• Two-person constant-sum games
• Two-person variable-sum games
  • The prisoner’s dilemma
• N-person games

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The Connect 4 game is a solved strategy game: the first player (Red) has a winning strategy allowing him to always win. The game has been independently solved by James Dow Allen and Victor Allis in 1988. Connect 4 Solver. This Connect 4 solver computes the exact outcome of any position assuming both players play perfectly. Game theory, branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent. This interdependence causes each player to consider the other player’s possible decisions, or strategies, in formulating strategy.

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Game theory, branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent. This interdependence causes each player to consider the other player’s possible decisions, or strategies, in formulating strategy. A solution to a game describes the optimal decisions of the players, who may have similar, opposed, or mixed interests, and the outcomes that may result from these decisions.

Although game theory can be and has been used to analyze parlour games, its applications are much broader. In fact, game theory was originally developed by the Hungarian-born American mathematician John von Neumann and his Princeton University colleague Oskar Morgenstern, a German-born American economist, to solve problems in economics. In their book The Theory of Games and Economic Behavior (1944), von Neumann and Morgenstern asserted that the mathematics developed for the physical sciences, which describes the workings of a disinterested nature, was a poor model for economics. They observed that economics is much like a game, wherein players anticipate each other’s moves, and therefore requires a new kind of mathematics, which they called game theory. (The name may be somewhat of a misnomer—game theory generally does not share the fun or frivolity associated with games.)

Game theory has been applied to a wide variety of situations in which the choices of players interact to affect the outcome. In stressing the strategic aspects of decision making, or aspects controlled by the players rather than by pure chance, the theory both supplements and goes beyond the classical theory of probability. It has been used, for example, to determine what political coalitions or business conglomerates are likely to form, the optimal price at which to sell products or services in the face of competition, the power of a voter or a bloc of voters, whom to select for a jury, the best site for a manufacturing plant, and the behaviour of certain animals and plants in their struggle for survival. It has even been used to challenge the legality of certain voting systems.

It would be surprising if any one theory could address such an enormous range of “games,” and in fact there is no single game theory. A number of theories have been proposed, each applicable to different situations and each with its own concepts of what constitutes a solution. This article describes some simple games, discusses different theories, and outlines principles underlying game theory. Additional concepts and methods that can be used to analyze and solve decision problems are treated in the article optimization.

Classification of games

Games can be classified according to certain significant features, the most obvious of which is the number of players. Thus, a game can be designated as being a one-person, two-person, or n-person (with n greater than two) game, with games in each category having their own distinctive features. In addition, a player need not be an individual; it may be a nation, a corporation, or a team comprising many people with shared interests.

In games of perfect information, such as chess, each player knows everything about the game at all times. Poker, on the other hand, is an example of a game of imperfect information because players do not know all of their opponents’ cards.

The extent to which the goals of the players coincide or conflict is another basis for classifying games. Constant-sum games are games of total conflict, which are also called games of pure competition. Poker, for example, is a constant-sum game because the combined wealth of the players remains constant, though its distribution shifts in the course of play.

Players in constant-sum games have completely opposed interests, whereas in variable-sum games they may all be winners or losers. In a labour-management dispute, for example, the two parties certainly have some conflicting interests, but both will benefit if a strike is averted.

Variable-sum games can be further distinguished as being either cooperative or noncooperative. In cooperative games players can communicate and, most important, make binding agreements; in noncooperative games players may communicate, but they cannot make binding agreements, such as an enforceable contract. An automobile salesperson and a potential customer will be engaged in a cooperative game if they agree on a price and sign a contract. However, the dickering that they do to reach this point will be noncooperative. Similarly, when people bid independently at an auction they are playing a noncooperative game, even though the high bidder agrees to complete the purchase.

Finally, a game is said to be finite when each player has a finite number of options, the number of players is finite, and the game cannot go on indefinitely. Chess, checkers, poker, and most parlour games are finite. Infinite games are more subtle and will only be touched upon in this article.

