Prisoner's Dilemma is characterized by the inequality
T > R > P > S
For any given other-player action, I'd rather defect than cooperate, but eliciting cooperation from the opponent is even more valuable: under almost any plausible scenario, R-S and T-P (the effects on me of the other player's choice) are larger than T-R and P-S (the effects on me of my own choice). When played iteratively for a long and unknown number of rounds, therefore, this game favors the development of strategies that are "nice" (not the first to defect), "punitive" (responding in kind to defection), and "forgiving" (responding in kind to a return to cooperation after defection). Since both players following such a strategy simultaneously is a stable equilibrium, the outcome feels "fair". (It is sometimes also stipulated that S + T < 2R, so successive rounds of (C,C) are preferable to an alternating series of mutual retribution: (C,D) and (D,C).)
The related game of Chicken is characterized by the inequality
T > R > S > P
Since S > P, you're better off being the sucker who swerves off the road than being one of two people who, both refusing to swerve, collide head-on. Again, my choice is between (T/P) and (R/S), but the implications are different: I'm not always better off defecting for a given other-player action, but I get to choose how large the other player's effect is on me (am I playing for large stakes or small?). No symmetric pair of deterministic strategies is a stable equilibrium for this game; it tends to favor the development of exploitative, dominant/submissive solutions in which one player consistently defects and the other consistently cooperates.
A less-studied variant might be called "Mine", and is characterized by the inequality
T > P > R > S
In this variant, my choices alone determine whether I'm in the better (T/P) part of the matrix or the worse (R/S) half of the matrix; the other player's choice only selects between T and P or between R and S. The effect on me of my own actions is unconditionally more important than the effect on me of the other player's actions. The obvious successful strategy is "always defect".
Which is where ideology and pragmatism come in. If you're a politician expecting to run for (re-)election on your record of accomplishments, you're playing a game in which eliciting cooperation from others is at least as important as your own unilateral actions, something like Prisoner's Dilemma. If, on the other hand, you're a politician expecting to run for (re-)election on your record of ideological purity, you're playing the game of Mine: you don't much care what anybody else does, and will always stand on principle (and seldom get anything accomplished).
For another example, suppose you're a government official charged with carrying out a law with which you personally disagree. If you believe in the rest of your job, and think you can make the world better on balance by doing it and maintaining a productive relationship with your co-workers, you'll find a way to live with it; if you're more interested in ideological purity than accomplishing anything, you'll stage a dramatic and futile protest (and set the stage for a future career as an ideological politician).
Now, what if I'm a pragmatist and you're an ideologue? I assume we're playing Prisoner's Dilemma, so I adopt a "tit for tat" strategy appropriate to that game, while you adopt an "always defect" strategy appropriate to the game of Mine. After cooperating once and getting screwed, I punish you by defecting. You are unchastened: my newfound intransigence only confirms you in your own, and we both continue defecting forever. If I'm feeling especially optimistic, I may occasionally send up another trial balloon of cooperation to see whether you've learned your lesson; you haven't, of course, and this only persuades you that I'm a sucker who can be taken advantage of, and that (because I sometimes cooperate) I have no integrity.
Of course, under some circumstances you CAN get something done by being ideologically pure. If your ideological position, while initially unpopular, is persuasive enough that you can develop a large voting bloc who believe the same way you do, you no longer need to elicit cooperation from the other side. Martin Luther King, by adopting strategies that initially looked dramatic and futile, captured the attention and conscience of the nation and gradually developed a solid majority in favor of racial integration and fairness (at least in theory), whereupon the remaining few segregationists could be marginalized and ignored. Today's Tea Partiers may anticipate something similar: by insisting on ideological purity, they hope to attract more and more people to their cause until the national consensus agrees with them. But as long as there's a large-but-not-majority ideological faction, not much will be accomplished.
Note that the ideologue/pragmatist distinction is almost independent from the left/right distinction. One can take an ideological or pragmatic approach to either left-leaning or right-leaning politics, although in 2015 most of the ideologues are on the right and most of the pragmatists are on the left.
At a meta-level, the whole thing is embedded in a larger game of Chicken: ideologues (whether left or right) have decided to play for the large stakes of eventually dominating the political discourse, while pragmatists (whether left or right) have decided to play for the smaller stakes of getting things done along the way.