The Pro Conference Premium

TL;DR. The conference premium is the share of a league’s in-division travel that exists only because of where it drew its conference line, the miles it could erase by ignoring that line and regrouping its teams on geography alone. To put a number on it, score three versions of each league by total within-division great-circle miles: the actual league \( A \), the same conferences with the divisions redrawn \( B \), and a free geographic redraw \( C \). The premium is \( (B - C)/A \). Baseball pays 28.87% and football 19.66%, because their conferences are boundaries kept from a rival-league merger, not maps. The NBA, the NHL, and MLS pay 0.00%, because their conferences already are the map, or there is no division layer beneath them to redraw. The premium tracks one piece of history: whether a league that merged with a rival kept the old line as a brand or dissolved it into geography. The NBA is the case that started this essay: its coming expansion to thirty-two teams, squared up to four divisions of four, leaves the premium at zero. The conference line stays exactly where geography would draw it.

This essay started with a basketball question. The NBA is widely expected to expand within a few years, adding two western teams, and I wanted to know whether that would pull its tidy East and West out of shape. Measuring it meant building a way to price any league’s conference line in travel, and that turned out to say something about all five.

Every major North American league, baseball’s MLB, football’s NFL, basketball’s NBA, hockey’s NHL, and soccer’s MLS, sorts its teams the same way. First into two conferences, then into divisions inside each conference, and a team plays its own division far more than anyone else. That nesting has a price measured in miles. A team locked into the wrong half of the country flies past closer rivals to reach the ones it is scheduled against. The question here is narrow and answerable: of all the travel a league spends inside its divisions, how much is the cost of where it drew the conference line, and how much is the unavoidable cost of geography? That first part is what this essay calls the conference premium: the travel a league could erase by forgetting its conference line and grouping teams on geography alone. It runs from nearly a third of all in-division miles down to zero, and which end a league lands on turns out to be a question of history.

The Scoreboard

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A boundary drawn for a league merger in 1970, or a rival circuit that folded a century ago, still routes a team’s bus today. The premium is the fare.

The rest of this essay builds that number one step at a time, then walks each league with its own maps, top of the table to bottom.

How the Number Is Built

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The method climbs in four steps, each one plain first and then exact.

Step one: what a division costs. Start as simply as possible. A division’s travel cost is the total distance between the teams in it. A division whose teams all sit near each other is cheap; one stretched across the country is expensive. Add that up over every division and you have the league’s score, where lower is tighter. Written out, with the set of divisions called \( \mathcal{P} \),

• \( \mathcal{P} \): one way of splitting the teams into divisions
• \( D \): a single division within that split
• \( d_{ij} \): the distance between teams \( i \) and \( j \)
• \( \mathrm{DIV}(\mathcal{P}) \): the league’s score, every teammate pair’s distance added up

The only ingredient is \( d_{ij} \), the distance between two teams, taken as the great-circle distance across the surface of the earth rather than a straight line on a flat map. The haversine formula gives it from latitude \( \phi \) and longitude \( \lambda \):

• \( R \): the earth’s radius, 3,958.8 miles
• \( \phi_i, \phi_j \): the two teams’ latitudes
• \( \lambda_i, \lambda_j \): the two teams’ longitudes
• \( d_{ij} \): the great-circle distance the formula returns

with \( R = 3{,}958.8 \) miles. Every pairwise distance is computed once into a matrix, so scoring any partition is a sum of lookups:

def haversine_miles(a, b):
    lat1, lon1, lat2, lon2 = map(math.radians, (a.lat, a.lon, b.lat, b.lon))
    d_lat, d_lon = lat2 - lat1, lon2 - lon1
    h = (math.sin(d_lat / 2) ** 2
         + math.cos(lat1) * math.cos(lat2) * math.sin(d_lon / 2) ** 2)
    return 2 * EARTH_RADIUS_MI * math.asin(math.sqrt(h))

def distance_matrix(teams):
    idx = list(teams)
    return {a: {b: haversine_miles(teams[a], teams[b]) for b in idx}
            for a in idx}