A game can be described in one of three ways: in extensive, normal, or characteristic-function form. (Sometimes these forms are combined, as described in the section Theory of moves.) Most parlour games, which progress step by step, one move at a time, can be modeled as games in extensive form. Extensive-form games can be described by a “game tree,” in which each turn is a vertex of the tree, with each branch indicating the players’ successive choices.

The normal (strategic) form is primarily used to describe two-person games. In this form a game is represented by a payoff matrix, wherein each row describes the strategy of one player and each column describes the strategy of the other player. The matrix entry at the intersection of each row and column gives the outcome of each player choosing the corresponding strategy. The payoffs to each player associated with this outcome are the basis for determining whether the strategies are “in equilibrium,” or stable.

The characteristic-function form is generally used to analyze games with more than two players. It indicates the minimum value that each coalition of players—including single-player coalitions—can guarantee for itself when playing against a coalition made up of all the other players.

One-person games

One-person games are also known as games against nature. With no opponents, the player only needs to list available options and then choose the optimal outcome. When chance is involved the game might seem to be more complicated, but in principle the decision is still relatively simple. For example, a person deciding whether to carry an umbrella weighs the costs and benefits of carrying or not carrying it. While this person may make the wrong decision, there does not exist a conscious opponent. That is, nature is presumed to be completely indifferent to the player’s decision, and the person can base his decision on simple probabilities. One-person games hold little interest for game theorists.

The prisoner’s dilemma is a popular introductory example of a game analyzed in game theory that demonstrates why “rational” individuals are unlikely to cooperate, even when it could be in both of their best interests to do so, a win-win scenario. The prisoners dilemma game above simulates both what we’ll call the classical example, two prisoners being interrogated and the iterated version of that game, where players can respond and change their strategy over time.

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Classical Prisoner’s Dilemma Game Simulation

The traditional prisoners dilemma works as follows, you and your accomplice get caught committing a crime. The police interrogate you separately. Before you are carted off, you promise not to snitch on each other.

Over the course of the “interrogations” by police the following things can happen:

• If you both remain silent then you each get 1 years in jail.
• If you confess you get 0 years in jail, but your friend gets 20, and vice versa for you.
• If you both confess you each get 5 years in jail

Play how you want to, and see what happens:

Source: Prisoner’s Dilemma by Jana Vembunarayanan

In a multiple-moves Prisoner’s Dilemma game you can play against the same partner many times. This simulates mutualistic interactions in social groups where individuals will frequently encounter each other. An example of cooperative behavior which appears to be a multiple-moves game is mutual grooming for parasites, which many animals do, including chimpanzees.

What is The Prisoner’s Dilemma?

Traditionally the Prisoner’s Dilemma game has a dominant strategy of betrayal. Given that you have two choices, defect or cooperate, you’re really choosing to:

1) Defect, and get the best or third best result (0 years, or 5 years of prison)

2) Cooperate, and get the second best or worst result (1 year in prison, or 20 years in prison)

Obviously you’d be ignoring your partner, but if the result of them defecting and you cooperating is 20 years of prison for you, it’s highly unlikely they, or you won’t take the 0 or 5 year option. Even though you’d both be better off cooperating with each other and each being in jail for one year.

Since betrayal rewards far better than cooperation (and avoids a massively worse result) all purely self-interested people will betray each other. The reason why the Prisoner’s Dilemma is so titillating is because rational choice would lead each prisoner to betray, even though both cooperating would be best.

TL;DR:

Over a single turn, or over a multiple turns between players, the Prisoner’s Dilemma framework can actually be applied to many social situations in life and business. Imagine that you had a friend who consistently relied on you for money and was always borrowing things, but never spots you when you need some cash? This might be an example of a strategy where they are always willing to defect from cooperating with you, and rarely decide to cooperate with you in a meaningful way. And that same principle can apply to sharing kills on the savannah or grooming between primates in large groups. Indeed within animal cultures and across animal ecosystems there are many opportunities for mutualistic behavior and cooperation.

