Is game theory empirically valid?

Game Theory: Strategic Action in Social Interactions

Transcript

1 Introduction A group of ten people goes to a restaurant for dinner. According to the agreement, the bill is shared and divided by the number of heads. Less civilized people will develop a greater appetite in this case than with individual settlement. Each additional expenditure is subsidized to nine tenths by the other table companions. If all guests follow this logic, the bill will be inflated to the detriment of all. The same principle can be observed with many problems: the yawning empty refrigerator in a shared apartment, the deforestation of forests, the overfishing of waters and the overexploitation of resources in general are examples. Individual rationality does not always increase the general good as in Adam Smith's theory, but often leads to collective self-harm. Social psychologists speak of social dilemmas or social traps, economists of collective goods problems, sociologists and political scientists of the problem of social order. There are many varieties of social traps. And there are antidotes: institutions, social norms, social sanctions and, under certain conditions, the self-organization of cooperation. The respective structure of a social dilemma can be precisely analyzed using the methods and models of game theory. Game theory does not deal exclusively with social dilemmas, but we encounter them very often in the relevant studies. Game theory analysis enables solutions to be found, and experimental game theory allows hypotheses resulting from the models to be empirically investigated. Game Theory: Strategic Action in Social Interactions Even if many applications of game theory refer to interactions between two or a few people, the models do not only deal with the micro-world of small groups. International conflicts, arms race, the exploitation of scarce environmental resources, trade relationships and cartels, the collapse of financial markets, the exploration of the possibility of social cooperation among selfish actors, exchange relationships, the emergence of social norms and conventions, the production of collective goods, trust and fraud in electronic markets , the credibility of announced threats and sanctions, and many other topics can be dealt with in more detail with game theory models. Often the theories lead to new and surprising predictions. The mightiest is not always the strongest, and under certain conditions the weak can succeed in putting the mighty in their place or even exempting them. The prerequisite for game theory models is always that the situation is strategic, that the fates of the actors are mutually linked, so to speak. Threats, bluff, asymmetrical information (only the players know their true preferences, not the other players), honest and false signals are the salt of refined game theory models. Hypotheses about the consequences of social interactions are not always obvious, and it is not uncommon for obvious hypotheses to turn out to be false. 2

2 It can be shown that 1. that more transparency in markets can result in consumers being more likely to be exploited by cartels through higher prices (Chapter 1). 2. The violation of social norms is also to be expected if the norm breaks are recognized with certainty and the sanctions outweigh the gain from the norm violation. The critical condition is the credibility of the execution of sanctions threats, in the language of game theory the question of the sub-game perfection of an equilibrium (Chapter 2). 3. Paradoxically, an additional relief road can lead to the fact that the driving time of drivers increases, although the number of cars “in the system” remains unchanged (Chapter 6). 4. Even among completely self-interested actors, cooperation can arise in dilemma situations if they repeatedly interact with one another. A notable historical example is the cooperation of enemy soldiers in World War I (Chapter 7). 5. In the case of trust problems (e.g. when renting out an apartment), the option of paying a deposit can have a desirable side effect: The extent of social discrimination is reduced (Chapter 9). The five examples relate to the results of investigations with game theory models. Game theory helps to analyze interaction structures and to work out conditions for the validity of the hypotheses. In this respect, game theory is an important tool for theory development in the social sciences. The examples also indicate what game theory is about: Game theory deals with decisions in situations of strategic interdependence. When you are spoiled for choice in the supermarket, choosing a yoghurt cup from a hundred different products (“Decision under Safety”, Chapter 4), or when you are wondering whether it might be better to put an umbrella under the umbrella for a walk when the clouds are gathering to clamp the arm («Decision under Risk», Chapter 4), then it is undoubtedly a decision problem. However, these do not fall into the field of game theory. The reason is that the outcome of the decision does not depend on the decisions of other actors. If you leave the umbrella at home and it rains, you will get wet, no matter how many other walkers have decided to take the umbrella with them. In the restaurant example mentioned at the beginning, however, the amount of the bill varies depending on the behavior of the table companions. The greater the number of uncooperative guests, the higher the bill. The actions of others have an impact on the outcome I get from my action. In a story by Arthur Conan Doyle, Sherlock Holmes flees his adversary Moriarty by train to Dover. On the way there is a stop in Canterbury. Moriarty follows Holmes with a special train. Should Holmes be caught by Moriarty in Dover, it should turn out badly for him. So Holmes is considering getting off the train in Canterbury. Of course, he knows Moriarty will consider this possibility as well. Moriarty will therefore consider stopping the special train in Canterbury. But then it would be better to continue the journey to Dover. But what if Moriarty also understands this conclusion? And so the consideration continues with an infinite recourse. This example (it comes from Oskar Morgenstern, see chapter 5 in detail) illustrates the interdependence of actions and the strategic 3

