User:Jacobhesster/sandbox

= Game Theory = "You treat world history as a mathematician does mathematics, in which nothing but laws and formulae exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal shallow, mathematical present." -Hermann Hesse

Watch here for introduction from Ben Polak's Game Theory class at Yale University. Write down the first five lessons that he gives.

What is Game Theory?
Everyday, humans go through millions of thoughts, perceptions, and sensations processed in their brain. Their attention sorts the information that is labelled as important for that person and makes decisions based on what is most important to them. The neurosciences could follow every electrical impulse and neuron in the network, but they have difficulty in understanding conscious decision-making. This leads to a social science approach to the topic called Game Theory.

Game theory can be associated with research in both economic and psychological studies. Economists study the strategic interaction between consumers, producers, firms, and just about anyone that participates in market activity like investing and spending. Psychologists study how people make decisions in social groups in certain situations with knowledge of preferences and information. Both sections share similar research and assumptions that are made when considering a decision maker in a set of given circumstances to make analysis simpler. Game theory focuses on the positive side of economics pinpointing an objective framework to measure an individual's outcomes and options against each other. Specific information regarding the differences in people's preferences or utility is kept constant and certain assumptions are made in order to keep analysis impartial to certain kinds of individuals. More plainly, game theory can provide a mathematical basis for decision making based on utility received from the outcomes of choices in a particular game.

Assumptions
There are many assumptions made when analyzing games, and the above are statements that well encompass the summary of things that are assumed The first is that players typically play in a rational manner, choosing options and strategies that end in a goal that should be clear to third party observers. For example, a spectator watching a basketball game should be able to discern the goals of the individuals players as attacking towards one basket and the opposition attacking towards the opposite basket. Players who follow these strategies are perceived as rational while a player that sometimes attacks their own basket would be considered irrational. The second is that, given that players are rational, they will look to maximize the utility that results from the possible outcomes. If the basketball game that we are watching has two goals, one that rewards two points and another that rewards three points, and both require the same amount of effort to reach, players will score in the three point hoop in order to maximize their utility. Different preferences of players means that utility maximization is not uniform for everyone. A basketball player may derive an extra amount of utility from scoring on the two point goal making it the choice for his personal utility maximization. This effect can be seen in the Battle of the Sexes game where the wife and the husband have distinct preferences.The last assumption is that players will use all information provided to make a decision that satisfies the first two assumptions. While rather unrealistic, this assumption allows game theorists to find the best responses in games using all the information given for a game. We will see how important this is when we discuss the effect of the absence of information. When playing games, it is important to keep these assumptions in mind so that our analysis is shown to be supported by the statements and the implications of them.
 * Players are rational (have a clear objective).
 * Players seek to maximize benefits for themselves.
 * For all knowledge and information given to a player, he or she will use it to the make the best decision.

Terminology

 * A game is a structured process where players choose strategies and end with payoffs assigned to those strategies. Games are always composed of players, strategies, and payoffs which are described in the following lines.
 * Players are the decision makers in games which choose between strategies. In tree structures, they are denoted by labelled nodes, and in matrices, they are show with their strategies on a specific side. Notation: P1, P2,...Pn.
 * Strategies are the decisions that players choose to reach a certain outcome. A game may have multiple stages of choices so a complete strategy would list all the choices chosen by players in a game. These are shown in tree diagrams as the branches, and the sides of the columns and rows in a matrix. Notation: S1, S2,...Sn.
 * Payoff is the outcome of a game that ends with a quantifiable reward defined in units like dollars or utiles. Notation: G1,G2,...Gn.