In a referendum on 23 June 2016, the British public voted to begin a process of withdrawing from the European Union. Since World War II, no developed country has decided to leave such a major international institution without a military defeat or political revolution. The Brexit negotiations are therefore likely to bring great uncertainty to both the UK and the EU. This study examines the processes and issues involved in the Brexit negotiations, and explores the implications of possible outcomes for the UK, the European Union and the United States. It draws upon formal negotiating positions and mandates adopted, but with the expectation that a wider array of concerns will come into play as the process develops. The study uses game theory insights to explore and create a better understanding of how a wide variety of factors might affect the outcome of the negotiations. The use of modelling and game theory modes of analysis illuminates the broader issues at play, at some remove from the day-to-day drama of the UKs negotiations with the European Union. We also model the potential economic implications of various options for a post-Brexit relationship between the UK and EU. Specifically, the study examines the economic effect of five hard Brexit options an arrangement in which UK trade is governed by WTO rules, including the application of most-favored-nation tariffs the successful negotiation of a UKEU free trade agreement FTA the creation of a UKEUUS FTA based on the proposed Transatlantic Trade and Investment Partnership the creation of a USUK FTA excluding the EU and an extended transition period during which EU and UK tariffs do not change but other non-tariff barriers to trade come into effect. In addition, we examine three soft Brexit scenarios the Norwegian model membership of the European Economic Area, the Swiss model a series of bilateral agreementsand a customs union.