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### MATHEMATICAL MODELLING & GAME THEORY

|-1| MATHEMATICAL MODELING

|-1.00| INTRODUCTION TO MODELING
Mathematical models describe our beliefs about how the world functions. In mathematical modeling, we translate those beliefs into the language of mathematics.

|-1.01| ADVANTAGES OF MODELLING
- Mathematics is a very precise language. This helps us to formulate ideas and identify underlying assumptions.

- Mathematics is a concise language, with well – defined rules for manipulations.

- Computers can be used to perform numerical calculations.

|-1.03| OBJECTIVES OF MODELLING
Mathematical modeling can be used for a number of different purposes. The following are the objectives of modelling :-
-Developing scientific understanding through quantitative expression of current knowledge of a system.

-Test the effect of changes in a system

-Aid decision making including tactical decisions by managers strategic by planners.

|-1.03| VARIABLES IN MODELING
Generally speaking, in any given model or equation, there are two types of variables:
-Independent variables: the values that can be changed in a given model or equation. They provide the “input” which is modified by the model to change the “output”.

-Dependent variables: The values that result from the independent variables.

|-1.04| USING INDEPENDENT AND DEPENDENT VARIABLES
The definition of an independent or dependent variable is more or  less universal in both statistical and scientific experiments and in mathematics; however, the way the variable is used varies slightly between experimental situations and mathematics.

|-1.05| EXAMPLE OF VARIABLES IN SCIENTIFIC EXPERIMENTS
If a scientist conducts an experiment to test the theory that a vitamin could extend a person’s life – expectancy, then:
-The independent variable is the amount of vitamin that is given to the subjects within the experiment. This is controlled by the experimenting scientist.

-The dependent variable, or the variable being affected by the affected by the independent variable, is life span.

-The independent variables and dependent variables can vary from person to person, and the variances are what are being tested; that is, whether the people given the vitamin live longer than the not given the vitamin.

The scientist might then conduct further experiments changing other independent variables gender, ethnicity, overall health, etc in order to evaluate the resulting dependent variables and to narrow down the effects of the vitamin on life span under difference circumstances.

Here are some other examples of independent and dependent variables In scientific experiments:

-A scientist studies the impact of a drug on cancer. The independent variables are the administration of the drug – the dosage and timing. The dependent variable is the impact the drugs have on cancer.

-A scientist studies the impact of withholding affection on rats. The independent variable is the amount of affection. The dependent variable is the reaction of the rats.

-A scientist studies how many days’ people can eat soup until they get sick. The independent variable is the number of days people can eat the soup. The dependent variable is the onset of illness.

|-2| GAME THEORY

|-2.00| INTRODUCTION TO GAME THEORY
Game theory is a study of strategic decision making. It is also the study of mathematical models of conflict and cooperation between intelligent rational decision makers.

Another term suggested game theory as a more descriptive name for the discipline that it is an interactive decision theory. Game theory is mainly used in economics, political science and psychology, as well as logic and biology. The theory first addressed zero sum games, such that one person’s gains exactly equal net losses of the other participant(s).

Today, however, game theory applies to a wide of behavioural relations, and has developed into an umbrella term for the logical side of decision science, to include both human and non-humans, like computers. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two person zero-sum games and its proof by a theorist named John Von Neumann.

He used Brownness’s fixed-point theorem on continuous mapping into compact convex sets in his original proof, which became a standard method in a game theory and mathematical economics. The second edition of his book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

This theory was developed extensively in the 1950s by many scholars. Games theory was later applied to biology in the 1970s.

|-2.01| DESCRIPTION OF TYPES OF GAMES
Cooperative or non-cooperative – a game is cooperative if the players are able to form binding commitments. For example the legal system requires them to adhere to their promises.

In non cooperative games this is not possible. Communication among players is allowed in cooperative games, but not in non-cooperative  ones.

Of the two types of games, non-cooperative games are able to model situations to the finest details, producing accurate results. Cooperative games focus on the games focus at large. Hybrid games coalitions of players are formed in a cooperative game, but these play in a non-cooperative fashion.

|-2.02| SYMMETRIC AND ASYMMETRIC
A symmetric game is a game where the pay offs for playing a particular strategy depend only on the other strategies employed, not on who is playing them.

If the identities of the players can be changed without changing the payoff to the strategies, then the game is symmetric. The standard representation of chicken, the prisoner’s dilemma, and the stag hunt are all symmetric games. Many of the commonly studied 2 x 2 games are symmetric.

Asymmetric games are game where there are identical strategy sets for both players. For examples, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible for a game to have identical strategies for both players, yet be asymmetric.

|-2.03| ZERO-SUM AND NON-SUM
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 (more informally, a player benefits only at the equal expense of others). Some theorists exemplifies a zero sum game (ignoring the possibility of the house’scut), because one wins exactly the amount one’s opponents lose.

Other zero-sum games include matching amount one’s opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and Chess.

Many games studied by game theorists including the infamous prisoner’s dilemma are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero sum games, a gain by one player does not necessarily correspond with a loss by another.

|-2.04| SIMULTANEOUS AND SEQUENTIAL
Simultaneous games are games where both players move simultaneously or if they do not move simultaneously, the later players are unaware of the earlier players’ actions (making them effectively simultaneously).

Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need be have some knowledge about earlier actions. This need be perfect information about every action of earlier players, it might be very little knowledge, for example, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed.