An integrated approach to evaluating sustainability in supply chains using evolutionary game theory

Accepted Manuscript

An Integrated Approach To Evaluating Sustainability In Supply Chains Using Evolutionary Game Theory Sujatha Babu, Usha Mohan PII: DOI: Reference:

S0305-0548(17)30008-4 10.1016/j.cor.2017.01.008 CAOR 4172

To appear in:

Computers and Operations Research

Received date: Revised date: Accepted date:

30 April 2015 12 July 2016 17 January 2017

Please cite this article as: Sujatha Babu, Usha Mohan, An Integrated Approach To Evaluating Sustainability In Supply Chains Using Evolutionary Game Theory, Computers and Operations Research (2017), doi: 10.1016/j.cor.2017.01.008

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Highlights • Addresses gap indicated by Brandenberg et al in holistic supply chain sustainability • Studied across multiple dimensions-environment,social,economic,cultural &governance • We use Evolutionary Game Theory to model stakeholders & payoff functions

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• Sustainability identified with equilibrium of system over finite period of time

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• Allows us to study factors that can cause shift in equilibrium over time

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An Integrated Approach To Evaluating Sustainability In Supply Chains Using Evolutionary Game Theory

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Sujatha Babu1∗and Usha Mohan† IIT Madras

Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, Tel.: +91.99625.92556,+91.44.22574576,+91.96000.56836,+91.44.2257.4552

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January 19, 2017

Abstract

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Sustainability in supply chains is typically studied across one or more dimensions such as environmental, social, economic, culture and governance. Traditionally sustainability in supply chains has focused on environmental dimensions, while a few have attempted to focus on social and economic dimensions without really integrating them. There has been only a small effort to define sustainability by integrating all relevant dimensions (a holistic approach). This paper proposes to fill this gap. We identify sustainability of a supply chain with the equilibrium of the system over a long (but finite) period of time after integrating the various dimensions. Thus it necessitates looking at factors that can cause a shift in the equilibrium. Towards this, we propose to build a strong theoretical framework to integrate, explain, and predict sustainability for supply chains using cross-disciplinary effort. In our theoretical framework, evolutionary game theory serves as the pure conceptual theory-building tool, the metrics are qualitative in nature and the indicators are quantitative statistical measures. The use of evolutionary game theory concepts allows us to understand how sometimes trivial actions by members of the supply chain can trigger cascading effects that can move the system away from equilibrium. One of the salient aspects of our model is its complete scalability in terms of changes to the dimensions and metrics. As an example, we explain and predict social and economic sustainability (in tandem) for a public health insurance supply chain using evolutionary game theory.

Keywords and phrases: Supply chain management, sustainability, evolutionary game theory, theoretical framework, public health insurance

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[email protected] [email protected]

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Introduction

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Operations research is applied in many areas of supply chain management including designing supply chains, coordinating decisions across members of the supply chain, optimization, logistics and sustainability. The focus of this paper is sustainability of supply chains. By its very definition, sustainability looks at the long-term effectiveness and continuation of the supply chain given the history of the supply chain and its future growth path. This allows the members of the supply chain to look beyond the present while making policy decisions. In this section, we highlight some of the literature on sustainability of supply chains to identify the gap that we hope to address through our model.

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Brandenburg et al. (2014) depict an aggregated research model to understand the notion of sustainable supply chain management solutions. As a part of this model, they define the various sustainability dimensions to include one or more of environment, social, and economic dimensions (often referred to as the triple bottom line approach). Some supply chains such as ecotourism may include an additional dimension referred to as either culture and society or governance, thus leading to the quadruple bottom line approach. The nature of the industry under study defines the actors, levels and process, and the dimensions to be considered. These serve towards modelling integrated sustainability to provide a sustainable supply chain management solution. Also in their literature surveys, Carter et al. (2008), Seuring (2013), and Brandenburg et al. (2014) amongst others indicate that sustainability in supply chains has been usually defined along environmental dimensions in terms of “green” concepts. According to the breakup provided in their reviews, very meagre literature looks at integrated sustainability in service supply chains. While there exists some literature that looks at sustainability in terms of either economic or social dimensions, their analysis shows a lack of integration across the dimensions (social, economic, and environmental) when studying sustainability of supply chains. According to them, most of the small number of literature on holistic sustainable supply chain models are more focused on industrial sectors and rarely focus on the microscopic analysis. Gunasekaran et al. (2012) review sustainable business development for manufacturing and services sector and provide a framework based on seven building blocks. Further, in their work with Ageron et al. (2012), they develop a conceptual model that looks at the factors influencing sustainable sourcing in a supply chain. They then study the impact of these factors on sustainability. In particular, they look at sustainable supply chain from a supplier selection perspective along with an empirical example. In both these papers, the need to develop a framework for performance measures and metrics for sustainable supply chains is identified as one of the future research directions. Carter et al. (2008) provide a visual representation of the three dimensions (also called the triple bottom line developed by Elkington (1998, 2004)) and indicate sustainability to lie in the intersection of the performance across all three dimensions. They further propose that firms that aim at integrated sustainability achieve the highest level of economic performance than firms that aim at only one or two of the dimensions. Some other relevant literature includes Tang et al. (2012) who review the operational research methodologies available to study the performance of supply chains along profit, 3

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planet, and people dimensions. However their review is aimed at identifying the metrics and gaps that are present in operational research and the arena of management decisionmaking, rather than an integrated look. De Brucker et al. (2013) use multi-criteria analysis to arrive at sustainable project evaluation across multiple stakeholders. They look at the endogenous creation of an institutional equilibria that in turn results in collective benefit. However their work imposes boundary conditions that may restrict the equilibrium choices available. Devika et al. (2014) look at designing a closed-loop supply chain network based on all three dimensions. However they aim at a Pareto-optimal solution rather than an equilibrium solution. Govindan et al. (2014) explore which of the lean, resilient, and green supply chain practices impact the sustainability of the supply chain. Varsei et al. (2014) develop a multidimensional assessment framework to study sustainability across the three dimensions. Kannegiesser et al. (2014) look at global supply chains with optimisation as the goal.

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As a continuation to the survey findings and to fill the research gap identified in the surveys by Seuring (2013), Brandenburg et al. (2014), Gunasekaran et al. (2012), Ageron et al. (2012) and others, we propose to define sustainability differently by posing the following questions while evaluating a supply chain. (1) What are the dimensions and metrics that concern and impact each stakeholder in the supply chain? (2) How does one study the effect of all these metrics as a cognizable whole on the entire supply chain at any point of time, instead of just looking at optimality? (3) If new factors come into play with the progress of time, are the objectives of the dimensions synthetically met across the supply chain? For this we turn towards a cross-disciplinary effort and look at a specific area of game theory (Owen, 1995). In fact, Brandenburg et al. (2014) specify game theory as an analytical modelling approach in sustainable supply chain management research.

