Allocation of Greenhouse Gas Emissions in Supply
Chains
Sanjith Gopalakrishnan† • Daniel Granot† • Frieda Granot† • Greys Sošić‡ • Hailong Cui‡
†
‡
Sauder School of Business, University of British Columbia, Vancouver, BC V6T 1Z2
Marshall School of Business, University of Southern California, Los Angeles, California 90089
[email protected] •[email protected] • [email protected] •
[email protected] • [email protected]
September 30, 2016
Abstract
In view of the challenges of meeting the goals set at the recent United Nations Climate Change
Conference in Paris, it should be noted that the 2,500 largest global corporations account for more
than 20% of global GHG emissions, and that companies’ direct emissions average only 14% of
their supply chain emissions prior to use and disposal. Therefore, rationalizing CO2 emissions in
supply chains could make an important contribution to the efforts of mitigating climate change.
Walmart has embraced its role to protect the environment and reduce emissions in its vast
supply chain, and in 2007 has started to collect data to assess GHG emissions of its supply chain.
However, since approximately 85% of all industrial use occurs in basic material manufacturing,
which is far upstream at the supply chain, Walmart needs to engage with upstream suppliers.
Indeed, aside for assigning responsibilities to suppliers for their own direct emissions, to improve
their environmental performance, supply chain leaders should be in a position to assign indirect
responsibilities to firms whose actions and decisions regarding, e.g., product design, packaging
design, material selection, or operating decisions, adversely affect GHG emissions by other firms
in the supply chain.
In this paper we consider supply chains with a motivated dominant leader, such as Walmart,
that has the power or authority to assign their suppliers responsibilities for both direct and
indirect GHG emissions. Given these responsibility assignments, we use cooperative game theory
methodology to derive an allocation scheme of responsibilities for the total GHG emissions in the
supply chain. The allocation scheme, which is the Shapley value of an associated cooperative
game, is shown to have several desirable properties. In particular, (i) it allocates responsibilities
for all the emissions in the supply chain without double counting, (ii) it is transparent and easy
to compute, (iii) it lends itself to several intuitive axiomatic characterizations which magnify
and clarify its appropriateness as a fair allocation of pollution responsibilities in a supply chain,
and (iv) it is shown to incentivize suppliers to exert pollution abatement efforts that, among all
footprint-balanced responsibility allocation schemes, minimize the maximum deviation from the
socially optimal pollution level.
1.
Introduction
At the United Nations Framework Convention on Climate Change concluded in Paris in December,
2015, 195 countries adopted the first-ever universal, legally binding global climate deal agreement.
The long term goal is to reduce global Greenhouse Gas (GHG) emissions by at least 60% below
2010 levels by 2050, and to set out clear, specific, ambitious and fair legally binding mitigation
commitments that would keep the increase in global average temperature to well below 2◦ C above preindustrial levels. The aim is to limit the increase to 1.5◦ C, since this would significantly reduce risks
and the impacts of climate change. The Agreement will be binding as soon as countries accounting
for more than 40 billion tons of CO2 equivalent emissions in 2015, representing approximately 80%
of current global emissions, have ratified it (Paris Agreement, 2015).
Unfortunately, current predictions suggest that the milestone objectives are quite challenging.
Limiting average global temperature rise to 2◦ C above pre-industrial levels is possible if cumulative
emissions in the 2000–2050 period do not exceed the target of 1,000 to 1,500 billion tonnes CO2 .
However, between 2000 and 2011, an estimated total of 420 billion tonnes CO2 was already cumulatively emitted due to human activities (including deforestation) (Olivier et al., 2012). If the current
global increase in CO2 emissions continues, cumulative emissions will surpass the target already by
2030 (Olivier et al., 2012). In fact, current emissions are tracking scenarios which project a mean
temperature increase of 4.2◦ C–5.0◦ C in 2100, with possible dire consequences (see, e.g., Peters et al.,
2012).
In view of the potential difficulties to meet the objectives of the Paris Agreement for reducing
CO2 global emissions to required levels, it should be noted that the 2,500 largest global corporations
account for more than 20% of global GHG emissions, and emissions resulting from corporate operations are typically exceeded by those associated with their supply chains (Carbon Disclosure Project,
2011; Jira and Toffel, 2013). In fact, Matthews et al. (2008) found that across all industries, companies’ direct emissions average only 14% of their supply chain emissions prior to use and disposal.
Therefore, rationalizing CO2 emissions in supply chains could make an important contribution to the
efforts of achieving the global objectives for emission reduction agreed upon by the Paris Agreement.
There are several reasons why firms should strive to reduce their CO2 supply chain emissions.
Indeed, in view of sustained interactions with various constituents, such as consumers, the media,
employees, shareholders, financial institutions, environmental NGOs, local communities, and governments, environmentalism has become central to the core objectives of the firm. (Hoffman, 2005).
1
Supply chains are vulnerable to the risks and costs stemming from possible new regulations related
to climate change (Van Bergen et al., 2008; Gunther, 2010; Halldorsson and Kovacs, 2010), and
poor environmental performance can reduce a firm’s market valuation (Klassen and McLaughlin,
1996; Konar and Cohen, 2001). On the other hand, a good environmental record can improve financial performance and overall competitiveness (Porter and van der Linde, 1995; Reinhardt, 1999),
enhance brand value, and increase market share (Hopkins, 2010; Kim and Lyon, 2011). In fact,
Corbett and Klassen (2006) argue that firms have always benefited from improving their environmental performance, though the precise nature or magnitude of these benefits were unpredictable in
advance.
Walmart has embraced its responsibility to protect the environment and reduce emissions in its
vast supply chain, and, as noted by Plambeck (2013), is profiting from its actions to reduce GHG
emissions in its supply chain. Indeed, in 2005, CEO Lee Scott announced that “Being a good steward
of the environment and being profitable are not mutually exclusive. They are one and the same”,
and committed Walmart “To be supplied 100 percent by renewable energy; to create zero waste; and
to sell products that sustain our resources and the environment” (Plambeck and Denend, 2007).
To act upon its challenging environmental goals, Walmart in 2007 has started to collect data to
assess GHG emissions of its supply chain. The United States federal government followed suit in
2009, when a new Presidential Executive Order required federal agencies to set reduction targets and
track the reduction of GHG emissions, including those associated with their supply chains (Obama,
2009), and many corporations have similarly joined the efforts to curb GHG emissions. Indeed,
the Carbon Disclosure Project’s (CDP) Supply Chain Program is a collaboration of multinational
corporations that have requested information about their key suppliers’ GHG emissions as well
as their vulnerabilities and opportunities associated with climate change. For a detailed related
discussion, see Jira and Toffel (2013).
Initially, prior to 2007, Walmart has asked only its tier-1 suppliers to measure and report their
GHG emission targets, and this simple request was reported to have improved environmental performance (Plambeck, 2013). The program has since evolved, and, in 2009, the company broadened its
supplier sustainability scorecard to 15 categories. The checklist used to evaluate suppliers, developed
in conjunction with the Sustainability Consortium1 , includes questions about GHG emissions, and
the responses are used to derive a total sustainability score for the suppliers.
However, since approximately 85% of all industrial use occurs in basic material manufacturing
1
https://www.sustainabilityconsortium.org/
2
(Intergovernmental Panel on Climate Change–IPCC, 2007; Plambeck, 2013), which is far upstream
at the supply chain, it is suggested that Walmart has to adopt a much more global perspective. It
must engage with upstream suppliers or motivate its tier-1 suppliers to do so (Plambeck, 2013).
Engaging all suppliers in a supply chain could be a monumental undertaking, even just due to
scope—Walmart, for example, has over 100,000 suppliers. Moreover, for a variety of reasons, as empirically investigated by Jira and Toffel (2013), suppliers may be reluctant to share information with
their buyers. Indeed, the response rate of inquiries addressed to nearly 8,000 suppliers about their
carbon emissions made by the 75 multinational companies participating in the CDP Global Supply
Chain Report 20162 was only 51%3 . Nevertheless, the difficulties of acquiring reliable information
about GHG emissions from suppliers need to be overcome in view of the urgency to mitigate climate
change.
To facilitate the rationalization of its supply chain, Walmart, for example, is collaborating with
academics and environmental third-party groups to identify processes in its supply chain that generate most of GHG emissions (Oshita, 2011; Plambeck, 2013). Indeed, to improve their environmental
performance, supply chain leaders, with the help of their suppliers, third-party environmental groups
and academics, must gain a deep insight into the causes of GHG emissions in their supply chain.
Aside for assigning responsibilities to suppliers for their own direct emissions, supply chain leaders
should be in a position to assign indirect responsibilities to firms in their supply chains whose actions
and decisions regarding, e.g., product design, packaging design, material selection (see, e.g., Gallego
and Lenzen, 2005; Wiedmann and Lenzen, 2006; Caro et al., 2013; HP 2014 Living Progress Report;
Herman Miller: Environmental Record, 2016), or operating decisions (e.g., Benjaafar et al., 2014),
adversely affect GHG emissions by other firms in the supply chain.
That is, the responsibility for GHG emissions in the supply chain attributed to a supplier should
incorporate both the responsibility for the emissions directly associated with the processes used by
that supplier in the production or distribution of the product, as well the responsibility for direct
emissions by other suppliers resulting from choices made by that supplier. The nature and size of
these responsibilities could be derived from the insight gained by supply chain leaders about GHG
emissions in their supply chain.
2
http://www.prnewswire.com/news-releases/companies-blind-to-climate-risks-in-half-their-supplychains-finds-largest-global-study-300209209.html
3
See also Plambeck and Taylor (2015) and references therein for other reasons why suppliers may not be forthcoming
with information about their operations, and, e.g., Guo et al. (2015) and Kalkanci and Plambeck (2015), who discuss
why firms may prefer to source from environmentally unreliable suppliers and why firms may prefer not to seek
environmental related information from their suppliers, respectively.
3
In this paper we consider supply chains with a motivated dominant leader, such as Walmart,
that has the power or authority to assign their suppliers responsibilities for both direct and indirect
GHG emissions in their supply chain. For example, adopting Caro et al. (2013) approach to our
setting, if firm i can exert abating efforts to reduce pollution by firm j, then the dominant supply
chain leader should assign4 firm i (some) indirect responsibility for the pollution emitted by firm
j. Given these responsibility assignments, we use cooperative game theory methodology to derive
a scheme for allocating the responsibilities of the total GHG emissions to the firms in the supply
chain. The allocation scheme, which is the Shapley value of an associated cooperative game, is
shown to have several desirable properties. In particular, (i) it is footprint-balanced, i.e., it allocates
responsibilities for all the emissions in the supply chain without double counting, (ii) it is transparent
and easy to compute (Theorem 1), (iii) it lends itself to several intuitive and insightful axiomatic
characterizations (Theorems 2 and 3), and (iv) when the abatement cost functions of the firms are
private information, it is shown to incentivize suppliers to exert pollution abatement efforts that,
among all footprint-balanced allocation schemes, minimize the maximum deviation from the socially
optimal pollution level (Theorem 4).
Thus, we argue in this paper that when there is a dominant leader, such as Walmart, which
has the power to assign firms in their supply chain direct and indirect responsibilities for GHG
emissions, the Shapley value should be adopted as an allocation rule of responsibilities for GHG
emissions in the supply chain. Indeed, as reported by Plambeck (2013), the mere request by Walmart
from its suppliers to measure and report their GHG emissions has led to an improvement in their
environmental performance. Therefore, it is reasonable to expect that drawing the attention of firms
to their direct and indirect responsibilities for GHG emissions, coupled with an aggregation of these
responsibilities into a fair, transparent and easy to compute allocation—the Shapley allocation—
with respect to which the firms’ efforts to curb GHG emissions can be evaluated, would similarly
lead to improved environmental performance by firms in the supply chain. Moreover, to the extent
that supply chain leaders would like to introduce carbon pricing, as many of them are doing5 (see,
e.g., CDP Report Summary, 2014), the Shapley allocation can be used to incentivize firms in the
supply chain to exert pollution abatement efforts which, among all footprint–balanced allocation
4
Indeed, from the perspective of incentivizing pollution abatement efforts, we prove in Section 5 that, in some sense,
