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Enjoying cooperative games: The R package
GameTheory
Sebastián Cano Berlanga
José Manuel Giménez Gómez
Cori Vilella

Document de treball n.06- 2015

DEPARTAMENT D’ECONOMIA – CREIP
Facultat d’Economia i Empresa

UNIVERSITAT
ROVIRA I VIRGILI
DEPARTAMENT D’ECONOMIA

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Departament d’Economia
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Email: sde@urv.cat

CREIP
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Universitat Rovira i Virgili
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43204 Reus
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Email: creip@urv.cat

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Dipòsit Legal: T - 220 - 2015
ISSN edició en paper: 1576 - 3382
ISSN edició electrònica: 1988 - 0820

DEPARTAMENT D’ECONOMIA – CREIP
Facultat d’Economia i Empresa

Enjoying cooperative games:
The R package GameTheory
Cano-Berlanga, S.

Gim´nez-G´mez, J.M.
e
o

Vilella, C.

Universitat Rovira i Virgili
GRODE

Universitat Rovira i Virgili
CREIP

Universitat Rovira i Virgili
CREIP

Abstract
This paper focuses on cooperative games with transferable utility. We propose the
computation of two solutions, the Shapley value for n agents and the nucleolus with a
maximum of four agents. The current approach is also focused on conflicting claims
problems, a particular case of coalitional games. We provide the computation of the most
well-known and used claims solutions: the proportional, the constrained equal awards, the
constrained equal losses, the Talmud and the random arrival rules.

Keywords: Cooperative game, Shapley value, nucleolus, claims problem, claims rule, bankruptcy.

1. Introduction
Game theory is the discipline that studies how agents make strategic decisions. It was initially
developed in economics to understand a large collection of economic behaviors, including
firms, markets and consumers. Specifically, a game is the mathematical formalization of such
conflicts, originated by Antoine Augustine Cournot (1801-1877) in 1838 with his solution of
the Cournot duopoly.
Modern game theory begins with the publication of the book “Theory of Games and Economic
Behavior” written by Morgenstern and Von Neumann (1953), who considered cooperative
games with several players. Indeed, according to Maschler (1992) after this initial point,
game theory was developed extensively in the 1950s by numerous authors. Later on, the
application field of game theory was not unique to Economics and we may find game theory
in social network formation, behavioral economics, ethical behavior and biology, among others.
Game theory is divided into two branches, called the non-cooperative and cooperative branches.
Actually, in the words of Aumann (1989, pp. 8-9):
“Cooperative theory starts with a formalization of games that abstracts away altogether from procedures and [. . . ] concentrates, instead, on the possibilities for
agreement [. . . ] There are several reasons that explain why cooperative games
came to be treated separately. One is that when one does build negotiation
and enforcement procedures explicitly into the model, then the results of a noncooperative analysis depend very strongly on the precise form of the procedures,
on the order of making offers and counter-offers and so on. This may be appropriate in voting situations in which precise rules of parliamentary order prevail,

2

The R package GameTheory
where a good strategist can indeed carry the day. But problems of negotiation
are usually more amorphous; it is difficult to pin down just what the procedures
are. More fundamentally, there is a feeling that procedures are not really all that
relevant; that it is the possibilities for coalition forming, promising and threatening that are decisive, rather than whose turn it is to speak [. . . ] Detail distracts
attention from essentials. Some things are seen better from a distance; the Roman
camps around Metzada are indiscernible when one is in them, but easily visible
from the top of the mountain.”

These two branches of game theory differ in how they formalize interdependence among the
players. In non-cooperative game theory, a game is a detailed model of all the moves available
to the players. In contrast, cooperative game theory abstracts away from this level of detail,
and describes only the outcomes that result when the players come together in different
combinations. This research usually centers its interest on particular sets of strategies known
as “solution concepts” or “equilibria” based on what is required by norms of (ideal) rationality.
Among the several types of games, this paper focuses on cooperative games with transferable
utility.
A coalitional game with transferable utility involving a set of agents (hereinafter a coalitional
game) is a cooperative game that can be described as a function that associates with each
group of agent (or coalition), a real number which the worth of the coalition. If a coalition
forms, then it can divide its worth in any possible way among its members. This is possible if
money is available as a medium of exchange, and if each player’s utility for money is linear (see
Aumann (1960)). A solution on coalitional games is a correspondence that associates with
each game a non-empty set of payoff vectors in RN whose coordinates add up to v(N ). One
of the most important solutions is the core and it selects for each game all the payoff vectors
such that no coalition could simultaneously provide a higher payoff to each of its members.
The core is a multi-valued solution but the ones we present here, the Shapley value (Shapley
1953) and the nucleolus (Schmeidler 1969), are single-valued. We propose the computation
of the Shapley value for n agents and the nucleolus with a maximum of four agents. As noted
by Guajardo and J¨rnsten (2015) it is usual to find mistakes in computing the nucleolus, but
o
our results coincide with theirs.
The current approach is also focused on a particular case of coalitional games, the conflicting
claims problems. This model describes the situation faced by a court that has to distribute the
net worth of a bankrupt firm among its creditors. But, it also corresponds with cost-sharing,
taxation, or rationing problems. Given a conflicting claims problem, a rule associates within
each problem a distribution of the available resources among the agents. In this sense, we
provide the computation of the most well-known and used claims solutions: the proportional,
the constrained equal awards, the constrained equal losses, the Talmud and the random arrival
rules.
Finally, the aim of this paper is to provide a toolbox which includes common solutions to
cooperative games. Currently, there is no package available covering such algorithms. Lately,
Kenkel and Signorino (2014) have developed the package Games, which focus on models of
strategic interaction.
This paper is organized as follows. Section 2 has a methodology review for coalitional games
and for the conflicting claims problem. In Section 3 we have the library structure and Section
4 provides some examples and illustrations.

