Tag: Nash equilibrium

The Paris Climate Change Conference (COP21) is under way. At the opening on November 30, 150 Heads of State gathered in Paris, most of whom addressed the conference. With representatives from 195 countries and observers from a range of organisations, the conference is set to last until 11 December. Optimism is relatively high that a legally binding and universal agreement will be reached, with the aim of keeping global warming below 2°C – what is generally regarded as a ‘safe’ limit.

But although it is hoped that a successor to the Kyoto Protocol of 1997 will be put in place, there are many problems in getting so many countries to agree. They may all wish to reduce global warming, but there is disagreement on how it should be achieved and how the burden should be shared between countries.

There are several difficult economic issues in the negotiations. The first is the size and impact of the external costs of emissions. When a country burns fossil fuels, the benefits are almost entirely confined to residents of that county. However, the environmental costs are largely external to that country and only a relatively small fraction is borne by that country and hardly at all by the polluters themselves, unless there is a carbon tax or other form or penalty in place. The problem is that the atmosphere is a common resource and without collective action – national or international – it will be overused.

The second problem is one of distribution. Politicians may agree in principle that a solution is necessary which is equitable between nations, but there is considerable disagreement on what is meant by ‘equitable’ in this context. As the third Guardian article below puts it:

The most important hurdle could be over whether industrialised countries like the US, UK and Japan, which have contributed the most to the historical build-up of emissions, should be obliged to cut more than developing countries. India, on behalf of many poor countries, will argue that there must be “differentiation” between rich and poor; but the US wants targets that are applicable to all. A collision is inevitable.

A third problem is that of uncertainty. Although there is general agreement among scientists that human action is contributing to global warming, there is less agreement on the precise magnitude of the causal relationships. There is also uncertainty over the likely effects of specific emissions reductions. This uncertainty can then be used by governments which are unwilling to commit too much to emissions reductions.

A fourth difficulty arises from the intertemporal distribution of costs and benefits of emissions reductions. The costs are born immediately action is taken. Carbon taxes or charges, or subsidies to renewables, or caps on emissions, all involve higher energy prices and/or higher taxes. The flows of benefits (or lower costs), however, of reduced emissions are not likely to be fully experienced for a very long time. But governments, whether democratic or dictatorships, tend to have a relatively short time horizon, governed by the electoral cycle or the likelihood of staying in power. True, governments may not be solely concerned with power and many politicians may have genuine desires to tackle climate change, but their political survival is still likely to be a major determinant of their actions.

Of course, if there is strong public opinion in favour of action to reduce emissions, governments are likely to respond to this. Indeed, all the expressions of public support for action ahead of the conference from all around the world, do give some hope for a strong agreement at the Paris conference. Nevertheless, there is still widespread scepticism in many countries over the relationship between human action and climate change, and many argue that the costs of policies to tackle climate change exceed the benefits.

Game theory can shed some insights into the difficulties ahead for the negotiators. The global optimum may be for a strong agreement, binding on all countries. The Nash equilibrium, however, may be a situation where countries push for their own interests at the expense of others, with the final agreement being much more minimalistic.

There do, however, seem to be more reasons to be cheerful at this summit that at previous ones. But negotiations are likely to be hard and protracted over the coming days.

Videos and webcasts
Paris Climate Conference: The Big Picture Wall Street Journal on YouTube, Jason Bellini (30/11/15)
Why is the Paris UN climate summit important? PwC, Leo Johnson (14/10/15)
Paris climate change summit 2015: ‘the near impossible task’ Channel 4 News on YouTube, Tom Clarke (30/11/15)
COP21: Rallies mark start of Paris climate summit BBC News, David Shukman (29/11/15)
With climate at ‘breaking point’, leaders urge breakthrough in Paris Reuters, Bruce Wallace and Alister Doyle (1/12/15)
COP21: Paris conference could be climate turning point, says Obama BBC News (30/11/15)
Leaders meet to reach new agreement on climate change BBC News, David Shukman (30/11/15)
Poll: Growing Doubts Over Climate Change Causes Sky News, Thomas Moore (30/11/15)
Paris climate protesters banned but 10,000 shoes remain The Guardian (29/11/15)

