-b_{i,j\to
j^{\prime}}/a_{i,j\to j^{\prime}}$. Finally, if the total provision increases
($a_{i,j\to j^{\prime}}<0$) or the total provision stays constant ($a_{i,j\to
j^{\prime}}=0$) but $Y$ decreases ($b_{i,j\to j^{\prime}}>0$), we set
$f_{i,j\to j^{\prime}}=-\infty$ indicating that this action will not be
chosen. The update procedure is carried out iteratively, one action adjustment
at a time, until the total provision is no greater than $L$ as indicated in
Eq. (3). In the case $\epsilon=0$, there exists a theoretical bound to ensure
that the optimal $Y$ obtained from this procedure will not be less than half
of the global optimal value [27].

Figure 3: Statistics of model parameters obtained from Kasikornbank data set.
(a) Box plot of $h_{i,j}$ and $A_{i,i^{\prime}}$ showing their distributions
and quantiles. (b) Box plot of $l_{i,j}$ in a log scale. (c) The histogram of
the node degrees in $A_{i,i^{\prime}}$.  Figure 4: Performance of the hybrid
algorithm. (a)-(b) shows $Y$ and the total provision during the GPR algorithms
with $\epsilon=0.5$ as a function of the number of steps $k$ in GPR,
respectively. The solid green lines are when the solutions reconstructed from
the QAOA are used as an input and the dashed grey lines are when random
solutions are used as an input. (c)-(d) shows $Y$ and the total provision as a
function of $\epsilon$, respectively. The square green dots are from the
hybrid algorithm and the circle grey dots are from the standalone GPR.
($\nu=7$, $T=2$). Figure 5: Total net profit presented to the bank as a
function of the number of DPO actions being taken. The square green dots are
from the hybrid algorithm and the circle grey dots are from the standalone
classical GPR. Note that our hybrid quantum-classical always yield a better
total net profit presented to the bank compared to the standalone classical
GPR algorithm ($\nu=7$, $T=2$).

##  IV Results and Discussions

###  IV.1 data set

We use the data set of a personal loan product provided by Kasikornbank. The
data set consists of 600 loanees and $5$ possible actions per loanee.
$h_{i,j}$ is estimated from $\it{Q}-$learning using the historical data of the
loanees [28]. $l_{i,j}$ is related to historical data on repayments and
default, loan collection expenses, credit losses, economic conditions,
interest rate and tax policy [17]. Specific details of the derivations are
omitted here due to data privacy. $A_{i,i^{\prime}}$ is estimated from
transactions between two loanees that happen internally within Kasikornbank.
Note that the actual transactions $\tilde{A}_{i,i^{\prime}}$between the two
loanees could be higher than this value, i.e.,
$\tilde{A}_{i,i^{\prime}}=\alpha_{i,i^{\prime}}A_{i,i^{\prime}}$ with
$\alpha_{i,i^{\prime}}\geq 1$. However, it is not possible for a bank to
collect every transaction from a customer. Therefore, we assume that
$\alpha_{i,i^{\prime}}$ does not depend on $i$ and $i^{\prime}$, so that it
can be absorbed into the redefinition of $\epsilon$. The interpretation of
$\epsilon$ in practice will be discussed later in the text.

The distributions of $h_{i,j}$, $l_{i,j}$, and $A_{i,i^{\prime}}$ are depicted
in Fig. 3. The units are made arbitrary for privacy concerns. The value of
$h_{i,j}$ and $A_{i,i^{\prime}}$ range from $0$ to $0.7$, while the value of
$l_{i,j}$ ranges from $\sim 10^{-5}$ to $\sim 1$. The financial network, as
indicated by $A_{i,i^{\prime}}$, is sparse with node degrees ranging from $0$
to $7$. We also benchmark our analysis with a simulated data set in Appendix A
to ensure the generality of our results.

###  IV.2 Numerical Results

To analyze the behavior of the QAOA, we will compare our hybrid algorithm with
a purely classical algorithm. The latter is achieved by directly feeding a
random set of actions to the GPR algorithm. In Fig. 4(a), with $\epsilon=0.5$
where the optimization objective values the expected return to the bank and
the financial welfare among loanees equally, we see that the set of actions
reconstructed from the QAOA starts with $Y\sim 120$ and then increases to
$\sim 140$ during the GPR algorithm. On the other hand, a random set of
actions start with $Y\sim 10$ and then increases to $\sim 100$, lower than the
one with the QAOA. Fig. 4(b) shows that, in both cases, the provision reduces
from $\sim 5.4$ to $\sim 4.4$, resulting in $18.5\%$ reduction. We found that
for other values of $\epsilon$, $Y$ always monotonically increases, and the
provision always monotonically decreases during the GPR algorithm. Hence,
there is no need to introduce $L$ to truncate the process. The minimum LLP is
treated as a suggested LLP rather than a hard constraint.

Fig. 4(c) shows $Y$ after the GPR algorithm as a function of $\epsilon$. We
can see that the hybrid algorithm provides a higher $Y$ compared to the
standalone GPR algorithm for all values of $\epsilon$. Fig. 4(d) shows the
provision after the GPR algorithm as a function of $\epsilon$. At
$\epsilon\lesssim 0.1$, the standalone GPR gives around $17\%$ lower provision
compared to the hybrid algorithm. This number is reduced to $\sim 4.3\%$ as
$\epsilon$ goes towards unity. This result is expected as the QAOA attempts to
increase $Y$ alone without considering the LLPs. Nevertheless, the provision
obtained from the hybrid algorithm is kept at $\sim 4.6$ for all values of
$\epsilon$. This value is $57.4\%$ lower than the highest possible provision.

Finally, we note that $\epsilon$ may be hard to interpret in practice because
it is normalized by the unknown variable $\alpha_{i,i^{\prime}}$ as discussed
above. To circumvent this, in Fig. 5, we plot the total net profit presented
to the bank as a function of the optimal number of DPO actions $N_{f}$, i.e.
the number of loanees with action $j=1$. The latter is, in turn, varied by
$\epsilon$. Fig. 5 provides an intuitive interpretation of the optimization
results. The collector can choose the number of DPO actions based on his/her
experience, then read out the estimated net profit that will be made. We found
that, for the same $N_{f}$, the hybrid algorithm always gives a higher
expected return. Specifically, for $N_{f}\sim 30-100$, the expected profit
from the hybrid algorithm is $\sim 70\%$ higher than the standalone GPR. In
addition, with the hybrid algorithm, we find that the expected return shows a
plateau at $N_{f}\sim 20-100$. This implies that the collector may choose to
apply, say, 30 DPOs to get the profit of $\sim 175$, instead of 120 DPOs
($300\%$ more) to get a profit of $\sim 200$ (only $14.3\%$ more).

