Game Theory
From Math2033
Game Theory is any theory designed to calculate the maximum efficacy of a players strategy. It is also believed to be applicable to real world situations.
History
Game Theory can trace its history back to the early 18th century with letters describing a minimax strategy for a card game between two British nobles. However, despite several ventures into the topic at various times, it wasn't distinguished as a discreet field until mathematician John von Neumann published a paper in his 1928 paper "Zur Theorie der Gesellschaftsspiele," in "Mathematische Annalen." Neumann later expanded these ideas in a book published in 1944 and co-authored with Oskar Morgenstern titled Theory of Games and Economic Behavior. Shortly thereafter, in 1950, Game Theory gained its most well known discovery with the rise of John Nash and his contribution of the so called Nash Equilibrium which eventually won him the title of Nobel Laureate in 1994. In the 1970's, due to the work of John Maynard Smith, Game theory was applied to the field of biology. In 2005, Thomas Schelling joined John Nash as Nobel Laureate, with the advent of has since come to be known as Evolutionary Game Theory.
Game Theory Defined
According to the Stanford Encyclopedia of Philosophy online, game theory is "the study of the ways in which strategic interactions among rational players produce outcomes with respect to the preferences (or utilities) of those players, none of which might have been intended by any of them."
According to Wikipedia, game theory "attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others."
According to an online source, game theory is "a concept that deals with the formulation of the correct strategy that will enable an individual or entity, when confronted by a complex challenge, to succeed in addressing that challenge."
It involves the study of how the final outcome of a competitive situation is dictated by interactions among people involved based on the goal and preferences of those players, and on the strategy that each player employs. It was developed on the premise that for whatever circumstance, or for whatever game, there is a strategy that will allow one to win.
Decision Making Process of Game Theory
"Decision theory can be viewed as a theory of one person games, or a game of a single player against nature. The focus is on preferences and the formation of beliefs. The most widely used form of decision theory argues that preferences among risky alternatives can be described by the maximization of the expected value of a numerical utility function, where utility may depend on a number of things, but in situations of interest to economists often depends on money income. Probability theory is heavily used in order to represent the uncertainty of outcomes, decision theory is often used in the form of decision analysis, which shows how best to acquire information before making a decision."
An Instructive Example
One way to describe a game is by listing the players (or individuals) participating in the game, and for each player, listing the alternative choices (called actions or strategies) available to that player. In the case of a two-player game, the actions of the first player form the rows, and the actions of the second player the columns, of a matrix. The entries in the matrix are two numbers representing the utility or payoff to the first and second player respectively. A very famous game is the Prisoner's Dilemma game. In this game the two players are partners in a crime who have been captured by the police. Each suspect is placed in a separate cell, and offered the opportunity to confess to the crime. The game can be represented by the following matrix of payoffs
not confess confess
not confess 5,5 -4,10 confess 10,-4 1,1
Note that higher numbers are better (more utility). If neither suspect confesses, they go free, and split the proceeds of their crime which we represent by 5 units of utility for each suspect. However, if one prisoner confesses and the other does not, the prisoner who confesses testifies against the other in exchange for going free and gets the entire 10 units of utility, while the prisoner who did not confess goes to prison and which results in the low utility of -4. If both prisoners confess, then both are given a reduced term, but both are convicted, which we represent by giving each 1 unit of utility: better than having the other prisoner confess, but not so good as going free.
Simple Examples
Martial Arts
An explanation of Game Theory and the Martial Arts can be accessed here:
http://math2033.uark.edu/wiki/index.php/Kumite
Football
Game theory can be applied to football. The defense has to react to the offense and vice-versa. Both teams must adjust their strategies in order to experience the best outcomes.
Click to open a pdf file explaining game theory in terms of Razorback football. Media:GAME_THEORY.pdf
Rock, Paper, Scissors
Rock, Paper, Scissors has been a game enjoyed by all generations for many, many years. It's an alternate way to choose a person other than flipping a penny, so how can you be guaranteed not to lose? |
See also the Rock Paper Scissors page.
The Minimax Theorem and Zero Sum Games
Cooperative Games and the Nash Equilibrium
A cooperative game is a game where groups of players ("coalitions") may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players. The Nash Equilibrium are sets of strategies for players in a noncooperative game such that no single one of them would be better off switching strategies unless others did. |
Altruism
Altruism is a complete concern for other's well-being, without any concern about oneself. |
How Game Theory Economics Explain Human Altruism
Homo Economicus and Game theories
In Economics, the concept of 'homo economicus' is to deal with situations which would, by economic perspective, lead to selfish, yet effective behavior or psychology. 'Homo economicus' is the concept of an agent which behaves as it does in order to constantly maximize its utility. Utility in this case can be anything from material to social purpose and interest.
