Introduction to Game Prediction
Greek Horse Racing Vase
History of Game Playing
A very short introduction of game theory for non-math majors. Game theory is used to logical and strategically decide on the best outcome.
The most familiar example happens every four years during a presidential election when most people vote for the "lesser of the two evils."
Since the dawn of humankind, people have gambled with dice. In ancient Greek religion and mythology, many a final decision was made with the roll of the bone dice. Dicé was the goddess of a fair and just decisions but her brother Ares throws the dice. Roman guards rolled the dice for Christ’s robe in the book of Matthew.
A good example of using logic is about the horse race story of General Tian Ji. Another good horse racing logic game is in the book "The predictioneer's game: using the logic of brazen self-interest", by Bruce Bueno de Mesquita. On page 140 he wrote, "Why did Sparta lose its hegemonic position in Greece just thirty-three years after victory in the Peloponnesian War?" Mesquita explains as the Spartan's Gini Index grew larger - the richest Spartans were saving their best horses to make money on racing rather than sending them to the war front.
Game theory started out about gambling and playing the best odds to win. Founders of the theory of probability, Blaise Pascal and Pierre Fermat were inspired by gambling. Pascal is famous for making one of the earliest calculating machines and Fermat also for his work in analytic geometry and precursor work towards differential calculus.
25 year Henry Rhind from Scotland traveled to Egypt because he was sick, during the winter of 1858. He bought a scroll in a Luxor market.
The Rhind mathematical papyrus, (RMP), circa 1650BC, during the Egyptian Middle Kingdom period of Thebes contains a compilation of 85 math problems and describes one of the earliest number puzzles recorded involving multiplication.
Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats of wheat. What is the total of all of these?
T = Temptation to defect
R = Reward for mutual cooperation
P = Punishment
S = Sucker's payoff
Modern Game Theory
John von Neumann, who earned his Phd at the young age of 23, first wrote a paper on game theory in 1923. In 1944, he and Oskar Morgenstern wrote the book, 'Theory of Games and Economics Behavior'.
In 1950, John Nash published his Equilibrium theory, followed by the Bargaining Solution and then the Nash Programme. Nash's idea was much broader than Neumann's two-person zero-sum games. Nash formulation used with any number of people, n-players having n-tuple strategies that are self-countering is called the equilibrium point.
This is the most popularized game theory because of the movie, 'A Beautiful Mind' (2001) starring Russell Crowe who played John Nash. It's just too bad that the blonde and brunette example shown is the movie is NOT a Nash Equilibrium because as soon as the blonde is left alone, everybody would want to dump their brunette and go for the blonde. Nash shared the Nobel prize with Harsanyi and Selten in universal, recursive rationality - and a little bit of irrationality
"The Nash equilibrium tells us what we might expect to see in a world where no one does anything wrong." Robert Weber, Northwestern University
In 1953 Lloyd Shapley formulated the theory of fair allocation of gains within a set of cooperative players known as the Shapley Value. Quantum game theory takes into account superposed quantum entanglement of initial states and strategies.
Take a familiar game such a poker and describe it in modern game theory terms. Poker is a strategic interactive decision making sequential with incomplete information. You have a plan for every possible move, take turns and hide your cards from one other. What each player does with their turn - effects the decisions of the rest of the players afterwards. Everyone knows the rules of the game - and everybody is rational in the sense that each player wants to win that pile of money.
An ordinal payoff is one only of ranking, first choice, second choice, etc. If measuring how much each player prefers one option over another than that is called cardinal payoff. Finite games have finite players. Pareto optimal means; "A state of affairs where it is not possible to improve the economic lot of some people without making others worse off." - answers.com
Iterated elimination of dominated strategies (IEDS) means just dropping out all worser/dominated choices for the better/dominating choices. If a game has an IEDS solution, then there is a unique Nash equilibrium. A strict domination is when choosing Y is always better than choosing Z - no matter what the other players decide. A weak domination is when where Y is much better on at least one choice of an opponents, and for all other opponents choices Y is no worse off than Z.
Thomas Kuhn wrote a theorem about a game with no chances and no hidden information with perfect recall by the players which can be rolled back using a tree diagram. The movie 'Memento' (2000), starring Guy Pearce with short term memory, challenges the most basic premise of game theory - that the players have perfect recall.
A game of pure strategy game has no luck involved in it - such as chess. A mixed strategy game includes some level of chance.
A non-cooperative game such as the Stag Hunt is a coordination game or trust dilemma. The Prisoner's Dilemma is a thought experiment when the final outcome ends up being the worst of all for the hapless prisoners.
The Prisoner's Dilemma is an artificial construct to focus the mind. It's a thought experiment but in real life most people familiar with the justice system would not believe a cop's empty promise of leniency with the judge other that a weak recommendation of, "putting in a good word." There are more versions of the Prisoner's Dilemma than the one most people are familiar with from the movies that shows second choices with more cooperation.
The Hamlet Effect also called 'rational irrationality' can be used in making credible threats, promises or commitments. Making a commitment is like when the Spanish Conquistador Hernando Cortez burnt his ships after landing, he and his men had to conquer or die. The Somali pirates' attack boat was scuttled during the hijacking of the Maersk Alabama for much of the same reason.
When making a threat it has to be believed by your opponent. If your opponent knows that you're lying - then it's useless. There is no better way to make your threats credible than to behave in an irrational manner. Having a reputation for ruthlessness in the past will make currents threats credible. Threats cost nothing and that is why threats are used more often that the costlier 'carrot and stick' promise approach.
A Nash Equilibrium occurs where a player maximises their payoff given what they anticipate their opponent is doing.
The BIG Questions
In the book by Steven E. Landsburg, 'The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics ', he describes the economic case for promiscuity that safer sex - is more sex. "Professor Michael Kremer of MIT's economics department,[said], 'The spread of AIDS in England could plausibly be retarded if everyone with fewer than about 2.25 partners per year were to take additional partners more frequently.'"
Science Math Probability