For most Major League Baseball teams, the trade deadline is a chance to step back and take stock of the franchise’s trajectory. Although only a small fraction of rumored deals actually end up happening, a team’s willingness to swap assets — as either a buyer or a seller — says a lot about where it is in the cycle between contending for a World Series and playing for the future.

For a few teams, the choice has already been made. These are the clubs on the ends of the baseball spectrum: the bottom dwellers already committed to punting the present in order to stockpile young talent and the clear front-runners who can begin fine-tuning their playoff rosters in July.

But the bulk of the league faces a fork in the road and doesn’t have the luxury of soul-searching with the trade deadline less than two weeks away. The decision to buy or sell is both critical — botched maneuvers can cripple a franchise for years — and further complicated by whether teams are getting a “rental” player (with an expiring contract) or someone who can help them for the next few years. But fear not, baseball general managers, we are here to help.

A few years ago, my colleague Nate Silver and I developed a statistical framework for trade-deadline strategy: the Doyle Number (named for a certain pitcher the Detroit Tigers mortgaged their future to acquire at the 1987 deadline). Doyle represents the number of future wins a team should be willing to part with in exchange for adding an extra win of talent this season. So a Doyle of 1.00 means a team should be indifferent to buying or selling — a one-win improvement this year adds as much to its current World Series odds as a future win would add over the long term.Specifically, its odds over the next six seasons.

“>¹ If its Doyle rises any higher, it should probably be buying (since wins this year are more valuable than future wins); any lower, and it should be selling.

For example, the Cleveland Indians currently have a Doyle Number of 1.48. With a good (though not quite great) roster and decent (but not quite ironclad) division-series odds,Doyle focuses on the division series rather than the wild-card playoff, because the latter’s single-elimination format truly is a crapshoot.

“>² they should probably be trying to add talent over the next few weeks to bolster their chances of returning to the World Series. Meanwhile, the New York Mets’ Doyle is 0.08; their injury-riddled talent base is mediocre, and they have very little shot at the division series, so they should be selling off anyone that isn’t nailed down.

With those ground rules in place, here’s every team’s Doyle number as of July 16:The only change I’ve made to the model for this season is that it now uses the future wins (per 162 games) implied by a team’s Elo rating to assess a team’s talent at the deadline, rather than its rest-of-season projected winning percentage from FanGraphs (which is slower to incorporate changes in a team’s play than Elo).

‘>³

The Doyle topples one of the most common perceptions of the deadline: The team most in need of a trade is the team that is one bat (or one arm) away from making a postseason run. By contrast, Doyle shows that the the teams who should be most willing to buy are the teams having the best seasons — not teams merely on the cusp of the playoffs. It’s a consequence of how random the MLB playoffs are: When even the best teams have long odds of winning, there’s practically no amount of talent a team can add that will cause its World Series probability to hit diminishing returns.

This year, the top Doyle teams are the historically dominant Los Angeles Dodgers and Houston Astros — and, to a lesser extent, the Indians, Washington Nationals and Boston Red Sox. With the possible exception of Houston, each team has at least one position where it can substantially improve, and Doyle indicates they should focus on shoring up those weaknesses in preparation for a World Series run.

More interesting, however, are the clubs near the threshold between buying and selling. These are teams for whom there is less of a clear-cut direction to take — but some decision must be made, since any direction would add more total future championships than merely standing pat. One archetype for that group is the unexpected contender: Think of the Milwaukee Brewers, who find themselves in first place in the National League Central division despite a relatively unimpressive collection of talent. Milwaukee’s 1.26 Doyle suggests it should lean toward buying, since an improved core will become much more valuable in the postseason.

The opposite model might be that of Milwaukee’s division rival, the Chicago Cubs: an expected favorite to whom Doyle gives a disappointingly low World Series probability. The defending champs are having a well-documented down year, and although they’re talented enough to have decent title odds if they make the playoffs, that’s far from guaranteed no matter what deadline moves they make. As a result, their 0.66 Doyle suggests they should lean toward punting on this season.

The Cubs, however, don’t seem willing to give up just yet, trading for starter Jose Quintana last week. They weren’t necessarily wrong to do it, either; it’s important to remember that the Doyle Numbers above mostly apply to rental players. After I tweaked the model to account for the remaining years on Quintana’s contract,Specifically, I gave Chicago 3.2 wins above replacement of talent this season — Quintana’s current talent level, per Tom Tango’s WAR projection system — with an annual half-win decline over each of the next three seasons. Quintana’s total four-year contribution to Chicago’s talent level (9.8 WAR) was then subtracted, spread evenly over the three seasons after his contract expires.

‘>⁴ Chicago’s Doyle for this specific trade became 1.31 — meaning it was probably worth it to give up top prospects in exchange for improving its talent base over multiple seasons.

