When we hear the term “game,” we usually start thinking about amusements or sports. But in a branch of mathematics called “Game Theory,” the word ‘game’ has a much broader connotation. Game Theory is, in fact, the study of mathematical models and their interaction with the decision-makers. The game theory includes strategic thinking in which players make decisions by viewing various perspectives and by looking at the viewpoint of other participant players; also by analyzing their actions and reactions in particular situations.

## Significance

The best use of game theory is to figure out the optimal solution from the best possible choices through the analysis of costs and benefits to each participant who competes with each other. The theory is applicable in different fields, like business, psychology, biology, economics, political science, computers, etc. From the business point of view, game theory can be utilized by business managers to predict the strategic planning or thought process of their competitors and collaborators. It is considered a very powerful medium for forecasting or predicting the output of interactions between different participants or competitors in which the reaction of one depends on the action of others.

## Understanding the Game Theory

The game theory focuses on formulating a model of decision-making by identifying the players’ preferences, and possible strategies. The game theory proposes that the outcome of a game is influenced by the actions and decisions of all the players involved in the game, and each player thinks rationally to get the maximum payoff. A person can utilize game theory to find an effective solution in any situation involving two or more players, wherein he/she is aware of the predictable consequences and payoffs. The game theory is based on the concept that the payoff of one player is dependent upon the strategies or the decisions of the other player/players. There are two main ways by which the game theory can be used – simultaneous games in which the players go ahead with their moves or actions simultaneously, irrespective of looking for information related to the moves or actions chosen by other players, and sequential games, which includes the dependency of the player’s moves or actions on the previous action’s results or another player’s choice.

## Five Types of Games in Game Theory

### 1. Cooperative and Non-Cooperative Games

In cooperative games, the players have to follow a specific strategy by negotiating with the other players. The problems are solved by following the agreement based on mutual discussion among the players. The cooperative game theory focuses on the interaction of cooperative groups and coalitions when the players are only aware of the payoffs. The cooperative game theory also questions how do the groups formulate and how the payoffs are distributed among the players. Let us understand this type through an example, suppose company X, which manufactures alcohol, has to spend a huge amount on the marketing of its product, hence the management decides to reduce the high ad expenditure. But, the problem is that there is not any surety that the other companies producing the same product will also do the same and spend less on marketing. Hence, a dilemma is created among the cigarette manufacturing companies that whether they should lower their budget on marketing or not because if one company does not follow the agreement and spends more on marketing then that company may get the profit, while others following the agreement are likely to face the losses. However, if the authorities or the government passes a rule that restricts cigarette advertisements, it would be helpful for Company A in reducing its advertisement budget. Here, even though the government may take the decision for the health of the people, it worked as cooperation for company A, hence it is an example of a cooperative game. Contrary to cooperative games, in non-cooperative games, the player focuses only on their profits and makes the strategies for their own maximum profit. The non-cooperative theory deals with how do the players think rationally and deal with the other players to achieve their targets. Rock-paper-scissors are a prominent example of a non-cooperative game. It is a strategic game, wherein the outcome is based on the combination of various sets of choices.

### 2. Normal Form and Extensive Form Game

When the description of the game, i.e., the strategies of the players and the payoffs, are represented in the tabular or matrix form, it is called a normal form game. These types of games help to find the Nash equilibrium and the strategies that can dominate the other players. The matrix in the normal game form represents the type of strategies used by the players and the possible outcomes associated with those strategies. While in the case of the extensive game forms, the description of the game is given in the form of a decision tree. In the case of the extensive game, the naturally occurring event in the game can be easily represented. The representation forms a tree-like structure, and the players’ names are defined on the nodes. Also, the possible actions and the payoffs of the players are also represented in the structure. To understand the extensive form of games, let us consider an example, suppose company A is an older organization with around a decade of experience, and company B is in the initial phase and wants to enter the market. Company B has two strategies, the first is that enter the market and face a strong challenge from the existing company B, and the second option is that avoid the chance to earn more profit by not entering the new market. Company A also has two strategies, i.e., either to face the competition from company A and risk its existence or help organization B in entering the new market and both companies cooperate with each other. The figure below represents that company B takes the first step and this step is later followed by company A. The payoff of Company B will be zero if it does not enter the market, but if the company decides to enter the market, the payoff would depend upon the situation of the market which is controlled by Company A. If both companies decide to compete with each other, both of them are likely to face a loss. But, if company A cooperates with company B, then both companies can make an equal profit. Hence, the best suitable strategy, in this case, would be that Company B enters the market and Company A helps Company B.