Articles

COP-21 climate deal in Paris spells end of the fossil era The Telegraph, Ambrose Evans-Pritchard (29/11/15)
Is there an economic case for tackling climate change? BBC News, Andrew Walker (28/11/15)
World Leaders in Paris Vow to Overcome Divisions on Climate Change Wall Street Journal, William Horobin and William Mauldin (30/11/15)
Experts discuss how to build a carbon-free energy industry The Guardian, Tim Smedley (25/11/15)
Africa could lead world on green energy, says IEA head The Guardian, Anna Leach (11/11/15)
Climate change talks: five reasons to be cheerful or fearful The Guardian, John Vidal (30/11/15)
The Paris climate change summit, explained in 4 charts The Washington Post, Philip Bump (30/11/15)
Why This Goal To Curb Climate Change ‘Is Not Ideal’ Huffington Post, Jacqueline Howard (30/11/15)
Paris climate change talks: What the different groups attending expect from these crucial meetings Independent, Tom Bawden (29/11/15)
UN Climate Change Conference: World Leaders Call For Price On CO2 Emissions Despite Uphill Battle At Paris Summit International Business Times, Maria Gallucci (30/11/15)
World Bank, six nations call for a price on carbon SBS (Australia) (1/12/15)
Uruguay makes dramatic shift to nearly 95% electricity from clean energy The Guardian, Jonathan Watts (3/12/15)

Questions

  1. Why is COP21 considered to be so significant?
  2. For what reasons is there hope for a binding agreement to limit global warming to 2°C?
  3. What would be the effect on global warming of the commitments made by more than 180 countries prior to the conference?
  4. What market failings contribute towards the problem of global warming?
  5. Why, if all countries want to achieve a binding agreement at the Paris conference, is it likely to be so difficult to achieve?
  6. Explain what is meant by a ‘Nash equilibrium’ and how the concept is relevant to international negotiations.
  7. Why is China investing heavily in solar power?
  8. Could Africa lead the world in green energy?
  9. Is a ‘cap and trade’ (tradable permits) system (a) an effective means of reducing emissions; (b) an efficient system?
  10. What is the best way of financing investment in renewable energy?
  11. How does the structure/order of the Paris conference differ from previous COPs? Is such a structure more likely to achieve substantial results?

John Nash was one of the pioneers of game theory and in 1994 was awarded the Nobel prize in economics for his work in this field. He was also the subject of the 2001 film, A beautiful mind, where the young John Nash was played by Russell Crowe, for which he won an Oscar.

Tragically, John Nash and his wife were killed in a car crash on May 23: he was 86 years old. Since his death there have been many tributes paid to him and his work.

As a student of economics you will almost certainly have studied the concept of a Nash equilibrium: a situation where everyone makes their best choice, given the choices of the other ‘players’ in a ‘game’. But a Nash equilibrium is not in the collective best interests of the participants in a non-cooperative game.

This is a very different concept of equilibrium from the simple equilibrium in a perfectly competitive market. In fact, in the complex world of strategic decision making, where firms are constantly looking at their rivals’ behaviour and possible reactions to their own behaviour, there are all sorts of Nash equilibria which are clearly sub-optimal. Competition may be highly destructive.

The following obituaries look at Nash’s contribution to the development of economics. As the Bloomberg article states:

The game theory concepts that Nash’s math[s] brought to the field were a true paradigm shift in economics. Macroeconomists, who continue to use the old Walrasian notion of equilibrium typically engage in hand-waving about how the macroeconomy is too big for strategic interactions to matter. But most of the economics profession has gradually shifted toward Nash equilibrium. The 2014 Nobel winner, Jean Tirole, is emblematic of the new economics. And many of the biggest successes in applied economics, like the auctions that power Google’s advertising, rely on Nash’s technique.

Do we owe him as much as Adam Smith?

Articles

Death of John Nash and His Beautiful Ideas Bloomberg, Noah Smith (26/5/15)
John Nash, economist and mathematician, 1928–2015 Financial Times, Ferdinando Giugliano (25/5/15)
Obituary: John Nash: Lost and found The Economist (28/5/15)
Remembering John Nash: Finding equations to explain the world The Economist (28/5/15)
John F. Nash Jr., Math Genius Defined by a ‘Beautiful Mind,’ Dies at 86 New York Times, Erica Goode (24/5/15)
Explaining a Cornerstone of Game Theory: John Nash’s Equilibrium New York Times, Kenneth Chang (24/5/15)
John Nash’s Indelible Contribution To Economic Analysis Forbes, Jon Hartley (25/5/15)
John Nash’s ground-breaking contributions to maths BBC News, John Moriarty (24/5/15)
From the archives: Nash’s Nobel prize The Economist (24/5/15)
A beautiful strategy: John Nash’s ‘game theory’ explained Hindustan Times, Gaurav Choudhury (25/5/15)

Interview with John Nash

Interview with John Nash Nobelprize.org (Sept 2004)

Questions

  1. What is meant by the ‘prisoners’ dilemma’? How is this a demonstration of a sub-optimal Nash equilibrium in a non-cooperative game?
  2. Why is it difficult to predict the outcome of a non-cooperative game?
  3. Give some examples of non-cooperative games played by companies competing with each other.
  4. Give some examples of non-cooperative games played by nations competing with each other.
  5. Why do competition authorities try to prevent cooperation between businesses?
  6. Why may a cooperative equilibrium between firms be an unstable one?

