1.2 Game Theory and Archetypical Coordination Problems Flashcards
- Please, distinguish cooperative and non-cooperative games.
In cooperative games the players can make costless fully binding and enforceable (e.g legal sanction) commitments and in a non-cooperative game the players CANNOT make fully binding and socially enforceable commitments. E.g Pure conflict game: football game – non cooperative game.
An example the cooperative game could be purchase of a house, interests of buyers are against interests of sellers, but if a deal is struck, it is generally enforceable. Non-cooperative game: bargaining between employer and employee about working hours and salary - the other part of the same interaction may be noncooperative because of the impossibility of writing or enforcing the relevant contracts.
- What difference does infinite repetition of the Prisoner’s dilemma make? What will be the outcome of this coordination game in this case and why, explain.
If the Prisoner’s dilemma game is played just one time, the strategy of defecting is the rational one. There is no way for the two prisoners to coordinate and influence each other’s behaviour. The situation is different if the game is played repeatedly indefinite number of times. It opens new strategic possibilities. The players can be punished – if one player choose to defect on one round, other can defect on the next round. Also players can establish a reputation, which helps to cooperate. As long as players are interested in the future payoffs, they can be convinced to play Pareto efficient strategy. If the game is going to be repeated an infinite number of times, then you do have a way of influencing your opponent’s behaviour: if he refuses to cooperate this time, you can refuse to cooperate next time. It calls “tot for tat” strategy – do whatever the other player did in the last round. It is a good mechanism for achieving the efficient outcome in a prisoner’s dilemma that will be played infinite times.Conclusion: If the prisoner’s dilemma is repeated indefinitely, then it is possible that the Pareto efficient outcome may result from rational play.
- Provide empirical examples of the prisoner’s dilemma, a pure coordination game, a pure conflict game, an assurance game, a chicken game, a battle of the sexes’ game.
Prisoner’s dilemma
Two fishers share a lake and catch fish for their own consumption. There is a lot of fish in the lake so that more fishing would always yield more fish for the two. However, the more one catches the fewer the other catches. For each there is a option that would yield higher payoffs when taken, in this case to increase their hours of fishing a day. However, this would make the outcome worse for both by overexploiting the resources.
Pure coordination game
A firm consists of an employer and an employee. If the firm succeeds both win (1) if they fail both lose (0). The probability of success depends on their actions (employer: invest, employee: work hard). The firm will definitely succeed if the employer invests and the employee works hard. If only the employee works hard or the employer invests the form will survive with a probability of p1 or p2.
Pure conflict game
A dollar is to be divided between two individuals; without prior communication they claim an amount. If they amount to more than 1 each gets 0. Assurance game Farmers in a rural area sow their crops several weeks after the optimal sowing date at which yield could be maximised. The reason is that farmers do not want to be the first to sow because the seeds could quickly be eaten by birds. The optimal situation would be if all farmers sow their seeds early, because every farmer would increase their yield. Another Nash equilibrium would be that all famers sow late, this is however not a social optimum as the yields are reduced. Only one farmer planting earlier would lead to most seeds being eaten and a bad crop production in that year for that specific farmer.
Chicken game
Two people in their cars are sitting in their cars facing each other. They start driving towards each other until one person “chickens out” and swerves.
Battle of the sexes’
A boy and a girls want to go the movies together. The guy prefers to see an action and the girl and art movie, but they both prefer to see a movie together. Both forgot their cell phone and have no option to coordinate their date night. Both would rather see any type of movie than not to go at all.
- Describe pure conflict and pure coordination games and provide examples. Compare key features of the two types of games referring specifically to the concepts of social optimum and Nash Equilibrium.
A pure conflict game can be good described by a soccer game when someone is kicking (left or right) a penalty and the other is defending (left or right). It is a game of competition. Because of that there is no dominant strategy (because the best strategies change depending on other‘s choice, no Nash equilibrium and No socially optimal scenarios (social payoffs=0), the result is in each of them one wins what the other loses.
- A pure conflict game is an interaction if all outcomes are Pareto optimum. Pure conflict game can be illustrated by the example of the Division Game. A dollar is to be divided between two players. Each player claims any amount, if the claims sum to one or less, the claims are met, if not each gets zero. Other example of pure conflict game is labour discipline, re payment of loans, and crop shares.