Sebasti´n Cano-Berlanga, Jos´-Manuel Gim´nez-G´mez, Cori Vilella
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2. Methodology Review
2.1. T U -cooperative games
TU-cooperative games are used to model situations where cooperation benefits the agents (in
terms of profits or costs). The different solutions given for the model propose distributions of
the profits obtained after cooperation. Some examples of situations where this model applies
are the construction of a motorway in which different agents are involved, the neighbors who
must pay costs of an elevator, common cables, antennas, etc., cooperation between countries
(European Union, UN, etc.) or between political parties to form governments (governments
in coalitions).
The situations where conflicts of interest arise are called games and the agents involved in
the game are called players, who may be individuals, nations, political parties, companies,
firms, etc. In these models we assume that players can make binding agreements about
the distribution of the payoffs or the choice of strategies. In addition, players are able to
compensate each other by transferring utility, for example, through a perfectly divisible good,
which is usually identified with money.
A TU-game involving a set of agents N ∈ N can be described as a function v, known as the
characteristic function, which associates a real number to each subset of agents, or coalition,
S contained in N . Formally, for each N ∈ N , a TU-game is a pair (N, v), where v : 2N → R.
For each coalition S ⊆ N , v(S) is commonly called its worth and denotes the quantity
that agents in S can guarantee for themselves if they cooperate. Therefore, it is assumed
that v(∅) = 0. It is also often supposed that (N, v) is superadditive, i.e., for any pair of
coalitions S, T ⊂ N such that S ∩ T = ∅, v(S ∪ T ) ≥ v(S) + v(T ), so that there is incentive
for the grand coalition forms. Let G N denote the family of TU-games with agents set N.
A solution for TU-games is a correspondence which for each N ∈ N and each (N, v) ∈ G N ,
selects a set of allocations of the worth of the grand coalition among the agents. If a TUgame solution consists of a unique allocation, it is called a TU-value. Next we introduce the
Shapely value (plus its natural extension as a power index) and the nucleolus.
The Shapley value (Shapley 1953). To present this solution, we need to define the marginal
contribution of an agent. Given (N, v) ∈ G N , for each i ∈ N and each S ⊂ N , we call the
marginal contribution of agent i to coalition S, denoted by ∆i v(S), the amount which
his adherence contributes to the value of the coalition, that is, ∆i v (S) = v(SU {i}) − v(S).
According to this solution the worth of the grand coalition is distributed assuming that all
orders of agents’ arrivals to the grand coalition are equally probable and in each order, each
agent gets his marginal contribution from the coalition that he joins. Formally, for each
Sh
(N, v) ∈ G N , the Shapley value, γ Sh , associates to each i ∈ N , the amount γi (N, v) =
[(s!(n − s − 1)!)/n!]∆i v (S) .
S⊆N \{i}

The Shapley and Shubik index (Shapley and Shubik 1954). This solution proposes the
specialization of the Shapley value to voting games that measures the real power of a coalition.1
1

Voting games are modeled by simple games. Those are cooperative games that can model various voting
systems and the characteristic function is v(S) ∈ {0, 1}, for all coalitions S ⊆ N , where v(N ) = 1 and
v(S) ≤ v(T ) if S ⊆ T .

