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Nobel Prize Winners

Game Theory

This simple game reveals a great deal about human nature. For some time now, economists have been interested in game theory and its application to a wide range of situations that are highly relevant to economic analysis. Economists like to think they belong to a tradition that values scientific methodology but there is one big problem facing the subject, which is that economics is a social science and as such is dealing with human behaviour. Human behaviour is not predictable in the same way that force, momentum or light might be.

This does not stop economists looking what is called the law of large numbers and offering some considerable insight into the way in which economies work and how humans behave. The latest winners of the Nobel Prize for Economics have spent much of their lives investigating game theory and their research has some fascinating implications for policy makers and for analysis a series of economic issues.

Robert J. Aumann (left) and Thomas C. Schelling (right) received their awards for the Bank of Sweden Prize in Economic Sciences in memory of Alfred Nobel at the Stockholm Concert Hall, December 10th 2005. Copyright: Nobel Web AB 2005

Robert J. Aumann was born in Germany in 1930 and is the Emeritus Professor at the Centre for Rationality at the Hebrew University of Jerusalem. His background is in mathematics and he gained his PhD in mathematics at the Massachusetts Institute of Technology in 1955. His work has been in the field of game theory and specifically the effect of how cooperation can be of benefit, as a result of the research carried out into repeated games. In such circumstances, the short run benefits may be outweighed by the longer-term benefits of cooperation rather than individual optimisation.

Thomas C. Schelling was born in 1921 and was Emeritus Professor in the Department of Economics and School of Public Affairs at the University of Maryland. Schelling's contribution to game theory was set against the backdrop of the Cold War and the tensions between the West and the East and the nuclear arms race. His research on strategies has helped build understanding of conflict and conflict resolution in spheres as diverse as nuclear war and market structure.

Both Prize winners' ideas and research have helped provide a framework for a greater understanding of human behaviour. We see the potential for applying their ideas every day, as well as in high-profile events such as the G8 summit, the trade talks and Mr Blair's performance at the EU budget summit in December 2005.

Conflict and Strategy

When we look at human behaviour, there is a potential to think of it in terms of evolutionary biology - the survival of the fittest - and extrapolate from this an assumption that humans are selfish individuals who all act to maximise their own self-interest. Adam Smith, of course, suggested that if this were the case, then by definition the welfare of society as a whole would improve - the so-called invisible hand.

Conflict has been a feature of human life ever since humans first walked the planet. However, it is equally the case that cooperation between humans can lead to benefits to the participants that are greater than the cost of the cooperation.

In essence, therefore, every decision we make involves some potential pay-off but might also involve some sacrifice. If we are all acting as rational beings, then we might seek to maximise our pay-offs and minimise our sacrifices. The classic scenario on which this analysis is based is the 'Prisoners' Dilemma'. The following is a summary of this scenario.

The Prisoners' Dilemma

Two people are detained by the authorities on suspicion of committing a crime. The two are separated and then offered a series of choices by the authorities.

• If one attributes the crime to the other but the other says nothing, then the one staying silent will get 20 years in jail but the one offering the evidence goes free. This represents a betrayal strategy.
• If they both refuse to say anything then they will both receive only six months in jail due to the lack of evidence.
• If both say it was the other then they will each get 10 years.

Put yourself in this situation - which choice would you make?

For Player A, select either 'confess' or 'Do not confess'. The animation will then select a response for Player B. When that response is highlighted, you will be given the result on the screen! Try again to see all the options.

(Nb. if nothing appears below you should make sure your Flash player is up to date.)

If you are behaving rationally, you will need to think about your options to maximise your well-being.

• If you think your colleague is also going to behave rationally, then s/he will also be seeking to maximise their outcome.
• If you think your colleague will stay silent, then the best option is to betray your colleague because you will go free.
• If you assume that your colleague will also seek to maximise his/her situation, then you should still betray them because the outcome will only be a 10-year term instead of a 20-year one.

This is referred to as a non-zero sum game. The decision of one protagonist does not mean there can be no benefit for the other. In a zero-sum game, increasing the benefit for one would lead to a corresponding loss for the other - there is always a winner and a corresponding loser.

