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A Beautiful Mind Game

As a committee, we’ve had some rather passionate conversations on how individuals, especially politicians, make decisions based on their available information. In our dire effort to understand the mechanism of strategic decision-making, many of such ideas have been articulated into articles that represent some of the most interesting reads of the past academic year of our magazine. The contract theory, Trump’s strategic reaction, and even the game of chess were added to our list of game-theory related articles. However, this article would attempt to show a distinct perspective of game theory: the focus will be rather on its chronological development. So, let us travel back in time to see how game theory has established itself as being one of the most discussed (and somewhat controversial) topics of modern-day sciences.

Before we kick off our journey, I put forward a slight cautionary note to the part of our audience who is not fully aware of the comprehensive interpretation of economic concepts. Game theory, as many would understand correctly, is the strategic decision-making process of a “game”, a systematic description of a strategic situation. However, to be qualified as a game, that situation needs to have multi-party interaction, and their respective decisions directly affect the consequences of each individual in the game. This is very different to our casual (independent) decision-making process that we make every day, which rather seems to have a negligible influence on others.

Historical recordings before the Industrial Revolution practically do not serve as a reliable indicator to determine how long game theory has been around, nor do they reveal any explicit evidence whether humans had recognized and formally termed it in a way that is similar to “game theory” now. Fragments of game-theoretic analytical ways of thinking can be found in readings such as the Bible, scholar works during Greek or Roman history, in addition to myriads of its applications during war times throughout history. But one might suggest that the earliest findings on game theory, at least applied to decision-making in maximizing one self’s returns, traces back to the Babylonian civilization. The problem is as follows:

Suppose that there is a wealthy merchant who earns a fortune from his expertise in trading. Suddenly, he passes away and leaves behind his three wives, who are the direct inheritors of his assets. Assuming that every asset is worth 100 during this period, how shall these assets be divided to the creditors? The scholars at the time provided a rather strange and contentious answer to the problem: for example, if the accumulated value of his assets is 100 (meaning 1 unit), then the asset is borne equally by the wives (each has to pay 33.3); if their value is determined at 200, the first wife would only be liable for one-fourth of the asset, and the rest is divided equally to the other two (meaning that the first wife is liable for 50, the latter two have an equal obligations of 75); if the value is at 300, the division is even more skewed: the first wife is still liable for 50, but the second wife now owns  for 100, so that leaves exactly half of the asset attributing to the third wife (150). Why is this the case and what is the rationale behind it?

It might sound a bit confusing at first, but the answer to this problem is quite simple. If we leave out other factors such as cultural attitudes or any social custom at the time, we could arrive at a very logical answer. Here, we will look at the case of three assets. In this case, we further assume that the man marries the first wife after he purchases his first asset, the second asset comes after his second wife, and so on. So, the first wife would claim the first 100 of the asset, the second wife would contest her rights to two assets at 200, and the final wife claimed all of the available assets at 300. Here’s how the mystery unfolds.

We divide the situation into three two-player pairs, so (100,300), (100,200) and (200,300) respectively. Depending on the claims that are made by the wives, the officials would decide on how much of the asset each wife would be designated to have. Firstly, they took the average of the two values of the claims. Take the pair (100,300) for example, so the average of the two values is 200. Since the first 100 are contested by both parties, they would be equally shared, so each would receive 50. For the remaining 100, only the third wife has the claim on the asset, so all 100 would be awarded to the third wife. Ultimately, the first wife would get 50 and the third wife would get 150. In a similar manner, you could do it for the remaining two pairs and you would be able to find that the amount that the second wife could claim is 100. Surprisingly, not only the results are actually fair (each wife would be granted half of their claims), but they are also consistent between pairs! There are also various extensions to the game, such that in each case, there is always a desirable outcome (albeit much more complicated to mention here). This solution to this problem is uncovered by Robert Aumann (1982) as he formally named the method “equal division of contested sum”.

The game with a two-player interaction was not conceptualized until the 18th century, when Francis Waldegrave provided a framework, which is now referred to as “minimax strategy”,  to decompose the problem of a card game called le Her. The problem is described as a situation where two separate players are given a card by the dealer from a 52-card deck; each player is allowed to exchange one card with the dealer. The winner of the game will be determined by the level of the card, and whoever keeps the card with the higher level (King is determined as high and Ace is deemed as low) is the ultimate winner of the game. With further discussions with other scholars at the time, Waldegrave concluded in his work the game has a mixed-strategy equilibrium (where players randomize their strategies by assigning a probability in each case).

The strategic decision-making between humans is also deeply enclosed in one of the most popular economic belief at that time. Adam Smith, with his analogy of the “invisible hand”, strongly advocated that if people act in their self-interest so as to increase his own benefits as much as possible, then society would benefit as a whole. We can actually interpret his idea in game theory terms: maximizing individual payoffs would eventually lead to the socially optimal outcome for every player involved. Although so far this idea has not been true empirically, this ideology was successful in the way that it sparked debates among academics at the time.

It was not until 1928 when von Neumann, who had numerous academic contributions in mathematics, physics, computer science, and of course, economics, published a highly technical paper on deciphering the minimax strategy game. However, he lost his interest in game theory for a while and rather chose to devote himself to mathematics. Fortunately, Morgenstern, an economic professor originally taught at the University of Vienna, and then was appointed to lecture for three years at the Princeton University, had an encounter with von Neumann and was able to renew his interest in game theory. Morgenstern and von Neumann later worked extensively for their new research, “Theory of Games and Economic Behavior” (1944), which attempted to form a compelling connection between these subjects. The publication focused on the most primitive aspect of economics: the analysis of interaction between buyers and sellers. The welcoming success of the work in the next decades could be seen as a revolutionary step towards making economics as a true matter of concern for academic disciplines, which is comparable to that of Newton’s Laws of Motion for physics.

However, there was one major problem with their work. While taking into account that an individual would maximize their strategy regardless of what others do in a non-cooperative game, both authors did not give any indication to how cooperative behavior would alternate the outcome of a game. John Nash, who was also a graduate from Princeton, decided to publish his work on the bargaining position and asserted that there exists a possible scenario where every player participating in the game benefits not at the expense of others. At the same time, he also continued researching into non-cooperative games. He later developed the most famous concept of game theory, “Nash equilibrium”, the state in which a player chooses the best possible outcome given that others are committed to one certain action, and that goes for every possible action for the remaining players in a non-cooperative game.

Soon after the game theory publications by John Nash, many academics (especially economic researchers and mathematicians) were deeply interested in developing advanced methods in understanding and interpreting economic behaviors. Alternative models of game theory such as the extensive game and games with repeated interaction were also developed during the period. For games with repeated interaction, the degree of patience of two players with respect to future periods of the game would potentially make the pursuit of self-interest of both players yielding optimal outcomes for the games. From these proposed models, related academic publications are put forward solutions to these games; some of which include the introduction of the concept of “subgame perfect equilibrium”, a refinement of the Nash equilibrium, and Bayesian games.

It comes to no surprise that game theory has an important position in everyday’s lives. Every strategic decision being conducted on a managerial level is based on game-theoretic analytical reasoning. The cases of the public good problem, auctions, or Cournot duopoly are typical economic examples of game theory that are familiar to most of our readers. 18 out of the last 25 Economic Nobel Prize award-winning publications are related to the principles of game theory. Game theory has also escaped far beyond the world of business and economics. Psychology, biology, or even computer science are only a few examples of how game theory disseminates to other academic disciplines. That is just to infer that game theory is far from fully discovered. As we attempt to acquire a seamless understanding of the human nature, game theory might just be the answer to all problems.


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