  Wireless secret sharing is crucial to information security in the era of
Internet of Things. One method is to utilize the effect of the randomness of
the wireless channel in the data link layer to generate the common secret
between two legitimate users Alice and Bob. This paper studies this secret
sharing mechanism from the perspective of game theory. In particular, we
formulate a non-cooperative zero-sum game between the legitimate users and an
eavesdropper Eve. In a symmetrical game where Eve has the same probability of
successfully receiving a packet from Alice and Bob when the transmission
distance is the same, we show that both pure and mixed strategy Nash equilibria
exist. In an asymmetric game where Eve has different probabilities of
successfully receiving a packet from Alice and Bob, a pure strategy may not
exist; in this case, we show how a mixed strategy Nash equilibrium can be
found.
