Friday, 11 December 2015

What is the difference between a Nash equilibrium and evolutionary stable strategy?

Nash equilibrium is the main game theoretic concept used in economics. Evolutionary stable strategy, or ESS, is the main game theoretic concept using in biology. The issue I want to look at here is the relationship between the two. This is an issue that appears to cause much confusion amongst students (and some academics).
         The main thing that I would suggest needs to be recognised is that Nash equilibrium and ESS are fundamentally different. They were conceived completely independently with very contrasting objectives in mind. The Nash equilibrium was developed by John Nash in the 1940/50s as an equilibrium concept for non-cooperative games. And note that the genius of Nash was not so much to develop the Nash equilibrium but to recognise the importance of non-cooperative games. The ESS, by contrast, was developed by John Maynard Smith and George Price in the 1970s as an equilibrium concept for evolutionary games. Again, the genius was not so much the ESS but the recognition that evolutionary forces can be modelled using game theory.
        Even though Nash equilibrium and ESS were designed with contrasting objectives in mind they end up looking similar in certain respects. It is here that the confusion starts. So, let us define each concept in turn and then make sense of the similarities.
         The basic idea with a non-cooperative game is that we have set of players who each, independently, choose a strategy. The payoff of each player is then determined by the strategies that each player has chosen. To illustrate consider the hawk-dove game depicted below. In this game, Gary and Dawn have to simultaneously choose whether to play hawk or dove. The numbers give the respective payoffs to all combinations of outcome. For instance, if Gary chooses Dove and Dawn chooses Hawk then Gary gets 0 and Dawn gets 4.

          A Nash equilibrium is a list of strategies for each player such that no player has any incentive to change their strategy (given the strategies of others). In the hawk-dove game above there are three Nash equilibria. (1) If Gary plays Dove then Dawn does best to play Hawk. Similarly, if Dawn plays Hawk then Gary does best to play Dove. So, Gary plays Dove and Dawn plays Hawk is a Nash equilibrium. (2) By a symmetric argument, Gary plays Hawk and Dawn plays Dove is a Nash equilibrium. (3) Suppose that Dawn tosses a coin to decide what to do. If the coin comes down heads she plays Hawk, if tails she plays Dove. Then the expected payoff of Gary if he plays Hawk is (0.5)(-2) + (0.5)(4) = 1. His expected payoff if he plays Dove is (0.5)(0) + (0.5)(2) = 1. So, Gary is indifferent what he chooses. By a similar logic, Dawn is indifferent if Gary tosses a coin to decide what to do. This means there is a Nash equilibrium where Gary and Dawn independently toss a coin and randomize between Hawk and Dove.
          It is worth noting that the first two Nash equilibrium we looked at are asymmetric - Gary chooses a different strategy to Dawn. The third equilibria is symmetric - both Gary and Dawn choose the same strategy, namely to toss a coin and choose Hawk if and only if their coin comes down heads.
         Let us look now at an evolutionary game. The most common formulation (there are many alternatives) goes something like this: Over time, individuals from a large population randomly meet each other and play a game. For instance, individuals might meet in pairs and play the hawk-dove game. Gary and Dawn are now just two of the many people we are interested in. Keeping track of everyone's strategy would be burdensome and so we focus on a population strategy. You can think of the population strategy as detailing the probability that a randomly chosen member of the population will choose Hawk, and the probability they play Dove.
           An evolutionary stable strategy is a population strategy that is immune to invasion. This means that if any small subset of the population deviate from the ESS then they will get a lower payoff than those who did not deviate. To illustrate, consider the population strategy in which everyone plays Dove and gets payoff 2. A mutant who plays Hawk would get a payoff of  4, because he only ever plays against individuals choosing Dove. This beats the population strategy and so everyone play Dove is not an ESS. In fact, the only ESS in the hawk-dove game, given above, is the population strategy in which Hawk is played with probability one half and Dove with probability one half.
           You might have noticed that this ESS is 'similar' to the third Nash equilibrium. Let us think what that means. In the evolutionary game we have a population strategy - Hawk is played with probability one half and Dove with probability one half. In the two player game between Gary and Dawn we have a symmetric Nash equilibrium - Gary and Dawn independently choose Hawk with probability one half and Dove with one half. More generally, if a strategy, call it X, is an ESS of an evolutionary game, where players are matched to play some game G, then the strategy profile where both players play X is a symmetric Nash equilibrium of game G. So, there is a sense in which ESS 'must be' a Nash equilibrium.
          Note that the reciprocal is not true. In the hawk-dove game there are three Nash equilibria and only one of those maps into an ESS. The basic issue here is one of symmetry. A population strategy has to be symmetric, by definition, because it treats all individuals in the population equally. For instance, if some play Hawk and some Dove then we cannot rule out the possibility that two individuals will meet who both play Hawk or both play Dove. The Nash equilibrium where Gary plays Hawk and Dawn plays Dove cannot, therefore, be equated with a population strategy.
           Textbooks and journals are full of results connecting ESS and Nash equilibrium. And from a theoretical point of view it is clearly of interest to study the relationship between the two. On a practical level, though, it is not really clear to me that we can learn much from knowing the two are related. This doubt comes from the fundamentally differences between a non-cooperative and evolutionary game. In the evolutionary hawk-dove game, for instance, forces will push towards the ESS. So, the ESS is the natural thing to study. But in the non-cooperative game between Gary and Dawn the equilibrium where both toss a coin to decide what to does not seem very appealing. It gives low payoffs. And it is hard to imagine this is how Gary and Dawn would think about the game. The two asymmetric equilibria seem more likely to be most relevant.
           My claim, therefore, is that we get too caught up in the similarities between ESS and Nash equilibrium. They are different concepts and there is no harm in thinking of them as such.  

Saturday, 28 November 2015

Why the rationale for nuclear weapons requires a little madness

The UK will soon have to decide whether to maintain its Trident nuclear weapon programme. Clearly, the nuclear capability will be maintained. This has not, though, stopped a fairly vociferous debate on the issue. The basic argument in favour of nuclear weapons, and one that we have heard time and time again in the debate, is that nuclear weapons are to deter attack and not be used. This is encapsulated in the concept of mutual assured destruction or MAD. But, just how solid is the MAD argument?
           A standard logic goes something like this: If the UK has nuclear weapons then Russia would not attack the UK because the UK would have the capability to destroy Russia. Thomas Schelling, in the Strategy of Conflict, pointed out that there is a basic flaw in this logic. To see why let us set out a hypothetical game tree, see below. Russia moves first by deciding whether to attack the UK. Then the UK decides whether to retaliate in the case of attack. We can see that if the UK will retaliate then Russia does best to not attack (and get payoff 100) than attack (and get payoff 0). Not attack, and retaliate if attacked, is indeed a Nash equilibrium of this game.

          But, here is the problem: the threat to retaliate is not credible. If Russia attacks then the UK is going to be destroyed. Nothing can stop this from happening once Russia has pressed the go button. Retaliation serves, therefore, purely as an act of revenge. And one that will kill millions of innocent people. Would the UK prime-minister press the retaliation button if the sole consequence of doing so would be the death of millions of innocent people? Probably not. Which is why not retaliate earns a higher payoff (you at least die with the moral high ground) than retaliate (where you die with the guilt of killing millions). Thus, the sub-game perfect Nash equilibrium in this game is for Russia to attack and the UK to not retaliate.
          So, how can nuclear weapons work as a deterrent? One possibility is to put an automatic trigger in the system so that the UK has no choice but to retaliate in the case of attack. This though is not compatible with the basic notion that a human being should ultimately be responsible for such an act. Another possibility is to have a prime-minister who would be 'mad enough' to exact revenge. To illustrate, suppose the payoffs are given as below. Notice that the UK now does best to retaliate. Indeed, the unique Nash equilibrium now sees Russia not attack because the threat of retaliation is credible.  

