This article covers the basic features to the model of games within game theory. We talk about the two basic forms of representation, that of the normal form in a matrix model and that of the extensive form as a tree graph which unfolds over time. We look at the important role of information, where games may be defined as having imperfect or perfect information and how agents may use information to their advantage. We talk about symmetrical and asymmetrical payoffs in games. We briefly touch on zero-sum games and non-zero sum games where the payoffs can get larger given cooperation. Finally, we talk about the distinction between a cooperative and non-cooperative game and some of the factors that create these different types of games.
A game within game theory is any situation involving the interactivity between adaptive agents.1 Games are fundamentally different from decisions made in a context with only one adaptive agent. all players engaged in a game must recognize their interaction with other intelligent and purposeful agents. Their own choices must allow both for conflict and for possibilities for cooperation. Thus a game really tries to capture the dynamic where autonomous agents that have their own goals are interdependent in effecting some joint outcome.2 A game has three major components: players, strategies, and payoffs.3 A player is a decision maker in a game. A strategy is a specification of a decision for each possible situation in which a player may find themselves. A payoff is a reward or loss that players experience when all the players follow their respective strategies.
Games are represented in either a matrix form or as a tree graph. The matrix form models a game without time involved where players must choose their strategies simultaneously. A tree graph model involves time as an element allowing for choices to be made in a sequential process over a course of time, thus forming a tree-like representation that captures the choices made by agents at each stage in the game.
The matrix model is the most common method for representing a game and is called in game theory normal-form representation. The normal-form representation to a game associates the players with the axes to the matrix, with each column or row along the axis corresponding to one unique strategy for the player, where the player’s different strategies interact in the matrix a value is placed to represent the associated payoffs for each player if that given strategy is played. In simultaneous games, the players do not have to move at the same time; the only restriction is that no players can know the other players’ decisions when they make their own choice. The normal form is a condensed form of the game, stripped of all features but the possible options of each player and their payoffs during one iteration of the game, this makes it more convenient to analyze.4
A game where choices are made in a sequential fashion over time is represented as a decision tree graph that branches out with each iteration of the game as time goes forward and players have to make choices. An example of this extensive form of game would be chess where players move in a sequential process with each move of one player creating a multiplicity for possible new moves of the other as they branch out into the future. Players engaged in a sequential game then have to look ahead and reason back as each player tries to figure out how the other players will respond to his current move, how he will respond in turn, and so on.5 The player anticipates where his initial decisions will ultimately lead and uses this information to calculate his current best choice.
Agents within a game are making their choices based on the information available to them, thus we can identify information as a second important factor in the makeup of the game. In any given game agents can have complete information meaning each player has knowledge of the payoffs and possible strategies of other players, or incomplete information referring to situations in which the payoffs and strategies of other players are not completely known.6
An example of a game of perfect information would be one that is called the ultimatum game where one player receives a sum of money and proposes how to divide the sum with the other player. The second player chooses to either accept or reject this proposal. If the second player accepts, the money is split according to the proposal. If the second player rejects, neither player receives any money. In this game, all information is available to all players. In contrast many real world games involve imperfect information, for example, prisoner dilemma games only make sense if given imperfect information where you are choosing without knowing how the other has chosen.
Information plays an important role in real-world games and it can work as an advantage or disadvantage to the players. When one player knows something that others do not, sometimes the player will wish to conceal this, for example in playing poker, and at other times they will want to reveal it, for example, companies offering guarantees for their products is a display of the information that they have that their product is not going to break down soon, and they want customers to know this information.
This reveals also how games can be asymmetrical, meaning the payoffs to individuals for the different possible actions may not be the same. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric.7 Many of the commonly studied 2 × 2 games are symmetric. Games of coordination are typically symmetrical. Take for example, the case of people choosing the sides of the road upon which to drive. In a simplified example, assume that two drivers meet on a narrow dirt road. Both have to swerve in order to avoid a head-on collision. If both execute the same swerving maneuver they will manage to pass each other, but if they choose differing maneuvers they will collide. In the payoff matrix successful passing is represented by a payoff of 10, and a collision by a payoff of 0 and we can see how the payoff to each player are symmetrical.
Zero-sum / Non-zero-sum
Games are played over some mutually desired resource, what every we are defining as value within that game. For example, countries go to war over territory, businesses compete for market share, creatures for the resources within an ecosystem, political parties for decision making power, athletes from prizes and prestige etc. In all of these situations, there is some shared conception of what the agents’ value and some interdependence in how that value is distributed out depending on the actions of the agents. But the question is whether the total value distributed out to all agents remains constant irrespective of their actions or can it grow or decrease depending on their capacity to cooperate.
Constant-sum games are games in which the sum of the players’ payoffs sum to the same number. These games are games of pure competition of the type “my gain is your loss”. Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources.8 In zero-sum games, the total benefit to all players in the game, for every combination of strategies, always adds to zero. One can see this in the game paper, rock, scissors or in most sporting events. In zero-sum games, the relationship between the agents’ payoffs are negatively correlated, which is called negative interdependence meaning individuals can only achieve their goal via the failure of another agent and this creates an attractor towards competition, there is no incentive to cooperate and thus these games are called strictly competitive.
Non-constant games or non-zero sum games are those in which the total value to be distributed can increase or decrease depending on the degree of cooperation between actors. For example, through the members of a business working together they can create more value than working separately, thus the whole payoff gets bigger. Equally, the total payoff may get smaller through conflict, like in an arms race between two gangs in a city. In non-zero sum games, the outcome for agents is positively correlated, if one gets more the other will too if one gets less the other will too. With non-zero sum games, we can get positive interdependence between the agents, meaning members of a group come to share common goals and perceive that working together is individually and collectively beneficial, and success depends on the participation of all the members leading to cooperation.
Cooperative / Noncooperative
A cooperative game is one in which there can be cooperation between the players and they have the same cost. Thus cooperative games are an example of non-zero sum games. This is because in cooperative games, either every player wins or loses.9 Cooperation may be achieved through a number of different possibilities, it may be built into the dynamics of the game as would be the case with a positive-sum game where payoffs are positively correlated. In such a case the innate structure of the game creates an attractor towards cooperation because it is both in the interest of the individuals and the whole organization. A good example of this are the mutually beneficial gains from trade in goods and services between nations. If businesses or countries can find terms of trade in which both parties benefit then specialization and trade can lead to an overall improvement in the economic welfare of both countries, with both sides seeing it as in their interest to cooperate in this organization, because of the extra value that is being generated. Equally, cooperation may be achieved by external enforcement by some authoritative third party such as governments and contract law. Where we cooperate in a transaction because the third party is ensuring that it is in our interests to do so by creating punishments or rewards. Likewise, cooperation may be achieved through peer-to-peer interaction and feedback mechanisms.
A noncooperative game is one where an element of competition exists and there are limited mechanisms for creating institutions for cooperation. This may be because of the inherent nature of the game we are playing, that is to say, it is a zero-sum game which is strictly competitive and thus cooperation will add no value. Noncooperation may be a function of isolation, lack of communication and interaction with which to build up the trust that enables cooperation.10 We see this within modern societies, as these societies have grown in size they have transited from communal cooperative systems based on the frequent interaction of members to requiring formal third parties to ensure cooperation because of the anonymity and lack of interaction between members of large societies. Lastly, there may simply be a lack of formal institutions to support cooperation between members, an example of this might be what we call a failed state where the government’s authority is insufficiently strong to impose sanctions and thus can not work as the supporting institutional framework for cooperation.11