Those are exactly the kinds of extenuating circumstances a team in Chicago’s current situation needs in order to justify buying instead of selling. Any team with a Doyle north of 0.60 or so could probably do a similar calculation, which means 11 clubs — the Dodgers, Astros, Nationals, Red Sox, Indians, Brewers, Diamondbacks, Yankees, Rays, Cubs and Rockies — could reasonably call themselves buyers this season under the right circumstances.

So we know who’s at the restaurant, and we know who’s on the menu — but what is everyone ordering? We can also use Doyle to build a trade deadline plan for each team, pairing them with players who fit a need and make sense given how realistic a club’s World Series chances are. For each of the 11 teams above, I gathered their current startersThe cutoff for pitchers was either the No. 4 slot in the rotation (for starters) or the setup man role (for relievers).

“>⁵ and tracked how good each is this season, according to Tom Tango’s WARcel projections. I also pulled a list of deadline rental targetsSo, expiring contracts only.

“>⁶ from the excellent RosterResource.com, calculating their WAR talent as well. Multiplying a team’s Doyle Number by the difference in WAR talent between a rental target and its current starter at the same position, we came up with a “deadline index” that indicates how good of a match the player is for the team. After assigning duplicated targets to the team whose index for the player was highest, here are the best pairings between team needs and available players, according to Doyle:

Obviously, here are other layers of complexity involved in actually pulling off these deadline deals, including the quality of the trading team’s farm system, which of its existing players might return from injury before the playoffs, and the possibility of a contract extension with the player being acquired. But the general idea of Doyle is that it provides a flexible framework for trade-deadline decisions, based on how valuable it is to add or shed current talent with an eye on the future.

Keep that in mind as we watch whatever deals unfold over the next couple of weeks. A team’s Doyle Number is a rough guideline, the starting point for thinking about trade possibilities. What happens after that is a combination of reading the market, picking the right moment to strike and then making endless phone calls until that forgettable middle reliever is finally yours.

This week on the FiveThirtyEight Politics podcast, the team examines the most recent version of the Senate Republicans’ health care bill. Then, the group revisits Nate’s “14 versions of Trump’s presidency” and judges which path Trump is most likely on. Plus, Gallup conducted a poll asking respondents who disapprove of the president’s job performance why they disapprove. Was that a good use of polling or a bad use of polling?

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After the latest chapter of Aaron Judge’s breakout season — a dominant Home Run Derby performance that was as impressive in its ease as it was in its raw power — the New York Yankees right fielder seems poised to take over as baseball’s elusive superstar ambassador.

“[He] can become the face of the game,” MLB commissioner Rob Manfred told reporters at the All-Star Game. “He is a tremendous talent on the field and really appealing off the field.”

Perhaps. Like the previous “face of baseball,” Derek Jeter, Judge plays in the biggest media market, for baseball’s highest-profile team. (One that happens to be having a surprisingly good season, its recent slide notwithstanding.) There’s a reason more people have Googled Judge in each month this season than have searched for possible future GOAT Mike Trout in any month of his career.

That’s an awful lot of pressure to heap on a guy who’s still technically an MLB rookie, of course, and his game still has flaws. Although he’s having one of baseball’s all-time great rookie seasons, even a player as gargantuan as Judge might be dwarfed by regression to the mean before the season is over.

Even if Judge gets swallowed whole by the regression monster, though, his phenomenal first half tells us a lot. Just as former Knicks guard Jeremy Lin’s dominant early games suggested he would become at least a pretty good player — which has been (kinda) true — it’s almost impossible to have even a half-season like Judge’s without at least panning out as a good ballplayer, if not something much more.

Let’s put some of his numbers to date in context: His 30 home runs at the All-Star break tied for the 26th most by any player — rookie or not — since 1913;The earliest season for which Baseball-Reference.com’s split finder has data.

‘>¹ his .691 slugging percentage was tied for 75th. He was one of only 44 players to post a first-half batting average of at least .320 and an isolated power of at least .350. Overall, his on-base plus slugging (OPS) was 101 percent better than league average, the 90th-best first half in that department that any hitter has enjoyed since 1913.

Most impressive is Judge’s company on that top-100 list: Babe Ruth (10 times), Ted Williams (eight), Lou Gehrig (five) and so forth. Twenty-five of the 52 players who appeared on the list are in the Hall of Fame, and many of those who aren’t either will be (Albert Pujols, Miguel Cabrera), should be (Larry Walker, Edgar Martinez), would be if not for a scandal (Barry Bonds, Shoeless Joe Jackson) or could eventually be (Bryce Harper).

And among rookies, Judge’s first half was virtually peerless. According to FanGraphs.com, he generated 5.1 wins above replacement (WAR) in the months of April, May and June. Since 1974,The first year FanGraphs.com tracked monthly WAR splits.

“>² no other rookie has broken 5 WAR in the first three months of a season, and only one — Chris Sabo in 1988 — has even earned 4 WAR. Judge is easily off to the best first half of a rookie season in modern history.