Step two: drawing the best divisions. The easiest way to picture this is brute force: list every way to sort the teams into groups of the right sizes, score each arrangement, and keep the cheapest. That is the answer we want; the catch is that the number of arrangements explodes, so a solver finds the same answer without trying all of them, and proves it found the best. The model is a set-partition. Treat every group of teams of an allowed size as a candidate division with a cost equal to its own within-group travel, \( c_g = \sum_{\{i,j\} \subseteq g} d_{ij} \), then pick a set of candidates that covers every team exactly once and costs the least. With a yes-or-no variable \( z_g \in \{0,1\} \) for whether candidate \( g \) is chosen,

• \( g \): a candidate division, any group of teams of an allowed size
• \( c_g \): that group’s own within-group distance
• \( z_g \): 1 if the group is chosen as a division, 0 if not
• first constraint: every team \( i \) sits in exactly one chosen division
• second constraint: the chosen divisions match the league’s shape, \( n_s \) of them at each size \( s \)

The first constraint is the partition: every team in exactly one chosen division. The second fixes the shape, exactly \( n_s \) divisions of each size \( s \), so the model matches a league whose divisions are not all the same size.

def partition(members, sizes, distances):
    cand = [g for s in set(sizes)
              for g in itertools.combinations(members, s)]
    cost = [sum(distances[a][b] for a, b in itertools.combinations(g, 2))
            for g in cand]
    solver = pywraplp.Solver.CreateSolver("CBC")
    z = [solver.BoolVar(f"z_{k}") for k in range(len(cand))]
    for i in members:                                  # cover: one division each
        solver.Add(sum(z[k] for k, g in enumerate(cand) if i in g) == 1)
    for s, n_s in Counter(sizes).items():              # shape: n_s divisions of size s
        solver.Add(sum(z[k] for k, g in enumerate(cand) if len(g) == s) == n_s)
    solver.Minimize(sum(cost[k] * z[k] for k in range(len(cand))))
    solver.Solve()
    return [cand[k] for k in range(len(cand)) if z[k].solution_value() > 0.5]

For a conference of fifteen or sixteen teams the candidate list is small enough to enumerate in full, so the result is exact. For the whole league at once the count is too large, so the same program runs over a candidate set restricted to each team’s nearest neighbors, which keeps the partition optimal against a strong field.

Step three: three versions of each league. Score each league three ways. \( A \) is the league as it stands today. \( B \) keeps the real conferences but redraws the divisions inside them for minimum travel, at the real sizes. \( C \) throws out the conference line, redraws every division across the whole league, then splits the result back into two halves by longitude. The longitude re-split only sorts the finished divisions into two conferences so the \( C \) map reads like the others; it places no constraint on the solve, so \( C \)’s mileage is the free optimum either way. The conference premium is how much worse the conference-bound best is than the free best, against what the league spends now:

• \( A \): the actual divisions’ within-division distance
• \( B \): the same conferences, divisions redrawn for least travel
• \( C \): the free redraw with the conference line erased
• \( \mathrm{premium} \): the gap \( B - C \) as a share of \( A \); \( \mathrm{recover}_B \) and \( \mathrm{recover}_C \) are how much each redraw saves

If a league’s conferences are nothing but a clean east and west cut, \( B \) and \( C \) land together and the premium is zero. If the conferences follow some other logic, the premium is what that logic costs.

Step four: the schedule does not change the answer. Teams play division rivals most, so it is fair to worry the best partition by raw distance is not the best by miles flown. It is the same partition. Let \( g_\bullet, g_\circ, g_\times \) be games per opponent against a division rival, a non-division conference team, and a cross-conference team, and group the season’s pair-distances into \( \mathrm{DIV} \), \( \mathrm{CONF} \), and \( \mathrm{CROSS} \). Season trip-miles are

• \( \mathrm{DIV}, \mathrm{CONF}, \mathrm{CROSS} \): season distance against division, conference, and cross-conference opponents
• \( g_\bullet, g_\circ, g_\times \): games played per division, conference, and cross-conference opponent
• \( W \): total season trip-miles; rearranged, only the \( \mathrm{DIV} \) term moves with the partition