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The study of game theory looks at strategic interaction between one or more players, played in a cooperative or non-cooperative environment, as a single shot or repeated over finite or infinite time horizon, with pre-defined rewards for each player. Each player tries to select an action to play so as to maximise or minimise his or her payoff. Nash equilibrium is one of the most well-known solution concepts in game theory. General game theory solution concepts, by themselves, are insufficient to answer our questions completely on the holistic sustainability of supply chains. We turn towards evolutionary game theory (Hofbauer et al., 1998) for the same. It had its beginnings in the seminal paper by two theoretical biologists, Maynard Smith and Price (Maynard Smith et al., 1973) who introduced a new solution concept, evolutionarily stable strategy, to define stability and equilibrium in a population over a period of time when the population is invaded by a small number of mutants. Later Taylor et al. (1978) gave a dynamic approach (known as replicator dynamics) to the static evolutionary stability concept of Maynard Smith et al. Hence it is meaningful to view the long-term sustainability of supply chains from an evolutionary game theory perspective. It is important to look at the dynamic characterisation and detection of all evolutionarily stable strategies (Bomze et al., 1992; Bomze, 1992). It is interesting to note that the concept of replicator dynamics has been used in some economic application studies such as portfolio selection (Bomze, 2005). There is very scarce literature on this inter-disciplinary topic. For example, in the insurance sector, Brown (2001) looks at the co-evolution of the 4

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public/private insurance in the public and private hospital market in Australia. Other literature (Zhu et al., 2007; Barari et al., 2012) studies greening supply chains by modelling as an evolutionary two-person game between the government and enterprise. Though Naini et al. (2011) use evolutionary game theory to design a mixed performance management system, they focus only on the environmental dimension. Mota et al. (2014) use generic multi-objective mathematical programming for economic, environmental and social design and planning of supply chains. However they do not focus on the actual evaluation of the supply chain for sustainability. The above papers restrict themselves in one or more of the following aspects: (1) They restrict themselves to two player games while there can be any number of players in reality; (2) They only address matrix games restricted to a simple payoff based on cost and reward, as opposed to payoff functions that consider various metrics; (3) They do not study the effects of mutations and learning; (4) They do not study dynamics that can exist within a population. To the best of our knowledge, there is no known literature that addresses the question of supply chain sustainability in a holistic manner using evolutionary game theory.

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We propose to build a strong theoretical framework to explain and predict environmental, social, economic and cultural/governance (quadruple bottom line) sustainability in tandem for supply chains. In our theoretical framework, evolutionary games serves as the pure conceptual theory-building tool, the dimensions are qualitative in nature, and the metrics/indicators are quantitative statistical measures. We propose to go beyond the standard “green” definition of sustainability by looking at the equilibrium of the evolutionary game to be the sustainable point for the supply chain. Our theoretical framework provides a way to identify sustainability by depicting how the supply chains move into equilibrium over a long (but finite) period of time, as opposed to the traditional optimality conditions. Our model also reflects the fact that sometimes trivial choices by members of the supply chain can trigger cascading effects that can move the system away from equilibrium. We show that our model in no way restricts the dimensions or the metrics to be defined for each player or the methodology used to arrive at the measure. This theoretical framework offers the flexibility of expanding the model and makes it completely scalable. The scalability of our theoretical framework and its adaptation to any supply chain where we need to look at sustainability across multiple integrated metrics is a critical success factor for our model.

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Towards this, we identify the members of the supply chain, the strategic stakeholders, the actions available to the members, the states that the supply chain can be in, the metrics for each member population, the nature of interactions, the payoff functions and equilibrium concepts. We also touch on various revision protocols that may come into play such as learning and invasion. In Section 2, we detail how to model a supply chain using evolutionary game theory concepts to study sustainability by identifying its components. We then build an evolutionary game model for a generic supply chain. We also briefly touch upon revision policies and possible invaders. In Section 3, we provide a detailed example of building an evolutionary game model to evaluate the sustainability of a public health insurance (PbHI) supply chain. In Section 4, we wrap up with a brief conclusion and identify possible future directions. 5

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Modelling as an Evolutionary Game

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We define sustainability in terms of the equilibrium of the supply chain across integrated dimensions over a finite period of time. As current literature does not provide a way of looking at sustainability this way, we model it as an evolutionary game. This gives us the flexibility in developing a theoretical framework that does not restrict itself to any particular supply chain, choice of dimensions, or choice of metrics. If equilibrium cannot be reached in the implementation of this model, it helps identify the factors that caused this. If instead the implementation of the model throws up multiple equilibria, it allows an analysis of the equilibrium that the supply chain would like to be in. We now give a brief overview of evolutionary games. The Technical Appendix A gives formal definitions of some of the relevant concepts as needed.

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An evolutionary game involves players from one or more populations, who interact with each other over a period of time and receive a reward for each interaction. Each population tries to maximise its reward by allowing the players to modify their strategies as the game progresses. This modification can arise due to multiple reasons such as learning, imitation, or even random choice. Studying the dynamics of the game allows us to see if an equilibrium can be reached, and if it is a unique equilibrium. To model as an evolutionary game, the principal components of the game need to be identified. For the game, this comprises of the population of players and the states that the game can be in. For each population, the components are the actions that can be taken by the players, the dimensions to be studied, metrics and indicators for each dimension, and the payoff function for each population that integrates all the dimensions. The final critical component is a single payoff function for the supply chain based on these population-based functions that hence spans all dimensions. Tracing the trajectory of this composite function, thereby relying on the implicit dynamics modelled as a differentiable function on the solution space (Hofbauer et al., 1998), will indicate if equilibrium exists. In other words, we identify the rate of change of the composite function over time (i.e. the concept of replicator dynamics as explained in the Technical Appendix A) when the supply chain starts at an initial state. There are many solution concepts that have been defined as equilibrium and we look at stability as the equilibrium solution concept. This choice is in agreement with our notion of sustainability since a stable strategy cannot be invaded by mutations in small proportions. Not all evolutionary games have evolutionary stable strategies (Hofbauer et al., 1998). In fact, a single evolutionarily stable strategy indicates a mixed optimal strategy and multiple such strategies indicate that they are pure strategies. In this section, we detail the components of the game to be identified, build a generic theoretical model, list some important conditions that guarantee the existence of a solution, and the concept of revision policies and invaders.

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2.1

Identifying Components of the Game

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The members/stakeholders of the supply chain constitute the populations that interact in the game. Existing supply chain literature details the value chain and network for the supply chain being studied. This is a good starting point in identifying the stakeholders and hence the populations. Some of the stakeholders may be strategic stakeholders (such as the government). Some populations may compose of players in different roles (e.g. stakeholders could include both government and regulatory bodies) who may be treated as a single player without limiting our theoretical framework. The number of players in a population is treated as finite for purposes of our model.

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Members of the populations may show within-population interactions, across-population interactions, or both. The interaction can be one-way or both ways. These interactions determine the actions available to the players in each population. The interactions also determine the state of the game in terms of the proportion of players in each population in that state.

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The dimensions are specific to the supply chain being considered and commonly include social, economic, environmental, governance and cultural dimensions. For example, environmental dimension may be more relevant to manufacturing supply chains. Social dimension may be more relevant to service supply chains. Cultural dimensions may be more relevant to ecotourism supply chain. Our framework in no way restricts the number or nature of dimensions to be studied. In fact, the dimensions need not be the same across all populations. For each dimension for a population, it becomes critical to identify the metrics to measure the various aspects of that dimension. Seuring (2013) and Brandenburg et al. (2014) offer a list of widely used metrics in each dimension as shown in Figure 1. For example, economic dimension has microeconomic metrics such as total cost and net revenue, and macroeconomic metrics such as gross development product and growth rate. Metrics for environmental dimension can be input-oriented (energy demand, consumption, natural resources) and output-oriented (CO2 -emissions, waste). Social metrics may focus on internal factors (corporate social responsibility, employment gender ratio, income distribution) and external influences (social acceptance, population growth). While Brandenburg et al. (2014) restrict themselves to the triple bottom line approach when looking at dimensions, our model in no way restricts addition of the fourth dimension, namely cultural/governance.