Researchers have traditionally been very interested in this field and have created a huge amount of experimental data researching it, mostly derived from different variations of well known game theories. Through those that exhibit 'homo economicus,' researchers can reveal what strategies these players use in order to leverage the most utility for themselves. Games which are used are typically so simple that almost anyone could figure out what to do in order to win them. Here are some examples:
Ultimatum
Ultimatum is a simple game where the first player is given an amount of real money–-say $10. Then this player is allowed to give an amount of money to the another player, who can accept or decline it. If declined either one of them will get nothing. If humans are really inherently 'homo economicus', the other player should take any amount of money because no matter how much it is, it’s still a profit.
Dictator
Dictator is simplified version of Ultimatum. In dictator the second player is passive; it doesn't matter whether they accept or decline as it has no effect on the first player.
Public Goods
Public Goods is game arrangement involving several players. Every player has an initial sum of money and they can put an amount in a mutual 'stash' – after this first round the money is taken from the stash and is distributed between all the players equally. Because all the players had the same amount of money, they are aware that not everyone has given any money. If played anonymously, sums which are given to the 'stash' start to decline. The interesting aspect is that the mutually distributed amount almost never declines to zero. This means that there is always some 'hard core givers' who continue to put their money in the stash, no matter what everyone else does.
The Tragedy of the Commons
Tragedy of the commons is a type of social trap that involves a conflict over finite resources, between individual interests and the common good. |
Voting schemes and Arrow's Theorem
Arrow's Theorem (also known as Arrow's paradox) states that there is no way to accurately project the individual's preference into a community setting while still following a set of guidelines with more than three options. This idea is used to show how all systems of voting have flaws. |
Fair Division
Fair Division divides a part equally amongst all participating players. |
Cake Cutting
The Cake Cutting problem illustrates the theory of Fair Division. The cake is cut in a way in which the players believe everyone has received an equal size piece. |
Knaster Inheritance
A means of allocating items or issues to more than two parties in an equitable manner. |
Game Theory in Films
Game Theory on TV
Game Theory in Pop-Culture
http://www.gametheory.net/popular/
Game Theory in Video Games
“The Javelin Glitch” in Call of Duty: Modern Warfare 2
Soon after its release, some players of the online first person shooter Call of Duty: Modern Warfare 2 discovered what became known as “the javelin glitch.” It comprises of a bizarre sequence of button presses that enable the player to "glitch" the game so that when he/she dies in multiplayer, the player would self destruct and murder everyone within 30 feet.
In other words, the javelin glitch presents players with a pragmatic dilemma: they could either abuse the glitch to boost their own scores or they could withhold and maintain the game’s integrity . The problem, in a classical, game theory sense, is that if they withhold, someone else may exploit the glitch to overcome the match. The middle ground is when everyone glitches.
From the table above, each player has what’s called a “dominating strategy" wherein they’re most likely to win (in this example, abusing the glitch.) Of course, unleashing such strategy may cause serious distrust and mayhem among the players, not to mention a suspensions from XBOX LIVE."
Game Theory Lectures
Game Theory Introduction
A lecture from Yale itroducing Game Theory:
Nash Equilibrium
A lecture explaining the Nash equilibrium:
Games People Play
Games People Play: Game Theory in Life,Business, and Beyond
Basic Probability Rules
Basic Probability Rules
Rock, Paper, Scissors
Rock, Paper, Scissors
Weak Dominance
Weak Dominance
How To Catch A Baseball During Batting Practice
How to catch a baseball during batting practice
Strictly Dominant Mixed Strategies
Strictly Dominant Mixed Strategies
Backward Induction and the Escalation Game
Backward Induction and the Escalation Game
Problems With Backward Induction
Problems With Backward Induction
Stag Hunt
Matching Pennies and Mixed Strategy Nash Equilibrium
Matching Pennies and Mixed Strategy Nash Equilibrium
Safety In Numbers
Safety In Numbers
Soccer Penalty Kick
Soccer Penalty Kick
Iterated Elimination Of Strictly Dominated Strategies
Iterated Elimination Of Strictly Dominated Strategies
The Ultimatum Game (Continuous)
The Ultimatum Game (Continuous)
Double Ultimatum
Double Ultimatum
Bargaining & Costly Delays
Bargaining & Costly Delays
Hostage's Dilemma
Hostage's Dilemma
References
Game theory. (n.d.). Retrieved from http://en.wikipedia.org/wiki/Game_theory
Game theory. (2006, March 10). Retrieved from http://plato.stanford.edu/entries/game-theory/