Introduction to Game Theory
Game Theory is a branch of mathematics that deals with the analysis of decision making processes in competitive situations where the outcome for each participant depends on the actions of other participants.
Game theory has far-reaching applications including biology, business, politics, and war.
Game Theory, also known as Multiperson Decision Theory, is the analysis of the decision making process in a situation with more than one decision maker and where each actor's payoff depends on the choices made by the other actors. Game Theory can be used to study behaviors in competitive and cooperative situations comprising actors with opposed, mixed, or similar interests.
Under game theory, each agent's preferences of action depend on the actions of the other parties involved. When this is unknown, the agent's actions depend on their belief of how the other agents will act. Similarly, the other agent's actions will depend on their belief of the other participant's actions. This process can continue ad infinitum.
If this is hard to intuit, think of scenarios in movies or TV shows where a character says something along the lines of, "He knows... and I know that he knows... But what if he knows that I know he knows?" This is an example of a decision-maker trying to determine the best course of action while understanding the probable actions of another agent.
Here is another example from Charles Shultz' classic Peanuts cartoon. This strip was originally published on Sept 30th, 1962.
As another example, think about how consumers and retailers interact regarding price.
Rational consumers try to make the best purchases possible within the confines of their individual budget, regardless of the choices of other consumers. Additionally, consumers make decisions based on their expectations of future prices, even when future prices are not known. On the flip-side, future prices are decided by consumer decisions today.
As an example, think about the prices of TV's. Many consumers aren't willing to pay the full price for a TV with the most brand-new technology. But unless consumers purchase these products companies won't be incentivized to produce the product and fight over price, lowering the price for all consumers. 
What is Game Theory?
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Eric Roberts is a Charles Simonyi Professor of Computer Science at Stanford University. Roberts teaches a variety of courses around computer programming and organized a capstone around game theory. He defined game theory as follows:
Game theory is concerned with the decision-making process in situations where outcomes depend upon choices made by one or more players. The word "game" is not used in the conventional sense but describes any situation involving positive or negative outcomes determined by the players' choices and, in some cases, chance.
In order for game theory to apply, certain assumptions must be made. The first is that each player is rational, acting in his self-interest. In addition, the players' choices determine the outcome of the game, but each player has only partial control of the outcome. 
In other words, game theory formalizes the language and structure of the analysis of decision making in competitive environments.
David K. Levine, a professor of economics at UCLA, continues:
What economists call game theory psychologists call the theory of social situations, which is an accurate description of what game theory is about. Although game theory is relevant to parlor games such as poker or bridge, most research in game theory focuses on how groups of people interact. There are two main branches of game theory: cooperative and noncooperative game theory. 
Lluís Bru Martínez at The University of the Balearic Islands defines cooperative and non cooperative game theory:
In non-cooperative game theory, we focus on the individual players’ strategies and their influence on payoffs, and try to predict what strategies players will choose (equilibrium concept).
In cooperative game theory, we abstract from individual players’ strategies and instead focus on the coalition players may form. We assume each coalition may attain some payoffs, and then we try to predict which coalitions will form (and hence the payoffs agents obtain). 
Game theory has become increasingly popular in the last few decades and has been applied in a wide range of applications, including psychology, evolutionary biology, war and politics, and economics and business.
The formal application of game theory requires knowledge of the following details: the identity of independent actors, their preferences, what they know, which strategic acts they are allowed to make, and how each decision influences the outcome of the game. Depending on the model, various other requirements or assumptions may be necessary. Finally, each independent actor is assumed to be rational. 
Despite its many advances, game theory is still a young and developing science.
The History of Game Theory
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John Von Neumann and Oskar Modenstern are widely credited with the invention of the mathematical theory of games.
Before them, there was a man named Emile Borel. Borel was a French politician and mathematician who published many papers on probability and measure theory. Borel wrote a series of papers in the 1920's on mathematic theory behind poker, one such study focused on the dilemma of bluffing and second-guessing bluffing in poker.
