Game theory and its limitations

Last week, I explored the idea of randomization in my former blog from a mathematical perspective. As the professor claims in the lecture, he believes that randomization is a great way for solving complex social problems, because it can always achieve fair results. However, there are limitations. These limitations intrigue me, for they remind me that math is not a perfect tool that can solve every real world problem. Therefore, in the process of doing math, not only do I need to search for a correct answer, but I also need to be aware of the fact that the eventual outcomes and inferences may not always provide perfect solutions in reality. While using multiple mathematical equations and models to generate a grand theory present us with a logical way of thinking, the extrapolated theory has limitations under the light of reality. Such is the case of Game theory—it is well established but is also somewhat idealized.

I was reading research papers on the effectiveness of randomization, hoping to get a sense of the differences between mathematicians’ way of thinking and historians’ or politicians’. In a mathematical world, or specifically, in Game Theory world, we assume that people are rational, and thus will always play their best strategies by anticipating their opponents’ strategies. Therefore, we conclude that randomization is a best response under every situation, because players do not even need to anticipate each other’s responses. When no one is considering others’ strategies, the situation is fair for everyone.

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In fact, the notion of randomization is more complicated. For example, suppose a group of scientists come to West Chester to perform a randomized control trial on the town’s population to check the effectiveness of a vaccine, and the effectiveness of the vaccine turns out to be 95%. This result, if interpreted from a mathematical perspective, should be published on paper and to convince the public to take the vaccination. However, in reality, will the vaccine have a 95% of working on my body? No, simply because I am not that part of the random draw: I am not a part of sample population. Under this hypothetical situation, we cannot conclude that randomization provides a fair result, because the randomization in a sample only provides impartiality in that specific sample. It does not necessarily grant fairness beyond that sample.

Dices on White

Let’s consider another hypothetical situation. In a town where scientists use a randomized sample to conduct the experiment, the effectiveness of vaccine turns out to be 95%; however, in a neighboring town where scientists use a biased sample (eg. The sample only includes early risers because the experiment is only conduced in the morning), the effectiveness of vaccine is only 50%. Should I take the vaccination? No. An average person will be tempted to trust to the biased sample and thus not take the vaccination, because, according to numerous experiments in psychological realms, human minds have a propensity to be risk-averse. Humans, in reality, are not rational. Regardless of whether results are biased or not, people tend to focus more on a negative result rather than a positive one. Therefore, concluding that randomization is the best strategy to every situation is not true, for human beings tend not to act in a way that either mathematicians or economists can predict.

As we can see, it is extremely hard to rationalize the idea of randomization in our daily life. In certain circumstances, it is safe to conclude that randomization can help us achieve fair results, as the case of breaking people up at a lunch table. Most of the time, however, randomization is only an ideology. This ideology does not hold up if humans are not rational or if the sample size is not big enough, etc.

I am thrilled to learn that my research on the topic of randomization gives me new insight into the study of mathematics. Now, I am going to shift the direction of my project to more research instead of only passive learning. I am looking forward to discovering more connections between mathematics and today’s society. I wish to see both the beauty and the limitation of math.

Love_math_1 2

 

 


 

Work Cited

Basu, Kaushik. The Method of Randomization and the Role of Reasoned Intuition.
Research rept. no. WPS6722. Mexico City: n.p., 2013. Print.

“Dices.” Study-Portal. Study-Portal, n.d. Web. 15 Oct. 2014.
<http://www.clinicaltrial-portal.com/services/patient-randomization/&gt;.

“Differential Equation.” University of Arkansas. University of Arkansas, n.d.
Web. 15 Oct. 2014. <http://ualr.edu/catalogs/undergraduate-catalog/
math/>.

Heckman, James J. Randomization and Social Policy Evaluation. Research rept. no.
107. N.p.: National Bureau of Economics Research, 1991. Print.

“I love math.” Before, During and After. Before, During and After, n.d. Web. 15
Oct. 2014. <http://madbageltrio.com/beforeduringandafterblog/wp-content/
uploads/2013/12/i-love-math.jpg>.

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