##  V Conclusions

We devise a hybrid quantum-classical algorithm to solve loan collection
problems with LLPs by formulating them as QCBO models. Our approach allows the
lender to maximize the expected net return while accounting for LLPs and the
financial well-being of the loanees, as measured by the association matrix.
Compared to a purely classical approach, our hybrid algorithm provides a
higher net profit, regardless of the amount of the network effect as measured
by $\epsilon$. The algorithm also suggests the collector with the lowest
number of DPO actions that still gives a relatively high expected return. Our
work paves a way to explore quantum advantage with near-term quantum devices
in the financial sector.

##  VI Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  VII Acknowledgement

We thank Vorapong Suppakitpaisarn for fruitful discussions. P.
Palittapongarnpim , P. Chaiwongkhot, and T. Chotibut are supported by the
Program Management Unit for Human Resources and Institutional Development,
Research and Innovation (Grant No. B05F630108). J.Tangpanitanon, P.
Prugsanapan, N. Raksasri and Y. Raksri are supported by Kasikorn Business-
Technology Group, Bangkok, Thailand.

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##  Appendix A Benchmarking with a Simulated Data Set

In this section, we apply the algorithms presented in the main text to a
simulated data set to show the generality of our results. We use $M=5$ and
$N=600$ as in the main text. We generate $h_{i,j}$ by sampling from the
distribution
$P(i,j)\equiv\left(\log_{10}(j+2)-(1-r_{i})\log_{10}(j+1)\right)/\mathcal{C}$,
which is a modified Benford’s law. Here, $\mathcal{C}$ is a normalization
constant, $r_{i}$ is a random number drawn from the Poisson distribution with
the mean value 0.7. This distribution has two peaks: one at $j=1$ (DPO action)
and $j=5$ (non-DPO action). The association $A_{i,i^{\prime}}$ is generated
from an Erdos-Renyi ensemble with $N$ nodes and mean degree 2, which has an
average path length of approximately $\log(N)/\log(2)$. In Fig. 6, we plot the
total net profit present to the bank as the number of DPO actions using
optimum solutions from the hybrid quantum-classical and the standalone
classical algorithm as in the main text. We found that the hybrid algorithm
gives a higher return compared to the classical part as expected.

Figure 6: Total net profit presented to the bank as a function of the number
of DPO actions being taken. The square green dots are from the hybrid
algorithm and the circle grey dots are from the standalone classical GPR.
($\nu=7$, $T=2$).

##  Appendix B Pseudo-codes for Hybrid Quantum-Classical Algorithms

In this section, we provide below pseudo-codes for four algorithms involved in
the hybrid quantum-classical algorithms. These are (i) recursive division in
the divide-and-conquer algorithm, (ii) QAOA with local constraints, (iii)
state reconstruction which is the second part of the divide-and-conquer
algorithm, and (iv) Greedy Provision Reduction. See the main text for an
overview discussion for each algorithm.

input : (i) Association matrix $A=(A_{i,i^{\prime}})$.
(ii) Maximum number of nodes per subgraphs $\nu\in\mathbb{Z}^{+}$.

output : A set of subgraphs. Each subgraph represents a group of loanees with
high association among them. Each node in each subgraph represents one loanee.

begin

$\mathcal{S}\leftarrow\\{\\}$

$G\leftarrow$ $\langle$a graph obtained from $A$ $\rangle$

$\mathcal{S}^{(1)}\leftarrow$ $\langle$a set of subgraphs of $G$ obtained from
greedy modularity$\rangle$

for  _$g^{(1)}\in\mathcal{S}^{(1)}$_ do

$S\leftarrow S\cup\textnormal{{RecLouvain}}(g^{(1)})$

return $\mathcal{S}$

function $\textnormal{{RecLouvain}}(g^{(1)})$

$\mathcal{S}^{(3)}\leftarrow\\{\\}$

$\mathcal{S}^{(2)}\leftarrow$ $\langle$a set of subgraphs of $g^{(1)}$
obtained from the Louvain method$\rangle$

for  _$g^{(2)}\in\mathcal{S}^{(2)}$_ do

% Introduce edge nodes

$\mathcal{W}\leftarrow[]$

$\mathcal{V}\leftarrow$ $\langle$a set of nodes in $g^{(2)}$$\rangle$

for  _$i\in\mathcal{V}$_ do

$\mathcal{E}_{i}\leftarrow$ $\langle$a set of edges in $G$ that are incident
to $i$$\rangle$

for  _$(i,i^{\prime})\in\mathcal{E}_{i}$_ do

if  _$i^{\prime}\notin\mathcal{V}$ and $i^{\prime}\notin\mathcal{W}$_ then

$\mathcal{W}\leftarrow\mathcal{W}\cup\\{i^{\prime}\\}$

$g^{(2)}\leftarrow$ $\langle$add nodes in $\mathcal{W}$ and corresponding
edges to $g^{(2)}$$\rangle$

% Recursive division

if  _$|g^{(2)}| >\nu$_ then

$\mathcal{S}^{(3)}\leftarrow\mathcal{S}^{(3)}\cup\textnormal{{RecLouvain}}(g^{(2)})$

else

$\mathcal{S}^{(3)}\leftarrow\mathcal{S}^{(3)}\cup\\{g^{(2)}\\}$

return $\mathcal{S}^{(3)}$

Algorithm 1 Recursive Division

input : (i) Local fields {$h_{i,j}\\}$ for subgraph $g$.
(ii) Association $\\{A_{i,i^{\prime}}\\}$ for subgraph $g$.
(iii) The number of driving cycles $T$ in QAOA.
(iv) The maximum number of iterations $\eta$ in the COBYLA.

output : A set of probabilities
$\\{p(\underline{x})|\forall\underline{x}\in\\{0,1\\}^{N_{g}\times M}\\}$,
where $N_{g}$ is the number of loanees in subgroup $g$. Configurations
$\underline{x}$ that have high values of the objective function will have high
$p(\underline{x})$.

begin

$\underline{\theta}\leftarrow$ $\langle$a vector of size $2T$ containing
randomized real numbers $\in[0,1)$$\rangle$