Borel was using mathematical theory in games with imperfect information, which is how we define game theory. However, even being the first to be recorded with a organized system for playing games he did not develop his ideas very far which is why the credit goes to Von Neumann and Modenstern. 
Anachronistic History of Game Theory
Socrates tells a story about the Battle of Delium (recounted in Plato's Latches and Symposium) as involving the following situation:
Consider a soldier at the front, waiting with his comrades to repulse an enemy attack. It may occur to him that if the defense is likely to be successful, then it isn't very probable that his own personal contribution will be essential. But if he stays, he runs the risk of being killed or wounded—apparently for no point. On the other hand, if the enemy is going to win the battle, then his chances of death or injury are higher still, and now quite clearly to no point, since the line will be overwhelmed anyway. Based on this reasoning, it would appear that the soldier is better off running away regardless of who is going to win the battle. Of course, if all of the soldiers reason this way—as they all apparently should, since they're all in identical situations—then this will certainly bring about the outcome in which the battle is lost.
Of course, this point, since it has occurred to us as analysts, can occur to the soldiers too. Does this give them a reason for staying at their posts? Just the contrary: the greater the soldiers' fear that the battle will be lost, the greater their incentive to get themselves out of harm's way. And the greater the soldiers' belief that the battle will be won, without the need of any particular individual's contributions, the less reason they have to stay and fight. If each soldier anticipates this sort of reasoning on the part of the others, all will quickly reason themselves into a panic, and their horrified commander will have a rout on his hands before the enemy has fired a shot. 
Just as Plato and Socrates had pondered what would later become part of game theory, military leaders also molded their strategies to approach battles more systematically. This most easily calls to mind Sun Tzu's Art of War but I'll share a different example. The following is the story of the Spanish conqueror Cortez's landing in the Aztec occupied Mexico:
The Spanish conqueror Cortez, when landing in Mexico with a small force who had good reason to fear their capacity to repel attack from the far more numerous Aztecs, removed the risk that his troops might think their way into a retreat by burning the ships on which they had landed. With retreat having thus been rendered physically impossible, the Spanish soldiers had no better course of action but to stand and fight—and, furthermore, to fight with as much determination as they could muster.
Better still, from Cortez's point of view, his action had a discouraging effect on the motivation of the Aztecs. He took care to burn his ships very visibly, so that the Aztecs would be sure to see what he had done. They then reasoned as follows: Any commander who could be so confident as to willfully destroy his own option to be prudent if the battle went badly for him must have good reasons for such extreme optimism. It cannot be wise to attack an opponent who has a good reason (whatever, exactly, it might be) for being sure that he can't lose. The Aztecs therefore retreated into the surrounding hills, and Cortez had his victory bloodlessly. 
The two situations above, the solider at Delium and the situation manipulated by Cortez, share a common logic.
Notice that the soldiers are not motivated to retreat just, or even mainly, by their rational assessment of the dangers of battle and by their self-interest. Rather, they discover a sound reason to run away by realizing that what it makes sense for them to do depends on what it will make sense for others to do, and that all of the others can notice this too. Even a quite brave soldier may prefer to run rather than heroically, but pointlessly, die trying to stem the oncoming tide all by himself. Thus we could imagine, without contradiction, a circumstance in which an army, all of whose members are brave, flees at top speed before the enemy makes a move. If the soldiers really are brave, then this surely isn't the outcome any of them wanted; each would have preferred that all stand and fight. What we have here, then, is a case in which the interaction of many individually rational decision-making processes—one process per soldier—produces an outcome intended by no one. (Most armies try to avoid this problem just as Cortez did. Since they can't usually make retreat physically impossible, they make it economically impossible: they shoot deserters. Then standing and fighting is each soldier's individually rational course of action after all, because the cost of running is sure to be at least as high as the cost of staying.) 
