Weeks 1 and 2

WEEK 1

Our term began on March 4th with an introduction to our class, Advanced Analytical Techniques. On the first day, Professor Kris Wheaton gave a brief overview of what to expect throughout the term and what was expected of us as students. He passed out a list of dozens of methods/modifiers that we could choose from to study for the term. As I was perusing the list, most of the methods/modifiers had brief descriptions of each and nothing seemed to catch my attention. That is, until I reached game theory (which, by the way, did not have a description underneath).

All that I knew about game theory was what a zero-sum game is (to be detailed in a later post) or at least I heard of it before. Actually, that's not entirely true; I also knew that it was mathematically complex, but I was confident in my ability to grasp the concepts by the end of the term. But, since we did not have to decide what we wanted to study until the end of week 2 (we only had one class the first week), I decided I would do a little background research to make sure I actually wanted to study it, so I read the Wikipedia article.

As soon as I read it, I knew I had chosen the right subject. However, it was not because I was overly excited about game theory itself. On the contrary, I was excited about the possible range of topics that I could apply it to, which was the other requirement of the course. Specifically, I learned that game theory is used extensively in international relations, which just happens to be what my undergraduate background is in. Essentially, that's what happened during the first week; nothing out of the ordinary, thus, so far so good.

WEEK 2

This week turned out to be a wash, at least when it came to my research of game theory. You see, I'm also a member of the Competitive Intelligence club here in the department and we were quite fortunate to receive an invitation from SCIP (Society of Competitive Intelligence Professionals) to come volunteer at their annual conference in Washington, DC. The conference lasted the entire second week of Spring Term (March 8-12). I had high hopes of getting some work done after the business day concluded, but, frankly, that was wishful thinking on my part. Let me just say that even though the conference was great, it was one of the most tiring weeks I have ever had to go through. And trust me, if you're enrolled in Mercyhurst College Institute of Intelligence Studies, you experience many weeks of extreme fatigue! But, at least I was able to lock down game theory as my subject for the term. Beginning in week three, that's when the real fun started...

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Zero-Sum and Non-Zero Sum Games

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A zero-sum game is a game where the total payoffs are fixed, where one player’s gain is another player’s loss. An excellent example of this is a poker game where the players contribute money into a pot and someone “wins” it after all the bets are tallied and the winning hand is revealed. However, nobody actually “won” the pot of money, rather all the other players lost money that another player gained. The total amount of money available will never change in this game. The simplest form of a zero-sum game consists of two players and two strategies because a one-player game is not a game and only having one choice of strategies is not really a choice. The only way for a player to win is for the other player to lose, no cooperation is possible. That is, in order to be a true zero-sum game, the expected payoff for one player must equal the expected cost for the other player (if I gain $1, you must lose $1) for a sum of zero.

NON-ZERO SUM GAMES

A non-zero sum game where one player's gain does not necessarily mean the other player's loss; these games are actually more complex because there is usually more than one rational strategy. They are referred to as "non-zero sum" because the sum of the two player's payoffs does not always equal zero. Furthermore, non-zero sum games are not forced to be non-cooperative. That is, sometimes cooperation between the players leads to the optimal solution. The greatest example of a non-zero sum game is the prisoner's dilemma (I'll explain it in a later post). Essentially, each player is acting in his own self-interest, but that doesn't necessarily mean that the one player's gain is the other player's loss. Depending on how much the prisoner's cooperate with each other, that will determine each player's individual strategy. Furthermore, examples of non-zero sum games are more prevalent in real world situations, which makes them more useful to game theorists.

John von Neumann and The Minimax Principle

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JOHN VON NEUMANN

John von Neumann was a Hungarian-American mathematician who is widely regarded as the father of game theory (although, a Frenchman named Emile Borel published seven years before von Neumann). He was born in Budapest, Hungary in 1903 and possessed an eidetic memory which allowed him to excel in his studies. Von Neumann's inspiration for developing game theory came from poker, which he played rather unsuccessfully. He quickly realized that poker was not guided by probabilities alone and that one needed to play against the players, not against the cards. Furthermore, he wanted to formalize the notion of deception against the other players in the game. It was in his 1928 paper "Theory of Parlor Games" where he first broached the subject of game theory and proved the minimax theorem. In fact, von Neumann is quoted as saying, "As far as I can see, there could be no theory of games on these bases without that theorem...throughout the period in question I though there was nothing worth publishing until the 'minimax theorem' was proved." Once he proved the theorem, he collaborated with Oskar Morgenstern, an Austrian mathematician, to work on game theory. In 1944, they published their seminal work "Theory of Games and Economic Behavior" which is widely considered one of the most important texts of modern economic theory. To illustrate the point, the intended audience for the book was originally only economists, but it was applied to other subjects such as politics, sociology, psychology, and many others. From that point on, John von Neumann focused much of his work on war and politics. In fact, he would go on to hold several positions within the United States government such as the RAND Corporation and the Atomic Energy Commission under the Eisenhower administration.

