In game theory, the strategic form (or normal form) is a way of describing a game using a matrix. The game is defined by exhibiting on each side of the matrix the different players (here players 1 and 2), each strategy or choice they can make (here strategies A and B) and sets of payoffs they will each receive for a given strategy (p_1A,p_2A ; p_1A,p_2B ; p_1B,p_2A ; p_1B,p_2B).

In game theory, the extensive form is away of describing a game using a game tree. It’s simply a diagram that shows that choices are made at different points in time (corresponding to each node). The payoffs are represented at the end of each branch. Since the extensive form represents decisions at different moments, it’s usually used to describe sequential games, while simultaneous games are described using the strategic form. Since sequential games imply making decisions at different moments for each player, information is perfect since each player can see the decision taken by the previous player, complete and the rules of the game and each player’s payoffs are common knowledge.

In the first game tree we can see how player 1 is the first to decide, while player 2 will make a decision after observing what player 1 has decided. The payoffs represented at the end of each brand represent all possible outcomes. For instance, if player 1 chooses strategy A and player 2 chooses strategy B, the set of payoffs will be p_1A,p_2B.

A good example of a sequential game described with the extensive form is when considering collusion agreements, as depicted in the second game tree.

Two firms share the market, colluding and maintaining high prices. Each firm can decide to stop colluding and start a price war, in order to increase their market share, even force the other to quit the market. Firm 1 can either keep colluding with firm 2, or start a price war. If firm 1 decides to keep colluding, firm 2 will need to make a decision. If they both agree to collude, they will get 5,5. However, if one of them decides to start a price war, the set of payoffs will be either 4,3 or 3,4, depending on which one starts the war (and therefore acquires a greater market share). It’s easy to see that collude-collude is both the Nash equilibrium and a Pareto optimum situation. This result may change when considering repeated games.

It’s worth mentioning that the extensive form can be used also to describe simultaneous games, by using information sets, as shown in the third game tree. These information sets, usually represented by a dashed line uniting two nodes or by encircling them, mean that the player does not know in which node he is, which implies imperfect information, like when using the strategic form.

The perfection of information is an important notion in game theory when considering sequential and simultaneous games. It is a key concept when analysing the possibility of punishment strategies in collusion agreements.

Perfect information refers to the fact that each player has the same information that would be available at the end of the game. This is, each player knows or can see other player’s moves. A good example would be chess, where ech player sees the other player’s pieces on the board.

Imperfect information appears when decisions have to be made simultaneously, and players need to balance all possible outcomes when making a decision. A good example of imperfect information games is a card game where each player’s card are hidden from the rest of the players.

Common knowledge is a condition usually required in game theory, so the model is completely specified and its analysis is coherently undertaken. It completes the notion of complete information, which requires all players to know the rules of the game and each other’s utility function. Indeed, all players should also be required to be aware of this fact; this is, all players must be aware of the awareness of the other players regarding the rules and each other’s utility functions. Furthermore, each player must be aware that each player is aware that each player is aware, and so on. To sum up, the awareness of the description of a given game must be part of that description.

This condition implies that each player will be fully rational (understood as rational agents, that form and adapt their expectations as rational expectations). It also implies that each player will have to take into consideration how other players face their maximisation problems, in order to make rational choices. Therefore, common knowledge is the starting point of Nash equilibriums.

Complete information and incomplete information are terms widely used in economics, especially game theory and behavioural economics. We say that there is complete information when each agent knows the other agent’s utility function and the rules of the game. As Luce and Raiffa put it in their “Games and Decisions: Introduction and Critical Survey“, 1957, complete information, understood as the situation where “each player is fully aware of the rules of the game and the utility functions of each of the players”, is a central assumption of game theory. However, this situation and its definition does not consider the awareness of each player, which is covered by the term “common knowledge“, which means that each player is aware that the other players know the rules and every utility function.

Incomplete information, also known as asymmetric information, refers to the contrary, where not all players know each other’s utility functions. John Harsanyi developed the theory of incomplete information in his “Games with Incomplete Information Played by ‘Bayesian’ Players”, 1967.