4

The R package GameTheory

The Shapley and Shubik index works as follows. There is a group of individuals all willing to
vote on a proposal. They vote in order and as soon as a majority has voted for the proposal, it
is declared passed and the member who voted last is given credit for having passed it. Let us
consider that the members are voting randomly. Then we compute the frequency with which
an individual is the one that gets the credit for passing the proposal. That measures the
number of times that the action of that individual joining the coalition of their predecessors
makes it a winning coalition. Note that if this index reaches the value of 0, then it means
that this player is a dummy. When the index reaches the value of 1, the player is a dictator.
The nucleolus (Schmeidler 1969). To introduce this solution, some additional notation is
needed. For each (N, v) ∈ G N , I(N, v) = {x ∈ Rn : i∈N xi = v(N ), xi ≥ v({i}) for all i ∈ N }
is the set of imputations. For each x ∈ Rn and each coalition S ⊆ N, e(x, S) = v(S) −
xi
i∈S

is the excess of coalition S with respect to x and represents a measure of dissatisfaction of
such a coalition. The vector e(x) = {e(x, S)}S⊆N provides the excesses of all the coalitions in
reference to x. Given x ∈ Rn , θ(x) is the vector that results from x by permuting coordinates
in decreasing order, θ1 (x) ≥ θ2 (x) ≥ ... ≥ θn (x). Finally, ≤L stands for the lexicographic
order, that is, given x, y ∈ Rn , x ≤L y if there is k ∈ N such that for all j ≤ k, xj = yj and
xk+1 ≤ yk+1 .
The nucleolus looks for an individually rational distribution of the worth of the grand coalition
in which the maximum dissatisfaction is minimized. Formally, for each (N, v) ∈ G N , the
nucleolus γ nu is the vector γ nu (N, v) = x ∈ I(N, v) such that θ(e(x)) ≤L θ(e(y)) for all y ∈
I(N, v). That is, the nucleolus selects the element in the core, if this is nonempty, that
lexicographically minimizes the vector of non-increasing ordered excesses of coalitions. In
order to compute this solution we consider the following linear programming model, which
looks for an imputation that minimizes the maximum excess ε among all coalitions. Formally,
min ε
x

subject to v(S) −

i∈S

i∈N

xi ≤ ε,

∀S ⊂ N, S = ∅

xi = v(N )

ε ∈ R, xj ∈ R; ∀j ∈ N

2.2. The conflicting claims problem
A conflicting claims problem is a particular case of the distribution problem, in which the
amount to be distributed, the endowment E, is not enough to satisfy the agents’ claims on
it. This model describes the situation faced by a court that has to distribute the net worth of
a bankrupt firm among its creditors. But, it also corresponds with cost-sharing, taxation, or
rationing problems. The formal analysis of situations like these, which originates in a seminal
paper by O’Neill (1982), shows that a vast number of well-behaved solutions have been defined
for solving conflicting claims problems, being the proportional, the constrained equal awards,

Sebasti´n Cano-Berlanga, Jos´-Manuel Gim´nez-G´mez, Cori Vilella
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the constrained equal losses, the Talmud and the random arrival rules the prominent concepts
used.2
An illustrative example of conflicting claims problems is the fishing quotas reduction, in which
the agent’s claim can be understood as the previous captures, and the endowment is the new
(lower) level of joint captures (Gallastegui, I˜arra, and Prellezo (2003); I˜arra and Skonhof
n
n
(2008)). A similar example is given by milk quotas among European Union (EU) members.3
In both examples, proportionality is the main principle used. Another example of conflicting
claims situations is the September 11th Victim Compensation Fund (VCF), where the income
each victim would have earned in a full lifetime was estimated and the individual claim is the
legal right to be compensated. Similarly, bankruptcy laws consider the claimants identity to
establish a priority rule. Specifically, bankruptcy codes normally list all claims that should be
treated identically in various categories and assigns to them lexicographic priorities (Kamiski
2006). Pulido, Borm, Hendrickx, Llorca, and S´nchez-Soriano (2002, 2008) analyze, under the
a
name of bankruptcy problems with references, the real-life case of allocating a given amount of
money among the various degree courses that are offered at a (public) Spanish university. The
(verifiable) monetary needs of each course constitute their claims. Additionally, there exist
reference values for each course, which are set by the government independently, below their
claims. Other relevant practical cases also involving more complex rationing situations could
be protocols for the reduction of pollution (Gim´nez-G´mez, Teixid´-Figueras, and Vilella
e
o
o
2014), water distribution in drought periods, or even some resource allocation procedures in
the public health care sector, in which past consumption could be considered as an entitlement,
and current needs as a claim (see, for instance, Hougaard, J., and Osterdal 2012 and MorenoTernero and Roemer 2012). The formalization of such problems is as follows.
Consider a set of agents N = {1, 2, ..., n} and amount E ∈ R+ of an infinite divisible resource,
the endowment, that has to be allocated among them. Each agent has a claim, ci ∈ R+ on
it. Let c ≡ (ci )i∈N be the claims vector.
n

