Posts Tagged ‘Game Theory’

Repeated games may help climate change negotiations

November 22, 2016 Leave a comment

Reading a post from Michael Cameron’s blog reminded me of how repeated games of the prisoner’s dilemma may help climate change negotiations.

The Paris Agreement came in to effect on 4th November this year and it brings all nations into a common cause to undertake take ambitious efforts to combat climate change and adapt to its effects, with enhanced support to assist developing countries to do so.

The main issue with tackling climate change is the cost to countries of implementing it. To be successful it will need profound transformation of energy and transport organisations, and changes in the behaviours of billions of consumers. Research has suggested that it will likely cost 1% of GDP – even though it doesn’t seem much, it is double the amount currently spent on development aid worldwide.

A successor treaty? 

According to Michael Liebreich, the prospects don’t look good when you consider the following:

  • The US sees a cap on carbon emissions as a threat to competitiveness, and hence to its global supremacy. Add to this the rhetoric of President elect Donald Trump which has dismissed global warming.
  • The developing world denounces any calls for a cap on emissions as an effort by former colonial powers to hold back development;
  • Europe has been making encouraging though patchy progress towards targets, driven mainly by a one-off switch from coal to gas.

The issue here is how countries can expect to make cuts in emissions when their economic competitors refuse. This in turn leads to The Tragedy of the Commons which occurs when a group’s individual incentive lead them to take actions which, overall, lead to negative consequences for all group members.

Climate Change as Prisoner’s Dilemma
The initial impression from the discussions over climate change is that of a typical Prisoner’s Dilemma. As mentioned previously, the cost of tackling climate change is approximately 1% of annual per capita GDP. However, if nothing is done about the issue the cost is estimated to be between 5% to 20% of GDP. So that defines what happens at the extreme of cooperative or non-cooperative behaviour.

Climate Change - Pris Dil

Form the table above, a country that refuses to act, whilst the other cooperates, will experience a free-rider benefit – enjoying the advantage of limited climate change without the cost. On the flip side, any country that imposes limits, when its competitors do not, incurs not just the cost of limiting its own emissions, but also a further cost in terms of reduced competitiveness – estimated here at an additional 3.0%.

From the table it seems predictable that countries should prefer to be self-interested: the best national policy, if others reduce emissions, is to defect; likewise, if other countries are not taking action, then it is pointless to be the only sucker to take action, and one should again defect.

Repeated Prisoner’s Dilemma and Cooperation 

The dynamics of the prisoner’s dilemma do change if participants know that they will be playing the game more than once. In 1984 an American political scientist at the University of Michigan, Robert Axelrod, argued that if you play the game repeatedly you are likely to see emerging is cooperative rather than defective actions. He identified four elements to a successful strategy which is this case can be applied to climate negotiations:

1  Be Nice – sign up to unilateral cuts in emissions, as deep as your economy and financing capacity allows.

2  Be Retaliatory – single out countries that have not commenced action and, in collaboration, find ways of pressurising them until they do so.

3  Be Forgiving – when non-compliant countries come onboard give them generous applause; signal that good behaviour
will be rewarded with even deeper cuts in your own emissions.

4 Be Clear – let everyone know in advance exactly how you are going to behave – that you will work with them if they take action on emissions, and that you will retaliate if they do not.

It is the belief of Michael Liebreich that this research by Axelrod should be put into practice by the world’s climate negotiators. As treaties on climate change are on-going and therefore become part of the game.

Final thought 

Repeated Prisoner’s Dilemma provides valuable insight into how countries should act away from the negotiating table and over the longer term. This analysis also highlights the fact that the negotiations themselves are not the game. Diplomats and politicians don’t reduce emissions, engineers and consumers do. However, there are errors in the resemblance as governments can form alliances, which makes the dynamics of the game a great deal more complex. Furthermore, they can act inconsistently and irrationally, and their willingness to act is most probably associated with the harshness of global warming. Ultimately, for the planet’s sake, one hopes that everyone will play the game.