optimal abating efforts are generated if each firm is assigned (partial) responsibility precisely for all processes whose
pollution it can affect.
5
Note that if carbon pricing is implemented then the dominant supply chain leader in our model plays precisely the
same role as the social planner in Caro et al. (2013), who decides on a footprint allocation rule and imposes a cost on
the firms in proportion to the emissions allocated to them.
4
rules, minimize the maximum deviation from the socially optimal pollution level.
We initially consider a basic supply chain for a single product, represented by a directed tree,
in which the nodes represent the producers, suppliers, and end-consumer—the leaf nodes signify the
most downstream suppliers, such as, e.g., material or energy suppliers, and the unique root node of
the tree denotes the end-consumer. The weight associated with the unique arc emanating from node,
say, j, represents the pollution which was directly created by producer j in the process of producing
the final good. Indirectly, however, other producers may be also be responsible for the pollution
directly created by producer j.
We note, though, that a supply chain graph could be more complex than a tree graph. Indeed,
Gallego and Lenzen (2005) (hereafter G&L) observe that in many real-life models outputs of one
supply chain member can be inputs for both upstream and downstream supply chain partners (e.g.,
a nuclear power plant using fuel elements and providing electrical power to the fuel elements’ manufacturer), or that one supplier can deliver directly to its non-immediate downstream consumers (who
are themselves producers—e.g., a power plant can deliver electricity to a steel manufacturer and an
upholstery manufacturer). This creates feedback loops and can lead to double counting of emissions.
Therefore, in order to address the more general case, we extend the analysis from a basic tree graph
to an arbitrary supply chain directed graph which supports the possible production of several final
products.
In this paper we assume that the GHG emissions from all processes in the supply chain are
known, and we focus primarily on the allocation of responsibilities for the GHG emissions to
firms in the supply chain. We acknowledge that there are numerous difficulties involved in calculating GHG emissions from all supply chain members. We note, though, that consistent attempts are being made by firms to measure GHG emissions in their supply chains. For example,
in the Carbon Disclosure Project, 75 multinational companies, such as Walmart, Dell, Amazon,
Ford and Vivendi, have inquired with nearly 8,000 of their suppliers about their carbon emissions6 . Similarly, for instance, Apple provides environmental reports for their products (http:
//www.apple.com/environment/reports/) which detail GHG emissions from different stages in a
product’s lifecycle, lists materials used in their products, etc. Timberland provides “green index”
for their products (http://greenindex.timberland.com), which details GHG emissions, chemicals
used, and resource consumption; the Innovation Center for U.S. Dairy provides emissions for fluid
6
http://www.prnewswire.com/news-releases/companies-blind-to-climate-risks-in-half-their-supplychains-finds-largest-global-study-300209209.html
5
milk and other dairy products, etc. In general, in order to calculate their carbon footprint, companies need to understand the emissions at different parts in their supply chains. Good illustrative
examples of such calculation are provided by the New Belgium Brewing Company for a 6-pack of
their Fat Tire Amber Ale7 and Levi’s and their 501 jeans8 .
In addition, while we believe that determining the proper carbon-based payments, which can
take the form of a carbon tax, permit costs in cap-and-trade systems, costs of carbon offsets, etc.,
is very important, analyzing the features of different payments lies beyond the scope of this paper.
However, as previously noted, the Shapley allocation, which incorporates both direct and indirect
responsibilities for emissions in the supply chain, can be used to incentivize suppliers to exert pollution abatement efforts that minimize the maximum deviation from the socially optimal pollution
level.
The plan of the paper is as follows. In Section 2 we provide a brief literature review. In Section
3 we introduce the basic model wherein the supply chain graph is a directed tree, and we present
and analyze the associated GHG Responsibility–Emissions and Environment (GREEN) game. In
Section 4 we develop three axiomatic characterizations for the Shapley allocation in the class of
GREEN games. In Section 5 we reveal the ability of the Shapley allocation to incentivize suppliers
to exert, in some sense, optimal abatement efforts, and in Section 6 we illustrate our approach
by allocating GHG emissions in a newspaper publishing supply chain. In Section 7 we develop our
extension to a general supply chain structure and demonstrate therein that all the results for the case
of a tree supply chain are valid for the general case as well. Some concluding remarks are provided in
Section 8. All proofs are given in Appendix A. The calculations of GHG emissions for the newspaper
publishing supply chain are available in Appendix B, while their allocations to different supply chain
members are computed in Appendix C.
2.
Literature Review
We note that the development of appropriate schemes for the allocation of GHG pollution responsibilities, which is the main objective of our paper, can be viewed as a natural step towards convincing
supply chain members to reduce their overall GHG emissions. Nevertheless, this topic has received
relatively limited attention in the supply chain literature so far. In the area of logistics, Cholette and
7
http://www.newbelgium.com/Files/the-carbon-footprint-of-fat-tire-amber-ale-2008-public-dist-rfs.
pdf
8
http://levistrauss.com/wp-content/uploads/2015/03/Full-LCA-Results-Deck-FINAL.pdf
6
Venkat (2009) study the impact of transportation and storage choice on carbon emissions in wine
distribution, while Hoen et al. (2012) analyze the effect of environmental legislation on transportation choices in supply chains. Cachon (2014) investigates the effect of retail store density on GHG
gas emissions, and Belavina et al. (2015) compares the financial and environmental performance of
two revenue models for on-line retailing of groceries. Plambeck (2012) studies some challenges facing
firms that try to reduce their GHG emissions and suggests some ways for dealing with them, and
Plambeck (2013) discusses the potential of operations management methodologies to reduce GHG
emissions. Sunar and Plambeck (2013) investigate three allocation methods of carbon emissions
among co-products. Chen et al. (2013) study simple supply chain models and their extensions to
incorporate carbon emissions, Benjaafar et al. (2013) investigate the potential synergies and emission
reductions from cooperation in a supply chain, and Benjaafar and Chen (2014) demonstrate that,
under some conditions, imposing emission penalties on firms in a decentralized supply chain could
lead to higher overall supply chain emissions. Corbett and DeCroix (2001) and Corbett et al. (2005)
study shared-savings contracts and their impact on the environment.
G&L have extended the traditional model with final-demand-driven industry by adding intermediate demand, and are primarily concerned with allocations of GHG emission responsibilities which
are efficient, i.e., allocate precisely the entire pollution among the supply chain members. Their
non-game-theoretic model has some features in common with our model. Specifically, similar to us,
they suggest that GHG emission responsibilities should be shared among all supply chain members
who have directly or indirectly created these emissions, and it can be shown that their allocations
belong to the core of our GREEN game. G&L suggest, however, that the parties further away from
the source have to bear a proportionally smaller share of responsibility for emissions, while according
to the Shapley allocation, emissions created by a firm are equally shared among all supply chain
members that are directly or indirectly responsible for it. Further, by contrast with the G&L’s
allocations, the Shapley allocation can be easily computed. Moreover, it lends itself to intuitive
axiomatic characterizations and it can be used to incentivize firms in the supply chain to exert, in
some sense, as is explained in Section 5, optimal efforts to mitigate CO2 emissions.
Caro et al. (2013) (hereafter CCTZ) study a model of joint pollution of GHG in a supply chain
where the total emissions can be decomposed into processes, and each process possibly influenced by
a number of firms. When a central planner allocates emissions to individual firms and imposes a cost
on them proportional to the emissions allocated, CCTZ find that even if the carbon tax is the true
social cost of carbon, emissions need to be over-allocated to induce optimal effort levels. However, as
7
noted by the authors, double counting may not be always feasible or appropriate. For example, to
avoid double counting, the GHG Protocol advises that “companies should take care to identify and
exclude from reporting any scope 2 or scope 3 emissions that are also reported as scope 1 emissions
by other facilities, business units, or companies included in the emissions inventory consolidation.”
Our approach to induce optimal abating efforts by firms in a supply chain can be viewed as
complementary to CCTZ. That is, we investigate the impact on abating efforts of allocation rules
which, by contrast with CCTZ, are restricted to be footprint-balanced (i.e., no over allocation).
Specifically, we show that when firms’ abating cost functions are private information, then, restricted
to footprint-balanced linear allocation rules9 , the Shapley value allocation induces abating efforts
which minimize the maximum deviation from the socially optimal pollution level (Theorem 4).
CCTZ do not consider specific methods for the allocation of GHG emissions, but propose, as
potential future work, building on game-theoretic models from Shubik (1962) and others to find
appropriate allocation rules. We note that Shubik (1962) investigates allocation methods of joint
costs that have desirable incentive and organizational properties, expressed through a set of axioms,
and has pointed out therein that the Shapley value (Shapley, 1953) is the unique allocation method
which satisfies these axioms. Thus, our work can also be viewed as being in the spirit of CCTZ
recommendation for future research regarding GHG emissions allocations.
Our very basic GREEN game model, corresponding to a directed tree, resembles Megiddo’s (1978)
tree model, which is an extension of the classical airport game model (Littlechild and Owen, 1973),
and, in turn, is a special case of the minimum cost spanning tree (mcst) game model (Bird, 1976;
Granot and Huberman, 1981, 1984). In all three models, as in the GREEN game model, the players10
are represented by nodes in a graph (directed path in the airport game), and the cost associated
with edges in that graph designate connection, maintenance or construction costs. The players in
all three models need to be connected to a special node, referred to as a central supplier, and the
cost for each set of players, S, is the cost of an mcst which spans S and the central supplier node.
By contrast, in the GREEN game defined on a directed tree, each player is “responsible” for the
cost (i.e., GHG pollution) created directly by her and indirectly created by an arbitrary set of other
producers/suppliers in the supply chain.
9
10
Note that CCTZ also restrict their analysis to linear allocation rules
In the airport game the players are landings by different size airplanes.
8
3.
The Basic GREEN Game Model
Consider a supply chain consisting of several entities, such as suppliers, manufacturers, assemblers,
etc. (hereafter referred to as players), which are cooperating in the production of a final product. The
supply chain is represented by a directed graph, G = (V (G), E(G)), which initially will be assumed to
be a directed tree T , T = (V (T ), E(T )), as shown11 in Figure 1. Extension to a general supply chain
directed graph, GT = (V (T ), E(GT )), will be considered in Section 7. The set of nodes, V (T ), aside
for the root node, denoted as node 0, represent the players, and the (directed) arc emanating from
each node i towards node 0 represents the process/activity by player i contributing to the creation
of the final product. We assume that only one arc enters node 0, and this arc emanates from node
1, which can represent the end consumer, in the cradle-to-grave life-cycle assessment (LCA) model,
or the most downstream manufacturer/assembler, in the cradle-to-gate LCA model. On the other
hand, the leaf nodes of T (or, in general, GT ), i.e., nodes which are not end nodes of any arc in T
or GT , represent the most upstream suppliers in the supply chain. We can also view node 1 as the
consuming country of the final product being manufactured by the supply chain, and all other nodes
as the producing countries/regions.
The numbers (weights) along the arcs (in italics) in Figure 1 represent the GHGs emitted by the
players in order to produce the final product. For each arc j ∈ E(GT ), we denote by aj the pollution
associated with arc j, and for each node i ∈ V (T ), we denote by Ei the set of arcs which emanate
from node i towards node 0 in G. (Clearly, if G is a tree, Ei is a singleton for each i ∈ V .) Thus,
P
a(Ei ) ≡ j∈Ei aj is the total pollution created directly by player i.
The total GHG emission in the process of producing the final product is the sum of the arc weights,
and we consider here the issue of how to allocate the responsibility of the total GHG emission among
the players. Two obvious and somewhat extreme allocation methods are as follows:
• Full Producer Responsibility—Each member in the supply chain is responsible for the emission
she directly creates.
• Full Consumer Responsibility—The most downstream manufacturer is responsible for all the
pollution created by the supply chain.
Both methods might be used in practice (e.g., in product labeling or in national GHG accounting;
11
Note that it is possible that a producer in a supply chain supplies several other producers in the process of producing