### 3. Symmetric and Asymmetric Games

The main feature of symmetric games is that all the players in these games adopt the same strategies. This is usually applicable in short-duration games because, in long-duration games, the players get more options. In symmetric games, the decisions do not depend upon the player, in fact, it is based on the type of strategies used. The decisions in symmetric games remain the same even if the players are interchanged in the game. The prisoner’s dilemma is a prominent example of the symmetric game. This example is discussed further in this article. In the case of asymmetric games, the decisions depend upon the players. In these games, if a particular strategy provides benefits to one player, it does not assure that the other player will also get equal benefits. A prominent example of asymmetric games is the decision of the company to enter a new market.

### 4. Simultaneous Games and Sequential Games

Those games in which two or more players simultaneously play the move (adopt strategies) are known as simultaneous games. In simultaneous games, the players are not aware of the moves of the other players, while in the case of sequential games, the players know about the moves of the other players if they already have applied any particular strategy. However, the players do not have a deep understanding of the type of strategies that may be used by the other players, for example, the player knows that the other player may not use only a single strategy, but he/she does not know that how many strategies the other player may adopt. We can represent the sequential games in the extensive form, and the simultaneous games in the normal form. The application of the simultaneous move game can be understood through the following example. Suppose both company A and company B want to outsource their marketing budget. However, the companies have a fear that if they outsource their marketing budget, the other companies that are spending huge on marketing will gain more profit. Hence companies A and B have two strategies, i.e., either to outsource their marketing budget or not. In the following table, it can be observed that companies A and B are not aware of each other’s strategies. Both companies operate on the assumption that the other company will choose the best strategy for itself, hence both companies will choose the strategy that benefits themselves.

This example can also be used to represent the sequential move games. Suppose company A makes the first move and decides either to outsource its marketing strategies or not (as shown in the image below). It can be seen in the image below that the first strategy was decided by Company A, and the decision of Company B is based on the decision of Company A. However, the outcome of the game is dependent upon the strategy of company B. In this example, the second player (company Y) has knowledge of the strategy of the first player (company X).

### 5. Constant Sum, Zero-sum, and Non-Zero-Sum Games

In the constant sum game, even if the players have received different outcomes, the sum of all the outcomes will always remain constant. In the zero-sum game, the net of all the player’s outcomes is zero. In a zero-sum game, the available resources can not be affected by the strategies of the different players. Also, the important feature of this game is that the gains and losses are proportional, i.e., the loss of one player is equal to the gains of the other player. On the contrary to this, in non-zero games, the net of all the outcomes of the players is not zero. The non-zero-sum games can be converted to zero-sum games if we add an extra player (dummy player). The dummy player’s losses are proportional to the net earnings of the other players. The main examples of zero-sum games are gambling and chess, in these games clearly if one player losses the other player wins, i.e., the loss of one player is the gain of the other player. Also, ‘matching coin’ is the most used type of zero-sum game. It consists of two players. In this game, each player simultaneously puts the coins on the table, and the payoff is dependent upon whether the coins match or not. If both the coins show head or tails, then Player A will win the game and gets the coin of Player B. If the coins do not match, player B will win the game and he/she will keep the coin of Player A.

