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#### Should The U.S. Be Worried About The Next Generation Of Women’s Soccer?

Wherever it lands in the upcoming World Cup draw — which will be held in Paris on Saturday at noon ET — the United States Women’s National Team will go into next year’s tournament in its familiar perch as favorites. The U.S. has been the most dominant team in the history of the event (which began in 1991), capturing the Cup three times and winning more matches (33) than any other country. Along the way, Team USA has given us multiple generations of superstars, the latest of which includes names like Carli Lloyd, Alex Morgan, Megan Rapinoe and Tobin Heath — all of whom figure to represent the Stars and Stripes in France next summer.

But this won’t quite be the same roster as the one last seen hoisting the World Cup in 2015 (or even the one America sent to the 2016 Olympics). Likely gone are goalkeeper Hope Solo, forward Abby Wambach, midfielder Lauren Holiday and defenders Meghan Klingenberg, Christie Rampone and Ali Krieger, among others. And while the U.S. has perpetually been able to retool on the fly with the emergence of even greater young talent than before (remember Morgan’s breakout performance as a 22-year-old at the 2011 World Cup?), there’s growing concern that the next generation won’t be as ready to carry the torch.

Specifically, the U.S. results at the youth level — including both the Under-17 and Under-20 Women’s World Cups — have been pretty mediocre in recent years. The U.S. didn’t advance out of its group in either tournament this year and has advanced past the quarterfinals just twice at the youth level since 2008.1

Obviously, those 2018 youth players won’t be old enough in 2019 to figure into the senior team’s World Cup fortunes. But the lack of earlier success could suggest a down period coming for the Americans in upcoming World Cups. At least, that’s assuming youth-level results are predictive of future senior-team outcomes. But is that true?

Fortunately for the U.S., the relationship there is only a moderate one. To measure this, I’m using a system called “Dynasty Points,” which I’ve used before to track a team’s postseason success over multiple years. In each tournament, you get 1,000 points for winning the whole thing, 500 for finishing second, 250 for losing in the semifinals (with a 100-point bonus for finishing third), 115 for losing in the quarters and 40 for losing in the Round of 16 (when applicable).2 If a team’s performance at the youth level was cause for concern or optimism for later iterations of the senior team, we’d expect a strong relationship between World Cup dynasty points and dynasty points produced by the youth teams in the decade between five and 15 years before.

But looking at the 2007, 2011 and 2015 World Cups (and weighting by the number of youth tournaments that led up to each — meaning 2015 had more of a sample, since the era of two youth tournaments didn’t begin until 2008), the correlation coefficient between youth performance and World Cup success is 0.49. That’s not a nonexistent relationship by any means, but it also indicates that about three-quarters of the variation between countries in World Cup success is explained by something other than youth-level results in the years beforehand.

Broadly speaking, if a team consistently finishes well in the Under-17 and Under-20 events, it tends to do better at the World Cup as well. Many of the most successful youth-level countries — such as Germany, the U.S. and Japan — are also among the best World Cup teams. But the relationship isn’t perfect. According to Dynasty Points, North Korea is the most successful youth-level team since 2002, winning four tournaments and finishing second on two other occasions. Yet North Korea has advanced out of the group stage just once in its World Cup history.3 Meanwhile, Norway and Sweden have been incredibly successful at the World Cup level (they rank third and fifth in all-time Dynasty Points, respectively) and have combined to advance out of the group stage of a youth tournament just twice since 2002.

The U.S.’s own history is instructive in the imperfect relationship between youth results and World Cup outcomes. Although it dominated junior tournaments from 2002 to 2008 — winning two events, finishing second in another and never losing before the semifinals — it has also made two World Cup Finals and won Olympic gold with the senior squad since its ongoing youth-level dry spell began in 2010.

