Texas Computer Science Ranks 8th in Alumni Who Have Founded Venture-capital Backed Firms. Abstract. This essay introduces the symposium on computer science and economic theory. Evolutionary game theory (EGT) is the application of game theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a.
Game theory is the study of the ways in which interacting choices of economic agents produce outcomes with respect to the preferences (or utilities) of. Game theory is included in the JEL classification codes as JEL: C7: Game theory is 'the study of mathematical models of conflict and cooperation between intelligent.
Game theory - Wikipedia, the free encyclopedia. Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision- makers."[1] Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science, biology and Poker (Texas No Limit Hold'em).[2] Originally, it addressed zero- sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. Modern game theory began with the idea regarding the existence of mixed- strategy equilibria in two- person zero- sum games and its proof by John von Neumann. Von Neumann's original proof used 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 1.
Game Theory Computer Science Application
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. This theory was developed extensively in the 1. Game theory was later explicitly applied to biology in the 1.
Game theory has been widely recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2. Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Representation of games[edit]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. Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".)[3] A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. 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. Extensive form[edit]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. The game pictured consists of two players. The way this particular game is structured (i. Player 1 "moves" first by choosing either F or U (letters are assigned arbitrarily for mathematical purposes). 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 his/ her 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.)Normal form[edit]Player 2chooses Left. Player 2chooses Right.
Player 1chooses Up. Player 1chooses Down. Normal form or payoff matrix of a 2- player, 2- strategy game. 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.
Characteristic function form[edit]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 origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union appears, it works against the fraction as if two individuals were playing a normal game. 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.
Formally, a characteristic function is seen as: (N,v), where N represents the group of people and is a normal utility. Such characteristic functions have expanded to describe games where there is no removable utility.
General and applied uses[edit]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 1. 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 1. 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.[citation needed]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.[6] In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior.
Game- theoretic arguments of this type can be found as far back as Plato.[7]Description and modeling[edit]The primary use of game theory is to describe and model how human populations behave. Some[who?] scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has been criticized. First, it 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. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. 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.[8]Some game theorists, following the work of John Maynard Smith and George R. 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).
Prescriptive or normative analysis[edit]Cooperate. Defect. Cooperate- 1, - 1- 1. Defect. 0, - 1. 0- 5, - 5. The Prisoner's Dilemma.
Introduction to computer science and economic theory. Abstract. This essay introduces the symposium on computer science and economic theory. JEL classification. Keywords. Algorithmic game theory; Implementation; Learning in games; Mechanism design; Networks. Introduction. Computer scientists and economists share interests in several areas of economic theory, and individuals from both groups have been working together and in parallel for approximately three decades. Interest in the interaction of computer science and economics has intensified in the last 1. Web and the Internet.
The emergence of these new systems has caused a profound expansion of the questions that computer science has had to address, since these networks operate through the cooperation and competition of many participants, leading inevitably to underlying social and economic issues. At the same time, the Internet has also made possible the development of more overtly economic structures, through the creation of new kinds of markets. The purpose of this symposium is to introduce economists to recent work in these areas.
The interaction of computer science and economics has had an impact on economic theory in three ways. It has introduced new problems — novel kinds of markets including those arising in the search industry, and new applications including network management and routing, on- line social systems, and platforms for the production and sharing of content.
It has raised new issues in areas already popular in economics, including learning, decision theory, market design, network- structured interaction, the analysis of equilibrium quality, and the computational complexity of equilibria. And it has brought new methods to existing problems, including efficient algorithms, lower bounds based on computational hardness, and techniques from discrete mathematics and graph theory. We focus in this introduction on the topic of market design as it provides compelling examples of new problems, methods, and techniques in a fundamental economic context, and because mechanism design is at the moment the most active area of joint interest. The next section discusses mechanism design. Subsequent sections provide brief introductions to other aspects of algorithmic game theory, learning in games, and networks.
Mechanisms and market design. Marshallian and Walrasian equilibrium analysis are not theories of how markets function. Their institution- free approach to predicting market outcomes precludes them from asking questions such as: When do market institutions fail? How do they behave when they fail?
How should markets be designed to minimize failure, and what tradeoffs with market efficiency arise in doing so? Research in economics arising from general equilibrium and welfare economics has been concentrated on market imperfections.
