Live Updates: Wyoming and Alaska Voters Weigh Fates of 3 Prominent G.O.P. Women

Representative Liz Cheney of Wyoming appears all but certain to lose her seat in the face of a furious Republican backlash against her role as co-chairwoman of the committee investigating the attack on the Capitol on Jan. 6, 2021, and against her vote to impeach former President Donald J. Trump over his role in inciting that riot.

In Alaska, former Gov. Sarah Palin is trying for a comeback in the open race for the state’s sole seat in Congress, last held by Don Young, who died in March. Ms. Palin, who was John McCain’s notoriously game-changing running mate in 2008, is running both in a special election runoff for the remainder of Mr. Young’s term and in a primary for a full term of her own.

Both Ms. Palin, whose rivals say she has been more visible in recent years on right-wing television in the Lower 48 than in her home state, and Ms. Cheney have encountered deep skepticism from voters who feel that the national spotlight has come to matter more to them than heeding and tending to the people who elected them.

Ms. Palin, however, was an early supporter of Mr. Trump’s and earned his endorsement.

Also in Alaska, Senator Lisa Murkowski, who was one of seven Republicans to vote to convict Mr. Trump of incitement of insurrection, is in a re-election fight against a field led by Kelly Tshibaka, a former state official whom Mr. Trump endorsed and called “MAGA all the way.”

Race calls in Alaska could be days or even weeks away, as mail-in ballots will be collected and counted through at least Aug. 31. It’s possible that by the end of the night, we’ll know only who has an initial lead in early and in-person voting.

Ms. Palin and Ms. Murkowski can take solace from this, though: They only need to be among the top four finishers in their primaries to secure a place on the general-election ballot in November.

Wyoming’s House Race

Ms. Cheney may have the admiration of a certain portion of the political spectrum for her actions on the House committee investigating Jan. 6 and her devotion to democracy and the rule of law. But in Wyoming, a state Mr. Trump won with 70 percent of the vote in 2020, her crusade to hold him accountable for inciting the Capitol mob, and for sitting idly by for hours as rioters threatened lawmakers and his own vice president, all but doomed her re-election chances from the outset of her campaign.

Ms. Cheney, who lost her job in House leadership after her vote for impeachment, has no regrets. “If the cost of standing up for the Constitution is losing the House seat, then that’s a price I’m willing to pay,” she told The New York Times this month.

The favorite in the Republican primary is Harriet Hageman, whom Mr. Trump handpicked as his instrument of revenge against Ms. Cheney.

Ms. Hageman, a former Republican National Committee official, led Ms. Cheney by nearly 30 percentage points in a poll this month by the University of Wyoming.

The survey included Democratic voters who plan to cross over and vote for Ms. Cheney, as Wyoming law allows and the Cheney campaign has encouraged. In fact, nearly half of Democratic likely voters said they would vote in the G.O.P. primary, nearly all for Ms. Cheney. But Democrats in Wyoming are vastly outnumbered.

Among likely Republican primary voters, the poll found, 45 percent said President Biden’s election was “not legitimate,” and 60 percent said the House Jan. 6 committee was not “fair and impartial.”

The conservative group Club for Growth is piling on with a TV ad saying, “Liz Cheney is wrong about Trump, and she’s wrong for Wyoming.”

Ms. Hageman, a trial lawyer who has fought against federal land-use regulations, has tailored a subtler message to voters, painting Ms. Cheney as more interested in national fame than in representing Wyoming. “Liz Cheney. She’s made her time in Congress and this election all about her,” one of her closing ads says. “Well, it’s not about her, it’s about you.”

First elected in 2016 as Wyoming’s lone member of Congress, Ms. Cheney, 56, occupies the seat her father, former Vice President Dick Cheney, held for a decade. She made almost no appearances at public gatherings in the state this year, in part because of death threats, according to her office.

In her own final campaign message, Ms. Cheney seemed resigned to whatever may come.

“History has shown us over and over again how these types of poisonous lies destroy free nations,” she said of Mr. Trump’s false claims of a stolen election, and of those who repeat them. “No one who understands our nation’s laws, no one with an honest, honorable, genuine commitment to our Constitution, would say that. It is a cancer that threatens our great republic.”

Alaska’s House Race

Ms. Palin, now 58, is one of only three candidates in the special election runoff for the rest of Mr. Young’s term as Alaska’s only House member. But she is among more than 20 in the primary.

Two years ago, Alaskans decided by ballot initiative to overhaul the state’s elections. Partisan primaries were replaced with a free-for-all or “jungle” primary open to candidates from all parties. The top four finishers advance to the general election, whose winner is determined by ranked choice.

Opponents of the changes argued that ranked choice was intended to give Democrats, with only half as many registered voters in the state as Republicans, a better shot in the general election.

Ms. Palin denounced ranked choice this month at the Conservative Political Action Conference in Dallas. “In Alaska, we have this bizarre system,” she said. “It’s convoluted, it’s complicated, and it results in voter suppression.”

Voters in ranked-choice contests prioritize their choices. To arrive at a winner, each candidate’s first-choice votes are counted, the last-place finisher is eliminated, and the eliminated candidate’s second-choice votes are added to those candidates’ totals — and so on, until someone surpasses 50 percent of the vote.

Ms. Palin, who enjoys almost universal name recognition, finished first in the June primary for the remainder of Mr. Young’s term. But she faces serious competition in Tuesday’s three-way contest with Nick Begich III, a Republican businessman from a prominent Alaska Democratic family, and Mary Peltola, a Democrat who served in the Alaska Legislature.

Public polling has been scarce. Alaska Survey Research in late July found that in a three-way race, Ms. Peltola led with 42 percent, followed by Mr. Begich and Ms. Palin with 29 percent each.

The poll also suggested why Ms. Palin may be struggling. She was viewed positively by 31 percent of registered Alaska voters and negatively by 61 percent. With Ms. Palin knocked off and voters’ second choices ranked, the poll found, Mr. Begich edged out Ms. Peltola.

Alaska voters have been skeptical of Ms. Palin since she resigned in 2009, midway through her first term as governor, to pursue rising fame as a national Tea Party star. She flirted with a 2012 presidential run and has more recently appeared as a TV commentator and occasional reality television star.

Alaska’s Senate Race

Alaska’s new election system will most likely work to Ms. Murkowski’s benefit. A traditional partisan primary could well have ended her career.

A moderate Republican, Ms. Murkowski is often a player in bipartisan deal-making in the Senate. She was defeated in a Republican primary in 2010 but won in the general election as a write-in candidate.

She is a strong favorite to come through the top-four primary on Tuesday, along with Ms. Tshibaka, who shared a stage with Mr. Trump in Anchorage last month. A Democrat, Pat Chesbro, is seen as another likely top-four finisher.

In all, there are 19 candidates in the Senate primary, including eight Republicans, three Democrats, two from the Alaskan Independence Party and many with no party affiliation — illustrating that Alaska’s politics are as varied as its famous landscape.