Decisions, decisions

Of all the more intractable problems I have encountered, Brexit is top of the list. Every time we think we have found a way forward, reality intervenes to cast us back to square one. It strikes me that one reason for this is that we are thinking about the problem in a binary way, which is inappropriate given the complexity of the issue. As it is conventionally presented in the media – and indeed in parliament – Brexit is a simple case of in or out. The prime minister’s ill-judged attack on MPs last week was a product of this kind of simplistic thinking. But it is wrong. There is a cost associated with each choice and the optimal strategy is to choose the one with the lowest costs. It is thus wrong to think simply of “in” or “out.” The real choice has always ever been between “in” and “what kind of out?”

Rather than trying to solve the problem by looking forward, we can use the method of backward induction which begins by looking at the end point and working back to determine the path necessary to get there. One of the great advantages of this approach is that it allows us to abstract from a lot of the political noise surrounding the current debate. Thus, to answer the question of how to leave the EU with an agreement that minimises economic costs to the UK, we can work out the sequence of events designed to get us to that point. By sequentially going through the outcomes, we end up gradually eliminating all the impossible options until only the possible ones remain. None of them accord with the plans put forward by the most fervent Brexit supporters.

But whilst this is an approach which allows us to look at desired outcomes, methods of voter choice help us to assess how we actually arrive at our choices, however unlikely they may be. Consider a system of single transferable votes in which 650 MPs face four options, A, B, C and D. Suppose they rank their preferences from 1 to 4. If no option commands a majority, the lowest-ranked first choice is eliminated from the ballot and the remainder are subject to a vote in the next round. The attached chart shows the sequence of how this might pan out.

In the first round, options A and D have an equally low number of first preferences but A is eliminated because it has a smaller number of second preferences. In the second round, D again scrapes through on the basis of having more second round votes than C and in the final round it ends up on top because more voters switched their allegiance to D than B, despite the fact that D was never a first choice winner in either of the first two rounds. Imagine now that option D is either a hard Brexit or revocation of Article 50. Although these are not plausible outcomes today, such a voting system demonstrates how they could end up as being the favoured choice depending what other choices are available.

An alternative voting system is the Condorcet method which attempts to force a decision by holding a series of one-on-one votes to determine whether there is one preference that comes out on top. In our example, we thus run six votes (A vs B); (A vs C); (A vs D); (B vs C); (B vs D) and (C vs D). If preferences are transitive (i.e. if A is preferred to B and B is preferred to C, then A must be preferred to C), it is possible to derive a winner. But if they are non-transitive it is not. Imagine, for example, the case where MPs are asked to choose between accepting the Withdrawal Agreement and revoking Article 50 and opt for the former. In a second vote, MPs prefer accepting the Withdrawal Agreement over a hard Brexit but in a third vote they express a preference for a hard Brexit over revoking Article 50. It is thus impossible to derive a series of ordinal preferences. This is known as the Condorcet paradox which we can liken to the game rock-paper-scissors, to which there is no obvious solution.

The work of Nobel Laureate Kenneth Arrow highlighted the problems involved in arriving at optimal choices. He gave his name to Arrow’s impossibility theorem which suggests that when there are three or more options, no ranked voting system can convert the ranked preferences of each individual into an overall ranking which meets a number of specific criteria. Perhaps the most important of these is that one person or group of people cannot be made better off without making others worse off (the Pareto criterion). This is an accurate description of where we are in the Brexit debate given the sentiments expressed on the streets of London at the weekend.

Continuing to put a series of votes to parliament, none of which commands a majority, can ever be guaranteed to find a resolution to the Brexit problem. The very fact that the electoral split in favour of leaving the EU was not much more than 50-50 ought to make us question why MPs can find a resolution when the electorate could not find a solution to the Brexit impossibility conundrum. The government’s approach has been to treat the outcome as a zero-sum game. But as the estimated one million people marching through London at the weekend highlighted, this approach is far from satisfactory. The only resolution to the problem is to buy more time: Kick the can down the road in the hope that society is able to agree on an acceptable compromise. Theresa May gained only an additional three weeks. It’s not enough!