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Game Theory Help

game theory help

“Game Theory Help” is a blog that uses game theory to solve problems. In short, game theory is the study of strategic interaction in conflict situations. In this blog post, the author will explain the basics of the term game theory and then apply it to a problem.

Game Theory Help

Game theory has been regarded as a theoretical framework that is used for conceiving social situations among various competing players. Some experts may define game theory as the science of strategies. This theory can be applied to make an optimal decision through a systematic and strategic approach. 

If you are looking for game theory help, this article comes with the necessary information, including your access to the list of math resources. Reputed economist Oskar Morgenstern and mathematician John von Neumann are the pioneers of the game theory. In 1940s, the game theory came into being. Since the inception, the theory has become gradually popular. Today, the theory has many applications, especially in the field of economics.

The Fundamental of Game Theory Help

Before you know how to solve game theory matrix, it is important to have a clear idea on the fundamentals of game theory. The focus of the game theory is the game, which acts as a model in among various players in an interactive situation. A player’s payoff depends on the strategies implemented by other players. 

The game must identify preferences, identities, and available strategies of the players. Various other assumptions and requirements could be there depending on the gaming model. So, what are the major applications of game theory? It has been applied widely in evolution biology, economics, politics, psychology, businesses, and many more.

Quick Definitions of Game Theory Components

For game theory homework solutions, you need to know the definitions of various components of game theory. So, in the following section, the crucial components of the game theory have been listed.

  • Players: In a game, the persons who make strategic decisions have been termed as players.
  • Game: The definition of game is the circumstance that results into actions of two or more players.
  • Payoff: In many circumstances, a strategic decision-maker wins reward for making the right decision at the right moment. The reward has been termed as payoff.
  • Strategy: A strategy is the decision that a player makes. In a circumstance, two or more players make well-planned actions plans to make the circumstances favorable for them. These action plans have been known as strategies.
  • Equilibrium: Equilibrium is a point in the game, when both players have taken their respective decisions.
  • Information Set: The information available for a game. Game rules, terms and conditions, and other things can be regarded as information.

Associated Theories with game Theory

You game theory tutor will help you to understand a few more theories along with the game theory. What are those factors? For the game theory assignment help, it is essential to know about associated theories with game theory. In the following section, you can find more details in this regard.

1. Decision Theory

Along with game theory, it is essential to know the decision theory. The decision theory is the theory of a person involved in a game. There are many games which involve only one player. The decision theory gives focus on the formation of beliefs and preferences. The player will find many alternatives, though they are risky options. Hence, decision theory helps a player to manage the risk and choose the right decision. In decision theory, the theory of probability has been used extensively. When it comes to game theory examples, studying decision theories will help you to deal with many examples.

2. General Equilibrium Theory

If you want game theory homework help, you should also know the general equilibrium theory. This theory has been viewed as the specialized branch of game theory that is related to the production and trading. When there are a large number of players, the equilibrium theory will come into the action. It conducts a macroeconomic analysis for check tax policies and monetary conditions. In stock market trading, many investors try this theory for safe and rewarding stock investment.

3. Mechanism Design Theory

Do you know the use of game theory in international relations? In various bilateral relations, countries and experts use game theory. For understanding the deep concept of game theory, it is essential to understand the mechanism design theory. This theory deals with the consequences of the different rules. It relies heavily on the game theory. 

The mechanism design theory incorporates the wage agreements and compensation designs. Using this theory, businesses can escalate their revenues. In auctions, the theory can be used for maximizing revenues.

Types of Game Theory

If you want to know game theory applications, you have to know the types of game theory. There are different variations. Some of those variations are symmetrical, asymmetrical, simultaneous, etc.

What is game theory in economics? Those who are searching for answer to this question should know about cooperative and non-cooperative game theories. These are the two commonest game theories. In cooperative theory, interactions, coalitions, and cooperative groups have been discussed. On the other hand, non-cooperative theory deals with the rational economics agents. Such theory ensures applications of game theory in operations research.

Assumptions of game Theory

After knowing the types of game theory, you need to know about the assumptions and examples of game theory. In an economic model, you will find many scopes for assumptions. People assume something about future analyzing the present or past before investing on a fund or stock. 

Businesses assume future before investing more money in their businesses. In game theory, there is a high scope of assumptions. Possible outcomes are always unknown, and thus players have to assume the outcome correctly. The player how assumes the outcome with perfection will be termed as the winner.