            As previously discussed, it is hard to imagine that the payoffs really would be like this. The prime-minister must be extremely vengeful if he would kill millions of innocent people. Crucially, though, the mere possibility that the UK might be willing to exact revenge can be enough to deter attack. To explain, let p denote the probability that the UK has a 'mad' prime-minister who would retaliate. If Russia does not attack then it gets 100 for sure. If it attacks then it gets 0 with probability p and 200 with probability 1 - p. If p > 0.5 then, unless Russia is a risk seeker, it is safer to not attack. A 50% chance the UK prime-minister is mad seems, however, unlikely.
            But, the payoffs we have been using are completely arbitrary. Suppose we let W denote the payoff to Russia if it attacks and the UK does not retaliate. (So far we have assumed that W = 200.) Now Russia will find it safer to not attack if 100 > W(1 - p). If W is close to 100, say 105, then a small chance of madness, say p = 0.05, is enough to deter an attack. The success of the nuclear deterrent depends, therefore, on two crucial things: (i) that the gain from 'winning' is relatively small, and (ii) there is some chance the UK will be mad enough to retaliate.
           I would suggest that point (i) was actually the main reason the Cold War ended peacefully. Russia, the US, UK, France etc. had little to gain from destroying the other. We should not, however, neglect point (ii). Critics of nuclear weapons often claim that they are of no use because the UK would simply not be attacked in the first place, nuclear weapons or not. This view seems na├»ve: history tells us that wars are a sad reality of life. Nuclear weapons are important in deterring conflict, provided you let others think that you might be mad enough to use them.    

Friday, 30 October 2015

Why the flu vaccine illustrates all that is wrong with the NHS

The UK's National Health Service is nudging ever closer to collapse. This fact is blatantly obvious to many. The political will to do anything about it is, however, sadly lacking. It is particularly disappointing that the Conservatives, in a position of strength, seem more interested in tackling the immigration 'problem' that isn't a problem, than getting to grips with the huge and pressing problem that is the health of the nation.
         As I have discussed before in this blog the NHS principle of free health care is simply unworkable in the modern world. That inevitably means some people are going to have to pay for treatment. This is already happening with the slow growth of the private system. Things would be much, much better, however, if the NHS would embrace the willingness of many to pay for better treatment. The flu vaccine provides a small but useful illustration of this issue.
          The flu vaccine is available free of charge on the NHS for young children and people aged over 65. What about the rest of us? More and more people want the vaccine and so a market has emerged to satisfy that demand. Indeed, the jab is now available at most major supermarkets for around £9-12 a shot. This means that many people are paying for a little bit of health care.  
          I don't think anyone sees anything wrong with the fact that some people get the jab for free and some don't. But, here is the crucial point: the NHS is denied the opportunity of making any money on the willingness of people to pay for this service. In our family, for instance, the kids go off to the local GP surgery to get their jabs while my wife and I go off to the local supermarket and pay £9. I, for one, would rather we just pay the NHS £18 and all get the jab at the same time.
        Clearly there is not a great deal of money to be made in flu jabs (although I doubt the supermarkets are doing it for the good of humanity). If the NHS did 1 million jabs at a profit of £1 at time then they still only make £1 million. This is not going to save an NHS short of billions of pounds. It is still, though, extra money that the NHS could make. If there was a willingness to sell other services then we might find the billions that are needed.
          The most common criticism of the NHS charging for some services is that it would create a two tier system. But, what is the problem with a two tier system? The beauty of a two tier system is that it can benefit both rich and poor. The rich gain because they can use their wealth to purchase a better quality service. The poor gain because the extra money coming into the NHS can improve services. Sure, there will be inequality. Everyone, though, gains. 
        As an example, consider waiting times to see a doctor. In the current climate a patient can consider themselves lucky if they get seen within an hour of the allotted time. Some people would be willing to pay to reduce that waiting time. Clearly, an option of fee for timely appointments would benefit the person who gets the 'better' service. The money that person spends can, however, be reinvested into the system to provide a better service to others.
      Indeed, those in the second tier may actually benefit most because they get the improvement for free! For instance, suppose it costs £100 for a timely appointment. Then the rich person gets seen on time but has to pay £100. Suppose that the extra revenue in the system reduces standard waiting times to 20 minutes. Then poor people get a better service and pay nothing for it. Everyone is a winner.
          So, rather than 'accept' that fee for service is 'necessary' for the NHS to survive, why not start to embrace it as something that can reinvigorate health care in the UK for everyone

Sunday, 4 October 2015

Tragedy of the commons and population growth

Garrett Hardin's 1968 Science article on the Tragedy of the Commons can easily lay claim to be one of the most cited 'game theory' articles of all time. According to Google Scholar it has a mighty impressive 27,362 citations, and counting. But, as with all well cited articles, it is not entirely clear how many people have actually read the paper. As someone who has neither cited nor read the paper I thought it was about time I educated myself. And, I was surprised by what I found.
        Let us start with the modern, textbook conception of the tragedy of the commons. The focus is on common resource goods. These goods are characterised by being non-excludable and (to some extent) rivalrous. Hardin, himself, gave the example of pastureland that can be used for cattle. The pastureland is non-excludable - everyone is free to graze their cattle - and rivalrous - the more cattle that graze the less grass available for others. Another example is fishing in the Atlantic Ocean. The Ocean is non-excludable - you cannot stop people fishing - and rivalrous - the more fish are caught the lower are fish-stocks.
        One person's consumption or use of a common resource good creates a negative externality for all other users. This means that common resource goods are likely to be over-used. The cattle grazer, for instance, has no incentive to take into account the harm it will do others if he grazes his cattle. Over-fishing, climate change, depletion of underground reservoirs, long queues at hospital accident and emergency departments, are all commonly cited as examples of the tragedy of the commons. A typical response to the tragedy of the commons is to advocate policy intervention. Elinor Ostrom (and others), however, suggested that groups can often avoid the tragedy, especially when communication between potential users of a common resource is possible. 
        Hardin's article does cover this, now, textbook account of the tragedy of the commons. But, it is only a very small part of his article. The big focus is instead on the problems of overpopulation. Hardin proposes that 'Freedom to breed is intolerable'. In particular he argues:
If each human family were dependent only on its own resources; if the children of improvident parents starved to death; if, thus, overbreeding brought its own "punishment" to the germ line--then there would be no public interest in controlling the breeding of families. But our society is deeply committed to the welfare state, and hence is confronted with another aspect of the tragedy of the commons.
          In a welfare state, how shall we deal with the family, the religion, the race, or the class (or indeed any distinguishable and cohesive group) that adopts overbreeding as a policy to secure its own aggrandizement? To couple the concept of freedom to breed with the belief that everyone born has an equal right to the commons is to lock the world into a tragic course of action.
This reasoning leads to a strong conclusion:
The only way we can preserve and nurture other and more precious freedoms is by relinquishing the freedom to breed, and that very soon. "Freedom is the recognition of necessity"--and it is the role of education to reveal to all the necessity of abandoning the freedom to breed. Only so, can we put an end to this aspect of the tragedy of the commons.
This is pretty extreme stuff! Little surprise, therefore, that modern accounts of the tragedy of the commons completely drop all mention of population. See, for instance, Elinor Ostrom's entry in the New Palgrave Dictionary of Economics.
        But is there any sense in Hardin's argument? In short, no. To elaborate further, it's worth clarifying that Hardin's argument is very different to the well known one of Thomas Malthus. Malthus argued that the law of diminishing returns means planet Earth cannot sustain an arbitrarily large population. This logic is sound. Malthus merely underestimated the huge advances in productivity that have been possible. Hardin's arguments is more one that the poor will 'overbreed' because they can exploit the benevolence of the rich.
         Now it is true that a welfare state creates moral hazard. A poor couple might, for instance, be more likely to have a child if they know there is a free health and education system waiting for their child. This, however, does not generate a tragedy of the commons. First, despite what the Daily Mail may have us believe, the standard of living for those who rely on the welfare state is not high. Few poor people, therefore, desire to rely on the benevolence of the rich. Second, the benevolence of the rich is not a resource that can be used freely. If the rich feel they are being exploited then they have the power to reduce the size of the welfare state. In other words, the benevolence of the rich is an excludable resource.
          This all makes it difficult to fix Hardin's contribution. Mancur Olson's 1965 book, The Logic of Collective Action, for example, provides a far more convincing and rigorous analysis of public goods, including common resource goods. Thankfully, this is just about reflected in the citation counts with Olson's book claiming 31,223 citations.