For a fixed conference split, \( \mathrm{CROSS} \) and \( \mathrm{CONF} \) are constants, so only \( \mathrm{DIV} \) moves with the partition. Since \( g_\bullet \ge g_\circ \), minimizing \( W \) is the same as minimizing \( \mathrm{DIV} \), for any schedule at all. This argument holds the conference split fixed, so it licenses the conferences-kept redraw \( B \) directly; the premium’s other half, the free optimum \( C \), erases the conference line, so it is treated as a pure-distance construct with no schedule attached, and the premium compares distance against distance throughout. The weighting only matters when comparing how finely to divide, through the frozen-cross rates:

• \( S \): total games on the schedule
• \( n, c, s \): teams in the league, in a conference, and in a division
• \( \rho \): how many times more a team plays division rivals than other conference teams
• \( g_\circ, g_\bullet \): the per-opponent conference and division game counts these fix

def schedule_games(division_size, conference_size, n_teams, ratio):
    g_cross = 2.0
    inconf_budget = SEASON_GAMES - (n_teams - conference_size) * g_cross
    n_div  = division_size - 1                  # division rivals
    n_conf = conference_size - division_size     # other conference teams
    g_conf = inconf_budget / (n_div * ratio + n_conf)
    g_div  = ratio * g_conf
    return g_div, g_conf, g_cross

With the method in hand, the walk goes top of the scoreboard to bottom, one league at a time, each map followed by what it shows: the current divisions, the same conferences with the divisions redrawn, and the full geographic optimum.

MLB, the Steepest Bill

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Start with the league as it stands: thirty teams, two leagues, three divisions each, the divisions cut so close rivals share one.

These are the actual divisions, and they cost 35,727.63 miles of within-division travel, the figure every later map is measured against. The clusters look tidy because the American and National Leagues are each already a compact set of teams; whatever slack exists is inside them, in how the divisions are drawn.

Every division has a geographic heart, the average position of its teams. Reverse-geocode that point and you land on a real, usually very small American town. Here is the center of each current division, with 2020 Census populations.

In plain terms, the solver tries every legal way to cut the teams into divisions of the right size and keeps the one with the least total within-division distance. Keeping the leagues means re-cutting each fifteen into three fives; dropping them means re-cutting all thirty into six:

• \( B \): keep the two leagues, cut each 15-team league into 3 divisions of 5
• \( C \): drop the league line, cut all 30 teams into 6 divisions of 5

Keep the two leagues exactly as they are and redraw only the divisions inside them, and almost nothing moves. The two West divisions do not change at all; the only shuffling is between East and Central, where the Braves slide over to sit with the four Central clubs and the Pirates cross the other way, with the Rays and Guardians making the mirror-image swap in the American League. That is the whole difference, and it recovers a rounding error, 1.39%, because each league is already locally tight.

And the hearts of those same divisions once the conferences are kept but the lines inside them are redrawn for least travel:

Now erase the American and National line and redraw across all thirty teams, and every single division turns mixed. The Yankees, Red Sox, and Orioles share a group with the Mets and Phillies; the Dodgers and Giants join the Angels and Athletics; geography pays no attention to which league a team plays in. Travel drops 30.26%, and the gap between that and the 1.39% above is the premium: 28.87%, the steepest bill in the study. Almost the entire cost is the league line itself.

That gap, as a share of the actual travel:

• \( A = 35{,}727.63 \): the actual divisions’ within-division miles
• \( B = 35{,}232.75 \): conferences kept, divisions redrawn
• \( C = 24{,}916.65 \): conference line erased, free redraw
• \( (B - C)/A \): the gap as a share of actual travel, the premium

And once the conference line is erased and every division is drawn from scratch:

The reason is history, not geography. The National League dates to 1876 and the American League to 1901, and the two ran as separate businesses for nearly a century before merging into a single organization in 2000.² The split down the middle of the sport was never about longitude, and the 162-game season multiplies every extra mile. Division count matters too: across one to three divisions per conference the travel swings 6.61%, and the optimum lands on three, which is what the league already runs.

Two of those groups look wrong until you trace the geometry. The Mariners sit in the Laramie group with Arizona, Colorado, Kansas City, and Minnesota, not with the California teams that are their nearest neighbors: the five California-area clubs already fill a division tightly, so Seattle, the leftover corner, attaches to the next cluster inland. The Marlins land in the Dauphin Island group with the two Texas teams because Miami is too isolated in the southeast to fill a division with close neighbors, so once Tampa and Atlanta are taken the group has to reach west to Texas to make five.

