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Mathematical Matters

Christopher Browne


Table of Contents
1. Mathematics and Computing

1. Mathematics and Computing

1.1. Matrix-oriented Math Software

1.3. Symbolic Math Software

1.5. Game Theory

This is the study of the strategy of conflicts; it was formalized in the analysis of strategies for nuclear war, but the analysis can be used to frame a wide scope of issues.

The common problems of Game Theory include The Prisoners' Dilemna and Zero sum games. Much contention arises over what sum a particular "game" comes to...

  • What is Game Theory?

  • The Game Theory Society

  • A History of Game Theory

  • Arrow's Theorem

    In 1948 and 1949 Kenneth J. Arrow conducted research on the theory of social choice (group decision making or voting). He investigated the possibility of a method of voting which would embody the following desirable features:

    • Universality.

      The voting method should provide a complete ranking of all alternatives from any set of individual preference ballots.

    • Monotonicity criterion.

      If one set of preference ballots preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed in such a way that the only alternative that has a higher ranking on any preference ballots is X, then the method should still rank X above Y.

    • Criterion of independence of irrelevant alternatives.

      If one set of preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed without changing the relative rank of X and Y, then the method should still rank X above Y.

    • Citizen Sovereignty.

      Every possible ranking of alternatives can be achieved from some set of individual preference ballots.

    • Non-dictatorship.

      There should not be one specific voter whose preference ballot is always adopted.

    Arrow concluded that it is not possible to have a voting method with all of these properties. The conditions are mutually contradictory.

    More simply put, there is no consistent method of making a fair choice among three or more candidates. This result assures us that there is no single election procedure that can always fairly decide the outcome of an election that involves more than two candidates or alternatives.

    The impact of this theorem is that people find themselves, whether from perspectives of moral obligation or reasonableness, wanting things that are downright impossible.

    In 1972 Arrow won the Nobel Prize in economics for his many contributions to that field, including this theorem.

  • Nash's Autobiography

  • Which Price is Right?

    Perhaps of somewhat nominal connection; pricing of products is a big, hairy, complex problem. There's a lot of " Prisoner's Dilemma" to it, and actually understanding why it is so troublesome likely requires some mathematical sophistication...

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Contact me at cbbrowne@acm.org