Tag Archives: Game Theory

New Zealand Supermarket Duopoly – maximax and maximin strategies

Michael Cameron did a very relevant post on his blog ‘Sex, Drugs and Economics’ which focused on the duopoly market structure of New Zealand’s supermarkets. Part of the NCEA Level 3 and the CIE A2 economics courses look at market structures – monopolistic, oligopoly, duopoly, monopoly, and monopsony. A duopoly refers to two firms in a market whilst an oligopoly has a small number of firms but greater than two. Therefore we can say that Oligopoly and Duopoly are very similar market structures and they can co-ordinate their behaviour to exploit the market by lowering competition which in turn leads to greater profits for all.

Using the example of the supermarket duopoly Foodstuffs and Woolworths. Each company has two options – high price or low price. Obviously if they both price low they stand to be worse off and if they price high they are both set to gain. The outcome and payoffs are illustrated below:

Maximax – riskier strategy
A maximax strategy is one where the player attempts to earn the maximum possible benefit available. This means they will prefer the alternative which includes the chance of achieving the best possible outcome – even if a highly unfavourable outcome is possible. This strategy, often referred to as the best of the best is often seen as ‘naive’ and overly optimistic strategy, in that it assumes a highly favourable environment for decision making.

In this case, for both food providers, the aggressive maximax strategy is $140m from a low price and $120m from a high price, so a low price gives the maximax pay-off.

Maximin – conservative strategy
A maximin strategy is where a player chooses the best of the worst pay-off. This is commonly chosen when a player cannot rely on the other party to keep any agreement that has been made – for example, to deny.

In terms of the pessimistic maximin strategy, the worst outcome from a low price is $100m, and from a high price is $70m – hence a low price provides the best of the worst outcomes.

Again, lowering price is the dominant strategy, and the only way to increase the pay-off would be to collude and increase price together. Of course, this requires an agreement, and collusion, and this creates two further risks – one of the food companies reneges on the agreement and ‘rats’, and the competition authorities investigate the food companies, and impose a penalty.

Nash equilibrium
Nash equilibrium, named after Nobel winning economist, John Nash, is a solution to a game involving two or more players who want the best outcome for themselves and must take the actions of others into account. When Nash equilibrium is reached, players cannot improve their payoff by independently changing their strategy. This means that it is the best strategy assuming the other has chosen a strategy and will not change it. For example, in the Prisoner’s Dilemma game, confessing is a Nash equilibrium because it is the best outcome, taking into account the likely actions of others.

Game Theory and online cheating in tests

Michael Cameron’s blog Sex, Drugs and Economics had an interesting post regarding game theory and cheating in online assessment. He mentions a paper by Eren Bilena Alexander Matros entitled ‘Online cheating amid COVID-19’ in the Journal of Economic Behavior and Organization. They use a simple game theory model below to show the payoffs of the student and the professor with the student cheating or being honest.

Sequential-move game
In the sequential-move game, the student chooses to either cheat or be honest. The professor observes the student choice and decides either to report the student for cheating or not. There are four outcomes in this game, but the professor and the student rank these outcomes differently – see table below.

The table (right) gives an example of players’ payoffs. This game has a unique mixed-strategy equilibrium, which means that the student and the professor should randomise between their two actions in equilibrium. Thus cheating as well as reporting is a part of the equilibrium.

To find the Nash equilibriums you use the ‘best response method – for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the textbook definition of Nash equilibrium). In this game, the best responses are:

  1. If the student chooses to cheat, the professor’s best response is to report the student (since 3 is a better payoff than 2);
  2. If the student chooses not to cheat, the professor’s best response is not to report the student (since 4 is a better payoff than 1);
  3. If the professor chooses to report the student, the student’s best response is to not cheat (since 2 is a better payoff than 1); and
  4. If the professor chooses not to report the student, the student’s best response is to cheat (since 4 is a better payoff than 3).

A Nash equilibrium occurs where both players’ best responses coincide – note that there isn’t actually any case where both players are playing a best response.

In cases such as this, we say that there is no Nash equilibrium in pure strategy. However, there will be a mixed strategy equilibrium, where the players randomise their choices of strategy. The student should cheat with some probability, and the professor should report the student with some probability.

Source: Sex, Drugs and Economics – Combating cheating in online tests

Oligopoly – Game Theory and Dominant Strategy

Been covering this topic with my A2 class and found Jason Welker’s video very good at explaining Prisoner’s Dilemma and the dominant strategy.

In recent years game theory has become a popular way of examining the strategies that oligopolists may adopt in a market. Game theory involves studying the alternative strategies oligopolists may choose to adopt depending on their assumptions about their rivals’ behaviour. Put at its simplest, if a firm is considering reducing its price, in making its decision it will need to take into account how its rival oligopolists might react and how it will affect them. Firms can choose high risk or low risk strategies in what is very similar to a game of poker between four or five players.

Although game theory strategy involves some extremely difficult maths the A2 Economics course concentrates on relatively straightforward two-player zero-sum games where one player’s gain must equal the other player’s loss.