## Examples of Game Theory Strategies

Game theory is used to analyse the number of games. Let us understand the analyses of some games through the following examples,

### 1. Prisoner’s Dilemma

The prisoner’s dilemma is one of the most popular examples of game theory. Let us suppose two people, Prisoner A, and Prisoner B, are arrested for committing a crime; however, the prosecutor still needs a shred of proper evidence to decide their conviction. To get the confession from the prisoners, they question the two in different rooms. Both prisoners are separated, and there are not any means of communication between them. The officials proposed some deals to both prisoners. According to the deal, if both the prisoners confess the crime, each of the prisoners will receive a prison sentence of five years. If Prisoner A confesses the crime, but Prisoner B denies it, then Prisoner A will receive a three years prison sentence, and Prisoner B will be freed. If Prisoner B confesses, but Prisoner A denies it, then Prisoner B will get the three years prison sentence, and Prisoner A will be freed from the prison sentence. If both prisoners do not confess, they both will receive two years of prison confession. Now, from all of the given deals, the most favourable one is that neither of the prisoners confesses as they are receiving the lesser punishment (two years of imprisonment) in this case only. However, the problem is that the prisoners do not know the strategies of each other; they are not certain whether the other prisoner will confess or not. According to the Nash equilibrium, the prisoners are more likely to choose the deal that is best for their own but is worst for both of them collectively, i.e., the expression ‘tit for tat’ is considered the best solution as per the game theory. The concept of tit-for-tat was presented by an American mathematical psychologist, Anatol Rapoport, who stated that the players are likely to not cooperate if provoked, while they are likely to cooperate if unprovoked.

### 2. Deadlock

The deadlock is a social dilemma similar to the example of the prisoner’s dilemma in which the players either cooperate with each other or do not cooperate (defect). In a deadlock, if organization X and organization Y both cooperate, then both organizations will receive the payoff of 1, and if they do not cooperate, they will receive the payoff of 2. If Organization X cooperates and Organization Y defects, the organization will get a payoff of zero, while Organization Y will get a payoff of 3. Let us understand the deadlock through an example. Suppose the two powerful nations are trying to settle for an agreement that manufacturing nuclear weapons should be prohibited. In this case, cooperation means that both nations stick to the agreement and do not manufacture nuclear weapons, while defections imply that either one or both nations secretly break the agreement and manufacture the nuclear weapons. Unfortunately, in this case, the dominant outcome for either nation would be secretly breaking the agreement and manufacturing nuclear weapons because it will give that nation a benefit over the other nation in case the war occurs. The other effective option is if both nations do not cooperate and keep on manufacturing nuclear weapons. The following image represents the example of a deadlock between Organization X and Organization Y.

### 3. Cournot Competition

This game is named after its inventor Augustin Cournot, a French mathematician who presented this model in 1838. Its concept is similar to the prisoner’s dilemma. The Cournot model is utilized in explaining the duopoly, i.e., the competition between the two main organizations in a market. For example, suppose the company X and Y both manufacture similar products, and they can manufacture the product in large and small quantities. If both companies corporate with each other and decide to manufacture the product in lower quantity, then the limited supply of that product in the market will result in a rise in the price of that product, hence both companies will get the benefit. However, if both companies revoke and decide to defeat each other and produce a product of higher quality, it will result in a low price for that product and both companies will face the drawbacks. Also, if one company cooperates and produces the product in a lower quantity while the other company does not, i.e., produce the product in a high quantity, then the former company will suffer the loss, while the latter will earn the maximum profit. The payoff matrix of Company X and Company Y is shown in image-A below (profits are represented in millions of dollars).

### 4. Dictator Game

In this game, Player X has to decide the division of the payoff with Player Y, who has zero contribution to the decision of Player X. According to the results revealed by different experiments, it was found that in this scenario, 50 percent of people keep all the payoff to themselves, 45 percent of people give a small share of the payoff to the other, while only 5 percent of people split the payoff equally. The dictator game is almost similar to the ultimatum game; in the ultimatum game, a certain amount of cash is given to Player X, part of which he/she has to give to Player Y, who has to either accept or reject the given amount. If Player Y rejects the amount offered, then both Players will get nothing.