##### Is the USWNT’s youth-level talent pipeline drying up?

Results for the United States at both the World Cup and youth-level tournaments

Year Tournament Type Matches Wins* Dynasty Points
1991 World Cup Senior 6 6.0 1000
1995 World Cup Senior 6 4.5 350
1999 World Cup Senior 6 6.0 1000
2002 U-19 WC Youth 6 6.0 1000
2003 World Cup Senior 6 5.0 350
2004 U-19 WC Youth 6 5.0 350
2006 U-20 WC Youth 6 4.0 250
2007 World Cup Senior 6 4.5 350
2008 U-17 WC Youth 6 3.5 500
2008 U-20 WC Youth 6 5.0 1000
2010 U-20 WC Youth 4 2.5 115
2011 World Cup Senior 6 4.0 500
2012 U-17 WC Youth 3 2.0 0
2012 U-20 WC Youth 6 4.5 1000
2014 U-20 WC Youth 4 2.0 115
2015 World Cup Senior 7 6.5 1000
2016 U-17 WC Youth 3 1.0 0
2016 U-20 WC Youth 6 3.0 250
2018 U-17 WC Youth 3 1.0 0
2018 U-20 WC Youth 3 1.5 0

Dynasty Points are awarded for success in a tournament. A team gets 1,000 points for winning the tournament, 250 for losing in the semifinals (with a 100-point bonus for finishing third), 115 for losing in the quarters and 40 for losing in the Round of 16.

* Ties are counted as a half-win.

Source: FIFA

It bears mentioning that the 2012 Under-20 squad also won gold, so the U.S.’s recent youth results haven’t been completely devoid of success. And it’s also important to note that the senior team can poach promising young stars from the youth level, inherently limiting the U.S.’s performance in U-17 and (especially) U-20 tournaments in favor of bolstering the main roster.

It’s true that next summer’s team will be asking more of some players who hadn’t taken on full-fledged starring roles in previous major tournaments. But a number of those have drawn rave reviews during friendlies and qualifiers with the national team this year, including Lindsey Horan and Crystal Dunn, both of whom also logged time on the 2016 Olympic team. And forward Mallory Pugh, who will turn 21 just a few months before the World Cup, is as promising as any member of the U.S.’s next generation. Pugh returned from injury to score six goals and set up three others in 10 international games this year — and could be just scratching the surface of her talent.

According to ESPN’s Stats & Information Group, the average age of Team USA in its 2018 games was 26.4 years old, compared with the average age of 32.6 for the group that took the field in the 2015 World Cup. A few standbys do remain from the rosters of old — Lloyd and Rapinoe are still the anchors of this team, while Morgan remains at the top of her game, having scored or assisted on 14 goals in 15 games with the national team this year. But the 2019 World Cup will also serve as a transition of sorts into a new era for the USWNT. And although America’s youth-level results leading up to next year aren’t great, we shouldn’t assume that the U.S.’s senior-team dominance will end any time soon.