Examples of Game Theory

Game theory has been used in different gaming models. You can easily find an array of game theory economics examples. Apart from the economics, the theory has been used in many other departments or spheres of life. For example, game theory can be used in auctions, gambling, etc. In sports betting, you can also find the applications of the game theory.

For game theory help, you need to have clear idea on the basics of assumptions. You must find a tutor, who can guide you with perfection. The tutor should have extensive knowledge of game theory in different spheres. It helps the students to understand the wide range of usages of game theory.

Game Theory

When you’re hanging out with your friends, you probably don’t think too hard about the math behind your making decisions, but there is a whole field of math and science that applies to social interactions. It’s called game theory. Game theory was pioneered in the 1950s by mathematician John Nash, Russell Crowe’s guy in A Beautiful Mind. But game theory isn’t about games the way we usually think about them. Instead, a game is any interaction between multiple people in which each person’s payoff is affected by the decisions made by others. So sure, that could apply to a game of poker, but it could also apply to practically any situation where people get together and get up in each other’s business. Like, did you interact with anyone today? Well, you can probably analyze the decisions you made using game theory. Game theory is incredibly wide-ranging, and it’s used all the time by: 

  • Economists
  • Political Scientists 
  • Biologists
  • Military tacticians
  •  Psychologists to name a few. 

Game theory has two main branches, cooperative and non-cooperative or competitive game theory.

Non-Cooperative Game Theory

The non-cooperative game theory covers competitive social interactions where there will be some winners and some losers. Probably the most famous thought, experiment, and competitive game theory is the prisoner’s dilemma. 

Prison Dilemma

The prisoner’s dilemma describes a game, a social interaction that involves two prisoners. We’ll call them Wanda and Fred. Wanda and Fred were arrested, fleeing from the scene of a crime. And based on the evidence the police have already collected, they will have to spend two years in jail. But the D.A. wants more, so he offers them both a deal. If you confess to the crime and your partner does not, you will be granted immunity for cooperating. You’ll be free to go. Your partner, though, will serve ten years in jail. If you both confess and dish up loads of dirt about each other, then you will both end up spending five years in jail. But if neither of you confesses, you will both spend only two years in jail. Those are their options. Then Wanda and Fred are split up. They don’t know what their partner is going to do. They have to make their decisions independently. Now, Wanda and Fred, they’ve had some rough time stealing diamonds, but they don’t have any special loyalty to each other. They’re not brother and sister. They’re hardened criminals. Fred has no reason to think that Wanda won’t stab him in the back and vice versa. 

Competitive game theory arranges their choices and their potential consequences into a grid that looks like this. 

Confess/Don’t Confess

If both Wanda and Fred choose not to confess, they’ll both serve two years. In theory, this is the best overall outcome. Combined, they would spend as little time in prison as possible, but that immunity sounds pretty good. If one of them chooses to confess and the other one doesn’t, the snitch gets to walk. Then the math looks like this. That’s the problem. Wanda and Fred have no reason to trust each other. Wanda might consider not confessing because if Fred doesn’t confess either, both only serve two years. If they could trust each other, that would be their best bet. But Wanda can’t be sure that Fred won’t snitch. He has a lot to gain by confessing. If he decides to confess and keeps silent, she’s risking ten years in jail while he goes free. Compared to that, the five years they both get for turning on each other doesn’t sound so bad. And that is a game theory solution. They should both confess and read each other out. 

Nash Equilibrium

So right now, you’re thinking, game theory is a jerk, but it makes sense that the square in the grid where they both confess is the only outcome that has reached what’s known as Nash equilibrium. This is a key concept in competitive game theory. A player in a game has found Nash equilibrium. When they make a choice that leaves them better off no matter what their opponents decide to do, if one that confesses and Fred does not confess, she’s better off. She gets to walk by, confessing she went from serving two years in prison to serving none. If Fred does confess, she’s still better off. If she’d kept her mouth shut, she’d be spending ten years in prison. Now she only has to serve five. Sure, if she decides not to confess and Fred keeps his pinky promise too, they’ll both get out in two years. But that is an unstable state because Wanda can’t trust Fred. She doesn’t know what he’s going to do. This is not a cooperative game. All of the players stand to gain from stabbing each other in the back. 

