Games have been around since 3000 BCE and since the invention of probability theory, probabilists have studied games especially games of chance. These games of chance, or strategic games as they are more commonly known, gained in popularity among mathematicians and economists when the first edition of John von Neumann and Oskar Morgenstern's book Theory of Games and Economic Behavior was published in 1944. This book was based on past work done on the Minimax Theorem. The minimax theorem is based off two-person zero-sum games, which is a where everything that someone wins must be lost by someone else. These are also referred to as strictly competitive games.

The earliest minimax solution to a game was proposed by Pierre-Remond de Montmort. He wrote a letter to Nicolas Bernoulli, a famous Swiss mathematician, on November 13, 1713. In the letter he told about a solution to a two-person version of a game proposed by James Waldegrave the First Earl of Waldegrave entitled le Her. In the game the two players are named Peter and Paul. Peter holds a common package of cards and he gives a card at random to Paul and subsequently takes one himself. The object of the game is to obtain the higher card than your opponent. The hierarchy of value is Ace, two, three, ... , Jack, Queen, King. If Paul doesn't like the card, he was dealt he can try to compel Peter change with him; but if Peter has a King then Peter is allowed to keep his card. If Paul is doesn't like his first card or the card that he got from Peter, Paul is allowed to get a new card chosen at random from the deck of cards. If the card that Paul draws at random from the deck is a King then he must keep his previous card, whether it was his original or obtained from Peter. If Paul and Peter finish with cards of equal value Paul is considered the loser.

The strategy that Waldegrave came up with was that Paul would change any card lower than seven and hold any card higher than seven, Peter meanwhile would change any card lower than eight and hold any card higher than eight. Waldegrave then wanted to find a way to maximize a player's probability of winning, no matter what strategy the opposing player was using. What resulted was a matrix of probabilities which persuaded Waldegrave that a player would be able to select a strategy and be able to be assured a certain outcome. In the game of le Her, Waldegrave discovered that Peter should keep cards of value 8 and over (change if lower) with the probability 5/8 and should change cards of 8 and under with the probability of 3/8. Paul should hold 7 and over with the probability of 3/8 and change 7 and under with probability of 5/8. This solution that Waldegrave came up with was a minimax solution. He however did not apply this result to other such mixed strategy games saying, "does not seem to be in the usual rules of play" of standard games of chance.

Although Waldegrave later gave up the study of mathematics to pursue a life of diplomacy, de Montmort published the letters he had written between him and Nicolas Bernoulli. Included in the published letters was the work that Waldegrave had done on le Her as an appendix in the second edition. This published work came to be known as Essai d' Analyse sur les Jeux d' Hasard which was published that same year of 1713. This appendix became popular for a letter it contained Bernoulli writing to de Montmort that stated the St. Petersburg paradox. The minimax solution that Waldegrave came up with went unnoticed, but the work that he did of "maximization of expected utility, diminishing marginal utility, and risk aversion put forward by Daniel Bernoulli (1738) in his analysis of the St. Petersburg paradox, lie at the core of game theory."

None of these great works would be interconnected for over 200 years until von Neumann and Morgenstern worked together. In 1944, while working together at Princeton, they discovered that Daniel Bernoulli's moral expectation aided in proving conditions of rationality for their axiomatic utility theory. They built off Waldegrave's game where "players attempt to maximize their probability of winning rather than either their expected monetary gain on each play or their 'moral expectation' (expected utility) of such gains." This discovery catapulted the idea of game theory to new heights in the field of economics and generated interest by other mathematicians who would later build upon it namely John Nash, who added upon von Neumann's theories, with the introduction of the Nash Equilibrium.

In order to understand how Game Theory works we need to understand the Minimax Theorem, which is the most important theorem in Game Theory. However diving right into the Minimax Theorem would be very difficult without understanding some of the elements that make it up. These elements are Zero-Sum Games, The Normal Form, and Mixed Strategies. We will explore each of these and then discover how they work with each other to give us the Minimax Theorem.

