Game theory deals with analyzing different methods and logical processes in the case of situations where the outcome of an individual’s decision largely depends on the actions of others. Game theory serves an important purpose in today’s world as different variations of game theory, such as traveler’s dilemma and prisoners dilemma, are applied in various fields of study and occupations such as biology, business, politics, etc. Foundational Aspects: We must look back at the previous resemblances in order to understand the development of game theory into the game theory we know today.
We must travel all the way back to the Greek Classical Age where famous philosopher Socrates and his student Plato wrote about the thought process of soldiers in battle in terms of whether to fight or run away depending on the circumstance. What they concluded was that based solely on whether or not the individual cared about their own safety or if they were more fearful of losing a battle. If the individual simply cared for their own survival, they would opt to run away no matter the circumstance.
However, if the individual takes into account that all the other soldiers may run away as well, they may stay for the sake of thinking that the rest of the soldiers had the same thought process as they did and therefore they would all stay. This would become the first known published example of game theory ever in the 4th century BCE. Fast forward to the 18th and 19th centuries and figures such as James Madison unintentionally used game theory in order to justify systems of taxation. The first time game theory was outright defined and considered a unique branch of mathematics was in 1944 when John von Neumann, along with his collegue Oskar Morgenstern, published Theory of Games and Economic Behavior which focused on the economic applications of this newly discovered theory. However, it was found that the applications that the book had discussed were very specific and didn’t display the versatility game theory had.
A key contribution of this work however was the introduction of the expected utility, which is one of the most critical components of game theory. The expected utility hypothesis is defined as decision theory that is in reference to people’s preferences in respect to choices that do not have certain outcomes. This is the mathematical equation for the expected utility where: the left side is the value of the gamble as a whole, xi is the ith possible outcome, u(xi) is its apparent value, and pi is its probability of occuring. There could be either a finite set of possible values or xivalues, in which case the right side of this equation has a finite number of terms. There can also be an infinite set of discrete values, in which case the right side has an infinite number of terms.
The publication of this book also set into motion the entire future of game theory as most mathematicians and economists began to focus on cooperative game theory, or in other words games that involve groups of people rather than individual decision-making. However, in order this theory to correctly represent rational decision making, both Neumann and Morgenstern decided to implement four requirements for the individual in order for this theory to work. They follow as; completeness, transitivity, independence and continuity.
Completeness is when an individual has well defined preferences and can always decide between any two alternatives with three separate decision of the individual either preferring X to Y, being indifferent between X and Y, or preferring Y to X. Transitivity is the property that, as an individual decides according to completeness, the individual also decides in a consistent manner. Independence of unnecessary alternatives also pertain to having specific preferences. It assumes that two choices in combination with an additional but irrelevant third one (Z) will maintain the same exact extent of preference as when the two are presented independently of the third one, essentially canceling out the effect of the third choice. The independence axiom is the most controversial axiom.
Continuity assumes that when there are three lotteries (A, B and C) and the individual prefers A to B and B to C, then there should be a possible combination of A and C in which the individual is then indifferent between this mix and the lottery B. If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function. This was the historical significance of the publishing of Theory of Games and Economic Behavior.
The final part of history that must be mentioned before moving onto modern game theory are the contributions by John Nash. John Nash is the founder of Nash’s equilibrium and an early version of the Nash equilibrium concept was first known to be used in 1838 by Antoine Augustin Cournot in his theory of oligopoly. In Cournot’s theory, firms choose how much output to produce to maximize their own profit. However, the best output for one firm depends on the outputs of others. However, Nash’s definition of equilibrium is broader than Cournot’s, since the Nash definition makes no judgements about the optimality of the equilibrium being generated. The modern game-theoretic concept of Nash equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible actions. The concept of the mixed-strategy Nash equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior.
However, their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions. The contribution of Nash in his 1951 article ‘Non-Cooperative Games’ was to define a mixed-strategy Nash equilibrium for any game with a finite set of actions and prove that at least one (mixed-strategy) Nash equilibrium must exist in such a game. The key to Nash’s ability to prove existence far more generally than von Neumann lay in his definition of equilibrium.
According to Nash, ‘an equilibrium point is an n-tuple such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each player’s strategy is optimal against those of the others. However, issues arise because Since the development of the Nash equilibrium concept, game theorists have discovered that it makes misleading predictions (or fails to make a unique prediction) in certain circumstances. They have proposed many related solution concepts designed to overcome perceived flaws in the Nash concept.
As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838. However, the use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well, allowing it to become one of the most versatile mathematical theories in today’s real world applications. Specific examples of the application of game theory in the business world is given the current day situation of neoclassical economics struggled to understand entrepreneurial anticipation and could not handle imperfect competition.
Game theory turned attention away from steady-state equilibrium toward the market process. In business, game theory is beneficial for modeling competing behaviors between economic agents. Businesses often have several strategic choices that affect their ability to realize economic gain. For example, businesses may face dilemmas such as whether to retire existing products or develop new ones, lower prices relative to the competition, or employ new marketing strategies. Economists often use game theory to understand oligopoly firm behavior. It helps to predict likely outcomes when firms engage in certain behaviors, such as price-fixing and collusion.
The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians. It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects.
Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime.
Thus, in a process that can be modeled by variants of the prisoner’s dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively. Prisoner’s Dilemma can best be explained as a example of a paradox in decision analysis that illustrates individuals acting upon their own self-interests ends up harming both parties. This situation not only displays why two people may not cooperate, but how cooperation between them can fulfill both individuals’ best interests. These individuals can be represented by various people and groups such as countries and corporations.
A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of non-democratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.