The Father of Game Theory: John Von Neumann
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John von Neumann, born János Neumann in Budapest, Hungary in 1903, was a Hungarian-American mathematician, physicist, and computer scientist.
The von Neumann's were an affluent Jewish family in Budapest. John was a remarkable child and showed signs of genius from an early age. By age 8, John could joke in Classical Greek, and, as a party trick, memorize and recite the names and numbers from a random page in the telephone book. 
John published his first paper in mathematics at the age of 18 (it was co-published with his tutor). John spent the next five years in school studying mathematics at the University of Budapest, the University of Berlin, and the Swiss Federal Institute of Technology in Zurich. By 1926, roughly 23 years old, he received his Ph.D. in mathematics with minors in physics and chemistry.
Within a few years he had garnered world-wide acclaim as a young mathematical genius. In 1929 he accepted a position at Princeton University.
Von Neumann was commonly described as "a practical joker and always the life of the party." John and (his second wife) Klara held parties at their house almost every week. John used his remarkable memory "to compile an immense library of jokes which he used to liven up a conversation. Von Neumann loved games and toys, which probably contributed in great part to his work in Game Theory." 
Inspiration for Game Theory
John Von Neumann's inspiration for game theory ostensibly stemmed from the game of poker. A game Von Neumann rarely played and one in which he was not particularly skilled. Von Neumann's insight was that poker was not guided by probability theory alone. Von Neumann wanted to formalize the idea of "bluffing," the strategy of deceiving other players by hiding information from them.
John von Neumann first approached game theory in his 1928 article, "Theory of Parlor Games," in which he proved the famous Minimax theorum. After that paper he teamed up with Oskar Morgenstern, an Austrian economist at Princeton, to develop his theory.
The pair published the book, Theory of Games and Economic Behavior, which revolutionized the field of economics. Game Theory was originally intended solely for economists but after the books publication, it became clear that the applications were far reaching, including psychology, sociology, politics, warfare, and recreational games. In fact, von Neumann himself was most interested in the applications to politics and warfare.
von Neumann's Involvement in Politics and Warfare
von Neumann created mathematical models to interpret the Cold War. He viewed the U.S. and the USSR as two players in a zero-sum game. From the onset of WWII, von Neumann was confident of a Allied victory, based on the output of his mathematical models.
In 1943, Von Neumann was invited to work on the Manhattan Project. Von Neumann did crucial calculations on the implosion design of the atomic bomb, allowing for a more efficient, and more deadly, weapon. Von Neumann's mathematical models were also used to plan out the path the bombers carrying the bombs would take to minimize their chances of being shot down.
Von Neumann was, at the time, a strong supporter of "preventive war." Confident even during World War II that the Russian spy network had obtained many of the details of the atom bomb design, Von Neumann knew that it was only a matter of time before the Soviet Union became a nuclear power. He predicted that were Russia allowed to build a nuclear arsenal, a war against the U.S. would be inevitable. He therefore recommended that the U.S. launch a nuclear strike at Moscow, destroying its enemy and becoming a dominant world power, so as to avoid a more destructive nuclear war later on. "With the Russians it is not a question of whether but of when," he would say. An oft-quoted remark of his is, "If you say why not bomb them tomorrow, I say why not today? If you say today at 5 o'clock, I say why not one o'clock?" 
The idea of "preventative war" only lasted a few years after it was first advocated. By 1953 it became an impossibility; the Soviets had three to four hundred warheads, meaning that any nuclear strike from the U.S. would be effectively retaliated.
Von Neumann served on the Atomic Energy Commission in 1954 and one year later was diagnosed with bone cancer. It's commonly believed that von Neumann developed the cancer from the radiation he received as a witness to the atomic tests on Bikini atoll. In fact, a number of physicists associated with the bomb developed cancer at relatively early ages.
Despite his sickness he maintained a busy schedule, even when he became confined to a wheelchair.