#### The Nationals Don’t Need Bryce Harper

The best thing to happen to the Washington Nationals this offseason might be Bryce Harper turning down the $300 million contract offered by the club at the end of the season. Rather than allocating vast resources to one free agent superstar, the Nationals made smaller moves to shore up their weak spots, signing starting pitcher Patrick Corbin on Tuesday to bolster their rotation and adding two catchers earlier in the offseason. It wasn’t clear if the Nationals would be able to compete without Harper. But according to some forecasts, they are already better without him. They have improved themselves in a National League East where every team — save for the Miami Marlins — seems intent on trying to dramatically improve this offseason. After the Nationals finished 82-80 last season, FanGraphs projects them as the sixth-best team in the majors at the moment — the best team in the division — with a 91-71 forecasted record. While it’s unclear whether the Nationals still want to compete for Harper, they don’t need him to improve over last season. Baseball is a weak-link sport, meaning that the quality of the minor contributors on a team is more important than in a sport like basketball, where star power is paramount. Baseball, by rule and nature, spreads around opportunity more uniformly. A slugger hits only once every turn through a lineup, an ace pitcher pitches once every five days. The same is true for the light-hitting starting shortstop and the back-of-the-rotation starting pitcher. While it’s possible to win with a stars-and-scrubs approach, that’s inherently riskier, given that one injury can derail a club’s entire season. That’s one reason that a successful MLB team rarely allocates more than 16 percent of its payroll to one player. With this in mind, the most efficient way to improve may be to strengthen the weakest links of a roster, not to target brand-name stars as saviors. A team can improve itself quickly and often more efficiently by bringing the most underperforming areas of its roster closer to, or exceeding, average production. It’s an idea the NEIFI analytics company attempted to quantify. (NEIFI co-founder Adam Guttridge was hired by the Mets to lead their data-science department.) While Corbin wasn’t cheap — he signed a six-year,$140 million contract — he upgrades what was a top-heavy Nationals staff that had question marks after Max Scherzer and Stephen Strasburg. Perhaps the Corbin signing was made possible by payroll flexibility coming from the club moving away from resigning Harper. Corbin could provide a similar production bump as Harper for a lesser cost. With his 3.5 forecasted WAR, Corbin projects as a 2.8-WAR upgrade over the Nats’ incumbent fifth-best starting pitching option, Joe Ross. (The Nationals project to receive 2.7 WAR from right fielders without Harper. Harper projects to produce 4.9 WAR in 2019.)

But even before signing Corbin, the Nationals had cheaply and significantly improved upon their weakest link.

The Nationals’ catchers have been among the most ineffective positional groups in recent seasons. Since 2015, the club ranks 27th in catcher wins above replacement, according to FanGraphs. That mark doesn’t even include catcher framing metrics, in which the Nationals were also one of the worst teams in 2017 (-10.7 framing runs) and below-average last season (-4.5 framing runs), as reported by Baseball Prospectus. And according to weighted runs created plus, a measure of offensive ability that adjusts for park and run-scoring environments,1 Nationals catchers were 27th in the majors, with a wRC+ mark of only 64.2

So it makes sense that the Nationals made upgrading their catcher group a priority in acquiring Yan Gomes in a trade with the Cleveland Indians last week and signing Kurt Suzuki to a two-year, $10 million deal in mid-November. At a time when catcher is the weakest offensive position in baseball (84 wRC+), Suzuki and Gomes are average performers compared with all hitters but virtual stars compared with the rest of the catching field. Suzuki posted a 108 wRC+ with the Braves last season (and a 127 mark in 2017). Gomes had a 101 wRC+ in Cleveland (on top of a 92 career mark). The players represent a major upgrade over Matt Wieters, who entered the past two seasons as the club’s primary catcher. Gomes, who will likely be the starter, is an above-average defender for his career. Consider the upgrade Gomes and Suzuki would have provided over the past two seasons. Nationals catchers combined for 0.5 WAR IN 640 plate appearances last year, according to FanGraphs. Gomes and Suzuki combined for 4.2 WAR (3.3 WAR when adjusted for 640 plate appearances). In 2017, Nationals catchers produced an MLB-worst -1.2 WAR. In 692 plate appearances in 2017, Gomez and Suzuki combined for 4.3 WAR. Catcher depth is key as the position requires more off days and the risk of injury is higher. For comparison, Harper has averaged 3.8 WAR over the past three seasons. Harper has superstar upside, but he could require a record contract commitment — and the Nationals have an outfield loaded with talent in Adam Eaton and young stars Juan Soto and Victor Robles. (A warning to clubs pursuing Harper and Manny Machado: Of the five highest-paid players in MLB history, three — Alex Rodriguez three years into his deal, Giancarlo Stanton three years into his deal and Robinson Cano five years into deal — were eventually traded to New York clubs, and the other two, Albert Pujols and Miguel Cabrera, are nearly untradeable.) The Nationals are paying Suzuki a relatively modest$4 million in 2019, and Gomes is owed a club-friendly $7 million. In other words, the Nationals are paying a relative bargain if Gomes and Suzuki upgrade the position by 3 or more WAR, roughly what Harper has brought to the outfield in recent years. Without Harper, they had payroll flexibility to improve elsewhere. For every win above replacement, a player on the open market in 2017 could be expected to receive more than$10 million. By that measure, the Nationals have done well. With Corbin and the catcher upgrades, they project to have improved by 6 WAR at a cost of $34.3 million in 2019 when combining Gomes and Suzuki’s salaries with Corbin’s. The Nats are not the only NL East team to have filled voids. Josh Donaldson signed a one-year,$23 million deal with the Braves after averaging 3.2 WAR per season the past two years. The defending NL East champs elected to take on short-term risk to seek reward in adding the talented but injury-prone Donaldson. With the addition, the Braves strengthened one of their few weak links and now project to be average or better at every position except left field.

