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2015 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Game Theory

A Multi-Leveled Approach

verfasst von: Hans Peters

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Texts in Business and Economics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

This textbook presents the basics of game theory both on an undergraduate level and on a more advanced mathematical level. It is the second, revised version of the successful 2008 edition. The book covers most topics of interest in game theory, including cooperative game theory. Part I presents introductions to all these topics on a basic yet formally precise level. It includes chapters on repeated games, social choice theory, and selected topics such as bargaining theory, exchange economies, and matching. Part II goes deeper into noncooperative theory and treats the theory of zerosum games, refinements of Nash equilibrium in strategic as well as extensive form games, and evolutionary games. Part III covers basic concepts in the theory of transferable utility games, such as core and balancedness, Shapley value and variations, and nucleolus. Some mathematical tools on duality and convexity are collected in Part IV. Every chapter in the book contains a problem section. Hints, answers and solutions are included.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
The best introduction to game theory is by way of examples. This chapter starts with a global definition in Sect. 1.1, collects some historical facts in Sect. 1.2, and presents examples in Sect. 1.3. Section 1.4 briefly comments on the distinction between cooperative and noncooperative game theory.
Hans Peters

Thinking Strategically

Frontmatter
2. Finite Two-Person Zero-Sum Games
Abstract
This chapter deals with two-player games in which each player chooses from finitely many pure strategies or randomizes among these strategies, and the sum of the players’ payoffs or expected payoffs is always equal to zero. Games like the Battle of the Bismarck Sea and Matching Pennies, discussed in Sect. 1.​3.​1 belong to this class.
Hans Peters
3. Finite Two-Person Games
Abstract
In this chapter we consider two-player games where each player chooses from finitely many pure strategies or randomizes among these strategies. In contrast to Chap.​ 2 it is no longer required that the sum of the players’ payoffs is zero (or, equivalently, constant). This allows for a much larger class of games, including many games relevant for economic or other applications. Famous examples are the Prisoners’ Dilemma and the Battle of the Sexes discussed in Sect. 1.​3.​2
Hans Peters
4. Finite Extensive Form Games
Abstract
Most games derived from economic or political situations have in common with most parlor games (like card games and board games) that they are not ‘one-shot’: players move sequentially, and one and the same player may move more often than once. Such games are best described by drawing a decision tree which tells us whose move it is and what a player’s information is when that player has to make a move.
Hans Peters
5. Finite Games with Incomplete Information
Abstract
In a game of imperfect information players may be uninformed about the moves made by other players. Every one-shot, simultaneous move game is a game of imperfect information. In a game of incomplete information players may be uninformed about certain characteristics of the game or of the players. For instance, a player may have incomplete information about actions available to some other player, or about payoffs of other players. Incomplete information is modelled by assuming that every player can be of a number of different types. A type of a player summarizes all relevant information (in particular, actions and payoffs) about that player. Furthermore, it is assumed that each player knows his own type and, given his own type, has a probability distribution over the types of the other players. Often, these probability distributions are assumed to be consistent in the sense that they are the marginal probability distributions derived from a basic commonly known distribution over all combinations of player types.
Hans Peters
6. Noncooperative Games: Extensions
Abstract
In Chaps. 2–5 we have studied noncooperative games in which the players have finitely many (pure) strategies. The reason for the finiteness restriction is that in such games special results hold, such as the existence of a value and optimal strategies for two-person zero-sum games, and the existence of a Nash equilibrium in mixed strategies for finite nonzero-sum games.
Hans Peters
7. Repeated Games
Abstract
In the famous prisoners’ dilemma game the bad (Pareto inferior) outcome, resulting from each player playing his dominant action, cannot be avoided in a Nash equilibrium or subgame perfect Nash equilibrium even if the game is repeated a finite number of times, cf. Problem 4.10. As we will see in this chapter, this bad outcome can be avoided if the game is repeated an infinite number of times. This, however, is coming at a price, namely the existence of a multitude of outcomes attainable in equilibrium. Such an embarrassment of riches is expressed by a so-called folk theorem.
Hans Peters
8. An Introduction to Evolutionary Games
Abstract
In an evolutionary game, players are interpreted as populations—of animals or individuals. The probabilities in a mixed strategy of a player in a bimatrix game are interpreted as shares of the population. Individuals within the same part of the population play the same pure strategy. The main ‘solution’ concept is the concept of an evolutionary stable strategy.
Hans Peters
9. Cooperative Games with Transferable Utility
Abstract
The implicit assumption in a cooperative game is that players can form coalitions and make binding agreements on how to distribute the proceeds of these coalitions. A cooperative game is more abstract than a noncooperative game in the sense that strategies are not explicitly modelled: rather, the game describes what each possible coalition can earn by cooperation. In a cooperative game with transferable utility it is assumed that the earnings of a coalition can be expressed by one number. One may think of this number as an amount of money, which can be distributed among the players in any conceivable way—including negative payments—if the coalition is actually formed. More generally, it is an amount of utility and the implicit assumption is that it makes sense to transfer this utility among the players—for instance, due to the presence of a medium like money, assuming that individual utilities can be expressed in monetary terms.
Hans Peters
10. Cooperative Game Models
Abstract
The common features of a cooperative game model—such as the model of a game with transferable utility in Chap.​ 9—include: the abstraction from a detailed description of the strategic possibilities of a player; instead, a detailed description of what players and coalitions can attain in terms of outcomes or utilities; solution concepts based on strategic considerations and/or considerations of fairness, equity, efficiency, etc.; if possible, an axiomatic characterization of such solution concepts. For instance, one can argue that the core for TU-games is based on strategic considerations whereas the Shapley value is based on a combination of efficiency and symmetry or fairness with respect to contributions. The latter is made precise by an axiomatic characterization as in Problem 9.17.
Hans Peters
11. Social Choice
Abstract
Social choice theory studies the aggregation of individual preferences into a common or social preference. It overlaps with several social science disciplines, such as political theory (e.g., voting for Parliament, or for a president) and game theory (e.g., voters may vote strategically, or candidates may choose positions strategically).
Hans Peters

