# Game Theory - An Austrian Invention

**by Rudolf Taschner**

The game called Ultimatum is one of the most trivial and bizarre games one can think of. It takes two players, who need not know one another and can even play in separate rooms, and one banker. A toss of the coin decides which of the two players goes first. Then the banker places 100 euros in coins on the table in front of the player who goes first and offers him to take a portion of it and give the rest to the other player. The catch is this: The other player can either accept or reject the offer. If accepted, the two players split the money in the amounts agreed upon and the game is over. If rejected, then both players lose, the money remains with the banker and the game is over.

What’s the purpose of this game that, when tried out on many players more than once, becomes very expensive for the banker? The answer? It proves that, from an economic perspective, man acts irrationally.

Among the many thousands of rounds of games played in the most diverse parts of the world, experience has revealed that beyond gender, religion, levels of education, or differences in age, the second player will accept the offer of the first player only when it does not deviate all too much from a fifty-fifty split. In other words, the second player is, generally speaking, willing to accept forty euros. But if the suggested split is felt to be unfair, as in the ratio of ninety to ten, then it will be abruptly rejected and taken to be almost as a form of humiliation.

Rudolf Taschner

A rational homo oeconomicus, however, would argue that the player who goes first should suggest to the other a split of ninety-nine euros for himself and one euro for the other, the reason being that one euro is more than no euro. That is, so thinks the homo oeconomicus - the other would rather treat himself to this small profit than end up totally empty handed. Far from it! Man is not a homo oeconomicus.

But what is he then? In order to answer this, one is compelled to turn to philosophy. If, however, one wishes to take a more modest position, asking questions about patterns of human behavior, then mathematics can help us.

All of this began some eighty years ago in Vienna. Karl Menger, son of the renowned economist, Carl Menger, wanted to study physics at the university but was cast under the spell of mathematics by a lecture given by the inspiring mathematician, Hans Hahn, on the meaning of the concept of a “curve.” If during the 1920s Austrian science was, on the one hand, a progressive and innovative force - (Hans Hahn, himself, belonged to the Vienna Circle) - the political context of the University of Vienna was overshadowed by rivalries of dull and unreflective students. Menger, however, was a man of finer sensibilities.

Motivated by the working style of his analytically-inclined mentor, he came up with the idea of studying human interaction as a mathematical object. Perhaps, he thought, one doesn’t necessarily make mistakes in ethical questions as easily as in scientific ones, but it might be possible to prove certain decisions as rational or irrational and, thus, create a basis for a rational approach to ethical problems. Menger worked these thoughts into a book entitled, Morality, Decision and Social Organization.

Few if any of his colleagues placed much value in this work. They felt that he who had achieved such groundbreaking findings in the Theory of Curves and Dimensions had undersold himself. On the other hand, the director of the Austrian Institute for Economic Research, Oskar Morgenstern, sensed that Menger’s mathematics of social relations might contain some important, hidden insights. Morgenstern fled from the Nazis and found his way to Princeton where he showed Menger’s work to John von Neumann, the mathematical guru at Princeton’s Institute for Advanced Study and, some few years later, the two of them published the book, Game Theory and Economic Behavior. This book was to bring about the birth of mathematical game theory.

The title conceals the fact that it deals with serious, applied mathematics. Initially, the military took an interest in it in order to better understand the dynamics of the Cold War and to assure an equilibrium of deterrence. Only later did economists become interested in it.

John von Neumann only looked at so-called “zero sum games” - what the one wins, the other has to lose - and was convinced that basically every game can be understood on the basis of zero sum games. John Nash, a young ambitious genius at Princeton at the time, went beyond zero sum games and built mathematical models of games whereby all of the participants can win something. This theory, already very abstract, became even more complex.

One example expressing the essence of human behavior is the famous prisoners’ dilemma. Again there are two players who are not allowed to communicate with one another during the game. Across from them sits the banker, who offers the option to each one of the two players to either put 100 euros into a pot or no money at all. He also promises to add 50% to the amount collected in the pot and to split the total amount 50-50 between the two players.

What should one do? If both players put 100 euros into the pot, there is a sum of 300 euros after the banker has added to the pot, which is then split between the two and each player has a net profit of 50 euros. But, the first player could be thinking that if he turns down the donation and the honest second player spends 100 euros, then there is a sum of 150 euros after the banker has added to the pot which is then split. The first player pockets the net profit of 75 euros - more than before - and the second player has retracted his loss of 25 euros. The first player thinks: what if the other player thinks like he does? Then both turn it down, give up on a possible profit and leave the game as losers. So, cooperation is still more beneficial than pure egoism. But how can one rely upon the cooperation of the other?