But the history of great rookie seasons also shows how strong the pull of regression can be: The average player on this list ended up producing only two-thirds as many WAR in the second half of his rookie campaign as he did in the first. For every Trout, who started strong (despite a delayed call-up) and then decimated the league down the stretch, there were more cases like Devon White, Joc Pederson and even Sabo, all of whom struggled to recapture the magic of their first-half performances.

Judge will likely face a similar fate. Based on his batted-ball numbers, we’d expect Judge to be hitting .303 right now, not .329. Although nobody hits the ball harder, Judge’s .426 batting average on balls in play (BABIP) stands out even next to the game’s other hard hitters, suggesting some kind of second-half downturn is probably in order.Although, as my ESPN colleague Paul Hembekides pointed out, BABIP doesn’t include home runs. If you add Judge’s homers into his BABIP, he’s hitting a ridiculous .516 when he doesn’t strike out — the best single-season mark in history.

“>³ Likewise, Judge has hit a staggering 42 percent of his fly balls out of the park, another relatively luck-driven number that will likely come back down to earth. (Even the most powerful hitters can sustain a homers-per-fly rate of only 25 to 30 percent.)Sometimes this particular stat can be more sustainable than we’d expect on its face because of park effects, but Yankee Stadium is much more of a haven for left-handed power hitters than righties like Judge.

‘>⁴ And while Judge has drawn plenty of walks and shown good plate discipline for a rookie, he’s also whiffed a lot. His low contact rate might become problematic as pitchers study his weaknesses and develop more sophisticated strategies against him.

Given all that, it would be very surprising if the regression bug didn’t bite Judge during the season’s second half.

If and when that does happen, some folks will surely blame it on the dreaded “Home Run Derby curse” — the idea that participating in the contest (much less winning it as Judge did) mucks up a player’s swing over the rest of the season. This topic has been researched to death over the years, with various studies finding it to be a myth … or maybe slightly real. While it’s true that far more Derby participants have fallen off in the second half than have improved, there’s an obvious selection bias at work, too: In order to be picked for the contest in the first place, players need great first-half power numbers — most likely well above their career norms. By regression to the mean alone, we’d expect a group of players selected specifically because they had an abnormally great first half to decline in the second.

For what it’s worth, I tried to account for this effect by measuring Derby contestants against a control group of hitters who had similar first-half power numbers (as measured by at-bats per home run) but didn’t participate in the contest.In each season, the control group had to include players who logged as many plate appearances, as many at-bats per homer and as high an OPS as the lowest-ranked Derby participant in each category. That left me with a sample of 416 hitters in the control group since 2002, versus 122 Derby contestants.

“>⁵ Since 2002, the average Derby participant saw his OPS decline by 56 points in the second half of the season — but the average member of the control group also saw his OPS fall by 49 points, a negligible difference.

However, there is one area where Derby participants did drop off a bit more than we’d expect: raw power. While the control group’s average isolated power (or slugging percentage minus batting average) fell 25 points, the Derby group’s average dropped by 36 points. Both groups saw an identical 0.5 percentage-point dip in hard-hit ball rate, but Derby hitters hit slightly fewer fly balls than we’d expect based on the control group,A decline of 1 percentage point, versus a 0.6-percentage-point dip for the control group.

“>⁶ and their rate of home runs per fly fell by 2.8 percentage points, compared with a decline of 1.7 percentage points for the control group.

Either way, the overall cost of participating in the Derby is (at most) quite small compared with the toll exacted by regression to the mean.

And said regression will probably be Aaron Judge’s biggest enemy in the second half of the season. Sadly, no player can escape its clutches for very long — not even the most wildly entertaining hitter on the planet.

On this emergency podcast, the FiveThirtyEight Politics podcast team looks at recent revelations about the Trump campaign’s relationship to Russia. Donald Trump Jr.’s emails showed that he, along with President Trump’s son-in-law Jared Kushner and former Trump campaign manager Paul Manafort, took a meeting with a Russian lawyer who claimed to be offering incriminating information about Hillary Clinton. The emails said the information was part of the Russian government’s support of then-candidate Trump.

You can listen to the episode by clicking the “play” button above or by downloading it in iTunes, the ESPN App or your favorite podcast platform. If you are new to podcasts, learn how to listen.

The FiveThirtyEight Politics podcast publishes Monday evenings, with occasional special episodes throughout the week. Help new listeners discover the show by leaving us a rating and review on iTunes. Have a comment, question or suggestion for “good polling vs. bad polling”? Get in touch by email, on Twitter or in the comments.

WIMBLEDON, England — No one has any idea which of the four semifinalists will win the women’s singles title at Wimbledon on Saturday. But there is one prediction you can make with confidence: The remaining matches will be close, hard-fought contests that could easily last three sets.