This same application can be true in groups and across companies. Have you ever worked on a group project where you felt like you were doing all the work. It could be said that your colleagues were “defecting” from cooperative behavior. In a multiple-moves Prisoner’s Dilemma game you can play against the same partner many times. This simulates mutualistic interactions in social groups where individuals will frequently encounter each other.

In the Prisoner’s Dilemma game simulation the prisoner’s are not allowed to communicate, and thus collude or reassure each other over time. Since they are making their decisions at the same time it would be considered a simultaneous game, and can be analyzed using the four square game payoff matrix referenced above.

Some Drawbacks of the Traditional Prisoner’s Dilemma

1) You have no preplanning with the other player, and no way to communicate.

2) The best and second best outcomes can occur if you do snitch.

3) The worst outcome will happen to the person who doesn’t snitch.

Depending on how you set-up your payoff matrix the game’s mechanics could actually get flipped on it’s head. If the best result and worst result is for both prisoners to cooperate, and the second and third best results are for cooperating, then cooperation becomes more appealing. And in reality thatis typically how the payoff matrix actually exists (in democracies or states with good legal systems) So if the DA has no case if no one snitches, then the best outcome is for each person is to always cooperate, and never snitch. But people usually do, since you have no guarantee at the end of the day as to what will happen. That said, as we’ll read in the next sections, there are many successful cartels, wherein companies cooperate on pricing instead of defecting, and they manage to maintain these systems for years.

Iterated Prisoner’s Dilemma Applications in the Market:

Monopolistic Competition Explanation:

Price Competition, Collusion, Cartels, & Adam Smith

Without collusion prices will be continually driven down, in what Adam Smith described as competing away profits. As you can imagine in a market, cooperating with the police is essentially equivalent to cutting your prices or defacing with your “competitors.” You’re likely to see significant top line revenue growth if you can cut prices and the competition can’t. But once they do so the competition does so as well and everyone ends up with less. As you could imagine this type of competition which is quite common eventually leads to a situation in industries where only a few firms are able to compete and capture any value. And as they become more concentrated they are able to collude more efficiently and capture more value from the market. Rather than competing the M&A spree and concentrating more…

Source: “To Much of a Good Thing” by The Economist.

Background to the Evolution of Cooperation

While this may not apply to markets (necessarily) the prisoners dilemma and game theory have a wide ranging impact on social constructs and every animal culture in the world.

Social Behavior

Any animal that lives in a groups evolves social behaviors like cooperative hunting and defense against predators. Since animals hunt together they have to opportunity to display cooperative behavior, or “altruism” wherein they at some cost to themselves can do something that benefits another member of the group. This “charitable” behavior would go against a baseline reading of Darwinian “survival of the fittest” but since cooperation among the group increases their total chances of survival it actually reduces the pressures of natural selection on those who are charitable.

Reciprocity

While the idea of reciprocal altruism in kin selection intuitively makes a lot of sense (after all you want your whole family to be successful and “have a good name” as much as you want your kids to be successful) But what about individuals that aren’t related? In this case you behave altruistically to these non-members of your family, or group, or even species (think about those fish that follow sharks around) with the expectation that they will help you. These type of cooperation/defection games are non-zero-sum games. Both parties can win by cooperating. This differs from other types of social and business interactions like predator-prey relationships, that are zero-sum game. So you setting beneficial prices with a direct competitor can be a non-zero-sum game. For example if people like playing 3D console games, that will grow the entire 3D console game market, it’s a non-zero-sum game, you can both grow the market with your marketing dollars because you’re generating a new demand. But fi you’re both killing older non 3D immersive games through the proliferation of your consoles, then you’re dealing with a zero-sum game. You putting those out of business is a zero-sum gain, because they lose as you start to win.

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