T . Quint, M. Shubik Mathematical Social Sciences 41 2001 89 –102
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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 n 2
uSu 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
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