the final product (see, e.g., G&L and Lenzen, 2009). Thus, in general, the supply chain graph G is not necessarily a
tree graph.
9
Figure 1: Supply chain with tree structure
see Munksgaard and Pedersen, 2001).
In this paper we use cooperative game theory methodology to formally consider responsibility
allocation of GHG emission among the firms in the supply chain. Specifically, for each player i, let
Ei denote the set of arcs whose associate pollution is the direct responsibility of firm i. In addition,
let Ei denote the set of arcs whose associated pollution is both the direct and indirect responsibility
of player i. Naturally, by definition, Ei ⊆ Ei . Otherwise, the choice of Ei is unconstrained and is
determined by the supply chain leader. As discussed earlier, Ei may reflect the (possibly partial)
responsibility of firm i, as understood by the supply chain leader, for pollution in other processes
in the supply chain stemming from firm i’s “myopic“ decisions related to, e.g., material, design, or
process choices, as well as operational decisions12 .
In the absence of motivated dominant supply chain leaders, it is hoped that institutions such as
ISO, which provides some general standards for GHG accounting on an organizational and project
level, or the World Resources Institute (WRI) and the World Business Council for Sustainable Development (WBCSD), which came up with the GHG protocol, would propose some methods to
assign indirect responsibilities for GHG emissions for different supply chain members. Such a gen12
See also Proposition 2 where it is proven that abating efforts by firms in the supply chain are maximized, in some
well defined sense, when the responsibility sets, Ei , are chosen to coincide with the set of processes (i.e., arcs) whose
pollution level are affected by actions or decisions by firm i.
10
eral method could require, for example, that firms designing the product assume both upstream and
downstream emission responsibility, component suppliers are responsible for upstream emissions,
transportation/distribution/warehousing companies are responsible only for their direct emissions,
customers are responsible for all upstream emissions, etc. A somewhat similar example for such an
approach from the Extended Producer Responsibility (EPR) domain is the UK regulations regarding packaging, which divides producer responsibility into five categories: manufacturers, converters,
packers/fillers, sellers, and importers (if products are not manufactured in the UK), and then apportion the recycling obligations, 6%, 9%, 37%, 48%, (6% + 9%), respectively, to each group (UK
Government, 2007; Walls, 2006; for a related discussion, see also Jacobs and Subramanian, 2011).
Global supply chains without a motivated dominant leader would naturally face more difficulties to rationalize GHG emissions in their supply chains. Numerous countries use GHG accounting
methodologies that make them responsible only for the emissions they create within their own borders. However, for instance, according to Porter (2013), about a fifth of China’s emissions are for
products consumed outside its borders, and while Europe emitted only 3.6 billion metric tons of
CO2 in 2011, 4.8 billion tons of CO2 were created to make the products Europeans consumed in that
year. Unfortunately, production abroad is not in the sphere of influence of a country’s legislation
and countries have few possibilities to restrict imports on environmental criteria because of international trade agreements under the terms of WTO. It is hoped that recent trends of cooperation
between different jurisdictions, such as those between California and Quebec and between the US and
China13 , and especially the Paris Agreement in 2015, at which 195 countries adopted the first-ever
universal, legally binding global climate deal agreement on climate change, and the active role taken
up recently by the World Bank and the International Monetary Fund to impose carbon tax14 , would
lead to universally accepted approaches to assign indirect responsibilities in global supply chains.
3.1
The GREEN game: Definition and Basic Results
To present our game-theoretic formulation of responsibility allocation for GHG emissions in a supply
chain we first need to introduce some definitions and notation. A (cost) cooperative game in a
characteristic function form is the pair (N, c), where N is the set of players and c is the characteristic
function such that for each S ⊆ N , c(S) is the cost that can be “attributed” to S, often representing
13
On January 1, 2014, California and Quebec signed an agreement outlining steps and procedures to fully integrate
their cap-and-trade programs and enable carbon allowances and offset credits to be exchanged between their respective
programs, and on November 11, 2014, the U.S. and China made a Joint Announcement on Climate Change and Clean
Energy Cooperation.
14
“Carbon Pricing Becomes a Cause for the World Bank and I.M.F.”, NYT, 4/23/2016
11
the cost that S would incur if it severed its cooperation with the rest of the players and acted alone.
One of the main issues addressed by cooperative game theory is how should the cost, c(N ),
of the grand coalition be allocated among all the players. Various solution concepts have been
proposed, based, for example, on fairness, equality, or stability criteria, and in this paper we propose
to adopt some of these solution concepts, in particular, the Shapley value, as a scheme to allocate
responsibilities for GHG emissions in a supply chain.
The core of a game (N, c), C((N, c)), is one of the most basic solution concepts. It consists of
all vectors x = (x1 , x2 , . . . , xn ) which allocate the total cost, c(N ), among all players in N such
that no subset of players, S, is allocated more than the cost, c(S), “associated” with it. That is,
P
C((N, c)) = {x ∈ IRn : x(S) ≤ c(S), ∀S ⊂ N, x(N ) = c(N )}, where x(S) = j∈S xj . The core of a
game could be empty, and if non-empty, it usually does not consist of a unique allocation vector. The
characteristic function of a game (N, c) is said to be convex if c(S ∪ {i}) − c(S) ≤ c(R ∪ {i}) − c(R) for
all i ∈
/ S and R ⊆ S ⊆ N . The game (N, c) is said to be monotone if for all Q ⊆ S ⊆ N, c(Q) ≤ c(S),
and is said to be convex if its characteristic function, c, is convex. The core of a monotone game, if not
empty, consists only of non negative allocation vectors and the core of a convex game is non-empty
and contains its Shapley value15 (Shapley, 1971).
Let us specialize the analysis to responsibility allocation for GHG emissions in a supply chain.
Then, the set of players, N , consists of all members of the supply chain (i.e., all nodes, V (G), in the
tree graph representing the supply chain). Let c({i}) denote the total pollution emission that player
i is directly or indirectly responsible for. Then, recalling that the sets, Ei , represent the set of arcs
P
whose associated pollution is the direct and indirect responsiblity of firm i, c({i}) = a(Ei ) ≡ j∈Ei aj .
For a subset of players S, let ES denote the collection of arcs whose associated pollution is the direct
or indirect responsibility of players in S. Thus, ES = ∪i∈S Ei , and the pollution which S is directly
P
or indirectly responsible for is c(S) ≡ a(ES ) = j∈ES aj .
We refer to (N, c) as the GHG Responsibility–Emissions and Environment (GREEN) game associated with the supply chain, where N is the set of players represented by the nodes of T or GT ,
aside for the root node 0, and the characteristic function, c(S), is as defined above for all S ⊆ N .
One can show that the full producer responsibility allocation, i.e., each member in the supply
chain is responsible for the emission she directly created, as well as the full consumer responsibility
allocation belong to the core of the GREEN game. However, both these allocations are extreme, in
the sense that they allocate all or nothing. Indeed, it can be shown that they are extreme points in
15
The Shapley value is formally introduced in the next subsection.
12
the core of the GREEN game and as such, do not possess the fairness property exhibited, for example,
by the Shapley value. However, the Shapley value, in general, is not a core member. Nevertheless,
as we will demonstrate below, in GREEN games, the Shapley value does belong to the core.
Proposition 1 The GREEN game (N, c) is convex. That is, c(S ∪ {i}) − c(S) ≤ c(Q ∪ {i}) − c(Q)
for all Q ⊆ S ⊆ N and i ∈ N .
As a result of Proposition 1, we conclude that Shapley value of the GREEN game belongs to the
core. Indeed, it is the barycenter of the core (Shapley, 1971). Thus, at the Shapley allocation, no
subset of supply chain firms is allocated more pollution responsibility than what they have directly
or indirectly created.
3.2
The Shapley Value of the GREEN game
We provide in this section a simple characterization of the Shapley value for GREEN games. In general, the Shapley value (Shapley, 1953), Φ(c), of a cooperative game, (N, c), is the unique allocation
which satisfies the following axioms:
1. Symmetry: If players i and j are such that for each coalition S not containing i and j, c(S ∪
{i}) − c(S) = c(S ∪ {j}) − c(S), then Φi (c) = Φj (c).
2. Null Player: If i is a null player, i.e., c(S ∪ {i}) = c(S) for all S ⊂ N , then Φi (c) = 0.
3. Efficiency: ΣN Φi (c) = c(N ).
4. Additivity: Φ(c1 + c2 ) = Φ(c1 ) + Φ(c2 ) for any pair of cooperative games (N, c1 ) and (N, c2 ).
An interpretation of Shapley value is given as follows. Consider all possible orderings of the
players, and define a marginal contribution of player i with respect to a given ordering as his marginal
cost to the coalition formed by the players before him in the order, c({1, 2, . . . , i−1, i})−c({1, 2, . . . , i−
1}), where 1, 2, . . . , i − 1 are the players preceding i in the given ordering. Shapley value is obtained
by averaging the marginal contributions for all possible orderings. This average is given by
Φi (c) =
X (|S| − 1)!(n − |S|)!
(c(S) − c(S \ {i})).
n!
(1)
{S:i∈S}
It was shown by Shapley (1953) that (Φi (c)), given by (1), is the unique allocation rule which
satisfies the above four axioms. Note that the Shapley allocation provides a direct link between
players marginal contributions and their corresponding allocations. Similarly, Hart and Mas-Colell
13
(1989), using a different axiomatic approach, have also demonstrated that the Shapley value can be
viewed as reflecting the players’ marginal contributions to the associated game, and Young (1985)
has provided an alternative axiomatization of the Shapley value wherein the additivity axiom is
replaced with a compelling monotonicity axiom. Thus, the Shapley value can be perceived as a
fair and “justifiable” allocation method, and it is not surprising that it was extensively considered
as an allocation method in a variety of problems arising, e.g., in economics, management, and cost
accounting. For example, it was considered as an allocation method of pollution reduction costs
(Petrosjan and Zaccour, 2003), for generating airport landing fees (Littlechild and Owen, 1973), for
allocation of transmission costs (Tan and Lie, 2002), for economic distributional analysis (Shorrocks,
2013), and in the non-atomic game formulation framework, e.g., to generate internal telephone billing
rates (Billera et al., 1978). In Section 4 we provide additional original axiomatizations of the Shapley
value in the class of GREEN games.
To characterize the Shapley value for GREEN games, let ej denote the arc emanating from node
j in T , with an associated GHG emission weight aj . Denote by N j the set of players who are directly
or indirectly responsible for aj . That is, player i ∈ N j if and only if ej ∈ Ei . We can now provide an
explicit and intuitive characterization of the Shapley value of this game that can be generated very
efficiently.
Theorem 1 The allocation according to which aj is allocated equally among members in N j for each
j ∈ N is the Shapley value of (N, c).
Thus, the Shapley value has an intuitive interpretation and is easy to compute, as pollution is
equally allocated among all supply chain members who are directly or indirectly responsible for it.
Finally, note that so far it was assumed that indirect responsibility of, say, player i, for the
pollution, aj , associated with player j, entails the responsibility for the entire pollution aj . However,
conceivably, it could be more appropriate in some scenarios to assign player i indirect responsibility
only for part of the pollution aj . In that regard we have:
Comment 1 The GREEN game, (N, c), can be extended to allow for partial indirect pollution responsibility by producers. Namely, each producer i can be assumed to have direct responsibility for
the pollution associated with all arcs in Ei , and, in addition, partial (i.e., fractional), instead of
complete, indirect responsibility for the pollution associated with other arcs in the graph. Indeed, all
the results that were derived, including convexity of the GREEN game and the explicit expression for
Shapley value, i.e., Theorem 1 (with the natural modification) hold for this extension of the GREEN
game. For the sake of a simpler exposition we have elected not to present the more general model.
14
4.
Axiomatic Basis for the Shapley Value in GREEN Games
As previously demonstrated, the Shapley value of a GREEN game incorporates notions of fairness
and can be easily computed. We take below a complementary approach to the Shapley value of a
GREEN game. That is, we first introduce several natural properties that an allocation rule should
satisfy in the context of allocation of pollution responsibilities, and we will subsequently demonstrate
that the Shapley rule is the unique pollution allocation rule satisfying these properties. A similar
approach in the context of environmental responsibilities is adopted by Rodrigues et al. (2006)
who consider an input-output framework and impose six properties to derive a unique indicator
of environmental responsibility. Other axiomatic characterizations of the Shapley value in related
domains have been derived, e.g., by Dubey (1982), providing an axiomatic basis for the Shapley value
in airport games (Littlechild and Owen, 1973), Aadland and Kolpin (1998), in irrigation situations
which are similar to airport games, and in minimum cost spanning tree games (Kar, 2002).
We consider a supply chain graph G = (V (G), E(G)). Suppose that there are n firms in the