The Phillies had too many weak links last season, ranking 22nd or worse in six position groups according to Baseball-Reference.com: shortstop, second base, third base, left field, center field and right field. They addressed arguably their weakest link entering 2019, shortstop — ranking 30th in WAR last year — in landing Jean Segura from the rebuilding Seattle Mariners.

Segura averaged 4 WAR the past three seasons, according to FanGraphs, and is signed to a club-friendly five-year, $70 million deal through 2022, with a club option for 2023. (The Phillies also shed the contract of Carlos Santana in the deal and added bullpen arms Juan Nicasio and James Pazos.) Segura is a borderline star. The Phillies have cash to fill other voids, and they need to do just that to beat their 78 projected wins. They could use that cash on either Harper or Machado, both of whom have reportedly been in their sights. As the for the Mets, new general manager Brodie Van Wagenen has been in search of brand-name star talent. In Cano, the Mets added a well-known star. But Cano does not directly address the Mets’ weaknesses. Second base tied with left field as the Mets’ most productive position group last season. Jeff McNeil had projected to be the club’s most valuable position player (3.0 WAR) — just shy of Cano (3.4 WAR projection), according to FanGraphs projections.3 While Cano has aged well, he did turn 36 in October. He’s going to decline at some point. After receiving cash contributions from the Mariners, Van Wagenen — a former agent with no front office experience — gave a deal to Cano that amounts to a five-year,$60 million contract. Cano and elite reliever Edwin Diaz — who will upgrade a need area — cost the Mets dollars and two top 100 overall prospects in their No. 3 (Jarred Kelenic) and No. 4 ranked prospects (Justin Dunn) in the trade.

The Mets ranked 20th or worse in the majors in production at catcher, first base, shortstop, third base and in their bullpen last season. The Mets didn’t need a star second baseman. They needed to strengthen their weakest links.

#### The Republican Party Has Changed Dramatically Since George H.W. Bush Ran It

George H.W. Bush, whose death at 94 was announced on Friday by his family, was a hugely influential figure in the Republican Party: chairman of the Republican National Committee, vice president, president and father of another GOP president. But the GOP has changed dramatically since it nominated Bush for the presidency in 1988 — a fact reflected in the ex-president’s strained relationship with the GOP’s new standard-bearer, President Trump.

So, I think Bush’s death is another moment to highlight what my colleague Clare Malone described in the summer of 2016 as “The End Of A Republican Party.” Let’s run through some of the big shifts that have occurred within the GOP:

## The GOP was once a more moderate party

Ideology is complicated to measure. By some standards, the Republican Party has moved to the left. In a poll conducted last year, 42 percent of Republicans backed same-sex unions; it’s safe to assume that number was far lower during George H.W. Bush’s presidency. In 1992, one of South Carolina’s senators was Republican Strom Thurmond, who ran a 1948 presidential campaign featuring his opposition to civil rights for blacks. Today, one of South Carolina’s senators is Republican Tim Scott, who is African-American.

