Category Archives: Game Theory

QUESTIONS FOR CLASSROOM DISCUSSION ON GAME THEORY

The basis of game theory is that rational behavior tends to be predictable (most of the time). After all, we form expectations of how firms are likely to behave in cases of interdependence by assuming rationality (maximization of payoffs). However, can you think of a scenario where IRRATIONAL behavior may actually be predictable?

Time permitting, our classroom discussion will take place on Thursday, April 26.

Posted by Prof. C-S

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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)

Game Theory

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

The 2004 article by Jordan Ellenberg, an associate professor of mathematics at the University of Wisconsin, uses the real-world example of the Bush v. Kerry presidential race to how game theory applies to politics. Specifically, Jordan discusses the three swing states, Pennsylvania, Florida, and Ohio and how Bush and Kerry should go about campaigning in these states for the best chance to win. Each candidate’s decision depends in part on the other (interdependence, the key point to making this an example of game theory as we discussed in class). Kerry needs Pennsylvania to win, but focusing his assets there would leave Ohio and Florida up for grabs. Bush could go pick off Pennsylvania, but then it leaves Florida open for Kerry. Game theory helps solve this dilemma. Jordan simplifies the problem by narrowing it down to just the three states (rather than include the other 47), and sets the percent chance Bush has of winning each state’s electoral votes. The first scenario that Jordan discusses leaves Bush with a dominant strategy in which the option is clear: campaign in Ohio for the best chances to win. Because campaigning in Ohio is also the dominant strategy for Kerry, that result is a Nash equilibrium. The second scenario that Jordan discusses is a bit more complex, one with a much more subtle Nash equilibrium, this one dependent on chance. Because of that, the second scenario finds equilibrium through a mixed strategy (rather than a dominant one). Jordan sums up the example by reminding the reader that the true situation is not this simple, and that finding the actual percent chances of victory in the separate states is difficult. However, he successfully describes how game theory can be applied to politics. As previously mentioned, connections from this article to our class discussions come through Jordan’s mention and explanation of dominant strategies, mixed strategies, and the Nash equilibrium.

To look at game theory in a different light, let’s comment on a quote Jordan used in the article: “The key is that rational behavior tends to be predictable, and in a game of strategy, predictability will leave you with a decided disadvantage.” When interdependence determines what strategy should be used, predictability leads to a party’s demise. If a party’s actions are predictable, his opponent is allowed the opportunity to counteract his predictable move in order to strategically win. Randomness allows a “player” to conceal his strategy so that his opponent cannot absolutely anticipate his move and act accordingly. This decreases the opponent’s certainty of “winning.” Thus, it is strategically to a party’s advantage to act in a spontaneous function. Consider a shootout in soccer. A kicker can position the ball to the left, to the right, in the corner, on the ground, etc. If a kicker clearly positions his body in a way that is indicative of where he intends to position the ball, the goalie’s chance of blocking the ball is increased. However, if the kicker approaches the ball straight on or in a position that is not consistent with how he will actually place the ball, the goalie may not have any indication as to where he will kick it, or will be misled. Here, the kicker has used the element of surprise to his advantage.

While Jordan uses the party system as one real world example of game theory application, another real world application of game theory is the prisoner’s dilemma. This is a thought experiment essential to game theory. An example of the prisoner’s dilemma is this: prisoner A and prisoner B have both been arrested for robbing a bank. They have been placed in two separate cells. Each of the prisoners has been given two options; they can either confess or remain silent. They are both told the same thing by the police: “If you confess and your partner remains silent, then you will go free and your testimony will be used to make sure that your partner will serve 15 years in jail. On the other hand, if you remain silent and your partner confesses, then they will go free and you will serve 15 years. If you both confess, then you will get early parole after 10 years. If you both remain silent, you will each get 2 years or less.

There is a dilemma here because each of the prisoners is better off confessing than remaining silent. However, the result of them each remaining silent is much better than the result of the both confessing. This creates a dilemma and a clash between decision-making based on self-interest versus the best interest of the group. There are many conflicts of interest that arrive from this.

Posted by Maddie, Dalton, and Jed (Section 3)

Resources:

http://plato.stanford.edu/entries/prisoner-dilemma/

Game Theory Blog Entry

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

Summary

Jordan Ellenberg’s article, “Game Theory for Swingers,” uses the 2004 election to explain game theory. Ellenberg describes a scenario in which each candidate believes on the last day of the election that Pennsylvania, Florida, and Ohio are key to winning the election. Each candidate can improve their chances of winning one of the states by visiting it on the last day before the election, but this is dependent on what the other candidate does. In one scenario we assume that Kerry will win Pennsylvania and the election comes down to Ohio and Florida. We assume that Bush has a 30% chance of winning Ohio and a 70% of winning Florida. Both candidates can increase their chances of winning by 10% by visiting one state on the final day before the election (unless both candidates visit the same state, since the increases would cancel each other out). In this scenario, both candidates should visit Ohio and ignore the other candidate since it increases both of their odds regardless of what the other does. This is an example of a Nash Equilibrium.

