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

This article uses the 2004 Presidential Election to explain game theory.  It uses Florida, Ohio, and Pennsylvania, three states that are thought to be up for grabs.  To simplify the situation, the model concedes Pennsylvania to Kerry.  Bush then must win both Florida and Ohio to win the election.  However, he only has enough time to visit one of the states before the election.  Which then must he choose?  For example Bush has a 30% chance of winning Ohio and a 70% chance of winning Florida.  Visiting a state will increase his chances of winning by 10%.  If Bush and Kerry both visit the same state, Bush’s chance for Ohio remains at 30%, and Florida at 70%, giving him a 21% chance at winning the election (0.3×0.7=21%).  If Bush visits Ohio and Kerry goes to Florida, Bush has a 24% chance of winning the election (0.4 in Ohio x 0.6 in Florida=24%.)  Finally, if Bush visits Florida and Kerry visits Ohio, Bush’s chance of winning is only 16% (0.2 in Ohio x 0.8 in Florida).  Because Bush is better off visiting Ohio no matter what Kerry does, visiting Ohio is Bush’s dominant strategy.  It, therefore, should not matter which state Kerry chooses to visit, Bush should always visit Ohio.  However, since Kerry’s dominant strategy is also to visit Ohio, he will also always visit Ohio and their efforts will cancel each other out. This combination of actions in which both Bush and Kerry would be satisfied with their decision even if they knew their opponent’s strategy, is referred to as the Nash equilibrium.

If the numbers are changed so that Bush has a 50-50 chance in each state, Bush prefers to visit the same state as Kerry and Kerry prefers to visit a different state than Bush. At first there seems to be no Nash Equilibrium. There is, however, a Nash Equilibrium in this situation too. If both candidates relied on a coin flip to make their decision, both players’ actions would be random and unpredictable. Bush has a 50-50 chance of ending up in the same state as Kerry and a 0.245 (0.5×0.25+ 0.5×0.24) chance of winning. Since the same statistics apply to Kerry, a Nash Equilibrium is once again established. When chance determines a player’s best strategy, it is a mixed strategy. In order for mixed strategies to work, the players’ actions must be unpredictable and simultaneous so that one player can’t wait for another’s actions.

Though this model is helpful, it is not foolproof. Bush’s predictions for winning each state are only based on estimations from polls. The model also excludes the fact that each state has a different amount of electoral votes, which leads some states to be much more important than others.  In addition, the other 47 states have been completely ignored in this analysis.  Though it has some faults, game theory is an excellent tool to use to aid in the Presidential Election process.

 Commentary

In his article Ellenberg says, “The key is that rational behavior tends to be predictable, and in a game of strategy, predictability will leave you with a decided disadvantage.” Unless there is a dominant strategy in which one action is the best action no matter what the opponent’s action is, it is important to remain unpredictable. For example if it is predictable that Kerry will put his efforts into Ohio, Bush can form the best reaction strategy based on this knowledge. Because it is so important to have unpredictable actions, the article suggests that when there is no dominant strategy it is best to leave a player’s actions to chance so that the actions are truly unpredictable. This is called a mixed strategy. To achieve a mixed strategy both candidates would have to flip a coin on the election eve to decide which state to campaign in. The actual coin flip makes the candidates’ actions random and therefore unpredictable and the fact that the coin flip is done on election eve ensures that the information remains unknown and therefore unpredictable. This inability to predict the opponent’s actions makes it impossible to properly strategize.

 Real-World Application of Game Theory

One real life application of game theory is seen in the prisoner’s dilemma. The prisoner’s dilemma depicts prison terms for when two guilty people convicted of a crime are separated and asked to sell each other out. If both people tell that the other one is guilty, then both people get 3 years in prison. If one person tells on the other, but the other one does not, then the one who told goes free and the other gets 5 years in prison. However, if both parties stay quiet then they both get 1 year in prison. What is best for the prisoner’s individually is to cooperate with law officials, in which case if the other person does not talk then the prisoner is let free. However if the other prisoner also cooperates it would have been better for the prisoners individually and as a group to stay quiet. The incentive to cooperate therefore arises out of both the fear that the other prisoner has cooperated and the draw to cooperate and hypothetically be let free.

The idea of using the prisoner’s dilemma to draw out admissions of guilt from prisoners is used in many criminal cases in today’s time. This usually takes place in the form of plea bargains. A famous plea bargain example involved a group of 5 men in La Jolla convicted of murder in 2007 (Perry 1). Instead of sticking to the scenario of all remaining silent and trying to claim innocence, these men all pled guilty, lessening their sentences from murder to manslaughter. They all proved unwilling to gamble possible murder charges should some members of the gang admit to the crime. Often in cases like these if the defendants are unwilling to admit guilt and the prosecution has a weak case against them, the prosecution will offer very attractive plea bargains to members of the group in order to obtain incriminating information against the other defendants.

Posted by Kelly, Kayla and Chris (Section 1)

Resources:

Perry, Tony. “Four Men Accept Plea Bargains in Killing of La Jolla Pro Surfer.” Los Angeles Times. Los Angeles Times, 28 June 2008. Web. 23 Apr. 2012. <http://articles.latimes.com/2008/jun/28/local/me-surfer28>.

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