  • The Economist – Economics Focus: Playing with the planet. 29th September 2007 
  • New Energy Finance – How to Save the Planet – Michael Liebreich– 11th September 2007 

Greek Crisis – A game of chicken

June 6, 2015 Leave a comment

Chicken gameThe negotiations between Greece and the Eurozone financial chiefs represent a typical game of ‘Chicken’. Chicken readily translates into an abstract game. Strictly speaking game theory’s chicken dilemma occurs at the last possible moment of a game of highway chicken. Each driver has calculated his reaction time and his car’s turning radius, which is assumed to be the same in both cars. There comes a time when each driver must decide to either swerve or keep going straight towards the other car. This decision is irrevocable and must be made in ignorance of the other driver’s decision. There is no time for one driver’s last-minute decision to influence the other driver’s decision. In its simulations, life or death simplicity, chicken is one of the purest examples of John von Neumann’s concept of a game. The way players rank outcomes in highway chicken is obvious. The worst scenario is for both players not to swerve – they crash and both are killed. The best thing that can happen is for you to keep driving straight letting the other driver swerving. The cooperative outcome is not so bad as both drivers are still alive although no one can call the other chicken.

As in the game of Chicken, both Greece and the Eurozone have the option to make concessions (Swerve) or hold firm in negotiations (Drive Straight). As with most negotiations, the best outcome for a party is to stand their ground while the other party makes the concessions. However, as both parties want this outcome, this raises the possibility of both sides holding firm and no settlement being reached. In the Greek-Eurozone crisis, this would mean a Greek default and the associated consequences that would ensue for the rest of the Eurozone.

Fortunately there is a third outcome that can prevail in Chicken – both parties can swerve their car at the same time. If both sides are willing to make concessions, then the second best outcome in this game can be attained for everyone. This co-operative outcome could be reached if the Eurozone extended further concessions to Greece, while Greece made binding promises to implement meaningful reforms to get their economy back on track.

However this is unlikely as each player achieves their best outcome by doing the opposite of their opponent. For example, if Greece believes the Eurozone will make concessions, it will achieve the best outcome by standing firm; if it believes the Eurozone will stand firm in negotiations, it’s best option is to make concessions to avoid the dire consequences of a full-blown default.

Chicken - Greece Germany

From the beginning of June until the end of December Greece needs to find another EUR28bn in total. After that point repayments drop off – one reason why Greece’s creditors are keen to ensure new reforms are enacted ASAP.
The inference however is clear: Greece won’t make it that far without a new deal. Greece is waiting on further funding from the IMF and the ECB (EUR 7.2bn) in order to meet some of these payments, but with both sides digging in, it isn’t a given that Greece will receive the funds. See graph below.

Greece repayments

Sources: NAB Australian Markets Weekly, Christoph Schumacher Massey University, Open Economy – Open minded Economics, Prisoner’s Dilemma – William Poundstone

John Nash – 1928-2015

May 25, 2015 Leave a comment

Sadly John Nash, the Nobel Prize-winning mathematician whose life story was the subject of the Academy Award-winning film “A Beautiful Mind” died yesterday in a car crash – his wife Alicia was also killed. Best known for advances in game theory, which is essentially the study of how to come up with a winning strategy in the game of life — especially when you do not know what your competitors are doing and the choices do not always look promising.

From 1930 to 1940 mathematician John von Neumann did the pioneering work to establish the field of game theory although Nash extended the analysis beyond zero-sum, I-win-you-lose types of games to more complex situations in which all of the players could gain, or all could lose.

Nash Equilibrium
In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

The film “A Beautiful Mind,” based on Dr. Nash’s life, tries to explain game theory in a scene in which Russell Crowe, playing Dr. Nash, is at a bar with three friends, and they are all enraptured by a beautiful blond woman who walks in with four brunette friends – see video below.

While his friends banter about which of them would successfully woo the blonde, Dr. Nash concludes they should do the opposite: Ignore her. “If we all go for the blonde,” he says, “we block each other and not a single one of us is going to get her. So then we go for her friends, but they will all give us the cold shoulder because nobody likes to be second choice. But what if no one goes to the blonde? We don’t get in each other’s way and we don’t insult the other girls. That’s the only way we win.”

