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The game theoretical version of the social dilemma is called the public goods game. Public goods games are usually employed to model the behavior of groups of individuals achieving a common goal. The public goods game has the same properties as the prisoner’s dilemma game, but describes a public good or a resource from which all may benefit regardless of whether or not they contributed to the good.

PUBLIC GOODS GAMES

In the past ten to twenty years interest in social dilemmas has grown dramatically within many domains – particularly those resulting from overpopulation, resource depletion, and pollution.

The study of social dilemmas is one of the most interdisciplinary research fields, with the participation of researchers from anthropology, biology, economics, mathematics, neuroscience, political science, and psychology among others.

The social dilemma captures the core dynamics within groups requiring collective action. Where there is a conflict between an individual’s immediate personal or selfish interests and the actions that maximize the interests of the group.

At the heart of social dilemmas lies a disjunction between the costs to the individual and the cost to the whole, or benefits to the individual and benefits to the whole.

We call this value that is not factored into the cost-benefit equation of the individual an externality. And it is externalities that create the disjunction between the parts and whole and result in the social dilemma.

As an example of the social dilemma, we can think about the voting process within democratic political systems. In such a system citizens are called upon to make informed decisions about who should manage their country’s government. Many people choose not to vote, but of those who do, in order to make an informed decision, they have to gather and process information about the candidates.

Since each person’s vote is unlikely to affect the outcome of an election, and everyone knows this, there is little incentive to the individual to increase their knowledge about the relevant issue and overcome misinformation and preconceptions. The benefits of collecting and processing the information are diffused to the whole population. But the cost of doing such action is carried by the individual. Informed voting is then what we would call a public good.

As another example, we can think of a situation where during winter people in a village are asked to keep their thermostats low to conserve the limited amount of energy available. This would though require them to suffer from the cold without appreciably conserving the fuel supply by their individual sacrifice; yet if all keep their thermostats high, all may run out of fuel and freeze.

What is happening in this game is that there is a positive externality. Some of the value that is being generated by the individual is being externalized to the whole organization. This external value can not be immediately factored into the cost-benefit analysis of the individual. If the system is simply operated according to this immediate cost benefit analysis of the individuals then there will likely be an undersupply of it.

Equally, we can have negative externalities. The classical example being air pollution and traffic jams. An important part of the social dilemma is that at any given decision point, individuals receive higher payoffs for making selfish choices than they do for making cooperative choices, regardless of the choices made by those with whom they interact. And everyone involved receives lower payoffs if everyone makes selfish choices than if everyone makes cooperative choices.

What is interesting about the social dilemma is that it is not a feature of the agents, but of the structure of the game. It has nothing to do with the personalities and motivation of the individuals. It is a tragedy because you know what the outcome will be, but you can not individually do anything to avoid it, given only the self-interests of the individual.

No one has an individual motive to change their behavior even though everyone will be worse off.

Our traditional tools of non-cooperative game theory that are focused on the immediate costs and benefits to the individual, only really work when the externalities are limited. As soon as the externalities both positive or negative go above a certain level we have to think and operate collectively.

Many private goods are rivalrous, meaning they can be only consumed by one agent, and excludable meaning it is possible to exclude others from their use. This reduces the externalities from the item and thus makes it possible to associate the value of that item with an agent and factor it into their cost-benefit analysis.

Goods are public when they are considered both nonrivalrous and nonexcludable, meaning that there will likely be many externalities and thus they can not be effectively managed by the immediate cost benefit analysis of the individual agents. In such a case the social dilemma can arise. When people can benefit from the positive externalities, and produce more of the negative externalities without paying, the result is a macro level imbalance that leads to the system being rendered unsustainable. As we will discuss further in a coming video, there are essentially two different approaches to solving this imbalance and managing collective action.

The organization can try to create some top-down structure that regulates the system to ensure that those who create negative externalities pay for them and those who create positive externalities get paid for them. This works to reintegrate the positive externalities into the cost-benefit analysis of the agents and maintain a macro-level balance. This is the traditional approach taken, the classical example of which being governments, that use force and incentives to regulate the system towards these ends.

