www.elsevier.nl / locate / econbase
The core of endo-status games and one-to-one ordinal
preference games
a ,* b
Thomas Quint , Martin Shubik a
Department of Mathematics, University of Nevada, Reno, NV 89557, USA b
Cowles Foundation, Yale University, New Haven, CT 06520, USA
Received 28 March 1998; received in revised form 16 January 2000; accepted 24 January 2000
Abstract
We consider status games [Quint, Thomas and Martin Shubik, ‘Games of Status’, technical report, University of Nevada and Yale University (1999)]. These are n-player ordinal preference cooperative games in which the outcomes are orderings of the players within a hierarchy. In particular we study ‘endo-status’ games. Here each coalition S has an exogenously given setPSof allocations of positions to its members that it can enforce. For such games, we define a condition of ‘balance’ on the setP*; hPS Sj #N. IfP* is balanced, the core of the associated status game is nonempty. Conversely, ifP* is not balanced, and the game is ‘exchangeable’, we can find an instance where the strict core is empty. Finally, we define a more general class of one-to-one
ordinal preference (OOP) games, which include both ‘exo-status’ and ‘endo-status’ games [Quint,
T., Shubik, M., 1999. Games of status. Technical report, University of Nevada and Yale University], as well as the class of restricted houseswapping games with ordinal preferences (RHGOPs) [Quint, T., 1997. Restricted houseswapping games. Journal of Mathematical Econ-omics 27, 451–470]. We again define a condition of ‘balancedness’ for these games, which (a) guarantees core existence, (b) reduces to the above condition for status games, and (c) reduces to ‘weak balancedness’ [Quint, T., 1997. Restricted houseswapping games. Journal of Mathematical Economics 27, 451–470] in the case of RHGOPs. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Core; Matching; Status game; Ordinal preferences; Cardinal preferences
JEL classification: C71; C78
*Corresponding author. Tel.: 11-775-784-1366; fax: 11-775-784-6378.
E-mail address: [email protected] (T. Quint).
1. Introduction
The role of ‘status’ (as opposed to wealth) is important in the social sciences. Indeed, there are many instances where one’s position or rank in relation to others is more important than the actual amount of consumption. A common example in our political system concerns the ‘election game’, in which payoffs are votes. One does not care how many votes one obtains, only how one’s vote total compares with that of the other candidates. Sometimes ‘first place’ is the only position worth anything (like in an election for a governor in a US state), but in other elections ‘second place’ or even lower positions are important. An example here would be the Russian presidential elections of
1996, in which out of 101 candidates the top two advanced to ‘the final round’.
Other examples of the importance of ‘position relative to others’ would be in sports (the ‘third-ranked tennis player’ or ‘winner of the Boston Marathon’, are honors stated in terms of status), and in rigid hierarchies, such as in the upper echelons of totalitarian governments or of the Catholic Church. Indeed, Hermann Goering reveled in his status
1
as ‘the Second Man’ of the Third Reich (Irving, 1989, p. 153). Finally, we have the example of the baseball player Barry Bonds, who allegedly was upset because his team, the San Francisco Giants, reneged on their promise to make him the highest paid player in the game (Associated Press, 1997). We doubt that Mr. Bonds really cares how many millions he makes, so long as it is more than any other player.
So far as we know, there have not been many attempts to analyze status (as opposed
2
to wealth) using the mathematical machinery of game theory . Recently (see Quint and Shubik, 1999), we have endeavored to change this by introducing status games. A status game is an n-player cooperative game in which the outcomes are orderings of the players.
Notationally, suppose the player set is N5h1, . . . ,nj. Then, outcomes are represented
by permutations of N, where if i occurs at position j in the permutation, this is taken to
mean that player i attains the jth best position. For example, if n54, the outcome in
which player 3 comes in ‘first place’, player 1 comes in ‘second place’, player 4 comes
3
in ‘third place’, and player 2 is ‘last’ is represented by the permutation [3 1 4 2] . We assume that players will always desire to placed as far ‘up’ in the hierarchy as possible, i.e., as close as possible to ‘first place’.