While this never-happened-in-real-life episode illustrates some of the machinations that game theorists consider, it is not an example of a Nash equilibrium.

Game theory and drug taking in sport- Athletes, Organisers and Spectators

May 11, 2015 Leave a comment

Lance ArmstrongDrug-taking in professional sport has long been a major concern and there is no better example than seven times Tour de France winner Lance Armstrong who admitted to doping. Furthermore in the period from 1997 until 2002 among 64 world class 100 metre sprinters 25% have been convicted of doping and this doesn’t include two American sprinters who tested positive this year. So why do athletes continue to dope eventhough you could get banned for life if caught?

Prisoners Dilemma to Inspection Game to Metagame

Game theory deals with differences of opinion between groups who know each other’s inclination but not their genuine objective or choice. It then concludes the optimum course of action for any rational player. In this scenario the parties involved are the competing athletes and although both are better off if neither takes drugs, they cannot trust each other so both engage in doping – Prisoners Dilemma. If you introduce an authoritative figure – the organisers – to test the athletes the fear of getting caught should ensure that athletes remain clean, referred to as the Inspection Game. However this cannot be said to happen at leading sports events as athletes, on the whole, don’t think they will get caught. Researchers from the University of Hamburg have introduced yet another party which they refer to as the customer (sponsor and spectators). Their critical role is the potential withdrawal of support which could see the sport’s demise. A withdrawal of one of these three parties can trigger the withdrawal of the other two. Sports events cannot survive without sponsors, withdrawal of the media restricts the access to the customers, and finally sport is only attractive for sponsors as long as there are customers. Therefore the strategies of the three parties looks like this:

Athletes – Dope or Clean (D C)
Organisers – Test or No Test (T N)
Customer – Stay or Leave (S L)

In the figure below organisers decide on the testing the athletes whether there was doping or not. But more importantly customers are to be informed about doping tests that turn out to be negative as well as positive. The customers then decide whether to stay or leave.

Tree - Game Theory

The assumptions are as follows:

D-N-S > C-N-S = athletes prefer to dope if not tested.
C-T-S > D-T-L = athletes prefer to be clean and tested = customers stay, over being doped and tested = customers leave (assuming that customers don’t like doping *scandals)

D-N-S > D-T-L = a scandal combined with a loss of customers is worse for organisers than undetected doping where customers stay.
C-T-S > C-N-L = testing clean athletes with customer support is better for the organisers than not testing clean athletes when customers leave.

D-T-L > D-T-S = customers prefer to withdraw support after a scandal
D-N-S > D-N-L = customers prefer to stay if there is no scandal.
C-T-S > C-T-L = customers prefer to stay if there is no scandal.
C-N-S > C-N-L = customers prefer to stay if there is no scandal.

*Dope & Test = Scandal
Dope & No Test, Clean & Test, Clean & No Test = No scandal

In reality customers who are ready to leave after doping scandals undermine the incentives to test athletes and find them guilty of doping. Consequently this encourages athletes to use performance- enhancing drugs and organisers to reduce their anti-doping methods in order to preserve the economic worth of the event – eg the Olympic Games.

Most athletes that have been found guilty of doping are not delinquent exceptions, but just unlucky scapegoats because the probability of being caught is low. The solution suggested would be to establish transparency so that the customer would know the results for all tests whether they were positive or negative. This allows the customer to condition their support on the presence of serious anti-doping tests. In practical terms this transparency could create a rating for each event based on how rigorous their anti-doping policy is.

Final thought

The vast majority of authorities in today’s sports events would state that their testing regimes were very stringent. However the likelihood of human deceitfulness is very realistic and in some cases it is not those that take the performance enhancing drugs who are the real cheats, but those who have generated an environment where athletes would be foolish not to.