Each day many investors anxiously watch the stock market to see if their shares have gone up or down. They may also speculate: buying if they think share prices are likely to go up; selling if they think their shares will fall. But what drives these expectations?

To some extent, people will look at real factors, such as company sales and profits or macroeconomic indicators, such as the rate of economic growth or changes in public-sector borrowing. But to a large extent people are trying to predict what other people will do: how other people will react to changes in various indicators.

John Maynard Keynes observed this phenomenon in Chapter 12 of his General Theory of Employment, Interest and Money of 1936. He likened this process of anticipating what other people will do to a newspaper beauty contest, popular at the time. In fact, behaviour of this kind has become known as a Keynesian beauty contest (see also). Keynes wrote that:

professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one’s judgement, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.

When investors focus on people’s likely reactions, it can make markets very unstable. A relatively minor piece of news can cause people to buy or sell in anticipation that others will do the same and that others will realise this and do the same themselves. Markets can overshoot, until, when prices have got out of line with fundamentals, buying can turn into selling, or vice versa. Prices can then move rapidly in the other direction, again driven by what people think other people will do. Sometimes, markets can react to very trivial news indeed. As the New York Times article below states:

On days without much news, the market is simply reacting to itself. And because anxiety is running high, investors make quick, sometimes impulsive, responses to relatively minor events.

The rise of the machine
In recent years there is a new factor to account for growing stock market volatility. The Keynesian beauty contest is increasingly being played by computers. They are programmed to buy and sell when certain conditions are met. The hundreds of human traders of the past who packed trading floors of stock markets, have been largely replaced by just a few programmers, trained to adjust the algorithms of the computers their finance companies use as trading conditions change.

And these computers react in milliseconds to what other computers are doing, which in turn react to what others are doing. Markets can, as a result, suddenly soar or plummet, until the algorithms kick the market into reverse as computers sell over-priced stock or buy under-priced stock, which triggers other computers to do the same.

Robot trading is here to stay. The articles and podcast consider the implications of the ‘games’ they are playing – for savers, companies and the economy.

The Beauty Contest That’s Shaking Wall St. New York Times, Robert J. Schiller (3/9/11)
Speculative bubbles don’t just pop – they may deflate and reflate The Guardian, Robert Shiller (19/7/13)
A dark magic: The rise of the robot traders BBC News, Laurence Knight (8/7/13)
Stock markets under computer control BBC News, Robert Peston
Eunuchs of the Universe: Wall Street Today The Daily Beast, Tom Wolfe (4/1/13)

Questions

  1. Give some other examples of human behaviour which is in the form of a Keynesian beauty contest.
  2. Why may playing a Keynesian beauty contest lead to an undesirable Nash equilibrium?
  3. Does robot trading do anything other than simply increase the speed at which markets adjust?
  4. Can destabilising speculation continue indefinitely? Explain.
  5. Explain what is meant by ‘overshooting’? Why is overshooting likely to occur in stock markets and foreign exchange markets?
  6. In what ways does robot trading (a) benefit and (b) damage the interests of savers?

John Von Neumann was a mathematician and one of his many accolades was applying mathematics and his observations of traditional games to create a new discipline – Game Theory. This involves a mathematical approach to decision making whereby different strategies can be assessed. It a tool that not only can be used in Economics, but also can be applied to a broad range of areas and fields of study.

Just as I arrived at work, I was listening to Radio 4 and heard the introduction to the programme In our Time. This one in particular caught my attention because of the name mentioned – Von Neumann, and after arriving in my office I then listened to the discussions surrounding game theory.

The main link is to the discussion from BBC Radio 4, led by Melvyn Bragg, with guests: Ian Stewart, a Professor of Mathematics from the University of Warwick; Andrew Colman, a Professor of Psychology at the University of Leicester and Richard Bradley, a Professor of Philosophy from the LSE. I’ll keep it brief and simply say enjoy!