NFL, History Over Geography

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Football runs eight divisions across thirty-two teams. The first map is the league as it plays today.

The actual divisions cost 27,324.58 miles. The four-team groups are looser than baseball’s, with several that reach across a time zone, so there is real slack to recover here before the conference line even comes up.

The center of each division, with its 2020 Census population:

The same idea on a looser grid. With conferences kept, each sixteen is re-cut into four fours; with the line gone, all thirty-two into eight fours:

• \( B \): keep the two conferences, cut each 16-team conference into 4 divisions of 4
• \( C \): drop the conference line, cut all 32 teams into 8 divisions of 4

Hold the AFC and NFC fixed and redraw their divisions, and the loose four-team groups tighten. The two West divisions hold, but elsewhere teams trade places: the Cowboys leave the NFC East to sit with the southern clubs while the Panthers take their spot, and on the AFC side the Colts join the old Browns and Steelers grouping while the Ravens slide east. Travel falls 16.57%, and this part of the bill is the league’s own division-drawing, not its conference line.

And the hearts of those same divisions once the conferences are kept but the lines inside them are redrawn for least travel:

Drop the conference line and redraw freely, and every division mixes the two conferences. The Raiders and Chargers join the Rams and 49ers in an all-California group; the Patriots and Jets pair with the Giants and Eagles in the northeast. Travel falls 36.23%, and the difference from the conference-bound redraw is the premium: 19.66%, second only to baseball.

The same subtraction:

• \( A = 27{,}324.58 \): the actual divisions’ within-division miles
• \( B = 22{,}798.07 \): conferences kept, divisions redrawn
• \( C = 17{,}424.84 \): conference line erased, free redraw
• \( (B - C)/A \): the gap as a share of actual travel, the premium

And once the conference line is erased and every division is drawn from scratch:

The AFC and NFC are a fossil. The American Football League launched in 1960 to challenge the NFL, the two agreed to merge in 1966, and to balance the new conferences three old NFL clubs crossed over to join the ten AFL teams for the 1970 season.³ So like baseball, the boundary is a brand, not a map. Football is also the league where division count matters most: across one to four divisions per conference the travel swings 14.50%, and the optimum prefers two divisions per conference against the four the league runs, because tight local rivalries are worth more to it than miles.

Two groups here look wrong for the same kinds of reasons. Seattle lands inland again, the Seahawks in the Rock Springs group with Denver and Arizona, and the Vikings beside them, rather than with the California four: the corner team and its nearest interior neighbor get absorbed by the mountain cluster. And the Cleveland Browns land in Group A, the western of the two halves, even though Cleveland is an eastern city. The free optimum sorts its finished divisions into two halves by longitude only so the map reads like the others, and the Great Lakes group, the Bears, Browns, Lions, and Packers, has a centroid just west of that median. The halves are labeled Group A and Group B rather than West and East precisely because a division can sit on one side while a team inside it, like Cleveland, belongs to the other.

NBA, Geographic After All

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The NBA is the league this essay started from. Its conferences look like a brand, an East and a West, and its coming expansion seemed like the kind of thing that might bend a geographic line out of shape. It does not. The NBA is geographic today, and it stays geographic through the expansion, which is the quiet result under all three maps that follow.

Start with the thirty teams that play now. The actual divisions, three of five in each conference, cost 25,571.22 miles of within-division travel.

The geographic center of each actual division, with its 2020 Census population:

Redraw those divisions for the least travel and the league recovers 2.61%, the ordinary slack in any real alignment. But keeping the conference line or erasing it changes nothing: turned loose on all thirty teams, the optimizer never once puts an Eastern and a Western team in the same division. The conferences already are the line a map would draw, so the premium is zero.

• \( A = 25{,}571.22 \): the actual divisions’ within-division miles
• \( B = 24{,}903.11 \): conferences kept, divisions redrawn
• \( C = 24{,}903.11 \): conference line erased, free redraw
• \( B = C \): the two optima tie, so the premium is zero

The center of each division once they are redrawn for least travel:

Now the part that prompted the essay. The league is widely expected to add two western franchises, a revived Seattle SuperSonics and a Las Vegas team, with play targeted to begin in 2028.⁴ Thirty-two teams will not hold three divisions of five, which is thirty, so the league would square up to four divisions of four, sixteen per conference:

• today: thirty teams, two conferences of three divisions of five
• projected: thirty-two teams, two conferences of four divisions of four

Solve for the best projected alignment and the conference line falls exactly where the free optimum already draws it. Not one of the eight divisions crosses it, so the premium is still zero.