A conflicting claims problem is a pair (E, c) with
we will order the agents according to their claims
the set of all conflicting claims problems.

ci > E. Without loss of generality,
i=1
c1 ≤ c2 ≤ . . . ≤ cn and we will denote by B

Given a conflicting claims problem, a rule associates within each problem a distribution of the
endowment among the agents. A rule is a single-valued function ϕ : B → Rn such that 0 ≤
+
n

ϕi (E, c) ≤ ci , for all i ∈ N (non-negativity and claim-boundedness); and

ϕi (E, c) = E
i=1

(efficiency). Those rules used throughout the present approach are introduced below.
The proportional (P) rule recommends a distribution of the endowment which is proporE
tional to the claims: for each (E, c) ∈ B and each i ∈ N , Pi (E, c) ≡ λci , where λ =
.
ci
i∈N
2

The reader is referred to Moulin (2002) and Thomson (2003, 2013) for reviews of this literature.
Quotas were introduced in 1984. Each member state was given a reference quantity which was then
allocated to individual producers. The initial quotas were not sufficiently restrictive to remedy the surplus
situation and so the quotas were cut in the late 1980s and early 1990s. Quotas will end on April 1, 2015.
3

6

The R package GameTheory

The constrained equal awards (CEA) rule (Maimonides, 12th century), proposes equal
awards to all agents subject to no one receiving more than his claim: for each (E, c) ∈ B and
each i ∈ N, CEAi (E, c) ≡ min {ci , µ} , where µ is such that
min {ci , µ} = E.
i∈N

The constrained equal losses (CEL) rule (Maimonides, 12th century (Aumann and Maschler
1985), chooses the awards vector at which all agents incur equal losses, subject to no one receiving a negative amount: for each (E, c) ∈ B and each i ∈ N , CELi (E, c) ≡ max {0, ci − µ} ,
where µ is such that
max {0, ci − µ} = E.
i∈N

The Talmud (T) rule (Aumann and Maschler 1985) proposes to apply the constrained equal
awards rule, if the endowment is not enough to satisfy the half-sum of the claims. Otherwise,
each agent receives the half of his claim and the constrained equal losses rule is applied
to distribute the remaining endowment: for each (E, c) ∈ B, and each i ∈ N, Ti (E, c) ≡
CEAi (E, c/2) if E ≤
ci /2; or Ti (E, c) ≡ ci /2 + CELi (E −
ci /2, c/2), otherwise.
i∈N

i∈N

The random arrival (RA) rule (O’Neill 1982). Suppose that each claim is fully honored
until the endowment runs out following the order of the claimants’ arrival. In order to remove
the unfairness of the first-come first-served scheme associated with any particular order of
arrival, the rule proposes to take the average of the awards vectors calculated in this way
when all orders are equally probable: for each (E, c) ∈ B, and each i ∈ N, RAi (E, c) ≡
1
j∈N,j i cj , 0}}.
∈RN min{ci , max{E −
|N |!

3. Library structure
The GameTheory package is designed to implement common solutions to cooperative games.
In particular, we focus on transferable utility games, conflicting claims problems and voting
power index. GameTheory is available in source code format.4 In order to install the library,
download the compressed file in your current working directory and execute,
> system("R CMD build GameTheory")
> system("R CMD INSTALL GameTheory_1.0.tar.gz")
> library("GameTheory")
GameTheory package depends on lpSolveAPI to perform linear programming optimization,
kappalab, combinat, and ineq. The results presented in this paper have been obtained using
R version 3.1.2 on a Mac running OS X 10.10.2. The main commands (a brief summary of
all available instructions is displayed in Table 1) of the library are :
Nucleolus(N,coalitions): Obtains the nucleolus of TU-game with a maximum of 4 players. The needed arguments are number of players and values of every coalition. This
command is designed to be used with a gains game. To calculate the nucleolus of a cost
game execute NucleolusCost().
4
The package will be available in the near future in Comprehensive R Archive Network. If you compile the
code from the source, be sure to install first all dependencies mentioned in the DESCRIPTION file.

Sebasti´n Cano-Berlanga, Jos´-Manuel Gim´nez-G´mez, Cori Vilella
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ShapleyValue(coalitions,names): Performs the calculation of the Shapley value for an
N-player TU-game. The extension to voting power index is made by ShapleyShubik().
AllRules(E,claims,names): Obtains the allocations for a N-agents following the all the conflicting claims rules presented in Section 2.2. This command performs Proportional(),
CEA(), CEL(), Talmud() and RandomArrival() simultaneously and includes the Gini
Index to check inequality among them. Results can be displayed with PlotAll() and
LorenzRules().