3 considerations of the actors. Each player has to look at the situation through the eyes of the other players, so to speak, and then agree on his own strategy. Game theory also has a solution for the infinite regress (Chapter 5). Tasks and goals of game theory The focus of all social sciences is on social interactions. They form the basis for the formulation of theories and hypotheses about social relationships and processes on the micro and macro level of society. Game theory is, so to speak, the mathematics of social interactions. It provides the methods and formal models to precisely describe social interactions. So there are z. B. Many different types of social dilemmas. Depending on the structure of a “social trap”, the solution will also turn out differently. Often there are only small differences that have a significant effect on the structure of the action. For example, the game of chick emerges from the prisoner's dilemma by interchanging only two neighboring preference values ​​(Chapter 1). Or a small change in the information area turns a simultaneous game into a sequential one. This fundamentally changes the structure of the game and its solution (Chapter 2). With the verbal colloquial or technical language, the different structures would hardly be noticed. Only with finely tuned instruments, with a formal language of social interactions, is it possible to recognize the differentiations. In the words of Gintis (2000), "game theory is a universal language for standardizing the social and behavioral sciences". Even if one does not make any demands on the prognostic power of game theory models, the analysis and classification of different interaction structures is an important achievement of game theory. Game theory also contributes to the formation of theories in the social sciences. Here, too, it is not the descriptive content, the predictive power of game theory models alone that is decisive. When an interaction structure is formalized with a game theory model, one first receives a kind of “benchmark”, a point of reference that records: How would the actors decide if they obeyed strict criteria of rationality? If one observes systematic deviations in practice (which often, but by no means always occurs), the first research question arises: How does it come about that actors systematically violate the assumptions of rationality? Or maybe they act “rationally”, but have different goals (preferences) than the model claims? Binmore (1992) argues that the following minimum condition must be met for game theory models to make accurate predictions: First, the game structure must be simple. Second, the actors must have experience with the structure of the game. And third, the incentives have to be high enough so that optimal strategies are worthwhile. However, the minimum requirements are not a guarantee. Even if all three prerequisites are met, discrepancies between game theory hypotheses and the behavior of test subjects can be demonstrated in experiments. Possibly the “rationality gap” or the misjudgment of preferences in the model can be remedied by additional assumptions, so that one can come to predictable theories. This is the research program of the 4

4 tensor-oriented (behavioral) game theory, which will be discussed in Chapter 10. In evolutionary game theory (Chapter 8) the problem is different. It is of no interest here whether the actors act purposefully and strictly rationally. If there is competition between different behavioral patterns, rational solutions will develop through evolution. Organizational structures and strategies of firms, cultural patterns and institutions develop as successful models are mimicked and spread, while unsuccessful patterns disappear (Chapters 4 and 8). The normative game theory is completely untouched by the dispute over the explanatory power of descriptive game theory. The aim here is not to formulate empirical hypotheses and behavioral predictions. Rather, it is about finding optimal decisions in a given interaction structure. Normative theory poses the question of how a rational actor should act if all other actors are also based on rational decision-making criteria. Another task of game theory is the analysis and planning of institutions. An institution is understood to be a permanent, predictable incentive mechanism, e.g. B. a wage system, the product liability of companies, laws on the equalization of benefits after a divorce or the rules for CO2 emission certificates in the EU. Nobody will deny that such institutions have an impact on behavior. A simple example is the rule: A divides the cake, B chooses. Two rational and completely selfish actors with an appetite for cake will always come to a fair division in this way. According to the Old Testament (first book of kings, 3, 16-18), King Solomon invented an institution to obtain a true testimony in a court case. Two women are fighting over a child in court; both claim motherhood. Solomon proposes that the child be divided among the plaintiffs. This reveals which of the two women is the real mother. Of course, she refuses to agree to the cruel plan and prefers to leave her child to the evil opponent. Wise Solomon, who foresaw the mother's behavior, can now award her the child. Was he really that wise? In any case, the process cannot be repeated. Institutions that game theorists have analyzed in detail are auction processes. When auctioning telephone frequencies, game theory, with its suggestions, successes and failures, was briefly in the spotlight (Chapter 1). If institutions are planned using game theory means, one speaks of «mechanism design». Institutional analysis is important in law. In Anglo-Saxon law, the interdisciplinary field of “Law and Economics” is cultivated. A core of the research program is the analysis of legal institutions, also using game theory. In several chapters of this book, especially in chapters 6 and 7, problems of social cooperation are dealt with. The terms “cooperative” and “non-cooperative” game theory should not be confused with cooperative actions. The cooperative game theory deals with problems of negotiation, coalition building and the allocation of resources. Closed contracts, so the assumption, are valid and enforceable. Social dilemmas, e.g. B. the prisoner's dilemma or the "chick game" (Chapter 1) are not a problem in cooperative theory. Players will negotiate and one for every 5