supply chain denoted by N = {1, 2, ..., n}. Denote by M = {1, 2, ..., m} the set of all processes in
the supply chain where each process j corresponds to an activity by some player in the supply chain
contributing towards the production of the final product. The process j with pollution fj has an
associated set of firms Nj which bear responsibility for it. Thus each process, j, could be thought
of as corresponding to some arc (j1 , j2 ), with an associated set of firms Nj that bear responsibility
for the pollution, fj = aj ≥ 0, emitted at arc (j1 , j2 ). The Shapley value was shown to divide
the pollution, fj , equally among the firms responsible for it. However, the processes could also
correspond more generally to portions of the arcs as shall be seen in Section 7, where each arc is
divided into several parts and each part is assigned as the responsibility of the correct set of players.
Thus henceforth, the discussion shall only deal with the set of processes M in the supply chain, and
not the arcs themselves.
Let firm i be responsible for the pollution associated with the set of processes Pi , and let P̃i
denote the set of processes with non-zero pollution that i is responsible for. We can consider a
responsibility matrix B, such that firm i is responsible for the pollution of process j if and only if
m
P
bi,j = 1, and 0 otherwise. Thus, a firm i is responsible for the total pollution,
fj bi,j . Similarly, for
j=1
a set of firms S, let bS,j = 1 if at least one firm in S is responsible for the pollution of process j and 0
m
P
otherwise. Then, the set of firms S is responsible for the total pollution given by f (PS ) =
fj bS,j .
j=1
Let f = (f1 , f2 , ..., fm ) be the total footprint set consisting of the pollution of all processes in the
15
supply chain.
Definition 1 A pollution allocation rule φ is defined on a supply chain with n firms, set of processes
n
M and a responsibility matrix B, as a mapping φ : Rm
+ → R+ which allocates to each firm its
n
m
P
P
responsibility towards the total pollution such that
φi (f ) =
fj .
i=1
j=1
Thus, a pollution allocation rule allocates the total pollution responsibility among the firms
without double counting. Next, let us consider certain intuitive properties that we believe that any
rule should possess in the context of pollution allocation.
1. Equal sharing of extra pollution: If f 0 ≥ f such that fj0 = fj for all processes j ∈ M \{k}
and fk0 > fk , then for p, q ∈ N such that bp,k = bq,k = 1, φp (f 0 ) − φp (f ) = φq (f 0 ) − φq (f ). This
property implies that if the pollution of some process increases and all the others remain the
same, then any two firms which are held responsible for the pollution of that process should
bear the extra burden equally.
2. No free riding: If for any firm i and footprints f 0 ≥ f , such that fj0 = fj for all processes j for
which bi,j = 1, then φi (f 0 ) = φi (f ). This property requires that if the total pollution increases,
but for some firm, the pollution of the processes it is responsible for are unchanged, then the
firm’s allocation remains the same. In other words, the increase in pollution allocation for a firm
is justifiable only if the pollution of the processes it is responsible for increases. Equivalently,
it also prevents free-riding of firm j on the pollution abatement improvements of other firms
on processes it is not responsible for. Chun (1989) invokes a similar principle of not rewarding
a player for the technology improvements of others and provides an alternate axiomatization
of the Shapley value. Lange (2006) also discusses the negative effects of free-riding and notes
that preventing free-riding improves the chances of cooperation in environmental agreements.
3. Firm equivalence: If for two firms i and j, P̃i = P̃j , then φi (f ) = φj (f ). This property
states that if two firms are equivalent in that they are responsible for the exact same set of
polluting processes, then they must be allocated an equal share of the total responsibility. Firm
equivalence is a fundamental equity principle that enhances the acceptability of a pollution
responsibility allocation mechanism.
4. Firm nullity: If for a firm i, P̃i = ∅, then φi (f ) = 0. This elementary property implies that
if a firm is not responsible for any polluting process, then it is allocated zero responsibility.
16
5. Process history independence: Let f 0 ≥ f and f˜0 ≥ f˜ be footprints such that fj0 = fj
and f˜j0 = f˜j for all processes j ∈ M \{k}, fk0 = f˜k0 and fk = f˜k . Then, for any firm i,
φi (f 0 ) − φi (f ) = φi (f˜0 ) − φi (f˜). This property states that the change in responsibilities of
the firms due to a change in pollution of any process is independent of the pollution levels
of other processes. Process history independence emphasizes the ease of interpretation of an
allocation rule. It implies that the firms can find out the effect of an increase or decrease
in the pollution of a process independent of the pollution levels of other processes. Thus
it enhances the transparency of the effects on pollution responsibility due to investment in
pollution abatement technologies.
6. Disaggregation invariance: Consider a supply chain graph G = (V (G), E(G)) with n
firms, and set of processes M . Let firm i be responsible for the set of processes Pi . Suppose
firm i chooses to disaggregate and represent itself as firms i1 and i2 , with an arc (i1 , i2 ) joining
them, which corresponds to a process with zero pollution (for example, a distribution process
with no pollution). Each of i1 and i2 is responsible for disjoint sets of processes Pi1 and Pi2 ,
respectively, such that Pi1 ∪ Pi2 = Pi ∪ {(i1 , i2 )}. The disaggregated supply chain G0 has n + 1
firms, and process set M ∪ {(i1 , i2 )}. Then, φ is said to be disaggregation invariant if for a firm
j 6= i, φj (G) = φj (G0 ), and φi1 (G0 ) + φi2 (G0 ) = φi (G).
Invariance of a pollution responsibility allocation to disaggregation of the supply chain is discussed by Lenzen et al. (2007) and also by Rodrigues and Domingos (2008). They argue that,
for example, if a manufacturer instead of selling directly, decides to disaggregate and sell via
a distributor who creates no additional pollution, it should not change the pollution allocations to the firms. If a pollution allocation rule is not invariant under disaggregation, it might
provide incentives for firms to resort to manipulation by de-merging while reporting emissions.
The following result provides an axiomatic basis for the Shapley allocation rule, which allocates
equally the pollution of each process to all the firms held responsible for it.
Theorem 2 The Shapley pollution allocation rule is uniquely characterized by each of the following
sets of independent properties:
i. Equal sharing of extra pollution and no free riding.
ii. Firm equivalence, firm nullity and process history independence.
iii. Firm equivalence, no free riding, and disaggregation invariance.
17
The first characterization is based solely on fairness considerations. The second is based on the
milder fairness properties of firm equivalence and firm nullity, and the process history independence
property that emphasizes the ease of interpreting the effect of a change in pollution of a process
on the allocation of responsibilities. The third characterization shows that subject to the natural
fairness properties of firm equivalence and no free riding, the Shapley allocation rule is the unique
pollution allocation rule that is disaggregation invariant and is thus strategy proof in that sense.
This makes it an attractive pollution allocation rule for regulators who wish to prevent firms from
falsifying by de-merging while reporting emissions. It also addresses fundamental equity considerations that the firms might expect from a pollution responsibility allocation mechanism. The three
distinct characterizations provide an axiomatic basis for adopting the Shapley value as a pollution
allocation rule in terms of fairness, ease of interpretation, and strategy proofness. From the practical perspective, it appears that the third axiomatization carries the most weight, as it prevents
free-riding and importantly discourages gaming of the system.
4.1
Partial pollution responsibilities
The above discussion can be extended to account for partial indirect pollution responsibility for some
processes, by modifying the definition of the responsibility matrix B suitably. A firm i bears bi,j
responsibility towards the pollution of process j, where bi,j ∈ [0, 1]: it bears full responsibility for j
if and only if bi,j = 1, and bears no responsibility if it is 0; otherwise, it is partially responsible. If
m
P
fj is the pollution of process j, then firm i is responsible for the total footprint
fj bi,j .
j=1
The explicit expression for the Shapley value obtained in the previous setting also holds after incorporating partial responsibilities with the appropriate modification. The Shapley rule can again be
characterized uniquely as above, with the equal sharing property suitably modified to a proportional
sharing of extra pollution property formalized below.
1. Proportional sharing of extra pollution: If f 0 ≥ f such that fj0 = fj for all processes
j ∈ M \{k} and fk0 > fk , then for p, q ∈ N such that bp,k , bq,k > 0,
φp (f 0 )−φp (f )
φq (f 0 )−φq (f )
=
bp,k
bq,k .
This
property implies that if the pollution of some process increases and all the others remain the
same, then the allocation to any two firms responsible (partially or fully) for that process
increases in proportion to their individual responsibility towards that process. Note that, if we
consider only full responsibility, proportional sharing coincides with the equal sharing of extra
pollution property.
18
2. Firm proportionality: If for two firms p and q and any process j,
α ≥ 0, then
φp (f )
φq (f )
bp,j
bq,j
= α for some constant
= α. This property implies that if two firms are proportionally responsible
for the same set of polluting processes, then the pollution allocation is also in proportion to
their responsibilities.
Theorem 3 Allowing for partial pollution responsibilities, the Shapley pollution allocation rule is
uniquely characterized by each of the following sets of independent properties:
i. Proportional sharing of extra pollution and no free riding.
ii. Firm proportionality, firm nullity and process history independence.
iii. Firm proportionality, no free riding and disaggregation invariance.
The proof of the above result is similar to the case previously discussed with only full responsibilities.
5.
Footprint Balanced Allocations and Pollution Abatement Incentives
CCTZ have shown that footprint balanced allocation rules, in general, cannot achieve first-best
emission reduction efforts. However, as previously discussed, footprint balancedness may be an
intrinsic constraint while designing a pollution allocation mechanism. Accordingly, we investigate in
this section the pollution abatement incentives that can be generated by footprint balanced emission
responsibility allocations. In particular, we prove that under some assumptions on the abating
cost functions, the Shapley allocation induce suppliers to employ abating efforts that minimize the
maximum deviation from the socially optimal pollution level.
Consider the supply chain model introduced in Section 3. Suppose that the firms can exert costly
pollution abatement efforts so as to jointly reduce pollution in the supply chain. As in Section 4,
suppose that firm i can influence the emissions at the set of processes (or arcs) denoted by Pi , and
that each firm i can introduce emission reduction efforts eij ∈ [0, 1] towards arc j ∈ Pi . For clarity, eij
could also correspond to indirect efforts and actions such as component design or material selection
which affects the direct emissions by other firms.
Following CCTZ, the emissions at arc j, aj , is assumed to be a symmetric decreasing convex
function of the emission reduction efforts and, in addition, to be additive separable in the efforts by
P
all firms, Nj , who are held responsible for the pollution in arc j, aj =
aij (eij ). The additive
i∈Nj
19
separability assumption of carbon reduction efforts is made to simplify the analysis, and it is valid
in several settings. For example, the design of a more efficient component by one supplier may not
affect the emissions due to another component in the same product sourced from a different supplier.
Further, as we note below, the commonly assumed pollution abatement models in the literature
satisfy the additive separable assumption. Finally, without loss of generality, the emissions are
assumed to be symmetric in efforts by folding up any asymmetry into the corresponding abatement
cost functions.
The abatement cost function cij (eij ) : [0, 1] → [0, C] is assumed to be convex and strictly increasing as in CCTZ. For an arc j, let cj denote the vector of cost functions cij over all the firms i ∈ Nj
and let c denote the collective vector of cost functions over all the arcs. Similarly, let ej denote
the vector of carbon reduction efforts by the firms in Nj and let e be the collective effort vector
for all firms over all the arcs. We also assume that aj and cij have non-negative third derivatives
for all firms i and arcs j. The commonly assumed model of pollution abatement in the literature is
of linear emission reduction with quadratic or cubic abatement costs (e.g., see Subramanian et al.
2007; Parry and Toman 2002) and it is easy to see that these models satisfy all our assumptions.
Numerical results for these particular models are provided at the end of this section.
The social first-best pollution emissions value for the supply chain, a∗ , is obtained by solving the
following optimization problem16 , where pS > 0 denotes the societal cost per unit of emissions,
P
min
e
cij (eij ) +
i∈N, j∈Pi
P
pS aj (ej ).
j∈E(GT )
CCTZ note that linear sharing rules are the only type of allocation rules observed in practice and,
moreover, non-linear differentiable rules could be replaced by linear sharing rules which would lead
to the same outcome in the decentralized pollution reduction game (Bhattacharya and Lafontaine,
1995). Therefore, we limit our attention to linear allocation rules, φ, that are footprint balanced,
P
P
given by
λij aj = aj ,
λij = 1 and λij ≥ 0, for all arcs j. Each player i faces the following
i∈Nj
i∈Nj
optimization problem in the decentralized game
min
P
eij : j∈Pi j∈Pi
cij (eij ) +
P
pS λij aj (ej ).
j∈Ei
For linear allocation rules φ, let aφ be the supply chain emission supported by an equilibrium of