Game Theory and why we go to a coronavirus lockdown

Michael Cameron posted a nice piece on his blog about this – Sex, Drugs and Economics.

Game theory refers to the decision that a firm/individual makes depends on its assumptions about other firms/individuals. Ultimately this means that individuals will try and calculate the best course of action depending on how others behave. When we were in a Level 2 situation the advise was no mass gatherings, physical distance on public transport, limit non-essential travel etc. Although the announcements of the four levels were made clear on the Saturday it was inevitable that we would be moving to Level 4 very quickly – Wednesday. Being able to police Level 2 would have been near impossible and the risk of community transmission meant that complete lockdown was needed.

In a Level 2 situation, which was somewhat voluntary, people had two choices – Stay home (cooperate) or Act Normally (defect). The table below looks at the payoffs if you don’t lockdown early and already have community transmission.

As Cameron points out: For most people, acting normally is a dominant strategy, at least in the early stages of the coronavirus spreading. They are better off acting normally if everyone else stays home (because they mostly get to go on with their lives as normal, and have low risk of catching the coronavirus; whereas staying home they would be giving up on things they like to do), and they are better off acting normally if everyone else is acting normally (because life goes on as normal, rather than giving up on things they like to do). So, individually people are better off acting normally.

Any voluntary measure is subject to the prisoners’ dilemma which is why we went to an early full lockdown. As we are in lockdown repeated games requires trust and the correct behaviour outlined by the government – this is essential to eliminate the virus. Therefore the dominant strategy of ‘Act Normally’ is no longer an option. Cameron quoted Robert Frank whom I have blogged on here

‘smart for one, dumb for all’.

Stay safe everyone

Repeated Prisoner’s Dilemma Tournament

I have played this with my classes and although not in the CIE A2 syllabus, found it useful to go into greater detail as to the strategies available.

Robert Axelrod used an experimental method – the indefinitely repeated PD tournament – to investigate a series of questions: Can a cooperative strategy gain a foothold in a population of rational egoists? Can it survive better than its uncooperative rivals? Can it resist invasion and eventually dominate the system? Contestants submitted computer programs that select an action, Cooperate or Defect, in each round of the game, and each entry was matched against every other, itself, and a control, RANDOM.

Prisoner’s Dilemma in the classroom.

This game can be played over as many rounds as you wish and played between two players in the classroom environment. The pay offs shown – Win = 3 or 5, Lose = 0 or 1

Below shows the first 14 rounds of a 100 round PD game between John and Kate that includes the comments that were apparently written after each player had decided on strategy in that particular game, but before the other player’s choice was known.

Source: William Poundstone – Prisoner’s Dilemma 1992

Strategies – see below

  • D (ALWAYS DEFECT): Defect on every move.
  • C (ALWAYS COOPERATE): Cooperate on every move.
  • T (GRIM – TRIGGER): Cooperate on the first move, then cooperate after the other cooperates. If the other defects, then defect forever.
  • TFT (TIT FOR TAT) cooperates in the first round, and then does whatever the opponent did in the previous round.
  • PAVLOV: 1st round, Cooperate. Thereafter if you win use the same action on next round. If you lose switch to the other action.

PAVLOV v M5. Time-average payoffs can be calculated because any pair will achieve cycles, since each player takes as input only the actions in the previous period. Here there is an average of 2 per player per cycle.

Action bias and penalty kicks – is it best if the goalkeeper does nothing?

Action bias is a situation where we would rather be seen doing something than doing nothing. This has been the case in numerous government elections as the voting population like to see some action from politicians when in some cases the best option is to let the economy run its course. President Nixon (US President 1969-74) was a great one for doing something even though it would have been better to do nothing – I refer to the wage and price controls introduced in 1971 – the controls produced food shortages, as meat disappeared from supermarket shelves and farmers drowned chickens rather than sell them at a loss. So when the economy is doing badly the government maybe tempted to intervene, even if the risks associated with the changes not necessarily outweigh the possible benefits. Furthermore if an economy is doing well policy makers may feel that they shouldn’t do anything even though the changes could improve the economy further.

According to classical assumptions in economics, when a people face decision problems involving uncertainty, they should choose what to do according to their utility from the possible outcomes and the probability distribution of outcomes that follows each possible action. Bar-Eli, Azar, Ritov, Keidar-Levin, & Schein, 2007

In a 2007 study, Michael Bari-Eli at the Ben Gurion University of the Negev, Israel, analyzed 286 professional soccer penalty kicks. They discovered that goalkeepers almost always jump right or left because the norm is to jump — a preference for action (”action bias”). The goalkeepers jumped to the left 49.3% of the time, to the right 44.4% of the time, but stayed in the centre only 6.3% of the time. Analysis revealed that the kicks went to the left 32.2%, to the right 39.2% and to the centre 28.7% of the time. This means that the goalkeepers were much more likely to stop a kick if they had just stayed put – see table below.

The table above suggests that the decisions taken by the kicker and goalkeeper are made roughly simultaneously. The fact that the directions of the kick and the jump match in 43% of kicks rather than in 0% or 100% of the kicks suggests that neither kicker nor goalkeeper can clearly observe what the other chose when choosing their action.

