The classic Prisoner s Dilemma situation is a static game. Zero-sum games mean that for one person to win, another has to lose. Complete information is exactly what it seems to be, complete knowledge of the situation. Poker is a common example of a game with incomplete information. There are good reasons that game theory puzzles are usually this particular type of game.
Payoffs are easy to calculate in a zero-sum game, particularly one with only two players.
Dynamic games are easier to follow than static games. If played as a dynamic game, the strategy is clear and produces a win for the second player every time. Most importantly, finite games of complete information are always solvable. The most complete, though by no means simplest, means of solving a game is by diagramming a game tree. It sets up every possible move in a branching tree, then repeatedly eliminates those branches that a player would not make until it has pruned back to the base of the tree.