Theory of Games and Economic Behavior

1. Game Theory Provides a Rigorous Approach to Economic Behavior

Our major interest is, of course, in the economic and sociological direction. Here we can approach only the simplest questions. However, these questions are of a fundamental character.

Beyond Traditional Economics. Traditional economic models often fall short in capturing the complexities of strategic interactions. Game theory offers a more rigorous framework for analyzing situations involving parallel or opposing interests, perfect or imperfect information, and rational decision-making under uncertainty.

Addressing Fundamental Questions. While game theory may initially tackle simplified scenarios, it provides a foundation for understanding more complex economic and sociological problems. It allows for a structured analysis of how individuals and groups make decisions when their outcomes are interdependent.

Mathematical Precision. The application of mathematical methods, including set theory, linear geometry, and functional analysis, enables a more precise and nuanced understanding of economic behavior. This approach moves beyond qualitative discussions to quantitative models that can be rigorously analyzed.

2. Rationality in Economics Requires Understanding Conflicting Interests

Every participant is guided by another principle and neither determines all variables which affect his interest.

Beyond Simple Maximization. The traditional economic assumption of individuals seeking to maximize utility or profit is insufficient in social exchange economies. Participants must consider the actions and reactions of others, leading to a complex interplay of conflicting maximum problems.

Interdependent Outcomes. In social economies, the outcome for each participant depends not only on their own actions but also on the actions of others. This creates a situation where individuals attempt to maximize a function they do not fully control.

Strategic Considerations. Rational behavior in such scenarios requires understanding the principles guiding others' actions and the interactions of conflicting interests. Game theory provides the tools to analyze these strategic considerations and develop models for rational decision-making.

3. Utility Measurement Extends Beyond Simple Preference

Even if utilities look very unnumerical today, the history of the experience in the theory of heat may repeat itself, and nobody can foretell with what ramifications and variations.

Numerical Representation of Preferences. While initially conceived as quantitatively measurable, utility has faced objections due to its subjective nature. However, by considering choices involving probabilities, a numerical utility can be derived from preferences.

Probability and Utility. Introducing probabilities allows for comparing preferences between events and combinations of events. This approach, similar to that used in physics for measuring heat, provides a basis for assigning numerical values to utilities.

Axiomatic Treatment of Utility. The axiomatic method can be applied to formalize the relationship between preferences, probabilities, and numerical utilities. This approach, involving axioms such as completeness, transitivity, and the algebra of combining, leads to a more rigorous understanding of utility measurement.

4. Solutions in Game Theory Represent Stable Standards of Behavior

A valuable qualitative preliminary description of the behavior of the individual is offered by the Austrian School, particularly in analyzing the economy of the isolated “Robinson Crusoe.”

Beyond Individual Optimization. While the "Robinson Crusoe" model provides insights into individual decision-making, it fails to capture the complexities of social exchange economies. In social settings, individuals must interact with others, leading to strategic considerations and the need for a broader concept of a solution.

Rules for Every Situation. A solution in game theory should provide rules for each participant, specifying how to behave in every conceivable situation. This includes accounting for irrational conduct by others and the potential for chance events.

Games as Models. Games of strategy serve as models for economic activities, providing a framework for analyzing rational behavior in complex social settings. These models must be precise, exhaustive, and similar to reality in essential respects.

5. Games of Strategy Offer a Model for Economic Activities

We hope to establish satisfactorily, after developing a few plausible schematizations, that the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy.

From Economics to Games. The study of games of strategy is essential for understanding rational behavior and the foundations of economics. Economic problems can be strictly identified with mathematical notions of suitable games of strategy.

Emphasis Shift. The focus shifts from economics to games, treating games as independent subjects. This approach allows for a more thorough investigation of strategic principles and their application to economic problems.

Common Elements. Identifying common elements between economic theory and game theory is crucial. This involves understanding the nature of economic problems and how they relate to the mathematical concepts of games of strategy.

6. Zero-Sum Two-Person Games: A Foundation for Understanding Strategy

The first phases of this work were published: J.von Neumann, “Zur Theorie der Gesellschaftsspiele,” Math. Annalen, vol. 100 (1928), pp. 295-320. The subsequent completion of the theory, as well as the more detailed elaboration of the considerations of loc. cit. above, are published here for the first time.

Building Blocks. The theory of zero-sum two-person games serves as a foundation for understanding more complex games. These games, where one player's gain is another's loss, provide insights into strategic decision-making and the concept of a strategy.

Key Questions. Essential questions in zero-sum two-person games include:

• How does each player plan their course of action (strategy)?
• What information is available to each player at every stage?
• What is the role of information about the other player's strategy?

Mathematical Precision. The application of mathematical methods, including functional calculus, linearity, and convexity, enables a more precise and nuanced understanding of strategic interactions. This approach moves beyond qualitative discussions to quantitative models that can be rigorously analyzed.

7. Coalitions and Compensations are Essential in n-Person Games

The classical definitions of free competition all involve further postulates besides the greatness of that number.

Beyond Two-Person Games. The transition from two-person to n-person games introduces the possibility of coalitions and compensations. These factors significantly complicate the analysis of rational behavior and require new theoretical tools.

Coalition Formation. In n-person games, participants may form coalitions to secure an advantage. The problem lies in determining how the proceeds of a coalition should be divided among its partners, considering the potential for defections and alternative alliances.

Compensatory Payments. The apportionment within a coalition depends not only on the rules of the game but also on the principles guiding the participants and the influence of alternative coalitions. This may involve compensatory payments between coalition partners to maintain stability.

8. The Characteristic Function Quantifies Coalition Power

The characteristic function is the proper instrument with which to develop a theory of economic behavior.

Measuring Coalition Strength. The characteristic function provides a quantitative measure of the power of a coalition. It represents the minimum amount a coalition can guarantee itself, regardless of the actions of other players.

Strategic Equivalence. Games with different rules can be strategically equivalent if they have the same characteristic function. This means that the essential strategic possibilities and incentives for coalition formation are the same, even if the specific details of the game differ.

Groups, Symmetry, and Fairness. The characteristic function can be used to analyze groups, symmetry, and fairness in games. By examining how permutations of players affect the characteristic function, insights can be gained into the fairness and stability of different game structures.

9. Simple Games: Decision-Making by Coalitions

The field covered in this book is very limited, and we approach it in that sense of modesty.

Winning and Losing. Simple games are characterized by the existence of winning and losing coalitions. These games provide a framework for understanding decision-making based on coalition formation.

Characterization of Simple Games. Simple games can be characterized by their winning and losing coalitions. The special role of one-element sets and the properties of winning and losing coalitions are essential for understanding the structure of these games.

Majority Games. Majority games are examples of simple games where decisions are made by coalitions. Homogeneity and the concept of imputation play a direct role in forming solutions in these games.

10. General Non-Zero-Sum Games: Cooperation and Competition

The theory finally obtained must be mathematically rigorous and conceptually general.

Extension of the Theory. The theory of games can be extended to general non-zero-sum games. This involves introducing a fictitious player and analyzing the concept of domination in the new setup.

The Characteristic Function. The characteristic function plays a crucial role in the theory of general non-zero-sum games. The extended and restricted forms of the characteristic function provide insights into the strategic possibilities and incentives for coalition formation.

Economic Interpretation. The results for n=1, 2, and 3 can be interpreted in economic terms, providing insights into markets, divisible goods, and the role of monopoly and monopsony.

Last updated: February 27, 2025