Women’s tennis matches keep getting longer and more grueling. Since the 2014 Australian Open, the four Grand Slams — including this year’s Wimbledon, which is not yet complete — have averaged 40.9 three-set women’s matches per tournament. That’s up from an average of 36.8 between 1988 and 2013. Last month’s French Open saw 46 women’s matches go the distance, which was the most ever at Roland Garros and tied for the fourth-highest tally at any Slam since 1988, according to the WTA. So far this Wimbledon, there have been 43 three-setters, with three matches left to play. One more would put this year’s competition in a three-way tie for the most three-set matches at Wimbledon since 1988 — the 2011 and 2008 tournaments each saw 44.

Tennis’s current crop of women are, as a group, extremely talented but frequently flawed players, which means anyone can win almost any match. This is especially true while Serena Williams, the sport’s most dominant player, is off the circuit during her pregnancy. For evidence that anyone can win, look no further than last month’s French Open, where an unseeded 20-year-old — Jelena Ostapenko, who at the time was ranked 47th in the world — took home the trophy. Five of her seven matches there lasted three sets, including the semifinal and final. In the final, Ostapenko trailed by a set and 3-0 before charging to victory.

Agnieszka Radwanska, seeded 9th at Wimbledon, saved two match points in the second round and beat Christina McHale, an unseeded American, in three sets. She needed three sets in the next round too, against Timea Bacsinszky, seeded 19th. Radwanska, who became a professional in 2005, said that today’s tennis is far more tense than it was in the past.

“A couple of years ago, those first two rounds when you didn’t play against seeded players, it was easy,” Radwanska said. “You don’t have to play 100 percent and you’re gonna win. Obviously it’s not gonna happen anymore in tennis right now. You can play … those players that you really don’t want to play in the first round, and that’s why I think we can also see a lot of upsets in early rounds.”

In men’s tennis, the late rounds of many tournaments remain compelling — it’s hard to be bored when greats like Roger Federer, Rafael Nadal, Novak Djokovic, Andy Murray and Stan Wawrinka are duking it out — but in early rounds, the excitement has fizzled. The men’s side rarely sees any early upsets, and players are more likely to retire in the middle of matches; nine men bowed out during the first three rounds at Wimbledon this year, compared to only two women. We don’t yet know if the women will have any 30-something superstars in the final this year, but the women’s matches are competitive and entertaining from the first round to the last. CoCo Vandeweghe, an American currently ranked 25th, described the men’s field as “top-heavy” compared to the women’s side.

“In the women’s game, as you’ll see, there is more upsets along the way with the seeded players, because I think there is more depth in the 20s to 30s to 40s,” she said. “I think there is some very solid depth of players that can make an impact against a top player. I mean, I’m an example of that myself.”

British star Johanna Konta says that anything can happen and happen quickly. She has won three three-set matches so far, including one that required 18 games in the third set. She said she worries about everyone she faces.

“I don’t underestimate any opponents,” she said. “I respect each and every opponent that I’m playing because I’m fully aware of the challenges that they will bring.”

Of course, no one will underestimate Konta’s next opponent. She plays five-time Wimbledon champion Venus Williams in the semifinals on Thursday.

During their free-agency period, NHL teams haven’t engaged in an all-out superpower arms race like their basketball brethren. Rather, NHL free agency has been a methodological redistribution of assets — one that can be traced to the addition of the Vegas Golden Knights and the expansion draft that ensued. All that extra movement of bodies and money, coupled with every general manager’s rush to add or retain a free-agent gem, means there will be a lot of familiar faces in unfamiliar jerseys next season.

Not every free agent has signed — in particular, this year’s rather elderly free-agent class has a few graybeards who remain without suitors — but most of the league’s big targets have decided where they’ll play next season. So without further ado, here’s our unofficial and admittedly incomplete guide to the moves that will affect the 2017-18 season.

Kevin Shattenkirk, New York Rangers

Come trade deadline time the past several seasons, Kevin Shattenkirk’s name has been linked seemingly to every team in the NHL (including the Rangers in February), and for good reason — since his rookie season in 2010-11, Shattenkirk ranks 10th among defensemen in goals scored, tied for eighth in assists and 10th in total points. His 35 power play goals are tied for fifth over that same stretch, and his power play points are good for third.

The New York Rangers ranked 10th in the league in power-play percentage last season, and that should only improve with the addition of hometownHe’s from nearby New Rochelle.

“>¹ boy Shattenkirk. The newly minted Rangers defenseman chews up big minutes, and he’s never finished a full season with a Corsi For percentage of less than 51.2. For the uninitiated, a Corsi For score of more than 50 percent means the player’s team was controlling the puck more often than not when he was on the ice, so the addition of Shattenkirk should make the Rangers a stronger possession team from the jump.

Over the past five seasons, the bulk of the blame for each Rangers playoff ouster seems to have pointed back to the blue line. With a mishmash of overworked veteran defensemen and disappointing acquisitions, the Rangers sit roughly in the center of the pack in shots allowed per game over the past five seasons. Adding a puck-moving defenseman who is entering his prime with better-than-average possession numbers should help decrease those shot totals — and take some of the burden off the suddenly human-appearing goaltender Henrik Lundqvist’s shoulders.