Friday, 25 September 2015

Why charities need stricter rules on fundraising

A government commissioned review is currently looking into the way UK charities raise money. Reports this week suggest it will be tough. And it should be! The review follows the high profile case of 92 year old Olive Cooke who took her own life having become exhausted by requests for money from charities. Such requests for money are all too familiar to many. In our household I would say that we average a charity letter per day, a phone call per week, and a fundraiser at the door every month. The Prime Minister was stating the obvious when he said that the behaviour of some charities was 'frankly unacceptable'.
          Against this public backlash the charity sector has stood firm. Time and again I have heard spokespeople arguing that charities only do good. The argument essentially seems to be that the funds they raise are spent on worthy causes and so the ends justify the means. The charity sector, therefore, can claim the moral high ground. This argument, however, is flawed.
          Charities provide public goods. It is textbook economics that public goods will be under-provided. This means that voluntary donations to charity will not be enough to provide an efficient amount of the public good - children will go hungry, old people will be lonely, and so on. In order to obtain an efficient amount a public good some form of coercion seems necessary. That's why we don't have any choice to pay taxes. And this is why charities could argue that the ends justify the means. Even if their fundraising tactics do include a heavy dose of coercion, so what - society is better off as a result.
          The flaw in this argument is one of asymmetry. We all have to pay taxes. Not all of us are targeted by charities. We know from lab experiments that approximately 50% of people can be classified as free-riders and 50% as conditional co-operators. Free-riders will not voluntarily give to charity. Conditional co-operators will. Who do charities go after? It is, of course, the conditional co-operators. The more a person gives to charity the more they are asked to give. So, conditional co-operators are coerced into giving more while free-riders get off scot-free. Where is the moral high ground in this? It seems neither fair nor just.
           It is also likely to backfire. That's because of the conditional in conditional co-operator. A conditional co-operator will give only if they feel they are not being exploited. And the recent behaviour of charities may well make them feel as though they are being exploited. Charities could well, therefore, see cracks appearing in their core support.
          The charity sector has over-stepped the mark. It would be nice to think that they could fix the problem themselves. If they do start to see cracks in their core support then they may. More realistically, however, we are going to need legislation. Otherwise we will end up with even more children going hungry and old people lonely.

Sunday, 23 August 2015

Why do centre right parties win elections?

The labour party is seemingly about to appoint a leader, Jeremy Corbyn, who almost everyone considers unelectable as prime minister. The apparent 'problem' with Mr Corbyn is that he is too left wing. But, according to simple political choice theory this should be an asset rather than a problem. So, where is the catch?
      Let us look first at the basic theory. To win an election a candidate (or party) needs majority support. Now, we all know that wealth is highly asymmetrically distributed - the top 1% own most of our wealth, the top 10% own even more, and so on. The flip side of this asymmetry is that the poorest 50% are a relatively homogeneous bunch that should, in principle, easily be able to gang up on the rich. To be a more specific, they could vote for redistribution of wealth from the rich to the poor and the rich would be powerless to do anything about it.
       In some countries the theory seems to work pretty well. Both Vladimir Putin, in Russia, and Cristina Fernandez de Kirchner, in Argentina, have, for example, done a pretty good job of ruining their respective economies - inflation is rampant, growth low, international relations shattered, markets far from open, and so on. Yet both Putin and Fernandez de Kirchner are hugely popular. There popularity comes from keeping enough poor people, like pensioners, happy.
        Overall, though, the theory that left is best seems woefully wide of the mark. Centre right parties dominate in Europe and most Western countries. (And the Democrats in the US are surely centre right relative to the norm elsewhere.) So, where does the theory go wrong?
      We can all agree that being poor is undesirable. But what is the fix for that? One option is to redistribute from rich to the poor and essentially equalise things by making the rich poorer. Another option is to give opportunities for the poor to become rich. In most countries the second option seems more appealing. So, the fact that 50% of people are poor does not stand for much because those same people aspire (or aspired) to be rich. They want a party that will back them in their quest to move up in the world, not one that will give them more cash now. 
        Social mobility, therefore, makes it as if 50% of the population are rich because at least 50% of the population are voting as if they were rich. This turns the tables in favour of centre right parties. But, only if enough people believe in social mobility. In Russia and Argentina they presumably don't, and who came blame them. In the UK there is a much stronger belief that anyone can succeed if they try hard enough. Which is why the Labour party would be shouting themselves in the foot big-time by appointing Jeremy Corbyn.   

Saturday, 25 July 2015

Nash equilibria in the ultimatum game

The ultimatum game is one of the most well known and well studied games. Yet there still seems much confusion and misunderstanding over the basic theory behind the game. This is worrying because it means one of the key lessons we can learn from the ultimatum game goes unrecognised.
       In the ultimatum game a proposer and responder need to decide how to split, say, $10. The proposer moves first by making a take-it-or-leave-it-offer. The responder then either accepts the offer or rejects it. If he accepts it the deal is done. If he rejects both get nothing. For example, the proposer could offer $3. If the offer is accepted the proposer gets $7 and the receiver $3. If it is rejected both get $0. 
      What is going to happen? In the experimental lab the almost universal outcome is that the proposer offers $5 and this is accepted. We also know, although the evidence for this is less unequivocal, that offers of less than $5 can be rejected.  
       What does theory say should happen? Here is the standard version you can find in the textbooks: The responder should accept any positive offer - because something is better than nothing. The proposer should, therefore, offer $0.01.
        The $0.01 'prediction' is clearly well wide of the mark. That leads many to the basic claim that 'game theory is wrong!'.
      Things, however, are far more complicated than that (even if we ignore social preferences). The standard prediction corresponds to the sub-game perfect Nash equilibrium of the game. There are though many other Nash equilibria. Indeed any offer is consistent with Nash equilibrium.
      To illustrate, suppose the responder will reject any offer less than, say, $5 and the proposer knows this. Then the proposer should offer $5. In other words, 'offer $5 and reject any offer less than $5' is a Nash equilibrium. This equilibrium is not sub-game perfect because it involves an incredible threat from the responder that he will reject an offer below $5. There is, though, no necessity that threats be credible.
       The behaviour we observe in the ultimatum game is, therefore, consistent with Nash equilibrium. Game theory is not wrong! We do need to question the relevance of sub-game perfection. But, sub-game perfection is only one of many equilibrium refinements that have been suggested by game theorists over the years. Sub-game perfection seems ill-suited to the ultimatum game. More suitable seems a refinement based on fairness principles or focal points (as advocated by Thomas Schelling in the The Strategy of Conflict way back in 1960).
       The ultimatum game is not the only challenge to sub-game perfection. Somewhat worryingly, however, sub-game perfection is still far and away the most common refinement concept used in economics. For instance, virtually every theoretical paper in industrial organization solves for the sub-game perfect Nash equilibrium. How reliable are the predictions from such an approach likely to be? Probably not very reliable at all.
          The way that game theory is applied may, therefore, need something of an overhaul. We certainly need to give consideration to Nash equilibria that are not sub-game perfect.