Noncooperative Games

Frontmatter
12. Matrix Games
Abstract
In this chapter we study finite two-person zero-sum games—matrix games—more rigorously. In particular, von Neumann’s Minimax Theorem is proved. The chapter extends Chap.​ 2 in Part I. Although it is self-contained, it may be useful to (re)read Chap.​ 2 first.
Hans Peters
13. Finite Games
Abstract
This chapter builds on Chap.​ 3, where we studied finite two person games—bimatrix games. (Re)reading Chap.​ 3 may serve as a good preparation for the present chapter, which offers a more rigorous treatment of finite games, i.e., games with finitely many players—often two—who have finitely many pure strategies over which they can randomize. We only discuss games with complete information. In the terminology of Chap.​ 5, each player has only one type.
Hans Peters
14. Extensive Form Games
Abstract
A game in extensive form specifies when each player in the game has to move, what his information is about the sequence of previous moves, which chance moves occur, and what the final payoffs are. Such games are discussed in Chaps. 4 and 5, and also occur in Chaps. 6 and 7. The present chapter extends the material introduced in Chaps. 4 and 5, and it may be useful to (re)read these chapters before continuing.
Hans Peters
15. Evolutionary Games
Abstract
In this chapter we go deeper into evolutionary game theory. The concepts of evolutionary stable strategy and replicator dynamics, introduced in Chap.​ 8, are further explored. It may be helpful to study Chap.​ 8 first, although the present chapter is largely self-contained.
Hans Peters

Cooperative Games

Frontmatter
16. TU-Games: Domination, Stable Sets, and the Core
Abstract
In a game with transferable utility (TU-game) each coalition (subset of players) is characterized by its worth, i.e., a real number representing the payoff or utility that the coalition can achieve if it forms. It is assumed that this payoff can be freely distributed among the members of the coalition in any way desired.
Hans Peters
17. The Shapley Value
Abstract
In Chap. 16 set-valued solution concepts for games with transferable utilities were studied: the imputation set, core, domination core, and stable sets. In this chapter, a one-point (single-valued) solution concept is discussed: the Shapley value. It may again be helpful to first study the relevant parts of Chaps. 1 and 9.
Hans Peters
18. Core, Shapley Value, and Weber Set
Abstract
In Chap. 17 we have seen that the Shapley value of a game does not have to be in the core of the game, nor even an imputation (Problem 17.5). In this chapter we introduce a set-valued extension of the Shapley value, the Weber set, and show that it always contains the core (Sect. 18.1). Next, we study so-called convex games and show that these are exactly those games for which the core and the Weber set coincide. Hence, for such games the Shapley value is an attractive core selection (Sect. 18.2). Finally, we study random order values (Sect. 18.3), which fill out the Weber set, and the subset of weighted Shapley values, which still cover the core (Sect. 18.4).
Hans Peters
19. The Nucleolus
Abstract
The core of a game with transferable utility can be a large set, but it can also be empty. The Shapley value assigns to each game a unique point, which, however, does not have to be in the core. The nucleolus assigns to each game with a nonempty imputation set a unique element of that imputation set; moreover, this element is in the core if the core of the game is nonempty. The pre-nucleolus exists for every essential game (and does not have to be an imputation, even if the imputation set is nonempty), but for balanced games it coincides with the nucleolus.
Hans Peters
20. Special Transferable Utility Games
Abstract
In this chapter we consider a few classes of games with transferable utility which are derived from specific economic (or political) models or combinatorial problems. In particular, we study assignment and permutation games, flow games, and voting games.
Hans Peters
21. Bargaining Problems
Abstract
The game-theoretic literature on bargaining can be divided in two strands: the cooperative and the noncooperative approach. Here, the focus is on the cooperative approach, which was initiated by Nash (1950) and which is axiomatic in nature; see Sect. 10.​1 for a first discussion. A seminal article on noncooperative bargaining is Rubinstein (1982). The basic idea of that paper is briefly repeated below, but see Sect. 6.​7 for a more elaborate discussion. We conclude the chapter with a few remarks on games with nontransferable utility (NTU-games).
Hans Peters

Tools, Hints and Solutions

Frontmatter
22. Tools
Abstract
This chapter collects some mathematical tools used in this book: (direct) convex separation results in Sects. 22.2 and 22.6; Lemmas of the Alternative, in particular Farkas’ Lemma in Sect. 22.3; the Linear Duality Theorem in Sect. 22.4; the Brouwer and Kakutani Fixed Point Theorems in Sect. 22.5; the Krein–Milman Theorem and the Birkhoff–von Neumann Theorem in Sect. 22.6; and the Max Flow Min Cut Theorem of Ford and Fulkerson in Sect. 22.7.
Hans Peters
23. Review Problems for Part I
Abstract
This chapter contains Review Problems to Chaps. 2–10, organized per chapter.
Hans Peters
24. Hints, Answers and Solutions
Abstract
There are saddlepoint(s) if and only if x ≤ −1.
Hans Peters
Backmatter
Metadaten
Titel
Game Theory
verfasst von
Hans Peters
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-46950-7
Print ISBN
978-3-662-46949-1
DOI
https://doi.org/10.1007/978-3-662-46950-7

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