Game theory is the science of strategic reasoning, in such a way that it studies the behaviour of rational game players who are trying to maximise their utility, profits, gains, etc., in interaction with other players, and therefore in a context of strategic interdependence.

Historical Framework

The initial game theory discussions and analysis can be traced back a long time before the 20^th century, with works from authors such as James Waldegrave and his minimax mixed strategy solution for a two-person game in 1713, James Madison and his theoretical analysis of what would the effect of different tax systems be, or Antoine Cournot’s solution to a duopoly that resembles to what would later be known as Nash equilibria. However, it is not until the 20^th century, that game theory is broadly developed. We can differentiate several periods in the game theory evolution through the 20^th century. However, many concepts were continuously developed through the years.

During the earliest years of the century, from 1910 to 1930, the main focus of game theory was on  strictly competitive games commonly referred to as two-person zero-sum games. In this kind of games, the preferences (and payoffs) of a player will always be opposite to the other player’s and therefore cooperation between both players will be pointless. This kind of games has been extremely productive, as they have set the bases for future development of game theory, becoming its first main milestones. During these years, John von Neumann’s contribution was especially important, and thus, he is considered as one of the fathers of game theory.  In his “On the Theory of Parlor Games”, 1928, he introduced concepts such as the extensive (or tree) form for the description of sequential games and the minimax theorem, providing empirical support to everything he wrote. Another important contribution during these years is the introduction of the strategic form (game matrix) of a game that represents each player’s profile in a matrix form.

The following 20 years were heavily influenced by von Neumann and Oskar Morgenstern’s work, which would culminate in the joint publication of their book “Theory of Games and Economic Behaviour”, 1944.  With this book, game theory gained the status of an independent scientific discipline. One of the major contributions of this book is the application of strategy game concepts instead of random ones, the introduction of a notion of a cooperative game, its coalitional form, and the ‘Neumann-Morgenstern stable sets’. They were the first to extensively apply game theory for practical reasons, especially for analysing economic behaviour. It is also important to stress that the impact of this book has gone beyond its own time.  Many future developments emerged from it, such as the notion of core and transferable utility, and the further development of expected utility theory. Other developments from this period include games with continuum of pure strategies, the computation of minimax strategies and advances in mathematical methods that would be instrumental to later work.

The 1950s were a key period for game theory. The initial phase of incubation had reached its end and the economic perspective and application took the lead, overtaking other perspectives such as the military one. It was during this period when John Nash laid the essential groundwork that would result in the general non-cooperative theory and for cooperative bargaining theory; Lloyd Shapley was another key figure as he defined the value for coalitional games (cooperative games), he initiated the theory of stochastic games, co-invented the core with D.B. Gillies, and, along with John Milnor, developed the first game models with a continuum of players; Harold Kuhn worked on behaviour strategies and perfect recall. The prisoner’s dilemma, which is probably the best known game, was formalized by Al Tucker also during this period. It was also during these years, when the Folk theorem appeared as a result of the interest and study of repeated games, used to analyse punishment strategies in collusion agreements.

The period from 1960 to 1970 was also important in the development of game theory. The discipline expanded, not only theoretically, but also geographically. With the extension of games such as those with incomplete information and non-transferable utility coalitional games, game theory became more widely applicable, and new research centres were established outside the U. S.

From 1970 on, game theory acquired maturity. Up till then, most of game theory concepts and discoveries had been spread word to mouth. However, during this period economic theory journals published many articles related with game theory and even, new journals entirely dedicated to game theory became popular. The interest and study in politics and political economic models grew. The use of non-cooperative game theory to a large variety of economic models brought also the study and refinement of the equilibrium concept. Other advances were made in other areas of research such as complete and incomplete information repeated games, stochastic games, value, core, nucleolus bargaining theory, games with many players and etc. The use of game theory proved to be useful in a wide diversity of areas that include biology, computer science, moral philosophy and cost allocation, to name a few.

It is worth to note, that game theory should be regarded as a tool to find where incentives will lead, but it makes no moral recommendation to what choices should be taken. This science could be pictured as the study of selfishness, but without recommending it.  Welfare economics and its main tools (such as Pareto optimality and compensation criteria) study these problems outside the selfishness condition.