the decentralized game. If multiple equilibria exist, then let it denote the minimum emission level.
16
As in CCTZ, we assume that abatement efforts, though costly, don’t affect the firms’ revenues from their core
operations.
20
Given the emission and abatement cost functions, let ã = min aφ denote the minimum decentralized
φ
supply chain emission level supported in equilibrium over all linear sharing rules. Though our model
differs from that adopted by CCTZ, it can still be shown that footprint balanced emission sharing
rules cannot achieve first-best efforts in the decentralized game and further that ã ≥ a∗ .
In Section 3, we characterized the Shapley allocation of the GREEN game with a complete flexibility regarding the definition of responsibility sets, Ei , for each firm i. From the perspective of
incentivizing pollution abatement efforts, the following proposition makes a definitive recommendation regarding the composition of these sets.
Proposition 2 ã is minimized when the responsibility sets Ei = Pi for all firms i in the supply
chain.
Thus, minimum possible supply chain emissions in the decentralized setting, supported by linear
sharing rules, is achieved when the responsibility sets of firms consist of all arcs (processes) whose
pollution they can influence.
We next address the question of identifying a unique best allocation mechanism, in terms of
incentivizing optimal emission reduction efforts, amongst the class of linear sharing rules which
are footprint balanced. Since the pollution abatement technologies available to a firm and the
corresponding cost functions, cij , are typically private information, the metric we adopt to evaluate
a footprint-balanced allocation is its worst-case performance over all possible private abatement cost
functions.
Definition 2 i. Loss of efficiency due to a sharing rule φ with a collective cost vector c is given by
δ(φ, c) = aφ − a∗ > 0.
ii. The worst-case loss of efficiency is defined as ∆(φ) = max δ(φ, c) for all i ∈ N , j ∈ Pi , where C
cij ∈C
is any subspace of all functions satisfying the abatement cost assumptions.
The following result identifies the Shapley allocation as a footprint-balanced linear sharing rule
that minimizes the worst-case loss of efficiency17 .
Theorem 4 For every linear allocation rule φ that is different from the Shapley allocation, Φ,
∆(φ) ≥ ∆(Φ).
17
Moulin and Shenker (2001) have derived a similar result to Theorem 4 but in a completely different setting
concerning participation games, where a cost sharing mechanism decides upon the subset of players that will receive
some common service and the cost allocated to each player, and players’ utilities from the service is a private information.
Gkatzelis et al. (2016) study general resource selection games where players choose to use a set of resources and share
the joint cost, and show that the Shapley cost-sharing method minimizes the worst-case price of anarchy (ratio of the
total cost in equilibrium to the socially optimal total cost).
21
We would like to point out that the symmetry of the Shapley allocation, as embodied by the
symmetry axiom described in Section 3, is not the driver of Theorem 4, since there are other allocation
mechanisms that satisfy this symmetry axiom. We further note that if the loss of efficiency were
to be measured with respect to the total social cost, instead of the socially optimal supply chain
emission level, then the worst-case loss of efficiency may no longer be minimized by the Shapley
allocation.
Comment 2 Theorem 4 and Proposition 2 can be extended to a setting with partial or fractional
responsibilities. For example, suppose that the supply chain leader has information about the relative
effectiveness of emission reduction efforts by two firms towards a process belonging to both of their
respective responsibility sets, such that identical efforts from the two firms result in proportional
emission reductions. This information can be incorporated by assigning partial responsibilities to the
firms. Then, Theorem 4 continues to hold true with the suitable modification for the expression of
the Shapley value.
5.1
Numerical Examples
We briefly consider the commonly assumed model of pollution abatement in the literature, of a linear
emission reduction with quadratic or cubic abatement costs, to exemplify the above discussion. The
first example is a two player supply chain with players 1 and 2. While the pollution at arc (2, 1), a2 , is
affected solely by the efforts of player 2, the pollution at arc (1, 0), a1 , can be reduced by joint efforts
of both players 1 and 2. Suppose that a1 (e11 , e21 ) = a1 − α1 (e11 + e21 ) and a2 (e22 ) = a2 − α2 e22 .
The abatement costs are given by c1 (e11 ) = β11 eh11 and c2 (e21 , e22 ) = β21 eh21 + β22 eh22 with h = 3.
A linear allocation is parametrized by the fraction of pollution at arc (1, 0) that firm 1 is allocated
responsibility for, denoted by λ in Fig. 2. Fig. 2 depicts the loss of efficiency at the two permutations
of the cost functions for a set of parameter values, and it is clearly seen that the worst-case loss of
efficiency is minimized at λ =
1
2
which corresponds to the Shapley allocation.
The second example is a three player supply chain with similar emission reduction and abatement
cost structures, where only players 2 and 3 are responsible for arcs (2, 1) and (3, 1), respectively, but
all players are jointly responsible for arc (1, 0). Suppose that a1 (e11 , e21 , e31 ) = a1 −α1 (e11 +e21 +e31 ),
a2 (e22 ) = a2 − α2 e22 and a3 (e33 ) = a3 − α3 e33 . The costs are given as before, c1 (e11 ) = β11 eh11 and
c2 (e21 , e22 ) = β21 eh21 + β22 eh22 and c3 (e31 , e33 ) = β31 eh31 + β33 eh33 , where h = 2, 3. Fig. 3a and Fig. 3b
are surface plots of the worst-case loss of efficiency over all permutations of a chosen set of cost
parameters. It is again seen that the worst-case loss of efficiency is minimized when each firm is
allocated one-third of the the pollution at arc (1, 0) which coincides with the Shapley allocation for
22
0.4
β11 = 2, β21 = 4
β11 = 4, β21 = 2
∆(φ)
δ(φ, c)
0.35
0.3
0.25
0.2
0
0.2
0.4
0.6
0.8
1
λ
Figure 2: Loss of efficiency in a 2-player supply chain with cubic abatement costs (α11 = 1, [β11 , β21 ]
are equal to the two permutations of [2, 4], and the rest of the model parameters are irrelevant for
the display).
the associated game.
0.6
∆(φ)
∆(φ)
0.36
0.34
0.32
0.5
0.4
0.8
0.6
λ2
0.8
0.4
0.2
0.2
0.4
0.6
0.8
0.6
λ2
λ1
(a) Quadratic abatement costs
0.4
0.2
0.2
0.4
0.6
0.8
λ1
(b) Cubic abatement costs
Figure 3: Worst-case loss of efficiency in a 3-player supply chain with quadratic and cubic abatement
costs is minimized at the Shapley allocation (α1 = 1, [β11 , β21 , β31 ] are equal to the six permutations
of [2, 4, 5], and the rest of the model parameters are irrelevant for the display).
6.
Illustrative Example: Newspaper Publishing
A typical newspaper publishing process involves paper and ink manufacturing and their transportation to the newspaper printing facility. Once newspapers are printed, they are delivered to the
retailers, libraries, or end consumers directly. Toffel and Sice (2011) discuss this process and provide
23
an example of aggregate carbon footprint calculation at the product-level. The corresponding supply
chain is illustrated in Figure 4.
0
Paper Transportation
1
Consumer
2
Newspaper Delivery
3
Newspaper Publishing
4
8 Ink Transportation
7
5
Paper Manufacturing
Energy for Paper
Manufacturing
Engery for
Newspaper
Publishing
9 Ink Manufacturing
6
10 Engergy for Ink
Manufacturing
Figure 4: A newspaper supply chain
Our analysis focuses on supply chains of two publishers, the Los Angeles Times (LAT) and the
New York Times (NYT), selling their newspapers to an average consumer in the Los Angeles area
who receives a daily delivery of two newspapers. Thus, Figure 4 can be viewed as an identical
supply chain representing two different products corresponding to LAT and NYT. In this simple
supply chain, we separate the GHG emissions due to manufacturing from transportation and show
the flexibility of our GREEN model by the choice of Ei as discussed above.
The direct emissions from each process (i.e, arc in Figure 4), represented as an edge weight in
our model, were calculated in Appendix B and are provided in Table 1. Thus, by Table 1, the total
newspaper carbon footprint for, say, the NYT in Los Angeles is 439 kg CO2 Eq. By comparison, the
total footprint of the newspaper supply chain associated with the NYT in Berkeley, California, is
788 kg CO2 Eq (Toffel and Sice, 2011). The main difference between these two figures (788 kg vs.
439 kg) apparently stems from the different estimates of the annual NYT newspaper weight.18
18
Specifically, Toffel and Sice, 2011) estimate the weight to be 236 kg vs. 101.5 kg in our case. Since both Toffel
and Sice (2011) and we use the same emissions factor of 2.8 kg CO2Eq/kg of newspaper, this difference alone accounts
24
In our analysis we consider a cradle-to-gate model (that is, we do not consider paper recycling),
so we set to zero GHG emissions for the end consumer. While the NYT has more pages than the
LAT, the difference in the GHG emissions between LAT and NYT is mostly caused by the difference
in the pricing of the newspapers, as publishing emission allocations are based on the dollar amount
of economic activity.
Supply chain member
arc ej
Emission on arc ej
LAT
NYT
Consumer
Newspaper Delivery
Newspaper Publishing
Paper Transportation
Paper Manufacturing
Energy for Paper Manufacturing
Energy for Newspaper Publishing
Ink Transportation
Ink Manufacturing
Energy for Ink Manufacturing
(1,0)
(2,1)
(3,2)
(4,3)
(5,4)
(6,5)
(7,3)
(8,3)
(9,8)
(10,9)
0
1.08
50.82
40.24
94.80
95.55
51.25
0.0002
0.0060
0.0012
0
1.69
97.02
42.24
99.51
100.30
97.84
0.0005
0.0115
0.0023
Table 1: GHG emissions estimation for each supply chain process (unit: kg CO2 Eq)
In line with our earlier discussion, we assume that the publisher (NYT or LAT) is a dominant supply chain leader who is seeking a better understanding of the sources of GHG emissions
and is motivated to reduce emissions in his supply chain. According to the Green Press Initiative
(http://www.greenpressinitiative.org/impacts/climateimpacts.htm), the paper industry is
the third or fourth largest source of industrial GHG emissions in most developed countries. In the
US, production of newsprints in 2009 emitted 26 million tons of GHGs, and publishers have an
opportunity to curb emissions by a careful selection of their supply chain partners. Indeed, recent
innovations in the paper industry have been reported. For instance, International Paper has reduced
its GHG emissions in 2013 by 5.8% by, among other things, using a biomass derived from a waste
product that would otherwise have been discarded instead of fossil fuels19 , and Cascades Papers has
for (236 kg - 101.5 kg) * 2.8kg CO2Eq/kg = 376.6 kg CO2 Eq. We estimated the NYT annual newspaper weight by
estimating the average weekly number of pages, multiplying it by a weight factor of 0.0039 kg/page, which was derived
by FedEx shipping scale, and multiplying it further by 52 weeks; see Appendix B for more details. We suspect that
the reason the weight estimate of Toffel and Sice (2011) is much higher is because they may have included everything
from NYT delivered to customers, including, e.g., Sunday magazines, miscellaneous advertisement, etc., while we only
used the weight of the actual newspapers.
19
http://www.triplepundit.com/special/sustainable-forestry-ip/international-paper-cuts-greenhousegas-emissions-by-5-8-percent/#
25
invested more than 1.8 million dollars in a system that captures methane from a landfill that would
otherwise be burned at the site of the landfill20 . The use of soy-based color ink and water-based
inks emit less volatile organic compounds (VOC) (EPA considers VOCs to be a contributor to air
pollution because they lead to the destruction of the ozone)21 . Thus, there are ample opportunities for publishers to gain insight into GHG emissions in their supply chains, as well as evaluate the
performance of their supply chain members and help them make better and more informed decisions.
As mentioned above, we assume that the publisher assigns direct and (partial) indirect emission
responsibilities to members of its supply chain. The assignment of these responsibilities require
insight into the processes in the newspaper supply chain and their possible externalities on other
processes in the supply chain. For illustration, we assign such responsibilities as follows. Nodes
2, 4, 8 represent supply chain members who only provide transportation or delivery services. Thus,
it may be appropriate to set Ei = Ei for these nodes. As there is no emission from consumption of a
newspaper, the newspaper publisher is not responsible for downstream emissions from its consumers.
All other allocations of direct and indirect pollution responsibilities for all players are given in Table
2. Note that Table 2 reflects the choice of assigning indirect pollution responsibility of the entire
supply chain to the publisher.
i
1
2
3
4
5
6
7
8
9
10
Ei
{(1, 0)}
{(2, 1)}
{(2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 3), (8, 3), (9, 8), (10, 9)}
{(4, 3)}
{(5, 4), (6, 5)}
{(6, 5)}
{(7, 3)}
{(8, 3)}
{(9, 8), (10, 9)}
{(10, 9)}
Table 2: GHG responsibilities for all nodes
Table 2 allows us to calculate, for each arc j, the set of producers N j which are directly or
indirectly responsible for the pollution aj . These sets, and their cardinalities |N j |, which are used to
calculate the Shapley value, are given in Table 3. The Shapley value allocations of GHG emissions
20
21
http://www.greenpressinitiative.org/documents/climateguide.pdf
http://www.naa.org/About-NAA/Newspapers-and-Sustainability/Environmental-Policy.aspx
26
Node
Arc
Nx
Cardinality
1
2
3
4
5
6
7
8
9
10
(1, 0)
(2, 1)
(3, 2)
(4, 3)
(5, 4)
(6, 5)
(7, 3)
(8, 3)
(9, 8)
(10, 9)
N 1 = {1}
N 2 = {2, 3}
N 3 = {3}
N 4 = {3, 4}
N 5 = {3, 5}
N 6 = {3, 5, 6}
N 7 = {3, 7}
N 8 = {3, 8}
N 9 = {3, 9}
N 10 = {3, 9, 10}
|N 1 | = 1
|N 2 | = 2
|N 3 | = 1
|N 4 | = 2
|N 5 | = 2
|N 6 | = 3
|N 7 | = 2
|N 8 | = 2
|N 9 | = 2
|N 10 | = 3
Table 3: GHG emissions responsibilities for all arcs
responsibility for the supply chain members and the end consumer are displayed in Table 4. Details
of our calculations are provided in Appendix C.
Supply chain member
node i
Consumer
Newspaper Delivery
Newspaper Publishing