Monday, 22 June 2015

Nash bargaining solution

Following the tragic death of John Nash in May I thought it would be good to explain some of his main contributions to game theory. Where better to start than the Nash bargaining solution. This is surely one of the most beautiful results in game theory and was completely unprecedented. All the more remarkable that Nash came up with the idea at the start of his graduate studies!
         The Nash solution is a 'solution' to a two-person bargaining problem. To illustrate, suppose we have Adam and Beth bargaining over how to split some surplus. If they fail to reach agreement they get payoffs €a and €b respectively. The pair (a, b) is called the disagreement point. If they agree then they can achieve any pair of payoffs within some set F of feasible payoff points. I'll give some examples later. For the problem to be interesting we need there to be some point (A, B) in F such that A > a and B > b. In other words Adam and Beth should be able to gain from agreeing.
          The solution is a pair of payoffs (A*, B*) that Adam and Beth should agree on. Nash gave a list of axioms he suggested that a solution should satisfy. Informally, these are: 
(1) Pareto efficiency: There must be no feasible point that would make both Adam and Beth better off.
(2) Individual rationality: Adam and Beth must do at least as well as the disagreement point.
(3) Scale invariance: If we do a linear transformation of payoffs, e.g. converting Adam's payoff from euros to dollars, then the solution merely needs to be transformed as well. In other words, the solution is 'the same' whether we use euros, dollars, roubles or anything else. 
(4) Symmetry: If Adam and Beth have symmetrical bargaining positions they should get the same payoff.
(5) Independence of irrelevant alternatives: If we eliminate some feasible points that were not a solution then the solution should stay the same.
Nash showed that there is a unique solution that satisfies all these five axioms. The solution is to find the pair A* and B* that maximize the Nash product (A - a)(B - b). This is a remarkably succinct result. It is also easy to apply. A beautiful theorem! To understand the result better we need to delve into the proof.
            Consider, first, a very simple example. Suppose that Adam and Beth are bargaining over how to split €100. If they disagree they each get €0. The feasible set is the blue triangle in the figure below. What is the solution? The Pareto efficiency axiom requires A* + B* = 100. The symmetry axiom requires A* = B*. Only one point satisfies both these axioms and that is A* = B* = €50. To see if this is the Nash product we need to maximize AB. Plugging in B = 100 - A we get A(100 - A). Maximizing this we respect to A does indeed give A* = 50 as desired.

          Given the symmetry of the first example there is nothing particularly controversial or exciting about the solution that Adam and Beth should split things equally. So, let us consider a slightly more complex example. Suppose that there are 100 tokens up for grabs. Any token given to Beth is worth €1 and a token given to Adam is worth €2. Also, suppose that if they disagree Beth gets €10 and Adam gets €30. The figure below depicts this scenario.

         Things are now asymmetric. What shall we do? We can apply the scale invariance axiom to convert the bargaining problem into one that is symmetric. It is necessary to solve some simple equations to do so. I'll illustrate with one possibility. Suppose for Beth, given any amount €B, we take off 10 and then multiply by 4/3. For Adam we take off 30 and then multiple by 2/3. Do this and we obtain the bargaining problem depicted below. This problem is symmetric and very similar to the first example. (The only difference is that negative payoffs are now possible meaning that we need to apply individual rationality).

            This new bargaining problem has a unique point satisfying Pareto efficiency and symmetry. Namely, €50 each. This means that to satisfy Pareto efficiency, symmetry and scale invariance the original problem must also have a unique solution. Moreover, we just need to reverse the scaling to find the answer. Doing so we get €105 for Adam and €47.50 for Beth. You can check that this maximizes the Nash product as desired.
           Consider a final example depicted below. This is similar to the first example except that Adam cannot get more than €40. No amount of scaling is going to make this example symmetric. So, we need a new trick. 
         Suppose we scale the respective payoffs as follows. For Beth we multiply any amount €B by 15/12 and for Adam we multiply any amount by 5/6. Then we obtain the bargaining problem depicted below. It may not be immediately clear that this has helped much. We can, however, apply independence of irrelevant alternatives. In particular, consider our first example of splitting €100 (orange dotted line). The feasible set of the current bargaining problem is a subset of the split €100 problem. Moreover, the €50 each solution to the split €100 problem is in the feasible set of this new problem. 

             So, in order to satisfy Pareto efficiency, symmetry, scale invariance and independence of irrelevant alternatives there must be a unique solution to the original problem. The solution of the split €100 bargaining problem is, of course, €50 each. Scaling this back to the original problem we get a solution of €40 for Adam and €60 for Beth. You can again check this is consistent with maximizing the Nash product.
            This last example shows that the solution of any bargaining problem can, with the use of scale invariance and independence of irrelevant alternatives, be equated with the solution of a symmetric bargaining problem. And a symmetric bargaining problem has an obvious solution. Herein lies the beauty of Nash's bargaining solution.
             The Nash solution has stood the test of time and remains in common use. Many, however, have questioned the independence of irrelevant alternatives axiom. In our third example, for instance, we might ask whether Beth could hope to do better than €60 given her strong bargaining position. The experimental evidence, I would say, is largely supportive of the axiom Nash proposed. That though is a topic for another time. 

Saturday, 13 June 2015

Is it easier to provide a threshold public good if potential contributors are poor?

A threshold (or step-level) public good is a good that would benefit members of a group but can only be provided if there are sufficient contributions to cover the cost of the good. A local community raising funds for a new community centre is one example. Flatmates trying to get together enough money to buy a new TV is another.
         There are some fundamental strategic parameters in any threshold public good game: number of group members (n), the threshold amount of money needed (T), the value of the good if provided (V), and the endowment of money that group members have available to contribute (E). Early experimental studies looked at the role of n, T and V but had little to say about E. This raised the intriguing question of whether E matters. Is it 'easier' to provide a threshold public good if group members are relatively rich or poor?
         To find out we needed to run some experiments. In two recently published papers, with Federica Alberti and Anna Stepanova, we report our findings. In this blog entry I will talk about a study with Federica Alberti published in Finanz Archiv. This study is distinguished by the fact we looked at games with a refund or money-back guarantee if contributions fall short of the threshold.
         The benchmark game in the experimental literature has n = 5, T = 125, V = 50 and E = 55. This means that if group members split the cost of providing the public good they would each have 55 - 125/5 = 30 left. Call this the endowment remainder. We compared this benchmark to games where, everything else the same, E = 30 and E = 70. This results in an endowment remainder of 5 and 45. You can think of these as games where group members are poor and rich, respectively. We also looked at two further games where the endowment remainder was 5 and 45 but T and V varied. This allowed us to check that our results are robust.  
          The figure below summarizes our overall results. You can see that there is some weak evidence of a lower success rate for the intermediate level of endowment. The differences, though, are hardly large. So, does the endowment make any difference? We claim it does.
         To see why, first note that successfully providing the public good requires coordination amongst group members. They want to contribute just enough to pay for the good. If group members  are relatively poor then it should be simple to coordinate because the good is only provided if all of them contribute most of their endowment - so there is nothing to disagree about. If group members are relatively rich then it should also be simple to coordinate because they have lots of money to spare - it is not worth disagreeing. The intermediate case may be more tricky.
          If this conjecture is correct we would expect groups to learn to coordinate when the endowment remainder is small or large. Our subjects played the game 25 times and so we can check for this. The next figure summarizes what happened in the last five plays of the game. As we would expect the success rate is now significantly higher with a low or high endowment.
          Another way to check our conjecture is to look at the variance in contributions around the threshold. The better group members are at coordinating, the lower should be the variance. The next figure summarizes the variance of contributions in the last five plays of the game. Here there is a big difference.  
         Our results, therefore, suggest that it is 'easier' for groups to provide the public good when group members are relatively poor or rich. The important caveat is that there should be sufficient time to learn how to coordinate. For an intermediate endowment group members seemingly could not coordinate no matter how much time they had. So, the endowment does matter.
          To turn this into practical guidance for fundraisers would suggest donors need to feel either critical - the poor case - or that giving will not cost them much - the rich case. It would be dangerous to enter the intermediate territory where, say, the donor is led to believe that the good could be provided without them - they are not critical - and that there donation would involve personal sacrifice - they are not rich.

