Optimal Game Theory

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Optimal Game Theory

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So, during review sessions you ought to ask yourself what you might have done with different holdings. If you are betting for value in certain situations, you should also be betting as a bluff with other hands in your range so that your opponent is unsure whether you are betting for value or as a bluff.

If you are only betting value hands on a certain river, your opponent can fold profitably every time knowing you have the goods.

On the other hand, if you are bluffing too much in certain situations, then your opponent can call profitably every time knowing that you are less likely to have a strong hand.

This frequency is optimal because it allows you to win the pot most often without the possibility of being countered. For simplicity, assume that we always win when our value bets are called, and always lose when our bluffs are called.

This win-win scenario is only achievable with a perfectly balanced range. Your opponent is indifferent to calling and folding because no matter which option she chooses, your range profits the same amount.

Adjusting this ratio in order to exploit weak players can be even more profitable, but that requires careful and correct adjustments based on reliable evidence.

If you want to move up in stakes and really crush the game long-term, understanding a GTO-influenced strategy is essential.

But the reality is that this process is unreliable against weak players. We can avoid ending up in this sort of situation by using a GTO-influenced bluffing strategy, which keeps us from confusing ourselves and getting into leveling wars on the flop with no equity.

Another benefit of a GTO-based approach to poker is that it forestalls potentially incorrect assumptions of other players.

A well-constructed GTO strategy eliminates confusion, and helps you make the long-term profitable play. Many players incorrectly judge how they played a hand by its outcome.

Yet, thinking objectively can be tough, especially when the result of a hand is either really bad or really good.

Game theory provides a foundation for discerning mistakes more easily. Why is theory important when it comes to making badass adjustments in your strategy?

Suppose you just forgot everything you know about poker strategy, except the rudimentary knowledge of the game, and you are just about to play your first hand ever.

Hero calls. How could you adjust to exploit him in the future? Well, without understanding the theoretically correct way of playing his specific hand, you would not know where to start.

This knowledge makes it easy to deduce ways to exploit this opponent. Striving for a perfect GTO strategy might seem like the logical conclusion, but the truth is nobody plays an entirely game theory optimal strategy.

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability of moves by other players have also been studied.

The " gold standard " is considered to be partially observable stochastic game POSG , but few realistic problems are computationally feasible in POSG representation.

These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed.

The theory of metagames is related to mechanism design theory. The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard.

Subsequent developments have led to the formulation of confrontation analysis. These are games prevailing over all forms of society.

Pooling games are repeated plays with changing payoff table in general over an experienced path, and their equilibrium strategies usually take a form of evolutionary social convention and economic convention.

Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time.

The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.

Mean field game theory is the study of strategic decision making in very large populations of small interacting agents.

This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal , in the engineering literature by Peter E.

The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game , the information and actions available to each player at each decision point, and the payoffs for each outcome.

These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.

Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees as pictured here. Here each vertex or node represents a point of choice for a player.

The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player.

The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.

The game pictured consists of two players. The way this particular game is structured i. Next in the sequence, Player 2 , who has now seen Player 1 ' s move, chooses to play either A or R.

Once Player 2 has made their choice, the game is considered finished and each player gets their respective payoff. Suppose that Player 1 chooses U and then Player 2 chooses A : Player 1 then gets a payoff of "eight" which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players and Player 2 gets a payoff of "two".

The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set i.

See example in the imperfect information section. The normal or strategic form game is usually represented by a matrix which shows the players, strategies, and payoffs see the example to the right.

More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions.

In the accompanying example there are two players; one chooses the row and the other chooses the column.

Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior.

The first number is the payoff received by the row player Player 1 in our example ; the second is the payoff for the column player Player 2 in our example.

Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other.

If players have some information about the choices of other players, the game is usually presented in extensive form.

Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.

In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity.

The idea is that the unity that is 'empty', so to speak, does not receive a reward at all. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.

Such characteristic functions have expanded to describe games where there is no removable utility.

Alternative game representation forms exist and are used for some subclasses of games or adjusted to the needs of interdisciplinary research.

As a method of applied mathematics , game theory has been used to study a wide variety of human and animal behaviors.

It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers.

The first use of game-theoretic analysis was by Antoine Augustin Cournot in with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.

Although pre-twentieth century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher 's studies of animal behavior during the s.

This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.

Game-theoretic arguments of this type can be found as far back as Plato. The primary use of game theory is to describe and model how human populations behave.

This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations.

Game theorists usually assume players act rationally, but in practice human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics.

Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists.

There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.

Price , have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players.

Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning for example, fictitious play dynamics.

Some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave.

Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players — provided they are in the same Nash equilibrium — playing a strategy that is part of a Nash equilibrium seems appropriate.