Paper Transportation
Paper Manufacturing
Energy for Paper Manufacturing
Energy for Newspaper Publishing
Ink Transportation
Ink Manufacturing
Energy for Ink Manufacturing
1
2
3
4
5
6
7
8
9
10
Emission allocated to node i
LAT
NYT
0.0000
0.5400
176.3585
20.1200
79.2500
31.8500
25.6250
0.0001
0.0034
0.0004
0.0000
0.8450
251.1001
21.1200
83.1883
33.4333
48.9200
0.0002
0.0065
0.0008
Table 4: GHG emissions allocated to each supply chain member (unit: kg CO2 Eq)
Table 4 provides the publisher with the emissions that can be attributed to its suppliers and
distributors, inform them about the consequences of their choices, and motivate them to reduce
their emission levels. Moreover, to the extent that the publisher would like to introduce carbon
pricing, the Shapley allocation displayed in Table 4 would motivate the firms in the supply chain to
exert, in some sense, optimal abating efforts.
As mentioned earlier, Table 3 represents one possible allocation of GHG responsibilities, in which
we consider a cradle-to-gate LCA model. Another application of our model can be used in the case
27
where the government aims to reduce carbon emissions and implements carbon tax on consumers,
based on their allocation of the emissions, which may be collected by the vendor and paid to the
government (similar to the model in which retailers currently collect CRV–California Refund Value–
from costumers purchasing beverage in recyclable containers). Our model can be used to estimate
the portion of carbon fees they should be charged. In that instance, a consumer would be responsible
for some or all of the upstream GHG emissions. As a result, his responsibility would increase from
0 to, say, 111.94 (LAT) or 154.30 (NYT), if he was indirectly responsible for all of the upstream
emissions, while the other supply chain members would see a reduction in their responsibilities. For
carbon cost of $100 per metric ton22 , charging a NYT subscriber in Los Angeles for 154.30 kg CO2 Eq
amounts to an annual increase of $15.43, or about 1.4% increase in annual subscription cost.
7.
Extension to a General Supply Chain Structure
We extend the analysis in this section to a more general supply chain structure. To elaborate on
the challenge encountered in a general structure, consider the supply chain graph G displayed in
Figure 5, in which the supply chain is a directed path, P , from node 7, the leaf node, to node 0, the
root of the path, and an additional arc (5, 2) from node 5 to node 2. Each player, j, directly created
the pollution, aj , j = 1, . . . , 7, associated with edge ej emanating from node j towards node 0 in
Figure 4. In addition, player 5 has directly created the pollution a(5,2) by supplying directly player
2 via arc (5, 2).
Now, for each player j, let us examine the choice of selecting Ej to consist of, aside from ej ,
all arcs (u, v) such that the path in G from node u to node 0 traverses node j. Then, player 5 is
responsible for the pollution she directly created, a5 and a(5,2) , and she is also indirectly responsible
for a7 and a6 . But, for which pollution is, say, player 4 responsible? Clearly, player 4 is directly
responsible for a4 and is also indirectly responsible for a5 . However, is she also indirectly responsible
for the entire pollution, a6 and a7 , created by players 6 and 7? Indeed, player 4 may argue that
some of these emissions were created in supply chain stages which were eventually used by player
5 to supply player 2 directly, and player 4 should not be responsible for them. Thus, we suggest
that, for the above specific choice of the sets Ej , both arc pollutions a6 and a7 should be split into
two parts—one part which was created to eventually supply player 4 via player 5, while the other
22
We use this number for illustrative purpose as there seem to be no consensus on the true social cost of carbon
emissions. EPA currently considers four possible scenarios, which estimate 2015 emission cost to be $11, $36, $56, or
$105; for more details see https://www.whitehouse.gov/sites/default/files/omb/inforeg/scc-tsd-final-july2015.pd
28
Figure 5: Supply chain with a more general structure
part that was created so as to enable player 5 to supply player 2 directly. Specifically, we split the
pollution a6 (resp., a7 ) to non negative components a6 (P ) and a6 ((5, 2)) (resp., a7 (P ) and a7 ((5, 2)),
such that a6 (P ) + a6 ((5, 2)) = a6 , a7 (P ) + a7 ((5, 2)) = a7 , and, e.g., player 4 (resp., 3) is responsible,
directly or indirectly, for a7 (P ), a6 (P ), a5 and a4 (resp., a7 (P ), a6 (P ), a5 , a4 and a3 .)
We note that a number of environmentally cautious companies do attempt to evaluate their
carbon footprints by considering all the “ingredients” that go into their final products, including the
raw material, transportation, processing, etc. Timberland Co., a shoe company, found that despite
their offshore manufacturing, transportation accounts for less than 5% of their carbon footprint,
while leather is the biggest contributor (Ball, 2009). However, Timberland’s leather suppliers argued
that the GHG emissions from a cow should not be allocated to them, but should be entirely the
responsibility of the beef producers. Their argument was that cows are grown mainly for meat, with
leather as a byproduct, so that growing leather does not yield emissions beyond those that would
have occurred anyway. Indeed, upon realizing that 7% of the financial value of a cow stems from its
leather, Timberland adopted guidelines requiring that the company should apply that percentage to
compute the share of a cow’s total emissions attributable to leather. Likewise, Nike Inc. finds that
about 56% of their emissions come from materials used to make their products (Nike, 2013), and,
29
in a conversation that one of the authors had with representatives from Nike, it was revealed that
Nike applies about 8% of a cow’s total GHG emissions to the leather used in their shoes. Similarly,
G&L consider a matrix of direct requirements whose elements are inter-industrial flows from an
industry i to an industry j per gross output of sector j, based on the Leontief model from inputoutput theory. Thus, as the examples above indicate, data for splitting arc pollution to its various
relevant components can be possibly obtained, and the suggested approach for allocating pollution
responsibilities seems to be appropriate.
In general, in view of the above discussion, we extend our model and analysis to a general
directed graph G = (V (G), E(G)), whose graphical structure is consistent with the topology of a
general supply chain structure. Thus, G must contain nodes without entering arcs, representing the
most downstream suppliers, and G must also contain root nodes, corresponding to the various final
products produced by the supply chain. There are no outgoing arcs from the root nodes, and we
will further assume that there is only one arc (u, v) entering any root node v. The node u could
represent either the final producer/assembler of a final product (in the cradle-to-gate LCA model)
or the end consumer of that product (in the cradle-to-grave LCA model).
To extend the analysis we may need to divide each arc pollution to several parts and allocate the responsibility of each part to the correct set of players (see, e.g., Keskin and Plambeck, 2011, and Sunar and Plambeck, 2015, for some challenges in allocating pollution responsibility of a process to the various co-products). Specifically, for each arc (j1 , j2 ) in G denote by
o
n
(j ,j )
(j ,j )
(j ,j )
N (j1 ,j2 ) = N1 1 2 , N2 1 2 , . . . , Nn(j11 ,j22 ) the set of all distinct subsets of N , such that the pollu(j1 ,j2 )
tion a(j1 ,j2 ) (Nj
) ≥ 0 was directly created by player j1 in order to support production processes
Pn(j ,j )
(j ,j )
(j ,j )
needed by Nj 1 2 , j = 1, . . . , n(j1 , j2 ), where `=11 2 a(j1 ,j2 ) N` 1 2 = a(j1 ,j2 ) . For example, the
(j1 ,j2 )
various distinct subsets of N (j1 ,j2 ) , {N1
(j1 ,j2 )
, N2
(j ,j )
, . . . , Nn(j11 ,j22 ) }, associated with arc (j1 , j2 ), could
correspond to the node subsets of all distinct pseudo-paths from node j1 to a root node in G whose
first arc is (j1 , j2 ), where a pseudo-path is a simple path whose arcs are distinct. Such a choice for the
subsets of N (j1 ,j2 ) would correspond to the convention that each producer i is indirectly responsible
for all the pollution created by upstream suppliers for processes needed by i. On the other hand,
a choice of N (j1 ,j2 ) = {j1 } for arc (j1 , j2 ) would imply that producer j1 is solely responsible for the
pollution she directly created. Indeed, as was the case with the choice of the sets Ei , the choice of
(j1 ,j2 )
the subsets of N (j1 ,j2 ) , {N1
(j1 ,j2 )
, N2
(j ,j )
, . . . , Nn(j11 ,j22 ) }, which determines the allocation of pollution
responsibility associated with arc (j1 , j2 ), (j1 , j2 ) ∈ E(G), could be guided by fairness or incentive
considerations, and could be designed to fit the situation at hand and incorporate any idiosyncrasy
30
of relationships in the supply chain.
(j ,j )
It follows that a(j1 ,j2 ) N` 1 2 can be attributed, perhaps not exclusively, to player i if i ∈
(j1 ,j2 )
N`
, and the total pollution, cG ({i}), attributable to player i can be expressed as
cG ({i}) =
Xh
i
(j ,j )
(j ,j )
(j ,j )
a(j1 ,j2 ) N` 1 2 : for all (j1 , j2 ) ∈ E(G), N` 1 2 ∈ N (j1 ,j2 ) such that i ∈ N` 1 2 .
Similarly,
cG (S) =
i
Xh
(j ,j )
(j ,j )
(j ,j )
a(j1 ,j2 ) N` 1 2 : for all (j1 , j2 ) ∈ E(G), N` 1 2 ∈ N (j1 ,j2 ) such that S ∩ N` 1 2 6= ∅ .
Proposition 3 The GREEN game (N, cG ), associated with a general supply chain structure G, is
convex.
Proposition 3 implies that the Shapley value of (N, cG ) is contained in its core. Further, as
we demonstrate in Theorem 5 below, the Shapley value for the general GREEN game model can
be easily computed and it has the same intuitive expression—the pollution for each process being
equally divided among all supply chain members who are directly or indirectly responsible for its
creation—as was the case in the basic model studied in Section 3.
(j ,j )
1 2
Theorem
∈ N (j1 ,j2 ) ,
5 The
allocation according to which for each (j1 , j2 ) ∈ E(G) and N`
(j ,j )
(j ,j )
a(j1 ,j2 ) N` 1 2 is allocated equally among members in N` 1 2 is the Shapley value of (N, cG ).
Comment 3 Similar to Comment 1 and the GREEN game model (N, c), the game model (N, cG )
studied in this section can be extended to allow partial indirect pollution responsibility. That is, if
(j ,j )
i ∈ N` 1 2 , and i 6= j1 , then player
i
is indirectly responsible, possibly not exclusively, to only a
(j1 ,j2 )
fraction of the pollution a(j1 ,j2 ) N`
, rather than its entirety. All the results, with the natural
modification for the expression of Shapley value, hold for the more general model. For simplicity of
exposition, the more general model was not presented.
Finally, recall that the axiomatizations results (i.e., Theorems 2 and 3) and the incentive capabilities of the Shapley value (Theorem 4) were given in terms of processes, which correspond to
activities by some firms in the supply chain, and that there could be several processes associated
with a single arc. That is, these theorems were originally derived in the environment of a general
supply chain, and therefore they are valid for the general case considered in this section.
8.
Concluding Remarks
In view of the challenges of mitigating climate change, rationalizing CO2 emissions in supply chains,
which account for more than 20% of global GHG emissions, could make an important contribution to
31
the efforts of achieving the global objectives for emission reduction agreed upon in the recent United
Nations Climate Change Conference in Paris.
Walmart has embraced its role to protect the environment, and in 2007 has started to collect
data from its tier-1 suppliers to assess GHG emissions in its supply chain. However, to rationalize
its CO2 emissions, it is suggested that Walmart needs to engage with many more of its suppliers,
especially those involved with basic material manufacturing, which are far upstream at the supply
chain. Indeed, to improve their environmental performance, supply chain leaders need to gain insight
into the causes of GHG emissions in their supply chain. They should be in a position to assign
responsibility to firms for their direct GHG emissions, as well assigning indirect responsibilities to
firms whose actions and decisions regarding, e.g., product design, packaging, material selection, or
operating decisions, adversely affect GHG emissions by other firms in the supply chain.
In this paper we consider supply chains with a motivated dominant leader, such as Walmart, that
has the power or authority to assign their suppliers responsibilities for both direct and indirect GHG
emissions. Given these responsibility assignments, we formulate the problem of allocating pollution
responsibilities among the suppliers as a cooperative game, referred to as the GREEN game, and
propose the Shapley value of the GREEN game as a scheme to allocate responsibilities for GHG
emissions in the supply chain. In view of the reluctance of suppliers to share information about
their GHG emission (see, e.g., Jira and Toffel (2013)), it is important to note that the Shapley value
of the GREEN game is both transparent and fair. For example, it allocates the responsibility for
the pollution generated by a process equally among all supply chain members who are directly or
indirectly responsible for it, and it lends itself to intuitive axiomatic characterizations, which further
magnify its fairness. In view of the reported improved environmental performance, stemming merely
from requests to suppliers to measure and report their GHG emissions, it is reasonable to expect that
drawing the attention of firms to their direct and indirect responsibilities for GHG emissions, coupled
with a fair and transparent allocation – the Shapley allocation—with respect to which the firms’
efforts to curb GHG emissions can be evaluated, would similarly lead to improved environmental
performance by firms in the supply chain. Moreover, we have shown that when suppliers’ abatement
cost functions are private information, the Shapley allocation induces suppliers to exert abatement
efforts which minimize the maximum deviation from the socially optimal pollution level.
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38
Appendix A: Proofs
Proof of Theorem 1: For each j ∈ N and S ⊆ N , let N, cj denote the game where

