Wednesday, 19 October 2016

Rescuing Doreen and the Kitty Genovese case

A few days ago we heard the story of how a waitress rescued an 86 year old lady who been stuck in her bath for four days. The waitress contacted the police after becoming concerned that Doreen had not come in for her usual lunch and wine. A story with a happy ending.
         A story with a not so happy ending is the infamous murder of Kitty Genovese in 1964 in New York. This murder caught the public's attention because of the supposed number of witnesses who did nothing to stop the crime. The exact details of what happened are debated. One thing is, however, for certain: Several people must have seen or heard the attack and none of them called the police.
        To try and make sense of these conflicting stories let us look at simple game theoretic model. Suppose that there is someone called Doreen that needs rescuing and there are n witnesses who can call the police. If (at least) one person calls the police then Doreen is rescued and all the witnesses feel relief equal to B. Calling the police incurs a cost of c < B. Note that calling the police is a public good because everyone, not just the caller, benefits.

If n = 1 and so there is only one witness then it is a simple decision. The witness should call the police because the benefit of doing so exceeds the cost. So, Doreen is saved!

If n > 1 and so there are multiple witnesses, then things become trickier because of a free-rider problem. In short, each witness would prefer that another witness calls the police. That way they get benefit B without incurring the cost c. If we assume that all of the witnesses think alike then we need to look for a symmetric Nash equilibrium where each witness independently calls the police with some probability p. In equilibrium we require p to be at a level where all the witnesses are indifferent between calling or not calling the police. (To see why we need this indifference, suppose witnesses prefer calling. Then everyone will call. But this cannot be an equilibrium because we only need one person to call. Similarly, if witnesses prefer not calling then nobody calls. But this cannot be an equilibrium because each witness would want to rescue Doreen.) Let us look at the incentives of a typical witness called Sonia.

If Sonia calls then she is guaranteed payoff B - c because Doreen is rescued.

If Sonia does not call the police then she avoids cost c but relies on someone else calling. There are n - 1 other witnesses and so the probability that none of them call is (1 – p)n – 1 . This means the probability that at least one calls is 1 – (1 – p)n – 1 . The expected payoff from not calling is, therefore, B(1 – (1 – p)n – 1).

Equating the payoff from calling with the payoff from not calling we get an equilibrium probability of calling:




Unsurprisingly, this probability is decreasing in n. In other words the more witnesses there are then the lower the probability that any one witness will call. The following graph illustrates what happens when c/B = 0.1.

The really crucial question, though, is what happens to the overall probability of someone calling. What chance does Doreen have of being rescued? The probability that at least one person calls is:
This is also decreasing in n. So, the more witnesses there are the less likely it is that Doreen gets saved! The following graph plots the probability of her being saved when c/B = 0.1.

To many this seems like a counter-intuitive result. It shows, though, the dangers of free-riding. In terms of producing public goods, more is not necessarily better.

        

Saturday, 8 October 2016

I am your leader - wherever you want to go I will follow

In economics there are two diametrically opposed ways of viewing politicians. For the most part we assume the benevolent social planner who acts to maximize social welfare. But, when it comes to specifically analysing political decision making we typically assume that politicians are just like everyone else - out to maximize their own payoff. If a politician's objectives coincide with those of society then we have no problem. But, there are, of course, lots of reasons to suppose that political and societal objectives do not coincide.
          A particularly important issue is that of electoral survival. Clearly, a politician needs to get elected in order to make a living. That means it is in a politicians interest to do things that go down well with the electorate. At first sight you might think that this aligns the incentives of the politician with those of society because the politician needs to do good things to get elected. There is, however, a problem of asymmetric information. In short, voters may not know what is good for them. This is not to say that voters a dumb. It is merely to reflect that voters have busy lives and cannot be expected to be informed about everything. That is why we have experts to advise and politicians to make informed decisions.
          So, what happens if a gap emerges between what voters want and what is good for them? We can hope for the benevolent social planner who does what is best. More realistically, however, we might have to accept that politicians are going to do what is popular. This, unfortunately, seems to explain why the United Kingdom is slipping ever further into disaster/farce territory. 
         The big issue in the UK is that of immigration. Just about every report or bit of research on the topic has shown that immigration is good for the UK. Immigrants create jobs, pay taxes, provide vital skills; international students work hard for a better future etc. etc. The popular perception, however, is that immigration is bad. Indeed, immigration seemingly has to take the blame for just about all of society's ills. We have, therefore, a worrying gap between reality and popular perception. In such circumstances, we might hope for politicians who do what is right and defend immigration. Unfortunately, though, the UK seems to be veering even more towards popularity politics. The Conservative Party Conference last week, for instance, appeared to have a strong anti-immigration vibe.
          In her conference speech, the Prime Minister, Theresa May, attacked the liberal elite who don't get Brexit. Surely, however, this is to miss the point. That 52 percent of voters wanted to leave the EU is a signal that something is wrong. But what? Given that immigration is good for this country then we surely we need to find the actual problem? The actual problem, I would suggest, has more to do with underfunding of public services and a basic miss-match between what people want and what they can realistically expect (given that money does not grow on trees). 'Tackling the immigration problem' is just going to make things worse. In particular, you do not tackle inequality with policies that will ultimately make the poor poorer.
         A more nuanced view of things is, however, very hard to sell to the electorate. Moreover, Theresa May's popularity seems sky high at the moment and so who can blame her for playing popularity politics. From an economic perspective she is doing exactly what we would expect her to - maximizing her own payoff. More important is how she can use this popularity. The comparison with Margaret Thatcher is particularly interesting. Margaret Thatcher was quite clever in mixing popular politics (reclaiming the Falklands) with unpopular but sensible policies (taking on the unions). This allowed her to square the circle of winning elections and good policy. Let's hope for something similar again.