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The final component is defining the payoff function for each population that integrates across all dimensions and the consolidated payoff function for the supply chain. Each of these must be a C 1 function (i.e. a continuous function whose first order partial derivative exists) with the following parameters: (1) metrics identified for each dimension for that population, and (2) time. One simple way to build the population payoff function involves understanding the relation between the various metrics, collecting data relating to the metrics for a finite time period, and fitting a C 1 equation after ignoring outliers. The entire supply-chain’s payoff function is then assembled as yet another C 1 function of the individual population’s payoff function with the state of the supply chain as an additional variable. We define sustainability based on the long-term equilibrium reached and the impact of new factors on

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Sustainability Metrics Population 1 Economic Dimension Microeconomic Metrics Macroeconomic Metrics

Social Dimension

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Metrics For Internal Factors Metrics For External Influences

Environmental Dimension

Input-Oriented Metrics

Output-Oriented Metrics

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Population N Economic Dimension

Microeconomic Metrics Macroeconomic Metrics

Social Dimension

Metrics For Internal Factors Metrics For External Influences

Environmental Dimension

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Figure 1: Sustainability Metrics For A Supply Chain

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the stability of the equilibrium.

Building the Model

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Let there be N finite size populations in the supply chain representing each member type. (jl) For each population, let the finite set of actions available be indicated by ai representing th th the i action available to players in the j population interacting with players in the lth population. When j = l, we are looking at within population interactions. It is interesting to note that if there is only one strategic stakeholder in the supply chain, then any interaction with the stakeholder is similar to a “single player control game”. Let the game have k P k states given by si such that si ∈ [0, 1] and si = 1. Here the value of si indicates the

proportion of the population in that state.

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As we are building a generic theoretical model, we define np to be the number of dimensions considered for population p (p = 1, 2, . . . , N ). Let t = time and the ith dimension for popu(p) lation p be given by di , i = 1, . . . , np . Let the metrics for dimension i of population p be 8

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(p,i)

(p,i)

(p,i)

(p,i)

given by m1 , m2 , . . . , mnp,i . Without loss of generality, mk is closed and bounded. Then the real-valued C 1 payoff function for population p, fp : Rnp +1 → R is given by

(p)

(p,2)

, . . . , m(p,1) np1 ), d2 (m1

, . . . , m(p,2) np2 ), . . . ,

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d(p) np (m1

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, . . . , mnp,npp ), t)

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(p,1)

fp (dp1 (m1

In the above function, the parameters other than time could either be a composite value given to the dimension based on the values for its metrics, or the individual metric values themselves. So equation (1) can be equivalently stated as (p,1)

fp (m1

(p,2)

, . . . , m(p,1) np1 , m1

(p,np )

, . . . , m(p,2) np2 , . . . , m1

(p,n )

, . . . , mnp,npp , t)

(2)

(p)

(p,1)

fp (d1 (m1 (p,1)

fp (m1

(p,1)

(p)

(p,2)

, . . . , mn1 ), d2 (m1 (p,1)

(p,2)

, . . . , mn1 , m1

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Hence our model in no way restricts the number of dimensions or metrics. For example, if we were looking at only 3 dimensions (social, cultural, and environmental), then the payoff function for population p is given by (p,2)

(p)

(p,3)

, . . . , mn2 ), d3 (m1

(p,2)

(p,3)

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, . . . , mn3 ), t) or

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where d1 refers to social dimension, d2 for cultural dimension, and d3 for environmental dimension. The integrated payoff function for the supply chain is then given by the composite function f (s, f1 , f2 , . . . , fp , t), where f : Rp+2 → R and s is the state of the supply chain.

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Let xji indicate the ith metric for population p. Without loss of generality, we can assume xji closed and bounded. As replicator dynamics uses techniques of differential calculus, we need to ensure that fj is C 1 to study the population dynamics. If fj were a polynomial function, then it is automatically C 1 . Else, if fj was defined such that the following two conditions are satisfied, then the function would be C 1 : (1) Small changes to the value of the parameters of fj causes only small changes to the value of fj (i.e.) continuity condition is satisfied. (2) fj does not contain any modulus component. So it is possible to always find functions fj that is C 1 . Replicator dynamics can then be applied to arrive at the stable points for the payoff function. Also when using time series data, as long as data is available for the entire period under study we are guaranteed that the payoff function remains C 1 .

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Existence of Equilibrium

As we have identified sustainability with the long term equilibrium of the supply chain viewed as a dynamical system, we list some propositions that guarantee the existence of one or more stable points (without technical proof).

Proposition 1 (Hofbauer et al., 1998). If the evolutionary game has a strict Nash equilibrium, then the equilibrium is evolutionarily stable. Also a symmetric Nash equilibrium 9

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solution to the replicator equation is an evolutionarily stable state if and only if the payoff function is a strict local Lyapunov function. This is equivalent to saying that a strategy is evolutionarily stable if and only if the corresponding point in the replicator dynamics is asymptotically stable.

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Proposition 2 (van Damme, 2002). In a two-population setting, a quasi-strict symmetric equilibrium where the payoff matrix is negative definite is evolutionarily stable. Proposition 3 (Hirsch et al., 2004: Existence and uniqueness theorem). Given a replicator system that traces the rate of change in the composite C 1 payoff function and the initial state for the system, there exists a unique solution for the replicator equation satisfying the initial conditions.

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For sake of completeness, we have stated the first two propositions that look at equilibrium through evolutionary stability, and also links to the stable equilibrium concept in replicator dynamics. The last proposition is of immense relevance given that our theoretical framework allows us to build C 1 composite payoff functions guaranteeing the existence of equilibrium.

Revision Policies

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A supply chain is not static. Indeed for a supply chain to sustain, it has to be reactive and proactive to changes in the market dynamics. Hence its members may choose to implement new policy decisions, renegotiate supply chain contracts, revise prices, drop or add product lines, and many more such actions depending on changes in the market. Any modelling environment has to effectively allow for such temporal revisions.

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In evolutionary games, players can randomly revise their actions at any point of time. This randomness allows the players to experiment and try new strategies through multiple methods including learning, replication, and mutation. These actions may create a new path for the game, and may move it away or towards stable equilibrium at a slower or faster pace. Our model not only allows us to study the stability of the supply chain based on past data, but to also use forecast data relating to these revision policies to see how the stability of the supply chain changes. This allows the members of the supply chain to evaluate the impact of possible courses of actions on sustainability of the supply chain.

2.5

Possible Invaders

Any supply chain is susceptible to factors such as competition, government regulations, new technology, changes in rules and guidelines and peer influence that in turn impact the sustainability of the supply chain. So it becomes important to study the impact of factors that can adversely or advantageously affect sustainability. The main advantage in modelling the supply chain as an evolutionary game is to understand the effect of factors that may arise at later points of time that affect the dynamics of the system. This can be done through 10

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identifying the metrics for these new factors, creating the payoff function, and viewing the revised evolutionary game to study its stability.

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Also in a supply chain setting, small changes can sometimes trigger cascading effects. Evolutionary game theory allows the study of the effect of introducing a small proportion of invaders on the equilibrium of the system.

Scalability of the Model

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As demonstrated through Equations 1 and 2 in Section 2.2, the model is easily scalable as it allows us to incorporate any number of dimensions, metrics, populations, actions, and states. If a supply chain takes too long to reach sustainability, an analysis of the factors that prevent the game from reaching the equilibrium gives a good indication of what needs to be done by the members. Also some games may have multiple evolutionarily stable pure strategies, which throws interesting light on which of them the supply chain moves towards and why. This can be done using replicator dynamics. The performance of this model can be easily demonstrated similar to any forecasting model through a number of techniques including prediction errors. This allows us to compare observed stability with predicted stability of the supply chain. Any deviations indicate a gap in the input parameters for the model and become a part of the learning strategy for the model.

Public Health Insurance Supply Chain: A Numerical Example

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In the next section, we showcase our theoretical framework and demonstrate scalability through a numerical example for a public health insurance (PbHI) supply chain. We look at an existing supply chain and identify its stable strategies. In addition, we show the evolution of the sustainability of the supply chain when it starts at different states. We also look at a scenario where decisions resulting in better payoffs affect the stability of the supply chain.

