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We study the role of costly information in non-cooperative two-player gameswhen an extrinsic third party information broker is introduced asymmetrically,allowing one player to obtain information about the other player's action. Thisbroker or "oracle" is defined by a probability of response, supplying correctinformation randomly; the informed player can pay more for a higher probabilityof response. We determine the necessary and sufficient conditions for strategyprofiles to be equilibria, in terms of how both players change their strategiesin response to the existence of the oracle, as determined by its cost ofinformation function. For mixed strategy equilibria, there is a continuouschange as information becomes cheaper, with clear transitions occuring atcritical {\it nodes} at which pure strategies become dominated (orundominated). These nodes separate distinct responses to the information forsale, alternating between regions where the paying player increases the amountof information purchased, and regions where the other player moves away fromriskier strategies, in favor of safer bets that minimize losses. We deriveconditions for these responses by defining a value of information.

Simultaneous games with purchase of randomly supplied perfect information: Oracle Games