Brexiteers hope to convince us that Britain will, after exit, be able to retain its status as a global trading nation with special deals with its largest trading partners in the EU and elsewhere. Remain has painted a bleak picture of what will happen to the economy if Britain  exits. Applying some elementary game theory to post-Brexit negotiations suggests that things might be worse for an EU trade deal than even Remain's so-called "project fear" is arguing.

The OECD's recently published analysis of the economic consequences of Brexit concluded that by 2020, UK GDP will be 3.1% smaller than it would be with continued EU membership, and more than 5% smaller by 2030. [The Economic Consequences of Brexit: a Taxing Decision, April 2016]. Other analyses have arrived at similar conclusions. Such estimates have, unsurprisingly, been cited by Remain campaigners as a powerful argument against the UK leaving the EU. 

The OECD estimates that UK growth will be 0.5% lower in 2017 and 2018 following Brexit, and that a "trade shock" in 2019, the end of the two-year negotiation period, will reduce growth by a further 1.5% in that year.

The OECD assumes that the UK will not reach a new trade deal with the EU until 2023, implying that trade after 2018 will be conducted under WTO rules, which will raise costs for UK exporters. It also assumes that the UK will not sign any new free trade deals with non-EU countries before 2030.

Brexiteers contend that these estimates are based on a flawed account of the UK's bargaining position in negotiating (or renegotiating) trade deals post-Brexit.  They argue that the UK will be in a strong position to negotiate with the EU, as EU countries have as much to lose from a cessation or worsening of trade relations as the UK does.  As the world's fifth largest economy, they argue that countries would be queuing up to do deals with Britain if it left the bloc.

Brexiteers further contend that Britain could use its clout, as Europe's second-biggest economy, to get an even better deal than it currently has. They point to Britain's trade deficit with the rest of Europe to argue that EU countries need access to the British market more than Britain needs access to Europe. They also imply that if no deal with Europe is forthcoming, relying on WTO rules or having a free-trade deal like Canada's might be good enough.

The Economist points out that the Brexiteers' trade deficit argument may not make much sense, however. [Brexit brief: Unfavourable trade winds, The Economist, March 26 2016]. What probably matters more is the share of exports, and roughly 45% of British exports go to other EU countries, while only around 7% of their exports come to Britain. While it is no doubt true that German carmakers will wish to continue selling into the British market, several EU countries barely trade with the UK at all. A new trade deal will likely be of little interest them. 

Moreover, the WTO rules do not remove tariffs on all products, so exports of cars to the EU would attract a tariff of about 10%, and both the WTO and Canadian arrangements exclude financial services which make up Britain's biggest exports to the EU.

These details aside, the key area of contention remains the likelihood of reaching advantageous trade agreements quickly after Brexit. Evaluating the competing claims and counterclaims is not straightforward, as all are based on uncertain and untestable assumptions about the ease of concluding new trade deals. There is a well-established body of economic theory, however, which can be used to shed some light on this issue, as well as on the likely outcomes of any negotiations, at least in relatively simple cases.  Nash bargaining theory (named after the mathematician and Nobel prize winning economist who's remarkable life was portrayed in the movie A Beautiful Mind), is the only fully developed economic theory for analysing bargaining situations and identifying the key factors which determine their outcomes.

How can Nash bargaining theory be used to shed light on the post-Brexit UK/EU trade negotiations? A key factor in the negotiations is the fact that any trade deal with the EU requires the approval of all 27 other member countries (plus the European Parliament). This means that any member state, even if it has only negligible trading ties with the UK, can use its veto power to extort concessions before approving an agreement.

To see this in its simplest and starkest form, let us suppose that the post-Brexit EU consists of just two states, Germany and "Small". Also suppose, as the Brexiteers claim, that Germany places a great deal of importance on its trade with the UK, leaving the UK and Germany in symmetrical negotiating positions, or in positions of equal “bargaining power”. If trade between the two countries creates a total surplus (or what economists call "gains from trade") of 2X million Euros over the status quo of restricted or no trade, the Nash Bargaining Solution would allocate X to each side. That is, the gains from trade will be split equally between the two countries.

"Small" on the other hand, has almost no trade with the UK, which to keep things stark we may assume to have a value of 1Euro. One might presume that in this case "Small" would have no important role to play in the negotiations. But since "Small" has veto power over the deal which allows Britain to obtain X from trade with Germany, the Nash Bargaining Solution gives "Small" a payoff of (1+X)/2. That is, "Small" is able to “extort” half of X from the UK simply by wielding its veto power.

This is an extreme example based on the simplest imaginable scenario. Clearly countries don't directly bargain over the division of the gains from trade when negotiating trade agreements. But the point is that EU countries with little or nothing to gain from a trade deal with the UK, but with veto power, will have every incentive to use their veto power to scupper any deal that does not offer them sufficient benefits. This fact alone could make Brexiteers' hopes of reaching an advantageous trade deal quickly with the EU post Brexit a wild pipe dream.

The agreements bargainers will reach under the Nash Bargaining Solution typically depend upon a number of other factors, including the bargainers' payoffs in the event that an agreement is not reached (the “disagreement payoffs”) and their degree of Impatience, or how important it is to each side that a deal is reached sooner rather than later.

Both of these factors seem to weigh against the Brexiteers' arguments. Since it will be common knowledge that the UK's trading position will worsen after the initial two-year negotiation period is up, even large EU trading partners like Germany will likely be in no hurry to negotiate an new trade agreement. After two years, Germany will continue trading within the EU as before, but the UK will trade under WTO rules. This increases Germany's disagreement payoff relative to the UK's and results in Germany receiving a higher share of the gains from trade in any agreement. The two-year deadline also makes reaching an agreement more urgent for the UK than for other the EU countries, which once again reduces the UK's expected share of the gains from trade. This is because the less impatient bargainers can credibly threaten to delay any agreement until the more impatient bargainer makes further concessions.

Finally, the UK will need to establish new trade agreements with countries that it currently has access to via EU treaties, such as Korea, Mexico and South Africa. These countries too will likely be aware that strategically delaying agreement might be to their advantage, as the UK's bargaining position deteriorates over time. Brexiteers may be wildly overly optimistic about the prospects of reaching new and better trade deals around the world soon after Brexit.