# Game Theory with Examples

Game theory is a branch of applied mathematics that provides tools to analyze situations where parties called participants make interdependent

# Game Theory with Examples

## Game Theory Definition

Whenever we encounter a situation involving two or more participants with known benefits or quantifiable consequences, we can use game theory to determine the most likely outcome. Nash equilibrium is a key concept in game theory: this happens when each participant is looking for the best possible strategy, knowing all the options available to other participants and what they might do.

In game theory, it can be used to describe thousands of situations that occur or may occur, such as an arms race or wage negotiations. The dominant strategy is the strategy that is most likely to bring players the best results, regardless of who the other players are. can do. Game theory is a branch of applied mathematics that studies the strategic situations of players choosing different actions in an attempt to maximize their income. The key point Game theory is to understand the social situation among competing participants and the theoretical framework for independent and competing participants to make the best decisions in the context of strategy. Game theory has been applied to various situations, in which the choices of players influence each other to affect the outcome.

By focusing on strategic aspects of decision-making or aspects controlled by participants, rather than pure opportunity, the theory complements and surpasses classical probability theory. This is called game theory, because the theory tries to understand the strategic actions of two or more "participants" in a given situation with established rules and outcomes. The core of game theory is game, which is a model of the interaction situation between rational players. Formal theory defines a game as consisting of two or more participants, each participant has a set of pure strategies, and the participant's profit function.

The game consists of a set of players, a set of actions (or strategies) available to these players, and the profit specification for each combination of strategies. The game consists of three main components: player, victory (such as victory, defeat, draw) and strategy. A normal game (or some form of strategy) is a matrix showing players, strategies, and payouts (see example on the right). The game determines the player’s personality, preferences, and available strategies, and how these strategies affect the outcome.

That is, the result for each participant depends on the choice (strategy) of each. In so-called zero-sum games, the interests of the players completely contradict each other, so that the gain of one person is always the loss of another. Game theorists might assume that players always act rationally to maximize their gains (the Homo economicus model), but real people often act irrationally or act rationally to maximize the gains of a larger group of people (altruism). Participants in the game are expected to be rational and will strive to maximize their income in the game.

The original concepts of the theory are the players (decision makers), strategies (the alternatives each player chooses), and returns (the numerical representation of the player's preferences in the possible outcome of the game). Part of the theory that deals with situations where players cannot force a strategy to choose is called non-cooperative game theory.

In each of these scenarios, the required decisions depend on the decisions of other participants. Their interests compete with those of the decision maker in some way. Therefore, ideally, game theory can be used to build them. mold. In many cases, there are competing interests among these participants, and sometimes their interests will directly harm other participants, which makes project management scenarios suitable for game theory modeling. In each of these areas, researchers have developed game-theoretic models in which the participants are usually voters, states, special interest groups, and politicians.

Game theory is a branch of applied mathematics that provides tools to analyze situations where parties called participants make interdependent decisions. In-game decisions describe the best decisions of players who may have similar, opposite, or mixed interests, and the possible outcomes of these decisions. Participants in the game make choices based on how they evaluate the potential outcomes of their choices.

First, all players are rational actors who maximize utility and have all the information about the game, the rules, and the consequences. A game is ideal information if all players know the moves previously made by all other players. Therefore, only sequential games can be games with complete information, since in simultaneous games not all players know the actions of others. Other theorists, most notably Reinhard Selten and John Harsagni, who shared the 1994 Nobel Prize from memory with Nash, have studied even more complex sequence games and games in which one player has more information than the others.

One way to explore the rationality of game theory is to reduce its decision-making concept to a more intuitive rational concept in the face of the uncertainty of decision-making theory. First, many games lack specific game-theoretic solutions, so psychological theories and empirical evidence are needed to discover and understand how people play them. For readers who are new to economics, game theory, decision theory, and action philosophy, this situation is naturally a problem.

Since game theory is about economically rational action with regard to the strategically important actions of others, you should not be surprised to hear that the fact that agents in games believe or disbelieve in the actions of others has a noticeable effect on the actions of others ... the logic of ours. analysis, as we shall see. The key to understanding strategic decision making is understanding your opponents' point of view and their likely reaction to your actions. Game = a situation in which companies make strategic decisions based on each other's actions and responses.

In the business world, competition between two companies can be analyzed as a game in which the participants strive for long-term competitive advantage, perhaps even complete domination - a monopoly. The problem of finding an optimal strategy in a differential game is closely related to the theory of optimal control. One or several solution concepts are selected, and the author demonstrates which sets of strategies in the presented game are equilibria of the corresponding type.

If all agents have optimal actions regardless of what others are doing, for example, in purely parametric situations, monopoly conditions, or perfect competition (see Section 1 above), we can model this without resorting to game theory; otherwise, we need it. Game theory cannot explain the fact that in some situations we can get into Nash equilibrium and sometimes not, depending on the social context and who the players are.

He opposes the traditional theory of non-cooperative games, which focuses on predicting the actions and returns of individual players and analyzing Nash equilibria. This is a branch of mathematics that deals with the analysis of strategies for working with competitive situations in which the result of a person's decision critically depends on the actions taken by other participants. The games he studies range from chess to parenting, tennis to practice.