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As per the World Health Organisation (WHO), the goal of universal health coverage is to ensure that people obtain needed health services without any financial hardship that may arise due to treatment-related payments. As PbHI schemes are designed with this in mind, it becomes important to view the sustainability of PbHI in terms of social, economic and governance dimensions rather than environmental or cultural dimension. In a typical PbHI supply chain, the government offers healthcare coverage to eligible individuals (hereafter called consumers). The government contracts with one or more insurance companies and with healthcare service providers (public and/or private). Hence the four stakeholders in the PbHI network are government or regulatory bodies, insurance companies, care providers, and consumers. We look at the PbHI related portion of two of the commonly cited value chains - namely Burns (Burns, 2002) shown in Figure 2 and Chakravarty (Chakravarty,

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2014) shown in Figure 3 .

Payers

Providers

Payers

Insurers

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Central State

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Government

Hospitals

Health Insurers

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Individuals

Physicians

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Figure 2: Modification To Healthcare Value Chain By Burns (2002)

Payments

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Rules, Payments

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Request Care

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Reports, Usage Bills Reports

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GOVERNMENT

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Figure 3: Modification To A Generic Healthcare Network By Chakravarty (2014)

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We restrict our attention to metrics relating social and economic dimension for each population for the above sample network, as shown in Figure 4.

3.1

Validating the Model - A Numerical Example

In order to validate our theoretical framework, we consider a subset of data for the National Health Services (NHS) in United Kingdom (a PbHI scheme already in existence for a length of time) to arrive at the sustainability factors for the PbHI scheme. For purposes of our illustration, we have restricted ourselves to two dimensions (economic and social), two stakeholders (consumers and providers), and a select few metrics as shown in Figure 5. The following sections detail the data used for each of these metrics. 12

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Sustainability Metrics Consumer Economic Reduction in financial risk Catastrophic expenditure due to OOP expense Impoverishment due to OOP Equity of access

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Social Modified Gini coefficient

Atkinson distributional measure Service level satisfaction

Insurance Company Economic

Care Provider Economic

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Reduction in moral hazard Welfare loss computation

Reimbursements claimed

Timeliness of reimbursements

Social

Prevention of contagious diseases

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Figure 4: Sustainability Metrics For A Generic PbHI Scheme

Defining The Evolutionary Game Parameters

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The state space for the provider population captures the proportion who have a high rating due to high productivity efficiency and high patient safety. Similarly state space for the consumer population captures the proportion of consumers who find the NHS satisfactory and hence use it. These states in turn define the strategies available to the players in each population.

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We start by building the payoff functions for each population and then the composite payoff function for the game. The payoff function for care providers is built based on the individual payoff matrix and the state function. The state function for care providers fH : R2 → R is a C 1 function across two dimensions with the following parameters : xH – Patient safety (social metric); t – Time, along with efficiency in productivity (economic metric). Based on the data available from NHS (Table 1), we integrate metrics across two dimensions resulting in the following function for providers (Figure 6). fH (xH , t) = −3 × 10−08 xH (t)3 + 4 × 10−05 xH (t)2 − 0.0133xH (t) + 96.58

R2

C1

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Let fC : → R be a function for the consumers with the following parameters: xC – Median inpatient waiting time (social metric); t – Time. The payoff function determines the 13

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Sustainability Metrics Provider Economic Efficiency in Productivity

Social Patient Safety

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Consumer Social

Median Inpatient Waiting Time Public Satisfaction With NHS

Figure 5: Sustainability Metrics Sample For NHS, UK

2004 2005 2006 2007 2008

Productivity Efficiency, Office For National Statistics 2010 96 96 98 97 97

Patient Safety (in ’000), NHS 2011

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Year

42 403 659 834 909

Figure 6: C 1 Function For Provider Population

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Table 1: Provider Related NHS Data

proportion of the population that is satisfied with NHS (social metric). Based on the data available from NHS (Table 2), the linear function for consumers is the following (Figure 7). fC (xC , t) = −1.3229 ∗ xC (t) + 57.795

(4)

The starting state for the two populations at any time t can be found from Equation 3 and Equation 4. The strategies for the provider are as follows: (1) provide low quality service leading to low rating, and (2) provide high quality service leading to high rating. The two 14

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1994 1995 1996 1997 1998 1999 2000

Inpatient Waiting Time (In Weeks)(1) 14 12 12 13 15 13 13

Public Satisfaction Survey (%)

Year

(2)

Inpatient Waiting Time (In Weeks) (1)

44 37 36 35 41 47 42

2001 2002 2003 2004 2005 2006 2007

13 13 12 10 8 7 6

Public faction (%)(2) 39 40 44 43 48 49 51

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Table 2: Consumer Related NHS Data (1) In Weeks, Department of Health 2013, (2) % of Population Surveyed, Appleby & Phillips (2009)

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Figure 7: C 1 Function For Consumer Population

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strategies available to the consumer are as follows: (1) use the services of NHS, and (2) do not use the services of NHS. Note that this in turn is indicated by the proportion of satisfied consumers. The composite C 1 function f : R4 → R is given by f (s0 , fH , fC , t) where s0 denotes the initial state of the supply chain. Tracing the rate of change in the trajectory of this composite average payoff function for the population identifies stability.

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The following sections detail the experimental setup and the results of the setup.

3.2

Experimental Setup

The experimental setup attempts to answer two questions: (1) In terms of the prediction error, does the predicted model become sustainable at a vastly different time from the actual system? (2) For a given a set of starting payoffs and population state, is the system sustainable by itself; and if yes, when? What is the effect of implementing a new policy (equivalent to introducing a new payoff) that will cause the system to reach sustainability

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faster?

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For both the questions, the experiment is setup as follows. Given a starting state and payoff matrix for the provider and consumer population, we trace the evolution of the population for 15,000 time periods. To study the effect of introduction of policy changes (reflected as new payoff matrix for either or both populations), a decision is taken to introduce a new payoff matrix at this point to study the evolution of the population for another 15,000 time periods. Thus the evolution of the population is tracked for a total of 30,000 time periods. Payoff Types

Providers

Consumers

Undesirable (U) Good (G) Best (B)

[2 0; 1 3] [2 0; 1 1] [1 2; 3 1]

[1 3; 2 0] [2 3; 1 0] [3 3; 1 0]

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Table 3: Input Payoffs

States

Providers

Consumers

Forecasted (F) Actual (A)

0.95, 0.05 0.97, 0.03

0.4986, 0.5014 0.51, 0.49

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(a) States For Computing Prediction Error

States

Providers

Consumers

S1 S2 S3

0.9, 0.1 0.5, 0.5 0.2, 0.8

0.2, 0.8 0.5, 0.5 0.9, 0.1

(b) States For Checking Sustainability

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Table 4: Input States

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Table 3 lists the input payoffs in our experimental setup. Three types of provider payoffs are considered: Undesirable (U ), Good (G), Best (B). Here G and B are payoffs that can incentivize the provider to offer better quality service for consumers who come through the NHS, while U does not. Similarly, three types of consumer payoffs are considered: Undesirable (U ), Good (G), Best (B). As before G and B are payoffs that can incentivize the consumer to avail of NHS while U does not. For instance, when the consumer payoff is U , the consumer is not motivated to use the provider in the NHS network. When the consumer payoff is G, he is indifferent to availing of medical services within or outside the NHS network. When the consumer payoff is B, he is incentivized to use the provider in the NHS network. The same holds for the provider payoff in terms of the quality of service provided. The entries in Table 3 are read as follows. For the provider payoff matrix B = [1 2; 3 1]: the provider receives 1 units when he operates under low efficiency and the consumer uses the network; 2 units when he operates under low efficiency and the consumer does not use the network; 3 units when he operates under high efficiency and the consumer uses the network; 1 unit when he operates under high efficiency and the consumer does not use the network. The other payoff matrices for the provider are read similarly. In a similar vein, the consumer payoff matrix B = [3 3; 1 0] looks at the payoff to the consumer using 16

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the network or not using the network when the provider provides high quality of service and low quality of service. The payoff for the consumer when they use the NHS network is higher than when they avail of services outside the network when the provider provides high quality service and low quality service. The other payoff matrices for the consumer are read similarly.