The Rangers’ defensive corps will look different with the retirement of Kevin Klein and longtime defenseman Dan Girardi going to Tampa Bay, but the franchise that hasn’t had a legitimate defensive star since Brian Leetch landed a big upgrade in Shattenkirk.

Justin Schultz, Pittsburgh Penguins (re-signed)

Justin Schultz’s career began with so much promise. In the lockout-shortened 2012-13 season, he scored 27 points from the blue line as a 22-year-old rookie. But in the 2.5 seasons that followed, he failed to match the impressive points-per-game numbers he put up as a rookie, and people in Edmonton began to sour on him. Oh what a difference a few seasons and some new scenery make.

Since joining the Penguins, Schultz has transformed into a top-line defenseman, and he was a key piece to their Stanley Cup-winning campaign last season. In less than two seasons with Pittsburgh, he’s scored 0.61 points per game from the back,He produced 0.41 points per game during his time in Edmonton.

“>² and he’s driven possession at a better clip, too.

Of the free agents in the 2017 class, Schultz’s goals above replacementGoals above replacement aims to estimate the number of net goals a player produces compared to a replacement-level player in the same amount of ice time. (The replacement level represents the amount of production a team could expect from a minimum-salary player at a given position.)

“>³ mark of 13.9 ranks at the top. Pittsburgh may have lost Trevor Daley to free agency, but a defensive core built around Schultz and a healthy Kris Letang is nothing to turn your nose up at.

Justin Williams, Carolina Hurricanes

The Carolina Hurricanes were pretty close to making the playoffs last season. They ended up falling eight points short of an eighth seed but were in the hunt until the last few weeks of the season — and if they make the 2018 postseason, they’ll be glad they signed Justin Williams. Williams is well-known around the league for being a playoff beast. His career playoff points per game rate of 0.67 is better than his regular-season rate of 0.63, and he’s tied for 14th among active NHL players in game-winning playoff goals — and he’ll be returning to the place where he lifted the first of his three Stanley Cups, and where he twice netted 30-plus goals.

Ten times in his career, Williams has scored 40 or more points, which should help a Carolina team that finished in the bottom third of goals scored last season. Oh, and Williams can do more than score timely playoff goals; his career Corsi For percentage of 56.9 isn’t too shabby either.

Alexander Radulov, Dallas Stars

Alexander Radulov is … mercurial. The Russian winger has been on alternating stints between the NHL and Russia’s KHL for his entire professional career: two seasons in Nashville, four in Salavat, a few more weeks in Nashville, four seasons in Moscow and then one in Montreal. So after his 54-point campaign with the Canadiens last season, he was bound to go anywhere from British Columbia to Siberia. He chose Texas. He’ll be there for a while, too, signing a long-term deal with the Dallas Stars. (Radulov’s signing in Dallas can also be read as a rebuff of KHL league chairman Arkady Rotenberg — and his close friend Vladimir Putin — who hopes to pry as many Russian players away from the NHL to ensure eligibility for the 2018 Winter Olympics in South Korea.)

But Radulov is a world-class scorer — he’s tallied 156 points in 230 NHL games and scored 492 points in 391 KHL games — and was by all accounts a great teammate in Montreal, so the Stars were lucky to land the big sharp-shooting right winger. Pair Radulov’s propensity for burying the puck with an insanely talented top line of Tyler Seguin and Jamie Benn, and the Stars might have on their hands the highest-scoring top line in the NHL next season.

Patrick Marleau, Toronto Maple Leafs

The Toronto Maple Leafs were a formidable offensive force last season, ranking fifth in total goals scored. The Leafs got a ton of scoring — 32 percent of their 250 goals — from their three rookie sensations: Calder Trophy-winning Auston Matthews, Mitch Marner and William Nylander. Patrick Marleau will make the Leafs’ offense only more dangerous, and you could do worse as far as mentor figures go.

Marleau, 37, has been in the NHL for 19 seasons, and until now he’s never worn anything but “deep Pacific” San Jose teal. In his 19 seasons, he’s made a point of scoring at least 19 goals — he’s only failed to do so three times, and one of those seasons was shortened to 48 games by a lockout. Marleau has also played in 177 playoff games, experience that should help the upstart Leafs — who gave the mighty Washington Capitals a serious scare in the first round of last season’s playoffs — immensely.

Joe Thornton, San Jose Sharks (re-signed)

“Jumbo” Joe Thornton is easily one of the best passers in NHL history. And he is certainly the best passer in NHL history who has a beard that is probably sentient. One of the premier centers of his generation, Thornton trails only Jaromir Jagr — who himself remains mysteriously unsigned — on the active assists leaderboard.He also ranks 13th all time and needs just 43 more to leapfrog Gordie Howe as the ninth-highest dime-disher in NHL history.

