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.
• Burn Favor: Your faction might choose to go all-out, sacrificing a significant share of its accumulated Favor to markedly improve odds of success on an especially important petition. (Think of this as calling in favors, making impassioned speeches, and otherwise pulling strings in a way that you can't afford to do every week.)

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.

A faction can choose to burn Favor on an especially critical petition. This costs two-thirds of its current Favor, and dramatically increases its odds of success.

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.)