John von Neumann would have been considered an extreme hawk by today's standards. He openly advocated preventive war against the Soviet Union. Another famous quote attributed to him, "If you say why not bomb them tomorrow, I say why not today? If you say today at 5 o'clock, I say why not one o'clock?" Of course, by the mid-1950s, the USSR had amassed enough of a nuclear arsenal to sustain a more than credible deterrent against a first strike by the United States.

Unfortunately, von Neumann was diagnosed with bone cancer in 1955. Amazingly, he continued his work as a consultant even while receiving debilitating chemotherapy. In fact, he moved his office to Walter Reed Army Medical Center and received frequent visits from the Secretary of Defense and his colleagues in the U.S. Air Force. John von Neumann succumbed to his cancer on February 8, 1957 and would be remembered as one of the greatest minds of the twentieth century.

His other significant accomplishments include his development of the digital computer, basing computer calculations on binary numbers, and having computers store programs in a coded form instead of punch cards.

THE MINIMAX PRINCIPLE

To quote from William Poundstone, "the minimax theorem proves that every finite, two-person, zero-sum game has a rational solution in the form of a pure or mixed strategy." In other words, when there is a precisely defined conflict between two people whose interests are completely opposite from one another, there is always a rational solution. Essentially, a player is trying to minimize his potential loss while maximizing his potential gain. The solution is rational because each player cannot expect to do any better given the nature of the conflict. The principle is explained using an example of two kids and a cake.

The first kid cuts the cake into two slices and the second kid decides which slice he wants. The cutter expects to get the smaller piece because the chooser will select the larger piece. By cutting the cake as evenly as possible, the cutter guarantees himself almost half the cake. But, if he cuts the cake unevenly, he knows he will get the much smaller piece. Therefore, in order for the cutter to minimize his opponent’s maximum payoff, he will cut the cake as evenly as possible. This is a very basic example of the minimax principle, however, its proof demonstrated that two rational players, whose interests are completely opposed, can agree on a rational course of action confident that the opponent will follow suit (by cutting the cake as evenly as possible, the cutter can be sure that the opponent will leave about half the cake).

Von Neumann thought that the minimax principle could be applied to n-person (two or more) games as well. Take a three-person game for example. The preferences of Player 1 and Player 2 are completely opposed to each other, but Player 1 and Player 3 share similar (or the same) preferences. In that case, they could form a coalition and defeat Player 2. By allying with each other, Players 1 and 3 essentially constitute one player and Player 2 is the other. Now you've got two players with completely opposing preferences (sound familiar?) It doesn't have to stop there; using the minimax principle you could develop n-person games ad infinitum, discover all the possible winning coalitions, and reduce them to zero-sum games. However, the one problem with this line of reasoning is that you're assuming that rational actors would determine the results of every possible coalition and join the one with the maximum payoff. What about games where cooperation is outlawed? As I'll discuss in a later post, John Nash discovered a way to arrive at an equilibrium even when players cannot cooperate with each other.

I researched what the actual minimax proof looks like and found this one from Brigham Young University to be the least challenging (I still have trouble understanding it though). If you're mathematically gifted, I suggest you read it, because it is essentially the foundational principle of game theory.

The Nash Equilibrium

From gametheory.net the definition of a Nash Equilibrium, "A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. For games in which players randomize (mixed strategies), the expected or average payoff must be at least as large as that obtainable by any other strategy.

Whereas John von Neumann studied cooperative games, John Nash studied noncooperative games. This definition is more easily understood by saying that Player 1 would be satisfied with his decision given that he knows what Player 2's decision is; neither player has any regrets. However, that does not necessarily mean that each player earned the maximum possible payoff. It just means that each player is willing to live with the outcome that was achieved. Actually, in most cases if one player realizes his maximum payoff, it probably is not the rational outcome (prisoner's dilemma is an excellent example of this). The reason for this, Nash argued, is that if either player has a reason to change strategy (and would if given the chance), then that outcome is unstable and irrational. And it makes sense, why would you let your opponent reach his maximum payoff while you don't? Obviously, you wouldn't, and Nash proved it. This built upon von Neumann's minimax principle that the solution to zero-sum games is the equilibrium point; Nash proved that non-zero-sum games have equilibrium points as well.

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