A pure coordination game is just the opposite. –
-Pure coordination game or a pure common interests game is a game in which the payoffs to only one of the strategy profiles, the pay offs of other strategy’s can be Pareto ranked. The Pareto optimal outcome is the best one at list for one participant and not worse for any participant. A Pareto superior outcome is the second best outcome. There is no conflict because there is no outcome that any player would strictly prefer over and outcome preferred by other player
An example of pure coordination game could be illustrated by an interaction between employer and employee in a firm; in case of success of the firm both get one, in case if not, both get zero. The success depends on players. Employer may invest or not, employee may work hart or not. If the employer invest and the employee das not work then the success of the firm probability is p1, in the opposite case p2<p1.></p1.>
<p>A Nash equilibrium can be interpreted as a pair of expectations about each person's choice such that, when the other person's choice is revealed, neither individual wants to change his behaviour. In pure coordination game we have a dominant strategy and a Nash equilibrium in the same time as a payoff, meaning that the best strategy for both the players gives also the best payoff (a perfect coordination between players).</p>
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- What is the invisible hand game and what is its outcome, what type of constellation does it entail? Please, provide an example.
An invisible hand process is one in which the outcome to be explained is produced in a decentralised way, with no explicit agreements between the acting agents. The second essential component is that the process is not intentional. The agents’ aims are not coordinated nor identical with the actual outcome, which is a by-product of those aims. The process should work even without the agents having any knowledge of it. This is why the process is called invisible. In the Invisible Hand Game self interested actions of both players have outcome which maximises the well being of each. This is illustrated in the example of producing corn or tomatoes by two farmers. If player one grows tomatoes and player two grows corn, they each receives five, which is Pareto optimum and a Nash equilibrium. It is a symmetric two person game with two strategies (2x2 game). The invisible hand game is a game in which the social optimum matches with the dominant strategies of the players involved. In other words, is a game in which the social welfare can be reached when the participants follow their self interest and nobody have incentive to deviate from that point (Nash equilibrium). Because of the lack of participants ́incentives to deviate from that equilibrium point, the solution of this game is an stable solution. As mentioned above, this is a really rare game which can be hardly find in a real situation.
- What is the difference between pure strategies and mixed strategies?
Pure strategy is the continuously use of the same strategy (each player is making one choice and sticks to it)
Mixed strategy is the probabilistic choice of strategies, that means that the players built expectations about the behaviour of other (a probability that it will be played in that/this way).
- What does “cheap talk” in the case of the prisoners’ dilemma refer to?
There are some ways to solve the prisoner‘s dilemma through rules/norms. One solution is the communication but there is the problem of „cheap talk“ concerning cooperation because there is no way to credible enforce commitments as we are „in the world of“ non-cooperative game theory.
Cheap talk in the prisoner’s dilemma refers to the fact that cooperation does not take place because no legal enforcements for cooperation’s can be made. Hence, all agreements on cooperation are voluntarily and is referred to as cheap talk.
- What is the constitutional conundrum (problem) in a) the prisoner’s di-lemma, b) the assurance game, c) the battle of the sexes’ game, d) the chicken game? Each time, illustrate your explanation relying on a game representation in normal form and referring to the concepts of dominant strategies, Nash Equilibria and Social Optimum.
Explanation: The constitutional conundrum is the challenge of ensuring that the pursuit of individual interests does not lead to “outcomes that none would have chosen”.
• These undesirable outcomes are called coordination failures (= occur when noncooperative interaction of two or more people leads to a result which is not Pareto optimal).
Core Question of the Constitutional Conundrum: how can social interactions be structured so that people are free to chose their own actions while avoiding outcomes that none would have chosen?
The conundrum is relevant for environmental protection on a global scale, the determination of work effort, the production and distribution of information and the formation of neighborhoods.
- In relation to the previous question: How can the constitutional conundrum be overcome in each of the cases referred to? Refer to several options for each case.
Prisoner’s dilemma: Here the undesirable outcome is the only Nash equilibrium. The only way that any of the other outcomes can be supported is by a permanent intervention to change the payoffs or the rules of the game. —> how to get there and how to stay there.