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Table 4 lists the input states in our experimental setup. Five initial population states indicated by S1, S2, S3, F and A are considered based on the proportion of population engaging in a specific action. To understand the notation in this table, we take the example of the provider state S1 = [0.9, 0.1]. This indicates that 90% of the provider population provide low quality service while the remaining 10% provide high quality service. The consumer state S1 = [0.2, 0.8] indicates that 20% of the consumers use the NHS network while the remaining 80% do not. The other states are read similarly. In Table 4a, the two statesforecasted (F ) and actual (A), will be used to compute the prediction error. In Table 4b, S1 and S3 represent the different ends of the spectrum and S2 lies in-between. These states are used to answer our questions on sustainability of the system.

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Since the provider and consumer payoff can be changed at a selected point in the run, we iterate through 32 combinations of provider payoff, 32 combinations of consumer payoff, and 5 different starting population states, leading to a total of 32 ×32 ×5 combinations. As some of these payoff changes may not be feasible (for example, if the payoff for the provider in the first half of the run is the best payoff, there will be no reason to introduce an undesirable payoff in the second half of the run), these infeasible runs have not been considered and are indicated as ‘Infeasible’ in Table 6 in Section 3.3.

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The code was implemented using Matlab (version R2013a). The first part of the numerical experiment looks at prediction errors based on the actual starting state (A) versus the forecasted starting state (F ). At time t = 1, different feasible combinations of provider and consumer payoffs along with the starting states F and A are considered (32 × 32 × 2 combinations, of which some combinations may be infeasible as mentioned earlier). At time t = 15, 001, a decision is taken to introduce a new provider and/or consumer payoff to study the effect of a policy change on the system.

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The second part of the experiment studies whether the system reaches sustainability; and if yes, when. At time t = 1, different feasible combinations of provider payoffs, consumer payoffs and starting states (S1, S2 and S3) are considered (32 × 32 × 3 combinations, of which some combinations may be infeasible as mentioned earlier). The run proceeds as earlier.

3.3

Results & Analysis Using Replicator Dynamics

We use replicator dynamics to track the evolution of the population to answer the questions relating to prediction error and sustainability.

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3.3.1

Prediction Error

Sustainability and Evolutionarily Stable Strategies

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3.3.2

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The actual states for the two populations are given in Table 1 and Table 2. The forecasted starting state is derived from Equation 3 and Equation 4. For example, in 2007, 97% of the provider population showed high productivity efficiency (as per Table 1). However the fitted equation for providers (Equation 3) indicates that only 95% of the provider population showed high productivity efficiency that year. Hence the actual state of the provider population ([0.97, 0.3]) is different from the forecasted starting state ([0.95, 0.05]). This implies that the time taken by the system to reach ESS will be different for the actual starting state and the forecasted starting state, and hence the average payoff to the population will differ too. Prediction error techniques help identify the magnitude of this difference across various provider-consumer feasible payoff combinations. Table 5 provides the Mean Squared Error (MSE) for the feasible combinations of input parameters. It can be seen that the value of MSE is less than 0.05 across all the feasible combinations of payoffs.

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As mentioned earlier, Tables 3 and 4a list the different types of payoffs and states being considered to study the sustainability of the system. The output of the experimental run is listed in Table 6 for selected time periods. This table helps study the sustainability for various feasible combinations of provider and consumer payoff and starting state. As a sample, we provide the simplex and evolution graph of two runs - namely, (1) Figure 8: when the provider and consumer start in state S1 and have payoff U for the entire run. In this case, the system is not sustainable as there is no ESS. (2) Figure 9: when the provider and consumer start in state S2 with payoff U , and move to payoff G and B respectively at time 15,001. In this case, the system is sustainable and does reach an ESS.

Figure 8: Replicator Dynamics: Has No ESS

As seen in Figure 8, though the system starts with 97% of the consumers using NHS, the percentage dwindles and then rises again. This cycle continues for the entire run and indicates the effects of the consumer payoff, the provider payoff, and the provider’s own starting state. Thus it is clear that the system is not sustainable. In Figure 9, the system 18

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Provider

Consumer

Provider

Consumer

U U U U U U U U U U U U U U U U U U G G G G G G G G G G G G B B B B B B

U U U U U U U U U G G G G G G B B B U U U U U U G G G G B B U U U G G B

U U U G G G B B B U U G G B B U G B G G G B B B G G B B G B B B B B B B

U G B U G B U G B G B G B G B B B B U G B U G B G B G B B B U G B G B B

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MSE For Provider

For Consumer

0.03154 0.00959 0.00948 0.01619 0.00901 0.00896 0.00667 0.00667 0.00667 0.00006 0.00006 0.00006 0.00006 0.00074 0.00074 0.00003 0.00003 0.00095 0.02312 0.00453 0.00372 0.00282 0.00438 0.00437 0.00002 0.00002 0.00081 0.00081 0.00001 0.00084 0.00005 0.00005 0.00005 0.00092 0.00092 0.00076

0.02851 0.00821 0.0082 0.03056 0.00821 0.0082 0.00821 0.00821 0.0082 0 0 0 0 0 0 0 0 0 0.04475 0.00839 0.00855 0.00818 0.00856 0.00862 0 0 0 0 0 0 0.00002 0.00028 0.0002 0.00001 0.00001 0

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Payoff at time t = 15001

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Payoff at time t = 1

Table 5: Prediction Error (MSE)

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Figure 9: Replicator Dynamics: Has ESS

3.3.3

Influence In Policy Making

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does cycle without moving towards sustainability for the first 15000 time units. Then due to the new payoff that is implemented, the system moves towards the entire consumer population using NHS (i.e. sustainability). This is indicated by the small arc reaching the point (1, 1) in the simplex graph.

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The payoff matrix for the consumers and the providers and the starting state provide an indication of how far the supply chain is away from sustainability at any point of time. The three charts in Figure 10 indicate the time taken to reach ESS for various combinations of provider and consumer payoff for starting states S1 , S2 , and S3 respectively. The broken bar indicates that there is no ESS for the specific combination. Some of the combinations listed may lead to the entire population of consumers not using NHS. Thus the system does have an ESS, but the scheme is not sustainable as the consumers no longer are using the NHS network. In other cases, the entire population of the consumers use the NHS network. In this case, the system does have an ESS and the scheme is sustainable. Figure 10 can be used by the government and regulatory bodies to bring in policy changes and governance rules so as to reach stability earlier. For instance, let the system start in state S1 where no favorable policies are in place for the providers and consumers to use NHS (i.e. providers and consumers have undesirable (U ) payoffs). To make the system sustainable, the policy maker must create a positive policy aimed at benefiting consumers. That is, they must design a policy that will change the payoff matrix for the consumer to G or B. In addition the policy maker may also create positive policies benefiting providers to make the system reach ESS (and hence for NHS to be sustainable) faster. The figures for each of the three states represent how fast the system can reach ESS.

