Saturday, August 3, 2013

91. Games People Play

Game theory is the mathematical study of situations of conflict of interest. It is the analysis of individual and group decision-making processes in situations involving a diversity of individual goals and differing levels of strategic control over the environment. It provides a mathematical description of certain interactive phenomena among two or more members of a population.

Game theory and 'transactional analysis' were brought into public perception by Eric Berne's (1964) bestselling book Games People Play: The Psychology of Human Relationships.

Another popular-level book I read when I was young (and still impressionable!) was Thomas Harris's I'm OK – You're OK (1967). It taught me the importance of striving for 'game-free relationships': The most valuable personal relationships are those in which the people involved do not feel the need to play 'games' with one another.

John Maynard Smith (1972, 1974) introduced game-theory considerations into the theory of biological evolution, building on the foundations laid by R. A. Fisher in the 1930s. A game is a formal model of an interactive situation. The formal definition of a game specifies the players, their preferences, their information, the strategic actions available to them, and how these actions may influence the outcome(s). Naturally, game theory is very useful for understanding certain kinds of biological and cultural evolutions of complexity.

The publication of the book Theory of Games and Economic Behaviour by von Neumann and Morgenstern in 1944 created a major and still expanding interest in this subject. [von Neumann again! I have described some of his other work in earlier posts. Not many people have been as original and creative and productive as him.] This book included for the first time a proof of the so-called Fundamental Theorem in the theory of games, namely the minimax theorem (minimization of the maximum possible loss; or maximization of the minimum possible gain).

Classical game theory provides a mathematical formalism of strategy as an extension of individual rational behaviour. Since human beings are not exactly famous for rational behaviour, classical game theory plays a role complementary to social psychology and other behavioural sciences which model human actions in terms of 'limited rationality' and nonconscious behaviour. The initial work in game theory was directed towards economics, but was soon diversified to a theory of competitive games relevant not only to competitive economics, but also combat and warfare.

For understanding complex adaptive systems (CASs), the importance of game theory is obvious. Members of a CAS have to optimize their interaction not only with the ever-changing environmental conditions, but also with the survival strategies adopted by fellow members of the CAS, as well as other interacting species. Both competition and cooperation play a role in the survival strategies adopted by interacting species. Particularly notable in this context is the pioneering work done by Maynard Smith (1974, 1976) on evolutionary dynamics. He introduced a modification of the game theory existing at that time, and analysed the nature of competition and cooperation among species. The important idea he introduced in this connection was that of evolutionarily stable strategies (ESS). I shall discuss coevolution and ESS in a future post.

In game theory one generally assumes that the players are rational. A rational player is defined as one who always chooses an action which gives the outcome he most prefers, given that he expects his opponents to be rational too.

The first defining feature of a game is the number of players involved. In general, this number can be n, and the set of players can be represented by N = {1, 2, . . . n}.

If n = 0, we speak of a zero-player game, an example being a cellular automaton (described in Part 68). Once an automaton starts, it keeps going, without any decision-making imposed on it by a person.
When n = 1, game theory becomes decision theory. Games of solitaire are examples of one-person games.

n = 2 games, or two-player games, are the best investigated; the concepts and the conclusions are clearer for them.

For modelling in the field of macroeconomics, the number of players can be extremely large, and sometimes it is assumed to be infinite. One even speaks in terms of a continuum of players if the influence any one player has on the game is infinitesimally small.

Apart from the number of players in a game, a characteristic feature that can distinguish one form of game from another is the level of detail considered important or relevant for playing the game. One can distinguish three models or 'forms' of games by this criterion: extensive form of games; normal or strategic form of games; and coalitional form of games.

Extensive form of games

Maximum detail is available in the extensive form of games, or extensive games. One can speak of a position in the game, and of a move in the game. A move takes the game from one position to another. This later position can depend on which player’s turn it was to make a move. A player may even make random moves (e.g. rolling of dice, or shuffling of cards before dealing them in a game of cards).

Any game is played according to certain rules. For example, the rules may specify the probabilities of the outcome of random moves.

In an extensive model of a game, players may have information before making a move. A perfect-information game is that in which each player knows about all the past moves by the players and the results of all such moves, as also the results of all the past random moves.

Two-person perfect-information games, with no chance moves, and with complete knowledge of 'win' or 'lose' outcomes, are called combinatorial games. Such games may be either impartial or partisan. In an impartial game, the two players can make the same set of valid moves from each position. If this is not the case, it is a partisan game.

In the language of graph theory, an extensive game can be represented by a tree which depicts the order in which the players make moves, and the information each player has at each decision point.

I shall discuss strategic games in the next post.