But by most other measures, the GOP is far more conservative than it used to be. The General Social Survey, for example, shows self-identified Republicans moving far more toward the “extremely conservative” end of its scale (as opposed to “extremely liberal”) over the past several decades.1

Political scientists, using DW-Nominate scores,2 have concluded that the Republicans now in Congress are much further to the right of congressional Republicans in the 1970s and 1980s. And even anecdotally, figures like former House Speaker John Boehner, Ohio Gov. John Kasich and the late Arizona Sen. John McCain — considered solid conservatives in the George H.W. Bush era — found themselves cast as insufficiently right-wing by the party’s base in recent years.

In Bush’s era, Fox News did not exist. Deeply conservative billionaires Charles and David Koch and their allies had not created a huge network of right-wing groups that constitute basically an alternative political party. There was no tea party or House Freedom Caucus. Trump may be personally more conservative than Bush, but even if he weren’t, the forces that push a Republican president to the ideological right are stronger now than they were in the 1980s.

Bush himself famously signed a tax increase to help reduce the federal budget deficit, a move that angered the party’s conservative base. His two GOP successors (George W. Bush and Trump) never even really considered tax hikes, aware of the power of the party’s conservative coalition.

## The GOP used to be more in line with the nation demographically

Trump’s rallies include lots of older white people, as the stereotype goes. But that is the Republican Party of today. The country has become older, more diverse and more educated. The GOP, meanwhile, has grown even more disproportionately old. And while its voters grew more diverse along with the country through the 1980s (though at a bit of a lag), that shift stalled in the party in the 1990s. Same with education: The share of non-Hispanic white voters without a college degree fell throughout the 1950s, ’60s, ’70s and ’80s — in the electorate overall and the GOP. But beginning in the 1990s, it stopped falling among Republicans.

In 1992, according to the Pew Research Center, about 38 percent of registered voters who identified as Republicans were 50 years or older. By 2016, that number had grown to 58 percent. In 1992, 61 percent of Republicans were under 50, compared with 41 percent today.3

## Republicans were once competitive on the coasts but weak in the South

The ideological and demographic shifts described above have corresponded with big changes in the GOP’s geographic coalition. In 1988, Bush won California — the sixth straight election in which the Republican presidential candidate carried the Golden State. Bush also won Connecticut, Delaware, Illinois, New Jersey and Vermont. The Republican caucus in the U.S. Senate from that era included two members from New Hampshire, two from Oregon and one from both Delaware and New York. There were zero Senate Republicans from Alabama, Arkansas, Georgia, Louisiana and Tennessee.

All but one of the 16 Senate seats I just described are now held by the opposite party that controlled them in 1992, as the GOP gained ground in the South but lost power on the coasts. Trump, reflecting the growing weakness of Republicans in California, lost there in 2016 by 30 percentage points. Bush lost West Virginia in 1988 by 5 percentage points. Trump’s 42-point win in West Virginia in 2016 was his second-biggest margin in any state.4

In short, today’s Republican Party is centered in the South and almost completely out of power on the West Coast.

Let me not overstate the changes in the Grand Old Party. It still loves to cut taxes, like it did in the George H.W. Bush era. It’s still overwhelmingly white. The majority of its voters are still whites without college degrees. White evangelical Protestants are still about a third of the party. It still deploys negative racial stereotypes about non-white Americans to appeal to white voters. (Remember the Willie Horton ad of the 1988 Bush campaign?)

But the changes highlighted here have dramatically altered the power dynamics in Washington. Presidents Nixon, Reagan and George H.W. Bush always had to share power with Democrats, who controlled the House from 1949 to 1994.5 But since the GOP won the House in 1994, the party has held the chamber for all but four years.6 (They won’t be in control in 2019, of course.)

When George H.W. Bush won 53 percent of the national popular vote in 1988, it was not that remarkable. Richard Nixon had won more than 60 percent in 1972, and Ronald Reagan breached 50 percent in both 1980 and 1984. But 1988 was a watershed moment for the Republican Party — it was about to start a measurable decline in terms of its national standing. In the seven presidential elections since then (including Bush’s 1992 defeat), Republicans have won more votes nationally than Democrats just once (2004).