Podcast

Game Theory (also at) BBC Radio 4, In our Time, Melvyn Bragg (10/5/12) (Programme details)

Articles

Game Theory cannot predict broadcasting future Financial Times, Andrew Edgecliffe-Johnson (4/5/12)
Game Theory, in the real world Phys Org (2/5/12)

Questions

  1. What is game theory? How is mathematics relevant here?
  2. The discussion talks about co-operative and non co-operative games. What is the difference between them?
  3. In the game – walking down the street – draw out the matrix and show whether a Nash equilibrium exists.
  4. Draw out the matrix for the game ‘Rock, Paper, Scissors’. How can game theory be applied to this game? What is the best strategy to win this game? Can there be a winner?
  5. Draw out the matrix for the problem of littering when it is non co-operative. Is there a Nash equilibrium?
  6. What is the Prisoner’s Dilemma? Give some examples of it. Explain why it is an example of a dominant strategy game.
  7. How is game theory relevant to broadcasting? Think about the role of auctions and also the information given in the Financial Times article.
  8. Explain how game theory is relevant to the Cold War.

At the end of two weeks of often acrimonious wrangling between representatives from 193 countries, an agreement – of sorts – was reached at the climate change summit in Copenhagen. What was this agreement? It was an ‘accord’ brokered by the USA, China, India, Brazil and South Africa.

This Copenhagen Accord contains three elements. The first is a recognition of the need to prevent global temperatures rising by more than 2 degrees Celsius above pre-industrial levels. The second is a commitment by developed countries to give $30bn of aid between 2010 and 2012 to developing countries for investment in green technology and to mitigate the effects of climate change. In addition, a goal was set of providing $100bn a year by 2020. The third is for rich countries to give pledges on emissions reductions and for developing countries to give pledges on reducing emissions increases. Developed countries’ pledges will be scrutinised by the UN Framework Convention on Climate Change, while developing countries will merely be required to submit reports on their progress in meeting their pledges.

But this is only an accord. It has no legal status and was merely ‘recognised’ by the countries at the conference. What is more, the target of limiting temperature rises to 2C does not contain a date by which temperature rises should peak. Also, as countries are not required to submit targets for emissions until February 2010, it is not clear how these targets will be kept low enough to meet the temperature target and there is no identification of penalites that would apply to countries not meeting their pledges.

Not surprisingly, reactions around the world have been mixed. The following podcasts and articles look at these reactions and at the economic mechanisms that will be required to meet the 2C limit

Podcasts and videos

Recriminations after Copenhagen summit (video) BBC News, David Loyn (21/12/09)
Copenhagen special: Climate change talks end in failure Guardian podcast (19/12/09)
Where do we go after Copenhagen? BBC Today Programme (21/12/09)

Articles

What was agreed and left unfinished in U.N. climate deal Reuters of India Factbox (20/12/09)
Copenhagen deal: Key points BBC News (19/12/09)
Copenhagen deal reaction in quotes BBC News (19/12/09)
Copenhagen climate summit fails green investors BBC News, Damian Kahya (22/12/09)
Why did Copenhagen fail to deliver a climate deal? BBC News (22/12/09)
Copenhagen climate accord: Key issues BBC News (19/12/09)
Harrabin’s Notes: After Copenhagen BBC News, Roger Harrabin (19/12/09)
Copenhagen climate conference: Who is going to save the planet now? Telegraph, Louise Gray (21/12/09)
Copenhagen’s One Real Accomplishment: Getting Some Money Flowing New York Times, James Kanter (20/12/09)
Copenhagen climate summit: plan for EU to police countries’ emissions (including video) Telegraph, James Kirkup, and Louise Gray (19/12/09)
The road from Copenhagen Guardian, Ed Miliband (UK Secretary of State for Energy and Climate Change) (20/12/09)
Carbon Prices Tumble After ‘Modest’ Climate Deal Bloomberg, Mathew Carr and Ewa Krukowska (21/12/09)
Copenhagen deal causes EU carbon price fall BBC News (21/12/09)
Have the hopes of environmentalists been dashed? Financial Times, Clive Cookson (21/12/09)
EU reflects on climate ‘disaster’ Financial Times, Joshua Chaffinin (22/12/09)
China not to blame on climate China Daily, Zhang Jin (23/12/09)
Selling a low-carbon life just got harder Times Online, Jonathon Porritt (21/12/09)
Better than nothing The Economist (19/12/09)
Copenhagen has given us the chance to face climate change with honesty Observer, James Hansen (27/12/09)

Questions

  1. What incentives exist for countries to agree to tough pledges to reduce emissions?
  2. Was the very limited nature of the Copenhagen Accord a Nash equilibrium? Explain.
  3. Is the carbon price a good indicator of the effectiveness of measures to curb emissions?
  4. Must any agreement have verifiable targets for each country of the world if it is to be successful in curbing carbon emissions?
  5. Is a cap-and-trade system the best means of achieving emissions reductions? Explain.