Saturday, 6 June 2015

Leadership in the minimum effort game

The minimum effort (or weakest link) game is fascinating - simple, yet capable of yielding profound insight. The basic idea is that there is a group of people who individually have to decide how much effort to put into a group task. Effort is costly for the individual but beneficial for the group. Crucially, group output is determined by the minimum effort that any one group member puts in to the task. A classic example is an airline flight: If any person involved in the flight - pilot, fuel attendant, mechanic, luggage handler etc. - gets delayed, then the flight is delayed, no matter how hard others try.
           In experiments the minimum effort game is usually reduced to the matrix in Figure 1. Here subjects are asked to choose a number between 1 and 7 with the interpretation that a higher number is higher effort. Someone choosing effort 1 is guaranteed a payoff of 70. Someone who chooses 2 gets 80 if everyone else chooses 2 or more, but only gets 60 if someone in the group chooses 1. Someone who chooses 7 can get from 10 to 130 depending on the choice of others.
       Suppose the minimum choice of others is 5. What should you choose? You do best to choose 5 and get a payoff of 110. What if the minimum of others is 6? You do best to choose 6 and get payoff 120. Following this logic we can see that 'everyone choose the same number' is a Nash equilibrium. Now here are the key points: Everyone choose 1 is the worst equilibrium while everyone choose 7 is the best equilibrium; but choosing 7 is very risky because others might let you down. To return to the airline example. It is no use the pilot racing around to get the flight ready to go if the fuel attendant is having an after-lunch snooze.
         In experiments we typically observe that effort converges over time to 1 - the worst equilibrium. Most people start by choosing high effort. But it takes only one bad egg to ruin the team and that drags average effort down. This is a bad outcome! The airline is not going to be on time. So how to fix things?
          An obvious answer seemed to be leadership. So, together with Mark van Vugt and Joris Gillet we ran some experiments on leadership. The basic idea was that one person chooses first and then others follow. We reasoned that if the leader chose 7 this would signal to the others in the group to also choose 7. Problem solved!
          Things did not work out quite as well as expected. The figure below summarizes average effort over the 10 rounds subjects played the game. In a 4 player simultaneous choice version (Sim4) effort fell over time. In the leadership treatments, with either an endogenously or exogenously chosen leader (End and Exo), effort was higher but not by much. Moreover, it was no higher than we got if we just took one person out of the group (Sim3). This is not a ringing endorsement of leadership. We found that leadership failed to increase efficiency as much as expected because leaders were not bold enough. If a leader chose high effort followers responded but leaders were reluctant to choose high effort.

        A recent study by Selhan Garip Sahin, Catherine Eckel and Mana Komai adds a slightly different twist. The figure below shows the average contributions they observed in a 6 player version (where effort could go up to 9). They looked at leadership by example (Exemplar) and leadership by 'communication' (manager). There overall results are very similar to ours. Again, leadership merely seems to stabilize effort and stop it falling. Their results, though, suggested more blame should be placed on followers. Specifically, leader effort increased over the rounds while follower effort fell.    

        There is still a lot we can learn about leadership in the minimum effort game. The failure of leadership to push effort up to the efficient level does though clearly illustrate the difficulty of getting groups to coordinate. And note that this is not because of some social dilemma like incentive to free-ride. There is no way to free-ride in this game. The problem is one of group members trusting that others will put in high effort. Trust, it seems, does not come easy.

Thursday, 28 May 2015

Two part tarrifs: lump sum fee versus user fees

Think of how a car park typically works: you pick up a ticket when you drive in and then pay in rough proportion to how long you park your car. Now think of how a gym typically works: you pay a fee when you go in and can then stay as long as you like.
         There is no reason, in principle, why the car park could not charge a lump sum fee and the gym charge for how long you stay there. But they typically don't. And the gym may well even offer a membership package that allows year long unlimited use. How can we make sense of all this?
         We need to think in terms of two part tariffs. With a two part tariff the customer is charged a lump sum fee for access to the good and then charged a user fee for each unit of the good consumed. For example, on your mobile phone you may pay a monthly subscription fee and then a fixed fee per text message or call. The car park that charges for how long you stay is using a two part tariff where the lump sum fee is zero. Similarly the gym that allows unlimited access is using a two part tariff where the user fee is zero
         Two part tariffs are an example of second degree price discrimination in that different units of a good are priced differently. Specifically, the first unit of the good costs the lump sum fee and the user fee combined. The second and subsequent units only cost the user fee. For instance, if the subscription on your mobile phone is £5 a month and the cost of a text message is 10p then the first text message effectively costs £5.10 while the second message only costs 10p.
         The advantage of two part tariffs (from the point of view of the seller) is that they allow the seller to extract consumer surplus. In other words the seller can make more profit. To understand why we need to look at a consumer's demand for a good. The table below gives some hypothetical numbers to work with.
          Let us look at the car park example for now. Suppose the car park charges £0.75 per 20 minutes. Then this person, call him Fred, would park for 80 minutes and pay 4 x 0.75 = £3 in user fees. But, Fred's total benefit from parking for 80 minutes is £2 + 1.60 + 1.20 + 0.80 = £5.6. So Fred is up £5.6 - 3 = £2.60 on the deal. This is his surplus. The car park could eat away at that surplus by charging a lump sum fee. For instance, if they charge £2.00 to enter the car park then Fred would still park for 80 minutes but now pay a total of £5.00.
          This example illustrates how two-part tariffs can only work to extract surplus if there is a positive lump sum fee and user fee. Either on their own is not enough. So, the car park and gym are missing out by charging only a user fee and only a lump sum fee. They presumable believe that the added complication of having a lump sum and user fee is not worth the extra revenue. But, why does one opt for a user fee and the other a lump sum fee?
          One reason (I'm not saying it is the only reason) can be found in the demand functions. If most customers have relatively flat demand functions meaning that the first 20 minutes has roughly the same benefit as the second 20 minutes and so on, then there is little consumer surplus to extract. A user fee is best. By contrast if the demand functions are relatively steep meaning that the first 20 minutes is worth a lot more than the second 20 minutes and so on, then there is a lot of consumer surplus to extract. A lump sum fee is best.
          This difference is apparent in the example. Fred is willing to pay a total of £6 for both the car park and gym. What differs is the steepness of the demand curve. Suppose, for instance, there was a user fee of £0.75 per 20 minutes. The car park would make £3 and the gym only £1.50. Put another way, the gym misses out on £3 of surplus while the gym misses out on £4.50.
          You might criticise these numbers by saying that gym should not charge £0.75. But the other thing to keep in mind is that Fred will not be the only customer. The optimal price for Fred might not be the optimal price for anyone else and so we need a pricing structure that can cope with this. If the typical customer has a steep demand curve then no matter how the firm prices they are going to miss out on a lot of surplus.
         So, why would we expect demand curves to be steeper for a gym than car park? A gym is something we do for pleasure. Parking the car is something we are likely to do for necessity. As a rough rule of thumb there is more surplus to be had from experience goods than necessary goods. Hence amusement parks charge a lump sum fee and the hire company charges a user fee.