Equally a second approach is to increase the degree of connectivity between the agents and thus the potential for a higher degree of interdependence between them. Given that interdependence means that what happens to one agent also happens to another, as interdependence between agents and between the individual and the whole increases, externalities decrease – because there is nowhere for them to go and agents have to increasingly factor them into their own cost-benefit analysis.

Which is one approach to managing the system and potentially solving the problem.

Externalities are always a function of independence. If two things are one, then there is no possibility for externalities, the more independent they are the greater the possibility for externalities. Thus all solutions to externalities and the social dilemma will involve creating positive interdependence between the elements in the system, but different approaches will do this in different ways.

We will pick this theme up again in a future video when we go deeper into ways of solving social dilemmas. In the next video, we will look at what game theory can tell us about the social dilemma as we talk about public goods games.

COOPERATIVE GAMES

Game theory is the study of interdependent interactions between adaptive agents, and when we look at these interactions between agents in the world around us we see elements of both competition and cooperation.

We see the organelles within a biological cell work together to enable its overall functioning. We see organisms within ecosystems forming symbiotic cooperative relations. We see people form families, tribes, cities, and nations all of which involve high levels of cooperation.

Game theory should then be a tool that helps us to understand what actions an agent should take within non-cooperative situations and what outcomes are most likely in such games.

But it should also be a tool that helps us understand how cooperation works. The first thing for us to note is that the dynamics of cooperation are very different from those that we have been studying in games of noncooperation. Cooperative dynamics rewrite the rules of the games people play.

In non-cooperative games like the prisoner’s dilemma, we noted how individual rationality, in fact, led to collective irrationality. We called this a dilemma and lamented the fact that within the non-cooperative framework of self-interested individuals there was nothing we could do about it. However, we as humans do not just look out for our own interests but also those of others and in so doing we have evolved highly sophisticated means for cooperation in the process.

When we add this new element to the game, that of cooperation, we now have the possibility for the agents to solve this dilemma. The question then turns to how and when do coalitions for cooperation form, where we can achieve both stable and optimal outcomes for the individual and the whole organization.

With non-cooperative games, we are solely focused on the payoffs to the individuals and searching for stable situations. Cooperative game theory, however, adds an extra dimension to this in that we now have to think about the payoff to the whole organization. In such a case we can not simply look a the actions of the individuals and their payoffs but we have to also look at the positive or negative externalities that these actions may have on the whole system.

COOPERATION

Cooperation is a process by which the components of a system work together to achieve the global properties. In other words, individual components that appear independent work together to create a complex whole, greater-than-the-sum-of-its-parts system.

Virtually all of human civilization is a product of our capacity to work cooperatively. Indeed the complex systems that surround us, like our global economy and technologies like a jumbo jet are a testament to our extraordinary capacity for cooperation.

In most animal groups and even our closest relatives in the primate group, competition is the norm, and cooperation occurs largely only among kin, who have common genes and so have a biological incentive to do so, or else among a few individuals who cooperate reciprocally. But humans cooperate with each other in very large groups in a multiplicity of ways.

People risk their lives in war for their countrymen and we make sure that our less fortunate compatriots have enough food and medical care to survive. On a daily basis, we obey all kinds of prosocial norms. And when we do breach some prosocial norm, like not doing our part in a collective enterprise, we feel guilty or ashamed, in general, we are highly sensitive to cooperative behavior.

Human evolved capacity for cooperation is a cultural one that distinguishes us from other creatures who we may share up to 98% of our genes in common with. A large-scale study has recently provided some foundation to this hypothesis.

Researchers compared two-year-old children to their nearest primate relative chimpanzees and orangutans. They were all given 16 different tasks grouped into two overall categories. One concerning an understanding of the physical world, the other an understanding of the social world. The physical tests were related to space, quantity, and causality, and the social tests concerned capacities for social imitation, communication, and intention reading. The experiment revealed that the children were not across the board more intelligent than the primate animals, but in fact only with respect to social cognition were they more advanced. At just two years of age, the children were already about twice as high on this indicator as the other two creatures. This social capacity enables us to take advantage of the skills and knowledge of others within a social group through cooperation.