 0,
if S ∩ N j = ∅,
j
c (S) =
 a , otherwise.
j
P
One can easily verify that for each S ⊆ N , c(S) = j∈N cj (S). By its symmetry property, the
Shapley value for the game N, cj is

 aj , if i ∈ N j ,
|N j |
Φi =
 0,
otherwise,
and by its additivity property, Φ(c) =
P
i∈N
Φ ci . Thus, the Shapley value of the GREEN game
(N, c) allocates the cost of the pollution aj equally among all players who are directly or indirectly
responsible for its creation.
Proof of Proposition 1: Clearly, for Q ⊆ S, EQ ⊆ ES . Thus, c(Q) = a(EQ ) ≤ a(ES ) = c(S),
implying that the GREEN game (N, c) is monotone. Further, note that for i ∈ N and S ⊆ N ,
c(S ∪ {i}) − c(S) = a(Ei \ ES ). Thus, for i ∈ N and Q ⊆ S, c(S ∪ {i}) − c(S) ≤ c(Q ∪ {i}) − c(Q) and
the proof follows.
Proof of Theorem 2: Part i. It is easy to show that the Shapley rule satisfies both properties,
and we will show that if φ is another pollution allocation rule satisfying them, then it coincides with
the Shapley allocation.
Consider the total pollution footprint f = (f1 , f2 , ..., fm ) associated with the m processes, and
let f 0 , f 1 , ..., f m be defined such that f 0 = (0, 0, ..., 0), f 1 = (f1 , 0, ..., 0) and so forth until f m =
(f1 , f2 , ..., fm ) = f . The proof is by induction. By definition, any pollution allocation rule φ shall
naturally allocate (0, 0, ..., 0) to the firms in f 0 when all the processes have zero pollution. So, for
f 0 , φ(f 0 ) = φS (f 0 ), where φS is the Shapley allocation rule. Assume that φ(f j−1 ) = φS (f j−1 ),
and we will prove that φ(f j ) = φS (f j ). Let Sj be the set of firms that are responsible for the
pollution of process j, and let |Sj | = nj . Since φ prevents free riding, it follows that for i ∈ N \Sj ,
φi (f j ) = φi (f j−1 ). Equal sharing of extra pollution implies that for all firms p and q that are
responsible for the pollution of process j, φp (f j ) − φp (f j−1 ) = φq (f j ) − φq (f j−1 ). By definition, a
pollution allocation rule is efficient which implies that for all p ∈ Sj , φp (f j ) − φp (f j−1 ) =
φi (f j ) = φi (f j−1 ) +
fj
nj
fj
nj .
Thus,
for i ∈ Sj and φi (f j ) = φi (f j−1 ) for i ∈ N \Sj . Thus, if φ(f j−1 ) = φS (f j−1 ),
φ(f j ) also has to coincide with φS (f j ), the Shapley allocation, and inductively φ = φS .
1
We complete the proof of the first axiomatization by showing the independence of the two properties.
1. Let φ be a pollution rule that allocates to each firm the average pollution of all the processes,
m
P
φi (f ) =
fj
j=1
n
. φ satisfies efficiency and equal sharing of extra pollution but allows free riding.
2. Consider a pollution allocation rule that, for each process j, with a corresponding pollution
fj , allocates the entire responsibility for fj to just one of the possibly many firms associated with it,
while all other firms associated with process j are allocated zero responsibility for fj . Note that according to this allocation rule, some firms which are responsible for pollution of some of the processes
may not be allocated any pollution responsibility. Full producer responsibility is one such allocation
rule where the consumer is not allocated any pollution responsibility. Such an allocation rule is efficient, and satisfies the no free riding property but, in general, does not equally share extra pollution.
Part ii. The Shapley allocation rule is easily seen to satisfy the three properties. The proof of
uniqueness is again via induction. As noted, any pollution allocation rule allocates (0, 0, ..., 0) to the
firms in f 0 when all the processes have zero pollution. Let us assume that φ(f j−1 ) = φS (f j−1 ), and
we will prove that φ(f j ) = φS (f j ). Let Sj be the set of firms that are responsible for the pollution of
process j and let |Sj | = nj . Define f˜j = (0, ..., 0, fj , 0, ...0). For the footprint set f˜j , by firm nullity
any firm not in Sj must be allocated zero. By firm equivalence and efficiency, each firm in Sj is
allocated
fj
nj
when the pollution footprint set is f˜j . Now, the pollution of process j alone increases
by fj in the footprint sets f 0 and f j−1 to yield f˜j and f j , while the pollution of the other processes
fj
nj .
f
φi (f j−1 ) + njj
remains the same. For any firm i ∈ N \Sj , φi (f˜j ) − φi (f 0 ) = 0 and for i ∈ Sj , φi (f˜j ) − φi (f 0 ) =
Process history independence implies, φi (f j ) = φi (f j−1 ) for i ∈ N \Sj and φi (f j ) =
for i ∈ Sj . Thus, if φ(f j−1 ) = φS (f j−1 ), φ(f j ) also has to be φS (f j ), the Shapley allocation, and
inductively φ = φS .
We complete the proof of the second axiomatization by showing the independence of the three
properties.
1. Consider the pollution allocation rule φ defined previously, that allocates to each firm the
m
P
average pollution of all the processes, φi (f ) =
fj
j=1
n
. φ satisfies firm equivalence and process history
independence but not firm nullity.
2. Consider a pollution allocation rule φ that allocates to each null firm zero responsibility, and
shares equally the pollution among all the other firms. Such a rule naturally satisfies firm nullity
2
and firm equivalence but in general it does not satisfy process history independence.
3. Consider again the pollution allocation rule that allocates the entire responsibility for each
process to just one of the possibly many firms associated with it. It satisfies firm nullity and process
history independence, but not firm equivalence.
Part iii. Again, it is easy to see that the Shapley allocation rule satisfies the three properties.
Now, recall that firm i is responsible for |Pi | = mi processes, and consider a fully disaggregated supply chain, G0 , derived as a result of each firm i in G disaggregating into mi firms, each responsible
for a distinct process in Pi and a corresponding newly formed process of zero pollution. In the fully
disaggregated supply chain G0 , each firm is now responsible for at most one polluting process. For
each process j, let Sj denote the firms responsible for j in G0 with |Sj | = nj . Following an argument
similar to proof for part i, we have that firm-equivalence and no free riding imply that each firm in
Sj is responsible for
fj
nj .
Invariance to disaggregation asserts that in the aggregated supply chain G,
each firm is responsible for the cumulative responsibilities of its disaggregated firms. This implies
that again in G, the responsibility of each process is shared equally among the firms responsible for
it, which is the Shapley allocation. We complete the proof by showing the independence of the three
properties.
1. Consider a pollution allocation rule φ which allocates the pollution responsibility proportional
m
P
f (Pi )
to the individual responsibilities, φi (f ) = P
fj . This is similar to the responsibility allocan
f (Pk ) j=1
k=1
tion suggested by Lenzen et al. (2007) in order to obtain a disaggregation invariant allocation rule.
While φ is disaggregation invariant and firm equivalent, it does not prevent free riding.
2. Let φ be a pollution allocation rule that allocates the pollution of each process equally among
some firm responsible for it and all other firms equivalent to it. φ is firm equivalent by definition,
satisfies the no free riding property, but is not disaggregation invariant.
3. Given a supply chain graph G, for each process j, choose a firm vj that is responsible for j and
allocate the full responsibility for fj to that firm. In any disaggregated supply chain G0 arising from
G, continue to allocate the responsibility of fj to the possibly disaggregated firm corresponding to
vj in G0 that bears responsibility for j. Such a pollution allocation rule is disaggregation invariant
and prevents free riding. However, it is not firm equivalent.
Proof of Proposition 2. Consider any linear pollution allocation rule φ, and suppose that for arc
j, φ allocates the emissions aj according to the cost share vector λ = (λ1j , ..., λnj ). The first-order
3
∂cij (eij ) S ∂aij
+p λij
=
∂eij
∂eij
0, where aij is the separable part of aj due to the efforts of i. Applying the implicit function theorem,
φ
∂ 2 aij ∂ 2 cij ∂(eij )
∂aij
∂aij
S
we obtain that p λij 2 +
+ pS
= 0. Note that
< 0 since aij is decreasing
∂eij
∂eij
∂eij
∂e2ij ∂λij
condition for some player i who can influence the pollution at arc j is given by
∂(eφij )
∂ 2 aij
∂ 2 cij
in eij . Also,
> 0
≥ 0 and
≥ 0 since aij and cij are convex in eij . Thus,
∂λij
∂e2ij
∂e2ij
and therefore, we have that the equilibrium effort of i towards its pollution at arc j, eφij (λij ) is
monotonically increasing in λij (its allocated responsibility towards arc j).
i. Suppose that firm k cannot influence the pollution at an arc j but j belongs to its responsibility
set, that is, j ∈
/ Pk but j ∈ Ek . Then, clearly eφkj = 0 for any sharing rule φ. Let ã be the minimum
decentralized supply chain emissions. Next, remove arc j from Ek , that is, Ek0 = Ek \ {j} and
Nj0 = Nj \ {k}, and let ã0 be the new minimum decentralized supply chain emissions. For any sharing
φkj
rule φ in the original setting, let us define φ0 such that φ0kj = 0 and φ0ij = φij +
for all i ∈ Nj0 .
|Nj0 |
0
Let φ0 allocate the pollution of all other arcs identically as φ. Then, again eφkj = 0, but all other firms
0
are now assigned a larger share of the responsibility for the pollution at arc j. Thus eφij ≥ eφij since
the equilibrium effort is monotonically increasing in the allocated share of responsibility. Therefore,
ã0 ≤ ã.
ii. Suppose that firm k can influence the pollution at arc j but j is not in its responsibility set, that
is, j ∈ Pk but j ∈
/ Ek . Let ã0 be the new minimum decentralized supply chain emission upon adding j
to Ek . Then, any supply chain emission level supported by some linear sharing rule φ in the original
setting can be supported by a linear sharing rule φ0 which allocates the pollutions of all arcs in the
same proportion as φ and φ0kj = 0. Thus again, ã0 ≤ ã.
Thus, the minimum supply chain emissions over all linear sharing rules is minimized when the
responsibility set for each firm i is defined to be Pi . This completes the proof.
Proof of Theorem 4. Consider any linear emission allocation rule φ, such that for a given arc j, φ
allocates the emissions aj according to the cost share vector λ = (λ1j , ..., λnj ). Suppose that for the
vector of cost functions c, φ performs strictly better than the Shapley allocation rule, Φ, with respect
j
to the emissions at arc j in equilibrium. That is, aφj (cj ) < aΦ
j (c ). We will show that there exists
φ 0j
j
some permutation, c0j , of the functions in the cost function vector cj , at which aΦ
j (c ) ≤ aj (c ).
Note that this will complete the proof because it will contradict the existence of an allocation rule
with a corresponding worst-case loss of efficiency being strictly smaller than the worst-case loss of
efficiency corresponding to the Shapley value.
4
From the proof of Proposition 2, we have that the equilibrium effort of player i towards pollution
abatement at arc j, eφij (λij ), is monotonically increasing in λij (its allocated responsibility towards
arc j). Implicitly differentiating the first order condition a second time, we have,
pS λ
φ
φ
2 φ
∂ 2 aij
∂ 2 cij ∂ (eij )
∂ 3 cij ∂eij 2
∂ 2 aij ∂eij S ∂ 3 aij
S
+ p λij 3 +
= 0.
+
+ 2p
ij
∂λij
∂e2ij
∂e2ij
∂λ2ij
∂e2ij ∂λij
∂eij
∂e3ij
Further, noting the non-negativity of the third derivatives of aij and cij with respect to eij , and the
convexity of aij and cij and that eφij is monotonically increasing in λij , we conclude that,
φ
φ
h
∂ 3 cij ∂eij 2 i
∂ 2 aij ∂eij S ∂ 3 aij
S
+ p λij 3 +
2p
∂λij
∂ 2 (eφij )
∂e2ij ∂λij
∂eij
∂e3ij
< 0.
=
−
∂λ2ij
∂ 2 cij ∂ 2 aij
S
p λij 2 +
∂eij
∂e2ij
Thus, we have that eφij (λij ) is concave in λij .
Now, consider an arbitrary permutation, π, of the vector of cost functions cj , resulting with the
vector of cost functions c0j . That is, for any firm i, c0ij = cπ(i)j for some π(i) ∈ Nj . The symmetry
of the emission in arc j, aj , in efforts implies that the effect of a permutation of the abatement cost
functions on the equilibrium level of emissions is equivalent to the corresponding permutation of the
share vector, from λ to λ0 , such that player π(i), who originally was allocated the share λπ(i)j , is
now allocated λ0π(i)j = λij .
Note that for the Shapley allocation, the symmetry of the share vector implies that for any
j
Φ 0j
permutation of the cost functions, the equilibrium emission level remains the same, aΦ
j (c ) = aj (c ).
For any other linear emission allocation φ, different from Φ, and for the permuted vector of cost
functions, c0j , let aj (λ0 ) be the equilibrium emission level with λ0 being the equivalent corresponding
permutation of the share vector as described above.
P
Now, aj (λ0 ) =
aij (λ0ij ) and further, aij (λ0ij ) is convex in λ0ij because aij is convex decreasing
i∈Nj
in efforts and as shown above, the equilibrium efforts are concave increasing in the share vector.
P
1
aj (λ0 ), where π(λ) denotes all possible permutations of the share vector asConsider
|Nj |! λ0 ∈π(λ)
P
sociated with aj . The convexity of aij in λ0ij and footprint balancedness,
λij = 1, implies that
i∈Nj
P
1
1
1 1
0
, ...,
) = aΦ
aj (λ0 ) ≥ aj ( ,
j . Thus, there exists some permutation λ of the share
|Nj |! λ0 ∈π(λ)
Nj Nj
Nj
vector λ such that the equilibrium emission level aj (λ0 ) ≥ aΦ
j .
Let the permutation λ0 of the share vectors correspond to the permutation of the abatement cost
functions with the costs c0j as described before. Then, equivalently, the equilibrium emission level
5
with the vector of abatement costs given by c0j at the allocation φ is given by aφj (c0j ) = aj (λ0 ) ≥ aΦ
j .
1
Repeating the same argument over all the arcs proves equilibrium that the allocation of
of the
|Nj |
pollution aj to each firm in Nj , which is precisely the Shapley allocation, minimizes the worst-case
loss of efficiency.
(j1 ,j2 )
(j1 ,j2 )
Proof of Proposition 3: Clearly, for Q ⊆ S ⊆ N , if S ∩ N`
= ∅ then Q ∩ N`
= ∅, which
implies that the game (N, cG ) is monotone, where cG (S) is as defined above.
Further, note that for i ∈ N , and Q ⊆ S ⊆ N ,
X h
(j1 ,j2 )
(j ,j )
cG (S ∪ {i}) − c(S) =
a(j1 ,j2 ) N`
: for all (j1 , j2 ) ∈ E(G), N` 1 2 ∈ N (j1 ,j2 )
i
(j ,j )
(j ,j )
such that i ∈ N` 1 2 , S ∩ N` 1 2 = ∅ .
(j1 ,j2 )
Again, for Q ⊆ S, if S ∩ N`
(j1 ,j2 )
= ∅, then Q ∩ N`
= ∅. Therefore, for i ∈ N and Q ⊆ S ⊆ N ,
cG (S ∪ {i}) − c(S) ≤ cG (Q ∪ {i}) − c(Q), and the proof follows.
Proof of Theorem 5: For each (j1 , j2 ) ∈ E(G),
(j ,j )
N` 1 2
∈
N (j1 ,j2 )
(j ,j )
N` 1 2
and S ⊆ N , let N, cG
denote the game where
(j ,j )
N` 1 2
cG
(S) =