Another way out of the prisoner’s dilemma is to enlarge the game by adding new choices. In infinitely repeated games prisoners’ can reach effective outcomes by applying tit for tat, in which players are rewarded for cooperation and punished for a lack of cooperation through their future actions. Another way to solve the dilemma is through contracting. Both players could sign a contract that they will stick with the cooperative strategy.
Assurance Game: A desirable outcome is an equilibrium. The challenge to governance is limited to the less challenging “how to get there” problem rather than also having to solve the more demanding “how to stay there”. Assurance problems can be reasonably well addressed by one-time rather than permanent intervention. One way to achieve assurance is for one player to move first (sequential).
Battle of the Sexes’: Solutions can be found outside the formal descriptions. For example, by using a focal point e.g. the location of the movie theatre.
Chicken: the most important strategy is commitment. For example, if one party has a locked steering wheel the other party will choose to swerve.
Assurance, the battle of sexes’ and chicken can be resolved by having one player move first and committing oneself. The other player can then observe the choice and respond accordingly. Instead of sequential moves, repetition and contracting are major ways to “solve” the prisoner’s dilemma.
A common problem to averting coordination failures (conundrum) is to implement policies or constitutions that change the payoff matrix in such a way that a prisoner’s dilemma becomes and assurance game by making the mutual cooperate outcome a Nash equilibrium.
- What are policy implications of different game constellations, compare prisoners’ dilemma and, for example, the assurance game.
Assurance Policies:
The desirable outcome is the equilibrium (two or more symmetrical pure strategy equilibria) so the focus is on how to get to the equilibrium. The individual payoffs increase with the number of people taking the same action which increases the path of dependencies (strategic complementarity).
- sequential gaming that enables learning
- commitment to choice
- additional information or focal point
Prisoner’s Dilemma Policies:
In the prisoner’s dilemma the undesirable outcome is the Nash equilibrium. It requires a permanent intervention e.g. a permanent sanction or compensation to find a way to get and stay there.
- Binding Contract. A solution to solve the dilemma is through contracting. Both players could sign a contract that they will stick with the cooperative strategy. If either doesn’t stick to the contract, he or she will have to pay a fine or be punished in some way. Contracts are very helpful in achieving all sorts of outcomes, but they rely on the existence of a legal system that will enforce such contracts. The purpose of a contract is to establish rules that either guarantee compliance of both parties or establish penalties or sanctions for noncompliance. Then the individual actors can be more confident that their compliance will not meet with a defection response. Thus the contract facilitates the mutually beneficial outcome.
- repetition to solve dilemma (iteration) coupled with Tit for Tat
In the assurance game (and other coordination games) conventions can emerge which can be a signal to make a strategy clear. Assurance game just needs additional information to reach what is best for society. In the Prisoners dilemma, sanctions are needed because Nash and social optimum do not collide, so payoffs have to be changed.
Despite policy intervention, social optimal result may not emerge. And in policy making there is the problem to identify if it is an assurance game or a prisoners dilemma.
- What are the consequences of applying a Tit for Tat rule to the Chicken game and to the Prisoners’ dilemma?
The consequences of applying Tit for Tat rule to Prisoner’s offers an immediate punishment for defection. It is a good mechanism for achieving the efficient outcome in a prisoner’s dilemma that will be played infinite times.
In the Chicken game it is impossible because there is no dominant strategy; the choices are made to player’s preference outcome.
Prisoner’s Dilemma “Tit for Tat”: In the first round you comply (deny). On every round thereafter, if your opponent cooperated on the previous round, you cooperate. If your opponent defected on the previous round you defect. The strategy does very well because it immediately punishes defection. On the other hand, it is forgiving because it punishes the player only once. If he falls into line and cooperates he will be rewarded with immediate cooperation. —> good mechanism for achieving efficient outcome.
Chicken: The consequences of applying a tit for tat rule would be punishment (non-cooperation).
The tit for tat rule will worsen the situation of the punishing e.g. if you go straight I will go straight on the next round which could increase the chances of a crash. However, it is not necessarily a credible threat because if in the first round A goes straight, B would not go straight as well in the next round because the payoff is so low.
- What is the role of conventions in coordination games like chicken, battle of the sexes and assurance?