“>⁴ Jumbo hasn’t scored fewer than 50 points in a full season since he was a 19-year-old playing for the Boston Bruins in 1998-99.

He’s not going to score a ton of goals — never really has — but San Jose was probably smart to prioritize Thornton over his longtime teammate Marleau. (Fun fact: Thornton and Marleau went first and second, respectively, in the 1997 NHL draft.) Thornton isn’t the lights-out goal scorer Marleau is, but he creates more opportunities for his teammates than his longtime counterpart; 43 of Jumbo’s 50 points were assists, 18 of which came on the power play. For a team that struggled mightily on the power play last season — San Jose ranked 25th in the league — the Sharks will need all of the passing acumen they can muster with the man advantage next season.

Ryan Miller, Anaheim Ducks

Ryan Miller probably won’t play much in Anaheim — starter John Gibson is just 23 years old and had a spectacular 2016-17 season — but there aren’t many teams in the NHL that can say they’ve got a backup who was once considered among the best goalies on the planet. Miller’s numbers were down a bit last season, but you can’t really blame him — his Vancouver Canucks gave up the sixth-most shots of any team in the league. Despite his impressive résumé, Miller might have to battle with Reto Berra for the backup position in Anaheim. Still, though, the Ducks are in great shape between the pipes — there aren’t many teams in the NHL that have the luxury of choosing from three proven(ish) NHL goaltenders.

Karl Alzner, Montreal Canadiens

Andrei Markov, unsigned

The Washington Capitals probably didn’t have the cap space to re-sign longtime defenseman Karl Alzner, but they probably didn’t really care to either. No worries for Alzner, though, because the Habs delivered a pretty solid payday. But truth be told, the native of Burnaby, British Columbia, got an awful lot of money — a cap hit of $4.6 million a year, to be exact — for a guy whose Corsi For percentage ranked 184th out of 229 qualifiedWith a minimum of 20 games played.

“>⁵ defensemen in 2016-17.

Alzner actually doesn’t do a whole lot very well — in 82 games, he scored just 13 points, zero of which came on the power play. He’s still relatively young for a defenseman — he’ll turn 29 before the 2017-18 season begins — and so perhaps Les Habitants are hoping Alzner’s game has room to grow. But that doesn’t change the fact that they signed a guy whose possession stats are subpar to a lucrative deal while their longest-tenured player — Andrei Markov — remains seated on the bench, waiting to see if he’ll play in Montreal next season.

Markov is old, but if last season’s performance is any indicator, he still has some gas left in his tank. Markov was limited because of injury but put up impressive numbers in the 62 games he did play in. At 38, and still playing more than 21 minutes per game, he drove possession at the best rate of his career. He also scored 36 points from the blueline, good for .58 points per game.Markov’s career points per game rate? .58.

“>⁶

In 16 NHL seasons, Markov has only ever worn a Habs jersey. Montreal would be foolish not to re-sign their longest-tenured player.

This week on the FiveThirtyEight Politics podcast: The team is joined by senior editor and chief economics writer Ben Casselman to talk about President Trump’s economic vision. Then, senior science writer Maggie Koerth-Baker talks about the fight over voter data. Finally, is the latest Russia development smoke or fire?

You can listen to the episode by clicking the “play” button above or by downloading it in iTunes, the ESPN App or your favorite podcast platform. If you are new to podcasts, learn how to listen.

The FiveThirtyEight Politics podcast publishes Monday evenings, with occasional special episodes throughout the week. Help new listeners discover the show by leaving us a rating and review on iTunes. Have a comment, question or suggestion for “good polling vs. bad polling”? Get in touch by email, on Twitter or in the comments.

The Boston Celtics, after whiffing on trades for Jimmy Butler and Paul George, finally got some good news this week when Gordon Hayward announced that he was leaving the Utah Jazz to play for the Celtics. Although Isaiah Thomas would have some beef with this assessment, Hayward’s well-rounded combination of skills will probably make him the best player on the Celtics next year. He’s a good fit with the team’s approach.

And yet, projection systems (including our own CARMELO) are somewhat skeptical of the Celtics, not expecting them to improve on last year’s 53-29 performance or to seriously challenge the Cleveland Cavaliers for Eastern Conference supremacy. Hayward is also fairly expensive; he’ll cost the Celtics $128 million over four years (the fourth season, 2020-21, is a player option). So let’s ask a tough question of Boston and general manager Danny Ainge: If Hayward is the best player on your team, could that team plausibly be good enough to win a championship?

The answer is probably not. Hayward made the All-Star team last season, but he’s a long way from being a superstar. A handful of modern NBA teams — the 1988-89 and 1989-90 Detroit Pistons, the 2003-04 Pistons, and the 2013-14 San Antonio Spurs — won a title with someone about as good as Hayward as their best player. But this is unusual: It requires a team to be constructed almost perfectly, with above-average players at nearly every position, a deep bench, and a cohesive rotation. It sometimes also requires a fair amount of luck.The “Bad Boys” Pistons peaked just as the 1980s Celtics and Lakers were declining, but before Michael Jordan’s Bulls really took off, for example.