Say Bush and Kerry both visit Ohio. Their odds of winning both Florida and Ohio remain the same at (.7)x(.3)=.21. If Bush visits Florida and Kerry visits Ohio, both candidates’ odds of winning both decrease to (.8)x(.2)=.16. If Kerry visits Florida and Bush visits Ohio, both candidates’ odds of winning both states increase to (.4)x(.6)=.24. As mentioned above, both candidates are better off visiting Ohio, no matter what the opponent does. However, it is not always this simple. If each candidate had a 50% chance of winning each Florida and Ohio and each candidate’s odds increased 10% by visiting a state, there would be no Nash Equilibrium because each candidate will be better off if he chooses the state the other candidate did not select.  There would be a Nash Equilibrium if the odds of which candidate will go to which state are considered, however. If each candidate flips a coin to determine which state to go to, each of their odds of winning will be (.5)x(.5)x(.5) (the coin flip times a 50% chance of winning each state if both candidates go to the same state) + (.5)x(.6)x(.4) (the coin flip times a 60% chance of winning one state and a 40% chance of winning the other state if the candidates choose different states)=.245, the Nash Equilibrium.

The element of chance from the coin flip reduces predictability and is called a mixed strategy. In order for a mixed strategy to work, both candidates must act simultaneously. It should be noted that this simulation ignores 47 states in the election and the differences in electoral votes from each state.

Class-Room Relevance

Game theory applies to managerial economics because producers are constantly trying to predict how competitors are going to price their goods. Especially in a monopolistic competition, where there is a possibility of long-run profit, it is important for companies to accurately predict how the competition will price substitutes for the goods it is producing so that it can swiftly react to changes in the market.

Commentary

he article states that “rational behavior tends to be predictable, and in a game of strategy, predictability will leave you with a decided disadvantage.” This means that people expect you to act in a rational manner, so if you act rationally it is easier for your competitor to anticipate what you are going to do. For example, suppose Apple comes out with a new iPad. The market for iPads has boomed with the price set at $499 for an entry-level model. The competition has adjusted to this price point, and have even started selling tablets at a loss to try to undercut Apple’s price. Common sense would dictate that Apple would continue to sell the iPad at $499 even if costs go down since it is such a hot seller. But Apple could also choose to cut into its margins and drop the price to $399 just to take sales away from the competition. History and common sense tell us that Apple would not cut into its margins, especially with a product selling as well as the iPad, and that is why such a move would cause such a stir in the market.

The article states that an element of chance, such as flipping a coin, helps keep your moves random and therefore more difficult for your competition to predict. Apple would not leave a pricing decision such as this up to chance since it has so much market power, but Amazon has changed the market for tablets by offering a tablet for $199, a move many saw as unexpected. Even though they are selling the tablets at a loss, they make up for it by controlling the content sold on the devices. The Amazon Kindle Fire may not be outselling the iPad, but it has been more successful than any other iPad competitor because it offered a different and unexpected pricing strategy: a tablet nearly as good as the iPad for less than half of the price. In short, tablet makers should learn to expect the unexpected and make decisions that their competition will not be ready to combat.

Real World Application

Game theory exists in many different negotiations and interactions in the real world. For example, one can see the theory at work in salary negotiations. This situation will be very relevant to us since we will all be working and receiving offers soon. Suppose that a company offers you a starting salary that doesn’t meet your expectations. You can either A:Take the offer or B: Try to negotiate a better offer. If you believe your skills and what is expected of you deserves a higher salary, you can request a higher salary and the company can either refuse to go higher, or accept it.

Another example of the game theory at work is when cigarette advertising was legal in the United States. The cigarette market realized that any advertising they did was cancelled out when another firm responded with the same amount of advertising. This created a prisoner’s dilemma. The firm had to advertise or they would lose customers to the rival, however it was very expensive to. Interestingly enough, when the government decided to ban cigarette advertising on TV, the companies at first fought the decision. However, tobacco firm’s profits improved after the ban.

Posted by Chris, Andrew and Mariah (Section 2)

Resources:

http://www.huppi.com/kangaroo/Prisonerdilemma.htm

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>.