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0.2, 2.5 1, 2 1.6, 1.1 0.2, 2.5 1, 2 1.6, 1.1 0.2, 2.5 1, 2 1.6, 1.1 0.2, 0.9 1, 1 1.6, 0.9 0.2, 0.9 1, 1 1.6, 0.9 0.2, 0.9 1, 1 1.6, 0.9 1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5

0.67896, 0.35395 0.16201, 0.31981 0.07008, 0.0816 0.67896, 0.35395 0.16201, 0.31981 0.07008, 0.0816 0.67896, 0.35395 0.16201, 0.31981 0.07008, 0.0816 0.67896, 0.3443 0.16201, 0.16061 0.07008, 0.05056 0.67896, 0.3443 0.16201, 0.16061 0.07008, 0.05056 0.67896, 0.3443 0.16201, 0.16061 0.07008, 0.05056 0.34913, 1.02327 0.24021, 0.32261 0.06608, 0.12064 0.34913, 1.02327 0.24021, 0.32261 0.06608, 0.12064 0.34913, 1.02327 0.24021, 0.32261 0.06608, 0.12064

0.01178, 0.35716 0.98582, 2.01321 0.03669, 1.9799 0.01178, 0.35716 0.98582, 2.01321 0.03669, 1.9799 0.01178, 0.35716 0.98582, 2.01321 0.03669, 1.9799 0.01178, 0.12298 0.98582, 0.99968 0.03669, 0.6722 0.01178, 0.12298 0.98582, 0.99968 0.03669, 0.6722 0.01178, 0.12298 0.98582, 0.99968 0.03669, 0.6722 0.24007, 0.13477 1.50645, 1.9855 1.32605, 0.70889 0.24007, 0.13477 1.50645, 1.9855 1.32605, 0.70889 0.24007, 0.13477 1.50645, 1.9855 1.32605, 0.70889

0.2, 2.5 1, 2 1.6, 1.1 0.2, 2.5 1, 2 1.6, 1.1

0.08897, 0.05113, 0.06201, 0.08897, 0.05113, 0.06201,

0.05167 0.02687 0.03108 0.05167 0.02687 0.03108

0.00063, 0.00036, 0.00043, 0.00063, 0.00036, 0.00043,

0.00036 0.00019 0.00022 0.00036 0.00019 0.00022

0, 0, 0, 0, 0, 0,

0 0 0 0 0 0

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

0.2, 0.9 1, 1 1.6, 0.9 0.2, 0.9 1, 1 1.6, 0.9

0.08897, 0.04688 0.05113, 0.026 0.06201, 0.03103 0.08897, 0.04688 0.05113, 0.026 0.06201, 0.03103

0.00063, 0.00036, 0.00043, 0.00063, 0.00036, 0.00043,

0.00033 0.00019 0.00022 0.00033 0.00019 0.00022

0, 0, 0, 0, 0, 0,

0 0 0 0 0 0

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5

0.04927, 0.02644, 0.03105, 0.04927, 0.02644, 0.03105,

0.2, 2.5 1, 2 1.6, 1.1

1.35644, 2.74605 0.16479, 0.31716 1.11174, 0.58557 1.35644, 2.74605 0.16479, 0.31716 1.11174, 0.58557 1.35644, 2.74605 0.16479, 0.31716 1.11174, 0.58557 1.35644, 1.3675 0.16479, 0.16065 1.11174, 0.56577 1.35644, 1.3675 0.16479, 0.16065 1.11174, 0.56577 1.35644, 1.3675 0.16479, 0.16065 1.11174, 0.56577 2.05677, 2.72394 0.2389, 0.32544 0.57567, 1.67751 2.05677, 2.72394 0.2389, 0.32544 0.57567, 1.67751 2.05677, 2.72394 0.2389, 0.32544 0.57567, 1.67751

0.13586 0.07713 0.09303 0.13586 0.07713 0.09303

3.86232, 1.75944 1.97399, 1.0285 3.83267, 1.97821 4.83212, 0.91624 4.99693, 0.99918 4.98288, 0.99146 5.78343, 0.92782 5.99784, 0.99929 5.97451, 0.9915 3.64392, 1.8623 1.35238, 1.90799 3.82768, 1.98083 4.92334, 0.96234 4.99713, 0.99929 4.98306, 0.99155 5.90058, 0.96688 5.998, 0.99934 5.97471, 0.99157 3.00011, 0.00027 3.00084, 0.00171 3.00003, 0.00007 3.00017, 0.0001 3.00119, 0.0006 3.00008, 0.00004 3.00023, 0.00008 3.00171, 0.00057 3.00011, 0.00004

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U U U G G G B B B U U U G G G B B B U U U G G G B B B U G G G B B B U G G G B B B U G G G B B B U G B B B U G B B B U G B B B B U U U G G G

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U U U U U U U U U G G G G G G G G G B B B B B B B B B U U U U U U U G G G G G G G B B B B B B B U U U U U G G G G G B B B B B U G G G G G G

5

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S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 * S1 S2 S3 S1 S2 S3 * S1 S2 S3 S1 S2 S3 * S1 S2 S3 S1 S2 S3 * * S1 S2 S3 * * S1 S2 S3 * * S1 S2 S3 * S1 S2 S3 S1 S2 S3

Average Population Payoff At Time (in 1000s) 10 15 20

1

0.00096 0.00055 0.00065 0.00096 0.00055 0.00065

0, 0.00001 0, 0 0, 0 0, 0.00001 0, 0 0, 0

0.02737, 0.01398 0.03974, 0.01992 0.06098, 0.0305

0.00019, 0.0001 0.00028, 0.00014 0.00043, 0.00021

0.2, 0.9 1, 1 1.6, 0.9

0.02737, 0.01378 0.03974, 0.01988 0.06098, 0.03049

1.7, 1.1 1.5, 2 1, 2.5 0.2, 0.9 1, 1 1.6, 0.9 0.2, 0.9 1, 1 1.6, 0.9

2.99456, 1.99029 3.68522, 1.83234 1.29547, 1.53579 4.99877, 0.99939 4.99998, 0.99999 4.99988, 0.99994 5.99841, 0.99947 5.99998, 0.99999 5.99982, 0.99994 1.26927, 1.88132 1.66052, 0.59589 1.33031, 1.99014 4.99945, 0.99973 4.99998, 0.99999 4.99988, 0.99994 5.99931, 0.99977 5.99998, 1 5.99982, 0.99994 3.00005, 0 2.99992, 0 2.99987, 0 3.00005, 0 2.99999, 0 3.00019, 0 3, 0 3.00001, 0 2.99999, 0

2.88546, 0.39663 1.96258, 1.04197 3.73753, 1.50211 4.99999, 1 5, 1 5, 1 5.99999, 1 6, 1 6, 1 2.13818, 0.61679 3.60132, 1.8938 0.65251, 1.27388 5, 1 5, 1 5, 1 5.99999, 1 6, 1 6, 1 2.99998, 0 2.99987, 0 2.9999, 0 2.99998, 0 2.99991, 0 2.99997, 0 2.99998, 0 2.99996, 0 2.9998, 0

3, 0 3.00001, 0.00001 3, 0 3, 0 3.00002, 0.00001 3, 0

0, 0 0, 0 0, 0

6, 1 6, 1 6, 1

6, 1 6, 1 6, 1

6, 1 6, 1 6, 1

0.00019, 0.0001 0.00028, 0.00014 0.00043, 0.00021

0, 0 0, 0 0, 0

6, 1 6, 1 6, 1

6, 1 6, 1 6, 1

6, 1 6, 1 6, 1

0.01388, 0.04116 0.0199, 0.05962 0.0305, 0.09148

0.0001, 0.00029 0.00014, 0.00043 0.00021, 0.00064

0, 0 0, 0 0, 0

5.9571, 0.9857 5.93817, 0.97939 5.90457, 0.96819

3.00926, 0.00309 3.0064, 0.00213 3.00414, 0.00138

3.00001, 0 3.00001, 0 3, 0

1.02397, 0.17418, 0.06158, 1.02397, 0.17418, 0.06158,

0.11149, 0.77815, 0.00597, 0.11149, 0.77815, 0.00597,

1.25785 0.36652 1.18398 1.25785 0.36652 1.18398

3.37975, 1.74382 1.35026, 1.58538 3.01692, 0.45233 4.93602, 0.97074 4.99269, 0.99822 4.44625, 0.7389