Game class

Command

Output

Max players

TU-Cooperative

Nucleolus()
NucleolusCost()
ShapleyValue()

Numerical
Numerical
Numerical

4
4
n

Conflicting Claims

Proportional()
CEA()
CEL()
Talmud()
RandomArrival()
AllRules()
PlotAll()
LorenzRules()

Numerical
Numerical
Numerical
Numerical
Numerical
Numerical
Graphical
Graphical

n
n
n
n
n
n

Voting Power

ShapleyShubik()

Numerical

n

Table 1: Summary of available instructions.

4. Examples and Illustrations
4.1. T U -cooperative games
In order to illustrate T U -cooperative games we first take the example proposed by Lemaire
(1991) where three individuals can collaborate by investing in common funds. This particular
game is defined by the following function,
v(∅) = ∅
v(N ) = 90000

v(1) = 46125
v(2) = 17437.5
v(3) = 5812.5

v(12) = 69187.5
v(13) = 53812.5
v(23) = 30750

and with this data we can compute the Shapley value and the nucleolus solutions. We calculate
both solutions using the commands ShapleyValue() and Nucleolus(), respectively. We
proceed in the following manner,
> COALITIONS <- c(46125,17437.5,5812.5,69187.5,53812.5,30750,90000)
> NAMES <- c("Investor 1","Investor 2","Investor 3")

8

The R package GameTheory

> LEMAIRESHAPLEY <- ShapleyValue(COALITIONS,NAMES)
> LEMAIRESHAPLEY

Investor 1
Investor 2
Investor 3

Shapley Value
51750
25875
12375

> LEMAIRE <- Nucleolus(3,c(
46125, # v(1)
17437.5, # v(2)
69187.5, # v(12)
5812.5, # v(3)
53812.5, # v(13)
30750, # v(23)
90000 # v(123)
))
Model name: Nucleolus of a gains game
C1
C2
C3
C4
Minimize
0
0
0
-1
R1
0
0
0
1 >=
0
R2
1
0
0
-1 >=
46125
R3
0
1
0
-1 >= 17437.5
R4
1
1
0
-1 >= 69187.5
R5
0
0
1
-1 >=
5812.5
R6
1
0
1
-1 >= 53812.5
R7
0
1
1
-1 >=
30750
R8
1
1
1
0
=
90000
Kind
Std
Std
Std
Std
Type
Real Real Real Real
Upper
Inf
Inf
Inf
Inf
Lower
0
0
0
0
Model name:
Objective:
SUBMITTED
Model size:
Sets:

'Nucleolus of a gains game ' - run #1
Minimize(R0)

8 constraints,

4 variables,
0 GUB,

Using DUAL simplex for phase 1 and DUAL simplex for phase 2.
The primal and dual simplex pricing strategy set to 'Devex'.
Found feasibility by dual simplex after

4 iter.

19 non-zeros.
0 SOS.

Sebasti´n Cano-Berlanga, Jos´-Manuel Gim´nez-G´mez, Cori Vilella
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Optimal solution

-6562.5 after

9

5 iter.

Excellent numeric accuracy ||*|| = 0
MEMO: lp_solve version 5.5.2.0 for 64 bit OS, with 64 bit LPSREAL variables.
In the total iteration count 5, 0 (0.0) were bound flips.
There were 2 refactorizations, 0 triggered by time and 0 by density.
... on average 2.5 major pivots per refactorization.
The largest [LUSOL v2.2.1.0] fact(B) had 18 NZ entries, 1.0x largest basis.
The constraint matrix inf-norm is 1, with a dynamic range of 1.
Time to load data was 0.010 seconds, presolve used 0.002 seconds,
... 0.002 seconds in simplex solver, in total 0.014 seconds.
[...some output omitted...]
> LEMAIRE
v(S)
x(S)
Ei
1 46125.0 52687.50 6562.50
2 17437.5 24468.75 7031.25
3 5812.5 12843.75 7031.25
Next, by analyzing costs instead of gains, we introduce cost allocation problems, usually called
airport problems (Littlechild and Thompson 1977). Consider, for instance, several airlines
that are jointly using an airstrip. Obviously, different airlines will have different needs for the
airstrip. The larger the planes an airline flies, the longer the airstrip it needs. An airstrip
that accommodates a given plane accommodates any smaller airplane at no extra cost. The
airstrip is large enough to accommodate the largest plane any airline flies. How should its
cost be divided among the airlines?
Note that under this illustration, several situations may be considered. For instance, consider
farmers that are distributed along an irrigation drain. The farmer closest to the water gate
only needs that the segment to his field would be maintained. Accordingly, the second closest
farmer needs that the first two segments be maintained (the segment that goes from the water
gate and the first farmer, and that segment from the first farmer to his field), and so on. The
cost of maintaining a segment used by several farmers is incurred only once, independently
of how many farmers use it. How should the total cost of maintaining the ditch be shared?
In order to illustrate this, consider the following cost airport game,