Von Neumann's last public appearance was in February 1956, when President Eisenhower presented him with the Medal of Freedom at the White House. In April, Von Neumann checked into Walter Reed Hospital. He set up office in his room, and constantly received visitors from the Air Force and the Secretary of Defense office, still performing his duties as a consultant to many top political figures. 
John von Neumann died February 8, 1957.
Accusations of Sexual Advances
Standford professor Eric Roberts writes briefly in a biography of Von Neumann:
An occasional heavy drinker, Von Neumann was an aggressive and reckless driver, supposedly totaling a car every year or so. According to William Poundstone's Prisoner's Dilemma, "an intersection in Princeton was nicknamed "Von Neumann Corner" for all the auto accidents he had there."
His colleagues found it "disconcerting" that upon entering an office where a pretty secretary worked, von Neumann habitually would "bend way way over, more or less trying to look up her dress." (Steve J. Heims, John Von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death, 1980, quoted in Prisoner's Dilemma, p.26) Some secretaries were so bothered by Von Neumann that they put cardboard partitions at the front of their desks to block his view. 
Other Branches of Economic Theory
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As David Levine explained above, game theory falls under the overarching theme of economic theory.
Economic theory has three main branches in addition to game theory:
- Decision theory,
- General equilibrium theory and
- Mechanism design theory.
All three are closely connected to game theory.
Decision theory is the theory of one-person games (or a person against nature). The focus of decision theory is on the preferences and beliefs of the player.
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, and Bayes Law is frequently used to model the way in which new information is used to revise beliefs. 
Decision theory is frequently used in decision making and choice analysis which shows how to best acquire relevant and necessary information before making a decision.
General Equilibrium Theory
General equilibrium theory is a specialized branch of game theory that deals primarily with trade and production, and typically a large number of players (producers and consumers).
It is widely used in the macroeconomic analysis of broad based economic policies such as monetary or tax policy, in finance to analyze stock markets, to study interest and exchange rates and other prices. In recent years, political economy has emerged as a combination of general equilibrium theory and game theory in which the private sector of the economy is modeled by general equilibrium theory, while voting behavior and the incentive of governments is analyzed using game theory. 
General equilibrium theory is used in the analysis of tax policy, trade policy, and international trade agreements.
Mechanism Design Theory
Mechanism Design Theory relies heavily on game theory while focusing more specifically on the impact of the rules of the game.
Game theory takes the rules of a game as a given. Mechanism design theory differs from game theory in that game theory takes the rules of the game as given, while mechanism design theory asks about the consequences of different types of rules. Naturally this relies heavily on game theory. 
Mechanism design theory is used to answer questions about compensation and wage agreements that spread risk while maintaining incentives, and the design of auctions to maximize revenue.
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 Yildiz, Muhamet. “14.12 Economic Applications of Game Theory.” STELLAR, MIT, stellar.mit.edu/S/course/14/fa04/14.12/materials.html.
 Roberts, Eric. “Von Neumann and the Development of Game Theory.” Von Neumann and the Development of Game Theory, Stanford University, cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/neumann.html.
 Levine, David. “Economic and Game Theory What is Game Theory?” What is Game Theory?, UCLA, levine.sscnet.ucla.edu/general/whatis.htm.
 Bru, Lluís, and Carles Solà. Decisions and games, The University of the Balearic Islands, www.uib.cat/depart/deeweb/pdi/hdeelbm0/Decisions_and_games.html.
 “Game Theory.” Investopedia, 15 Apr. 2016, www.investopedia.com/terms/g/gametheory.asp.
 von Neumann, J., and Morgenstern, O., (1944). The Theory of Games and Economic Behavior. Princeton: Princeton University Press.
 Ross, Don. “Game Theory.” Stanford Encyclopedia of Philosophy, Stanford University, 25 Jan. 1997, plato.stanford.edu/entries/game-theory/.
 Poundstone, William. “John von Neumann.” Encyclopædia Britannica, Encyclopædia Britannica, inc., 15 Oct. 2017, www.britannica.com/biography/John-von-Neumann.