Some time ago, Jin Mitaka lost his memories when battling against Count Pythagoreas. Now that the Orpheus Order has been defeated, Kaito, Nonoha and Rook are determined to restore the man to his former self by bringing him to familiar locations, and soon arrive at the Fool’s Puzzle from their past. But things escalate when a mysterious girl named Raetsel ambushes them in the puzzle’s chambers; she works for an organization called Master Brain, and she’s come to steal Jin away. And not only does Jin recognize the girl, but she also claims that Master Brain’s goal is to rid the world of puzzlers at Jin’s request! Now, it’s up to the Children of Phi Brain to discover the truth and put a stop to the organization’s nefarious plans.
Kaiji Ito is as pathetic a person as they come; a man who gambles his days away, only winning enough to lose significantly more. He hates himself, is riddled with envy for others, but is ultimately too weak to think of a way out of his massive debts. Then one day he is approached by a strange man who offers him what seems the solution of a lifetime – to take a short journey on a ship called Espoir, during which time he will be given the chance to win more cash than he can dream of in a card game like no other. Ever the desperate, Kaiji takes the gamble of his life; however, the game turns out to be far darker than he expected and the hard lessons pile on thick and fast. Now stuck in a closed world of unsavory characters willing to do anything to destroy him, can Kaiji gather enough courage to outwit them all?
These are both high stakes games anime where the players often have to put their lives on the line. Both these anime involve players using strategy and tactics to try and outwit the traps their opponent has set. Both anime have a good deal of variety because the puzzles in Phi Brain are all very different and in Kaiji each arc is about a different game.
Phi Brain is a pretty basic shounen anime and there isn't much depth to either the characters or the strategies they use. kaiji is a much more intelligent take on the high stakes game genre with much more insight into the tactics and the psychology behind them. If you watched Phi Brain but want something a little more meaty in the high stakes game genre then you should definitely check out Kaiji.
Yugi Mutou was a boy who was constantly bullied – until one day he managed to put together an ancient and mysterious puzzle that no one has been able to solve for two hundred years. Now, whenever Yugi's friends are threatened, a strange transformation occurs, and Yugi seems to become an entirely different person. This transformed Yugi is a master of games, and uses these spontaneously-invented games to trap and to punish the bad guys. At the end of each game, his opponent appears to suffer a painful death; but it is soon revealed that this was all in their minds – or was it?
Both anime series feature various puzzles (= deadly games), where people are punished for failure and are mostly episodic. Do note that the YGO! series I have in mind is the original one with 27 episodes.
Ever since Yugi Mutou completed the Millennium Puzzle his life has changed; he's made friends, learned how to play the card game Duel Monsters and has gained more confidence in himself. After his grandfather is beaten in a duel he is challenged by Seto Kaiba, ranked number two in the world, but Yugi beats him regardless. Now, as a result of his victory, Yugi has attracted the attention of the game's creator, Pegasus. Yugi is then challenged to a dual in an alternate reality where magic brings the monsters to life, but is beaten and his grandfather's soul is taken. In order to retrieve it Yugi must now participate in a tournament against the best duelists in the world, defeat the game's creator, and learn more about the secrets within his Millennium Puzzle!
These are both shounen anime where the characters get involoved with games they enjoy on a much larger and more dangerous scale. Phi Brain is about solving puzzles and Yu-Gi-Oh is about a card game, but at their heart they are both shounen anime with all the usual cliches and character archetypes. Both anime have characters matching wits with villains through either puzzles or the card game.
In both these shows there is at least some attempt to have strategy and tactics play a role in the puzzles or the games, but don't expect anything too complicated from either of them. If you are looking for a stereotypical shounen anime then either of these should suit your needs.