 As we talked about in the previous module, a game within game theory is any situation involving their interdependence between adaptive agents. Games are fundamentally different from decisions made in a context with only one adaptive agent. To illustrate this point, think about the difference between the decisions of a bricklayer and those of a business person. When a bricklayer decides how he might go about building a house, he does not expect the bricks and mortar to fight back, we could call this a neutral environment. But when the business person tries to enter a new market, they must take note of the actions and the other actors in that market in order to best understand the viable options available to them. A situation that depends only on the actions of one actor is best understood as one of decision theory and not so much game theory. Like the business person, 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 the possibility of cooperation. So a game really tries to capture this dynamic where autonomous agents that have their own goals are independent in affecting some joint outcome. All games have three major components, players, strategies and payoffs. A player is a decision maker in the game. A strategy is a specification of a decision for each possible situation in which a player might find themselves. A payoff is a reward or loss that players experience when all the players follow their respective strategies. Games are typically 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 following 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 it is called in game theory the 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 axes corresponding to one unique strategy for the player. Where the players different strategies interact in the matrix, a value is placed to represent the associated payoffs for each player when 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 player's decisions when they make their choices. The normal form representation 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 simplification makes it more convenient to analyse. 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 forwards and players have to make their choices. One 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 of possibilities of moves for the other player as they branch out into the future. Players engaged in a sequential game then have to look ahead and reason backwards as each player tries to figure out how the other players will respond to his current move, how he will then respond in turn and so on. 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 make up of the game. In any given game agents can have complete information meaning each player has knowledge of the payoffs and possible strategies of the other players or we may have a situation of incomplete information referring to a situation in which the players and strategies of other players are not completely known. An example of a game of perfect information would be one that is called the ultimatum game. In this game one player receives a sum of money and proposes how to divide the sum with the other player involved. 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. One example of this would be playing poker and other times they will want to reveal it. For example a company offering a guarantee for its products is a display of the information they have that their products are 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 said to be symmetrical. Many of the commonly studied 2x2 games are symmetrical. For example games of coordination are typically symmetrical. Take for example the case of people choosing the side 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 manoeuvre they will manage to pass each other but if they choose different manoeuvres they will collide. In the payoff matrix successful passing is represented as a payoff of 10 and a collision by a payoff of 0 and we can see how the payoffs to each player are symmetrical. Games are played over some mutually desired resource. That is whatever 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 in an ecosystem. Political parties for decision making power. Athletes for prizes 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 for cooperation. Constant sum games are games in which the sum of the player's payoffs always adds up to the same. These interactions are games of pure competition where one person's gain is another person's 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. 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 competitive sports. In zero sum games the relationship between the agent's payoffs are negatively correlated and this is called negative interdependence meaning individuals can only achieve their goal via the failure of other agents and this creates an attractor towards competition. There is no incentive for cooperation and thus these games are called strictly competitive as competition is always the best strategy. Non constant sum 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 the actors involved. For example through the members of a business working together they can create more value than working separately and 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 agent gets more then the other agents will also get more. If one agent gets less then the others will get less also. And thus in 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 that success depends on the participation of all the members leading to a dynamic of cooperation. A cooperative game is one in which there can be cooperation between the players where they may have the same costs. So cooperative games are an example of non zero sum games. This is because in cooperative games either every player wins or loses. 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 in a positive sum game where payoffs are positively correlated. In such a case the innate structure to 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 through external enforcement by some authoritative third party such as governments and contract law. In such a case we might cooperate in a transaction because the third party is ensuring that it is in our interest to do so by creating punishments or rewards. Likewise cooperation may be achieved through peer-to-peer interaction and feedback mechanisms and we'll talk about this in future videos. A non-cooperative game is one where an element of competition exists and there are limited mechanisms for creating institutions of cooperation. This may be because of the inherent nature of the game we're playing that is to say it is a zero-sum game which is strictly competitive and thus cooperation will add no value. Non-cooperation may be a function of isolation lack of communication and interaction with which to build up the trust that enables cooperation. 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 now requiring formal third parties to ensure cooperation because of the anonymity and lack of interaction between members in a large society. Lastly there may simply be a lack of formal institutions to support cooperation between the 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 cannot work as the supporting institutional framework for cooperation. In this video we've looked at some of the basic features to games. We talked 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 that unfolds over time. We talked about the important role of information where games may be defined as having imperfect or perfect information and how agents may use this information to their advantage. We talked about symmetrical and asymmetrical payoffs in games. We briefly looked at zero-sum games and non-zero-sum games where the payoffs can get larger given cooperation. Finally we talked about the distinction between a cooperative and non-corruptive game and some of the factors that create these different types of games which we'll be discussing further throughout the course.