Within the rules, the game does not need an arbiter. Cheating is another matter.
The rules of the game, together with the card designs, make sure that cards whose abilities affect the game state or have to fulfill certain criteria will at some point be visible to all players, if only at the end of the game. Some casual (not tournament legal-legal) game variants that put more emphasis on the social aspect of Magic might have exceptions, such as the Un-sets or certain multiplayer variants, but these are not discussed in the Comprehensive Rules that describe typical tournament formats.
For example, when a tutor effect allows you to search your library for a creature card and put it in your hand, you will always be required to reveal that card before you do that.
For another example, if you play a card as a 2/2 morph creature and that creature survives until the end of the game, it has to be revealed as the game ends to make sure you were allowed to play it as a morph.
708.9. If a face-down permanent or a face-down component of a merged permanent moves from the battlefield to any other zone, its owner must reveal it to all players as they move it. If a face-down spell moves from the stack to any zone other than the battlefield, its owner must reveal it to all players as they move it. If a player leaves the game, all face-down permanents, face-down components of merged permanents, and face-down spells owned by that player must be revealed to all players. At the end of each game, all face-down permanents, face-down components of merged permanents, and face-down spells must be revealed to all players.
That being said, rules violations both intentional and not do happen, of course. For that reason, each official tournament is required to have a Head Judge, an WotC-accredited person with sufficient knowledge of the game rules and tournament organization. The Head Judge assists players and ensure the orderly conduct of the tournament, including checking allegations of cheating, as described in the Magic Tournament Rules (MTR). The MTR further define the types of information in a game of Magic and how players are required to keep track of them.
This Head Judge requirement, however, is not part of the Comprehensive Rules that govern an individual game of Magic. Any game of Magic can correctly finish without any outside supervision at all. If cheating occurs, then the rules are not to blame, because by definition, they have been broken by the cheater.