Computer scientists have paid relatively more attention to the nuts and bolts of market mechanisms and the robustness of market institutions. These two distinct approaches to the challenges of modeling markets at a detailed level meet in the field of mechanism design.
Leo Hurwicz's research program was a response to the Lange–Lerner–Hayek debate about the virtues of markets versus central planning. To clarify Hayek's claim that the virtue of the market is its ability to harness widely dispersed information to achieve social goals, Hurwicz [1] defined a mechanism to be a communication system in which participants send messages to a center, and a function (or correspondence) which assigns to each profile of messages an allocation of commodities (or a set thereof).
Modern mechanism design began with Hurwicz's [2] introduction of incentive compatibility, and with Gibbard's [3] introduction of the revelation principle for dominant strategy equilibria and its subsequent extension to Bayes–Nash equilibria. Subsequent work has addressed two general questions. The Implementation problem is concerned with objectives: Can a particular social choice function be implemented over some rich class of environments? The Design problem is concerned with perhaps more practical institutional questions, including: How do particular mechanisms (e. Which auction mechanisms are optimal from a welfare- maximizing point of view, or from a revenue maximizing point of view, or some other point of view?
These two questions are not distinct. Although the early implementation theorists (e. Hurwicz [4], Mount and Reiter [5]) concluded that markets perform well in “neoclassical” environments, their abstract description of markets did not enable the analysis of particular methods of market organization, and they had little to say about non- neoclassical environments. These questions are the locus of most mechanism design today.
Outside of neoclassical environments, there is yet no general theory of implementation failure, and so the theory develops one problem at a time. In extending received implementation theory, computer scientists have raised the issue of complexity in mechanism design in two different ways: Computational complexity, the difficulty of computing the function that maps message profiles to allocations and of finding an equilibrium, and communication complexity, the amount of information that must be communicated by the mechanism's participants. In practical mechanism design both are clear concerns; one needs to have confidence that the algorithm employed in an online market will reach its conclusion in a reasonable amount of time, and market participants need to be able to participate without extensive communication with the mechanism. Computer science has developed a range of ways to measure the efficiency of a procedure or algorithm to solve a problem, based on the resources used by the algorithm; these include the running time, the space or memory required, and the amount of communication required. This also leads to natural measures for the inherent computational complexity of an underlying problem, by considering the minimum resources required by any algorithm to solve the problem. These requirements can be evaluated in either the worst case over all possible initial conditions, or in the average case in a Bayesian environment; for some well- known algorithms, such as the simplex method for linear programming, the difference between worst case and natural measures of average- case performance can be very large. In a completely analogous way, one can study the amount of computational or communication resources required by a mechanism, and one can define the inherent computational complexity of implementing a social choice function or correspondence as the minimum resources required by any mechanism to implement that function or correspondence.
Of particular interest is the question — first raised by Nisan and Ronen [9] — of whether a social choice correspondence which is efficiently computable can always be implemented by a mechanism whose outcome is efficiently computable. Such a result, if true, would imply that the only obstacles to achieving computational efficiency of mechanisms stem from the inherent computational complexity of the problem being solved, not from the game- theoretic challenge of implementing the solution in equilibrium.
Unfortunately, the result is false, as can be seen by considering the problem of implementing social welfare maximization with quasi- linear utility in dominant- strategy equilibrium. The Vickrey–Clarke–Groves (VCG) mechanism is unique for implementing social welfare maximization, and requires solving an NP- hard problem (a widely adopted criterion for computational intractability) in settings with an exponentially large discrete set of alternatives, as was observed by Kfir- Dahav, Monderer, and Tennenholtz [1. In light of this negative result, attention naturally turned to implementing outcomes that achieve close to maximal welfare. The answer to whether the computational complexity of implementing this social choice correspondence differs significantly from the complexity of merely computing it depends on the details of the problem.
In some cases there exist computationally efficient algorithms implementing an outcome whose efficiency is within a small constant factor of the maximal social welfare. In other cases, computational constraints can be arbitrarily costly for the efficiency of outcome. An example of this latter case is the work of Papadimitriou, Schapira and Singer [1. Selecting an outcome that achieves fraction 1−1/e. The same phenomenon, whereby incentive- compatibility constraints lead to an exponential blow- up in the computational resources required to solve a problem, occurs even for the seemingly innocuous problem of allocating identical, indivisible goods to a set of bidders, as shown by Dobzinski and Nisan [1.