### 5. Coordination Game

In coordination games, the players get the maximum benefits if they choose the same strategy. For example, suppose the two giant tech companies are in the dilemma of whether to introduce new technology in the computers that would make them earn huge or advance the old technology that would make them earn only a few. The problem is if only one company decides to introduce the new technology in computers, the customers would find it difficult to adapt to the new technology and prefer to buy the systems with the technology that the customers are familiar to use, hence as a result the company will face losses. On the other hand, if both companies decide to introduce the new technology, the customers will be left with no other option than to buy the computers with the new technology. Hence, in this case, both companies will earn more and get profit.

### 6. The Centipede Game

In the centipede game, the two players get the alternative turns to get the larger share of the money stash, which is increasing slowly. It is a sequential game as players do not make the move simultaneously, in fact, they make the moves one after the other, hence they are aware of the strategy of the player who made the move before them. The game ends the moment when the players get the stash, where one player gets the larger portion and the other gets the smaller portion. For example, suppose Player X starts the game and has to either ‘take’ or ‘pass’ the stash, which amounts to two dollars. If Player X takes the stash, then both Player X and Player Y will get one dollar each, but if Player X passes the stash, then Player Y has to decide whether to pass or take the stash. If Player Y takes this stash, he will get three dollars (previous 2 dollars + 1 dollar), while Player X will get zero dollars. However, if Player Y passes, Player A gets the chance to decide. In case both players always choose to pass the stash, they will receive the payoff of 100 dollars at the last. The catch here is that the players will get the maximum payoff if both players keep on passing until the last of the game, but, as predicted by the Nash equilibrium, if any of the players disrupt the game and choose to ‘take the stash’ the players are likely to get the lowest payoff of one dollar. However, some experimental studies have revealed that this type of rational behaviour, which is anticipated by the game theory, is rarely observed in real life.

### 7. Volunteer’s Dilemma

In the case of the volunteer’s dilemma, someone has to take responsibility for the benefit of a large number of people. According to the game theory, the worst outcome would be, if nobody dares to volunteer. For example, suppose some junior members of the recruitment agency assigned by the government to hire people for various state-level vacancies are frauds and are responsible for the malfunctioning of the recruitment drives. The higher authorities are unaware of this malfunction. Some other members who know about this fraud are hesitant to tell the higher authorities about this fraud as they are scared to lose their job or be called whistleblowers by the members involved in the fraud. But, if nobody dares to volunteer, the fraud will result in a large disruption in the state.

## Examples of Game Theory

Let’s have some real-life examples of Game Theory.

### 1. Bidding at Auction

In an auction, different bidders bid for purchasing any good or service, and the same is sold to the bidder having the highest bid. Game theory is applicable in bid auctions, especially for analysis of the first price-sealed auction bidding. In this type of auction, bidders are supposed to submit bids in a safe and sealed way. Different players are unaware of the value of goods or services to each other and try to devise a bidding strategy. The behavior of bidders and different other factors are analyzed while making decisions during bid preparation.

### 2. Collective Bargaining or Negotiation Between Parties

Game theory plays an important role in different collective bargaining or negotiation activities among different parties or participants. For example, different negotiations take place between worker unions and the management during the situation of strikes of workers or lockout periods to increase wages. Using game theory, both parties can arrive at the optimum solution to the issue, i.e., to increase wages by examining different options available for wages and benefits, which can maximize the welfare of both workers and management. Salary negotiation is also an example of the game theory application. The concept of game theory is used in other negotiations also like negotiations with suppliers while purchasing, compensation or incentive negotiations between management and suppliers or business partners, etc.

### 3. Decisions Related to New Products

Product-based decisions related to launching a new product in the market or to exit the launch of the product are also executed using game theory by businesses. Using game theory, business people can understand if the first-mover advantage is there or not, the competitor’s possible moves related to new products, and also decide upon the strategies for defense, etc. Similarly, the concept of game theory is also used in deciding whether to enter a new market or exit it.