A goalkeepers’ decision making.

In order to suggest a best option for goalkeepers it is necessary to examine the probability of stopping the ball following each combination of kick and jump directions. The table below presents the average saving chances using the formula

Number of penalty kicks saved ÷ Number of penalty kicks x 100

Jumping left = 20 ÷ 141 x 100 = 14.2%
Staying Centre = 6 ÷ 18 x 100 = 33.3%
Jumping right = 16 ÷ 127 x 100 = 12.6%

The research conclusions state that goalkeepers jump to the right or the left during penalty kicks more than they should. In analysing the 286 kicks Bar-Eli et al show that while the utility-maximising behaviour for goalkeepers is to stay in the goal’s centre during the kick, in 93.7% of the kicks the goalkeepers chose to jump to their right or left. This non-optimal behaviour suggests that a bias in goalkeeper’ decision making might be present. The reason that they suggest is ‘action bias’. However you also need to look at the psychological aspects of a goalkeeper. Arsenal and former Chelsea goalkeeper Petr Cech said that he never liked to stay in the centre as it might look to the fans that he wasn’t trying. Although he would be in a good position to save a penalty that was kicked down the centre, he would feel a lot worse if he stayed in the centre and the ball went into the goal either side of him.

Sources:

Bar-Eli, M., Azar, O. H., Ritov, I., Keidar-Levin, Y., & Schein, G. (2007). Action Bias Among Elite Soccer Goalkeepers: The Case of Penalty Kicks. Journal of Economic Psychology, 28(5), 606-621.

An “Action Bias” Can Be Counterproductive

Aussie Cricketers, Sandpaper and Game Theory

You will no doubt have heard of the ball tampering episode at the third cricket test in Cape Town between South Africa and Australia. Australian cricketer Cameron Bancroft was caught by a TV camera roughening up the ball with some yellow sandpaper that was kept in his pocket. He was seen later getting rid of the sandpaper down his trousers so that when the umpires asked him about he produced a sunglasses bag. Later on that day at the post play press conference Australian captain Steve Smith came clean with what was a premeditated plan to roughen one side of the ball and so that the Australian bowlers could take advantage of reverse swing. Tampering with the ball is illegal in cricket and the ICC (cricket’s governing body) banned Steve Smith for one game

I have blogged previously on game theory in sport looking at – Penalty Kicks in Football and How doping impacts Athletes, Organisers and Supporters. As prize money and sponsorship deals get bigger, so do the incentives for coaches and players to find ingenious ways to cheat. So how would the sandpaper incident at the third cricket test in Cape Town lend itself to game theory?

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 cricketers and, although both are better off if neither tampers with the ball, they cannot trust each other so both engage in ball tampering – Prisoners Dilemma. If you introduce an authoritative figure – the International Cricket Council (ICC) – to observe cricketers with many camera angles, the fear of getting caught should ensure that no ball tampering takes place. This is referred to as the inspection game. However as you know it wasn’t the ICC who took strong action over the video footage but Cricket Australia.

Another party that is crucial to cricket is sponsors and the spectators. Their critical role is the potential withdrawal of support which could see the cricket’s demise. Wealth-management company Magellan has terminated its three-year sponsorship agreement with Cricket Australia in response to the ball-tampering scandal worth around AUS$20 million.

A withdrawal of one of these three parties can trigger the withdrawal of the other two. Cricket cannot survive without sponsors, withdrawal of the media restricts the access to the customers, and finally cricket is only attractive for sponsors as long as there are customers. Therefore the strategies of the three parties looks like this:

Cricketers – Ball tamper or Don’t ball tamper (B D)
ICC – Video or No Video (V N)
Sponsors / Spectators – Stay or Leave (S L)

The assumptions are as follows:

Cricketers

B-N-S > D-N-S = cricketers prefer to ball tamper if not videoed.
D-V-S > B-V-L = cricketer prefer not to ball tamper and be videoed = customers stay, over being ball tampering and videoed = customers leave (assuming that customers don’t like ball tampering scandals)

Organisers

B-N-S > D-V-L = a scandal combined with a loss of customers is worse for organisers than undetected video where customers stay.
D-V-S > D-N-L = videoing and non ball tampering actions with customer support is better for the organisers than not videoing other cricketers when customers leave.

Sponsors / Spectators
B-V-L > B-V-S = customers prefer to withdraw support after a scandal
B-N-S > B-N-L = customers prefer to stay if there is no scandal.
D-V-S > D-T-L = customers prefer to stay if there is no scandal.
D-N-S > D-N-L = customers prefer to stay if there is no scandal.

Ball tampering & Video = Scandal
Ball tampering & No Video, Don’t ball tamper & Video, Don’t ball tamper & No Video = No scandal

Final thought
The vast majority of authorities in today’s sports events would state that their regimes to combat cheating are very stringent. However the likelihood of human deceitfulness is very realistic and in some cases it’s not those that tamper with the cricket ball who are the real cheats but those who have generated an environment where players would be foolish not to.

Repeated games may help climate change negotiations

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.

Sources: 

  • 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

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

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.