Nobody’s Innocent – The Role of Customers in the Doping Dilemma. Berno Buechel et al. University of Hamburg. January 30th 2013

Athlete’s dilemma – The Economist Print Edition July 20th 2013

Somali Pirates – Protection and the hijack v re-hijacking dilemma

October 1, 2014 Leave a comment

You may have seen the movie “Captain Phillips” about the hijacking of a shipping container off the coast off Somalia by pirates. Recent research by Anja Shortland and Federico Varese mapped the locations of hijacked ships between 2005 and 2012 and found that:

1. Hijacked vessels were always anchored far away from regional trading routes
2. Big ports were not prone to piracy.

The rationale for this is that Somali clans directly influence local trade by issuing licences and also imposing informal taxes on imports and exports. Clans refuse to protect pirates because the income they get from trade is safer and more lucrative that those they can get from pirates. Clans that have little dealings in formal trade tend to offer protection to pirates in order to get a share of their spoils.

This was evident during the ban on Somali livestock imports imposed by Saudi Arabia between 2000 and 2009. With most Somalis being farmers and Saudi Arabia being their main export destination they were hit hard. Therefore in order to secure income the farmers in the coastal regions offered protection to pirates. However once the embargo was lifted and income once again flowed in from trade. Subsequently pirates found it much harder escaping arrest.

Hijack and re-hijack dilemma

The same authors looked at the hijack and re-hijack dilemma.

Consider two groups of pirates. Each group has one ship to ransom, but can in addition re-hijack the other group’s ship and negotiate a further ransom. The optimal outcome is achieved if both commit to not hijacking each others’ ship. This lowers protection costs and raises the confidence of ship-owners that the ransom payment will release the ship immediately. If players engage in re-hijacking, protection costs rise and the ship-owners would reduce the ransom to make provisions for an additional ransom and a longer loss of hire. However, each group has an incentive to re-hijack the ship from the other group to share in this (reduced) ransom. Mutual re-hijacking would therefore be the dominant strategy in the game illustrated below: with finite time horizons, the co-operative outcome would need to be enforced.

The hijack /re-hijack dilemma
Co-op rehijack

Avoid the Germans in a Penalty Shootout

July 7, 2014 Leave a comment

A particular interest of mine is game theory and penalty shootouts – see Game Theory Lesson: Man Utd v Chelsea Penalty Shootout. This involves studying the alternative strategies a person may choose to adopt depending on their assumptions about their rivals’ behaviour. The most significant research into penalty kicks has been done by Steven Levitt (co-author of Freakanomics) who co-authored a paper on mixed strategies when players are diverse in their decision making and studied 459 penalties in the Spanish and Italian leagues. Another economist Ignacio Palacios-Huerta analysed 1417 penalty kicks from several European countries during the period 1995-2000.

Approaching the business end of the Football World Cup and you have to have some sympathy for the Costa Ricans who celebrated after extra-time with the scores being tied – assuming that this was their plan from the commencement of the game and that they fancied their chances in the shootout. However why do some teams do better at penalty shootouts than others? The Economist looked at this recently and came up with the following:

1. Defeat is habit forming – players who miss regularly become fatalistic
2. Countries that are collectivist in nature rather than individualistic do much better – mindful of their public image
3. Research shows that star players tend to miss more than your average player – they feel the pressure
4. Who goes first is important as players are far more likely to miss a penalty if it to stay in the contest rather than one that will win it. The majority of the time the team going first wins the shootout.

Below is a chart from The Economist which shows that England have struggled at this part of the game but the Germans, as you would expect, are very efficient. As for the Czech Republic they have yet to miss a penalty.

Penalty Shootout stats

Psychological Preparation for Penalty Shootouts – Academic Paper

A recent piece of research from the British Association of Sport and Exercise Sciences addresses how you should prepare for Penalty Shootouts.

Players who take less than one second to place the ball on the penalty spot score on about 58% of their penalties whereas those who take longer score on about 80% of their penalties (Jordet et al., 2009).

Similarly, taking about a second or more to respond to the referee’s whistle to initiate the shot is associated with a higher probability of scoring than immediately rushing towards the ball (Jordet et al., 2009).