## Game Theory Full Explanation

Game theory cannot explain the fact that in some cases we can enter the Nash equilibrium, sometimes we cannot, depending on the social background and who the participants are. This is called game theory, because the theory tries to understand the strategic actions of two or more "participants" in a given situation with established rules and outcomes. The key point Game theory is to understand the social situation among competing participants and the theoretical framework for independent and competing participants to make the best decisions in the context of strategy. In project management, game theory is used to shape the decision-making process of investors, project managers, contractors, subcontractors, governments, and customers.

In many cases, there are competing interests among these participants, and sometimes their interests will directly harm other participants, which makes project management scenarios suitable for game theory modeling. In each of these scenarios, the required decisions depend on the decisions of other participants, and their interests compete with those of the decision maker in some way. Therefore, ideally, game theory can be used to build them. mold. An instructive example One way to describe a game is to list the players (or individuals) participating in the game, and for each player list the alternatives (called actions or strategies) available to that player.

In the case of a game with two players, the actions of the first player form rows, and the actions of the second player form the columns of the matrix.

These games have a parameter to get the payout function and strategy for each player. Information about each player is modeled as a \ (\ upsigma \) subfield to achieve optimal strategies. Conditions are characterized by value and equilibrium strategies.

The game determines the personalities, preferences and available strategies of the players, as well as how these strategies affect the outcome. These concepts are used to formally predict the development of a game. The core of game theory is the game, which serves as a model for the interactive situation between rational players. Game theory is applied in various fields of research to understand why a person makes a particular decision and how decisions made by him affect others.

Decision theory can be thought of as a theory about single-player games or single-player against nature. Decision theory is usually used in the form of decision analysis, which shows how to best obtain information before making a decision. Game theory is a standard analysis tool for professionals in operations research, economics, finance, regulation, military, insurance, retail marketing, politics, analysis, conflict, and energy, to name a few.

Modern game theory, a branch of applied mathematics founded by Neumann and Nash, is the study of mathematical models in conflict and collaboration between intelligent and rational decision-makers. Unlike physics or chemistry, which have a well-defined and narrow scope of application, the principles of game theory are useful in a variety of areas, from everyday social interactions and sports to business and economics, politics, law, diplomacy and war. Game theory models can be used to analyze many competitive aspects of animal behavior, including habitat selection, foraging, predator-prey interactions, communication, parent-child interactions, and sibling interactions.

Imperfect information - a game in which the players are not aware of the actions taken by other players; however, everything else, the type of player, strategies, payouts, etc. are well known. Comprehensive information - a game in which all participants have access to the knowledge of other players; payoff functions, strategies and "types" of players are in the public domain.

Nash Equilibrium is the optimal outcome of a game in which no player has any incentive to deviate from the chosen strategy; there is no additional benefit from changing actions if other players stay the same in their strategies. We determine the best strategy for each player, anticipating all possible outcomes.

Simple games such as tic-tac-toe can be solved in this way and are therefore not difficult. For many other games, such as chess, the calculations are too complex to be done in practice, even with computers. This is where modern mathematical game theory comes into play. A mathematical description of a zero-sum game for two people is easy to construct and determine optimal strategies, and the value of the game is computationally simple.

We can show that heads and tails are fair play, and that both players have the same optimal mix of strategies that randomize heads or tails 50% of the time for each. Stone scissors are fair game too, and both players have optimal strategies that use each choice a third of the time.

The participants in the constant sum game have completely opposite interests, while in the change sum game, they can all be winners or losers. In the so-called zero-sum game, the interests of the participants are completely contradictory, so that one person's gain is always the other's loss. Games with the potential for mutual benefit (positive numbers) or mutual losses (negative numbers) and some conflicts are more typical.

Other theorists, most notably Reinhard Selten and John Harsagni, who shared the 1994 Nobel Prize from memory with Nash, have studied even more complex sequence games and games in which one player has more information than the others.

By focusing on strategic aspects of decision making or aspects controlled by the players, rather than pure chance, the theory complements and goes beyond the classical theory of probability. He opposes the traditional theory of non-cooperative games, which focuses on predicting the actions and returns of individual players and analyzing Nash equilibria. It is a game between coalitions of players, not individuals, and it raises the question of how groups are formed and how they distribute profits among the players. Whenever we have a situation with two or more players that involves known payoffs or quantifiable consequences, we can use game theory to determine the most likely outcomes.

The most widely used form of decision theory is that preferences between risky alternatives can be described by maximizing the expected value of a numeric utility function, where utility can depend on a number of things, but in situations of interest to economists, it often depends on money. income.

Both relate to a state of economic or rational choice or subjective maximization of expected utility, a model of how people decide what to do (I will use rational choice below). Parsons and Wooldridge (2002) discussed both game theory and decision theory.

They used zero-sum discrete search games as the basis for their research. They obtained play value and derived utility at the same time using decision theory. They studied zero-sum games in which the interests of the two players were strictly opposed. They noticed that economics is very similar to a game in which players anticipate each other's moves, and therefore requires a new type of mathematics, which they call game theory.