Conventions are not rational but helps to coordinate. An imitation leads to convention in pure strategy. It usually starts with one person and when more people starts the same there is more advantage to coordinate in a particular situation. Fully rational players would not be able to solve those games without following conventions because they think not having enough data (knowledge) to come to a decision (Sudgen 1989: 89). It changes expectations of the players and is benefiting for the players because each of them has more information by following the convention.
- Does a convention help to resolve the prisoners’ dilemma? Please, explain your answer.
The prisoner’s dilemma is a coordination game and there are three possibilities to solve it (Varian 2006: 528)
1) Sequential moves
2) Repetition: indefinite
3) Contracting with sanction/ enforcement
Since conventions can only establish over a longer time ore if you have common knowledge with the other player a convention could be only possible solution. A “tit-for-tat” strategy might emerge as a convention when players repeatedly play Prisoner’s Dilemma over some indefinite period. You co-operate as long as your opponent cooperates; if your opponent defects, then you defect for some prescribed number of rounds as retaliation before cooperating again; if your opponent cooperates but you defect by mistake, then accept your opponent’s punishment in the next rounds without retaliating.
- What is the function of a focal point in coordination games? What qualities does it require to orient coordination?
Additional information/ focal point:
*The function of a focal point is to provide the players with additional information. For example when two people loose each other in a city and they cannot communicate, each person has to choose where to go in order to meet the other person. A focal point would be a place where the two players or one of them often goes or a place which one of them likes a lot. A focal point could be a way to solve coordination games: if you have two possible equilibria for example a focal point could be an initiator for one equilibrium to be chosen more likely than the other (e.g. if there are two possible places where you can meet it is more likely to choose the one which is closer to you actual position). “When players have good reasons to believe that one of the equilibira is more `natural´ than the other, it is called a focal point of the game” (Varian 2006: 525). The concept of the focal point (sometimes also called prominence or salience) provides an explanation for why people follow conventions (Sudgen 1989: 89). A focal point can only orient coordination if the players have common knowledge/ experience – in example above the one person has to know preferred places of the other player (Sudgen 1989: 90). Common knowledge emerges for example if we have a repeated game.
- What is an evolutionary stable strategy (ESS) and how does it relate to Nash equilibria and pareto optimal outcomes of coordination?
Evolutionary stable strategies (ESS)
A strategy is evolutionary stable if generally followed in a population, if people deviate from that they will do less well than others will, because majority adheres to the strategy. For example talking English at course in HU.
It is a Nash equilibrium because nobody who changes would be better off.
No Pareto optimum because the more people who adhere to that strategy the better.
Makes the whole population to be better off.
When decisions need to be made, a strategy will be applied that has evolved over time and is proved a good decision. Others who do not follow an evolutionary stable strategy will be worse off à hence it is a self-enforcing strategy.
To define if a strategy is evolutionary stable:
- Can any individual gain by deviating?
- Can any small group gain by deviating at the same time in the same way?
An ESS always has to correspond with a Nash equilibrium (otherwise individuals would gain by deviating), that is, a strategy cannot be evolutionary stable if it is no Nash equilibrium. However, not all Nash equilibria are evolutionary stable!
A strategy is not evolutionary stable if deviants to better by deviating: In this case, if deviants meet they will be able to coordinate with one another. Therefore, there is a tendency to imitate and repeat this deviation strategy.
How does ESS relate to parte optimal outcomes of coordination is explained by Sudgen (1989: 93 and 94): conventions can be evolutionary stable even if they are not Pareto efficient. An inefficient convention could probably be more prominent than an efficient one – for example, rules favoring first arrivals seem to have prominence, even
though they can lead to wasteful races.
- Please, describe a prisoners’ dilemma situation and draw the corresponding normal form representation. Please, describe a way to restructure the setting by means of introducing a sanction. How high does the sanction need to be in the example chosen? If you introduce a sanction, in what way do you transform the game. What is the implication if the sanction was only applied in one round and subsequently it was not applied anymore?
Example: pizza delivery. The sanction must be smaller than 3 and bigger the one. It will create a pure coordination game. Otherwise the pizza will be delivered and do not payed or be payed and do to be delivered. There will be two Nash equilibrium, but one Nash equilibrium will meets the social optimum (pareto efficiency) – Deliver (B) and Pay (A).
