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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

If a player pays {2}, that enables only them to ignore Leonin Arbiter's effect that turn. Each other player is still affected by it.

Paying {2} to ignore Leonin Arbiter's effect is a special action. Any player may take this special action any time they have priority. It doesn't use the stack and can't be responded to.

Paying {2} doesn't let a player pick up a library and search it — it just allows them to ignore Leonin Arbiter's effect that turn. That player can search a library only if another spell or ability instructs them to do so.

Since players who haven't paid {2} can't search, they won't be able to find any cards in a library. The effect applies to all players and all libraries. If a spell or ability's effect has other parts that don't depend on searching for or finding cards, they will still work normally.

If an effect says "You may search your library . . . If you do, shuffle your library" or "You may search your library . . . then shuffle your library," you can't choose to search, so you won't shuffle.

If an effect says "Search your library . . . then shuffle your library," you shuffle you libraries even though you can't search.

Effects that instruct you to reveal or look at cards from the top of you library will still work. Only effects that use the word "search" are affected.

Once a player has paid {2}, they may search libraries that turn as many times as they are instructed to do so.

If there are multiple Leonin Arbiters on the battlefield, a player must pay {2} for each one before being able to search libraries that turn.

A player who has paid {2} may search any library when instructed to do so that turn, not just their own library.