# What is Game Theory?

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Definition:

Game theory is the study of choices that people or actors make in different scenarios — Including what factors influence those choices, and what outcomes result from them.

## 🤔 Understanding game theory

Everyone makes choices. There are some choices we hardly think about, like what to eat for breakfast or which shoe to tie first. Other choices can have serious consequences; think of a doctor choosing which patient to save first, or a military general deciding whether or not to deploy troops into a battle. Game theory studies the choices people make in different scenarios, including what factors influence those choices and what outcomes they lead to. Some game theorists study the decision that produces the best outcome for each party in a given situation.

Example

One classic example of game theory is the prisoner’s dilemma. Let’s say that a police officer interrogates two prisoners, involved in the same alleged crime, at the same time. What are the possible outcomes?

• If neither prisoner blames the other, both will spend two years behind bars.
• If the first prisoner blames the second, and the second remains silent, the first will go free, while the second will spend ten years behind bars.
• If each prisoner blames the other, both will spend five years behind bars.

The goal of a game theorist is to figure out how each prisoner should act to optimize their outcome — In other words, the decision and outcome that will benefit each the most.

## Takeaway

Game theory is like trying to solve word problems in math class...

When you read a word problem, you need to break it down (often into an equation) so that you can calculate the answer. Similarly, game theory breaks down a scenario where someone faces a set of choices and solutions — And often tries to identify which choice will result in maximum benefits for the decision maker.

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## What is game theory?

At its core, game theory is the study of decisions and their possible outcomes. Game theory is typically used in situations where one or more actors (think people, governments, companies, etc.) must make a choice based on the information available to them, while aiming to produce the best result. You might use game theory in almost any game or real-world scenario that involves making a choice — Like a child trying to come up with the winning move in tic-tac-toe, or a government deciding whether or not to launch a nuclear missile.

The idea of game theory has been around for as long as people have been making decisions. There’s evidence of game theory analysis in the work of ancient philosophers like Plato. But it wasn’t until later that game theory was a named concept and field of study. The modern study of game theory traces back to the work of mathematician John von Neumann and economist Oskar Morgenstern, who published one of the first papers about the subject in the 1940s.

Game theory provides a language that people can use to describe scenarios that involve choices. Take a game like tic-tac-toe, which is structured so that all players in the game know their opponents’ possible moves — Game theory describes this kind of situation as having “perfect information.” By contrast, a poker game has imperfect information, because players do not know what cards their opponents have or how they’ll play their next hand. Zero-sum games, like tic-tac-toe or poker, create a winner and a loser — There is no middle ground.

Because you can apply it to almost any scenario involving choice, game theory has many different real-world applications. Economists can use game theory to predict the outcome of different policies. Military leaders use it to build battle strategies. Game theory also has its limitations. For one, it assumes that all actors in a scenario must be rational, meaning they act consistently and in their own interest, which may not always be the case.

Game theory also assumes that there’s some level of competition or outside influence in a given scenario. In other words, it assumes that the scenario has at least two parties involved. Game theory may not be as useful for a single farmer who is choosing the best field to harvest, because no one else is going to react to his decision. Game theory assumes that another actor’s actions could impact you or the choices you make — Like another farmer trying to beat his crops to market.

## What is the purpose of game theory?

Game theory is useful because it helps people make decisions in a wide range of scenarios, be it in a board game, a science lab, or a war room. Typically, game theory helps identify the best decisions or course of action that would produce the ideal outcome in a given scenario. For example, in a game of tic-tac-toe, you have nine possible moves (if you go first). Game theory can help determine the best first move if your goal is to avoid losing the game, or to determine which is the optimal move if you’re trying to win.

Game theory can also help you make choices in a more complex game like chess, where there are many more possible moves. In fact, in many of its real-world applications, game theory allows you to analyze hundreds, thousands, or even more potential decisions to determine which will lead to the outcome that fits the decision maker’s goal. Likewise, game theory can help predict how other people will make decisions in particular situations, and the possible outcomes.

## What are the types of game theory?

Game theory involves the study of many different types of scenarios. Here are some examples:

### Symmetric and Asymmetric

In symmetric games, all the parties involved use the same strategies. Symmetry tends to exist only in short-term games, like tic-tac-toe, because the number of options available to each player grows as the game increases in length. For example, a game of Go — Where each player tries to surround or “capture” the pieces placed by their opponent — can last hundreds of turns.

In asymmetric games, on the other hand, players take different approaches to accomplishing their goal. Usually, this occurs because different strategies don’t provide equal benefits to all players. For example, in a game of poker, replacing the cards in your hand offers less value to a player with a straight than a player a pair of twos.