• \( A = 16{,}625.25 \): the projected divisions’ within-division miles, all 32 teams
• \( B = 16{,}625.25 \): conferences kept, four divisions of four redrawn
• \( C = 16{,}625.25 \): conference line erased, free redraw
• \( B = C \): the optimal even split is already a clean East and West

The optimizer even moves a team across the old map without being told to. The Timberwolves, closer to the upper-Midwest East than to the southern West, land in an eastern division, which is the league’s own reported instinct too: shift a central team East and balance the conferences at sixteen apiece. Geography and the front office reach for the same team.

The center of each division in the projected thirty-two-team league:

The NBA can absorb all this because of how it grew. The American Basketball Association ran from 1967 to 1976, and when the NBA took in four of its teams it spread them across existing divisions rather than keeping an ABA bloc, then drew its conferences around geography in the mid-2000s.⁵ A merger only leaves a premium if the absorbed teams are never re-sorted by map. The NBA re-sorted, and even two more teams do not undo it.

NHL, Geography Already

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Hockey is the clean case the NBA’s real roster resembles. Here is the league as it stands.

The actual divisions cost 68,797.66 miles, the second-highest raw total in the study, because hockey simply covers a lot of ground, from Florida to western Canada. Its geographic center, marked on each map, gives that away: it sits nearly three degrees of latitude north of the other four leagues, pulled up by the Canadian clubs, so the whole league is stretched vertically in a way none of the others are. Raw miles and premium are different things, though, as the next two maps show.

The center of each division anyway, with its 2020 Census population:

With conferences kept, each sixteen is re-cut into two eights; with the line gone, all thirty-two into four eights:

• \( B \): keep the two conferences, cut each 16-team conference into 2 divisions of 8
• \( C \): drop the conference line, cut all 32 teams into 4 divisions of 8

Keep the conferences and redraw the divisions, and only the East changes: the optimizer splits the sixteen eastern teams north and south rather than along the old Atlantic and Metropolitan line, while the West stays exactly as it is. Travel falls 5.92%, all of it the league’s own division-drawing.

And the hearts of those same divisions once the conferences are kept but the lines inside them are redrawn for least travel:

Erase the conference line and redraw freely, and the same two camps rebuild themselves: not one division mixes an Eastern and a Western team. Travel falls by exactly the same 5.92%, so the premium is 0.00%. The line was already where a map would put it.

The two optimal maps tie, so the subtraction vanishes:

• \( A = 68{,}797.66 \): the actual divisions’ within-division miles
• \( B = 64{,}726.97 \): conferences kept, divisions redrawn
• \( C = 64{,}726.97 \): conference line erased, free redraw
• \( B = C \): the redraws tie, so the premium is zero

And once the conference line is erased and every division is drawn from scratch:

When the NHL absorbed four teams from the World Hockey Association in 1979, it placed the survivors by geography rather than keeping a rival bloc, and its conferences have followed the map since.⁶ Division count is nearly irrelevant here, a 1.10% swing, and the optimum agrees with the two divisions per conference the league uses.

MLS, Nothing to Recover

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Soccer has only two maps, because there is no division layer beneath its conferences to redraw.

This is the league as it stands: thirty teams split fifteen and fifteen into an Eastern and a Western Conference, and nothing finer.

Even a conference has a center, with its 2020 Census population:

There is only one cut to make, the split itself, because no divisions sit beneath it:

• \( C \): the only cut is the split itself, 30 teams into 2 conferences of 15
• no \( B \): there is no division layer beneath the conferences to redraw

Solve for the best fifteen and fifteen split and the optimizer hands back the identical two lists; not one team changes conference. The premium is 0.00%, but it is a different zero from hockey’s. Hockey earns its zero, since the optimizer would draw that same line. Soccer’s is structural: with no divisions to redraw, there is nothing to recover.

