By Akshay Agrawal
Photo by Jonathan Petersson on Unsplash
Should I write my English essay or finish my chemistry homework? Should I read “Harry Potter” or “The Murder on the Orient Express”? Should I first brush my teeth or take a shower? Every day, we have to make thousands of decisions—where some are as simple as choosing what side of the bed you wake up on. Although we generally make these decisions subconsciously, how do we actually decide which book to read? Well, our brain subconsciously evaluates the possible outcomes of each decision—if I’m going to study “Harry Potter” in class later in the year, it may make more sense for me to read “Murder on the Orient Express”, as the potential outcome of reading the same book twice isn’t enticing. The study of accurately evaluating the potential outcomes of different decisions in various scenarios or ‘games’, is known as Game Theory. At the end of this article, you should be able to take any scenario, whether that may be a poker hand or the imperative decision of which side of the bed you wake up on, and break down the decisions to find the objectively best one.
Our exploration of Game Theory concepts will be with the archetypal Game Theory scenario, The Prisoner’s Dilemma. The scenario is as follows: two friends have been arrested for the same crime, and in separate, they’re given some information—if they confess, but their friend doesn’t, the one who confesses will be freed while the one who stayed quiet will be sentenced to five years in prison. Rather, if both friends stay quiet, they will each be sentenced to only one year in prison. Finally, if both friends confess, they will each be sentenced to three years in prison.
Now…thinking about this situation with words is awfully complicated; therefore, we can construct a decision-outcome table, pictorially showing the various potential outcomes for each criminal's decision. That table would look like this:
As you can see, the rows and columns correspond to each prisoner's decisions, while the two values within each cell correspond to the possible outcomes. Therefore, in the bottom-left cell, the value is (0,5), as when prisoner 2 confesses and prisoner 1 doesn’t, prisoner 2 is freed while prisoner 1 is held captive for another five years.
But now what…what can we deduce from this table? Well, let’s say you’re prisoner 1. If you’re considering the possibility of staying quiet, you look down the column of staying quiet and take your outcomes as the second number in the ordered pair. Therefore, if you stay quiet, your possible outcomes are one year or five-year imprisonment. As you don’t know what your friend’s going to do, the likelihood of those outcomes can currently be thought of as equal, meaning your expected outcome when you stay quiet is 12*(5+1) = 3 years of imprisonment.
Similarly, the table shows that the potential outcomes if you confess are freedom (no imprisonment) and three years imprisonment. The same computation shows the expected outcome when you confess is 12*(0+3) = 1.5 years imprisonment.
So it’s simple—the expected outcome when you confess is half the years of imprisonment for when you stay quiet…so CONFESS! But unfortunately…Game Theory isn’t so straightforward. If both you and your friend reach the same conclusion, you’re both going to confess. And looking at the table above, both of you confessing is…the worst outcome, as both of you have to endure three years of imprisonment.
So something isn’t making sense. We analyzed the scenario and found the best decision, but if our friend follows the same logic, it actually becomes the worst decision. Why is that the case? Well, it’s a phenomenon in game theory known as the Nash equilibrium, named after mathematician John Nash. It says that sometimes, changing our actions won’t actually change the outcome, as it depends on someone else. In a more common scenario, poker, sometimes bluffing is beneficial if your opponent falls for it (although it’s objectively not correct).
Any game, any problem, and any situation in the world can be modeled and analyzed using the concepts of Game Theory. Here, we’ve explored the world of action vs outcome analysis, looking at Game Theory concepts such as the Nash equilibrium and decision-outcome tables. Hopefully, this should give you the motivation to learn about other, significant ideas in Game Theory, helping you have full awareness and control over the decisions in your everyday life.
References
Friedman, J. W. (1990). Game Theory with Applications to Economics. Oxford University Press, USA.
Hajek, B. (2017). An Introduction to Game Theory. http://hajek.ece.illinois.edu/Papers/GameTheoryDec2017.pdf
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