Tuesday, 19 May 2015

Why do people vote?

As the dust finally settles on the 2015 UK general election it is interesting to reflect on the big (game theory) question - why did over 30 million people turn up to vote?
       A simple model of voting would suggest that hardly anyone should vote. Basically there are non-negligible costs to voting in terms of time. But, the expected benefit of voting seems very, very small. Indeed, since universal suffrage in 1928 there is not a single constituency election in the UK that has been won by one vote. In other words everyone who has voted in a UK general election for the last 70 years or so could have stayed at home and the outcome would have been exactly the same.
        With such dismal prospects of making a difference why would anyone vote? Yet people do vote! This is the paradox of voting. And I saw the paradox in full swing at 7am on polling day - people were already turning up in numbers, eager to vote, smiles on their faces.
     Typical explanations that have been proposed to resolve the paradox appeal to social norms. There are two ways to look at this. One is to say that people vote because they feel an obligation to do so - they would feel guilty if they did not vote. The other is to say that people enjoy civic engagement - they feel good because they vote. The smiles I saw on people's faces on Thursday would suggest the latter explanation has some merit. There is, though, a problem with such explanations.
       If people simply wanted to vote to avoid guilt or feel good about themselves then they could vote for anyone. Remember they are not going to effect the outcome whatever they do. A typical voter, however, seems to put serious thought into who they are going to vote for. Clearly, the idea that someone would randomly decide how to vote is extreme. But, a person can put more or less effort into deciding and many people seem to put in more.
      One could counter this criticism by arguing that people want to vote for the 'right candidate' or maybe want to vote for the winning candidate. For instance, we might say there is social norm to engage in the whole election process and not just turning up to vote. That, though, doesn't really convince me. It seems that people put in effort deciding who to vote for because they think they 'can make a difference'.
      To fully explain the paradox I think we, therefore, need to appeal to 'irrational' beliefs. Realistically a voter is not going to make a difference but he or she may behave as though she can make a difference. To explain why this happens we can look at how people behave in 'large' games. A large game is a game with 'many' players. In the theoretical literature many is often read to mean an infinite number. But, a general election has enough players or voters to comfortably qualify as large.
       The experimental evidence suggests that people tend to think of a large game as a two player game where everyone else is grouped together. In other words people think in terms of me and them. To given one example, in a recent paper with Denise Lovett we look at leadership in public good games and find that subjects behaved the same whether one or three will follow their lead. From a theoretic perspective this is strange because there is more to be gained by influencing three than one. But, if a person thinks in terms of me and them it makes no difference if there are one, three, or a million followers.
     Seen through the lens of a two player game it is easy to see why a person may think their vote matters. Indeed, they have the casting vote! And note that politicians and media feed this bias by making ever vote feel critical. It is also interesting to look at what people who do not vote say. I typically hear things like 'I don't like any of the candidates' or 'all politicians are the same'. This is a complaint about the choices on offer and not a recognition that there vote is non-critical.
      What are the practical implications of this? If irrational beliefs really do explain the paradox of voting then we have an interesting ethical conundrum. Most seem to think that a high turnout at elections is a 'good thing'. So, we might want to encourage voting. But, then this would require us to miss-inform people by reinforcing an irrational belief that each vote counts. And, ethically, we surely want to reduce miss-information!    

Wednesday, 29 April 2015

Two different ways to charge for a good

The conventional way to charge consumers for something is pretty simple - they pay for every unit they buy. But, there are alternatives. One is to charge an up-front fee and then refund customers for every unit they purchase below some threshold. Google's new Project Fi has an element of this built into it, as they explain - 'Let's say you go with 3GB for $30 and only use 1.4GB one month. You'll get $16 back, so you only pay for what you use'.
          It is simple to design a pay and refund pricing policy that are theoretically equivalent. For instance, suppose you regularly buy movies from an online website. Also, suppose that you would never buy more than 20 movies a month. Then the following pricing policies are equivalent:
  • Pay: You pay £5 for every movie you download.
  • Refund: You pay a monthly fee of £100 and receive £5 back, per movie, if you download less than 20 movies. 
For example, if you buy 10 movies in a month then this either costs £10 x 5 = £50 or £100 - 10 x 5 = £50.
          To say, though,  that a pay policy and refund policy are equivalent in theory does not mean they are equivalent in practice. So, which type of policy would you prefer? And which type of policy do you think would lead to most downloads?
           One way of looking at this is to say that paying money for something is coded as a loss. So, under the pay policy there would be 10 times during the month where you 'lose' £5. The abundant evidence for loss aversion tells us that losses are bad. So, 10 losses are very bad. Under the refund policy there is one big loss of £100 and then a gain of £50. Now, a big loss is clearly worse than a small loss. Evidence suggests, however, that we don't consider a big loss as bad as lots of small losses. This, in itself, does not tell us that a £100 big loss is better than 10 losses of £5. But, it does suggest that people might prefer the refund policy once we factor in the additional £50 gain. So, expect people to opt for the refund policy.
           Consider now the number of downloads. With a pay policy each download is a loss of £5. With a refund policy each download reduces the gain at the end of the month by £5. Loss aversion clearly implies that people will download less with a pay policy than a refund policy. Interestingly, this poses something of a conundrum for the company deciding on a pricing policy - the refund policy may be more popular but lead to more usage.  
           If you are not convinced by the above arguments lets churn out some numbers consistent with prospect theory. Suppose that if you lose £x you suffer a psychological cost of 2log(x). Then losing £5 costs 3.22 while losing £100 costs 9.21. This is already enough to tell us that 10 losses of £5 is worse than a one-off loss of £100 (because the 10 losses costs 32.3 compared to only 9.21). Hence, you would prefer the refund policy. Suppose that if you gain £x you experience pleasure of log(x). Then gaining £5 adds at most 1.61 to pleasure which is nothing compared to the cost of paying £5. Hence, you will download more movies under a refund policy.

