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SUMMARY

Much of the literature of cooperative game theory relies strongly on simplifying assumptions that originate with von Neumann and Morgenstern. There are a number of properties one might like a solution to have: needless to say, no one solution will have all of them. One of the most common solution concepts is the core, which rests on the idea that no group can be denied the payoffs they could obtain if they formed a coalition and chose a joint strategy. The core may, however, be null or may have many potential solutions within it. Three solution concepts that are never null and are unique are the Nash bargaining solution, the Shapley value, and the nucleolus, though the bargaining solution is inapplicable to more than two agents. Applications of these models are largely in economics (and to some extent in political science) and include a theory of exchange, a theory of restrictions on competition, measurement of power and the allocation of shared costs.

NOTES

  • 1. This appears to be inconsistent with the previous sentence, as Telser (1978) notes in a similar context. We might say that the valuations are subjectively consistent in that each coalition is equally (and utterly) pessimistic about the decisions of those outside the coalition.
  • 2. Some of the literature would use the term preimputation at this point, and consider it as an imputation only if every agent obtains at least what he would get as a singleton. However, that distinction will not be made here.
  • 3. This may occur because the techniques of production involve division of labor (Smith, 1776/1994; Kaldor, 1934), so require a certain minimum workforce to be put into effect.
  • 4. This is demonstrated in the game theory literature by forming a set of axioms one of which is the property, and showing that these axioms are equivalent to the solution concept.
  • 5. These are technical terms from mathematical analysis and will not be discussed in detail here.
  • 6. This follows Forgo et al. (1999).
  • 7. This listing follows Peleg and Sudholter (2003).
  • 8. This listing follows Peleg and Sudholter (2003).
  • 9. Extension to divisible goods, drawing in the economic concept of a preference system, would demand a bit of mathematics.
  • 10. For this discussion, an allocation of available goods and services among individuals replaces an imputation of value to a coalition. The term “allocation” may replace “imputation” in some other applications, following the example of games of exchange.
  • 11. In some contributions this latter assumption is relaxed.
 
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