Random Ballot could be considered a prisoner's dilemma, depending on what your priorities are.
The Prisoner's Dilemma is a scenario where two people are making a binary choice where the stable equilibrium of the system (the strategic choice of all parties trying to maximize their personal gain) is less profitable for all players than if everyone had made the opposite choice. It assumes that neither participant is able to see how the other is choosing until after both choices have been made.
To analyze the Random Ballot, let's consider a simple system where there are two voters (Alex and Chris), and three politicians (A, B, and C). Alex wants A and dislikes C, Chris likes C and dislikes A, and both tolerate B.
If both voters pick their preferred candidate, then the odds are 50% A and 50% C, with one voter getting exactly what they want, and one player losing out.
If one player (we'll use Chris) decides to compromise and picks B, then the odds are 50% A and 50% B. This is obviously fantastic for Alex, but kind of sucks for Chris.
If both players compromise, then B is guaranteed to be elected.
The stable equilibrium here is for both players to choose their preferred candidate. The probability of your ballot being chosen is independent of what you write on it, so you might as well write your favorite candidate.
This is a prisoner's dilemma if (and only if) both voters consider the guaranteed election of B to be preferable over 50/50 odds of A or C being elected.
If both voters like B almost as much as they like their favorite candidate, then they'd be interested in reaching a compromise, and this becomes a prisoner's dilemma.
If both voters dislike B almost as much as they dislike the other candidate, then they'd probably prefer a 50/50 shot at their favorite over the election of B, and this is not a prisoner's dilemma.
If one voter likes B, but the other does not, then it is not a prisoner's dilemma, because while one voter might prefer B over the 50/50 odds, they know that the other voter does not and therefore has no incentive to vote for B.
Of course, really Random Ballot voting is more complicated than this because there are more than two voters and three candidates, and because you don't perfectly know the desires of the other voters.