Monday, 13 April 2015

Public good versus common resource dilemma: Framing in social dilemmas

In a social dilemma what is good for the group is not necessarily good for the individual. For instance, if Fred donates time and money to charity that costs Fred but benefits society. Similarly, if Fred cycles to work so as to not pollute that costs Fred but benefits society. We know in the lab that many people (typically around 50%) are willing to put the interests of the group ahead of their own. This gives hope when it comes to things like combating climate change. There is, however, an intriguing and unexplained framing effect regarding willingness to cooperate.
         Any social dilemma can be framed in two alternative ways. One can frame things in terms of Fred making a contribution to the group or in terms of Fred making a withdrawal from the group. Some things, like giving to charity, are more naturally thought of in terms of contribution. And others, like cycling to work, are more naturally thought of in terms of withdrawal. But, one can always reframe things. For instance, charities, when soliciting donations, have a wonderful way of making a donation seem already given; this means to say 'no I do not want to give' appears like a withdrawal.
        Framing effects are most pronounced in asymmetric social dilemmas. To illustrate, let us look at a simple example. Suppose four work colleagues have to do a project. For the project to be a success a total of 12 hours need to spent working on it. Fred and Fay each have 6 hours they could spend on the project while Max and Mary each have 12 hours they could spend. How should they split the workload?
        With a framing of how much should they contribute the typical outcome is that each worker contributes in proportion to the number of hours they have available. So, Fred and Fay would work 2 hours each while Max and Mary would work 4 hours each. While this may seem advantageous for Fred and Fay they still end up with less free time than Max and Mary.
         Consider now the withdrawal framing. In this case we think of Fred and Fay as provisionally working 6 hours on the project and Max and Mary as working 12 hours. This adds up to 36 hours which is clearly a lot more than necessary. The question thus becomes how many hours should they spend not working on the project. The typical outcome is that workers split equally the number of hours they save. Given that there are 24 hours to be saved they each save 6. This means that Max and Mary will work 6 hours each on the project while Fred and Fay do nothing.
         With a different framing we, therefore, get a very different outcome. So, why does framing matter? Put simply, we do not know. Some studies have looked carefully at the issue, such as one by Eric van Dijk and Henk Wilke on decision-induced focussing. Clear answers, though, have not been forthcoming. Why? The traditional approach has been to assume some 'gut instinct' effect - or system 1 processes - drive the framing effect. Choice could then be affected by whether contributing is perceived in terms of loss (of time) or gain (in free-time or success on the project). Inter-related is the issue of property rights - does Fred feel as though he 'owns' 6 hours free time or zero?
          In ongoing work with Federica Alberti we are exploring an alternative approach. One that puts the focus on more structured reasoning - system 2 processes. In particular, our approach, emphasizes that colleagues need to coordinate if the project is going to succeed and so they may be looking for focal or salient ways to coordinate. Hence, the focus becomes more 'what do I expect others to do' than 'what do I want to do'. It is well known that framing can effect focal points.
          Our work to date has primarily focussed on focal points with a contribution framing. For example, in one study we look at the effects of requiring full agreement in order for the project to work. It is relatively straightforward to extend our work to a withdrawal framing. Whether this will help solve the framing effect puzzle in social dilemmas is not yet clear. But, there is a definite sense that until we solve this puzzle our understanding of why people cooperate (or do not) in social dilemmas is worryingly incomplete!     

Monday, 16 March 2015

Why should you cooperate in the prisoners dilemma?

The prisoners dilemma is familiar to any student of game theory. One of its appealing properties is its seeming simplicity - there is an 'obvious' way to play the game. What is obvious in theory does not, however, always show up in reality. So, does that mean people are dumb or the theory is dumb? The conventional approach seems to be to say that people are dumb. But, I think its the theory that needs a rethink. Here is one reason why:
          The table below gives an example of the prisoners dilemma. Alice and Barney are two work colleagues who have to produce a project together. If they cooperate they will finish the project in one hour. If they do not cooperate it will take them 10 hours. If Barney cooperates and Alice defects then Alice gets away with doing nothing while Barney spends 15 hours. Vice versa if Alice cooperates and Barney defects.
   So, what should Alice do? If Barney cooperates then she does better to defect - she saves one hour. If Barney defects then she also does best to defect - she saves five hours. Either way Alice should defect. We get, therefore, our simple prediction. And if both Alice and Barney defect then they take 10 hours each rather than the one hour they would have spent if cooperating. This is the theory for a one-shot interaction. 
      What happens if they are going to do, say, 10 projects over the course of a year? On the last (tenth) project the incentive to defect remains. Moreover, nothing that happens in the ninth project will change the incentive to defect on the last project. That means there is an incentive to defect on the ninth project. This logic can be rolled back to the eighth project, seventh and so on. Standard theory predicts that Alice and Barney will defect from the first project onwards. So, where is the flaw in the logic?
      Suppose that Alice thinks Barney is a 'nice guy' who will cooperate on every project unless Alice has previously defected. On the last project Alice still has an incentive to defect. On the ninth project, though, things are already different. If Alice cooperates she will spend one hour on the current project and no time on the last. If she defects she will spend no time on the current project but 10 hours on the last. She does better to cooperate. This logic rolls all the way back to the first project. If, for instance, she cooperates on the first nine projects Alice will spend a total of nine hours working. If she defects on the first project she can expect a total of 90 hours work.
       The standard theory would reject this story on the basis that Alice should not expect Barney to be a nice guy. But, abundant experimental evidence shows that nice guys do exist. There is nothing, therefore, bizarre about Alice expecting Barney might be nice. And she only needs to put a sufficiently high probability, say 25%, on Barney being nice to justify cooperating. Indeed it may be that both Alice and Barney are 'nasty guys' but end up cooperating because they expect the other may be nice.
       The ideas sketched above are not new. Indeed Kreps and Wilson set out such ideas back in 1982 with a paper on 'Reputation and imperfect information'. According to Google this paper has a huge 3332 cites. I think it is fair to say, however, that the insights from that paper have not fed into conventional thinking on games like the prisoners dilemma. That's probably because of a reluctance by many to combine behavioural insight with game theory. So, don't trust a game theorist if he or she tells you it is always best to defect when playing the prisoners dilemma. Your intuition might be a safer guide.  

Friday, 27 February 2015

Time to 'privatize' the NHS?

Labour's pitch for the upcoming UK general election has been simple - let's talk about the National Health Service. Particularly headline grabbing was the '100 days until the election, 100 days to save the NHS as we know it' campaign. To focus on the NHS may seem like a simple winning strategy for Labour given the rollercoaster of 'NHS in crisis' stories hitting the news in recent months. But, I think labour strategists may have badly misjudged this one. After years of dodging the issue the British public may finally waking up to the idea that we can only preserve the 'NHS as we know it' with some pretty radical change.
       The NHS is a publicly provided health care system that is centrally funded and free at the point of delivery. Labour is broadly committed to maintaining that status quo. They reject private involvement in providing health care. And they reject anything other than a free health service. 'Save our NHS, privatization is putting our NHS at grave risk' sums up the position nicely. The truth is, though, the NHS has to change. There simply is not enough money to fund the standard of service that Britons have got used to. So, we either accept a low standard of care or move to something different.
       The main problem with the NHS model is one of moral hazard: If the government guarantee a free health service then there is less incentive to stay healthy. And if people take more health risks then health costs will be higher than they need to be. Simple though it is this chain of logic needs some justification. After all, no one wants to be ill and so there are some big incentives to stay healthy! The 'free at the point of delivery' mantra has, however, created a society that expects great things from its NHS. That means this is not just a money issue; expectations of the NHS have grown well beyond what it can reasonably deliver. This also feeds into another consequence of moral hazard: people expect service immediately even for relatively minor problems.
       Charging for health care has already begun with a largely privatized dentistry, charges for prescriptions, and a growing private sector for routine operations. This trend will surely have to continue. What's interesting though are attitudes towards paying for health care. The 'free at the point of delivery' mantra is deeply ingrained in the British psyche; anyone who dared question its merits faced criticism. That's why labour might think it is on safe ground defending the NHS as we know it. Things though appear to be changing.
         I think this change partly reflects a growing understanding of individual choice in health care: people choose to get drunk on a Friday night, they choose to smoke, they choose to not eat healthy foods or exercise. Why should the taxpayer pay for health care that was avoidable? Surely the individual should take some of the responsibility? Another thing changing attitudes is the growing recognition that something has to give. The previous labour government put a lot of money into the NHS but that merely delayed the inevitable. People want a high quality health service and the current system is creaking at the edges. Can we not do better?
        Various ideas have been proposed to put additional charges into the NHS. Maybe it will be a nominal fee to visit a GP or A&E. Maybe benefits will be withdrawn from those who refuse to abide advice on healthy eating. Maybe it will be increasing private options in the NHS. One thing, however, seems clear and is often overlooked. Once charges are unleashed there is likely to be a run away train of increased charges. This is what happened with dentistry and is the kind of thing that will scare plenty of people. So, there needs to be a proper debate on what a future health service will look like. That is clearly not going to happen in this election with the conservatives afraid to mention the issue. Time is running out to really save the NHS as we know it.               

