Tuesday, 28 March 2017

Brexit and the Condorcet Paradox

Tomorrow the government will trigger Article 50 and start the formal process of getting the UK out of the EU. So, how did we get in this mess in the first place? I think the Condorcet Paradox provides an interesting angle on the problem. In particular, I want to look at preferences for Remain versus Soft Brexit, i.e. leave the EU but still remain in the single market or other collaborations centered on the EU, and Hard Brexit, i.e. walk completely away from the EU. 
          The one thing we know for sure is that in the referendum last June around 52% of people voted Leave and 48% voted Remain. What does that tell us? In my recollection the referendum campaign primarily focused on the question of Soft Brexit versus Remain. No doubt some would disagree with that. But things like the customs union only started being talked about after the vote. Instead we heard a lot during the campaign about the Norway or Swiss model of Soft Brexit. True the Leave camp made promises like 'take back control of our borders' that inevitably mean hard Brexit. But, the Leave camp was far less pro-active in actually joining the dots and saying what hard Brexit would mean. The referendum vote tells us, therefore, that the British people prefer Soft Brexit to Remain.
         Once Theresa May took power the discussion very quickly turned to focus on Soft Brexit versus Hard Brexit. Now, the Brexiters were keen to join the dots and argue that 'taking back control' inevitably meant Hard Brexit. Soft Brexit, they argue, is essentially Remain in different clothes - if we are going to leave then it has to be Hard Brexit. We have no idea how the country would vote on this issue but I think there is a fair chance the country would prefer Hard Brexit to Soft Brexit
        If the country prefers Hard Brexit to Soft Brexit and prefers Soft Brexit to Remain then you might expect they would prefer Hard Brexit to Remain. But, I would be surprised if that was the case. If the original referendum campaign had been a tussle between Hard Brexit and Remain then Remain may well have won. The vote was close enough as it was and opinion polls have consistently shown that people want to remain part of the single market. Overall, therefore, we end up with a Condorcet Paradox:

Hard Brexit beats Soft Brexit
Soft Brexit beats Remain
Remain beats Hard Brexit.

          If there is a Condorcet Paradox then it is impossible to say which option is the most preferred. Note, however, that we are set to end up with an outcome, namely Hard Brexit, that is worse than we started with, namely Remain. That does not look like a good deal! Plaudits should, however, go to the Brexiteers for their strategic opportunism. In particular, we know that whenever there is a Condorcet Paradox the outcome depends on the voting procedures. The only way those favoring Hard Brexit were going to get what they wanted was to play off Soft Brexit versus Remain and Hard Brexit versus Soft Brexit. By design or good fortune that is exactly what has happened. 
          And are we going to get a vote on the final deal, pitting Remain versus Hard Brexit? Of course not. And how long before the UK votes to rejoin the EU because Join is better than Hard Brexit? Probably, a long, long while. 

Monday, 20 March 2017

How to get rid of an incompetent manager?

In a paper, recently published in the International Journal of Game Theory, my wife and I analyze a game called a forced contribution threshold public good game. A nice way to illustrate the game is to look at the difficulties of getting rid of an incompetent manager.
         So, consider a department with n workers who all want to get rid of the manager. If they don't get rid of him then there payoff will be L. If they do get rid of him then there payoff will be H > L. But, how to get rid of him? He will only be removed if at least t or more of the workers complain to senior management. For instance, if a majority of staff need to complain then t = n/2.
        If t or more complain then the manager is removed and everyone is happy. The crucial thing, though, is what happens if less than t complain. In this case the manager will remain and any workers that did complain will face recrimination. To be specific suppose that the cost of recrimination is C. Then potential payoffs to a worker called Jack are as follows:

If t or more complain then Jack gets payoff H.
If Jack complains but not enough others do then he gets payoff L - C.
If Jack does not complain and others don't either then he gets payoff L.

Note that this game is called a 'forced' contribution game because, if the manager is removed, Jack's payoff does not depend on whether or not he complained. This contrasts with a standard threshold public good game in which those who do not contribute (i.e. complain) always have a relative advantage. Hence, there is a sense in which every worker is 'forced to contribute' if the manager is removed.
         The fear of recrimination is key to the game and going to be the potential source of inefficiency. In particular, if Jack fears that others will not complain then it is not in his interest to complain either. Hence we can obtain an inefficient equilibrium in which nobody complains and the manager carries on before. This is not good for the workers and presumably not good for the firm either. So, how can this outcome be avoided?
      In our paper we compare the predictions of three theoretical models and then report an experiment designed to test the respective predictions. Our results suggest that the workers will struggle to get rid of the manager if
This means that the threshold t should not be set too high. For instance, if a simple majority is needed to get rid of the manager, and so t = n/2, then we need H to about 25% higher than L. If less than a majority is enough then H does not need to be as high. This result would suggest that it is relatively simple to have a corporate policy that would incentivise people like Jack to complain about his manager.
           Of course, in practice there are almost certainly going to be some who will defend the manager and so things become more complex. Moreover, there are likely to be significant inertia effects. In particular, the 'better the devil you know' attitude may lead workers to underestimate the difference between H and L. Also senior managers may set t relatively high because of a desire to back managers. These are all things that will make it less likely Jack complains and more likely the incompetent manager continues. Firms, therefore, need to strike the right balance to weed out inefficiency.