We feel that status games are a good model for some of the situations outlined above. For example, in parliamentary politics, it is often only possible to form a government via the coalition of two or more political parties. This coalition is then able to assign various ministrial positions, each of which carries a certain ‘ranking’. This occurred during the ‘second round’ of the Russian election mentioned above, when Yeltsin and Lebed joined forces in order to ensure Yeltsin’s victory over Zyganov; as part of the agreement, Lebed was made Security Minister in the new government.
1
Note a direct quote. 2
For one such early attempt, see Shubik (1972). 3
2. On status games vs. matching games
Below, we will define status games formally. Readers versed in the theory of matching games will see a strong resemblance between status games and the models presented by Shapley and Scarf (1974); Kaneko (1982); Quinzii (1984) and Quint (1997). Indeed, all of the above constructs use permutation matrices in order to define feasible distributions of a set of indivisible goods. In the matching games listed above, the indivisible goods are thought to be large objects such as houses, or else service-binding contracts. In status games, they are of course the different ‘positions’ that a society has to offer.
However, there are two important differences between status games and the matching models listed above. First, in matching games there is an initial ownership of goods, while in status games there is not. In this sense, status games are more general than matching games, because the capabilities of coalitions are not limited by their members’ initial endowments. On the other hand, in matching games the rankings of the players
over the objects is allowed to be more general than in status games. For example, in
Shapley and Scarf’s houseswapping game, the traders are each allowed to value different houses differently – while in status games, all players rank ‘first place’ first, ‘second place’ second, etc.
In Section 5, we will define one-to-one ordinal preference games, which are a large class of games including both status games and the above matching games.
3. Two models
We have distinguished two ways of modeling games of status, based on what is assumed about the capabilities of coalitions smaller than N. In ‘exo-status games’, we assume that such small coalitions are capable of enacting orderings over all n players. [The adjective ‘exo’ refers to the fact that a coalition S might have the power to directly determine the fates of players outside of S.] Alternatively, in ‘endo-status games’, we assume that a small coalition S is only capable of guaranteeing certain positions for its
own members within the n-player hierarchy, but has no control over how the other
n2uSu players are placed into the remaining positions. To see the difference between these two approaches, consider an example of an organization with a defacto leader. If modeled as an endo-status game, the most powerful he could be would be if he were able to guarantee himself ‘first place’. On the other hand, in an exo-status game, he would also be able to place the other players into any positions he wished in the hierarchy.
In this paper, we are concerned primarily with endo-status games. For a discussion of exo-status games, see Quint and Shubik (1999).
4. Endo-status games
4.1. The model
N
let 2 be the set of subsets of N. In this game, an outcome for a player is either (a) a
ranking (or status or position), i.e., an ordinal number from the seth1, . . . ,nj, or (b) to
4
be ‘unranked’ . The utility to any player of being unranked is defined to be zero. For
rankings, define the n3n matrix R, where r represents the utility to player i of endingij
up with ranking j. Since we assume that a player always weakly prefers a higher status
to a lower status, we have j,k⇒rij$r . We also assume that any player weaklyik
prefers any ranking over the ‘unranked state’, i.e. rij$0;i, j.
N
Let S be an element of 2 . Then, an S-ordering P is an nS 3n 0–1 matrix in which (a)
n n
if i[⁄ S, then pSuij50 for j51, . . . ,n, (b) for all j,oi51 pSuij#1, and (c) for all i,oj51
pSuij#1. In words, an S-ordering describes an outcome that the coalition S could
potentially effect, under the interpretation pij51⇔ player i receives position j.
Condition (a) is the characteristic assumption of endo-status games, that coalition S has
5
no power to determine the placement of players in N2S. Condition (b) states that two
6
players cannot be placed in the same position . Finally, (c) states that no player can be placed in more than one position. Note that there is no requirement that individual
members of S be given exactly one position, i.e., it is possible for there to exist an i[S
n
withoj51 pSuij50. Such a player would be unranked, and would receive payoff zero.
Let P be the set of all N-orderings, and, for any S let PS denote the set of all
S-orderings that S can actually effect. We assume the n3n zero-matrix Z is an element
of PS for all S. LetpS(i ) (i[S ) denote the position that i obtains under S-orderingpS,
with the convention pS(i )55meaning that i is not given a position under pS.