### Cooperative and Noncooperative

In a cooperative game, players can work together and select their strategies based on negotiations with the other players. Cooperative games require that the players have the option to communicate. Typically, cooperative games can produce outcomes where multiple parties benefit — There isn’t necessarily a single winner or a loser.

A noncooperative game involves players making decisions without discussing or making agreements with other players. Each player involved makes their own decisions based on what offers the most personal benefit. These games are more likely to produce a single winner or loser.

### Normal and Extensive Form

In a normal-form game, a player can often display their decisions and outcomes as a matrix. Typically, a normal-form game has fewer possible decisions and potential results compared to an extensive-form game.

Think of an extensive-form game or scenario as a decision tree, showing each situation, the options available, and the result that each decision leads to. An extensive form game or scenario typically takes longer to play out, involves more choices, and can accommodate more events caused by chance, such as natural disasters.

### Sequential Move and Simultaneous Move

In scenarios involving sequential moves, each player takes turns making decisions and bases future decisions on the choices made by other players. Chess is an example of a game that has sequential moves. While players might not know how many strategies the other player may be considering, they can make an educated guess about their opponent’s plans based on the moves they’ve already made. A sequential move game often uses a decision tree to show its possible outcomes, making it extensive in form.

A simultaneous-move game or scenario involves players who make decisions at the same time. The players don’t have a way to know what the other players are doing before they make their choices. Usually, you can describe a simultaneous game and its outcomes using a matrix, making it an example of a normal-form game.

### Constant Sum, Zero Sum, and Non-Zero Sum

The sum of a game describes the total value of the outcomes that the game produces. The value of a given outcome is usually arbitrary. For example, someone evaluating a board game might say the winner receives a value of 1 while the loser receives a value of 0. Or consider a scenario where an employer is willing to pay \$100,000 to hire someone and the potential employee negotiates a salary of \$90,000. In this scenario, one might assign the employee a value of 90,000 and the employer a value of 100,000.

A constant-sum game assigns each player the same value, even if the game distributes the outcomes differently based on the players’ choices. When flipping a coin, for example, there’s always one winner and one loser.

In a zero-sum game, the total value produced by the game equals zero. It’s like the saying, one person’s gain is another’s loss — Meaning, if one player benefits some amount in the game, the other players lose the same amount. In a non-zero sum game, however, players can benefit without negatively impacting the other parties involved.

Typically, a two-person, competitive game like checkers or tic-tac-toe, are considered zero-sum. Games where cooperation is an option are typically considered non-zero sum. In the real world, two countries warring over control of territory could be considered a zero-sum game. One country gets to control the land and the other does not. Countries working together to reduce global pollution could be described as a non-zero sum game.

## What is the Nash Equilibrium?

The Nash Equilibrium, named for mathematician John Nash, describes a scenario where no player in a game or scenario has an incentive to change their strategy based on other players’ choices. Let’s say you’re playing a game where each player can choose to receive one dollar or pay one dollar. Each player may choose to receive a dollar, regardless of the other player’s choices.

## What are some examples of games?

The prisoner’s dilemma is one of the most common example scenarios that shows how game theory works. In the prisoner’s dilemma scenario, the police capture two people suspected of committing the same crime and place them in separate cells. Neither suspect can speak to their accomplice. The police offer two options to each prisoner: Confess to the crime, or remain silent.

In this scenario, there are a few possible choices and outcomes:

• If both prisoners remain silent, they will both be sentenced to one year behind bars.
• If one confesses and the other remains silent, the confessor will go free while the silent one will get a three-year sentence.
• If both admit to the crime, each will receive a two-year sentence.

In game theory, the prisoner’s dilemma is considered a normal-form game, where the outcomes are few enough to display in a matrix, like so:

*The first number within the parentheses is Prisoner A’s sentence. The second is Prisoner B’s sentence.

Based on the matrix, you might look at both prisoners remaining silent as the optimal decision, because it leads to the lowest overall sentence. But neither prisoner can communicate with the other, so it’s impossible for them to negotiate and agree to remain silent. Assuming that each prisoner’s goal is to reduce their own sentence to zero, it’s likely that each will choose to confess, even though that will lead to the longest overall sentence.

Here’s an example of an extensive-form game: Let’s say you run a shipping company that is deciding whether or not to expand into a new market by selling supplies like boxes and containers, on top of its usual shipping and handling services. Other companies that sell shipping supplies must decide whether to compete with the newcomer by stepping up advertising or trying to draw more customers, or to accommodate them by not working to push them out of the market.

