We present a simple proof of the balancedness of permutation games. In the proof we use the existence of envy-free allocations in economies with indivisible objects, quasi-linear utility functions, and an amount of money.Permutation Games;Balancedness;Envy-free Allocations

Hamers, H.J.M.

Klijn, F.

Tijs, S.H.

English

Research Papers in Economics

Balancedness of Permutation Games and Envy-free
Allocations in Indivisible Good Economies
Flip Klijn , Stef Tijs, and Herbert Hamers
Department of Econometrics and CentER, Tilburg University,
P.O.Box 90153, 5000 LE Tilburg, The Netherlands
Abstract: We present a simple proof of the balancedness of permutation games. In the proof
we use the existence of envy-freeallocations in economies with indivisible objects, quasi-linear
utility functions, and an amount of money.
Classiﬁcation numbers (J.E.L.): C71 and D63.
Keywords: Permutation Games, Balancedness, Envy-free Allocations.
Running title: Balancedness of Permutation Games.
1 Introduction
In this paper we provide an alternative proof for the balancedness (i.e. the non-emptiness of
the core) of permutation games as introduced by Tijs et al. (1984). For this result some proofs
already exist. Tijs et al. (1984) obtained the result by using the extreme points theorem of
Birkhoff-von Neumann on doubly stochastic matrices, and Curiel and Tijs (1986) by using an
equilibrium existence theorem of Gale (1984) for a discrete exchange economy with money.
Our alternative proof is interesting because it relates the core conditions of permutation
games with the properties of envy-freeness and Pareto-efﬁciency in economies with indivisible
objects, quasi-linear utility functions, and an amount of money. Moreover, our proof does not
rely on deep mathematical theorems. In an economy an allocation is called envy-free(cf. Foley
(1967)) if every agent in the economy likes his own bundle at least as well as that of anyone
Correspondingauthor. Tel. (31) 13 466 2733. Fax. (31) 13 466 3280. e-mail: F.Klijn@kub.nl
1else. Alkan et al. (1991) and Svensson (1983) showed that in a large class of economies the
set of envy-free allocations is non-empty and that it is a subset of the set of Pareto-efﬁcient
allocations. Aragones (1995) gave another proof of the existence of envy-free allocations in
the particular case of economies with indivisible objects, a ﬁxed amount of money, and agents
with quasi-linear utility functions. Finally, Klijn (1999) provided an algorithm consisting of
two natural procedures for ﬁnding envy-free allocations in quasi-linear economies. To prove
that permutation games are balanced, we invoke the existence result on envy-free allocations in
quasi-linear economies to construct a core allocation.
2 The balancedness of permutation games
Let us start by recalling the deﬁnition of permutation games. After that, we will recall a result
on envy-free allocations for our particular class of economies. And ﬁnally, we will provide
the proof on the balancedness of permutation games by relating the core conditions with the
properties of envy-freeness and Pareto-efﬁciency.
Permutation games, introduced by Tijs et al. (1984) (cf. Curiel (1997) and Quint (1996)),
describe a situation in which n persons all have one job to be processed and one machine
on which each job can be processed. No machine is allowed to process more than one job.
Sidepayments between the players are allowed. If player i processes his job on the machine of
playerj theprocessing costsareaij.L e tN:= f1;:::;ngbethesetofplayers. Thepermutation
game (N;c) with costs aij is the cooperative TU-game deﬁned by
c(;): =0 and
c(S): = m i n
 S 2  S
X
i 2 S
a iS(i) for all S 2 2
Nnf;g;
where S is the class of all S-permutations and 2N the collection of all subsets of N.T h e
number c(S) denotes the minimum costs for coalition S to process its jobs on its own machines
under the restriction that each machine processes exactly one job.
Tijs et al. (1984) proved, using the Birkhoff-von Neumann theorem on doubly stochastic
matrices, that permutation games are balanced. This means that there is an efﬁcient payoff
vector x for which no coalition has an incentive to separate from the grand coalition. Formally,
there is a vector x 2 IR
N such that
P
i2S xi  c(S) for all S 2 2Nnf;g and
P
i2N xi = c(N).
Curiel and Tijs (1986) gaveanother proof of thebalancedness of permutationgames. They used