A game is said to be zero-sum if and only if at each vertex, the payoff function,

Essentially what this expression is saying is that a zero-sum game represents a closed system where everything that a player wins must be lost by the other player. These games are also known as strictly competitive games because of this reason. In a two-person zero-sum game there is a major difference than other two-player games in that there is no reason for negotiations between the two players. This is because of the payoff function above. There is only one winner and so the other is the complement, the loser. This equilibrium is illustrated by the following theorem by Owen Guillermo,

Theorem - In a two-person zero-sum game let,

Proof:

This theorem is only true for two-player zero-sum games.

The next element that we need to understand is that of the Normal Form. The Normal Form is in many aspects of mathematics. For the purposes of studying Game Theory we say that the Normal Form is a matrix, we will call this matrix A, which represents the finite zero-sum two-person game. The number of rows is equal to the number of strategies that Player 1 has, and the number of columns is equal to the number of strategies that Player 2 has. We can then say that the expected payoff of the game will be element a_ij of matrix A. This is a result of Player 1 choosing his or hers i^th strategy and Player 2 choosing his or her j^th strategy. This will be the equilibrium of the strategy pair if and only if the element, a_ij, is both the largest in its column and the smallest in its row. If such an element exists then the element is then referred to as the matrix's saddle point. This comes from the fact that the surface of a horses saddle curves upwards in one direction, but in the opposite direction it curves in a downwards fashion. Consider the follow matrix A. This matrix has a saddle point in the second row and the second column with element 2. The second matrix, B, has no saddle point because no one element is greatest in its column and smallest in its row.

Let's say that Player 1 and Player 2 are playing a game that involves matrix strategy. Player 1 chooses a strategy (row) and Player 2 chooses a strategy (column) the intersection element of their strategies will be the payout of Player 1. (meaning this is what Player 1 receives from Player 2) If the intersection element is negative then Player 1 is giving a payout to Player 2. Player 1 is trying to maximize the payout, while Player 2 is trying to minimize the payout (or have the payout be negative). The problem is that neither player knows what strategy the other player will employ. From our example matrix above, A, lets analyze what sort of strategy each player might employ if the players faced off. In this case, Player 1 will most likely want to employ their first or second strategy. But what if Player 2 figures out this strategy? Player 1 should then employ their second strategy. Even if Player 2 does guess correctly what Player 1 strategy will be using, it is still best for Player 1 to stick to their second strategy in order to maximize their gain and minimize their loses. This is why strategies need to be kept secretive when playing these types of games. Secretive strategies are the reason why we study Game Theory; it gives us the ability to quantitatively understand what is going through the mind of our opponent.

The last element that is essential for understanding the Minimax Theorem is the concept of Mixed Strategy. With Mixed Strategy we are dealing with games that were like our matrix B above that is defined as having no saddle point. For the purposes of illustrating this type of strategy consider the following matrix.
\begin{align*} C=\begin{pmatrix} 4 & 2 \\ 1 & 3 \\ \end{pmatrix} \end{align*}
We don't have anything that aids up in what will happen in this matrix based on our prior knowledge. However, let us suppose that our opponent is unpredictable and omniscient (all knowing: they will always guess correctly whatever strategy we decide to employ). In the case of this matrix we, Player 1, will choose strategy 1 because we cannot win less than 2, whereas if we chose strategy 2 we would win no less than 1. Our strategy of winning at least 2 is what is known as our "gain-floor". If our opponent were to be Player 1 and we are Player 2 then we would employ the recursive strategy known as "loss-ceiling". As Player 2 we would choose strategy 2 because we want to minimize our losses, so the maximum we would have to payout would be 3. It would be illogical to think that Player 1's gain-floor would be greater than Player 2's loss-ceiling. These two ideas are what make up Mixed Strategy and its associated games. In Mixed Strategy games sometimes it is hard for us to keep our strategy a secret because we have no way of stopping our opponent from reconstructing our reasoning. Logically we would want to choose our strategy rationally; but if our opponent can simply reason what our strategy will be then what is the point? The answer is to then choose our strategy irrationally, but we need to choose our randomization scheme rationally. This idea is the core of Mixed Strategy and the games that it can be adapted to.

Now that we understand the three elements that make up the Minimax Theorem we can now combine these ideas into giving the formal definition of the Minimax Theorem. The Minimax Theorem states that for every two-person, zero-sum game, there always exists a mixed strategy for each player such that the expected payoff for one player is the same as the expected cost for the other. Therefore, V is the best payoff each player can expect to receive from the game. More accurately, V represents the expected payoff to Player 1 and the expected cost to Player 2. This can also be thought inverse to say that -V is the expected cost to Player 1 and the expected payoff to Player 2. The following is a formal proof done by R. D. Luce and H. Raiffa in their published work, Games and Decisions.

