Game Theory and Public Policy
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Game Theory and Public Policy

Roger A. McCain

Game theory is useful in understanding collective human activity as the outcome of interactive decisions. In recent years it has become a more prominent aspect of research and applications in public policy disciplines such as economics, philosophy, management and political science, and in work within public policy itself. Here Roger McCain makes use of the analytical tools of game theory with the pragmatic purpose of identifying problems and exploring potential solutions in public policy.
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Chapter 13: Formal Aspects of Games in Partition Function Form

Roger A. McCain


This chapter will review some of the concepts of games in partition function form that have played a role in the previous chapters. The purpose of this chapter is to give a formal statement of some of the concepts and results that have been used. The reader who is not interested in the mathematical aspects should be able to skip this chapter without difficulty in following the argument in the remainder of the book. 13.1 FUNDAMENTALS Let N be an index set of agents in a game, aieN, i 5 1, . . ., n. A partition P is a set of subsets {Si} where Si 2 Sj 1 Si d Sj 5 [ and N 5 < Si. Si [ P Let PN be the set of all partitions of N and P [ PN. P 5 {C1, C2, . . ., Cr, [}. |Ci| will denote the number of members in Ci and |P | will denote the number of nonempty coalitions in P. A pair {P, Ci} with Ci [ P is called an embedded coalition. A coalition value function v(P, Ci) assigns a real number value to coalition Ci where Ci [ P. G 5 {N, v(P, Ci)} comprises a game in partition function form. For aj eN, CP(aj) 5 Ci[P ] aj e Ci. A game in partition function form is proper if v(P, S) 5 v(Q , S) 4P, Q, S ] P [ PN, Q [ PN, P ? Q , S , N, S [ P, S [ Q. Other games in partition function form are improper. For a proper...

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