10

The R package GameTheory
v(∅) = ∅
v(N ) = 110

v(1) = 26
v(2) = 27
v(3) = 55
v(4) = 57

v(12) = 53
v(13) = 81
v(14) = 83
v(23) = 82
v(24) = 84
v(34) = 110

v(123) = 108
v(124) = 110
v(134) = 110
v(234) = 110

first we compute the Shapley value,
>
>
>

COALITIONS <- c(26,27,55,57,53,81,83,82,84,110,108,110,110,110,110)
NAMES <- c("Airline 1","Airline 2","Airline 3","Airline 4")
ShapleyValue(COALITIONS,NAMES)
Shapley Value
Airline 1
17.33333
Airline 2
18.00000
Airline 3
36.33333
Airline 4
38.33333
and now we can see what would be the imputations using the nucleolus,
NucleolusCost(4,c(
26, # v(1)
27, # v(2)
53, # v(12)
55, # v(3)
81, # v(13)
82, # v(23)
108, # v(123)
57, # v(4)
83, # v(14)
84, # v(24)
110,# v(124)
110, # v(34)
110,# v(134)
110,# v(234)
110 # v(1234)
))

Model name: Nucleolus of a cost game
C1
C2
C3
C4
C5
Maximize
0
0
0
0
1
R1
0
0
0
0
0
R2
1
0
0
0
1
R3
0
1
0
0
1
R4
1
1
0
0
1

C6
-1
1
-1
-1
-1

>=
<=
<=
<=

0
26
27
53

Sebasti´n Cano-Berlanga, Jos´-Manuel Gim´nez-G´mez, Cori Vilella
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R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
R16
Kind
Type
Upper
Lower

0
1
0
1
0
1
0
1
0
1
0
1
Std
Real
Inf
0

Model name:
Objective:

0
0
1
1
0
0
1
1
0
0
1
1
Std
Real
Inf
0

1
1
1
1
0
0
0
0
1
1
1
1
Std
Real
Inf
0

0
0
0
0
1
1
1
1
1
1
1
1
Std
Real
Inf
0

1
1
1
1
1
1
1
1
1
1
1
0
Std
Real
Inf
0

-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
0
Std
Real
Inf
0

<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
=

11

55
81
82
108
57
83
84
110
110
110
110
110

'Nucleolus of a cost game ' - run #1
Maximize(R0)

SUBMITTED
Model size:
Sets:

16 constraints,

6 variables,
0 GUB,

61 non-zeros.
0 SOS.

Using PRIMAL simplex for phase 1 and DUAL simplex for phase 2.
The primal and dual simplex pricing strategy set to 'Devex'.
Found feasibility by primal simplex after

5 iter.

Optimal solution

7 iter.

13 after

Excellent numeric accuracy ||*|| = 0
MEMO: lp_solve version 5.5.2.0 for 64 bit OS, with 64 bit LPSREAL variables.
In the total iteration count 7, 0 (0.0) were bound flips.
There were 2 refactorizations, 0 triggered by time and 0 by density.
... on average 3.5 major pivots per refactorization.
The largest [LUSOL v2.2.1.0] fact(B) had 59 NZ entries, 1.0x largest basis.
The constraint matrix inf-norm is 1, with a dynamic range of 1.
Time to load data was 0.021 seconds, presolve used 0.000 seconds,
... 0.002 seconds in simplex solver, in total 0.023 seconds.
[1] 13 14 42 41 13 0
Using PRIMAL simplex for phase 1 and DUAL simplex for phase 2.
The primal and dual simplex pricing strategy set to 'Devex'.
Found feasibility by primal simplex after

5 iter.

12

The R package GameTheory

Optimal solution

13.5 after

8 iter.

Excellent numeric accuracy ||*|| = 0
[...some output omitted...]

1
2
3
4

v(S)
26
27
55
57

x(S)
13.00
13.50
40.75
42.75

Ei
13.00
13.50
14.25
14.25

Voting Power
During Autumm 2014 Artur Mas (member of the Democratic Party of Catalunya (CiU) and
President of Catalunya) said to Oriol Junqueras (leader of the Republican Party of Catalunya
(ERC)) that “alternative majorities are possible” after discussing the referendum proposal of
November 9.5 To conclude our paper we analyze these words trough Shapley-Shubik power
index. As mentioned in section 2, this voting power index often reveals surprising power
distribution that is not obvious on the surface. In order to compare the power index of CiU
and ERC we use the results of the elections of 2003, 2006 and 2012, whose results are displayed
in Table 2.
year