5 parties involved achieve a favorable contract solution. However, the solution is based on the assumption of “binding contracts”. In non-cooperative game theory, on the other hand, an institution that guarantees enforceable contracts is not required. If contracts are fulfilled and cooperative solutions are achieved, this must be explained without already assuming that the contract is in compliance. The non-cooperative theory is therefore more fundamental and requires fewer requirements. Nevertheless, cooperative game theory has also achieved important results. Today it has fallen behind, perhaps wrongly, compared to the non-cooperative theory. In addition, there is a research program to standardize the theory and to explain the results of cooperative game theory as "non-cooperative". In this book we will only deal with non-cooperative game theory. Applications in the social sciences Institutions, social interactions, social norms and sanctions are core concepts in sociology. Dealing with the problem of social order has also had a long tradition since Thomas Hobbes «Leviathan» with the classic sociological works of Émile Durkheim, Talcott Parsons and others in sociology and political science. With the discussion about “social capital”, studies on the problem of trust have gained in importance. These theoretically and empirically important research questions are examined today using game theory methods. The competition between parties, electoral processes, problems of the common good and conflicts between states are topics in political science, the processing of which can benefit from game theory methods. In economics, cartels, the overuse of resources, financial markets and auction processes are examples of application areas for game theory. Even more: since the microeconomic and game theory turn in economics, game theory has actually become the formal basis of economic modeling. Many of the research topics mentioned, in particular the investigation of the possibilities of cooperation in social dilemmas, are of interest to all social science disciplines. That is why game theory, such as statistics, probability theory and knowledge of empirical research methods, are part of the basic training in the social sciences. Game theory is not just a method like statistics. It is also a tool for theorizing. Anyone who deals with theories in the social sciences today, be it in political science, sociology or economics, cannot avoid basic knowledge of game theory. Development of game theory Game theory is used in social sciences, law, biology and even computer science (e.g. interacting computers). But like statistics and probability theory, it is a mathematical discipline. The first game theory analyzes were presented by the mathematicians Ernst Zermelo and Emile Borel. In 1913, Zermelo proved in his work "On an application of set theory to the theory of the chess game" that zero-sum games with a finite number of strategies and perfect information 6

6 have a saddle point (Chapter 5). Chess, mill, checkers, etc. fall into this category. Translated into German, Zermelo's proof of existence says that there is an optimal strategy in chess. Fortunately, no one has been able to find out what it is. Zermelo's theorem was extended by John von Neumann (1928) with the Minimax theorem. The von Neumann name is closely linked to the early development of game theory. Together with the economist Oskar Morgenstern, von Neumann published the first fundamental work on game theory in 1944, "Theory of Games and Economic Behavior" (2nd edition 1947). Zero-sum games and cooperative game theory occupy a preferred position in the work. Von Neumann is considered the mathematics genius of the 20th century.He allegedly invented game theory because he was interested in poker and because he also wanted to analytically deal with conflicts with his wife more precisely (Poundstone 1992) 1. As a cold warrior and military strategist, he also had a dark side. It is claimed that von Neumann was the model for the figure of Dr. Strangelove in the film by Stanley Kubrick with the German title «Dr. Strange or How I Learned to Love the Bomb ». Von Neumann's Minimax theorem relates to zero-sum games. Another generalization goes back to the mathematician John F. Nash. Nash initially defines the concept of equilibrium for all types of games, including non-zero-sum games. He then presents the proof of existence that in every game with a finite number of strategies there is at least one "Nash equilibrium" (1950, 1951). Furthermore, Nash made significant contributions to cooperative game theory. Nash developed schizophrenia in the late 1950s and recovered from the disease 25 years later. His life was described in the brilliant biography "A Beautiful Mind" by Nasar (1998) and filmed under the same title. With Nash's generalization, the basis of the theory is significantly expanded. Most conflicts in social, political and economic life do not have the radically antagonistic character of zero-sum games. However, with enlargement new problems arise. If several equilibria exist, the solution is no longer unique as in the world of zero-sum games. You need additional criteria to weed out less useful equilibria. One of the most important suggestions, the criterion of "partial game perfection" (Chapter 2), comes from the mathematician and economist Reinhard Selten (1965). In a further step, the philosopher and economist John C. Harsanyi succeeded in developing a strictly rational procedure for the treatment of games with imperfect information (Chapter 9). New applications of game theory were seen in biology in the early 1970s. The players here are organisms that carry out genetically fixed strategies. For the evolutionary analysis, however, the concept of the Nash equilibrium had to be modified. Maynard-Smiths and Price (1974) idea of ​​"evolutionarily stable equilibrium" (ESS) was the beginning of a game theory-inspired research program in biology, with evolutionary game theory (Chapter 8) retrospectively stimulating research and concepts in the social sciences. In parallel with the theoretical innovations, shortly after they appeared, von Neumann's wife, according to Poundstone, indicated to her husband that she would only be interested in game theory if it contained an elephant. In von Neumann and Morgenstern (1947) one finds an elephant hidden in a set diagram on page 64. 7th