Schelling's work was based on the premise that the vast majority of instances of social interaction involved multiple individuals or groups with each having a mix of common interest and conflict.

Schelling identified two aspects of game theory - non-cooperative game theory and cooperative game theory.

• Cooperative game theory assumes that there are a set of outcomes or agreements that is known to each player and that each player has preferences over these outcomes.
• Non-cooperative game theory assumes players have a series of strategies they could use to gain an outcome and that each player has a preference over their desired outcome.

The other main concept we need at this point is that of the Nash Equilibrium. John Nash was another Nobel Prize Winner in 1994 and carried out research into strategies and solutions in game theory. A Nash Equilibrium exists if, in a non-cooperative game, there is no incentive for any player to change his or her strategy to achieve any benefit to them. There would only be a reason to change strategy if the player thought they could make some gain whilst other players kept their choices unchanged. The resulting pay-offs and the set of strategies constitutes a Nash Equilibrium.

Deterrence

Schelling further included other complications to the analysis by looking at how the strategies of each player would change in light of threats and action. One of the key features of Cold War strategy was the use of the terms 'first strike' and 'second strike capability'. 'If you send one missile to my land, I will strike back with 20 missiles on yours'. Such a deterrence might influence decision-making - but only if it is credible. In establishing such a deterrence, Schelling also noted that there could be an impact on the perception by other players of the country's interests and intentions, i.e. is this a deterrence strategy or an intention to get a first strike capability where you act and succeed in knocking out the other country's capacity to retaliate, therefore gaining a positive payoff?

Having confidence in knowing what your opponent will do in different circumstances, therefore, is vital if some form of equilibrium outcome is to be secured. Country 1 adopts the strategy of not mobilising unless Country 2 chooses to do so. What does Country 2 now do? It knows that if it mobilises then Country 1 will also mobilise - the payoff is zero. Country 2 therefore would be better off choosing not to mobilise. Schelling referred to such a situation as 'a balance of terror'. This translated into the idea of mutually assured destruction (MAD) and inspired the establishment of a hotline between the Kremlin and the White House following the Cuban Missile Crisis.

Brinksmanship

In analysing intentions and outcome, each player will be basing their decisions on some form of probability assessment. What is the probability that your opponent will mobilise? What is the probability that your opponent will retaliate to your actions? In such cases, Schelling noted the use of 'brinkmanship' - providing your opponent with a gradually increasing probability of zero payoff. A key to successful brinkmanship is to up the ante each time by only a small step.

It pays, therefore, to keep your opponent guessing as to your intentions. This might certainly be an analysis that could be applied to the way in which Tony Blair approached the negotiations at the EU Budget meeting. At the heart of game theory is the necessity of assessing the costs and the benefits of a strategy - not too difficult a concept in essence, but extremely powerful in reality.

Schelling then turned his attention to the issue of what happens to strategic decision-making in the long term as opposed to just a short-term view. Decisions made now might have an impact on the future state of negotiations between different parties. Mr Blair suggested that getting a budget deal now was important because it helped set the scene for future negotiations about the Common Agricultural Policy (CAP).

The incentive to forge an agreement and then renege on that agreement i.e. cheat, might be strong. However, Schelling noted that parties will need to recognise that the costs to them of cheating and gaining some short term benefit (which may be much greater) is far outweighed by the costs to them in the longer term of the destruction of the trust that results from cheating. The relationships between players will need to be assessed in the context of repeated playing of the game over a period of time. Certainly, the negotiations on the EU budget have not ended with the agreement in December 2005. The issue will come up again and the parties, whilst not necessarily the same people, will remember what has gone on before and factor this into their decision making.

Schelling's Other Contributions

The work on conflict and aggression that Schelling has worked on established his reputation in the field. However, he continued to work on many other implications of his research, which had an impact on a range of different fields other than war. One such example is his analysis of 'tipping', described as the rapid movement from one equilibrium state to another.

One aspect of this is highlighted in immigration policy. Take the East End of London as an example. During the first half of the twentieth century, the East End was largely composed of white British families. The immigration of different ethnic groups and the tendency for such groups to be concentrated into a particular area meant a fairly rapid change in the social structure of the area. In the Tower Hamlets area, for example, the concentration of communities from Bangladesh, India and Pakistan is relatively high. In such circumstances, white communities can quickly find themselves in the minority and this might not be something that the white community would want.

