## Optimal Game Theory Wir empfehlen

Play Optimal Poker shatters the myth that game theory is only for elite poker players. Renowned poker pro and coach Andrew Brokos takes you step-by-step. No Limits Hold Em: Learn Game Theory Optimal! A Dominant Framework When It Comes To Finding An Ideal Poker Strategy | Harrington, Ryan | ISBN. Suche nach: Advertisement. News. Game Theory Optimal Finding Equilibrium: Chaos Theory in Poker (Simple 3-Way). 7. Januar | 0 Kommentare. This comprehensive work examines important recent developments and modern applications in the fields of optimization, control, game theory, and equilibrium. In this chapter, we first study different game theory models and their applications in power system. Then, an appropriate model is selected to formulate the.

The Game Theory uses a mathematical method to study optimal strategies in games. A game is understood as a process in which two or more parties participate. Suche nach: Advertisement. News. Game Theory Optimal Finding Equilibrium: Chaos Theory in Poker (Simple 3-Way). 7. Januar | 0 Kommentare. However, while game theory looks at optimal strategies for individual players in the interaction with others, economic ethics analyzes the institutional conditions.Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann.

Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets , which became a standard method in game theory and mathematical economics.

His paper was followed by the book Theory of Games and Economic Behavior , co-written with Oskar Morgenstern , which considered cooperative games of several players.

The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

Game theory was developed extensively in the s by many scholars. It was explicitly applied to biology in the s, although similar developments go back at least as far as the s.

Game theory has been widely recognized as an important tool in many fields. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

Discussions of two-person games began long before the rise of modern, mathematical game theory. In , a letter attributed to Charles Waldegraveto analyzed a game called "le her".

He was an active Jacobite and uncle to James Waldegrave , a British diplomat. One theory postulates Francis Waldegrave as the true correspondent, but this has yet to be proven.

This paved the way for more general theorems. In , the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem.

Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games , a conjecture that was proved false by von Neumann.

Game theory did not really exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in Von Neumann's work in game theory culminated in this book.

This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.

In , the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. RAND pursued the studies because of possible applications to global nuclear strategy.

Nash proved that every finite n-player, non-zero-sum not just two-player zero-sum non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.

Game theory experienced a flurry of activity in the s, during which the concepts of the core , the extensive form game , fictitious play , repeated games , and the Shapley value were developed.

The s also saw the first applications of game theory to philosophy and political science. In Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was often a simple "tit-for-tat" program--submitted by Anatol Rapoport --that cooperates on the first step, then, on subsequent steps, does whatever its opponent did on the previous step.

The same winner was also often obtained by natural selection; a fact that is widely taken to explain cooperation phenomena in evolutionary biology and the social sciences.

In , Reinhard Selten introduced his solution concept of subgame perfect equilibria , which further refined the Nash equilibrium. Later he would introduce trembling hand perfection as well.

In the s, game theory was extensively applied in biology , largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy.

In addition, the concepts of correlated equilibrium , trembling hand perfection, and common knowledge [a] were introduced and analyzed.

Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing equilibrium coarsening and correlated equilibria, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.

Myerson's contributions include the notion of proper equilibrium , and an important graduate text: Game Theory, Analysis of Conflict. In , Alvin E.

Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design".

In , the Nobel went to game theorist Jean Tirole. A game is cooperative if the players are able to form binding commitments externally enforced e.

A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing e. Cooperative games are often analyzed through the framework of cooperative game theory , which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs.

It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria.

Cooperative game theory provides a high-level approach as it describes only the structure, strategies, and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition.

As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory the converse does not hold provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation.

While it would thus be optimal to have all games expressed under a non-cooperative framework, in many instances insufficient information is available to accurately model the formal procedures available during the strategic bargaining process, or the resulting model would be too complex to offer a practical tool in the real world.

In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.

A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them.

That is, if the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. The standard representations of chicken , the prisoner's dilemma , and the stag hunt are all symmetric games.

Some [ who? However, the most common payoffs for each of these games are symmetric. Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.

For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric.

For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players. Zero-sum games are a special case of constant-sum games in which choices by players can neither increase nor decrease the available resources.

In zero-sum games, the total benefit to all players in the game, for every combination of strategies, always adds to zero more informally, a player benefits only at the equal expense of others.

Other zero-sum games include matching pennies and most classical board games including Go and chess.

Many games studied by game theorists including the famed prisoner's dilemma are non-zero-sum games, because the outcome has net results greater or less than zero.

Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another. Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade.

It is possible to transform any game into a possibly asymmetric zero-sum game by adding a dummy player often called "the board" whose losses compensate the players' net winnings.

Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions making them effectively simultaneous.

Sequential games or dynamic games are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge.

For instance, a player may know that an earlier player did not perform one particular action, while they do not know which of the other available actions the first player actually performed.

The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones.

The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form.

Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.

An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players.

Most games studied in game theory are imperfect-information games. Many card games are games of imperfect information, such as poker and bridge.

Games of incomplete information can be reduced, however, to games of imperfect information by introducing " moves by nature ".

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games.

Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon.

There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.

Games of perfect information have been studied in combinatorial game theory , which has developed novel representations, e.

