Game Theory

Discussion on “Game Theory for Swingers: What states should the candidates visit before Election Day?” by Jordan Ellenberg (Slate Magazine, October 2004).

 Summary

In the article “Game Theory for Swingers: What states should the candidates visit before Election Day?” author Jordan Ellenberg tackles the topic of game theory as it relates to a hypothetical political showdown between George Bush and John Kerry.  Ellenberg defines game theory as the division of mathematics that aids in solving the tactical problem that occurs when two players’ actions are dependent on one another.  The theory contains dominant solutions for each player where each player has an optimal outcome, independent of what the other player does.  In the article, this is painted as both Bush and Kerry deciding to visit Ohio to have their best chance at winning both Ohio and Florida.

In a case where both players’ optimal solutions are the same, Nash equilibrium occurs.  Nash equilibrium is defined as where, “B and K would each be satisfied with their current strategy, even if they knew in advance what their opponent’s strategy would be.”  Sometimes, it appears that two players cannot reach Nash equilibrium.  However, there is typically a more subtle way of arriving at this result: mixed strategy.

Mixed strategy occurs whenever chance is involved in determining a player’s action.  In the article, this is painted as both Bush and Kerry flipping a coin to determine which state to visit on their final day of campaigning.  This element of chance makes what would be a non-equilibrium outcome into Nash equilibrium.  However, mixed strategy only works when actors are acting simultaneously; if Kerry has a week to counter Bush’s chance-determined visits, he can execute an even more optimal pure strategy, thereby gaining an advantage over Bush.

This article on game theory relates to our classroom topics when it comes to competitive oligopolies.  A firm competing in a small market means that a firm’s choices have a large impact on other firms due to the interdependence present in these markets.  Game theory is extremely relevant in this situation, as it is the study of interdependent actions, and can aid in helping firms form rational expectations about what decisions other firms might make.

Commentary

At one point in the article, Ellenberg states, “The key is that rational behavior tends to be predictable, and in a game of strategy, predictability will leave you with a decided disadvantage.”  While predictability may be prized in certain economic pursuits, it can be detrimental for an opponent in a game of strategy. From an economist’s perspective, predictable behavior is the result of the “rational person”. The rational person will behave in such a way as to maximize their utility. As a result, all will logically proceed in this pursuit leading to a very predictable course of events. In the case of a game of strategy, this condition necessarily provides the opponents with each other’s decisions.

Consequently, rational behavior will ultimately lead to the predetermined demise of one opponent. However, this rationale simplifies a very complex issue. Recently, the importance of behavioral economics in finance has been recognized. It acknowledges the limited scope of the concept of the “rational person.” In our quest to solve economic questions with calculus and the manipulation of theoretical graphs, it is easy to neglect behavioral aspects which can be significant. The article references the game of rock, paper, scissors and the inevitable doom you face if your opponent can accurately guess your next move. In such a case, the most “random” act will be the most successful. While this is true, there are websites and groups devoted to the strategy of rock, paper, scissors. Some studies reference the behavioral implications associated with different “random” moves. Finally, it is prudent to recognize outlying factors in every situation. As proven in the real world, behavior cannot always be predicted because of human irrationality and biases. Not all situations can be analyzed by the delineation of a series of algebraic equations, as noted in the article.

Real World Application

Another real-world application of game theory is determining how to maximize one’s GPA going into final exams (which seems morbidly appropriate considering the current time of year).  The scenario here is that Jeff, a college student, has two exams left to take, and both are tomorrow.  Given his results on assignments and tests leading up to these finals, he is able to calculate his chances of getting certain grades in these two classes based on whether he splits up his study time evenly or spends an extra couple of hours on one of these two subjects.  However, since these classes’ grades are based on a curve, the time allocation decisions made by Jeff’s classmates are tied to his final grades, as well.  As a result, Jeff must decide on his appropriate amount of study time while keeping in mind that his final grades are dependent on the choices of his classmates, as well as his own choices.  This is where game theory comes into play, since Jeff must be able to predict his classmates’ decisions to the best of his abilities if he wants the best chance of getting an optimized GPA. 

Posted by Eric, Mary and Stefan (Section 4)

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