Making Friends in Networks 2040 Lectures using Game Theory
The concept of game theory is built upon mathematical and logical models that try to map the outcomes, probabilities, and strategies of players in a game. The Prisoner’s Dilemma is an infamous example of two players choosing strategies in order to maximize their payoff (or minimize their punishment, in this case). Where game theory gets interesting is when people – with flawed decision-making and differences among each other- get involved in this strict concept. Labeling payoffs and probabilities gets muddy and depends on a whole new set of factors. This area of game theory is called Behavioral Game Theory (BGT), which incorporates “a framework for modeling and predicting behavior in social situations of cooperation, coordination, and conflict” (Gachter). What if we were to apply BGT to a situation everyone in Networks 2040 is familiar with – a payoff matrix for a game between two people in a Networks 2040 lecture?
Game theory and Nash Equilibrium have been utilized to study social interactions in the past, but the pandemic has introduced new variables and different payoffs than in the past. Attending lectures today, where students are masked and this is the first time in two years they have stepped foot into a classroom, has its new challenges. Looking at an example payoff matrix for a game consisting of an awkward interaction between two people when walking by/with each other, this will be the foundation for constructing a game for making friends in a Networks 2040 lecture.
An awkward interaction game example is shown above for two people walking down a hallway. This is an example of a coordination game, where it is optimal for the two players to choose the same strategy (L,L) or (R,R). While this scenario is a little different, we will base our matrix off of these values. If there are two players that represent two students in the class who don’t know each other but are sitting next to each other. They have two options; either strike a conversation with the person next to them or don’t. We are assuming that if at least one player is not engaged, no conversation occurs (the only time a conversation occurs is (Engaged, Engaged). The matrix would look something like this:
Where the numbers came from: we are assuming that the conversation with both engaged students goes well and both players benefit. We are also assuming being “rejected”, where one player chooses to engage but the other doesn’t, has more of a negative impact (-2) than not pursuing a friendship when choosing to not be engaged (-1). If they choose not to engage and end up not engaging, there is no benefit or harm (payoff = 0).
As you can see from the graph, there are two Nash Equilibriums – the pair (E,E) and (DE, DE). In this unbalanced coordination game (the two pure equilibriums have different payoffs), the (E,E) equilibrium is payoff-dominant and the (D,D) equilibrium is risk-dominant. However, this is slightly contradictory – if the strategy is to not engage unless the other person engages, then there is a waiting game between the two and an engagement will never happen. This standoff holds true as long as the punishment for a player engaging while the other is not engaging is worse than both not engaging. The tricky part about this situation, and many games where behavior is a factor, is that the payoffs look different for everyone – some people are more social than others and tend to always engage, some may take rejection better than others, etc. It would be useful to utilize Schelling’s theory of focal points, where we can use different features of the situation to see if a player chooses to engage or not engage. For example, if they ask the person next to them for help with a problem they would need to engage, or if they are close to the professor and do not want to interrupt, they choose not to engage. But in my opinion, if you engage enough, you can change someone’s disinterest into interest and influence someone’s decision!
Sources: http://web.math.ucsb.edu/~padraic/ucsb_2014_15/math_honors_s2015/math_honors_s2015_lecture8.pdf
https://people.ict.usc.edu/~gratch/CSCI534/Readings/Behavioral%20Game%20Theory.pdf