Developing and practising a suitable pre-shot routine is a potentially useful way to guide these timings and help protect performance under pressure. Indeed, recent research by Wood and Wilson (2012) has suggested that learning a routine involving a gaze control element (look at the point where you want to shoot prior to the run-up) helped penalty takers in a shootout task to be more accurate, maintain effective visuomotor control and increase perceptions of psychological control and contingency.

In the Shootout:

1. Don’t rush: Place the ball properly on the spot and take a breath while focusing on where you intend to shoot, before starting the run-up. Taking a deep breath is likely to ease feelings of anxiety and provides a temporal cue to ensure that sufficient processing of target-related information is enabled.
2. Trust your technique and routine – pick a spot and hit it.
3. Celebrate! It will help your team-mates who have to take the subsequent penalty kicks.

With the semi-finals coming up you’ve got to fancy the Germans if it goes to a penalty shootout.

Game Theory Lesson: Man Utd v Chelsea Penalty Shootout

March 11, 2014 2 comments

Game theory involves studying the alternative strategies a person may choose to adopt depending on their assumptions about their rivals’ behaviour. The most significant research into penalty kicks has been done by Steven Levitt (co-author of Freakanomics) who co-authored a paper on mixed strategies when players are diverse in their decision making and studied 459 penalties in the Spanish and Italian leagues. Another economist Ignacio Palacios-Huerta analysed 1417 penalty kicks from several European countries during the period 1995-2000.

Technically the kicker and the goalkeeper play a zero-sum game – any gain for one player is exactly offset by the loss to the other side – plus one goal for me is minus one goal for you. The situation that kickers face in a penalty kick is a simultaneous-move game where they have three alternative strategies: shooting right, left, or centre. Similarly the goalkeeper also has three alternative strategies: dive to the right, dive to the left or remaining in the centre of the goal. In defining the sides of the goal researchers use the “natural side” of the kicker (which is the goalkeeper’s right, if the kicker is right-footed, and the goalkeeper’s left, if the kicker is left-footed) and the “opposite side”. Labeled like that, the strategies of both kicker and goalkeeper will be to choose the natural side of the kicker (NS), the centre (C) or the opposite side (OS).

2008 UEFA Champions League final – Chelsea v Manchester Utd.

A mutual friend of Ignacio and Chelsea manager, Avram Grant, brought the two men together and subsequently Ignacio sent Grant some facts regarding Man Utd, in particular about their goalkeeper Van der Sar. There were 4 main points:

1. Man Utd goalkeeper (Van der Sar) tended to dive to the kicker’s natural side (i.e. GK’s right for a right footed kicker)

2. Van der Sar tends to save penalties that are hit at mid-height

3. Man Utd midfielder Cristiano Ronaldo often stops in his run-up and if he does the ball is kicked towards the right hand side of the keeper. It was important that the Chelsea goalkeeper, Petr Cech, did not move early. When goalkeepers moved early Ronaldo always scored.

4. If you win the toss you take the first penalty. 60% of teams going first win the game.

Students watch the penalty shootout between Manchester United and Chelsea and complete the following table. They have to record where the penalty taker placed the ball and which way the goalkeeper dived – NS C OS. They then answer the questions below.

Man U Chelsea Penalty Shootout

What do you notice about the Chelsea penalty takers?



Was Ignacio correct about Cristiano Ronaldo?



How did Man Utd goalkeeper (Van der Sar) psych out Anelka the 7th penalty taker for Chelsea? Watch the video closely.



What did Anelka do which went against the advice of Ignacio?



Ignacio stated in an academic paper that generally the shooter should kick to the left 38% of the time and right 62% of the time. The keeper should dive to the shooter’s left 42% of the time and to the shooter’s right 58% of the time. When both strategies are played out, the shooter will score 80% of the time. This is the best strategy for both players because any deviation from the strategy by the keeper will result in the success rate increasing while any deviation from the strategy by the shooter will result in the success rate decreasing.

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