### 4. Product Pricing Decisions

Game theory is widely used in deciding upon the pricing strategies of both consumers and retailers. Retailers compete against each other to gain market share of customers for which they opt for different games or strategies, like offering attractive discounts on specific goods to increase sales of complementary products. For example, in the offseason, i.e., off-summer or off-winter season, garment shop vendors or retailers offer attractive sales on a certain stock of clothes in which they adopt optimal pricing strategies to attract maximum customers. In this, retailers use the game theory approach where retailers and consumers are the main players. The focus of retailers is on using the best pricing strategy, while the preference of consumers is to choose the best deal in terms of discount and variety.

### 5. Stock Market Decisions

Using game theory, decisions regarding buying and selling shares in the stock market can be taken wisely. Investors make different stock market decisions by using different strategies of investment and by considering different players or investors. Game theory helps in predicting the decisions of other players related to investments, and based on these decisions, they can decide upon the strategies for themselves which maximize profit.

### 5. Salary Negotiation

Asking for raise in salary is a complex task, it gets even more complex if more than one person is involved in the negotiation. The game theory helps in better negotiation with the manager. One should be aware of the kind of services he/she provides and the worth of those services in the company. One can not simply go and ask for a raise just because he/she wants a higher salary. According to the game theory, you must know the person well with whom you are making the salary negotiations and be prepared for the potential responses and the counter-arguments. If you know how to effectively respond to the person with whom you are negotiating about the salary, then chances are high that you will get the raise, considering you have the skill and values that add worth to the organization.

### 6. While Buying a Car

Game theory can help you buy a car at a much cheaper price than the amount it will cost you otherwise. The first step of car buying is looking for the dealers near your area that sell the car you want. Now, rather than simply visiting the dealers and bargaining with the salesperson, the better step is to call each dealer and mention to them that you are looking for a particular car, and you will buy it from the dealer that will give the best offer. The dealer may deny negotiating about the price, but you should respond confidently that you are only available for the negotiations over the phone and will consider buying if the offer is good. The dealer will then try to give the best offer to seal the deal. Hence, by using the strategies of the game theory you can buy the car at much lower prices.

### 7. Real Estate

You may have never noticed before, but the game theory finds its applications in real estate. Most of the real estate negotiations are already understood by the agents which makes it easier for sealing a deal. However, the situation is entirely different in multi-offer deals. You will only have three options if the bid you have submitted is announced as a multi-offer situation by the real estate agent.

- Keeping the original offer

- Withdrawing the current offer

- Raising the offer price

To win the bid, the person has to overbid the other bidder. If the game theory is followed, the person should bid the number he/she has calculated to win the bid according to his/her budget, rather than increasing the price in bidding over the budget. In this case, even if you lose the bid, you still made an effective decision because bidding at a price that is not in your budget is an incorrect decision that may cause a burden on you.

### 8. Rise and Fall of Bitcoin

Sometimes the decisions of the people in doing a certain task are not only based on their own perspective but it is influenced by the decisions of the majority of the population. Stock trading and crypto trading are prominent examples of decisions dependent upon other people and utilization of game theory. Suppose you want to buy a Bitcoin. What would be your first step? Of course, you will look at the chart of the recent fluctuation to decide whether to buy Bitcoin or not. You see that a large number of people are buying Bitcoin, and they are gaining profit and you also buy Bitcoin. Now, suppose a well-known financial expert posts a tweet that questions the stability of Bitcoin. Like the majority of people, you also think of selling the bitcoin as an expert has criticized it, and the value of the bitcoin gets lower. Hence, in this case, the game theory results in the rise and fall of Bitcoin.

### 9. Rock, Paper, and Scissor Game

Have you ever got into disputes with your friend and you couldn’t decide who is right or wrong? Then the game rock, paper, and scissors remain the only option, and the one who wins; wins the dispute. In this game, we know the consequences but are not aware of what another player is going to do.

### 10. Poker Card Game

Most of us have seen people losing huge amounts of money in poker clubs in movies as well as in real. The poker card game exemplifies the game theory correctly because one wins exactly the amount one’s opponents lose.