The prisoner’s dilemma is just one example of a competitive game. But the basic idea behind its solution applies to all kinds of situations. Generally, when you’re competing with others, it makes sense to choose the course of action that benefits you the most, no matter what everyone else decides to do. 

Cooperative Game Theory

Then there are cooperative games where every player has agreed to work together toward a common goal. This could be anything from a group of friends deciding how to split up that cost to pay the bill at a restaurant to a coalition of nations deciding how to divide up the burden of stopping climate change. In-game theory, a coalition is what you call a group of players in a cooperative game. Game theory’s main question is how much each player should contribute to the coalition and how much they should benefit from it when it comes to cooperative games. In other words, it tries to determine what’s fair. Where competitive game theory has the Nash Equilibrium, Cooperative game theory has the Shapley value. 

Shapley Value

The Shapley value is a method of dividing up gains or Costs among players according to the value of their individual contributions. It works by applying several axioms. 

  1. First, each player’s contribution is determined by what is gained or lost by removing them from the game. This is called their marginal contribution. Let’s say that every day this week, you and your friends are baking cookies when you get sick for a day, probably from eating too many cookies. The group produces 50 fewer cookies than they did on the days that you were there. So your marginal contribution to the coalition every day is 50 cookies. 
  1. Secondly, interchangeable players have equal value. If two parties bring the same things to the coalition, they should contribute the same amount and be rewarded for their contributions equally. If two people order the same thing at the restaurant, they should pay the same amount of the bill. If two workers have the same skills, they should receive the same wages. 
  1. Thirdly, dummy players have zero value. In other words, if a member of the coalition contributes nothing, then they should receive nothing. This one is controversial. It could mean that if you go to dinner with your friends, but you don’t order anything, then you shouldn’t have to chip in when the bill comes, which seems fair in that case. But it could also mean that if somebody can’t contribute to the workforce, they shouldn’t receive any compensation. The thing is, there are good reasons why somebody might not be able to contribute. Maybe they’re on maternity leave, or they got in an accident or have some disability in situations like that. The coalition might want to pay something out to them despite them not being able to contribute.
  1. The fourth axiom says that if a game has multiple parts, cost or payment should be decomposed across those parts. This means that, for example, if you did a lot of work for the group on Monday but you slacked off on Tuesday, your rewards on each day should be different. Or if you ordered a salad one night with a steak dinner the next, you probably should pay more on the second night. In other words, it’s not always fair to use the same solution every time. The numbers should be reviewed regularly so that the coalition can make adjustments. If you find a way of dividing up costs or dividing up payment to all players, that satisfies all of those axioms. That’s the Shapley value. 

The Shapley value can be expressed mathematically, which is complicated, but we can break down the concepts into something less mathematical. So let’s go back to looking at cookies.

Illustration

You’re baking cookies, and your friend is baking cookies in an hour. You can bake ten cookies when you’re working alone. Your friend, though, is like a cookie wizard. And in the same hour, working alone, he can bake 20 cookies. When you decide to team up, when you work together, you streamline your process. One person can mix up all the batter at once, which saves you a lot of time. So after an hour, you have 40 cookies. But if you’d been working alone, you’d only have made 30 cookies in the same hour. Then you sell each of those cookies for a dollar. Now you got forty dollars. How do you divide up the loot? 

The Shapley Value equation tells you to think about it like this. If you take the fact that you can make ten cookies an hour and subtract them from the total, that gives your friend credit for the other 30 cookies. That’s what happens when you remove your friend from the system. Their marginal contribution to you is thirty cookies. But if you take the fact that your friend can make 20 cookies an hour and subtract that from the total, that gives you credit for 20 cookies. Because if you’re removed from your friend’s cookie-making system, your marginal contribution to them is 20 cookies. In the first case, your value to the coalition was only ten cookies. But in the second case, your value to the coalition is 20 cookies. According to the Shapleigh value equation, you should average those two numbers together. Ten plus 20 is 30 divided by two is 15. So the Shapleigh value equation says that you should get fifteen dollars, and your friend should get twenty-five. 

Conclusion

This method can be scaled up to coalitions with hundreds of players by finding their marginal contributions to every other player and then calculating the average of all those numbers. Interactions can get much more complicated than the prisoner’s dilemma or baking cookies. So there’s a lot more to game theory. But it comes down to this. In a competitive situation, game theory can tell you how to be smart. And in a cooperative situation, game theory can tell you how to be fair.

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