We assume that the setshPS Sj #N satisfy the following assumptions:
(1) GRAND COALITION POWER: P 5 PN . The grand coalition can institute any
N-ordering of the players.
(2) ‘CAN DO NOTHING’: Z[PS for all S.
(3) RANK CONSISTENCY: Suppose S and T are disjoint, and letpS[PS,pT[PT.
Then the matrix p 1 pS T has all of its column sums equal to zero or one.
The idea of rank consistency is to forbid disjoint coalitions from having the ability to carry out mutually inconsistent outcomes. To wit, if the condition stated above did not hold, this would imply the existence of a pair of disjoint coalitions, each able to guarantee a common ‘‘position j’’ for one of its constituents.
(4) SUPERADDITIVITY: Suppose S and T are disjoint, and let pS[PS and
4
We need to allow for the ‘unranked’ outcome in this game, in order to ensure that the characteristic function (of the resultant NTU game) evaluated at any coalition is nonempty.
5
Except via the ‘filling up of positions’ with its own players. For instance, suppose that the coalition ]]]
S51, . . . ,n21 can guarantee position 1 for player 1, position 2 for player 2, . . . , position n21 for player
n21. Then necessarily S can limit player n to position n (even though n[⁄ S) because that position is the
only one left. 6
pT[PT. Then (given its column sums are all 0 or 1), the matrixp 1 pS T is an element of PS<T.
(5) MONOTONICITY: If S#T, then PS#PT. Note that (5) follows from (2) and
(4).
We now define a cooperative NTU game, called an endo-status game, in which the
N n 7
characteristic function V :2 →R is given by :
ui#rip(i ), for i[S:pS(i )±5
S
V(S )5h(u , . . . ,u )1 n j:'pS [PSsatisfying
H
ui#0, for i[S:pS(i )55
Note that we may define an endo-status game by the triple (N, hPS Sj #N, R).
4.2. The core of an endo-status game
n
The core of an endo-status game is defined as the set of vectors u[R such that (a)
u[V(N ) and (b) there does not exist (S, w) with w[V(S ) and wi.ui;i[S. The strict core is defined similarly, except condition (b) is amended to read ‘there does not exist
(S, w) with w[V(S ) and wi$ui;i[S, with wi.u only required for one i is S.’ Thesei
are the usual definition of core and strict core for a NTU game.
Example 4.1. Suppose n53, withP 5S hZjfor all S withuSu51 (Z is the three-by-three
the matrix R5hrijjbe any 3 by 3 matrix in which ri 1.ri 2.ri 3for i51,2,3 (i.e., each player strictly prefers ‘Rank 1’ to ‘Rank 2’ to ‘Rank 3’). Then we claim the core is
empty. This is because in any element ofPNthere will be one player who gets ‘Rank 2’
or worse, and another who gets ‘Rank 3’ or worse. These two players will be able to form a coalition in which the former player can get ‘Rank 1’ and the latter ‘Rank 2’.
]
Example 4.2. Let us modify Example 4.1, so thatP23 is equal tohZ, M5jinstead ofhZ, M , M5 6j. Then the outcome where player 1 gets ‘Rank 1’, player 3 gets ‘Rank 2’, and
8
player 2 gets ‘Rank 3’ is in the core. Hence here we have a nonempty core.
We next turn to classical game theory to find sufficient conditions for core nonemptiness. To some readers, the ‘balancedness’ conditions that we present below will seem complicated. To some extent, we agree; however, we also note that throughout the theory of matching games, such balancedness conditions have been used again and again to prove that the cores of certain games are empty. See Shapley and Scarf (1974), Kaneko (1982), Quinzii (1984) and Quint (1997).
Suppose G5(N,V ) is an NTU game, with N the player set and V(S ) the characteristic
n
function whose values are regions in R . Let T be a set of coalitions for which there
exist a set of nonnegative weightshdS Sj [TwithoS[T :S]id 5S 1 for all i[N. Then T is a
balanced family of coalitions, and hdS Sj [T its balancing weights. It is a minimal
balanced family if furthermore it has no proper subset which is balanced. Note that if T
is minimal balanced, thend .S 0 for all S[T. Then the core of the game is nonempty.