CiU

PSC

ERC

PP

ICV

C’s

CUP

2003
2006
2012

46
48
50

42
37
20

23
21
21

15
14
19

9
12
13

3
9

3

Table 2: Catalan seats distribution after elections of 2003, 2006 and 2012.
Having a look to the data of 2003 it might seem that PSC might have much more power than
ERC (19 less seats in the camera), and the same should apply to year 2006. After executing
ShapleyShubik() (results displayed in Table 3) one can see there are no differences in power
among ERC and PSC for the chosen years. Another interesting case is the dummy player,
both C’s (in 2006) and CUP (in 2012) parties, never become pivotal players. Furthermore,
one might consider that President Mas was right as there are two more parties with the same
Shapley - Shubik power index.
To perform the Shapley - Shubik power index one simply provides the number of members of
each party and the minimum amount of votes needed to pass a vote. The R session to obtain
magnitudes of Table 3 is as follows,
# 2003 Elections
> SEATS<-c(46,42,23,15,9)
5
Manch´n, M. (2014) “Mas a Junqueras: Other majorities are possible in the Parliament,” Econom´ Digital,
o
ıa
September 16, 2014 [on-line].

Sebasti´n Cano-Berlanga, Jos´-Manuel Gim´nez-G´mez, Cori Vilella
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year

CiU

PSC

ERC

PP

ICV

C’s

CUP

2003
2006
2012

0.400
0.400
0.533

0.233
0.233
0.133

0.233
0.233
0.133

0.067
0.067
0.133

0.067
0.067
0.033

0.000
0.033

13

0.000

Table 3: Shapley - Shubik power index of the Catalan parliament.
> PARTIES<-c("CiU","PSC","ERC","PP","ICV")
> ShapleyShubik(68,SEATS,PARTIES)
CiU
PSC
ERC
PP
ICV
Votes
46.000 42.000 23.000 15.0000 9.0000
Votes (%)
0.341 0.311 0.170 0.1111 0.0667
Shapley-Shubik 0.400 0.233 0.233 0.0667 0.0667

# 2006 Elections
> SEATS<-c(48,37,21,14,12,3)
> PARTIES<-c("CiU","PSC","ERC","PP","ICV","C's")
> ShapleyShubik(68,SEATS,PARTIES)
CiU
PSC
ERC
PP
ICV
C's
Votes
48.000 37.000 21.000 14.0000 12.0000 3.0000
Votes (%)
0.356 0.274 0.156 0.1037 0.0889 0.0222
Shapley-Shubik 0.400 0.233 0.233 0.0667 0.0667 0.0000

#
>
>
>

2012 Elections
SEATS<-c(50,20,21,19,13,9,3)
PARTIES<-c("CiU","PSC","ERC","PP","ICV","C's","CUP")
ShapleyShubik(68,SEATS,PARTIES)
CiU
PSC
ERC
PP
ICV
C's
CUP
Votes
50.000 20.000 21.000 19.000 13.0000 9.0000 3.0000
Votes (%)
0.370 0.148 0.156 0.141 0.0963 0.0667 0.0222
Shapley-Shubik 0.533 0.133 0.133 0.133 0.0333 0.0333 0.0000

4.2. Conflicting claims problem
The usefulness of the mechanisms proposed in the current paper covers several contexts,
as mentioned in the introduction. As an example we replicate Gallastegui et al. (2003).
They analyze the distribution of Northern European Anglerfish Fishery quotas among EU
countries in terms of the allocations recommended by different solutions and how this may
affect the sustainable growth of the fishing catches. Specifically, they consider seven countries
(France, Spain, U.K., Ireland, Belgium, Netherlands and Germany). Each country has a
claim, which depends on its historical fishing catches (13,952; 6,256; 4,348; 2,196; 927; 299;
158, respectively).
To replicate the study of Gallastegui et al. (2003) we can execute the commands one by one
or use Allrules() to run all of them at once. To do so, we create objects containing the

14

The R package GameTheory

Claim
Germany
Netherlands
Belgium
Ireland
UK
Spain
France

Proportional

Nucleolus

Shapley

158
299
927
2,196
4348
6,256
13,952

76
143
445
1,054
2,086
3,002
6,694

79
145
463
1,098
2,174
3,128
6,408

74
140
437
1,071
2,147
3,101
6,530

Table 4: Fishing captures allocations. E=13,500; claim = average catches 1986-93. Source:
Gallastegui et al. (2003).

