ON DECISION THEORY UNDER COMPETITION,
CALIFORNIA UNIV LOS ANGELES WESTERN MANAGEMENT SCIENCE INST
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Games are studied in which a decision maker chooses an information function and a decision function. The information function generates a message based on the decision makers opponents action. The decision function generates a counteraction based upon the message. The payoff depends upon the action, the counteraction, and the message. Twenty zero-sum games are defined, differing in the decision makers freedom to probability mix his choices and in his opponents information about these choices. Relationships among these games are established which reduce the maximin and minimax theories of all of them to those of three games one in which the decision maker cannot mix choices at all, and both his choices are observed by his opponent one in which he can mix freely, and neither choice is observed and one in which he can mix only decision functions, and only his information function is observed. The first two of these games are reduced, respectively, to the minorant and the rectangular game over a certain matrix, with rows corresponding to different decision functions. The third game is interpreted as a minorant game in which the decision maker chooses an information function and his opponent decides what message that function shall generate, followed by a small rectangular game varying with the choices in the minorant game. These results are improved under the assumption that the payoff can be separated into two parts, one depending on the two players actions and the other a message cost.
- Operations Research