The Darkin’s assistance comes at a steep price. Aatrox will be considered their gained champion for the week.

it also may not support or oppose another faction’s petitions this week. If only one faction attempts to recruit Aatrox in a given week, it automatically gets him; if more than one faction attempts to recruit him, the faction with the highest vote percentage for recruiting him gets him. Whoever does end up recruiting Aatrox instantly loses two-thirds of its Favor. (This penalty takes effect before its petition is processed: it will be hard to convince the League to do anything while you bring the avatar of war under your banner.) All Favor gains for the remainder of the arc are cut in half. Finally, the faction’s Danger Level is doubled.

“That is one costly top-laner,” you say. First, it is worth noting that these costs are less severe for a faction that is already down-and-out: it likely has little Favor left to lose. Second, Aatrox has a reputation for a reason. For the remainder of the arc, the faction rolls twice and uses the higher roll for all violent, military-related, or otherwise warlike options in lore events, islands, and the like. As for actual battles, Aatrox allows the faction to roll twice on the d6 random Strength modifier and use the higher result, increases the faction’s overall Strength by 20%, and inspires defeated units (lost, driven into retreat, etc.) to fight at full strength for the remainder of the battle.

More information on the Politics system is available here: http://leaguefactions.net/politics-system/

Failing a petition costs the faction 10 Favor. A smaller penalty of 5 Favor applies if another faction's petition fails despite their support, or succeeds despite their opposition.

• Request Research Help. Factions may request assistance from the League's sages and scholars. They gain about 75 points (3d50) of Research. (Difficulty: 100.)
• Enact Substitution of Field. Factions can demand that other Fields of Justice besides Summoner’s Rift get some love. If successful, they may select the map for the next two weekends of Featured Matches they participate in, for all participants. (Difficulty: 130)
• Launch Formal Inquiry. Launch a formal inquiry into another faction's affairs, potentially exposing some damaging or embarrassing Secrets. (See below for how Secrets work.) For each Secret that the targeted faction has, the targeted faction rolls Espionage (d100 plus their Espionage stat) while the investigating faction rolls d100 and adds their Favor. For each Secret, if the investigating faction rolls higher than the targeted faction, that Secret is exposed. (Difficulty: 135.)
• Disqualify Champion. Factions can try to twist the laws of the League to suit their ends, by pushing a specific Champion out of contention on a technicality. If successful, that Champion is banned from participation in the next two weekends of Featured Matches. (Difficulty: 140)
• Impose Spell-block. Factions may insist that a particular Summoner spell be disallowed, for all other participants, for the next weekend of Featured Matches. (Difficulty: 160)
• Support/Oppose. Alternatively, a faction may exchange one of its petition slots for an extra support/oppose opportunity.

Possible Core/Secondary/Tertiary Champions for each faction are listed here: http://leaguefactions.net/faction-rosters/

As a first-order approximation, the rolls work as follows: take your faction's current Favor, roll a d100, and compare the result to Difficulty. If you score above the option's difficulty, you succeed. So, if you have a Favor score of 50, you've got about a 50% chance of success on a Difficulty 100 petition.

Here are some additional factors that can come into play:
• Critical successes: You have about a 20% chance of scoring a critical success, and doubling your Favor.
• Support/Oppose: Other factions might support or oppose your petition, increasing or decreasing the odds of success.

Mortal Bargain: The Shadow Isles may not burn favor, Instead, Mortal Bargin allows you to roll twice and take the better result; however, doing so will create a secret of 10 severity each time it is used. If any of these secrets are found, the normal penalties will be applied. Additionally, they lose use of this ability for two weeks after. If one of the secrets is found, the others still remain.

This is the first round of voting. Select the petition (or petitions) you most strongly want to see made.

In this round, the question is not which petitions you most strongly support but which petitions you'd be alright with putting forward.

Effectively, this asks, for each option: "Given the choice between this petition or no petition, which would you prefer?" Remember that failing a petition costs the faction 10 Favor.

The Shadow Isles may not burn favor, Instead, Mortal Bargin allows you to roll twice and take the better result; however, doing so will create a secret of 10 severity each time it is used. If any of these secrets are found, the normal penalties will be applied. Additionally, they lose use of this ability for two weeks after. If one of the secrets is found, the others still remain.

If you select a "negative" option, like disqualifying a Champion, which faction will you target? (This gives them fair warning, and may help you gain support from allies.)