These methods address games with higher combinatorial complexity than those usually considered in traditional or "economic" game theory.

A related field of study, drawing from computational complexity theory , is game complexity , which is concerned with estimating the computational difficulty of finding optimal strategies.

Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures like chess, go, or backgammon for which no provable optimal strategies have been found.

The practical solutions involve computational heuristics, like alpha—beta pruning or use of artificial neural networks trained by reinforcement learning , which make games more tractable in computing practice.

Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner or other payoff not known until after all those moves are completed.

The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.

Much of game theory is concerned with finite, discrete games that have a finite number of players, moves, events, outcomes, etc.

Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set.

For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.

Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations.

The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory. In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method.

A particular case of differential games are the games with a random time horizon. Therefore, the players maximize the mathematical expectation of the cost function.

It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted. Such rules may feature imitation, optimization, or survival of the fittest.

In biology, such models can represent biological evolution , in which offspring adopt their parents' strategies and parents who play more successful strategies i.

In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.

Although these fields may have different motivators, the mathematics involved are substantially the same, e. Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" " moves by nature ".

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution.

The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely but costly events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.

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.

In a hand discussed on the show, a listener in a limit hold'em game held out of position on an ace-high flop. Heads-up, this is still a plus-EV hand, but there isn't much value in betting.

You're not getting an ace to fold, and by checking, you'll get more value from a bluff, as well as from a value bet with middle pair. More importantly, the value of check-calling might drop by less than then the value of betting out.

Estimating the value of your hand, assuming that both players play well and about even in the long run, is just another way of approximating GTO.

Once you're in three-handed or more games, there is no game theory optimal solution , strictly speaking.

This is because there is no stable equilibrium or too many equilibria to count, depending on whom you ask. The players can always adjust to each other, or take advantage of a player trying to execute a GTO poker strategy and not adjusting to them, through a process that Bill Chen and Jerrod Ankenman call "implicit collusion" in their book The Mathematics of Poker.

When playing heads up, if you or a bot follows a GTO poker strategy, an opponent can't beat you in the long run, no matter what he or she does.

This does not mean that you are winning the most against this opponent, but you are locking in a long-term tie, while still benefitting from some of your opponent's mistakes.

For example, the limit hold'em GTO bot will pay off on the river with bottom pair often enough so you can't bluff it effectively.

If you never bluff in this spot, the bot will still pay you off at the same rate. An exploitative player would stop paying you off after a while, and win even more.

Poker pro and poker training site founder Doug Polk spoke on the TwoPlusTwo Pokercast about this situation coming up during a man-vs.

It was such a relief to the players once they realized that while the computer played well "4 out of 10" compared to his regular opponents, according to Polk , it did not attempt to exploit their betting patterns.

If when you flop the nuts you bet 1. It just plays GTO poker. But in practice, if the players don't change their strategies too much from hand to hand and they don't , a lot of the heads-up Game Theory Optional principles apply.

A friend of mine went to graduate school with one of the best online poker players in the world, and had a chance to watch him play.

He was surprised that his classmate did not make any unusual plays, or really any "moves" at all. According to the poker pro :.

If you take Chen and Ankenman's ideas about "implicit collusion" to heart, one could also add that if the players were ganging up on him instead of trying to beat each other, the pro would just quit the game.

This is a non-issue in the nosebleed games, since everyone knows everyone else, and playing anonymously or collusively isn't really possible.

The point is, the best players in online poker play GTO. And then they actually do it. There's a lot to be said for good execution.

It's also easy to see why Polk in the same interview is pessimistic about humans' chances, once the bots learn all the right bet frequencies. Our silicon friends will always have that edge in execution, and they don't need room, food, or beverage.

I took some flack on Twitter for comparing the small-bet, check-down game happening between Daniel Negreanu , Dan Colman and Christoph Vogelsang to a nightly satellite at a local casino.

Folks fired back that these guys are the best poker players in the world , and I'm clearly an idiot. But it sure did look like none of the players were trying to pressure the others.

And why should they? On other hand, if one guy pushed, he knew the others knew how to fight back. So nobody pushed.

For about two hours, three of the best short-handed NLHE players in the world checked or small-bet every hand, until Vogelsang, the short stack, busted.

Do you need to play poker GTO in order to win? Or rather, how close do you need to get to optimal poker game theory in order to hold your own against a strong set of opponents?

In a recent article in Cigar Aficianado interviewing academics and enthusiasts at the Annual Computer Poker Championship , Sandholm was asked about his colleagues at the University of Alberta solving limit hold'em.

I think that counts," responded Sandholm. Once your baseline strategy can't be easily exploited, you can spend the rest of your time studying opponents' tendencies and adjusting to their weaknesses.

There will be plenty of opponents who don't think about ranges, who don't adjust to some of the game information, or who are just playing their own way.

GTO or Game Theory Optimal refers to an approach to the game of poker based on creating models about the players to then evaluate the game in terms of ranges and probabilities.

The GTO or Game Theory Optimal treats poker as a mathematical problem and uses analysis and calculations to 'solve' the game. The goal of this poker game theory is to lead a player to optimal play and make him impossible to beat.

A number of poker training sites offer courses and coaching programs to learn GTO poker. As a beginner, you might want to use this article to start with poker GTO.

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