$\underline{\theta}_{\rm opt}\leftarrow$ $\langle$minimizes
$\textnormal{{Energy}}(\underline{\theta})$ using COBYLA$\rangle$

$|\psi_{\rm
opt}\rangle\leftarrow\textnormal{{Evolve}}(|\psi_{0}\rangle,\underline{\theta}_{\rm
opt})$

return $\\{p(\underline{x})=\psi_{\rm opt}\rangle|^{2}\\}$

function $\textnormal{{Energy}}(\underline{\theta})$

$|\psi\rangle\leftarrow\textnormal{{Evolve}}(|\psi_{0}\rangle,\underline{\theta})$

return $\langle\psi|\hat{H}_{B}|\psi\rangle$

function $\textnormal{{Evolve}}(|\psi\rangle,\underline{\theta})$

for  _$t=0$ to $T-1$_ do

% Exploitation

$\hat{U}_{A}\leftarrow\exp\left(-i\hat{H}_{A}\cdot\underline{\theta}[2t]\right)$

% Exploration

$\hat{U}_{B}\leftarrow\exp\left(-i\hat{H}_{B}\cdot\underline{\theta}[2t+1]\right)$

$|\psi\rangle\leftarrow\hat{U}_{A}\hat{U}_{B}|\psi\rangle$

return $|\psi\rangle$

Algorithm 2 QAOA with local constraints

input : (i) A set of the subgraphs $\mathcal{S}$ obtained from the Recursive
Division algorithm.
(ii) A set of probabilities for every subgraphs
$\\{p_{g}(\underline{x})|\forall\underline{x}\in\\{0,1\\}^{N_{g}\times
M},\forall g\in\mathcal{S}\\}$ obtained from QAOA.
(iii) A hyper-parameter for the maximum number of candidates per subgraph
$\lambda$.

output : A set of actions that maximises $Y$.

begin

$g_{L}\leftarrow$ $\langle$a randomly chosen subgraph from
$\mathcal{S}$$\rangle$

$\mathcal{X}_{L}\leftarrow$ $\langle$a set of the first $\lambda$ bit-strings
$\\{\underline{x}\\}$ from subgraph $g_{L}$ that have the highest
probabilities $\\{p_{gL}(\underline{x})\\}$$\rangle$

for  _$g_{R}\in\mathcal{S}\cap\\{g_{L}\\}$_ do

$\mathcal{X}_{R}\leftarrow$ $\langle$a set of the first $\lambda$ bit-strings
$\\{\underline{x}\\}$ from subgraph $g_{R}$ that have the highest
probabilities $\\{p_{g_{R}}(\underline{x})\\}$$\rangle$

$\mathcal{W}_{LR}\leftarrow$ $\langle$a set of edge nodes between $g_{L}$ and
$g_{R}$$\rangle$

$\mathcal{X}_{LR}=\\{\\}$

for  _$\underline{x}_{L}\in\mathcal{X}_{L}$_ do

for  _$\underline{x}_{R}\in\mathcal{X}_{R}$_ do

$\mathcal{X}_{LR}\leftarrow\mathcal{X}_{LR}\cup\\{\textnormal{{Combine}}(\underline{x}_{L},\underline{x}_{R},\mathcal{W}_{LR})\\}$

while  _$|\mathcal{X}_{LR}|==0$_ do

$\underline{x}^{\prime}_{R}\leftarrow$ $\langle$the next-highest-probability
bit-string from $g_{R}$$\rangle$

for  _$\underline{x}_{L}\in\mathcal{X}_{L}$_ do

$\mathcal{X}_{LR}\leftarrow\mathcal{X}_{LR}\cup\\{\textnormal{{Combine}}(\underline{x}_{L},\underline{x}^{\prime}_{R},\mathcal{W}_{LR})\\}$

if  _$|\mathcal{X}_{LR}| >\lambda$_ then

$\mathcal{X}_{LR}\leftarrow$ $\langle$only keep the first $\lambda$ elements
that have the highest $Y$$\rangle$

$g_{L}\leftarrow$ $\langle$combine nodes and edges in $g_{L}$ and
$g_{R}$.$\rangle$

$\mathcal{X}_{L}\leftarrow\mathcal{X}_{LR}$

return $\mathcal{X}_{L}$

function
$\textnormal{{Combine}}(\underline{x}_{L},\underline{x}_{R},\mathcal{W}_{LR})$

if  _actions at the edge nodes from $\underline{x}_{L}$ and
$\underline{x}_{R}$ are the same_ then

return $\langle$a combined bit-string $\underline{x}_{LR}$ as explained in the
main text.$\rangle$

Algorithm 3 State Reconstruction

input : (i) A set of actions $\underline{j}$, where the element
$\underline{j}[i]\in\\{1,2,..,M\\}$ is the optimal action $j$ that is taken to
loanee $i$.
(ii) The maximum number of iteration $\eta$.

output : A new optimal actions that minimises the LLP.

begin

$\underline{j}_{\rm best}\leftarrow\underline{j}$

$y_{\rm best}\leftarrow$ $\langle$The objective function for
$\underline{j}$$\rangle$

$l_{\rm best}\leftarrow$ $\langle$The LLP of $\underline{j}$$\rangle$

for  _$k=1$ to $k=\eta$_ do

$F\leftarrow\\{\\}$

for  _$i=1$ to $i=N$_ do

for  _$j^{\prime}=1$ to $j^{\prime}=M$_ do

$\underline{j}_{\rm new}\leftarrow$$\langle$a new set of actions where
$\underline{j}[i]$ is changed to $j^{\prime}$$\rangle$

$y_{\rm new}\leftarrow$$\langle$the objective function for $\underline{j}_{\rm
new}$$\rangle$