In such circumstances there might be significant micro and macro social and economic impacts as a result of this 'tipping'. What might happen is that if there are fears that a community might end up in the minority, they seek to move quickly to get out before this happens. In such a case the impact on property prices, local businesses, how the existing area is left and what effect this has on future investment, schooling, health provision and so on as well as the knock-on effects of the movements elsewhere could be quite dramatic.

Schelling also suggested that his ideas could be applied to other areas of social policy. Take smoking for example. Smokers impose costs on the health care system and on the social security system (if they are ill and cannot work then they need support). Should the government try to put in place incentives to persuade smokers to give up - such as taxes?

If you apply strict economic theory to the issue, the answer is 'no'. The reason? Smokers tend to die younger because of their habit. In so doing, they impose less of a cost on the health system than if they did not smoke and stayed alive longer! It is a simple case of cost and benefit. The benefit to the state of people smoking in terms of the overall costs to the health and social security system is greater than the cost of them giving up.

Schelling, himself a smoker who tried to give up over a period of twenty years, argued that despite the apparent logic of this rational analysis, the government in the US should impose taxes on tobacco. His argument was based around the intrapersonal conflicts faced by humans as individuals who do things that they know is not good for them - eating too much and eating the wrong sorts of food, smoking, drinking, exercising too little and not saving enough. In this scenario we are facing a 'game' whereby we face strategic options - to smoke and receive both positive and negative payoffs, or to quit and again receive negative and positive payoffs. The analysis is equally applicable and has relevance to public policy in such circumstances as the research did to military conflict in the 1960s. Witness the current debate over the decision to ban smoking in public places.

Long-term Cooperation

Robert Aumann made his contribution to game theory in a slightly different way. Schelling was noted for the accessibility of his ideas; Aumann's contribution is no less valid but is rooted in a mathematical tradition and as such is not as accessible to the general public.

Aumann's most noted contribution is in the field of long-term cooperation around game theory. We have already seen how, in the prisoners' dilemma, the optimum short-term strategy for each player is to betray the other. In so doing, however, they receive a payoff that is worse than if they cooperated and said nothing. Aumann asked the question about what the equilibrium outcome would be if the game were repeated over and over again, with each prisoner trying to maximise the average payoff from each game.

In this case, Aumann showed that the equilibrium outcome was to cooperate because any cheating on the agreement in the short term would be punishable by a refusal to cooperate at some point in the future - and both players would know this. Any short-term gains, therefore, are outweighed by longer-term losses. Aumann expressed this through what he referred to as a 'supergame' - that is, looking at the collection of repeated games as a whole game in itself.

Aumann's work was extended to look at how groups of players might react in such situations. In a cartel, for example, there is always the tendency or incentive for one firm to break the cartel to seek to gain some advantage in the market. Aumann's work suggested that long-term cooperation could be 'enforced' by the many against the few who might be seeking to defect.

The work was extended in subsequent research to try to take into account the strategies players might adopt in repeated games with incomplete information. This provides an incentive to players to hide, or seek to conceal, information from their rivals. Firms are very keen to keep their costs to themselves! If one player does manage to access information about their rivals and has some form of strategic advantage therefore, what is the best way to utilise this knowledge? If this situation arose, would playing your hand to gain short-term benefit reveal that you did actually know more than you were letting on? For the player who does not have the information they would like, could they discover anything about the player's position by reviewing the strategies and decisions made by that player in the past?

Such scenarios are relevant to the world of financial markets where the issue of insider trading is always something that the authorities are keen to stamp out. The number of people with access to privileged information about market moves, business plans and strategies does mean that there is potential for many decisions that could have long-term implications for the markets and the businesses involved.

Aumann and Schelling's work has a great many applications in everyday life. Trade wars, price wars, negotiations over budgets, wage negotiations, discussions about environmental issues, merger discussions, social policy, monetary policy, fiscal policy, the work of the Competition Commission, EU fisheries debates and oligopolies are just some of the major areas where their work can be applied.