1.29088, 3.13772, 3.89942, 4.99955, 4.99995, 4.99506,

0.51332 1.16094 0.73538 0.51332 1.16094 0.73538

0.90562, 0.29909, 0.37953, 0.90562, 0.29909, 0.37953,

Table 6: Sustainability Of The System

4.90506, 4.94692, 4.93404, 5.8576, 5.92038, 5.90106,

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3.0018, 0.0009 3.00329, 0.00164 3.00266, 0.00133 3.00271, 0.0009 3.00493, 0.00164 3.00399, 0.00133

0.77934 0.46816 0.05175 0.77934 0.46816 0.05175

0.00034, 0.00019, 0.00022, 0.00034, 0.00019, 0.00022,

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0.95253 0.97346 0.96702 0.95253 0.97346 0.96702

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Payoff At Time = 15001 Provider Consumer

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Initial State

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Payoff At Time = 1 Provider Consumer

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1.69672 1.39774 1.94822 0.99979 0.99999 0.99759

3.09848, 1.18411 1.76567, 1.78766 1.69032, 1.99336 5, 1 5, 1 4.99996, 0.99998

Earliest Time To Reach ESS Infinity Infinity Infinity 29912 25157 27545 29745 25032 27520 Infinity Infinity Infinity 29053 25015 27534 28920 24957 27512 22134 22594 21015 22123 22452 20631 21602 22351 21158 * 14213 13642 13846 14213 13642 13846 * 14213 13642 13846 14213 13642 13846 * 14213 13642 13846 14213 13642 13846 * * 13023 13391 13841 * * 13023 13391 13841 * * 13023 13391 13841 * Infinity Infinity Infinity 28782 25946 31982

Remarks No ESS No ESS No ESS

No ESS No ESS No ESS

Infeasible

Infeasible

Infeasible

Infeasible Infeasible

Infeasible Infeasible

Infeasible Infeasible

Infeasible No ESS No ESS No ESS

0.2, 0.9 1, 1 1.6, 0.9 1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5

1.02397, 0.17418, 0.06158, 1.0467, 0.84924, 0.07271, 1.0467, 0.84924, 0.07271, 1.0467, 0.84924, 0.07271,

0.77934 0.46816 0.05175 1.80331 0.64234 0.11333 1.80331 0.64234 0.11333 1.80331 0.64234 0.11333

0.11149, 0.77815, 0.00597, 0.97089, 1.9328, 1.46777, 0.97089, 1.9328, 1.46777, 0.97089, 1.9328, 1.46777,

0.51332 1.16094 0.73538 0.62481 1.93909 0.74135 0.62481 1.93909 0.74135 0.62481 1.93909 0.74135

0.2, 0.9 1, 1 1.6, 0.9 0.2, 0.9 1, 1 1.6, 0.9

0.0199, 0.02804, 0.0571, 0.0199, 0.02804, 0.0571,

0.02203 0.01494 0.02858 0.02203 0.01494 0.02858

0.00014, 0.00015 0.0002, 0.00011 0.0004, 0.0002 0.00014, 0.00015 0.0002, 0.00011 0.0004, 0.0002

0, 0, 0, 0, 0, 0,

0 0 0 0 0 0

1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5

0.03411, 0.01586, 0.02861, 0.03411, 0.01586, 0.02861,

0.04192 0.04299 0.08568 0.04192 0.04299 0.08568

0.00023, 0.00011, 0.0002, 0.00023, 0.00011, 0.0002,

0, 0, 0, 0, 0, 0,

0 0 0 0 0 0

0.00029 0.00031 0.00059 0.00029 0.00031 0.00059

0.2, 0.9 1, 1 1.6, 0.9

0.00687, 0.00357 0.02345, 0.01174 0.05661, 0.02831

1.7, 1.1 1.5, 2 1, 2.5

0.00371, 0.01045 0.01176, 0.0352 0.02831, 0.08492

0.00002, 0.00007 0.00008, 0.00024 0.0002, 0.00059

1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5

1.99554, 0.99983, 0.99998, 1.99554, 0.99983, 0.99998, 1.99554, 0.99983, 0.99998,

1.99997, 1.00003, 0.99997, 1.99997, 1.00003, 0.99997, 1.99997, 1.00003, 0.99997,

1.7, 1.1 1.5, 2 1, 2.5 1.7, 1.1 1.5, 2 1, 2.5

1.7, 1.1 1.5, 2 1, 2.5

0.90562, 0.29909, 0.37953, 2.06289, 0.5835, 2.1782, 2.06289, 0.5835, 2.1782, 2.06289, 0.5835, 2.1782,

1.00219 2.99949 2.99995 1.00219 2.99949 2.99995 1.00219 2.99949 2.99995

0.98305, 0.99989, 0.99998, 0.98305, 0.99989, 0.99998,

2.94758 2.99965 2.99997 2.94758 2.99965 2.99997

0.9961, 2.98825 0.9999, 2.99968 0.99999, 2.99997

1.25785 0.36652 1.18398 2.16346 0.66561 1.56351 2.16346 0.66561 1.56351 2.16346 0.66561 1.56351

5.93813, 0.97943 5.9955, 0.99851 5.67783, 0.89395 3.00839, 0.03391 3.00081, 0.00171 0.99865, 1.99675 3.00127, 0.00065 3.00082, 0.00041 3.70773, 0.981 3.00119, 0.0004 3.00109, 0.00037 3.70348, 0.2375

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

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B B B U U U G G G B B B B U G G G B B B U G G G B B B B U G B B B U G B B B B B U U U G G G B B B B B U G G G B B B B B U G B B B

0.00005, 0.00002 0.00016, 0.00008 0.0004, 0.0002

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G G G B B B B B B B B B U G G G G G G G B B B B B B B U G G G G G B B B B B U G B B B B B B B B B U G B B B B B B B U G B B B B B

Average Population Payoff At Time (in 1000s) 10 15 20

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S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 * * S1 S2 S3 S1 S2 S3 * S1 S2 S3 S1 S2 S3 * * * S1 S2 S3 * * S1 S2 S3 * * S1 S2 S3 S1 S2 S3 S1 S2 S3 * * * S1 S2 S3 S1 S2 S3 * * * * S1 S2 S3

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U U U U U U U U U U U U G G G G G G G G G G G G G G G B B B B B B B B B B B U U U U U U U U U U U G G G G G G G G G B B B B B B B

Payoff At Time = 15001 Provider Consumer

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G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G B B B B B B B B B B B B B B B B B B B B B B B B B B B

Initial State

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Payoff At Time = 1 Provider Consumer

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1, 1.00004, 0.99997, 1, 1.00004, 0.99997,

1.00002 3.00001 2.99998 1.00002 3.00001 2.99998 1.00002 3.00001 2.99998

2.99999 3.00002 2.99998 2.99999 3.00002 2.99998

1, 3 1.00004, 3.00002 0.99999, 2.99999

4.97892, 4.97071, 4.93954, 5.96838, 5.95606, 5.90931,

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5.99957, 0.99986 5.99996, 0.99999 5.99758, 0.99919 3.00013, -0.00001 2.99992, 0.00001 0.99999, 1.99998 3.00005, 0 3.00005, 0 3.01858, 0.00935 2.99975, 0 2.99999, 0 3.00006, 0.00002