$l_{\rm new}\leftarrow$$\langle$the LLP of $\underline{j}_{\rm new}$$\rangle$

% Reward

$a_{i,j\to j^{\prime}}\leftarrow l_{\rm best}-l_{\rm new}$

% Penalty

$b_{i,j\to j^{\prime}}\leftarrow y_{\rm best}-y_{\rm new}$

% Finesse score

if  _$a_{i,j\to j^{\prime}} >0$ and $b_{i,j\to j^{\prime}}\leq 0$_ then

$f_{i,j\to j^{\prime}}\leftarrow a_{i,j\to j^{\prime}}$

if  _$a_{i,j\to j^{\prime}} >0$ and $b_{i,j\to j^{\prime}}>0$_ then

$f_{i,j\to j^{\prime}}\leftarrow-b_{i,j\to j^{\prime}}/a_{i,j\to j^{\prime}}$

if  _$a_{i,j\to j^{\prime}} <0$_ then

$f_{i,j\to j^{\prime}}\leftarrow-\infty$

if  _$a_{i,j\to j^{\prime}}=0$ and $b_{i,j\to j^{\prime}}\>0$_ then

$f_{i,j\to j^{\prime}}\leftarrow-\infty$

$F\leftarrow F\cup\\{f_{i,j\to j^{\prime}}\\}$

$\underline{j}_{\rm best}\leftarrow$$\langle$a new set of actions where
$\underline{j}[i]$ is changed to $j^{\prime}$ such that $f_{i,j\to
j^{\prime}}\in F$ is maximized$\rangle$

$y_{\rm best}\leftarrow$ $\langle$The objective function for
$\underline{j}$$\rangle$

$l_{\rm best}\leftarrow$ $\langle$The LLP for $\underline{j}$$\rangle$

return $\underline{u}_{\rm best}$

Algorithm 4 Greedy Provision Reduction


person games and three-person games, it can be observed that DCH provides a
significantly better fit to the data compared to the standard CH model (Vuong
test: p-value $<0.001$). However, there is no statistically significant
difference between DCH and AQRE (Vuong test p-value $=0.173$). Furthermore, I
conduct a likelihood ratio test on all three models to assess whether allowing
different parameters for two-person and three-person games can significantly
improve the model fit. The results indicate that the heterogeneous models are
significantly better than the homogeneous models. Taken together, these
findings lead to the conclusion that both the level of sophistication and the
precision vary with the complexity of the games.

To summarize, it is not surprising that DCH can provide a better explanation
for the data compared to the misspecified standard CH model in dynamic games.
However, what is surprising is that when the CH model is correctly specified,
the estimated average level of sophistication is $1.03$, which falls within
the expected range of a “regular” $\tau$ value between 1 and 2, as predicted
by Camerer et al. (2004).

##  Appendix E Supplementary Analysis for Experimental Data (For Online
Publication)

###  E.1 Supplementary Tables

This section includes all the supplementary tables from the experiment. Table
A.4 lists the empirical frequencies of choosing $C$ at each information set
for both treatments. Table A.5 presents the empirical frequencies of choosing
$C$ at information set $(2,X)$ for different payoff structures and treatments.

Table A.4: The Empirical Frequencies of $C$ at Each Information Set

Sequential Treatment Simultaneous Treatment $x_{-i}$ $O$ $X$ $O$ $X$ Obs Claim
% s.d. Obs Claim % s.d. Obs Claim % s.d. Obs Claim % s.d. Periods 1 148 0.845
0.364 572 0.313 0.464 148 0.811 0.393 548 0.263 0.441 2 16 0.438 0.512 210
0.600 0.491 28 0.250 0.441 404 0.223 0.417 3 4 0.000 0.000 34 0.206 0.410 21
0.190 0.402 314 0.172 0.378 4 3 0.000 0.000 21 0.190 0.402 16 0.250 0.447 259
0.131 0.338 5 2 0.500 0.707 14 0.214 0.426 14 0.143 0.363 227 0.172 0.378 •
Note: For the simultaneous treatment, the choice data at the information set
level are implied by the contingent strategies. For instance, choosing the
contingent strategy “claim at period 4” implies that the subject will wait
from period 1 to period 3 and claim in period 4.

Table A.5: The Empirical Frequencies of $C$ at Information Set $(2,X)$ for Different Games | Sequential Treatment |  | Simultaneous Treatment
---|---|---|---
$(\delta,\bar{\alpha})$ | Obs | Claim % | s.d. |  | Obs | Claim % | s.d.
Diagnostic Games |  |  |  |  |  |  |
(0.60, 0.45) | 39 | 0.564 | 0.502 |  | 78 | 0.256 | 0.439
(0.95, 0.80) | 36 | 0.667 | 0.479 |  | 59 | 0.237 | 0.429
Control Games |  |  |  |  |  |  |
(0.60, 0.80) | 38 | 0.789 | 0.413 |  | 61 | 0.361 | 0.484
(0.80, 0.45) | 35 | 0.543 | 0.505 |  | 67 | 0.134 | 0.344
(0.80, 0.80) | 24 | 0.542 | 0.509 |  | 63 | 0.190 | 0.396
(0.95, 0.45) | 38 | 0.474 | 0.506 |  | 76 | 0.171 | 0.379

To compute the measure of violation of invariance under strategic equivalence
of each payoff structure $(\delta,\bar{\alpha})$, I run the following
regression on the data of information set $(2,X)$:

$\displaystyle\mathds{1}\\{\mbox{claim}\\}_{i}=\alpha_{0}+\alpha_{1}\mathds{1}\\{\mbox{sequential}\\}_{i}+\epsilon_{i}$
(A.7)

where $\mathds{1}\\{\mbox{claim}\\}_{i}$ is the dummy variable for player $i$
choosing $C$ and $\mathds{1}\\{\mbox{sequential}\\}_{i}$ is the dummy variable
for the sequential treatment. Table A.6 reports the results for all payoff
structures. The standard errors are clustered at the session level.

Table A.6: The Magnitude of the Treatment Effect for Different Games

Payoff Structure $(\delta,\bar{\alpha})$ (0.60, 0.45) (0.60, 0.80) (0.80,
0.45) (0.80, 0.80) (0.95, 0.45) (0.95, 0.80) Sequential Treatment ***0.308***
*0.429* *0.409* **0.351** **0.303** **0.429** (0.066) (0.160) (0.160) (0.101)
(0.073) (0.110) Constant ***0.256*** ***0.361*** ***0.134*** ***0.190***
**0.171** **0.237** (0.057) (0.070) (0.018) (0.025) (0.049) (0.062) N 117 99
102 87 114 95 R-squared 0.0914 0.1744 0.1889 0.1203 0.1028 0.1808 • Note: The
standard errors are clustered at the session level. * $p<0.05$, ** $p<0.01$,
*** $p<0.001$.

Finally, Table A.7 and A.8 report the distributions of reaction times when
players see a dirty face in the sequential and simultaneous treatments,
respectively.

