11 October 1994
The Royal Swedish Academy of Sciences has decided to award the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 1994, jointly to
Professor John C. Harsanyi, University of California, Berkeley, CA, USA,
Dr. John F. Nash, Princeton University, Princeton, NJ, USA,
Professor Dr. Reinhard Selten, Rheinische Friedrich-Wilhelms-Universität, Bonn, Germany,
for their pioneering analysis of equilibria in the theory of non-cooperative games.
Games as the Foundation for Understanding Complex Economic Issues
Game theory emanates from studies of games such as chess or poker. Everyone knows that in these games, players have to think ahead – devise a strategy based on expected countermoves from the other player(s). Such strategic interaction also characterizes many economic situations, and game theory has therefore proved to be very useful in economic analysis.
The foundations for using game theory in economics were introduced in a monumental study by John von Neumann and Oskar Morgenstern entitled Theory of Games and Economic Behavior (1944). Today, 50 years later, game theory has become a dominant tool for analyzing economic issues. In particular, non-cooperative game theory, i.e., the branch of game theory which excludes binding agreements, has had great impact on economic research. The principal aspect of this theory is the concept of equilibrium, which is used to make predictions about the outcome of strategic interaction. John F. Nash, Reinhard Selten and John C. Harsanyi are three researchers who have made eminent contributions to this type of equilibrium analysis.
John F. Nash introduced the distinction between cooperative games, in which binding agreements can be made, and non-cooperative games, where binding agreements are not feasible. Nash developed an equilibrium concept for non-cooperative games that later came to be called Nash equilibrium.
Reinhard Selten was the first to refine the Nash equilibrium concept for analyzing dynamic strategic interaction. He has also applied these refined concepts to analyses of competition with only a few sellers.
John C. Harsanyi showed how games of incomplete information can be analyzed, thereby providing a theoretical foundation for a lively field of research – the economics of information – which focuses on strategic situations where different agents do not know each others’ objectives.
Game theory is a mathematical method for analyzing strategic interaction. Many classical analyses in economics presuppose such a large number of agents that each of them can disregard the others’ reactions to their own decision. In many cases, this assumption is a good description of reality, but in other cases it is misleading. When a few firms dominate a market, when countries have to make an agreement on trade policy or environmental policy, when parties on the labor market negotiate about wages, and when a government deregulates a market, privatizes companies or pursues economic policy, each agent in question has to consider other agents’ reactions and expectations regarding their own decisions, i.e., strategic interaction.
As far back as the early nineteenth century, beginning with Auguste Cournot in 1838, economists have developed methods for studying strategic interaction. But these methods focused on specific situations and, for a long time, no overall method existed. The game-theoretic approach now offers a general toolbox for analyzing strategic interaction.
Whereas mathematical probability theory ensued from the study of pure gambling without strategic interaction, games such as chess, cards, etc. became the basis of game theory. The latter are characterized by strategic interaction in the sense that the players are individuals who think rationally. In the early 1900s, mathematicians such as Zermelo, Borel and von Neumann had already begun to study mathematical formulations of games. It was not until the economist Oskar Morgenstern met the mathematician John von Neumann in 1939 that a plan originated to develop game theory so that it could be used in economic analysis.
The most important ideas set forth by von Neumann and Morgenstern in the present context may be found in their analysis of two-person zero-sum games. In a zero-sum game, the gains of one player are equal to the losses of the other player. As early as 1928, von Neumann introduced the minimax solution for a two-person zero-sum game. According to the minimax solution, each player tries to maximize his gain in the outcome which is most disadvantageous to him (where the worst outcome is determined by his opponent’s choice of strategy). By means of such a strategy, each player can guarantee himself a minimum gain. Of course, it is not certain that the players’ choices of strategy will be consistent with each other. von Neumann was able to show, however, that there is always a minimax solution, i.e., a consistent solution, if so-called mixed strategies are introduced. A mixed strategy is a probability distribution of a player’s available strategies, whereby a player is assumed to choose a certain “pure” strategy with some probability.
John F. Nash
John Nash arrived at Princeton University in 1948 as a young doctoral student in mathematics. The results of his studies are reported in his doctoral dissertation entitled Non-cooperative Games (1950). The thesis gave rise to Equilibrium Points in n-person Games (Proceedings of the National Academy of Sciences of the USA 1950), and to an article entitled Non-cooperative Games, (Annals of Mathematics 1951).
In his dissertation, Nash introduced the distinction between cooperative and non-cooperative games. His most important contribution to the theory of non-cooperative games was to formulate a universal solution concept with an arbitrary number of players and arbitrary preferences, i.e., not solely for two-person zero-sum games. This solution concept later came to be called Nash equilibrium. In a Nash equilibrium, all of the players’ expectations are fulfilled and their chosen strategies are optimal. Nash proposed two interpretations of the equilibrium concept: one based on rationality and the other on statistical populations. According to the rationalistic interpretation, the players are perceived as rational and they have complete information about the structure of the game, including all of the players’ preferences regarding possible outcomes, where this information is common knowledge. Since all players have complete information about each others’ strategic alternatives and preferences, they can also compute each others’ optimal choice of strategy for each set of expectations. If all of the players expect the same Nash equilibrium, then there are no incentives for anyone to change his strategy. Nash’s second interpretation – in terms of statistical populations – is useful in so-called evolutionary games. This type of game has also been developed in biology in order to understand how the principles of natural selection operate in strategic interaction within and among species. Moreover, Nash showed that for every game with a finite number of players, there exists an equilibrium in mixed strategies.