an equilibrium existence theorem of Gale (1984) for a discrete exchange economy with money,
2thereby showing a relation between assignment games (cf. Shapley and Shubik (1972)) and
permutation games.
Next, letusturntothespeciﬁcresultonenvy-freeallocationsthatwewillneedforourproof.
Here, an economy is represented by an ordered triple E =( N;Q;U),w h e r eN=f 1 ;:::;ng
is a ﬁnite set of agents and Q = f1;:::;nga set of indivisible objects, and U the utility matrix
which will be deﬁned next. Each agent i 2 N is assumed to be endowed with a quasi-linear
utility function ui : Q  IR ! IR :
u i( j;x)=u ij + x (j 2 Q;x 2 IR ) ;
where uij can be any real number. The number ui(j;x) is interpreted as the utility that agent
i 2 N derives when he receives an object j 2 Q and an amount of money x 2 IR .N o w , w e
deﬁne the utility matrix U by letting uij be its ij-th entry (i 2 N, j 2 Q). Let E =( N;Q;U)
be an economy. A feasible allocation for the economy E is a pair (;x) 2 N  IR
n such that
Pn
i=1 xi =0 . A feasible allocation (;x) gives object (i) and the amount x(i) of money to
agent i.
We are interested in so called envy-free allocations, which satisfy the following notion of
equity: no agent prefersthebundleof any other agent to his own. Formally,afeasibleallocation
(;x) is envy-free (cf. Foley (1967)) if
ui(i) + x(i)  ui(j) + x(j) for all i;j 2 N:
Another property that is often used in the selection of normatively appealing allocations
is Pareto-efﬁciency. In our model a feasible allocation is Pareto-efﬁcient if and only if there
is no other feasible allocation that makes all agents strictly better off. The proof of the next
result for a more general class of preferences can be found in Alkan et al. (1991) and Svensson
(1983). Proofsof theexistence of envy-freeallocationsforquasi-linear economies canbe found
in Aragones (1995) and Klijn (1999). These proofs are based on elementary mathematics.
Theorem 2.1 ForeveryeconomyE,thereexistsanenvy-freeallocation. Moreover, allenvy-free
allocations are Pareto-efﬁcient.
We are now ready to give our alternative proof of the balancedness of permutation games.
Theorem 2.2 Permutation games are balanced.
3Proof. Let (N;c) be a permutation game. We prove that (N;c) is balanced. Let aij be the
processing costs that correspond with (N;c).L e tUbe the utilitymatrixdeﬁnedby uij := −aij,
and let E be the economy deﬁned by E := (N;Q;U),w h e r eQ=N=f 1 ;:::;ng.F r o m
theorem 2.1 it follows that there is an envy-free allocation, say (;x).D e ﬁ n e
y i:= −ui(i) − x(i) + xi for all i 2 N:
Then y := (yi)i2N is a core element of (N;c). This can be seen as follows.
For S 2 2Nnf;g, with S 2 S such that
c(S)= m i n
 S2  S
X
i 2 S
− u iS(i) =
X
i2S
−uiS(i);
it holds that
X
i2S
yi =
X
i2S
(−ui(i) − x(i) + xi)=
X
i 2 S
( − u i(i) − x(i) + xS(i)) 
X
i2S
−uiS(i) = c(S);
where the inequality follows from the envy-freeness of (;x). Now the theorem follows from
the remark that for S = N the inequality is an equality since theorem 2.1 implies that (;x) is
Pareto-efﬁcient, i.e. maximizes the sum of utilities among all feasible allocations.
2
4References
Alkan, A., G. Demange, and D. Gale, 1991, Fair Allocation of Indivisible Objects and Criteria
of Justice, Econometrica 59, 1023-1039.
Aragones, E., 1995, A Derivation of the Money Rawlsian Solution, Social Choice and Welfare
12, 267-276.
Curiel, I., 1997, Cooperative Game Theory and Applications (Kluwers Academic Publish-
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Curiel, I., and S. Tijs, 1986, Assignment Games and Permutation Games, Methods of Op-
erations Research 54, 323-334.
Foley, D., 1967, Resource Allocation and the Public Sector, Yale Economics Essays 7, 45-
98.
Gale, D.,1984, EquilibriuminaDiscreteExchangeEconomywithMoney, InternationalJournal
of Game Theory 13, 61-64.
Klijn,F., 1999, AnAlgorithmfor Envy-freeAllocationsinanEconomywithIndivisibleObjects
and Money, forthcoming in Social Choice and Welfare.
Quint, T., 1996, On One-Sided versus Two-Sided Matching Games, Games and Economic
Behavior 16, 124-134.
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Tijs, S., T. Parthasarathy, J. Potters, and V. Rajendra Prassad, 1984, Permutation Games:
another Class of Totally Balanced Games, OR Spektrum 6, 119-123.
5

Balancedness of Permutation Games and Envy-Free Allocations in Indivisible Good Economies

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