The tensions between Trump and the Bush family, and between Trump and McCain, speak to this broader narrative. Trump is a different kind of Republican — and he is changing the party in his image in ways they don’t like. But he is also the product of a different Republican Party than the one that the Bushes and McCain ascended in. Trump got the GOP nomination in some ways by embracing what the Republican Party had become, not what the Bushes wished it were.

CORRECTION (Dec. 1, 2018, 1:30 p.m.): An earlier version of this article said President Trump’s largest margin of victory in any state in the 2016 election was in West Virginia. It was in Wyoming.

#### Alice And Bob Fall In Love

Welcome back to The Riddler — I hope you had a wonderful Thanksgiving. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-size and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,1 and you may get a shoutout in next week’s column. If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.

## Riddler Express

From Graydon Snider, road race intimidation tactics:

Two runners, Alice and Bob, are participating in a footrace. The route is a straight line out some distance and the same straight line back — the starting point and the finish line are the same. As the starting gun is about to go off, Alice hatches a race plan: Her legs feel good and she wants to run fast enough compared to Bob that after the U-turn, they are staring face-to-face for as long as possible. How much faster than Bob should Alice run to spend the maximum amount of time facing Bob before they pass each other going in opposite directions? Assume that, on the advice of their coaches, they’ve each committed to running at a constant speed the whole time, and that the turn-around time at the halfway point is negligible.

## Riddler Classic

From Dan Johnston, in which Alice and Bob then fall in love:

After their Riddler Express footrace, Alice and Bob fell in love and got married. Now they want lots of kids. However, as you may know, having one child, let alone many, is a lot of work. But Alice and Bob realized children require less of their parents’ time as they grow older. Alice and Bob, romantics that they are, decided to calculate how this relationship worked. They figured out that the work involved in having a child equals one divided by the age of the child in years. (Yes, that means the work is infinite for a child right after they are born. That may be true.)

Anyhow, since having a new child is a lot of work, Alice and Bob don’t want to have another child until the total work required by all their other children is 1 or less. Suppose they have their first child at time T=0. When T=1, their only child is turns 1, so the work involved is 1, and so they have their second child. After roughly another 1.61 years, their children are roughly 1.61 and 2.61, the work required has dropped back down to 1, and so they have their third child. And so on.

(Feel free to ignore twins, deaths, the real-world inability to decide exactly when you have a child, and so on.)

Five questions: Does it make sense for Alice and Bob to have an infinite number of children? Does the time between kids increase as they have more and more kids? What can we say about when they have their Nth child — can we predict it with a formula? Does the size of their brood over time show asymptotic behavior? If so, what are its bounds?

## Solution to the previous Riddler Express

Congratulations to 👏 Chris Sears 👏 of Maysville, Kentucky, winner of the previous Riddler Express!

The World Chess Championship ended this week after 12 straight draws followed by a series of speedier tie-breaking games. That’s fitting, because two weeks ago we posed this question: What are the chances that the better player wins a 12-game match? Specifically, suppose one of the players is better than his opponent to the degree that he wins 20 percent of all games and loses 15 percent of games; the other 65 percent end in draws.2 What are the chances the better player wins a 12-game match? How many games would a match have to be in order to give the better player a 75 percent chance of winning the match outright? A 90 percent chance? A 99 percent chance?

That better player wins a 12-game match about 52 percent of the time. The number of games required for those larger thresholds are, in order, 82, 248 and 773. (Call me crazy, but I’m totally game for a two-year-long World Chess Championship.)

So how do we get there? Solver Dan Swenson suggested this tidy approach. Consider the following expression:

\begin{equation*}\left(0.2x + 0.65 + 0.15x^{-1}\right)^{12}\end{equation*}

The coefficients 0.2, 0.65 and 0.15 are the probabilities of the three outcomes of an individual chess game, and the whole thing is raised to the 12th power because of the 12 games of the match. Expand that expression, multiplying it all out and grouping its like terms. Then, the coefficient on $$x^r$$, where the exponent $$r$$ is some integer, is the probability that the better player wins the match by $$r$$ games. Finally, add the coefficients on all the positive powers of $$x$$, which gives the probability that the better player wins the match. The result is about 0.5198, or about 52 percent.