The researchers noted that apes did have social cognitive skills, but they were mainly using their social understanding of others within contexts of competition. From this, the researchers proposed that on top of great apes skills for social cognition humans had evolved additional social cognitive capabilities for dynamics of cooperation, which involve greater complexity, but which can ultimately be seen as the foundations to advanced forms of civilization.

Thus they came to understand others as not just intentional goal seeking agents, but also as potential cooperative agents with whom they could work together to produce outcomes that neither could produce alone. This cognitive capacity along with communications enabled us to create the ever more complex social and cultural institutions for cooperation, that today form the foundation of our advanced systems of socio-economic coordination.

The researchers claimed that this distinction between apes and humans can be identified even in the earliest human economies. Noting how apes are individual foragers, where they will travel in small groups until they find a food source like a fig tree and then run up and grab the food separately without collaboratively producing it or sharing it. Humans, however, are collaborative foragers meaning that most traditional forager groups derive most of their daily nutrition from collaborative activities in different forms, such as hunting.

This is not to say that advanced forms of cooperation do not happen within other creatures. We just have to look at ant or bee colonies to see sophisticated coordination. However, these creatures have nowhere near the kind of individual cognitive capacity that apes and humans do, and thus we do not get the same kind of complex dynamic between the individual and the group that is at the heart of human social systems and the study of cooperation.

Dynamics of cooperation require more cognitive capabilities on the behalf of the individual because they are greatly more complex in nature than noncooperative situations.

Whereas non-cooperative dynamics are governed solely by the self-interest of the individual’s, cooperation involves a new level of organization, that of the group, and a complex dynamic between the individual and the overall group.

This central dynamic within cooperation is captured in the idea of the social dilemma. Social dilemmas are characterized by two properties: The social payoff to each individual for defecting behavior is higher than the payoff for cooperative behavior, regardless of what the other society members do, yet all individuals in the society receive a lower payoff if all defect than if all cooperate. It is a situation where individual rational behavior leads to a situation where everyone is worse off.

Social dilemmas are of interest to many because they reveal the core tension between the individual and the group, that is engendered in situations of cooperation. At their core, social dilemmas are situations in which self-interest is at odds with collective interests and they can be found in many situations of interdependence; from resource management to relationship development, to international politics, public goods provision and business management. In the next section, we will zoom in to look more closely at the workings of this social dilemma.

PARETO OPTIMALITY

In non-cooperative game theory, the focus is on the agents in the game and strategies that optimize their payoffs, which results in some form of equilibrium.

As we can see in the prisoner’s dilemma game the issue arises in that what turns out to be the equilibrium is suboptimal for all the agents when taken as a whole. One way of defining what we mean by suboptimal for all is the idea of Pareto Optimality.

Named after Vilfredo Pareto, Pareto optimality is a measure of efficiency.

Whereas Nash Equilibrium is a solution concept of non-cooperative games, Pareto optimality in game theory answers a very specific question of whether an outcome can be better than the other?

Pareto optimality is a notion of efficiency or optimality for all the members involved.

An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player strictly better off. That is to say, a Pareto optimal outcome cannot be improved upon without hurting at least one player.

To illustrate this let’s take the game called the stag hunt, wherein two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, they must have the cooperation of their partner in order to succeed. An individual can get a hare by themselves, but a hare is worth less than a stag.

In the stag hunt, there is a single outcome that is Pareto efficient, which is that they both hunt stags. With this outcome, both players receive a payoff of three, which is each player’s largest possible payoff for the game. In this case, we cannot switch to any other outcome and make at least one party better off without making anyone worse off. The stag option is here the only Pareto optimal outcome.

One of the features of a Nash equilibrium is that in general, it does not correspond to a socially optimal outcome. That is, for a given game it is possible for all the players to improve their payoffs by collectively agreeing to choose a strategy different from the Nash equilibrium. The reason for this is that some players may choose to deviate from the agreed-upon cooperative strategy after it is made in order to improve their payoffs further at the expense of the group. A Pareto optimal equilibrium describes a social optimum in the sense that no individual player can improve their payoff without making at least one other player worse off. Pareto optimality is not a solution concept, but it can be an important attribute in determining what solution the players should play, or learn to play over time.

