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In order to make a claim on another nation's throne, the claimant nation must have a royal marriage with the target of its claims. Additionally, the claimant nation must have more prestige than the target nation, and the target nation must be under the disputed succession status. A disputed succession occurs when a nation has no legal heir, or an heir with a weak claim to the throne (for example, a bastard). The player can tell which nations have a disputed succession by hovering over the disputed succession herald at the top of the screen. This will list all nations with a disputed succession, as well as their current prestige levels. Additionally, all nations with disputed successions that are royally married to the player will be highlighted in green.
Making a claim on a nation's throne has several immediate effects. The claimant nation loses 20 prestige and 20 legitimacy, as the action is generally frowned upon by the world and their own people. The claimant nation also loses 50 relations with all nations they are royally married to, including the target nation. Finally, the claimant nation receives the "Claim on Throne" casus belli, which allows them to force the target nation into a personal union through military action.
There are two ways for the claimant to receive the Personal Union of the target nation. One is when the target nation's monarch dies before producing a legal heir, then the target nation will automatically enter a personal union led by the claimant nation's monarch. The other is pressing the Claim Throne Casus Belli, and force the target nation into a personal union with 84% warscore.
Note that, because declaring war against a nation of which you have a royal marriage with will cost you 1 stability, while being the Papal controller removes the stability penalty for removing a royal marriage, it's best to wait till you have the Papal control as a catholic nation, and cancel the royal marriage prior to the war. Keep in mind that when the target nation manages to produce an heir of a strong claim, you have one month before the Claim Throne CB expire.
As a junior partner of this union, the target nation will be treated as a vassal nation, but with severely limited diplomatic options.
When the monarch dies, the union will either end (if the two nations have negative relations), continue, or the senior partner will inherit the junior. The chance of inheritance is based on the two nations' relations and the senior partner's prestige. The inherited nation is completely annexed, and if its culture is in the same culture group as the inheritor or both parties are part of the Holy Roman Empire, the inheritor will gain cores in all provinces where the inherited nation had cores.
If two or more nations have claimed the throne of the target nation, then they will automatically enter a succession war if the monarch dies before producing a legal heir. <details about succession war>. The winning nation will form a personal union with the target nation.
Claiming the throne of other nations is considered to be the best method of (mostly) peaceful expansion. It is particularly useful with nations that share the player's culture group. For example, the quickest way for Castille to form the Spanish nation is to force Aragon into a personal union (or vice versa). Since Castille and Aragon are both part of the Iberian culture group, inheriting the junior partner will grant cores in all the provinces necessary to form Spain. The Ottomans have the potential to form a massive intercontinental empire through royal claims, as the Turko-Semitic culture group extends to the Arabian peninsula and most of North Africa.
In order to use claims effectively, it is usually best to focus on one nation at a time, due to the penalties for making a claim. The best opportunity for making a claim is when the player gains a new monarch (as all their old royal marriages will end) and the target nation has a relatively new monarch (as it will likely not yet have a legal heir). Changing the message settings to show whenever another nation gains a new monarch is useful for finding these opportunities, as well as is checking the the disputed succession herald.