Game Theory

A thief and a guard. Is tonight a good time to commit a burglary? Or should the thief stay at home? Can the guard afford to sleep on the job or will that get him fired? What is most probable outcome and how do the decisions made by one party affect those of the other? This is both an amusing puzzle and complex mathematics –game theory. Whether in politics, sports or medicine, modeling problems as a strategic game helps in decision-making in a variety of fields. The players are individuals, companies or traders. The rules are the strictly laid out limiting parameters and options for action. Who is pursuing what interest? What are the players’ goals? The remarkable thing is that the decisions are not made primarily based on psychology but on the character of the game, on its rules. The success of any one player always depends on the decisions of the other players as well. To put it simply, game theory investigates what happens when each player tries to act more intelligently than the others and to maximize his or her own personal advantage. Let’s apply it to the scenario of the thief and the guard. How can a burglary best be prevented? By stricter punishments for the thief? The solutions are not determined on a game board but through a formal mathematical description. In fact, mathematics was the starting point of game theory. In 1944 John von Neumann and Oskar Morgenstern published their book “Theory of Games and Economic Behavior” and it was a pioneering milestone of game theory. In it von Neumann and Morgenstern were mainly concerned with zero-sum games for two players in which one player’s win is the other’s loss. But reality is often more complex. A more general solution was derived by the 21 year old graduate student John Nash. Decades later Nash received the Nobel Prize for the insights that he published in his dissertation, for example, that for all non-cooperative games with finite sets of strategies there exists at least one solution, the so-called Nash equilibrium. That remains at the heart of game theory today –so simple and yet so profound that the discussions about it are still continuing. First of all I’d like to pay my respects to John Nash. I mean he’s really the god of game theory. The parents of game theory were Von Neumann and Morgenstern, the god of game theory is John Nash, the high priest of game theory is Lloyd Shapley. So let me, you know, pay my respects to John Nash. A Nash equilibrium is established when each player chooses his or her own optimal strategy based on the expected actions of the others and has no reason to change it. That this is not necessarily the best result for all participants is shown by the “prisoners’ dilemma”, one of the best-known examples of game theory. Two gangsters have been arrested for a robbery, although there is no proof that they took part. The two separately receive the following offer. They can deny the deed or confess. If they both deny it, they will be put in jail for a year for possessing a weapon. If one of them confesses, he can turn state witness while the one who denies it will get a 5-year jail sentence. If both of them confess they both get four years in jail. According to the Nash equilibrium both would confess so as not to risk the highest sentence. As a consequence they go to jail for four years. If they had denied it, they would have only served one year. So the Nash equilibrium has its limitations. To conform more to reality and prevent economically unfeasible equilibrium situations Reinhard Selten and John Harsanyi refined Nash’s concept. The three of them shared the Nobel Prize in 1994. But why did it take so many years for game theory to attain the importance in economics and business practice that it has today? One reason may be its mathematical complexity. And even though von Neumann recognized its practical potential, the practitioners themselves long remained skeptical. It was only the successful use of game theory in the design of auctions to allocate radio frequencies, so-called spectrum auctions, that brought out a rethink. Humans do not always act as rationally as the concepts of classical economics assume. Feelings like pride, envy or self-esteem affect our decisions. The extension of game theory to include such human behavioral patterns won Robert Aumann and Thomas Schelling the 2005 Nobel Prize. At the center of their research are the concepts of conflict and cooperation. With the years numerous sub-fields of game theory have developed. One approach is based on desired goals. Here game theory investigates which incentives need to be created to make all players “voluntarily” act optimally, for example, so that an insurance company deals as fairly with its customers as they do with it: the company by offering the promised protection at a reasonable cost; the insured parties by not falsifying claims or withholding information. For their work in the area of market design Lloyd Shapely and Alvin Roth received the 2012 Nobel Prize. Among other work, Roth has studied markets in which regulation does not occur on the basis of price, for example organ transplants. A special case concerns the allocation of donated kidneys. In my work as a market designer one of the things that I’ve done is try to increase access to transplantation without having to change the law. And because you each have two kidneys, you can give a kidney to someone. But sometimes you’re healthy enough to give a kidney to someone you love but you can’t give them your kidney because it’s not a good match for them. And this is what opens up the possibility of exchange. And so a kidney exchange is an exchange that is an in-kind exchange. It’s like bringing wine to dinner instead of bringing money to dinner. Here is a simple kind of exchange where Donor 1 has blood type A and would like to give it to recipient one but can’t. And Donor 2 is blood type B and is incompatible with recipient two. But the blood type A-kidney can go into the patient who needs a blood type A- kidney. And the B-kidney can go into a B-patient. That’s a kidney exchange, that’s a way of getting two people transplants that they couldn’t otherwise have gotten. Game theory enables rational insights into the basic principles of social interaction and has therefore become indispensible for economics and social science. In 2014 Jean Tirole became the latest game theorist to win a Nobel Prize and he will certainly not be the last.

Abstract

Game theory enables rational insight into the basic principles of social interaction and has therefore become indispensable for economic and social sciences. Whether in politics, sports or medicine, modelling problems as a strategic game helps in decision-making in a variety of fields. With lecture snippets of Reinhard Selten, Robert Aumann and Alvin Roth, this Mini Lecture introduces to the mathematical beginnings of game theory, its socioscientific development and entrepreneurial integration.