A rational homo oeconomicus, however, would argue that the player who goes first should suggest to the other a split of ninety-nine euros for himself and one euro for the other, the reason being that one euro is more than no euro. That is, so thinks the homo oeconomicus - the other would rather treat himself to this small profit than end up totally empty handed. Far from it! Man is not a homo oeconomicus.

But what is he then? In order to answer this, one is compelled to turn to philosophy. If, however, one wishes to take a more modest position, asking questions about patterns of human behavior, then mathematics can help us.

All of this began some eighty years ago in Vienna. Karl Menger, son of the renowned economist, Carl Menger, wanted to study physics at the university but was cast under the spell of mathematics by a lecture given by the inspiring mathematician, Hans Hahn, on the meaning of the concept of a “curve.” If during the 1920s Austrian science was, on the one hand, a progressive and innovative force - (Hans Hahn, himself, belonged to the Vienna Circle) - the political context of the University of Vienna was overshadowed by rivalries of dull and unreflective students. Menger, however, was a man of finer sensibilities.

Motivated by the working style of his analytically-inclined mentor, he came up with the idea of studying human interaction as a mathematical object. Perhaps, he thought, one doesn’t necessarily make mistakes in ethical questions as easily as in scientific ones, but it might be possible to prove certain decisions as rational or irrational and, thus, create a basis for a rational approach to ethical problems. Menger worked these thoughts into a book entitled, Morality, Decision and Social Organization.

Few if any of his colleagues placed much value in this work. They felt that he who had achieved such groundbreaking findings in the Theory of Curves and Dimensions had undersold himself. On the other hand, the director of the Austrian Institute for Economic Research, Oskar Morgenstern, sensed that Menger’s mathematics of social relations might contain some important, hidden insights. Morgenstern fled from the Nazis and found his way to Princeton where he showed Menger’s work to John von Neumann, the mathematical guru at Princeton’s Institute for Advanced Study and, some few years later, the two of them published the book, Game Theory and Economic Behavior. This book was to bring about the birth of mathematical game theory.

The title conceals the fact that it deals with serious, applied mathematics. Initially, the military took an interest in it in order to better understand the dynamics of the Cold War and to assure an equilibrium of deterrence. Only later did economists become interested in it.

John von Neumann only looked at so-called “zero sum games” - what the one wins, the other has to lose - and was convinced that basically every game can be understood on the basis of zero sum games. John Nash, a young ambitious genius at Princeton at the time, went beyond zero sum games and built mathematical models of games whereby all of the participants can win something. This theory, already very abstract, became even more complex.

One example expressing the essence of human behavior is the famous prisoners’ dilemma. Again there are two players who are not allowed to communicate with one another during the game. Across from them sits the banker, who offers the option to each one of the two players to either put 100 euros into a pot or no money at all. He also promises to add 50% to the amount collected in the pot and to split the total amount 50-50 between the two players.

What should one do? If both players put 100 euros into the pot, there is a sum of 300 euros after the banker has added to the pot, which is then split between the two and each player has a net profit of 50 euros. But, the first player could be thinking that if he turns down the donation and the honest second player spends 100 euros, then there is a sum of 150 euros after the banker has added to the pot which is then split. The first player pockets the net profit of 75 euros - more than before - and the second player has retracted his loss of 25 euros. The first player thinks: what if the other player thinks like he does? Then both turn it down, give up on a possible profit and leave the game as losers. So, cooperation is still more beneficial than pure egoism. But how can one rely upon the cooperation of the other?

Menger's famous sponge, which inspired Game Theory

This ‘Mickey Mouse game’ is only one out of a wealth of games that become more and more elaborate and sophisticated. It schematically characterizes patterns of human behavior which are more precise than the out-dated-sounding concepts of “self”- versus “common interest.” These are patterns which can be astonishingly well subjected to methods of higher mathematics.

After decades of standstill, the mathematics of game plans are being pursued again and primarily by researchers in Vienna. Karl Sigmund and his colleagues have achieved impressive results over the last few years. Mathematics which fifty years ago was considered the kind of science that mainly served astronomers, physicists and engineers, now includes also biological systems and no longer refrains from the question of the “right ethical” decision.

Rudolf Taschner, born in 1953 south of Vienna, is Professor at the Institute for Analysis and Scientific Computing at the Technical University in Vienna. Together with other colleagues, he runs the “math.space” project which was started in 2003 (http://math.space.or.at), located in the MuseumsQuartier in Vienna. He is author of numerous scientific publications and non-fiction. The Club of Education and Science Journalists of Austria chose Rudolf Taschner as “Scientist of the Year 2004.”