“>¹

But Hayward can be a building block toward a championship. He’s roughly as good as the second-best player on a typical championship team. That might sound like faint praise, but it’s no small achievement.

Let’s develop some terminology to describe degrees of the stardom in the NBA. I’ll introduce three types of players: Alphas, Betas and Gammas.

• An Alpha is a player who’s as good as the best player on a typical championship-winning team. This is an MVP contender — one of the half-dozen best players in the league.
• A Beta is as good as the second-best player on a typical NBA champion. As I mentioned, Hayward is a good example of a Beta. Betas are usually All-Stars, perhaps even All-Star starters, and they’re among the best players at their position. But they’re not among the very best players in the league.
• And a Gamma is good as third-best player on a typical championship team. A Gamma might be an All-Star, but he usually won’t make one of the three All-NBA teams. He probably has one or two weaknesses (defense, shooting, etc.) along with his obvious strengths. But he’s still a very good player and might be the best player on a non-contending team. Thomas, although he’s somewhat difficult to evaluate because of his defense — various statistical systems rate it anywhere from mediocre to execrableThere’s a big difference between mediocre and execrable. If Thomas’s defense is merely somewhat below average, as the NBA’s opponents’ shooting data suggests, then his offense more than makes up for it and he’d probably qualify as a Beta. If it’s terrible, as RPM suggests, then Thomas is barely even an above-average player overall. Classifying Thomas as a Gamma, as CPM does, is something of a compromise between these assessments.

  ‘>² — is a reasonably good example of a Gamma.

At any given time, only a few dozen players in the league will rated as Alphas, Betas or Gammas. (CARMELO projects that there will be 35 of them in 2017-18, for example.) It’s these players who determine who competes for NBA championships. Doesn’t depth matter also? Well, sure. A well-rounded roster is often the difference between winning a title and losing one. But a team needs its share of star-level talent to compete for a championship in the first place. Otherwise, it’ll usually wind up like last year’s Celtics, a well-constructed team that was overmatched in the playoffs.

Below, you’ll find a table listing the top three players on NBA championship teams since 1984-85 — the first year the league used a salary cap — as rated by a statistic called Consensus Plus-Minus. Consensus Plus-Minus, or CPM, is a statistic I use when I don’t want to get into arguments about the value of individual players. It reflects a combination of four popular statistics — Real Plus-Minus,RPM is available from the 2000-01 season onward. For seasons prior to 2000-01, I use BPM twice in the average.

“>³ Box Plus/Minus, Win Shares and player efficiency rating — equally weighted and translated to the same scale. It also adjusts for the player’s position, which the other metrics do not,CPM measures a player’s value relative to the positional average, rather than relative to the league average. Right now, there’s an excess of good centers and point guards but less depth among small forwards and shooting guards. This adjustment helps Hayward, who plays both those positions.

“>⁴ and it regresses players’ ratings to replacement level if they fall below a certain threshold of playing time.If the player plays less than 20 minutes per team game — or 1640 minutes over an 82-game season — his actual CPM is blended with a replacement-level CPM of -2 points per 100 possessions.

“>⁵ For predictive purposes, we think CPM is liable to be slightly less accurate than the blend of statistics CARMELO uses (a combination of RPM and BPM), but CPM is still a perfectly reasonable stat and much more in line with the consensus view of NBA players. Like BPM and RPM, CPM is expressed in net points added or subtracted per 100 possessions. So a player with a CPM of +2.5, teamed with four average players, would help his team to outscore his opponents by 2.5 points per 100 possessions, for example.

To no one’s surprise, the best players on title-winning teams are usually extraordinary talents. Among the 33 NBA champions since 1985, the top-rated player on the team, according to CPM, was one of the three best players in the league that season on 23 occasions. And the team’s best player was among the top 10 in the league on all but three occasions. The exceptions were Bill Laimbeer of the 1988-89 and 1989-90 Pistons (CPM, perhaps dubiously, rates Laimbeer ahead of his teammate, the other Isiah Thomas) and Ben Wallace of the 2003-04 Pistons. Pretty much all the other No. 1 players are current or future Hall of Famers, however, with the possible exception of Manu Ginobili, who may be a borderline case.

No matter how brightly he shines, however, a superstar usually can’t deliver a title without a good sidekick or two.Take LeBron James’s 2012-13 season, for example, when he had a CPM of +10.6 and played almost 38 minutes per game — about as good as an NBA player can be under modern conditions. A team with that version of LeBron, plus a roster full of replacement-level players, would still project to go only 41-41, according to our forecasting method. A team with that version of LeBron plus a roster full of league-average players (as opposed to replacement-level players) would project to go 59-23, which is closer to being a title contender. League-average players don’t just grow on trees, however.