Table A.7: Reaction Times (seconds) when Seeing $X$ in the Sequential Treatment Periods | Obs | Mean | s.d. | Q1 | Median | Q3
---|---|---|---|---|---|---
1 | 572 | 11.29 | 9.845 | 5.359 | 7.574 | 13.72
2 | 210 | 8.172 | 6.531 | 4.495 | 6.106 | 9.745
3 | 34 | 7.530 | 3.816 | 4.785 | 6.277 | 11.44
4 | 21 | 6.901 | 4.135 | 3.150 | 6.299 | 9.767
5 | 14 | 7.663 | 5.597 | 3.231 | 6.677 | 8.856
All | 851 | 10.20 | 8.917 | 5.000 | 7.152 | 12.08
Table A.8: Reaction Times (seconds) when Seeing $X$ in the Simultaneous Treatment Stopping Strategies | Obs | Mean | s.d. | Q1 | Median | Q3
---|---|---|---|---|---|---
Claim at 1 | 144 | 11.93 | 8.804 | 5.811 | 8.958 | 15.91
Claim at 2 | 90 | 13.85 | 10.18 | 7.191 | 11.32 | 16.62
Claim at 3 | 54 | 17.64 | 12.81 | 8.095 | 14.30 | 24.61
Claim at 4 | 34 | 21.15 | 15.16 | 12.32 | 16.46 | 22.96
Claim at 5 | 39 | 23.34 | 14.43 | 13.84 | 19.76 | 27.83
Always Wait | 187 | 14.24 | 11.09 | 7.342 | 10.48 | 20.24
All | 548 | 15.04 | 11.58 | 7.413 | 11.32 | 19.48

###  E.2 Likelihood Functions

#### Quantal Cursed Sequential Equilibrium

The “Quantal Cursed Sequential Equilibrium (QCSE)” is a model applicable to
multi-stage games with observed actions. This model relaxes both the
requirements of best responses and Bayesian inference. Specifically, QCSE is a
hybrid model, combining the Agent Quantal Response Equilibrium (AQRE) proposed
by McKelvey and Palfrey (1998) and the Cursed Sequential Equilibrium
introduced by Fong et al. (2023a).

Consider an assessment $(\mu,\sigma)$. For any player $i$ and any history
$h^{t-1}$, the average behavioral strategy profile of $-i$ is defined as:

$\bar{\sigma}_{-i}(a_{-i}^{t}|\theta_{i},h^{t-1})=\sum_{\theta_{-i}\in\Theta_{-i}}\mu_{i}(\theta_{-i}|\theta_{i},h^{t-1})\sigma_{-i}(a_{-i}^{t}|\theta_{-i},h^{t-1}).$

In QCSE, players have incorrect perceptions about the behavioral strategies of
other players. Instead of thinking they are using $\sigma_{-i}$, a
$\chi$-cursed type $\theta_{i}$ player $i$ would believe the other players are
using a $\chi$-weighted average of the average behavioral strategy and the
true behavioral strategy:

$\sigma_{-i}^{\chi}(a_{-i}^{t}|\theta_{-i},\theta_{i},h^{t-1})=\chi\bar{\sigma}_{-i}(a_{-i}^{t}|\theta_{i},h^{t-1})+(1-\chi)\sigma_{-i}(a_{-i}^{t}|\theta_{-i},h^{t-1}).$

The beliefs of player $i$ about $\theta_{-i}$ in QCSE are updated via Bayes’
rule, whenever possible, assuming other players are using the $\chi$-cursed
behavioral strategy rather than the true behavioral strategy. This updating
rule is called the $\chi$_-cursed Bayes’ rule_. Specifically, an assessment
satisfies the $\chi$-cursed Bayes’ rule if the belief system is derived from
the Bayes’ rule while perceiving others are using $\sigma_{-i}^{\chi}$ rather
than $\sigma_{-i}$.

Consider any totally mixed strategy profile $\sigma\in\Sigma^{0}$. As shown by
Fong et al. (2023a), if the belief system $\mu$ is derived from the
$\chi$_-cursed Bayes’ rule_ , then player $i$’s cursed belief is simply a
linear combination of player $i$’s cursed belief at the beginning of that
stage (with $\chi$ weight) and the Bayesian posterior belief (with $1-\chi$
weight). That is, for any $h^{t}=(h^{t-1},a^{t})$,

$\displaystyle\mu_{i}(\theta_{-i}|\theta_{i},h^{t})=\chi\mu_{i}(\theta_{-i}|\theta_{i},h^{t-1})+(1-\chi)\left[\frac{\mu_{i}(\theta_{-i}|\theta_{i},h^{t-1})\sigma_{-i}(a_{-i}^{t}|\theta_{-i},h^{t-1})}{\sum_{\theta^{\prime}_{-i}}\mu_{i}(\theta^{\prime}_{-i}|\theta_{i},h^{t-1})\sigma_{-i}(a_{-i}^{t}|\theta^{\prime}_{-i},h^{t-1})}\right].$

For any player $i$, any $\chi\in[0,1]$, $\sigma\in\Sigma^{0}$, and type
profile $\theta\in\Theta$, let
$\rho_{i}^{\chi}(h^{T}|\theta,h^{t},\sigma_{-i}^{\chi},\sigma_{i})$ be $i$’s
perceived conditional realization probability of terminal history
$h^{T}\in\mathcal{H}^{T}$ at history
$h^{t}\in\mathcal{H}\backslash\mathcal{H}^{T}$ if the type profile is $\theta$
and $i$ uses the behavioral strategy $\sigma_{i}$ whereas perceives other
players’ using the cursed behavioral strategy $\sigma_{-i}^{\chi}$. At every
non-terminal history $h^{t}$, a $\chi$-cursed player in QCSE will use
$\chi$-cursed Bayes’ rule to derive the posterior belief about the other
players’ types. Accordingly, a type $\theta_{i}$ player $i$’s conditional
expected payoff at history $h^{t}$ is:

$\bar{u}_{i}(\sigma|\theta_{i},h^{t})\equiv\sum_{\theta_{-i}\in\Theta_{-i}}\sum_{h^{T}\in\mathcal{H}^{T}}\mu_{i}(\theta_{-i}|\theta_{i},h^{t})\rho_{i}^{\chi}(h^{T}|\theta,h^{t},\sigma_{-i}^{\chi},\sigma_{i})u_{i}(h^{T},\theta_{i},\theta_{-i}).$

Moreover, let $\bar{u}_{i}(a,\sigma|\theta_{i},h^{t})$ be the conditional
expected payoff of player $i$ of using $a\in A_{i}(h^{t})$ with probability
one, and using $\sigma_{i}$ elsewhere.