For the second part of the problem, we can use that same approach, and just change the “12” in the exponent of the expression, and then, the same as before, expand it and sum the coefficients on the positive powers of $$x$$. The smallest exponent that gives a 75 percent chance or better is 82, the smallest that gives a 90 percent chance or better is 248, and the smallest that gives a 99 percent chance or better is 773.

Solver Chris Sears illustrated how the probability that the better player wins the match increases as the number of games increases.

Somebody get FIDE on the phone!

## Solution to the previous Riddler Classic

Congratulations to 👏 Scott Wu 👏 of Baton Rouge, Louisiana, winner of the previous Riddler Classic!

Two weeks ago, we presented you with a sort of Riddlerified version of beer pong. You had an infinite supply of ping-pong balls, each labeled with some number 1 through N. There was also a group of N cups, labeled 1 through N, each of which could hold an unlimited number of ping-pong balls. The game was played in rounds that had two phases: throwing and pruning. During the throwing phase, you draw balls randomly, one at a time, from the supply and toss them at the cups. The phase ends when every cup contains at least one ball. Next comes the pruning phase, in which you go through all the cups and remove any ball whose number does not match the number on its cup. Every ball drawn had a uniformly random number, every ball landed in a uniformly random cup, and every throw landed in some cup. The game was over when, after a round was completed, there were no empty cups.

How many balls would you expect to need to draw and throw to finish this game? How many rounds would you expect to need before finishing this game?

The first question — how many throws would you expect to need — is like a “coupon collector’s problem,” and versions of it have appeared in this column before, for example in a problem about collecting Riddler League football cards. In this case, we’re filling cups instead of collecting coupons.The solution to this problem is well-known and stems from the fact that as we collect coupons — or fill cups — it becomes more and more difficult to collect or fill the ones that remain. The specific solution is quite mathy and involves something called the Euler-Mascheroni constant, but the main takeaway is that the more coupons or cups there are, the more and more time we can expect to spend collecting coupons or throwing balls.

Solver Laurent Lessard illustrated how the expected number of throws increases roughly quadratically as the number of cups (and numbers on the balls) increases:

The second question — how many rounds would you expect to play — turns out to be much trickier. Lessard also illustrated his mathematical approach when playing with three cups. The cups in his diagram are either empty (white), filled with a correct ball (green), or filled with an incorrect ball (red). The arrows and the numbers next to them represent transition probabilities — the chances a cup goes from empty to correctly filled, for example, are 1/N. Our goal, of course, appears in the bottom right of the diagram, where all three cups are correctly filled and the game is over.

From there, Laurent describes how to turn this diagrammatic approach into an answer. He uses a Markov chain, which describes probabilistic sequences of events that depend on the current state of events — such as the balls currently in our cups. And from there he invokes a holy Riddler trinity: absorbing states and limiting distributions and nilpotent matrices.

The result of all this is that while the number of throws required grows roughly quadratically, the number of rounds required grows roughly linearly.

Of course, this is also a problem that admits programmatic, simulation-based approaches. Solver Robin Chin shared Python code for a Monte Carlo simulation, and Michael Branicky shared his code as well. Finally, solver Tim Book shared the results of his computer simulations, which illustrate how the expected number of rounds increases as the number of cups (and the number of numbers on the balls) increases — roughly linearly:

Happy tossing!

## Want more riddles?

Well, aren’t you lucky? There’s a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. It’s called “The Riddler,” and it’s in stores now! Consider your holiday shopping done.

## Want to submit a riddle?

Email me at [email protected]

#### Mike Espy Needs An Unprecedented Swing To Win The Mississippi Runoff

We don’t often see a runoff in a general election, but if Sen. Cindy Hyde-Smith were to lose Mississippi’s Senate runoff on Tuesday, after the two Republican candidates combined to win a sizable majority of the initial vote, that would be even more unusual.