0

1000

2000

3000

4000

Allocations for UK

Claim

Proportional

CEA

CEL

Talmud

RA

Figure 1: Fishing captures allocations for United Kingdom.

individual claims and labels of the countries. After that, running Allrules() is straightforward. Displaying the output of the allocations is undertaken by running PlotAll() (Figure
1). Graphical analysis of the inequality among rules is performed by LorenzRules() (Figure
2).
> CLAIMS <- c(158,299,927,2196,4348,6256,13952)
> COUNTRIES <- c("Germany","Netherlands","Belgium","Ireland","UK","Spain","France")

> INARRA <- AllRules(13500,CLAIMS,COUNTRIES)
> INARRA
Claim Proportional
CEA
CEL
Germany
158
75.81 158.00
0.00
Netherlands
299
143.46 299.00
0.00

Talmud
79.00
149.50

RA
73.73
139.53

Sebasti´n Cano-Berlanga, Jos´-Manuel Gim´nez-G´mez, Cori Vilella
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Belgium
Ireland
UK
Spain
France
Gini Index

927
2196
4348
6256
13952
NA

444.79
1053.67
2086.22
3001.71
6694.34
0.58

927.00
0.00
2196.00
0.00
3306.67
662.67
3306.67 2570.67
3306.67 10266.67
0.38
0.77

463.50
1098.00
2174.00
3128.00
6408.00
0.56

15

436.92
1071.42
2147.42
3101.42
6529.57
0.57

> PlotAll(INARRA,5) ## Display allocations for UK
> LorenzRules(INARRA) ## Inequality graph

Lorenz curve
1.0
Proportional
CEA
CEL
Talmud
RA

0.8

L(p)

0.6

0.4

0.2

0.0
0.0

0.2

0.4

0.6

0.8

1.0

p

Figure 2: Inequality analysis for the different rules.

A note on the nucleolus computation
Validating the nucleolus solution might be an issue. Indeed, there is a fruitful discussion in the
literature on what strategy is optimal to obtain the nucleolus, how it should be calculated, or
what algorithm could achieve a better solution. For instance, Guajardo and J¨rnsten (2015)
o
discuss several nucleolus solutions among different papers, finding that the published results
might not be optimal. Therefore, for the sake of robustness, we check our nucleolus commands
through the correspondence between conflicting claims problems and bankruptcy games.
Following O’Neill (1982), a bankruptcy game is the T U -game associated with a conflicting
claims problem. Formally, for each (E, c) ∈ B, the cooperative game induced by (E, c) is the
pair (N, v), where the function v : 2N → R+ associates to each coalition S ⊆ N the real

16

number v(S) = max 0, E −

The R package GameTheory

ci
i∈N \ S

.

According to O’Neill (1982) and Aumann and Maschler (1985), for each conflicting claims
problem, the Talmud and the random arrival rules coincide with the nucleolus and the Shapley
value solutions of the associated bankruptcy game, respectively. A fact that we can replicate
by computing the Talmud and the random arrival rules to the considered conflicting claims
problem. For instance, the example of the airport cost game presented in Section 4.1, is
associated to the following conflicting claims problem (E, c) = (110, (26, 27, 55, 57)). Hence,
we can transform this particular problem into a gains game by applying the aforementioned
definition of O’Neill (1982). The new game adopts the form: v(1) = 0, v(2) = 0, v(3) = 0,
v(4) = 2, v(12) = 0, v(13) = 26, v(14) = 28, v(23) = 27, v(24) = 29, v(34) = 57, v(123) = 53,
v(124) = 55 v(134) = 83, v(234) = 84, v(N ) = 110. Applying Nucleolus() over this game
coincides with the result of Talmud(110,c(26,27,55,57)).
Acknowledgements
We are grateful to Generalitat de Catalunya under project 2014SGR631 and Ministerio de
Ciencia e Innovaci´n under project ECO2011-22765 for its support.
o

References
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Gallastegui M, I˜arra E, Prellezo R (2003). “Bankruptcy of Fishing Resources: The Northern
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European Anglerfish Fishery.” Marine Resource Economics, 17, 291–307.
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The R package GameTheory

Warnes GR, Bolker B, Lumley T (2014). gtools: Various R programming tools. R package
version 3.4.1, URL http://CRAN.R-project.org/package=gtools.
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Affiliation:
Sebasti´n Cano-Berlanga
a
Department d’Economia
Facultat d’Economia i Empresa
Universitat Rovira i Virgili / GRODE
43204 Reus, Spain
E-mail: sebastian.cano@urv.cat
Jos´-Manuel Gim´nez-G´mez
e
e
o
Department d’Economia
Facultat d’Economia i Empresa
Universitat Rovira i Virgili / CREIP
43204 Reus, Spain
E-mail: josemanuel.gimenez@urv.cat
Cori Vilella
Department de Gesti´ d’Empreses
o
Facultat d’Economia i Empresa
Universitat Rovira i Virgili / CREIP
43204 Reus, Spain
E-mail: cori.vilella@urv.cat