This normative use of game theory has also come under criticism. Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria".

A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium.

A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.

The payoffs of the game are generally taken to represent the utility of individual players. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation.

One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type.

Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses noted above : descriptive and prescriptive.

Sensible decision-making is critical for the success of projects. In project management, game theory is used to model the decision-making process of players, such as investors, project managers, contractors, sub-contractors, governments and customers.

Quite often, these players have competing interests, and sometimes their interests are directly detrimental to other players, making project management scenarios well-suited to be modeled by game theory.

Piraveenan [94] in his review provides several examples where game theory is used to model project management scenarios. For instance, an investor typically has several investment options, and each option will likely result in a different project, and thus one of the investment options has to be chosen before the project charter can be produced.

Similarly, any large project involving subcontractors, for instance, a construction project, has a complex interplay between the main contractor the project manager and subcontractors, or among the subcontractors themselves, which typically has several decision points.

For example, if there is an ambiguity in the contract between the contractor and subcontractor, each must decide how hard to push their case without jeopardizing the whole project, and thus their own stake in it.

Similarly, when projects from competing organizations are launched, the marketing personnel have to decide what is the best timing and strategy to market the project, or its resultant product or service, so that it can gain maximum traction in the face of competition.

In each of these scenarios, the required decisions depend on the decisions of other players who, in some way, have competing interests to the interests of the decision-maker, and thus can ideally be modeled using game theory.

Piraveenan [94] summarises that two-player games are predominantly used to model project management scenarios, and based on the identity of these players, five distinct types of games are used in project management.

In terms of types of games, both cooperative as well as non-cooperative games, normal-form as well as extensive-form games, and zero-sum as well as non-zero-sum games are used to model various project management scenarios.

The application of game theory to political science is focused in the overlapping areas of fair division , political economy , public choice , war bargaining , positive political theory , and social choice theory.

In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy , [95] he applies the Hotelling firm location model to the political process.

In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence.

It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects.

Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king or other established government as the person whose orders will be followed.

Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime.

Thus, in a process that can be modeled by variants of the prisoner's dilemma , during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.

A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states.

In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept.

Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy. On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting.

War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting.

Moreover, war may arise because of commitment problems: if two countries wish to settle a dispute via peaceful means, but each wishes to go back on the terms of that settlement, they may have no choice but to resort to warfare.

Finally, war may result from issue indivisibilities. Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation.

One example would be Peter John Wood's research when he looked into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions.

However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations. Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness.

In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces.

Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium , every ESS is a Nash equilibrium. In biology, game theory has been used as a model to understand many different phenomena.

It was first used to explain the evolution and stability of the approximate sex ratios. Fisher harv error: no target: CITEREFFisher help suggested that the sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.

Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication.

For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization.

Ants have also been shown to exhibit feed-forward behavior akin to fashion see Paul Ormerod 's Butterfly Economics.

Biologists have used the game of chicken to analyze fighting behavior and territoriality. According to Maynard Smith, in the preface to Evolution and the Theory of Games , "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed".

Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature. One such phenomenon is known as biological altruism.

This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself.

This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness.

Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival.

Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives.

The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles.

This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on.

Ensuring that enough of a sibling's offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.

Similarly if it is considered that information other than that of a genetic nature e. Game theory has come to play an increasingly important role in logic and in computer science.

Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations.

Also, game theory provides a theoretical basis to the field of multi-agent systems. Separately, game theory has played a role in online algorithms ; in particular, the k -server problem , which has in the past been referred to as games with moving costs and request-answer games.

The emergence of the Internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets.

Algorithmic game theory [] and within it algorithmic mechanism design [] combine computational algorithm design and analysis of complex systems with economic theory.

Game theory has been put to several uses in philosophy. Responding to two papers by W. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games.

In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis.

Game theory has also challenged philosophers to think in terms of interactive epistemology : what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from the interactions of agents.

Philosophers who have worked in this area include Bicchieri , , [] [] Skyrms , [] and Stalnaker Since games like the prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project.

This general strategy is a component of the general social contract view in political philosophy for examples, see Gauthier and Kavka harvtxt error: no target: CITEREFKavka help.

Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors.

These authors look at several games including the prisoner's dilemma, stag hunt , and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality see, e.

Game theory applications are used heavily in the pricing strategies of retail and consumer markets, particularly for the sale of inelastic goods.

With retailers constantly competing against one another for consumer market share, it has become a fairly common practice for retailers to discount certain goods, intermittently, in the hopes of increasing foot-traffic in brick and mortar locations websites visits for e-commerce retailers or increasing sales of ancillary or complimentary products.

Black Friday , a popular shopping holiday in the US, is when many retailers focus on optimal pricing strategies to capture the holiday shopping market.

