Games of Status and Wealth and Status Part II: A Game Theoretic Approach

In Part I we provide a heuristic discussion of the motivation for the investigation of games of status. Here we confine our remarks to several alternative formulations of games of status and to exploring the relationship between these games and the class of simple games, in part using the results from Quint and Shubik (1997). A Game of Status is an $n$-player cooperative game in which the outcomes are orderings of the players. Notationally, suppose the player set is $N = \{1,...,n\}$. 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 $j$th best position. For example, if $n=4$, the outcome in which player 3 comes in ``first place,'' player 1 comes in ``second place,'' player 4 comes in ``third place,'' and player 2 is ``last'' is represented by the permutation [3 1 4 2].$^1$ Let $r_{ij}$ denote the payoff that player $I$ obtains if he ends up in the $j$th position. We assume that $j \greaterthan k$ implies $r_{ij} \greaterthanor= r_{ik}$ for all $I$, i.e. players always desire to be placed as far ``up'' in the hierarchy as possible. Alternatively, a {\bf Game of Status with Ties} is a variant in which outcomes are allowed in which players ``tie'' for positions. For example, an outcome in the $n=4$ case could now be [3 1$T$4 2], which represents the situation in which player 3 again comes in ``first place'' and 2 again comes in ``last,'' but now players 1 and 4 come in tied for second place. In our current analysis, we consider only Games of Status without Ties, as we feel these should be eaiser to analyze. $^1$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.