Definition 4.4. Suppose N andhPS Sj #N are the player set and ‘feasible permutation set’
from an endo-status game. Suppose that for every minimal balanced family T and every
setb; hPS Sj [T (P is any element ofS PS), there is at least one player who gets ‘position
n or worse’ in some element ofb, at least two players who get ‘position n21 or worse’
in some element of b, . . . , and at least n21 players who get ‘position 2 or worse’ in
9
some element of b. Then we say (N,hPS Sj #N) is balanced.
Theorem 4.5. If G5(N,hPS Sj #N,R) is a status game in which (N,hPS Sj #N) is balanced, its core is nonempty ( for any R).
] ] ]
Example 4.6. In Example 4.1 above, consider T5h12, 13, 23j (with d 5(1 / 2, 1 / 2,
8
And no player disposes utility. See Footnote 7. 9
1 / 2)). Considerb 5hM , M , M1 3 5j. There is no player who gets ‘position n or worse’ in
any element of b, and so (N, hPS Sj #N) is not balanced. In fact, we have seen that the
core of this game is empty.
Example 4.7. Suppose we now consider the game of Example 4.2. Using the logic of
Example 4.6, we observe that again (N,hPS Sj #N) is not balanced. Yet, as we have seen,
the core of the game is nonempty. Hence we conclude that the converse of Theorem 4.5 will not hold.
Finally, before embarking on the proof of Theorem 4.5 we need two more pieces of
*
Essentially, the matrix PS is an extension of P in which an extra (the nS 11st) column
*
is added that represents the ‘unranked’ status. LetPS represent the set of members of
PS, written in this fashion.
In keeping with this interpretation, we define rin1150 for i51, . . . ,n.
Proof of Theorem 4.5. We use Theorem 4.3. So suppose G5(N, hPS Sj #N,R) is an
endo-status game in which hN, PSj is balanced, and consider any minimal balanced
family T with balancing weightshdS Sj [T. Suppose u[V(S );S[T. Our aim is to show
and Scarf, 1974). Hence it can be written as a convex combination of stochastic 0–1
k
m ˆ
*
matrices, i.e. oS[T dSPS 5ok51akP for some positive constants hakj and n3(n11)
10
k
ˆ
stochastic 0–1 matriceshP j. Continuing from (4.1), we have
m
In addition, due to the ‘at least one player gets position n or worse . . . ’ condition, in
*
matrixoS[TdSPS there is at least one row with a positive element in column n or n11,
at least two rows with a positive element in column n21, n, or n11, at least three
k
Remark 1. Unfortunately, we believe that the balancedness condition is much stronger
than core nonemptiness. In matching games, it is desirable at least to have the
‘balancedness condition’ be equivalent to guaranteed core nonemptiness in the case of
three players. This is in fact the case with restricted houseswapping games with ordinal
preferences (Quint, 1997). We have already seen in Example 4.7 that we do not have
this equivalence.
Remark 2. However, let us now consider the class of exchangeable endo-status games,
2 2
i.e., games in which if PS[PS and if P is another S-ordering withS oi PSuij5oi PSuij 2
for all j, then PS[PS as well. In words, an exchangeable endo-status game is one in
which, if a certain S can guarantee a certain set of statuses for its members, then it is possible for those members to freely distribute those rankings among themselves in any way they like.
Note that the game in Example 4.1 / 4.6 is exchangeable, while that of Example 4.2 / 4.7 is not. We also remark that exchangeability (like balancedness) is a property of
(N, hPS Sj #N) and not of R.
For exchangeable endo-status games, it is now true that for n53, the balancedness
condition is equivalent to guaranteed core nonemptiness. For higher n, we have the following ‘almost converse’ of Theorem 4.5:
Theorem 4.8. Suppose (N,hPS Sj #N) is exchangeable but not balanced. Then there is a ranking matrix R for which G5(N,hPS Sj #N,R) has an empty strict core.
1
Lemma 4.9. Let T be a balanced family of coalitions for an n-person game. Let S and
2 1 2
S be coalitions (not necessarily members of T ) satisfying (a) S >S 55 and (b)
1 2 1 2
uS u.uS u. Then there is an element S of T for which uS>S u.uS>S u.