This is the interesting thing about the prisoner’s dilemma, that all options are Pareto optimal except for the unique equilibrium, which is for both to defect. This strong contrast between Pareto optimality and Nash equilibrium is what makes the prisoner’s dilemma a central object of study in game theory. The fact that all of the overall efficient outcomes are the ones that do not occur in equilibrium, makes it a classical illustration of the core dynamic between cooperation and competition.

This is a good segway into the next section of the book where we will be talking about the dynamics of cooperation. Where we will be looking specifically at the overall outcomes trying to optimize them instead of just individual payoffs.

SOLUTION CONCEPT

As we talked about in the last video, the central aim in non-cooperative game theory is in trying to find the optimal strategy for agents to play within a game and trying to predict the outcomes of the game by finding points of equilibrium.

This equilibrium is called the Nash Equilibrium and is considered the best option given the absence of frameworks to support cooperation.

This is what we call a solution concept.

In game theory, a solution concept is a model or rule for predicting how a game will be played. These predictions are called “solutions”, and describe which strategies will be adopted by players and, therefore, the results of the game.

The most commonly used solution concepts are equilibrium concepts.

Where we look for a set of choices, one for each player, such that each person’s strategy is best for them when all others are playing their stipulated best response.

In other words, each picks their best response to what the others do.

In game theory the term best response refers to the strategy (or strategies) which produce the most favorable outcome for a player, taking other players’ strategies as given. Best response is when you know what others are going to do and you choose your best response.

DOMINANT STRATEGY

Sometimes one person’s best choice is the same no matter what the others do. This is called a “dominant strategy” for that player. Hence, a strategy is dominant if it is always better than any other strategy, for any profile of other players’ actions.

A strategy is termed strictly dominant if, regardless of what any other players do, the strategy earns a player a strictly higher payoff than any other. If a player has a strictly dominant strategy then they will always play it in equilibrium.

A strategy is weakly dominant if, regardless of what any other players do, the strategy earns a player a payoff at least as high as any other strategy.

If there are better strategies to take within a game then there must also be worse strategies to take and we call these worse strategies dominated. A dominated strategy in a game means that there is some other choice for the agent to make that will have a better payoff than that one.

When the game is non-cooperative and players are assumed to be rational, strictly dominated strategies are eliminated from the set of strategies that might feasibly be played. Thus the search for an equilibrium typically begins by looking for dominant strategies and eliminating dominated ones.

For example, in a single iteration of the prisoner’s dilemma game cooperation is strictly dominated by defect for both players. Because either player is always better off playing defect, regardless of what their opponent does. In searching for the equilibrium to this game we would simply look at each cell and ask is there a better option for the play? If so then the cell is dominated and we should not choose it. Once we have done this for both players we can identify a corresponding cell or number of cells that is optimal for each, giving us the equilibrium or possibly a number of different equilibria.

MINIMAX/MAXMINI

In games of conflict and competition, we are often interested in knowing what is the strategy that one can play that will reduce one’s exposure to some negative event.

For example, this might be a scenario of war, where we have a number of different options as to the route along which we will send our food supply to our troops. Along any of these routes, there is the possibility that they will get bombed. We would then try to choose the option that will minimize the amount of damage that might possibly be caused to the convoy. This is captured in the term minimax. Minimax is a decision rule for minimizing the possible loss for a worst case scenario. The minimax value of a player is the smallest value that the other players can force the player to receive, without knowing the agent’s actions.

A minimax strategy is commonly chosen when a player cannot rely on the other party to keep any agreement or they have in their interest that you gain the minimum payoff, such as in a zero-sum game.

Calculating the minimax value of a player is done in a worst-case approach: for each possible action of the player, we check all possible actions of the other players and determine the worst possible combination of actions – the one that gives the player the smallest value. Then, we determine which action the player can take in order to make sure that this smallest value is the largest possible.

A maximin strategy is one where the player attempts to earn the maximum possible benefit available. This means they will prefer the option which offers the chance of achieving the best possible outcome – even if a highly unfavorable outcome is possible when taking that strategy. This maximin strategy that is often referred to as the best of the best, is also seen as ‘naive’ and an overly optimistic strategy, in that it assumes a highly favorable environment for decision making. In contrast, the minimax strategy is a more realistic strategy in that it takes account of the worst case scenario and prepares for that eventuality.