Abstract
In Chapter 4 we introduced a decision tree. At each node of the tree, either the actor or chance made a choice. The end point of each path along the branches of the tree represented an outcome characterized by a payoff to the actor.
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Notes
A 2 x 2 game has no saddle point if and only if both entries in one of the diagonals are larger than either entry in the other diagonal. To show `only if’, suppose a minimal violation of the condition, e.g., a. b, a. c, d. b, d . c in Game 9.3. Since d is minimal in its row (d . c) and maximal in its column (d . b), d is a saddle point. Similarly all other violations of the condition can be shown to lead to the existence of a saddle point. Conversely, if the condition is satisfied, it is easy to see that no entry can be simultaneously minimal in its row and maximal in its column.
A convex mixture of two numbers, a and b,is pa+(1— p)b,where 0 . p 5 1. It follows that the convex mixture is always in the interval between the two numbers, including the extremities.
What was proved was the existence of equilibria in all such games (cf. Nash, 1951 ). Whether all such equilibria necessarily represent outcomes of rational strategy choices of both players is another question, which we will examine in Chapter 11.
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© 1989 Springer Science+Business Media Dordrecht
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Rapoport, A. (1989). Two-Person Constant Sum Games. In: Decision Theory and Decision Behaviour. Theory and Decision Library, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7840-0_10
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DOI: https://doi.org/10.1007/978-94-015-7840-0_10
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