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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.

# Chapter 8: Superadditive Games in Coalition Function Form

## Roger A. McCain

Subjects: economics and finance, game theory, politics and public policy, public policy

## Extract

This chapter reviews some concepts from what might be called nearconsensus cooperative game theory. The objective of the chapter is primarily expositional. Apart from expression, examples, arrangement, and some critical comments, the chapter is not intended to be original. For this chapter, the game is primarily represented in coalition or characteristic function form. That is, the game comprises a set N of players, a1,. . .,an; ai [ N; the enumeration of all subsets of that set, the potential coalitions, and a mapping from subsets to real numbers, the characteristic or coalition function, which gives us the value attainable by each coalition. The value of a coalition C will be denoted as v(C) or v{a1, a2, . . .} where a1, a2, . . . are the members of coalition C. As we recall, von Neumann and Morgenstern identified this with the assurance value. The key point is that the value of a coalition is well defined and depends only on the membership of the coalition. We also adopt the assumption, from von Neumann and Morgenstern, that the game in coalition function form is superadditive; that is, that the value of a merged coalition is no less than the sum of the values of the merged coalitions acting separately. 8.1 SOLUTION CONCEPTS For a superadditive game in coalition function form, the only rational arrangement is the grand coalition. If the grand coalition is formed, nothing can be lost (since the grand coalition must have a value no less than those of any proper coalitions into which...

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