In the first round, Republicans Hyde-Smith and Chris McDaniel combined for a bit less than 58 percent of the vote, while Democrat Mike Espy and one other candidate from his party together won a little more than 42 percent. For Espy to win, the runoff vote has to swing more than 15 points more Democratic than the initial vote margin. But if we look at the five Senate elections since 1990 where an initial round of voting was held on the national Election Day and two candidates advanced to a runoff,1 no challenger has ever come close to outperforming the previous round of voting by the kind of margin Espy would need to win. What’s more, no runoff has ever shifted that much in either party’s direction.

##### Democrats need a record-setting swing to win Mississippi

Shifts in vote share margin between first election and runoff, in Senate races since 1990 where the first election took place on the national November Election Day in a midterm or presidential cycle

vote share margin
Year State First Election Runoff Swing
1992 GA D+1.6 R+1.3 R+2.9
2002 LA R+2.9 D+3.4 D+6.3
2008 GA R+2.9 R+14.9 R+12.0
2014 LA R+12.3 R+11.9 D+0.4
2016 LA R+25.4 R+21.3 D+4.1
2018 MS R+15.3

In cases where the first election included multiple candidates from the same party (all Louisiana and Mississippi races in the table), the margin is based on the difference in the total vote share between all Democratic and Republican candidates. In some cases, parties had unequal numbers of candidates running, which may have exaggerated the differences in vote share. Some data may not add up due to rounding.

Sources: ABC News, Georgia Secretary of State, Louisiana Secretary of State, UVA Center for Politics, Federal Election Commission

In fact, the biggest swing recorded — 12 points in the 2008 Georgia Senate runoff — favored the incumbent and not the challenger. Of the five previous Senate elections that went to a runoff, four featured incumbents, and in those contests, the swing in party vote share moved in the challenger’s favor only once. In Georgia’s 1992 Senate election, Democratic Sen. Wyche Fowler missed a majority by 1 percentage point in the general election, and in the runoff, challenger Republican Paul Coverdell squeaked out a narrow victory as the margin shifted 3 points in the GOP’s direction.

As for that big swing in Georgia’s 2008 Senate election, it came about in part because of the unprecedented circumstances surrounding the general election. The initial vote took place during Barack Obama’s first presidential bid, which saw African-American voters turn out at much higher rates than usual, both nationally and in Georgia. That turnout did not repeat itself in the runoff, however, and Republican Sen. Saxby Chambliss won re-election by 15 points.

The next-largest shift (and the largest shift in a midterm election) also benefited an incumbent, although it was much smaller. In 2002, four GOP candidates combined for a narrow edge over two Democratic candidates in the initial vote, but Democratic Sen. Mary Landrieu narrowly won the Louisiana Senate runoff by shifting the margin 6 points in a Democratic direction. So a Democrat has moved the needle enough to win a runoff in the Deep South in the recent past, although the politics of Louisiana (and the rest of the region) have shifted further toward the GOP since the early 2000s. Moreover, Landrieu didn’t need nearly as large a swing as Espy does in 2018.

From this admittedly small sample of Senate runoffs, an Espy win looks like a long shot: These races rarely swing by nearly as much as he needs to win, and when they swing at all, it’s more often toward the incumbent than the challenger. Furthermore, the Nov. 6 result in Mississippi looks similar to the 2014 and 2016 contests in Louisiana, which also bodes poorly for Espy. In those races, the GOP candidates combined for a double-digit margin in the first election, and in the runoff, the Democrat only gained a little ground. In that 2014 race, Landrieu was voted out of office after nearly 20 years in the Senate (no incumbent ran in 2016). As Nathaniel Rakich wrote in our preview of the Mississippi runoff, Espy needs today’s electorate to be substantially different from the voters who turned out earlier this month; Espy likely needs a strong turnout from black voters, who lean very Democratic, combined with some apathy among white voters, who are strongly Republican in Mississippi.