Tuesday, 10 February 2015

Premier League TV rights: Winners curse?

The wait is final over to discover who will broadcasting Premier League football from 2016 onwards. There is no surprise this time in the successful bidders - Sky and BT. But there is surprise at the price they have paid - a huge £5.1 billion. That equates to over £10 million per game. This figure is 70% up on last time and above all analyst forecasts. Surely it is time to talk of the winners curse?
        The winners curse is the idea that a winner of an auction may well end up losing money. The intuition is simple enough in that the winner of an auction is likely to be the most optimistic as to how much the prize is worth; that optimism may be misplaced. In the past, TV rights have provided some textbook examples of the winners curse. So, do we have another example?
        One reason to doubt Sky and BT have overbid is the fact they know pretty well what they are bidding for. The winners curse is most likely to occur when the value of the prize is highly uncertain. But Sky have been broadcasting the Premier League for decades and so they surely know what they are doing. Moreover, football rights have come to be central to both Sky and BT's business plan. So, they may well be content to make a loss on football in order to protect their general image as top broadcasters.
        Even so, the latest numbers are shockingly high. It seems that Sky and BT are not so much betting that football fans will continue to pay huge amounts to watch football but that they will pay ever increasing amounts of money to watch football. That seems a dangerous presumption. The football market is already saturated with fans disgruntled at the cost of it all. Can they continue to pay more? I do not think so. Will Russian oligarchs and Arab sheiks continue to plough money into the Premier League? Who knows.
       So, what if Sky and BT have paid too much? Then, I'm afraid, the bubble may well burst. Since its inception the Premier League has hugely distorted the football market in the UK. And the Premier League also appears of the mind-set that ever increasing amounts of money can be expected. Take away the money and the whole thing might collapse. I think, therefore, that it is the Premier League who are taking a big risk by extracting so much money from Sky and BT. Sky and BT will survive the loss of £5.1 billion; I'm not sure the Premier League could!  

Sunday, 1 February 2015

Revealed preference, WARP, SARP and GARP

The basic idea behind revealed preference is incredibly simple: we try to infer something useful about a person's preferences by observing the choices they make. The topic, however, confuses many a student and academic alike, particularly when we get on to WARP, SARP and GARP. So, let us see if we can make some sense of it all.
          In trying to explain revealed preference I want to draw on a study by James Andreoni and John Miller published in Econometrica. They look at people's willingness to share money with another person. Specifically subjects were given questions like: 
Q1. Divide 60 tokens: Hold _____ at $1 each and Pass _____ at $1 each. 
In this case there were 60 tokens to split and each token was worth $1. So, for example, if they held 40 tokens and passed 20 then they would get $40 and the other person $20. Consider another question:
Q2. Divide 40 tokens: Hold _____ at $1 each and Pass ______ at $3 each.
In this case each token given to the other person was worth more than that kept. So, for example, if the subject held 20 tokens and passed 20 they would get $20 and the other person $60. The figure below plots the feasible set for each of the two questions.

           Suppose a subject called James chooses A when asked Q1 and B when asked Q2. In other words he passes all 60 tokens in question 1 and holds all 40 tokens in question 2. Is that 'sensible'? When answering Q1 James could have chosen B; he could have given $0 to the other person and kept $40 for himself. Instead he chose A; he chose to give $60 to the other person and keep $0 for himself. We say that A is directly revealed preferred to B. But, then look at question 2. In this case James could have chosen A but chose B. So B is directly revealed preferred to A. This seems inconsistent; how can A be preferred to B and B preferred to A. It is a violation of the Weak Axiom of Revealed Preference (WARP).
           Why would James or anyone else violate WARP? There are three basic reasons: (1) James does not have consistent preferences. Which is another way of saying he behaves irrationally. (2) He is indifferent between A and B. I'll come back to this later. (3) There are menu or framing effects. To illustrate how menu effects work it is easier to consider a slight variation on Q1 and Q1. Specifically, consider questions:

Q3. Would you choose: A. $0 for yourself and $60 for the other person, or B. $40 for yourself and $0 for the other person.

Suppose James chooses B. So, B is directly revealed preferred to A. Now consider an alternative question:

Q4. Would you choose: A. $0 for yourself and $60 for the other person, B. $40 for yourself and $0 for the other person, or C. $0 for yourself and $50 for the other person.

Clearly option C is a nonsense option because A beats C. But, the presence of C may sway James to choose A. This would be a violation of WARP caused by an attraction effect: option A looks best because it is clearly better than C. There is nothing inherently inconsistent, therefore, in some violations of WARP.
          Let's move on now to the Strong Axiom of Revealed Preference (SARP). Suppose, as before, that James chooses B when asked question 3. So, B is directly revealed preferred to A. Suppose also that when asked he would choose A over C. So, A is directly revealed preferred to C. Given that B 'beats' A and A 'beats' C we would say that B is indirectly revealed preferred to C. Suppose, however, that when asked James choses C over B. Then we have a violation of SARP.
          Note that there is nothing in the story of the previous paragraph that violates WARP. That, however, is because we only consider pairwise comparisons. If we ask James to choose between A, B and C then, whatever his answer, we would get a violation of WARP. More generally, if we elicit James' choices over all the possible combinations of options then WARP and SARP essentially become equivalent: if SARP is violated then so must be WARP and vice versa. The setting of Andreoni and Miller is like this. In order to observe a violation of SARP without a violation of WARP it must be that we only observe choices for some possible combinations of options. That, however, is not unlikely in applied settings. Preference reversals being one example.
         WARP and SARP get most of the attention. They have, however, a basic 'flaw'. Suppose James is indifferent between A and B. Or, to give a slightly more mundane example suppose that when he goes to the canteen for lunch, James is indifferent between ham sandwiches at $2 or cheese sandwiches at $1.50. If we observe James sometimes buying ham sandwiches and sometimes cheese we would say that his preferences violate WARP and SARP. But, there is nothing inconsistent in his preferences or choices. To get out of this problem we need the Generalized Axiom of Revealed Preference (GARP).
          To sometimes choose A and sometimes choose B, or to sometimes choose ham and sometimes choose cheese, is not a violation of GARP. What is? Suppose we observe James sometimes choosing ham and sometimes cheese. Then the price of a cheese sandwich increases from $1.50 to $1.75. If James is observed buying a cheese sandwich after the price rise we say that cheese is strictly directly revealed preferred to ham. This is a violation of GARP. The rise in price should have broken James' indifference between ham and cheese but it did not.
         In order to fit this into the setting of Andreoni and Miller we need a slight reframing of James' choice. Let us consider how much he is willing to give to the other person. With question 1 it costs James $60 to give $60 to the other person and reach point A. With question 2, how much would it cost James to get to point A? It would only cost $40. (Inefficient though it is he would hold 0 tokens and pass 20 tokens, losing 20 tokens in the process.) We said that James would say A when asked question 1. Given that it is 'cheaper' to say A when asked question 2 it would be a violation of GARP for James to now choose B. If he did so it would be the case that B is strictly directly revealed preferred to A.  
         Let me finish by briefly commenting on why WARP, SARP and GARP matter. If GARP is satisfied then preferences are 'well behaved' and we can model choice using a utility function. Given that utility maximization drives most of economics this is clearly crucial! WARP and SARP are less relevant because of the indifference issue. For example, preferences could satisfy GARP and not WARP or SARP.
          That just leaves the question of whether preferences satisfy GARP? There is abundant evidence of violations of GARP. So, in a strict sense people's preferences don't satisfy GARP; something many behavioural economists emphasize. But, to say that preferences don't satisfy GARP is different to saying we cannot make meaningful predictions by assuming they do.