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The winner must act quickly to address rising prices if the party is to have any chance of winning back support after election losses to the LDP and its junior partner Komeito in the past year cost their coalition a majority in both houses of parliament.

The new party president is still likely to become prime minister because the LDP still has the most seats in the lower house, which chooses the prime minister, and opposition groups are splintered.

The winner will immediately face a big test — hosting a possible summit with U.S. President . A meeting is reported to be in the works as Trump travels to the Asia-Pacific Economic Cooperation summit in South Korea that starts Oct. 31.

Whoever becomes the LDP leader must also gain cooperation from main opposition parties or risk a cycle of short-lived leadership.

Only members and LDP lawmakers are voting

The vote is being held within the LDP — its 295 parliamentarians and 1 million dues-paying grassroots party members. That’s less than 1% of Japan’s eligible voters.

If no one wins a majority in the first vote on Saturday, a runoff will quickly follow between the top two vote-getters.

The lower house will then choose a new prime minister in a leadership vote expected in mid-October. The new LDP leader will need votes from some opposition lawmakers to take office.

A woman, a young scion and a veteran moderate

All five candidates — two of them are currently in the government and three were in past Cabinets — have emphasized their willingness to work with opposition groups that are more centrist politically.

Surveys have suggested the front-runners are Sanae Takaichi, who could become the first female prime minister, Shinjiro Koizumi, who would be the youngest in more than a century, and Yoshimasa Hayashi, a veteran all-rounder.

Sanae Takaichi, 64, admires former British Prime Minister Margaret Thatcher and is a protégée of former Prime Minister Shinzo Abe. She is a wartime history revisionist and China hawk. She regularly visits Shrine, seen as a symbol of militarism. Takaichi has vowed toughness on immigration, a major issue in the race. Her staunch ultra-conservative stance is considered a major risk to Japan’s relations with Asian neighbors.

Shinjiro Koizumi, 44, is the son of a popular former prime minister, Junichiro Koizumi, and seeks to be the youngest prime minister in Japan’s modern history. As agriculture minister for Ishiba, Koizumi released an emergency rice stockpile to lower prices and stabilize supply as he promoted a reformist stance. Koizumi has made rebuilding his party a priority and pledges to listen to the people’s voices closely to address their concerns, such as rising prices, growing foreign population and public safety.

Yoshimasa Hayashi, 64, is the chief Cabinet secretary in Ishiba’s government. He is a moderate who promises steady wage increases, a strong economy and defense. He also advocates a universal credit system to provide assistance to low-income families for basic goods. Hayashi is a pragmatic, pro-China veteran with a reputation for being capable at damage control. He has also served as foreign, defense and education minister.

Toshimitsu Motegi, 69, has served in key posts including foreign and trade ministers and is known as a tough trade negotiator.

Takayuki Kobayashi, 50. The ultra-conservative former economic security minister promises strong growth, defense and national unity. He calls for tougher restrictions on foreigners.

Cooperation with opposition parties is vital but risky

The new LDP leader and prime minister will need help from either or both of the center-right opposition groups, the Japan Innovation Party, or Ishin, and the Democratic Party for the People, with which the LDP has collaborated on budget bills.

While it's still uncertain, cooperation or even expanding the coalition might contribute to political stability.

Koizumi has approached Ishin and in August visited Osaka Expo, where he was escorted by party leader and Osaka Gov. Hirofumi Yoshimura. Hayashi reportedly met with a senior Ishin lawmaker over dinner recently, while Motegi announced his willingness to form a coalition with the two parties.

Takaichi said the LDP's ongoing coalition with the Komeito is the foundation, but that she is open to cooperation with far-right groups such as the emerging anti-globalist Sanseito.