Will America see its first all-female ticket for the White House? The signs are strong. Hillary Clinton is the presumptive nominee of the Democratic Party, and Massachusetts Sen. Elizabeth Warren is most everyone’s favorite pick for the vice presidential slot, despite some very good reasons she shouldn’t be on the ticket. Not only is she under heavy consideration from the Clinton campaign—which is “vetting” her, along with Virginia Sen. Tim Kaine and Housing and Urban Development Secretary Julián Castro—but she’s been at the forefront of the Democratic push against Donald Trump and a strong advocate for the former secretary of state.

The same day President Obama endorsed Clinton as his successor, for instance, Warren did the same in an interview with MSNBC. But that was after she attacked Trump in a merciless address to the American Constitution Society, calling him a “disgrace” and concluding with a brutal slam. “[W]e will not allow a small, insecure, thin-skinned wannabe tyrant or his allies in the Senate to destroy the rule of law in the United States of America,” she said. Last week, Warren visited Team Clinton offices in Brooklyn to give a pep talk, and earlier Monday she could be found at a campaign event in the swing state of Ohio, joining hands with Clinton.

Rhetoric and optics aside, the substantive case for Warren is good. For Clinton, she helps bridge the gap with Bernie Sanders and his supporters, giving skeptical progressives a positive reason to back the Democratic Party’s nominee. Likewise, with her anti–Wall Street message and her work on consumer advocacy, she brings a populist edge to the Clinton ticket, offsetting a weakness that harmed Clinton in the primaries and could do the same in the general. And if part of the Democrats’ anti-Trump strategy is putting his misogyny into sharp focus, then a two-woman ticket is ideal. There’s also the sheer history of it all. To have two women on a major party ticket for president of the United States—after 228 years of only male presidents—would be a remarkable shift in American politics and an enormous symbolic victory for gender equality.* It’s difficult to say what it would mean for the nuts and bolts of the election. But it would stand as a powerful moment in our history, on par with Obama’s election.

But there are serious downsides to Warren, and just as critically, for Warren. To start, a Warren pick risks violating the cardinal rule of the “veepstakes”: Do no harm. A vice presidential pick should neither overshadow the top of the ticket nor jeopardize control of Congress. For all of her virtues, Warren might do both. On the former count, the Massachusetts senator is a star in her own right, with a substantial following independent of the larger Democratic Party. A Clinton–Warren ticket could easily become a Warren-Clinton one in practice, which is not something a presidential nominee wants. (Ask John McCain.)

This might be a risk worth taking if Warren could be replaced by another Democrat in the Senate, but that’s an open question. If chosen, Massachusetts’ Republican governor would pick Warren’s replacement. It wouldn’t be permanent—the state has to hold a special election after roughly five months—but it risks handing an otherwise safe Democrat seat to the GOP, which adds a serious impediment to the party’s agenda, should it win the White House in November without capturing the House of Representatives. In that (likely) world, President Clinton would be working her will through the executive branch and would need a Senate with as many Democrats as possible to confirm her nominees. And if, somehow, Democrats capture the House, they’ll need as large a Senate margin as they can manage to overcome any Republican filibuster.

Obama’s choice of Joe Biden is instructive. Biden was a senator from solid-blue Delaware, and there was no chance that picking him would jeopardize the Democrats’ hold on the chamber. Biden “balanced the ticket,” offsetting Obama’s youth; relative inexperience; and, frankly, his race. But most importantly, Biden was an able partner for Obama, managing key projects like the stimulus, advising the president on national security decisions, and standing as a strong advocate for the administration’s policies. Biden had no apparent trouble in the role of subordinate, and it made him more versatile and valuable in the role.

In Warren’s case, the vice presidency could be valuable if she’s empowered to make crucial bureaucratic and regulatory decisions. It’s not hard to imagine a Vice President Warren who acts as a watchdog within the administration, pushing Clinton to find and hire aggressive regulators and reject Wall Street–sourced appointees. But this would be an unusual amount of freedom for a vice president not named Dick Cheney. Given the extent to which the modern vice president is something like a chief advocate—responsible for pressing, negotiating, and enforcing the president’s agenda—Warren could find herself bound to Clinton’s agenda and priorities, even when she disagrees, leaving her with less autonomy than she had in the Senate. In which case, progressives will have lost one of their most able and vocal champions with little in return.

There’s real symbolism in “Warren for vice president,” but it may hurt progressives in the long term to have her in a presidential administration versus in the Senate, where her advocacy has long and impressive reach. (The same, incidentally, goes for another vice presidential contender, Ohio Sen. Sherrod Brown, a strong progressive choice who would be replaced by a Republican if he left the Senate.)

There are other options. The aforementioned Kaine has a vice presidential résumé (former governor, current senator), deep ties in the Democratic Party (he served as chairman of the Democratic National Committee), and is in step with the party and its liberals on most issues. He won’t excite Sanders or his supporters like Warren might, but he is a safe choice whose departure from the Senate wouldn’t harm the party’s majority (Virginia Gov. Terry McAuliffe would choose his replacement). Castro is young and talented with views on the liberal side of the mainstream. He adds a touch of youth (although that comes with the distinct downside of inexperience) and heightens the symbolism of the Democratic Party’s ticket: If elected, Clinton would serve as the first female president of the United States and Castro as its first Hispanic American vice president.

The “dark horse” choice, if there is one, is Labor Secretary Tom Perez, a longtime member of the Obama administration (he previously served as head of the civil rights division of the Justice Department) with strong bureaucratic skills and a real following among progressive activists and advocates. Of all the possible picks, he might be the closest thing to a middle ground, someone acceptable to all factions in the Democratic Party, a Latino politician with symbolic appeal who is amenable to skeptics on the left of the campaign.

In all of this gaming and strategizing, however, it’s worth remembering one important point: Whoever the vice presidential nominee is, he or she won’t change the game. In general, the vice presidential nominee has no more than a modest effect on the final outcome. Other than John Kennedy in the 1960 election (who in part owed his victory to Lyndon Johnson’s popularity in the South), presidents don’t win or lose on the basis of whom they picked to run on their tickets. Which means that, as Clinton weighs her choice, she shouldn’t spend too much on the politics. What counts is whether he or she can do the job, however defined, without leaving unfillable holes in his or her previous posts. Everything beyond that is just window dressing.

*Update, June 28, 2016: This sentence has been updated to clarify that the 228 years of only men refers to the U.S. presidency.

Read more Slate coverage of the Democratic primary.