In QCSE, there is a parameter $\lambda\in[0,\infty)$ that governs the
precision of choices. Given an assessment $(\mu,\sigma)$ where
$\sigma\in\Sigma^{0}$ and $\mu$ is derived from the $\chi$-cursed Bayes’ rule,
if $(\mu,\sigma)$ is a QCSE for any player $i$, history $h^{t}$, and type
$\theta_{i}$, then type $\theta_{i}$ player $i$ will have choice probabilities
at $h^{t}$ that follow a multinomial logit distribution. In particular, the
probability of player $i$ choosing $a\in A_{i}(h^{t})$ is

$\displaystyle\frac{e^{\lambda\bar{u}_{i}(a,\sigma|\theta_{i},h^{t})}}{\sum_{a^{\prime}\in
A_{i}(h^{t})}e^{\lambda\bar{u}_{i}(a^{\prime},\sigma|\theta_{i},h^{t})}}.$

In summary, for each $\lambda\in[0,\infty)$ and $\chi\in[0,1]$, an assessment
$(\mu,\sigma)$ is a QCSE if

  * 1.

The belief system is derived from the $\chi$-cursed Bayes’ rule, and

  * 2.

For any player $i$, type $\theta_{i}$, history $h^{t}$ and $a\in
A_{i}(h^{t})$,

$\displaystyle\sigma_{i}(a|\theta_{i},h^{t})=\frac{e^{\lambda\bar{u}_{i}(a,\sigma|\theta_{i},h^{t})}}{\sum_{a^{\prime}\in
A_{i}(h^{t})}e^{\lambda\bar{u}_{i}(a^{\prime},\sigma|\theta_{i},h^{t})}}.$

When estimating QCSE, constructing the likelihood function follows a similar
process as described in Appendix D.2. For each information set
$\mathcal{I}_{i}$, QCSE uniquely predicts the choice probability of each
$a_{i}$, denoted as $\bar{Q}(a_{i}|\mathcal{I}_{i},\lambda,\chi)$, given
$\lambda$ and $\chi$. The log-likelihood function can be formed by aggregating
over every subject $i$, action $a_{i}$, and information set $\mathcal{I}_{i}$:

$\ln
L^{\bar{Q}}(\lambda,\chi)=\sum_{i}\sum_{\mathcal{I}_{i}\in\Pi_{i}}\sum_{a_{i}\in
A_{i}(\mathcal{I}_{i})}\mathds{1}\\{a_{i},\mathcal{I}_{i}\\}\ln\left[\bar{Q}(a_{i}|\mathcal{I}_{i},\lambda,\chi)\right].$

#### Quantal Dynamic Cognitive Hierarchy Solution

The “Quantal Dynamic Cognitive Hierarchy Solution (QDCH)” is a natural
extension of DCH, where all strategic levels of players make quantal responses
instead of best responses. In particular, following previous notations, for
any $i\in N$, $\tau_{i}\geq 1$, $\theta\in\Theta$, $\sigma$, and $\tau_{-i}$
such that $\tau_{j}<\tau_{i}$ for any $j\neq i$, let
$P_{i}^{\tau_{i}}(h^{T}|\theta,h^{t-1},\tau_{-i},\sigma_{-i}^{-\tau_{i}},\sigma_{i}^{\tau_{i}})$
be level $\tau_{i}$ player $i$’s belief about the conditional realization
probability of $h^{T}\in\mathcal{H}^{T}$ at history
$h^{t-1}\in\mathcal{H}\backslash\mathcal{H}^{T}$ if the type profile is
$\theta$, the level profile is $\tau$, and player $i$ uses
$\sigma_{i}^{\tau_{i}}$. In this case, level $\tau_{i}$ player $i$’s expected
payoff at any $h^{t}\in\mathcal{H}\backslash\mathcal{H}^{T}$ is:

$\displaystyle\bar{u}_{i}^{\tau_{i}}(\sigma|\theta_{i},h^{t})\equiv$
$\displaystyle\;\;\;\sum_{h^{T}\in\mathcal{H}^{T}}\sum_{\theta_{-i}\in\Theta_{-i}}\sum_{\\{\tau_{-i}:\tau_{j}<k\;\forall
j\neq
i\\}}\mu_{i}^{\tau_{i}}(\theta_{-i},\tau_{-i}\mid\theta_{i},h^{t})P_{i}^{\tau_{i}}(h^{T}|\theta,h^{t},\tau_{-i},\sigma_{-i}^{-\tau_{i}},\sigma_{i}^{\tau_{i}})u_{i}(h^{T},\theta_{i},\theta_{-i}).$

Similar to QCSE, in QDCH, there is a parameter $\lambda\in[0,\infty)$ that
governs the precision of choices. Let
$\bar{u}_{i}^{\tau_{i}}(a,\sigma|\theta_{i},h^{t})$ be the conditional
expected payoff of level $\tau_{i}$ player $i$ of using $a\in A_{i}(h^{t})$
with probability one, and using $\sigma_{i}^{\tau_{i}}$ elsewhere. In QDCH,
players’ choice probabilities follow multinomial logit distributions. That is,
in QDCH, the probability of level $\tau_{i}$ player $i$ choosing $a\in
A_{i}(h^{t})$ at history $h^{t}$ is

$\displaystyle\sigma_{i}^{\tau_{i}}(a|\theta_{i},h^{t})=\frac{e^{\lambda\bar{u}_{i}^{\tau_{i}}(a,\sigma|\theta_{i},h^{t})}}{\sum_{a^{\prime}\in
A_{i}(h^{t})}e^{\lambda\bar{u}_{i}^{\tau_{i}}(a^{\prime},\sigma|\theta_{i},h^{t})}}.$

When estimating QDCH, I assume the prior distribution of levels follows
Poisson($\tau$). At any information set $\mathcal{I}_{i}$, let
$f(k|\mathcal{I}_{i},\lambda,\tau)$ be the posterior distribution of levels at
information set $\mathcal{I}_{i}$ given $\lambda$ and $\tau$. In this case,
the predicted choice probability for $a_{i}$ at $\mathcal{I}_{i}$ is the
aggregation of quantal responses from all levels weighted by the proportion
$f(k|\mathcal{I}_{i},\lambda,\tau)$:

$\bar{\mathcal{D}}(a_{i}|\mathcal{I}_{i},\lambda,\tau)\equiv\sum_{k=0}^{\infty}f(k|\mathcal{I}_{i},\lambda,\tau)P_{k}(a_{i}|\mathcal{I}_{i},\lambda,\tau),$

where $P_{k}(a_{i}|\mathcal{I}_{i},\lambda,\tau)$ is the probability of level
$k$ players choosing $a_{i}$ at $\mathcal{I}_{i}$. Consequently, the log-
likelihood function for QDCH can be formed by aggregating over every subject
$i$, actions $a_{i}$ and information set $\mathcal{I}_{i}$:

$\ln
L^{\bar{D}}(\lambda,\tau)=\sum_{i}\sum_{\mathcal{I}_{i}\in\Pi_{i}}\sum_{a_{i}\in
A_{i}(\mathcal{I}_{i})}\mathds{1}\\{a_{i},\mathcal{I}_{i}\\}\ln\left[\bar{\mathcal{D}}(a_{i}|\mathcal{I}_{i},\lambda,\tau)\right],$

where $\mathds{1}\\{a_{i},\mathcal{I}_{i}\\}$ is the indicator function which
is 1 when subject $i$ chooses $a_{i}$ at $\mathcal{I}_{i}$.

##  Appendix F Experimental Instructions (For Online Publication)

###  F.1 Sequential Treatment

General Instructions

Thank you for participating in the experiment. You are about to take part in a
decision-making experiment, in which your earnings will depend partly on your
decisions, partly on the decision of others, and partly on chance.

The entire session will take place through computer terminals, and all
interactions between participants will be conducted through the computers.
Please do not talk or in any way try to communicate with other participants
during the session.

The main task of the experiment consists of 12 matches. Before the main task,
you will be asked to complete some comprehension questions. If you have any
questions, please raise your hand and the question will be answered so that
everyone can hear.

In this experiment, you will earn “points” in each match. Your earnings will
be determined by the total points you earn in the 12 matches. Each point has a
value of $0.02. That is, every 100 points generates $2 in earnings for you. In
addition to your earnings from decisions, you will receive a show-up fee of
$10. At the end of the experiment, your earnings will be rounded up to the
nearest dollar amount. All your earnings will be paid in cash privately at the
end of the experiment.

Main Task

1\. In this experiment, you will be asked to make decisions in 12 matches. You
will be randomly matched with another participant into a group for every
separate match. This random pairing changes in every match.

2\. Each match in this experiment corresponds to a game with the following
rules.

  * •

At the beginning of each match, each of you and the other participant will be
randomly assigned a “color” (either Red or White). After the colors are
assigned, you will be able to see the color of the other participant who is
paired with you. However, you cannot see your own color!

  * •

There are 3 possible situations, and the probabilities of these situations are
summarized in the following table.

Situations | Probabilities
---|---
You are Red and the other participant is White. | $p$
You are White and the other participant is Red. | $p$
Both of you are Red. | $1-2p$

In other words, there is always at least one Red participant among each group.

  * •

Each match is played in rounds. In each match, there are at most 5 rounds.
Your color and the other participant’s color are fixed in the match. In each
round, you and the other participant will simultaneously choose either “I’m
Red” or “wait.” If both participants choose “wait,” then the match will
continue to the next round. The match will end:

    1. (1)

after round 5; or

    2. (2)

after some round where there is at least one participant choosing “I’m Red.”

This round is called the “terminal round.” Your payoff for this match depends
on which round the terminal round is, your action in the terminal round, and
your color. Important: your payoff does not depend on the other participant’s
color.

  * •

Payoffs:

    1. (1)

If you choose “wait” in the terminal round, you will get 0 points for this
match regardless of your color.

    2. (2)

If you choose “I’m Red” in the terminal round, your payoff for this match
depends on which round the terminal round is and your own color. The payoffs
are summarized in the following table. Notice that in each match, you and the
other participant will face the same payoff table.

Terminal Round |  1 |  2 |  3 |  4 |  5
---|---|---|---|---|---
Your payoff if your color is Red | $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$ | $p_{5}$
Your payoff if your color is White | $-w_{1}$ | $-w_{2}$ | $-w_{3}$ | $-w_{4}$ | $-w_{5}$
    3. (3)

Examples:

       * a.

If you choose “I’m Red” in round 3, you will earn $p_{3}$ points if your color
is Red and $-w_{3}$ if your color is White.

       * b.

If you choose “wait” in round 4 and the other participant chooses “I’m Red” in
the same round, you will get 0 points regardless of your color.

3\. Decisions:

  * •

After observing the other participant’s color, you and the other participant
matched with you will play the game according to the rules described above.

  * •

Therefore, your payoffs are summarized as below.

|  |  | You choose “I’m Red” in the
---
terminal round
| You choose “wait” in the
---
terminal round
Terminal Round | Your color |  Red | White | Red or White
1 |  | $p_{1}$ | $-w_{1}$ | 0
2 |  | $p_{2}$ | $-w_{2}$ | 0
3 |  | $p_{3}$ | $-w_{3}$ | 0
4 |  | $p_{4}$ | $-w_{4}$ | 0
5 |  | $p_{5}$ | $-w_{5}$ | 0
  * •

Each match starts from Round 1. You will make your decision in the following
screen.

After you make your decision, the following would happen: If either you or the
other participant chooses “I’m Red,” then this round is the terminal round,
and your payoff is determined by your action in this round. However, if both
you and the other participant choose “wait,” the match continues to the next
round, and you will make your decision in the following screen.

Like the previous round, if either you or the other participant chooses “I’m
Red,” the match will end after this round. Yet if both you and the other
participant choose “wait,” the match will proceed to the next round.

  * •

If the game proceeds to round 5, then the match will end after this round and
your payoff is determined by your action (and color) in round 5.

4\. At the end of each match, there will be a summary of the match which
includes both of your colors, actions in each round (whenever applicable) and
your own payoff for this match.

5\. At the beginning of the experiment, you will start from 900 points. You
will get paid in cash based on your total points earned from the 12 matches.
If your total point is negative, you will only receive the show-up fee.

6\. Important:

  * a.

After each match, you will be randomly paired with another participant in the
next match.

  * b.

Your color and the other participant’s color will also be randomly re-drawn in
each match. The colors in each match are independent of the colors in other
matches.

  * c.

The probability distribution of colors and the payoff table will change in
each match.

Please raise your hand if you have any questions. The question will be
answered so that everyone can hear.