 0,
(j1 ,j2 )
if S ∩ N`
(j1 ,j2 )
 a
,
(j1 ,j2 ) N`
and note that for each S ⊆ N , cG (S) =
P
(j ,j )∈E(G)
otherwise,
N
P
(j ,j )
N` 1 2 ∈N (j1 ,j2 )
1 2 (j ,j ) N 1 2
is
property, the Shapley value for the game N, cG`
Φi =



(j ,j )
a(j1 ,j2 ) N` 1 2
(j1 ,j2 ) N`


0,
and by its additivity property, Φ (cG ) =
= ∅,
(j1 ,j2 )
if i ∈ N`
(j1 ,j2 )
cG`
(S). By its symmetry
,
otherwise,
P
(j1 ,j2 )∈E(G)
(j1 ,j2 )
(j ,j ) N` 1 2
Φ
c
. Thus, for each
(j1 ,j2 )
(j
,j
)
G
N
∈N 1 2
P
`
∈ N (j1 ,j2 ) , the Shapley value of the GREEN game (N, cG ) allocates the cost of the pollution
(j ,j )
(j ,j )
a(j1 ,j2 ) N` 1 2 equally among all players in N` 1 2 , who are directly or indirectly responsible for
N`
its creation.
6
Appendix B: Estimation of GHG emissions in a newspaper supply
chain
Paper manufacturing and transportation
We estimate the annual weight of papers required for LAT and NYT to be 96.7 and 101.5 kg,
respectively, based on the weekly average numbers of pages 483 and 507, respectively. The number
of pages is estimated by counting pages in September and October of 2014, while the newspaper
weight is measured on the FedEx shipping scale.
We assume both newspapers use 100% virgin paper, and then utilize Paper Calculator1 provided
by Environment Paper Network to obtain a GHG emissions factor of 2.8 kg CO2 Eq per kg of paper.
Based on a GHG emissions report from the same agency, Ford (2012), we estimate 35.0% of the total
GHG emissions comes from manufacturing, 35.3% arises from energy and assume about half of the
rest—14.9%—is attributed to transportation. Therefore, we calculate GHG emissions as shown in
Table 1.
paper manufacturing
paper transportation
energy for paper manufacturing
LAT
NYT
96.7 · 2.8 · 0.35 = 94.80
96.7 · 2.8 · 0.149 = 40.24
96.7 · 2.8 · 0.353 = 95.55
101.5 · 2.8 · 0.35 = 99.51
101.5 · 2.8 · 0.149 = 42.24
101.5 · 2.8 · 0.353 = 100.30
Table 1: GHG emissions for paper manufacturing and transportation (unit: kg CO2 Eq)
Newspaper publishing and delivery
We use the Life Cycle Assessment tool from Carnegie Mellon University Green Design Institute2 to
estimate the GHG emissions for the newspaper publishers; this tool estimates GHG emissions by
the dollar amount of economic activity. Per $1000 of the newspaper publishing, the estimated total
emission factor is 317 kg CO2 Eq, from which we will subtract 62.06 kg CO2 Eq and 0.015 kg CO2 Eq
from the paper and ink related GHG emissions respectively, and 128 kg CO2 Eq from energy, which
leaves us with 126.925 kg CO2 Eq.
Since subscription prices fluctuate significantly by promotional offers, we use the more reliable
retail prices for this analysis. The annual cost for the LAT and NYT is $572 and $1092 based on the
weekly prices of $11 and $21. We assume about 30% of total cost is contributed by the delivery of
1
2
http://c.environmentalpaper.org/home
Economic Input-Output Life Cycle Assessment (EIO-LCA), US 2002 producer model, http://www.eiolca.net
7
newspaper (as in Toffel and Sice, 2011), so we weight the above factors by 70% to derive the GHG
emissions as below.
newspaper publishing
energy
LAT
NYT
572
1000 · 0.70 · 126.925 = 50.82
572
1000 · 0.70 · 128 = 51.25
1092
1000 · 0.70 · 126.925 = 97.02
1092
1000 · 0.70 · 128 = 97.84
Table 2: GHG emissions for newspaper publishing (unit: kg CO2 Eq)
At the time of this analysis, the LAT is published at two production facilities in Los Angeles area
(downtown Los Angeles and Orange Country), according to the LA Times Media Center3 . Google
Map (source: New York Times)4 shows that the NYT has one printing plant (Gardena) to serve the
Los Angeles area. Therefore, we assume a consumer has an average distance of 20 miles (32 km)
and 30 miles (48 km) away from the LAT and NYT printing facilities, respectively. The newspaper
delivery GHG emissions are calculated based on the delivery range: 64 km for LAT and 96 km for
NYT (round trip) and a GHG emissions factor of 0.1737 CO2 Eq per ton-km.
We find that the Greenhouse Gas Conversion Factor Repository by the UK Department for
Environment Food & Rural Affairs5 has the most comprehensive GHG emissions data on delivery
vehicles, so we use their GHG emissions factor for this part of the analysis. We assign the same
weight for both Diesel (0.1535 CO2 Eq per ton-km) and Petrol factors (0.1940 per ton-km) from the
Delivery Vehicles tab in the comprehensive DEFRA spreadsheet. Therefore, the GHG emissions can
be computed as below.
LAT
newspaper delivery
96.7
1000
· 64 · 0.1737 = 1.08
NYT
101.5
1000
· 96 · 0.1737 = 1.69
Table 3: GHG emissions for newspaper delivery (unit: kg CO2 Eq)
Ink manufacturing and transportation
From above we have an estimated ink manufacturing GHG emissions factor as 0.015 kg CO2 Eq
per $1000 of newspaper publishing. Using the same Life Cycle Assessment tool as before, this
time for printing ink manufacturing, we estimate that ink transportation emits less than 4% of the
manufacturing GHG emissions and the energy for ink manufacturing attributes about 20%.
3
http://www.latimes.com/services/newspaper/mediacenter/la-mediacenter-production-story.html
https://www.google.com/maps/d/u/0/viewer?oe=UTF8&ie=UTF8&msa=0&mid=zqNGJsNdyY4U.kD3_dO288jcM
5
http://www.ukconversionfactorscarbonsmart.co.uk
4
8
ink manufacturing
ink transportation
energy for ink manufacturing
LAT
NYT
572
1000 · 0.70 · 0.015 = 0.0060
572
1000 · 0.70 · 0.015 · 0.04 = 0.0002
572
1000 · 0.70 · 0.015 · 0.20 = 0.0012
1092
1000 · 0.70 · 0.015 = 0.0115
1092
1000 · 0.70 · 0.015 · 0.04 = 0.0005
1092
1000 · 0.70 · 0.015 · 0.20 = 0.0023
Table 4: GHG emissions for ink manufacturing and transportation (unit: kg CO2 Eq)
In our analysis we consider a cradle-to-gate model (that is, we do not consider paper recycling),
so we set to zero GHG emissions for the end consumer and summarize the estimated GHG emissions
for each supply chain process in Table 1 in the main document. The emissions from Table 1 can
be written in matrix form, with each row representing GHG emissions on the corresponding arc for
LAT and NYT:

0


 1.08


 50.82


 40.24


 94.80

C=
 95.55


 51.25



0.0002


0.0060

0.0012
0



1.69 


97.02 


42.24 


99.51 


100.30


97.84 



0.0005


0.0115

0.0023
Appendix C: The Shapley value allocation for a newspaper publishing supply chain using the GREEN model (unit: kg CO2 Eq)
To simplify presentation of our calculations, we define a conversion matrix B, where each column
vector shows how the GHG emissions should be allocated among the supply chain members: The
9
rows and columns represent arcs in

1 0


0 0.5


0 0.5


0 0


0 0

B=
0 0


0 0



0 0


0 0

0 0
the same order as in the above computation.

0 0
0
0
0
0
0
0


0 0
0
0
0
0
0
0 


1 0.5 0.5 0.33 0.5 0.5 0.5 0.33


0 0.5 0
0
0
0
0
0 


0 0 0.5 0.33 0
0
0
0 


0 0
0 0.33 0
0
0
0 


0 0
0
0
0.5 0
0
0 



0 0
0
0
0 0.5 0
0 


0 0
0
0
0
0 0.5 0.33

0 0
0
0
0
0
0 0.33
For example, the fourth column vector has values b34 = b44 = 0.5 and bk4 = 0 for k 6∈ {3, 4}, which
means the GHG emissions created by paper transportation, represented by the arc weight on (4, 3),
should be allocated equally to nodes (i.e., players) 3, 4; similarly, the sixth column shows that the
GHG emissions created by energy for paper manufacturing, a(6,5) , should be divided equally among
nodes 3, 5, 6 (because b36 = b56 = b66 = 0.33 and bk6 = 0 for k ∈
/ {3, 5, 6}). Finally, we define a third
matrix, D, as the product of matrices B and C—it gives us the Shapley allocation of GHG emissions
for each of the supply chain members for both the LAT and NYT.
D10×2

1


0


0


0


0

=
0


0



0


0

0
= B10×10 × C10×2 =
0
0
0
0
0
0
0
0
0


0
 
 

0 
  1.08
 

0.5 1 0.5 0.5 0.33 0.5 0.5 0.5 0.33
  50.82
 

0 0 0.5 0
0
0
0
0
0 
  40.24
 

0 0 0 0.5 0.33 0
0
0
0 
  94.80
×

0 0 0
0 0.33 0
0
0
0 
  95.55
 

0 0 0
0
0
0.5 0
0
0 
  51.25
 
 
0 0 0
0
0
0 0.5 0
0  0.0002
 
 
0 0 0
0
0
0
0 0.5 0.33 0.0060
 
0.0012
0 0 0
0
0
0
0
0 0.33
0.5 0
0
0
0
0
0
0
10
0


0
0


 

 

 0.5400
0.8450
1.69 

 

 


97.02  176.3585 251.1001


 


42.24   20.1200 21.1200 


 


99.51 
  79.2500 83.1883 
=



100.30
  31.8500 33.4333 

 


97.84 
  25.6250 48.9200 

 

 
0.0002 
0.0005  0.0001

 

 
0.0065 
0.0115  0.0034

 
0.0004
0.0008
0.0023