## Game Theory Examples In Real Life

For example, the Dinner with Friends game we reviewed earlier is an eight-player PD with a dominant strategy of “order a fancy burger” (rather than “confess”) and an overall negative outcome of “everyone pays more for their food.” "(not" everyone spends more time in jail. ") We can have competitive multi-person situations where players can form coalitions and collaborate with other players; you play with many people with non-zero sum; games with infinite strategies and games for two people with nonzero sum, just to name a few.The mathematical analysis of such games led to a generalization of the result of von Neumann's optimal solution for two-person games with zero sum, called the equilibrium solution.

Game solutions describe the best decisions of players who may have similar, opposite, or mixed interests, and the possible outcomes of these decisions. Game theory uses mathematical tools to find solutions when interdependent parties make strategic decisions. Game theory is used to study human behavior in strategic situations, and has applications in economics, politics, business strategy, law, entrepreneurship, and military science. In project management, game theory is used to shape the decision-making process of investors, project managers, contractors, subcontractors, governments, and customers.

In each of these scenarios, the decisions required depend on the decisions of other players, whose interests somehow compete with the interests of the decision maker, and therefore, ideally, they can be modeled using game theory. The game determines the personalities, preferences and available strategies of the players, as well as how these strategies affect the outcome. Using game theory, you can identify real-world scenarios for situations such as price competition and product launch (and more) and predict the results. Key Points Game theory is the theoretical framework for understanding social situations between competing players and for developing optimal decision making by independent and competing actors in a strategic context.

Game theory is a branch of applied mathematics that provides tools to analyze situations where parties called participants make interdependent decisions. This is called game theory, because the theory tries to understand the strategic actions of two or more "participants" in a given situation with established rules and outcomes. By focusing on strategic aspects of decision-making or aspects controlled by participants, rather than pure opportunity, the theory complements and surpasses classical probability theory. He opposed the traditional non-cooperative game theory, which focused on predicting the behavior and returns of individual players and analyzing the Nash equilibrium.

Each player has several strategies (in this example = {Home, Beach} two players). In this form, the game is represented by a payout matrix, where each row describes the strategy of one player, and each column describes the strategy of another player. There is a third kind of equilibrium in this game, including the so-called mixed strategy. Also known as the Nash equilibrium, the minimax solution is a strategy that provides players with the maximum guarantee and the minimum return.

In this example, if each player rolls a dice individually, if it rolls 1 or 2, it floats, and if it rolls 3, 4, 5, or 6, it rises, and the final expected utility (2/3 of each player) cannot Improve for any player because other players are using this strategy. In this example, going to the beach is the (strict) dominant strategy for each player, because it always gives the best results no matter what other players are doing. In this game, the defection strategy is a weak advantage for every player, which means that no matter what other players do, rejection is at least as good as what the remaining unity does, if not better.

Poker, for example, is a constant-sum game because the aggregate wealth of the players remains constant, even if its distribution changes over the course of a game. For example, the game on the right is asymmetrical, despite the same strategy sets for both players. The game is a prisoner's dilemma whenever (i) all players have a dominant strategy and (ii) all players will be better off if they all choose not to play their dominant strategies. A situation is a “game” where (i) more than one person makes a decision and (ii) people's decisions influence each other.

Game theory cannot explain the fact that in some situations we can get into Nash equilibrium and sometimes not, depending on the social context and who the players are. Whenever we have a situation with two or more players that involves known payoffs or quantifiable consequences, we can use game theory to determine the most likely outcomes.

The beauty of Nash equilibrium is that it shows how rational action from an individual point of view can lead to collective undesirable (or even catastrophic) results. The reason this example is so famous is that it refers to many real-life phenomena in which individualistic behavior leads to negative consequences for society. Well, in classical economic theory, Adam Smith argued that people, in pursuit of their own interests, maximize the collective welfare of society as a whole, a concept that developed through the allegory of the “invisible hand”. The theory of expected utility for one agent is sometimes called the theory of "play against nature."

Other examples of using game theory to make decisions in daily life include when to change lanes in traffic, when to make a request, and even when to wash the dishes. Games without accurate information, such as penny matching, rock-paper-scissors or poker, will pose challenges to players because there is no clear winning strategy. Rock-paper-scissors is also a fair game. Both players have the best strategy, and each choice has one-third of the time. For the zero-sum game of two players, John von Neumann, the most famous mathematician of the 20th century, showed that all these games have optimal strategies with corresponding expected values for both players.

The value of the game, denoted by v, is the value at which player, say, player 1, is guaranteed to win, at least if he adheres to the indicated optimal combination of strategies, regardless of the combination of strategies used by player 2.

"N-player games cannot be solved like a two-player game, as I proved with the running and passing balance problem. Nash proved that every n players has a finite non-zero sum (not just the zero-sum of two players). ). sum) Non-cooperation in mixed strategies has what is now called the Nash equilibrium. They point out that economics is very similar to a game in which players predict each other’s actions, so a new kind of mathematics is needed, which they call a game. In this article, We use ideas from the branch of mathematics "game theory" to study this situation. Known as the Prisoner's Dilemma (PJ), it illustrates why people often fail to cooperate well.