Game Theory despite its name has been used to understand one subject more than any other, economics. Game Theory started as an attempt to quantify a winning strategy to a two-player zero-sum game, but has since grown into finding the solution to the problems of duopoly, oligopoly and bilateral monopoly in the world of business. With all of these economic issues the difficulty is to determine a solution due to conflicting interests and strategies of the organizations and individuals that run them. Game Theory is utilized to attempt to find "equilibrium points" which help these individuals what the possible outcomes are given the participants (players) in their respective markets. The Minimax Theorem that was discussed above was the basis of most of the economic decisions made by corporations all over the world from its conception in 1928 and is still utilized in world markets today. However a new development came about that has changed the face of economic practices since its inception in 1951, known as the Nash Equilibrium. This breakthrough which was immortalized by actor Russell Crowe in the Hollywood production of, "A Beautiful Mind" changed the idea of payoffs. In applying the Nash Equilibrium, each player decides on a strategy that not only best for them as an individual but also best for the group as a whole. This concept revolutionized the previous notion of "survival of the fittest" and aided companies in making decisions based as a whole instead of individuals in competition.

Since Game Theory is so heavily used in the field of business here is a list developed by Lloyd Shapley, a Nobel Prize winning mathematician and economist, where he explains the importance of Game Theory to business. The following list was taken from a lecture that Lloyd Shapley gave at the University of Stockholm on December 8, 2012.

"1. Game theory shows the importance to duopolists of finding some way to agree. It helps to explain why duopoly prices tend to be administered in a rigid way. If prices were to change often, tacit agreements would not be found and would be difficult to enforce.
2. Game theory also highlights the importance of self-interest in the business world. In game theory, self-interest is routed through the mechanism of economic competition to bring the system to the saddle point. This shows the existence of the perfectly competitive market.
3. Game theory tries to explain how the duopoly problem cannot be determined. For this, it uses the solution without saddle point under constant-sum-two-person game. At the same time, the duopoly problem without a saddle point is solved by allowing each firm to adopt mixed strategies on a probability basis. In this way, the duopoly problem is shown to be always determined.
4. Further, game theory has been used to explain the market equilibrium when more than two firms are involved. The solution lies in either collusion or non-collusion. These are known as coopera­tive non-constant-sum game and non-cooperative non-constant-sum game respectively.
5. "Prisoner's Dilemma" in game theory points towards collective decision making and the need for cooperation and common rules of road.
6. A player in game theory may be regarded as a single person or an organization in the real world subject to decision making with a certain amount of resources. The strategy in game theory is a com­plete specification of what a player will do under each circumstance in the playing of the game. For example, the Director of a firm might tell his sales staff how he wants an advertising campaign to start and what should they do subsequently in response to various actions of competing firms.
7. The importance of the pay-off values lies in predicting the outcome of a series of alternative choices on the part of the player. Thus a perfect knowledge of the pay-off matrix to a player implies perfect predictions of all factors affecting the outcome of alternative strategies. Moreover, the minimax principle shows to the player the next course of action which would minimize the losses if the worst possible situation arose.
8. Again, game theory is helpful in solving the problems of business, labor and management. As a matter of fact, a businessman always tries to guess the strategy of his opponents so as to implement his plans more effectively. Similar is the case of management in trying to solve the problem of labor union's bargaining for higher wages. Management might adopt the most profitable counter-strategy to tackle such problems. Further, producers might make decisions in which estimation of profits were to be balanced against the cost of production.
9. Last but not the least, there are certain economic problems which involve risk and technical relations. They can be handled with the help of the mathematical theory of games. Problems of linear programming and activity analysis can provide the main basis for economic application of the theory of games."

Game Theory has applications to so many other fields there is no way to cover them all. Some of the popular topics are: Military Conflicts, Terrorism, Missile Defense, Sales Price Wars, Energy Regulation, Bidding, Advertising, Elections and Voting, Agricultural Crop Selection, Conflict Resolution, Sports, and Telecommunications.