Many interesting economic issues, such as the analysis of oligopoly, originate in non-cooperative games. In general, firms cannot enter into binding contracts regarding restrictive trade practices because such agreements are contrary to trade legislation. Correspondingly, the interaction among a government, special interest groups and the general public concerning, for instance, the design of tax policy is regarded as a non-cooperative game. Nash equilibrium has become a standard tool in almost all areas of economic theory. The most obvious is perhaps the study of competition between firms in the theory of industrial organization. But the concept has also been used in macroeconomic theory for economic policy, environmental and resource economics, foreign trade theory, the economics of information, etc. in order to improve our understanding of complex strategic interactions. Non-cooperative game theory has also generated new research areas. For example, in combination with the theory of repeated games, non-cooperative equilibrium concepts have been used successfully to explain the development of institutions and social norms. Despite its usefulness, there are problems associated with the concept of Nash equilibrium. If a game has several Nash equilibria, the equilibrium criterion cannot be used immediately to predict the outcome of the game. This has brought about the development of so-called refinements of the Nash equilibrium concept. Another problem is that when interpreted in terms of rationality, the equilibrium concept presupposes that each player has complete information about the other players’ situation. It was precisely these two problems that Selten and Harsanyi undertook to solve in their contributions.
The problem of numerous non-cooperative equilibria has generated a research program aimed at eliminating “uninteresting” Nash equilibria. The principal idea has been to use stronger conditions not only to reduce the number of possible equilibria, but also to avoid equilibria which are unreasonable in economic terms. By introducing the concept of subgame perfection, Selten provided the foundation for a systematic endeavor in Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit, (Zeitschrift für die Gesamte Staatswissenschaft 121, 301-24 and 667-89, 1965).
An example might help to explain this concept. Imagine a monopoly market where a potential competitor is deterred by threats of a price war. This may well be a Nash equilibrium – if the competitor takes the threat seriously, then it is optimal to stay out of the market – and the threat is of no cost to the monopolist because it is not carried out. But the threat is not credible if the monopolist faces high costs in a price war. A potential competitor who realizes this will establish himself on the market and the monopolist, confronted with fait accompli, will not start a price war. This is also a Nash equilibrium. In addition, however, it fulfills Selten’s requirement of subgame perfection, which thus implies systematic formalization of the requirement that only credible threats should be taken into account.
Selten’s subgame perfection has direct significance in discussions of credibility in economic policy, the analysis of oligopoly, the economics of information, etc. It is the most fundamental refinement of Nash equilibrium. Nevertheless, there are situations where not even the requirement of subgame perfection is sufficient. This prompted Selten to introduce a further refinement, usually called the “trembling-hand” equilibrium, in Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games (International Journal of Game Theory 4, 25-55, 1975). The analysis assumes that each player presupposes a small probability that a mistake will occur, that someone’s hand will tremble. A Nash equilibrium in a game is “trembling-hand perfect” if it is robust with respect to small probabilities of such mistakes. This and closely related concepts, such as sequential equilibrium (Kreps and Wilson, 1982), have turned out to be very fruitful in several areas, including the theory of industrial organization and macroeconomic theory for economic policy.
John C. Harsanyi
In games with complete information, all of the players know the other players’ preferences, whereas they wholly or partially lack this knowledge in games with incomplete information. Since the rationalistic interpretation of Nash equilibrium is based on the assumption that the players know each others’ preferences, no methods had been available for analyzing games with incomplete information, despite the fact that such games best reflect many strategic interactions in the real world.
This situation changed radically in 1967-68 when John Harsanyi published three articles entitled Games with Incomplete Information Played by Bayesian Players, (Management Science 14, 159-82, 320-34 and 486-502). Harsanyi’s approach to games with incomplete information may be viewed as the foundation for nearly all economic analysis involving information, regardless of whether it is asymmetric, completely private or public.
Harsanyi postulated that every player is one of several “types”, where each type corresponds to a set of possible preferences for the player and a (subjective) probability distribution over the other players’ types. Every player in a game with incomplete information chooses a strategy for each of his types. Under a consistency requirement on the players’ probability distributions, Harsanyi showed that for every game with incomplete information, there is an equivalent game with complete information. In the jargon of game theory, he thereby transformed games with incomplete information into games with imperfect information. Such games can be handled with standard methods.
An example of a situation with incomplete information is when private firms and financial markets do not exactly know the preferences of the central bank regarding the tradeoff between inflation and unemployment. The central bank’s policy for future interest rates is therefore unknown. The interactions between the formation of expectations and the policy of the central bank can be analyzed using the technique introduced by Harsanyi. In the most simple case, the central bank can be of two types, with adherent probabilities: Either it is oriented towards fighting inflation and thus prepared to pursue a restrictive policy with high rates, or it will try to combat unemployment by means of lower rates. Another example where similar methods can be applied is regulation of a monopoly firm. What regulatory or contractual solution will produce a desirable outcome when the regulator does not have perfect knowledge about the firm’s costs?
Other Contributions of the Laureates
In addition to his contributions to non-cooperative game theory, John Nash has developed a basic solution for cooperative games, usually referred to as Nash’s bargaining solution, which has been applied extensively in different branches of economic theory. He also initiated a project that subsequently came to be called the Nash program, a research program designed to base cooperative game theory on results from non-cooperative game theory. In addition to his prizewinning achievements, Reinhard Selten has contributed powerful new insights regarding evolutionary games and experimental game theory. John Harsanyi has also made significant contributions to the foundations of welfare economics and to the area on the boundary between economics and moral philosophy. Harsanyi and Selten have worked closely together for more than 20 years, sometimes in direct collaboration.
Through their contributions to equilibrium analysis in non-cooperative game theory, the three laureates constitute a natural combination: Nash provided the foundations for the analysis, while Selten developed it with respect to dynamics, and Harsanyi with respect to incomplete information.