###  F.2 Simultaneous Treatment

General Instructions

Thank you for participating in the experiment. You are about to take part in a
decision-making experiment, in which your earnings will depend partly on your
decisions, partly on the decision of others, and partly on chance.

The entire session will take place through computer terminals, and all
interactions between participants will be conducted through the computers.
Please do not talk or in any way try to communicate with other participants
during the session.

The main task of the experiment consists of 12 matches. Before the main task,
you will be asked to complete some comprehension questions. If you have any
questions, please raise your hand and the question will be answered so that
everyone can hear.

In this experiment, you will earn “points” in each match. Your earnings will
be determined by the total points you earn in the 12 matches. Each point has a
value of $0.02. That is, every 100 points generates $2 in earnings for you. In
addition to your earnings from decisions, you will receive a show-up fee of
$10. At the end of the experiment, your earnings will be rounded up to the
nearest dollar amount. All your earnings will be paid in cash privately at the
end of the experiment.

Main Task

1\. In this experiment, you will be asked to make decisions in 12 matches. You
will be randomly matched with another participant into a group for every
separate match. This random pairing changes in every match.

2\. Each match in this experiment corresponds to a game with the following
rules.

  * •

At the beginning of each match, each of you and the other participant will be
randomly assigned a “color” (either Red or White). After the colors are
assigned, you will be able to see the color of the other participant who is
paired with you. However, you cannot see your own color!

  * •

There are 3 possible situations, and the probabilities of these situations are
summarized in the following table.

Situations | Probabilities
---|---
You are Red and the other participant is White. | $p$
You are White and the other participant is Red. | $p$
Both of you are Red. | $1-2p$

In other words, there is always at least one Red participant among each group.

  * •

Each match is played in rounds. In each match, there are at most 5 rounds.
Your color and the other participant’s color are fixed in the match. In each
round, you and the other participant will simultaneously choose either “I’m
Red” or “wait.” If both participants choose “wait,” then the match will
continue to the next round. The match will end:

    1. (1)

after round 5; or

    2. (2)

after some round where there is at least one participant choosing “I’m Red.”

This round is called the “terminal round.” Your payoff for this match depends
on which round the terminal round is, your action in the terminal round, and
your color. Important: your payoff does not depend on the other participant’s
color.

  * •

Payoffs:

    1. (1)

If you choose “wait” in the terminal round, you will get 0 points for this
match regardless of your color.

    2. (2)

If you choose “I’m Red” in the terminal round, your payoff for this match
depends on which round the terminal round is and your own color. The payoffs
are summarized in the following table. Notice that in each match, you and the
other participant will face the same payoff table.

Terminal Round |  1 |  2 |  3 |  4 |  5
---|---|---|---|---|---
Your payoff if your color is Red | $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$ | $p_{5}$
Your payoff if your color is White | $-w_{1}$ | $-w_{2}$ | $-w_{3}$ | $-w_{4}$ | $-w_{5}$
    3. (3)

Examples:

       * a.

If you choose “I’m Red” in round 3, you will earn $p_{3}$ points if your color
is Red and $-w_{3}$ if your color is White.

       * b.

If you choose “wait” in round 4 and the other participant chooses “I’m Red” in
the same round, you will get 0 points regardless of your color.

3\. Decisions:

  * •

Instead of playing the game round by round, after observing the other
participant’s color, you and the other participant are asked to simultaneously
choose a “plan” which describes how you would commit to play the game if the
game were played round by round. After you and the other participant both
submit the plans, the computer will implement the plans and your payoff is
determined accordingly.

  * •

Since the game ends after some participant chooses “I’m Red,” there are six
possible plans corresponding to the earliest round you intend to choose “I’m
Red” or “always wait.” Specifically, the six plans are listed below.

    1. $\circ$

“I’m Red in Round 1” means you plan to choose “I’m Red” in Round 1.

    2. $\circ$

“I’m Red in Round 2” means you plan to choose “wait” in Round 1 and choose
“I’m Red” in Round 2.

    3. $\circ$

“I’m Red in Round 3” means you plan to choose “wait” in Round 1 and Round 2
and choose “I’m Red” in Round 3.

    4. $\circ$

“I’m Red in Round 4” means you plan to choose “wait” in Round 1 to Round 3 and
choose “I’m Red” in Round 4.

    5. $\circ$

“I’m Red in Round 5” means you plan to choose “wait” in Round 1 to Round 4 and
choose “I’m Red” in Round 5.

    6. $\circ$

“Always wait” means you plan to choose “wait” in Round 1 to Round 5.

  * •

In each match, you will be asked to choose your plan in the following screen.

  * •

Therefore, your payoffs are summarized as below.

You choose “I’m Red” no later than the other participant You choose “I’m Red”
later or choose “always wait” Terminal Round Your color Red White Red or White
1 $p_{1}$ $-w_{1}$ 0 2 $p_{2}$ $-w_{2}$ 0 3 $p_{3}$ $-w_{3}$ 0 4 $p_{4}$
$-w_{4}$ 0 5 $p_{5}$ $-w_{5}$ 0

4\. At the end of each match, there will be a summary of the match which
includes both of your colors, the terminal round, your action, and your own
payoff for this match. If you choose “I’m Red” later or at the same round as
the other participant, you will be informed the other participant’s exact
plan. Otherwise, you will be told that the other participant is “later than
you.”

5\. At the beginning of the experiment, you will start from 900 points. You
will get paid in cash based on your total points earned from the 12 matches.
If your total point is negative, you will only receive the show-up fee.

6\. Important:

  * a.

After each match, you will be randomly paired with another participant in the
next match.

  * b.

Your color and the other participant’s color will also be randomly re-drawn in
each match. The colors in each match are independent of the colors in other
matches.

  * c.

The probability distribution of colors and the payoff table will change in
each match.

Please raise your hand if you have any questions. The question will be
answered so that everyone can hear.

###  F.3 Screenshots

Figures A.4 and A.5 show the actual screenshots of the sequential treatment,
and Figures A.6 to A.8 display the actual screenshots of the simultaneous
treatment. Notice that Figure A.7 represents the feedback screen of a player
who selects “I’m Red” earlier than the opponent, and Figure A.8 provides the
perspective from the other player.

Figure A.4: The decision stage of the sequential treatment Figure A.5: The
feedback stage of the sequential treatment Figure A.6: The decision stage of
the simultaneous treatment Figure A.7: The feedback stage of the simultaneous
treatment Figure A.8: The feedback stage of the simultaneous treatment