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We present Colordag, a blockchain protocol where following the prescribed strategy is, with high probability, a best response as long as all miners have less than 1/2 of the mining power. We prove the correctness of Colordag even if there is an extremely powerful adversary who knows future actions of the scheduler: specifically, when agents will generate blocks and when messages will arrive. The state-of-the-art protocol, Fruitchain, is an ε-Nash equilibrium as long as all miners have less than 1/2 of the mining power. However, there is a simple deviation that guarantees that deviators are never worse off than they would be by following Fruitchain, and can sometimes do better. Thus, agents are motivated to deviate. Colordag implements a solution concept that we call ε-sure Nash equilibrium and does not suffer from this problem. Because it is an ε-sure Nash equilibrium, Colordag is an ε-Nash equilibrium and with probability 1-ε is a best response.

Traditionally, work in distributed computing has divided the agents into "good guys" and "bad guys". The good guys follow the protocol; the bad guys do everything in their power to make sure it does not work. By way of contrast, game theory has focused on "rational" agents, who try to maximize their utilities. Here I try to combine these viewpoints. Specifically, following the work of Abraham et al. [I. Abraham et al., 2006], I consider (k,t)-robust protocols/strategies, which tolerate coalitions of rational players of size up to k and up to t malicious players. I focus in particular on the problem that economists have called implementing a mediator. That is, can the players in the system, just talking among themselves (using what economists call "cheap talk") simulate the effects of the mediator (see, e.g., [I. Barany, 1992; E. Ben-Porath, 2003; Forges, 1990; D. Gerardi, 2004; Y. Heller, 2005; A. Urbano and J. E. Vila, 2002; A. Urbano and J. E. Vila, 2004]). In computer science, this essentially amounts to multiparty computation [O. Goldreich et al., 1987; A. Shamir et al., 1981; A. Yao, 1982]. Ideas from cryptography and distributed computing allow us to prove results on how many agents are required to implement a (k,t)-robust mediator just using cheap talk. These results subsume (and, in some cases, correct) results from the game theory literature. The results of Abraham et al. [I. Abraham et al., 2006] were proved for what are called synchronous systems in the distributed computing community; this is also the case for all the results in the economics literature cited above. In synchronous systems, communication proceeds in atomic rounds, and all messages sent during round r are received by round r + 1. But many systems in the real world are asynchronous. In an asynchronous setting, there are no rounds; messages sent by the players may take arbitrarily long to get to their recipients. Markets and the internet are best viewed as asynchronous. Blockchain implementations assume partial synchrony, where there is an upper bound on how long messages take to arrive. The partial synchronous setting already shows some of the difficulty of moving away from synchrony: An agent i can wait to take its action until it receives a message from j (on which its action can depend). This cannot happen in a synchronous setting. Abraham, Dolev, Geffner, abnd Halpern [I. Abraham et al., 2019] extend the results on implementing mediators to the asynchronous setting.