1
Proof. Suppose the Lemma was not true. Then there is a balanced family T with S and
2 1 2
By the definition of the set of balancing weights, the sum on the left is equal touS uand
2
the sum on the right is uS u. Hence we have violated our initial assumption that
1 2
uS u.uS u.
Proof of Theorem 4.8. Since (N, hPS Sj #N) is not balanced, there exist a T, a
b 5hPS Sj [T (PS[P ;S S[T ), and a k* such that at least k*11 players get position k*
or better in every element ofb that contains them. Let Yk *be the set of such players. We
have uYk *u5k*1p, where p$1.
Now suppose the ranking matrix is given by:
1, if j#k*;
rij5
H
0, otherwise.We claim the resultant status game has an empty strict core. For, suppose PN is any
11
Now, by Lemma 4.9 there is an element S*[T which contains more members of Yk *
1C 0 0
than it does of Yk *. Consider any i [S*>Y . [Such a player exists becausek *
0 1C
uS*>Yk *u.uS*>Yk *u$0.] This player gets payoff 0 out of P , but gets 1 out of P .N S *
11
12
Hence, the only possible way in which S* would not ‘block’ P via PN S *is if there were
0
another player j(i )[S* for which
0
Now form PS *from PS *by having i and j(i ) switch positions. By the
exchangeabili-1 0 1
ty property, we have PS *[PS *. Also note that j(i ) now gets a payoff of 1 out of P .S *
0 0
Next, we observe that there must be another element of S*>Y , other than i , becausek *
0 1C 0 1C 1
And again we observe that there must be a third element of S*>Y , because we havek *
1C 2
already seen two distinct elements of S*>Yk *. Call this element i . . .
m
Continuing in this fashion, we see that eventually we will find a i for which there
m m
will not exist a corresponding j(i ). In this case, S* will ‘block’ P via P .N S * h
5. One-to-one ordinal preference (OOP) games
In this section we define the class of one-to-one ordinal preference (OOP) games. Qualitatively this is to be ‘the largest possible’ class of games in which players may express their preferences ordinally over a set of single objects with which they are to be matched. There is no transferable resource (money) via which the characteristic function sets can be ‘smoothed’. Specifically, this class will include the restricted houseswapping
13
games with ordinal preferences (RHGOPs) (Quint, 1997), as well as both exo- and
endo-status games.
Formally, a OOP game is a quadruple (N, J, R, hPS Sj #N), in which the four
components are defined as follows. First, N5h1, . . . ,nj is the player set. Second,
J5h1, . . . ,nj is a set of indivisible objects. These objects may be indivisible physical
12
The phrase ‘S* blocks P via P ’ means that all the players in S* get at least as high a payoff out of PN S * S *as they do out of P , with at least one player getting a strictly higher payoff. Hence this would imply that theN payoff under P is not in the strict core.N
13
goods like houses, or else positions in a hierarchy; either way, we assume that no player has any use for more than one of them. Hence we will assume that in any feasible outcome no player ends up with more than one object; however, there may be one (or more) objects that end up unassigned. In this case there will be one or more players who end up with no object.
Next, R5hrijjis an n3(n11) matrix of rankings, in which r ( jij #n) represents the
valuation player i has for object j. In addition, rin11represents the valuation that player i
has for the prospect of ending up with no object. In the framework of OOP games, we
assume nothing about how r compares with r for j±n11.
in11 ij
Finally, define an ordering P as an n3(n11) 0–1 matrix in which (a)oiPij#1 for
n11
each j#n, and (b) oj51 Pij51 for each i[N. In words, an ordering describes an
outcome that a coalition could theoretically effect. Condition (a) reflects the requirement that the same object cannot be given to more than one player, while (b) states that every player must either be given an object, or remain in the state of having no object. Denote
by Z the special ‘nobody-gets-anything’ ordering, i.e., the ordering in which Zin1151
for all i. Finally, let P be the set of all orderings.