6, 1 6, 1 5.99998, 0.99999 2.99994, 0 2.99996, 0 1, 2 2.99983, 0 2.99999, 0 3.00001, 0 2.99994, 0 2.99998, 0 3, 0

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

5, 5, 5, 6, 6, 6,

1 1 1 1 1 1

3.00841, 0.0042 3.00603, 0.00301 3.00291, 0.00146 3.01261, 0.0042 3.00904, 0.00301 3.00437, 0.00146

3, 3, 3, 3.00001, 3.00001, 3,

0 0 0 0 0 0

0, 0 0, 0 0, 0

6, 1 6, 1 6, 1

6, 1 6, 1 6, 1

5.99996, 1 5.99996, 1 6, 1

0, 0 0, 0 0, 0

5.98916, 0.99639 5.96327, 0.98776 5.91121, 0.9704

3.03679, 0.01226 3.01083, 0.00361 3.00447, 0.00149

3.00001, 0 3.00001, 0 3, 0

2, 1 1.00003, 3.00001 1.00001, 3.00001 2, 1 1.00003, 3.00001 1.00001, 3.00001 2, 1 1.00003, 3.00001 1.00001, 3.00001

1.00002, 1.00002, 1.00002, 1.00002, 1.00002, 1.00002,

0.98946 0.98535 0.96977 0.98946 0.98535 0.96977

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3.00001 3.00001 3.00001 3.00001 3.00001 3.00001

1.00001, 3.00001 1.00007, 3.00003 1.00002, 3.00001

Table 6: Sustainability Of The System (Continued)

1, 2.99999, 2.99999, 2.00007, 2.99999, 2.99999, 3.01007, 2.99999, 2.99999,

2.99999, 2.99999, 2.99999, 2.99999, 2.99999, 2.99999,

2 0 0 1 0 0 1 0 0

0 0 0 0 0 0

2.99998, 0 3, 0 2.99999, 0

1, 2.99997, 3.00004, 2.01091, 2.99997, 3.00004, 5.96043, 2.99997, 3.00004,

2.99997, 3.00004, 3.00004, 2.99997, 3.00004, 3.00004,

2 0 0 1 0 0 1 0 0

0 0 0 0 0 0

3.00004, 0 2.99999, 0 2.99997, 0

1, 2 2.99999, 0 2.99969, 0 3.05428, 1 2.99999, 0 2.99969, 0 5.9998, 0.99993 2.99999, 0 2.99969, 0

2.99999, 2.99968, 2.99996, 2.99999, 2.99968, 2.99996,

0 0 0 0 0 0

2.99971, 0 2.99999, 0 2.99999, 0

Earliest Time To Reach ESS 28411 25764 30798 25268 22806 25872 22079 22152 25329 22321 22245 25329 * * 12857 13053 13769 12857 13053 13769 * 12857 13053 13769 12857 13053 13769 * * * 11592 12840 13752 * * 11592 12840 13752 * * 11544 7595 5707 11544 7595 5707 11544 7595 5707 * * * 9095 7305 5118 9095 7305 5118 * * * * 8003 6361 5109

Remarks

Infeasible Infeasible

Infeasible

Infeasible Infeasible Infeasible

Infeasible Infeasible

Infeasible Infeasible

Infeasible Infeasible Infeasible

Infeasible Infeasible Infeasible Infeasible

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Figure 10: Time To Reach ESS For Different Starting States

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3.3.4

Other Game Dynamics

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Other game dynamics can also be used to analyze evolutionarily stable strategies and hence the sustainability of the supply chain. For instance, let the provider and consumer population start in state S2 with payoff U , and move to payoff G and B respectively at time 15,001. As shown earlier, Figure 9 depicts the population evolution and the simplex for the game using replicator dynamics. Figures 11, 12, 13, and 14 depict the system evolution when other game dynamics are applied.

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Figure 11: Maynard Dynamics (Sandholm, 2011)

Figure 12: Brown - von Neumann - Nash (BNN) Dynamics (Sandholm, 2011)

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Conclusion

Our theoretical framework of modelling a supply chain as an evolutionary game allows us to study its sustainability in an integrated manner across multiple dimensions, while 24

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Figure 13: Smith Dynamics (Sandholm, 2011)

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Figure 14: Logit Dynamics (Sandholm, 2011)

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retaining scalability. Placing certain natural restrictions on the form of the payoff functions ensures the existence of an equilibrium for the supply chain. The challenge is in defining apt dimensions and metrics that in turn influence the correctness of the payoff function. Though we have focused on replicator dynamics, future work could include exploring other game dynamics for modelling as well as stochastic evolutionary games. Evaluation of the performance of the model within a supply chain through various prediction error techniques allows for learning. Incorporation of learning techniques (such as imitation and mutation) and epistemic game theory into our model is an avenue that we hope to explore. Also our numerical example only looks at a sample subset of metrics for NHS. A thorough analysis using all relevant metrics will be of immense interest and use.

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5

Abbreviations

OOP : Out-of-pocket PbHI : Public Health Insurance

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Technical Appendix A Here are definitions of some relevant game theory and evolutionary game theory concepts mentioned in this paper.

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Bimatrix Game: A game with two players, each of them having a finite set of strategies, is called a bimatrix game. If the first player chooses strategy xi and the second player choses strategy yj from their respective strategy sets, then the payoff the first player is given by aij = u1 (xi , yj ) and the payoff to the second player is given by bij = u2 (xi , yj ). Thus these payoff values can be easily represented as matrices A = (aij ) and B = (bij ), and hence the name “bimatrix games”.

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Nash Equilibrium: Consider a non-cooperative game involving more than one player. If each player’s strategy is an optimal response to the other player’s strategies, then this profile of strategies is a Nash equilibrium (i.e.) s∗ is a Nash equilibrium if ui (s∗i , s∗−i ) ≥ ui (si , s∗−i ) ∀si , ∀i.

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Single Player Control Game: If only one of the player’s actions control the transition to another state, the game is called a single player control game.

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Evolutionarily Stable Strategy: Let U be the payoff matrix for a population in an evolutionary game. A strategy p∗ is said to be evolutionarily stable if it can withstand the invasion by all other strategies p 6= p∗ as follows.

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p∗ U (p + (1 − )p∗ ) > pU (p + (1 − )p∗ ), ∀p 6= p∗ where  > 0 is termed as the invasion barrier. This is equivalent to the following two conditions: (1) p∗ U p ≥ pU p∗ (equilibrium condition); (2) If p 6= p∗ and p∗ U p∗ = pU p∗ , then p∗ U p > pU p (stability condition), (i.e.) p∗ does better against p than p does against itself.

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Replicator Dynamics: (Taylor et al., 1978) Replicator dynamics refers to a selection process in evolutionary games wherein more successful strategies spread in the population. They are described using replicator equations.

Replicator Equations: (Taylor et al., 1978) Consider a population of n types where xi is the frequency of type i leading to a state s ∈ Sn of the population. Let individuals in the population meet randomly and engage in symmetric games with payoff matrix A. Then, the replicator equation gives the per capita rate of growth as x˙ i = xi ((Ax)i − xt Ax). A solution to the replicator equation is referred to as a fixed point for the replicator system.

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Lyapunov function: (Barreira et al., 2010) A C 1 function is a Lyapunov function at a solution to the differential function if it takes positive values in every neighbourhood of the equilibrium except the equilibrium point itself, and is decreasing for all trajectories of the differentiable function. If it is strictly non-increasing for all trajectories, then it is a strict Lyapunov function. If the Lyapunov function exists, its fixed point is said to be asymptotically stable.

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