Quantum cryptography uses quantum mechanics to guarantee secure communication. BB84 is a widely used quantum key distribution that provides a way for two parties, a sender, Alice, and a receiver, Bob, to share an unconditionally secure key in the presence of an eavesdropper, Eve. Three di®erent criteria can be assumed to study the BB84 protocol. They are the e±ciency of the protocol, the probability that Eve remains undetected, and the amount of knowledge Eve has about Alice\\\'s bit sequence. In a previous approach, we only considered the probability that Eve remains undetected. We viewed this protocol as a three player static game in which Alice and Bob were two cooperative players and Eve was a competitive one. In our game model, Alice\\\'s and Bob\\\'s objective was to maximize the probability of detecting Eve, while Eve\\\'s objective was to minimize this probability. In this paper, our previous e®ort is extended and we also consider the other two criteria, i.e. the e±ciency of the protocol and the amount of knowledge Eve has about Alice\\\'s bit sequence. Using these models, we show how game theory can be used to ¯nd the strategies for Alice, Bob and Eve.

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