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. Quint, M. Shubik Mathematical Social Sciences 41 2001 89 –102
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
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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
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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 N 5 h1, . . . ,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 n 5 4, the outcome in
which player 3 comes in ‘first place’, player 1 comes in ‘second place’, player 4 comes
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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.
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Note a direct quote.
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For one such early attempt, see Shubik 1972.
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For now, we use the ‘vector notation’ for the permutation. Later on, when we formalize the model, we will write this using a permutation matrix.
T . Quint, M. Shubik Mathematical Social Sciences 41 2001 89 –102
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2. On status games vs. matching games
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