You can illustrate this game using a decision tree.

Based on all possible decisions, the new company can expect higher value from entering the new market than not. That means the optimal choice is to enter the market.

## What are the applications of game theory?

Game theory has several real-world applications, like divvying up resources or assessing market competition. Let’s say you’re responsible for allocating food at a soup kitchen. There’s a limited amount of food available in the kitchen and many people who want to eat. According to game theory, the patrons will compete to increase their share of food. You might use game theory to predict how to distribute the food so that each patron gets what they need.

Game theory also has applications in political science and economic theory. Let’s say a small number of businesses control a market (aka an oligopoly). When those companies go about setting prices for their product, the situation can look like the prisoner’s dilemma. If the companies cooperate, they might increase their prices and profits. But they might instead set prices in isolation, without knowing what the others are doing — Which could result in a price war as they compete for market share, lowering the overall revenue.

A computer engineer might use game theory when programming artificial intelligence. Since game theory is all about studying scenarios and decision theory, an engineer could use its principles to instruct a computer to extrapolate on a set of scenarios, and let it adapt to new or unforeseen scenarios that may emerge in the future.

## What are the limitations of game theory?

While it’s widely useful, game theory isn’t perfect. Like any theory involving humans, game theory can’t perfectly explain or predict how things will play out in the real world. One major limitation of game theory is that it assumes rationality by all players in a game or scenario, meaning everyone acts in their own interest. In reality, no one is perfectly rational — It’s possible that someone would choose to take an action that benefits someone else over themselves.

It’s also nearly impossible for game theory to account for every factor that may influence a decision or every choice available to a decision maker. While it’s easy to see how game theory would play out in a simple game like the prisoner’s dilemma, where there are few variables and choices to make, the real world is often much more complex. Think of a company that’s deciding whether or not it should expand into a new market. There may be thousands of factors that can lead to different outcomes. While game theory can help think through a particular set of choices and outcomes, it may not account for every factor.

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This information is educational, and is not an offer to sell or a solicitation of an offer to buy any security. This information is not a recommendation to buy, hold, or sell an investment or financial product, or take any action. This information is neither individualized nor a research report, and must not serve as the basis for any investment decision. All investments involve risk, including the possible loss of capital. Past performance does not guarantee future results or returns. Before making decisions with legal, tax, or accounting effects, you should consult appropriate professionals. Information is from sources deemed reliable on the date of publication, but Robinhood does not guarantee its accuracy.

Options trading entails significant risk and is not appropriate for all customers. Customers must read and understand the Characteristics and Risks of Standardized Options before engaging in any options trading strategies. Options transactions are often complex and may involve the potential of losing the entire investment in a relatively short period of time. Certain complex options strategies carry additional risk, including the potential for losses that may exceed the original investment amount.

Commission-free trading of stocks, ETFs and options refers to \$0 commissions for Robinhood Financial self-directed individual cash or margin brokerage accounts that trade U.S. listed securities and certain OTC securities electronically. Keep in mind, other fees such as trading (non-commission) fees, Gold subscription fees, wire transfer fees, and paper statement fees may apply to your brokerage account. Check out Robinhood Financial’s Fee Schedule for details.

Brokerage services are offered through Robinhood Financial LLC, (RHF) a registered broker dealer (member SIPC) and clearing services through Robinhood Securities, LLC, (RHS) a registered broker dealer (member SIPC). Cryptocurrency services are offered through Robinhood Crypto, LLC (RHC) (NMLS ID: 1702840). Robinhood Crypto is licensed to engage in virtual currency business activity by the New York State Department of Financial Services. The Robinhood spending account is offered through Robinhood Money, LLC (RHY) (NMLS ID: 1990968), a licensed money transmitter. A list of our licenses has more information. The Robinhood Cash Card is a prepaid card issued by Sutton Bank, Member FDIC, pursuant to a license from Mastercard®. Mastercard and the circles design are registered trademarks of Mastercard International Incorporated. RHF, RHY, RHC and RHS are affiliated entities and wholly owned subsidiaries of Robinhood Markets, Inc. RHF, RHY, RHC and RHS are not banks. Products offered by RHF are not FDIC insured and involve risk, including possible loss of principal. RHC is not a member of FINRA and accounts are not FDIC insured or protected by SIPC. RHY is not a member of FINRA, and products are not subject to SIPC protection, but funds held in the Robinhood spending account and Robinhood Cash Card account may be eligible for FDIC pass-through insurance (review the Robinhood Cash Card Agreement and the Robinhood Spending Account Agreement).

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