“>⁶ On average, the second-best player on these championship teams was the 14th-best player in the league, according to CPM. And although not every champ had a classic “Big Three” like LeBron James’s Miami Heat, the third-best player on the championship team rated as the 37th-best player in the league, on average — still very solid.

So let’s get back to the idea of Alpha, Beta and Gamma players, which were meant to correspond to a typical championship team’s best, second-best and third-best players. By looking at the historical data, we can define these classifications as follows:

• An Alpha has a CPM of +6.0 or higher.
• A Beta has a CPM of between +3.5 and +6.0.
• And a Gamma has a CPM of between +2.0 and +3.5.

I re-ran CARMELO using CPM instead of its usual blend of statistics, and it projected the following players to be Alphas, Betas and Gammas for the upcoming NBA season:

This makes for a fairly intuitive list. LeBron, Russell Westbrook, James Harden, Stephen Curry, Kevin Durant and Kawhi Leonard are the league’s six Alphas. Chris Paul falls just short of the Alpha category; instead, he joins players such as Hayward, Giannis Antetokounmpo, Rudy Gobert and Anthony Davis on the Beta list. Gammas include players like Isaiah Thomas, Kyrie Irving, Kevin Love, John Wall, DeAndre Jordan and Paul Millsap.

So then all you need is an Alpha, a Beta and a Gamma and — presto! — you win an NBA championship? Actually, your options are more flexible than that. A team with an Alpha and a Beta — say, this year’s Houston Rockets — could probably skip the Gamma if they had a deep rotation. A team with no Alphas but three Betas — say, Jimmy Butler, Antetokounmpo and Kyle Lowry — would more than likely be good enough to contend for a title. A team with a very strong Alpha could go without a Beta and make up for it with two or more Gammas instead — that’s sort of how the current Cavaliers are constructed.

To help teams think through these decisions, let’s invent one more statistic, which I’ll call star points. The formula is simple: A team gets three star points for each Alpha on its roster, two for each Beta, and one for each Gamma. Next year’s Warriors project to have 9 star points, for example: three each for Curry and Durant, two for Draymond Green and one for Klay Thompson.

Even having that much talent on your roster doesn’t necessarily guarantee a title. But historically, a team’s chances of winning a title are remote if it has four or fewer star points. It has a fighting chance with five or six star points, depending on how the rest of the roster is constructed. And its probability increases rapidly once it acquires seven or more star points.

This system isn’t perfect, but it lines up intuitively with how we evaluate teams. After the Warriors and their nine projected star points next season, the Cavaliers and Rockets are the closest thing the league has to ready-made title contenders, as they’re tied for second at five star points each. They’re followed by the Thunder, Timberwolves and Pelicans at four each; these four-point teams probably need at least one more thing to click (say, George taking the next step in Oklahoma City) to be title-worthy. The Celtics are one of several teams with three star points.

This measure can underrate the importance of team depth; the Spurs, who have only three star points, are rated too low, for instance. The Celtics — although they’re losing a few players to make room for Hayward — are also a deep team, with lots of average or slightly-above-average players and lots of draft picks to keep priming the pump. They could probably compete for a title with five star points, therefore, instead of needing six or seven. Adding another Beta-level player might be enough to do the trick.

It’s hard to see where that player comes from, however. The Celtics lost some of their financial flexibility in signing Hayward. And while they could develop a star player rather than acquiring one, giving more playing time to young players such as Jaylen Brown and rookie Jayson Tatum could make them less competitive in the short run.

If there’s one Celtics move that looks bad in retrospect, it isn’t necessarily trying and failing to acquire Butler or George, it’s trading the No. 1 draft pick for Philadelphia’s No. 3 pick, with which they chose Tatum. While Tatum has a fairly promising projection, he doesn’t have the upside of No. 1 pick Markelle Fultz, whose comparables include players such as Harden, Westbrook and Wall. The trade might have made sense for a team that already had its stars in place and wanted to develop complementary players around them, but the Celtics have plenty of complementary players and are short on stars.

At the same time, it would be easy to underestimate the challenge Ainge faced. The Celtics’ 53-29 record last year was deceptive, in that it came against a weak conference and relied on what were arguably career years from several players, including Thomas.The Celtics also outperformed their 48-34 Pythagorean record by several games.

‘>⁷ In many respects, they were a rebuilding team dressed up as a competing team. And precisely because the Celtics weren’t just one player away from contending for a title, Ainge needed to acquire a player like Hayward or Butler without compromising the Celtics’ ability to acquire or develop another such player down the road. Even if the Celtics are still a star away from seriously contending for a title — maybe even a superstar away — that’s closer than they were last week.

We’re back after a holiday break! This week the FiveThirtyEight Politics podcast team digs into President Trump’s trip abroad for the G20 Summit and how worldwide trust in U.S. leadership has changed since President Obama left office. Plus: Survey Monkey took a poll asking if Americans trust outlets like CNN and The New York Times more than President Trump. Was it a good use of polling or a bad use of polling?

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