The retailer is focused on an optimal pricing strategy, while the consumer is focused on the best deal. In this closed system, there often is no dominant strategy as both players have alternative options.

That is, retailers can find a different customer, and consumers can shop at a different retailer. The open system assumes multiple retailers selling similar goods, and a finite number of consumers demanding the goods at an optimal price.

Amazon made up part of the difference by increasing the price of HDMI cables, as it has been found that consumers are less price discriminatory when it comes to the sale of secondary items.

Retail markets continue to evolve strategies and applications of game theory when it comes to pricing consumer goods. The key insights found between simulations in a controlled environment and real-world retail experiences show that the applications of such strategies are more complex, as each retailer has to find an optimal balance between pricing , supplier relations , brand image , and the potential to cannibalize the sale of more profitable items.

From Wikipedia, the free encyclopedia. This article is about the mathematical study of optimizing agents. For the mathematical study of sequential games, see Combinatorial game theory.

For the study of playing games for entertainment, see Game studies. For other uses, see Game theory disambiguation.

Collective behaviour. Social dynamics Collective intelligence Collective action Self-organized criticality Herd mentality Phase transition Agent-based modelling Synchronization Ant colony optimization Particle swarm optimization Swarm behaviour Collective consciousness.

Evolution and adaptation. Artificial neural network Evolutionary computation Genetic algorithms Genetic programming Artificial life Machine learning Evolutionary developmental biology Artificial intelligence Evolutionary robotics Evolvability.

Pattern formation. Spatial fractals Reaction—diffusion systems Partial differential equations Dissipative structures Percolation Cellular automata Spatial ecology Self-replication Spatial evolutionary biology Geomorphology.

Systems theory. Nonlinear dynamics. Game theory. Prisoner's dilemma Rational choice theory Bounded rationality Irrational behaviour Evolutionary game theory.

The study of mathematical models of strategic interaction between rational decision-makers. Index Outline Category. History Branches Classification.

History of economics Schools of economics Mainstream economics Heterodox economics Economic methodology Economic theory Political economy Microeconomics Macroeconomics International economics Applied economics Mathematical economics Econometrics.

Concepts Theory Techniques. Economic systems Economic growth Market National accounting Experimental economics Computational economics Game theory Operations research Middle income trap.

By application. Notable economists. Glossary of economics. See also: List of games in game theory. Main articles: Cooperative game and Non-cooperative game.

Main article: Symmetric game. Main article: Zero-sum game. Main articles: Simultaneous game and Sequential game. Prior knowledge of opponent's move?

Extensive-form game Extensive game. Strategy game Strategic game. Main article: Perfect information. Main article: Determinacy.

Main article: Extensive form game. Main article: Normal-form game. Main article: Cooperative game. See also: Succinct game.

Main article: Evolutionary game theory. Applied ethics Chainstore paradox Chemical game theory Collective intentionality Combinatorial game theory Confrontation analysis Glossary of game theory Intra-household bargaining Kingmaker scenario Law and economics Outline of artificial intelligence Parrondo's paradox Precautionary principle Quantum game theory Quantum refereed game Rationality Reverse game theory Risk management Self-confirming equilibrium Tragedy of the commons Zermelo's theorem.

Chapter-preview links, pp. Statistical Science. Institute of Mathematical Statistics. Bibcode : arXivB. Hobson, E. Cambridge: Cambridge University Press.

Archived from the original PDF on October 23, Retrieved August 29, Game theory applications in network design. IGI Global. Mathematische Annalen [ Mathematical Annals ] in German.

In Tucker, A. Contributions to the Theory of Games. In Weintraub, E. Roy ed. Toward a History of Game Theory. Durham: Duke University Press.

Zalta, Edward N. Stanford Encyclopedia of Philosophy. Stanford University. Retrieved January 3, A New Kind of Science.

Wolfram Media. Retrieved September 15, University of Texas at Dallas. Archived from the original PDF on May 27, Game Theory: Third Edition.

Bingley: Emerald Group Publishing. Stack Exchange. June 24, Handbook of Game Theory with Economic Applications.

PBS Infinite Series. March 2, Perfect information defined at , with academic sources arXiv : Luck, logic, and white lies: the mathematics of games.

A K Peters, Ltd. Cambridge University Press. Tim Artificial Intelligence: A Systems Approach. Springer, Cham.

Cmu-Cs : 3—4. Games and Information 4th ed.

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Optimal Game Theory - Weitere Kapitel dieses Buchs durch Wischen aufrufen

Energy, 94, — Baldick, R. Zurück zum Zitat Lasseter, B. Botterrud, A. Zurück zum Zitat Bertsimas, D. Two game models based on deterministic and stochastic approaches are proposed, formulated, and investigated.

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