7 of the classical work of Neumanns and Morgensterns began to use experimental methods to investigate how people actually behave in conflict situations that can be described in game theory. Mainly social psychologists were involved in the company. One of the outstanding works is the study by Anatol Rapoport and Albert M. Chammah (1965) on cooperation in the repeated prisoner's dilemma (Chapter 7). The "tit-for-tat strategy", which later became widely known in the computer simulations of Robert Axelrod (1987), emerged from the research of Rapoport and Chammah. In economics, apart from pioneers such as Reinhard Selten in Germany or Vernon Smith in the USA, experimental game theory research only gained momentum in the 1980s. Since then, there has been an almost exponential growth in research on experimental game theory in economics (Chapter 10). In 1994, John Nash, Reinhard Selten and John Harsanyi received the Nobel Prize for Economics for their scientific discoveries, the first prize awarded for work on game theory (and the first and, to date, only Nobel Prize for Economics awarded to a German scientist, Reinhard Selten has been). In the following years, research on game theory was awarded four other prizes: 2001 George A. Akerlof, Michael Spence, Joseph E. Stiglitz for work on asymmetric information, 2002 Daniel Kahnemann for psychological-economic decision research and Vernon L. Smith for experimental economic research, 2005 Robert J. Aumann for research especially on repeated games and Thomas C. Schelling for work among others on the role of self-commitment in resolving conflicts, 2007 Leonard Hurwicz, Eric S. Maskin and Roger B. Myerson for work on “mechanism design”. Structure of the book This book introduces the basics of modern, non-cooperative game theory using numerous examples from sociology, politics and economics. The first two chapters explain central concepts such as the normal form and extensive form of games, the concept of strategy, the Nash equilibrium and the criterion of sub-game perfection. We start with the normal form because this representation makes the concept of Nash equilibrium easy to understand. In addition, the well-known 2 2 games with “mixed motifs” are explained using examples. In the following chapter, the extensive form is introduced, the concept of strategy is explained in more detail and subgame-perfect strategies are discussed. In Chapter 3 the concepts are applied to the trust problem and the problem of social norms and sanctions. The role of institutions is discussed and there is a brief discussion of the micro-macro problem from the point of view of game theory. Chapter 4 explains principles of decision theory and deals with basic concepts such as rational action, evolution and measurement of utility. This has laid the first foundations. The following five chapters cover specific types of games. We try to keep the notation simple in the representation. As a rule, algebraic knowledge from school mathematics is sufficient to understand the 8th grade

8 following chapter. Chapter 5 is about zero-sum games. The concept of mixed strategy is introduced using examples. Chapter 6 deals with different types of social dilemmas and Chapter 7 deals with solving social dilemmas through repetitive games. Robert Axelrod's computer tournament is the starting point here, with the historical case study of “living and letting live” in the First World War being discussed. “Shadow of the future”, discounting, some of the theorems that follow from it, the folk theorem and a critique of the results achieved by Axelrod are further topics of the chapter. The evolutionary game theory is the subject of Chapter 8. The concept of evolutionarily stable equilibrium (ESS) is explained using the examples of the "hawk-pigeon game" and the "wear and tear". Using different variants of trust games, Chapter 9 introduces the analysis of games with incomplete information, signal games and “Bayesian updating”. The final chapter is devoted to the methodology of experimental game theory, explains selected results on altruism, reciprocity and social norms and outlines a model of behavioral game theory to explain the findings. 9