For policymakers, game theory has two main functions. First, it provides a framework for taking a complex social situation and boiling it down to a model that is manageable. Second, it provides methods for extracting insights from that model regarding how people do behave or how they should behave.
Consider the following situation: it’s spring semester, companies are recruiting, and you’re thinking about applying for a summer internship. You’ve narrowed your choices down to two crypto exchanges: Coinbase and Robinhood. You, as well as everyone else from your school, prefer the Coinbase internship, but you are hesitant to apply for two reasons. First, there are only 6 summer intern positions at Coinbase, while there are 30 openings at Robinhood. Second, you know that everyone else finds the Coinbase internship more attractive, and this will likely make it harder to get hired. Suppose the application deadline is quickly approaching and you only have enough time to apply for one of the two internships. Which one should you choose, and how can game theory be used to decide?
In the context of game theory, a payoff is simply the result a player receives from arriving at a particular outcome of a strategy. The payoff is expressed as a number, regardless of whether we’re dealing with a qualitative unit of measure (e.g. value, utility) or quantitative unit of measure (e.g. dollars). In this scenario, suppose 100 students from UNC Chapel Hill are interested in a public policy internship at either Coinbase or Robinhood. Each student has the same preferences, and each assigns a value of 200 to a Coinbase internship and a value of 100 to a Robinhood internship. Our optimal payoff, called a Nash equilibrium, is the point at which none of the students, all of whom we assume are rational folks, have any incentive to change their strategy based on what another student does.
A student’s payoff from applying for a Coinbase internship is maxed out at 200 only if he is assured of getting the internship, which is the case exclusively if he is one of only 6 or fewer people to apply. The payoff is lower than 200 when more than 10 students apply for the internship at Coinbase, and decreases further as more students apply. Similarly, a student’s payoff from applying for a Robinhood internship is maxed out at 100 only if he is assured of getting the internship, which is the case only if no more than 30 people apply. When there are more than 30 applicants, the payoff from applying decreases as the number of applicants increases. The payoffs for the internship game are listed in Table 1 and plotted in Figure 2.
|Table 1: Payoffs to Applying for Coinbase and Robinhood|
|Number of applicants to Coinbase (x)||Payoff to Coinbase applicant||Payoff to Robinhood applicant|
As shown in the table, the more students who apply to Coinbase, the lower the payoff is to each of those additional applicants.Transitively, since more applicants to Coinbase means fewer applicants to Robinhood, the payoff for applying to Robinhood increases with the number of students competing for a position at Coinbase.
To determine the best strategy to pursue, let’s first suppose that absolutely no one applies to Coinbase and all 100 apply at Robinhood. Robinhood’s applicant pool becomes congested.The payoff to each of those 100 Robinhood applicants is 30, considerably less than the payoff for being the lone Coinbase applicant, which is 200. Therefore, all students applying to Robinhood is not an equilibrium.
Next, consider a strategy profile in which 10 students apply to Coinbase and the other 90 apply to Robinhood. Again, Robinhood’s applicant pool becomes congested, and any of those Robinhood applicants would do better by applying to Coinbase. Applying to Coinbase raises the payoff from 35 (the payoff to a Robinhood applicant when there are only 10 Coinbase applications) to 100 (the payoff to a Coinbase applicant when there are 20 Coinbase applicants). More generally, in considering a strategy profile in which x students apply to Coinbase, a student who is intending to apply to Robinhood is comparing the payoff from being one of the applicants to Robinhood and one of the applicants to Coinbase. As depicted in the Figure 2 chart, when the payoff for applying to Coinbase is higher, it is not optimal for this applicant to apply to Robinhood. As long as the number of applicants to Coinbase is less than 40, we do not have an equilibrium.
Now let’s start with the other extreme: suppose all 100 students apply to Coinbase. Then each has a payoff of 15, which falls well below the payoff of 100 from applying to Robinhood. Indeed, as long as more than 40 students apply to Coinbase, an applicant to Coinbase would do better by applying to Robinhood. If the strategy profile has x students applying to Coinbase, then, when the payoff for being one of x applicants to Coinbase is less than the payoff for being one of the applicants to Robinhood, a Coinbase applicant ought to switch his application.
From the chart, we can see that any strategy profile in which fewer than 40 or more than 40 students apply to Coinbase is not a Nash equilibrium.
This leaves one remaining possibility: exactly 40 students apply to Coinbase and 60 apply to Robinhood. In that case, the payoff to both a Coinbase applicant and a Robinhood applicant is 50. If 10 of the students intending to apply to Coinbase switch to Robinhood, then their payoff declines to 45 (see Table 1); and if ten of those students intending to apply to Robinhood switch to Coinbase, then the payoff declines to 40. Because any student is made worse off by changing his strategy, exactly 40 students applying to Coinbase is a Nash equilibrium.
Therefore, in this scenario, even though all students have identical options and preferences, they make different choices in the equilibrium situation. Your chances of getting hired are best when 40 students apply to Coinbase for its 6 available positions, while 60 apply to Robinhood, competing for the 30 available slots there.
In summary, we have used game theory to develop a useful model for making a strategic decision. Now, you may ask, “Hey, Victor – what can I do with this information?” Great question. Recruiters, for example, can use this model to bring in the best talent by focusing their efforts on areas where there are many high-quality candidates. Prospective employees can use this model to gauge the likelihood of getting an interview by taking a look at the number of applicants on a Linkedin job posting (one can even see how their resume compares to other applicants with a Premium subscription), and do what’s necessary to stand out from the crowd.
However you use this information, I wish you all the best.