Conceptualizing the GOP debate as a game
We can conceptualize CNN’s recent GOP debate as a game with two players: first-place candidate Donald Trump as player A and second-place candidate Ben Carson as Player B. Mentioning other candidates is associated with various risks and rewards depending on the position of the player and the position of the candidate being mentioned. The article explains: “Of course, merely speaking in a debate helps make a candidate stand out and potentially become the focal point. But Wednesday’s debate added a twist: A candidate got to speak if another candidate mentioned him or her. Thus, by mentioning another candidate, you’re helping make that candidate a temporary focal point”. The article also makes it clear that mentioning candidates that are closer to the player has a greater risk than those farther away (unless the candidate is ranked higher, in which case it is beneficial), and making yourself the subject of discussion as the greatest payoff.
I will assign arbitrary values to represent reward on an 11-point scale from -5 to 6, where a negative number represents a risk while a positive number represents a reward. The 11 points maps to the 11 candidates. More positive or negative numbers represent a greater reward or risk, respectively.
Using this game, we see a natural Nash equilibrium when mentioning themselves and a natural aversion to directly referencing those directly below the candidate.