Game theory is the domain of applied mathematics that studies situations of interdependence between adaptive agents. A game is a system wherein adaptive agents are interdependent in effecting each other and the overall outcome. Game theory is the mathematical modeling of such systems. These adaptive systems are pervasive in our world, from cities and traffic to economies, financial markets, social networks, ecosystems, politics, and business. The central ingredients of these systems is that of agents and interdependency. Without either of these elements, we don’t have a game. If the elements did not have agency and the capacity for adaptation they would have no choices and we would have a deterministic system. Likewise, if they were not interdependent then they would not form some combined organization and we would then study them in isolation in which case likewise we would not have a game.
Games are formed out of the interdependencies between adaptive agents. An agent is any entity that has what we call agency, agency is the capacity to make choices based upon information and act upon those choices autonomously to affect the state of their environment. Examples of agents include social agents, such as individual human beings, businesses, governments etc. they may be biological agents such as bacteria, plants, or mammals, they may also be technologies such as robots or algorithms of various kind. All adaptive systems regulate some process and they are designed to maintain and develop their structure and functioning. For example, plants process light and other nutrients and their adaptive capacity enable them to alter their state so as to intercept more of those resources. The same is true for bacteria and animals, the same is true for a basketball team or a business, they all have some conception of value that represents whatever is the resource that they require, whether that is sunlight, fuel, food, money etc.
This creates what we can call a value system, that is to say, whatever structure or process they are trying to develop forms the basis for their conception of value and they use their agency to act and make choices in the world to improve their status with respect to whatever it is they value. As we can see this concept of value is highly abstract and as we will discuss in a future module this value system can be very simple or very complex but it forms the foundations to what we are dealing with when talking about adaptive agents and games. One can not model a game without understanding what the agents value and the better you understand what they really value and incorporate it into the model the better the model will be. Thus agents can also be defined by what we call goal oriented behavior, they have some model as to what they value and they take actions to affect their environment in order to achieve more of whatever is defined as value.
In game theory, a game is any context within which adaptive agents interact and in so doing become interdependent. Interdependence means that the values associated with some property of one element become correlated with those of another. In this context, it means that the goal attainment of one agent becomes correlated with the others. The value or payoff to one agent in the interaction is associated with that of the others. This gives us a game, wherein agents have a value system, they can make choices and take actions that affect others and the outcome to those interactions will have a certain payoff for all the agents involved. A game then being a very abstract model can be applied to many circumstances of interest to researchers, and it has become a mainstream tool within the social sciences of economics, political science and sociology but also in biology and computer science. The trade negotiations between two nations can be modeled as a game, the interaction of businesses within a market is a game, the different strategies adopted by creatures in an ecosystem can be seen as a game, the interaction between a seller and buyer as they haggle over the price of an item is a form of game. The provision of public goods and the formation of organizations can be seen as games, likewise, the routing of internet traffic and the interaction between financial algorithms are games. As an example of a game one can think about the current situation with respect to international politics and climate change. In this game we have all of the world’s countries and all countries will benefit from a stable climate, but it requires them to cooperate and all pay the price of reducing emissions in order to achieve this. Although this cooperative outcome would be best for all, it is in fact in the interest of any nation state to defect on their commitments as then they would get the benefit of others reducing their pollution without having to pay the cost of reducing their own emissions. Because in this game it is in the private interests of each to defect, in the absence of some overall coordination mechanism the best strategy for an agent to adopt given only their own cost-benefit analysis is to defect and thus all will defect and we get the worst outcome for the overall system.
This game is called the prisoner’s dilemma and it is the classical example given of a game, because it captures in very simplified terms the core dynamic, between cooperation and competition, that is at the heart of almost all situations of interdependence between adaptive agents. In the interdependency between agents there comes to form two different levels to the system, the macro-level, wherein they are all combined and have to work cooperatively to achieve an overall successful outcome, and the micro-level, wherein we have individual agents pursuing their own agendas according to their own cost-benefit analysis. It is precisely because the rules and dynamics that govern the whole and those that govern the parts are not aligned that we get this core constraint between cooperation and competition. This is what is called the social dilemma and it can be stated very simply as what is rational for the individual is irrational for the whole. If you do what is rational according to the rules of the macro-level to achieve cooperation then you will be operating in a way that is irrational to the rules of the micro-level and vice versa. If either of these dimensions to the system was removed then we would not have this core constraint. If the agents were not interdependent within the whole then there would be no macro-level dynamic and the set of parts would be simply governed by the rules of the agents locally. Equally if each agent always acted in the interests of the whole without interest for their own cost-benefit analysis, then again we could do away with the rules governing the micro-level and we would simply have one set of rules governing the whole thus there would be no core dynamic of interest, things would be very simple and straightforward. The complexity arises out of the interaction between these two different rule sets and trying to resolve it by aligning the interests of the individuals with those of the whole.