subject: Game theory, Equilibrium (Economics) cover. Page III. Strategies and Games. Theory and Practice. Prajit K. Dutta. THE MIT PRESS CAMBRIDGE. The MIT Press , pages ISBN Game theory has become increasingly popular among undergraduate as well as business school students. Games Theory Notes. Based on Strategies and Games: theory and practice by Prajit K. Dutta. Introduction. Block I: Strategic Form Games. Theme 1: Strategic.

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Ben Polak, Professor of Economics and Management Description This course is an introduction to game theory and strategic thinking. Ideas such as dominance, backward induction, Nash equilibrium, evolutionary stability, commitment, credibility, asymmetric information, adverse selection, and signaling are discussed and applied to games played in class and to examples drawn from economics, politics, the movies, and elsewhere. Texts A. Dixit and B. Thinking Strategically, Norton J. This course is an introduction to game theory. Introductory microeconomics or equivalent is required. We will use calculus mostly one variable in this course. We will also refer to ideas like probability and expectation. Some may prefer to take the course next academic year once they have more background. Students who have already taken Econ b should not enroll in this class.

A zero-sum game 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. Poker exemplifies a zero-sum game ignoring the possibility of the house's cut , because one wins exactly the amount one's opponents lose.

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

Many games studied by game theorists including the famous prisoner's dilemma are non-zero-sum games, because some outcomes have 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 an additional dummy player often called "the board" , whose losses compensate the players' net winnings. Simultaneous and sequential 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 he does 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, and 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. Game theory 11 Perfect information and imperfect information 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. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect-information games, although there are some A game of imperfect information the dotted line represents interesting examples of perfect-information games, ignorance on the part of player 2, formally called an information set including the ultimatum game and centipede game.

Recreational games of perfect information games include chess, go, and mancala. Many card games are games of imperfect information, for instance poker or contract bridge. Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken. Combinatorial games 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 or incomplete 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 some general questions. 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. 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.

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 what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. It can be proven, using the axiom of choice, that there are games—even with perfect information, and where the only outcomes are "win" or "lose"—for which neither player has a winning strategy.

The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory. Discrete and continuous games 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. Evolutionary game theory considers games involving a population of decision makers, where the frequency with which a particular decision is made can change over time in response to the decisions made by all individuals in the population.

In biology, this is intended to model biological evolution, where genetically programmed organisms pass along some of their strategy programming to their offspring. In economics, the same theory is intended to capture population changes because people play the game many times within their lifetime, and consciously and perhaps rationally switch strategies Webb Stochastic outcomes and relation to other fields Individual decision problems with stochastic outcomes are sometimes considered "one-player games".

These situations are not considered game theoretical by some authors. They may be modeled using similar tools within the related disciplines of decision theory, operations research, and areas of artificial intelligence, particularly AI planning with uncertainty and multi-agent system.

Main article: Perfect information A game of imperfect information the dotted line represents ignorance on the part of player 2, formally called an information set 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. Games of incomplete information can be reduced, however, to games of imperfect information by introducing " moves by nature ". 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. 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.

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.

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.

Discrete and continuous games[ edit ] 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[ edit ] 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[ edit ] 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.

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. For some problems, different approaches to modeling stochastic outcomes may lead to different solutions.