Recent advances in Internet and electronic commerce have resulted in an explosive growth of cryptographic protocols. These protocols use cryptographic primitives and provide recipes for the participants, achieving certain desired guarantees of security and trust. The problem we are interested in is two-party exchange of digital signatures on pre-agreed texts, which may be used for electronic exchange of goods and money. Inherent asymmetry of network communication puts one of the participants at a disadvantage. Some useful properties of such protocols are \it{fairness}, which means either both parties get each other's signatures or neither does, and \it{timeliness}, which means that a signer has some recourse to avoid unbounded waiting. A popular solution to achieve fairness is to introduce a trusted third party (TTP) as a protocol participant. We formally analyze trusted third party protocols using a game-theoretic approach, give formal definitions of useful properties and establish relationships amongst them. In the process of establishing relationships, we obtain a fundamental impossibility result: in a certain family of fair, timely signature exchange protocols there is a point at which one player can unilaterally decide the outcome of the protocol. Since such an advantage cannot be completely eliminated, we argue that the strongest property attainable in such protocols is the absence of \it{demonstrable} advantage, \ie, abuse-freeness in the sense of Garay-Jakobsson-MacKenzie. We formalize abuse-freeness using the notion of knowledge borrowed from epistemic logic. Various aspects of this work were carried out with Max Kanovich, John Mitchell, Andre Scedrov and Vitaly Shmatikov.