Let PS be the set of all orderings that S can effect. Specifically, we mean that if
PS[PS and (i , j ), (i , j ), . . . ,(i , j ) are the positions in the first n columns of P1 1 2 2 k k S
which contain ‘1’s, then it is possible for S to give object j to i , . . . , j to i , while1 1 k k
being powerless to assign objects to the rest of the players (i.e., those i’s for whom
rin1151). Note that there is no requirement that any of the players i , . . . ,i1 k be
We call (a) and (b) the rank consistency assumptions. Condition (a) states that disjoint coalitions cannot both have the power to assign objects to the same player, while (b) states that they cannot both assign the same object. These restrictions make sense in light of the question ‘what happens if both coalitions simultaneously and independently assign
14
the same player / object?’.
1 2
Another assumption is superadditivity. Suppose P and P are orderings. Define their
1 2 1 2 1 2 1 2 n 1
sum P 1P by: (P 1P )ij5Pij1Pij if j#n; while (P 1P )in11512oj51 (P 1 2
P ) . Note that if S and S are disjoint coalitions with Pij 1 2 S1[PS1 and PS2[PS2, then
rank consistency implies PS11PS2is an ordering. The superadditivity assumption states
that this ordering is an element of PS1<S2. Formally, PS1[PS1, PS2[PS2, S1>S25
5⇒PS11PS2[PS1<S2.
We call any quadruple (N, J, R, hPS Sj #N) satisfying rank consistency and
superad-ditivity a one-to-one ordinal preference (OOP) game. At this point, we can model OOP
14
games as non-transferable utility (NTU) games. The characteristic function is given by
V(S )5h(u , . . . ,u )1 n j:'PS[PSwith ui#riP (i )S for i51, . . . ,n.
5.1. Two classes of OOP games
A restricted houseswapping game with ordinal preferences (RHGOP) is a OOP game
15
where
1. PS[PS⇒PSuin1150;S,;i[S.
2. PS[PS⇒PSuin1151;S,;i[⁄ S.
3. Phi *j[Phi *j for i*51, . . . ,n, where Phi *j is the ordering for which Phi *jui *i *51 and
P 51 for all i±i*.
hi *juin11
We remark that (3) accounts for the initial ownership of objects (houses) in a RHGOP, a feature not present in status games.
A status game is a OOP game in which
1. rij$rik if j,k, for all i, j,k and rin1150 for all i. 2. P 5 PN .
3. Z[P ;S S.
We note two things. First, we do not need to make a ‘monotonicity’ assumption, because monotonicity here is implied by superadditivity combined with (3). Second, both exo-and endo-status games are included under this definition. In fact, we can define the class of exo-status games as those OOP games satisfying (1), (2), (3) and the following
M1) PS[PS⇒PS5Z or PSuin1150;i.
Similarly, an endo-status game is a OOP game satisfying (1), (2), (3) and the following
M2) PS[PS⇒Pin1151 if i[⁄ S.
Another class of OOP games are the ‘exchangeable’ games. Suppose in a OOP game
ˆ ˆ
P is an ordering and that P is any other ordering withoi Pij5oi P for all jij #n and
ˆ ˆ
oj :j#nPij5oj :j#nP for all i. If (for any S ) Pij [PS implies any such P is an element of
PS as well, we say the OOP game is exchangeable. In words, an exchangeable OOP
game is one where any S with the power to assign a certain set of objects to a certain set of players may redistribute those objects amongst those players in any way it likes.
15
5.2. The core of a OOP game
Then the core of G is nonempty.
Proof. The proof follows that of Theorem 4.5. So again let T be a balanced family, and
k u k
Remark. In the case of RHGOPs, the ‘weak balancedness’ hypothesis in Quint (1997)
b b
is precisely assumptions (1) and (2) in the special case where P is equal to P .N
In future work, we suggest that the class of OOP games can be regarded as a subset of the even broader class of general ordinal preference (GOP) games. GOP games are the class of games with finite outcome spaces. They merit distinction as a third major division of cooperative games, in addition to the transferable utility (TU) and nontransferable utility (NTU) games.
Acknowledgements
Quint was supported by a JFRA grant from the University of Nevada.
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Quint, T., 1997. Restricted houseswapping games. Journal of Mathematical Economics 27, 451–470. Quint, T., Shubik, M., 1999. Games of status. Technical report, University of Nevada and Yale University. Quinzii, M., 1984. Core and competitive equilibria with indivisibilities. International Journal of Game Theory
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