With no division layer there is no \( B \) to compute; the best conference split just equals the actual one:

• \( A = 168{,}085.75 \): the actual split’s within-conference miles
• \( C = 168{,}085.75 \): the best split returns the identical miles
• \( A = C \): the optimal split is the actual one, so nothing is recovered

The optimal split returns the same two conferences, so the centers do not move: the East stays centered on Warm Springs and the West on Delta.

Major League Soccer was built from scratch in the 1990s rather than from a merger, and it has never carried a division layer beneath its two geographic conferences.⁷ There is simply no historical line for a premium to hide in.

The Spectrum, or the Lack of One

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Laid end to end, the five leagues do not fall on a tidy gradient. They fall into two groups, and the split tracks one piece of history: what each league did with a rival it absorbed. Baseball and football kept the old boundary as a brand and pay for it, 28.87 and 19.66%. The NBA and the NHL absorbed rivals too, then re-sorted them by geography, so at their real rosters they pay nothing. Soccer never merged and never drew a division layer, so its zero is structural. The premium is a single subtraction, but it sorts the leagues by whether a merged league kept its old line or dissolved it into the map.

The NBA is the league that prompted all this, and it is the cleanest case of the rule. Its conferences already are the map, so it pays zero; and its projected expansion, two western teams and a square-up to four divisions of four, leaves the optimal split a clean East and West, so it still pays zero. Geography that was already honest does not turn dishonest just because the league grows.

A league that pays this premium is choosing its line on purpose; nothing here disputes that choice, only measures what it costs in travel.

The Groupings That Look Wrong

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Across the maps the free optimum keeps drawing groups that look wrong: Seattle stranded inland, Miami reaching to Texas, Cleveland sorted into the western half. They share one cause. The optimum minimizes total within-division distance at fixed division sizes, which is not the same as putting every team with its single nearest neighbor, so a team near a corner or a seam gets sorted by what makes the whole map cheapest, not by what looks right for it alone.

Behind all of it is where each league’s weight actually sits. The open crosshair on every map marks the league’s geographic center, the average position of its teams, and not one of the five lands in the middle of the country. Because it depends only on who is in the league and not on how the divisions are cut, it sits in the identical spot on a league’s three maps; the optimizer regroups teams around the center, it never moves the center itself. Measured against the geographic center of the contiguous United States, near Lebanon, Kansas at about 98.6 degrees west, every league leans east. Football leans hardest, its center about 8 degrees of longitude east; baseball and hockey sit about 6 degrees east; the NBA’s real thirty teams about 5; soccer about 4 and a half. The eastern half of the map holds more teams, so the optimizer packs its tightest divisions there and the far western and far southeastern clubs, Seattle and Miami, are the ones left over to absorb. The center is why the corners look odd.

The NBA is the one league whose center moves on the page. Its real thirty teams center near 93 degrees west, but the projected panel adds Seattle and Las Vegas, both far west, and that pulls the center about a degree and a half back toward the middle, to near 95 degrees west. The expansion does not bend the conference line. It nudges an east-leaning league toward the balance the line already assumed, which is the same reason the projected split stays clean and the premium stays at zero.

Minnesota goes the other way, and goes both ways. The Vikings land in the western mountain group in football, the Timberwolves cross into an eastern division in the projected NBA, and the Twins sit in the interior group in baseball. Minneapolis sits near the seam between the eastern and western blocs of teams, so it is among the cheapest teams to move whenever one side needs a body. Whichever half is short, Minnesota tends to be the one that slides over, which is why the same city reads as West in one league and East in another.

What This Does Not Mean

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The premium is a geographic accounting, not a proposal, and a few limits keep it honest.

It measures dispersion, not itineraries. Each arena is a single point, and the score is the spread of teams within a division, not a literal flight schedule with road trips, layovers, and charter logistics. Two leagues with the same premium can fly very different real routes.

The NBA section carries a projection. Its first two maps are the real thirty-team league; the third is a projected thirty-two-team league with Seattle and Las Vegas added, an expansion the league voted in March 2026 to explore but has not yet approved. Because the league has not drawn those divisions yet, the projected panel shows them at the optimum, what a sensibly aligned expanded league would look like, rather than a forecast of the exact divisions.

The full-league optimum is strong, not provably global. Where exact enumeration is infeasible, \( C \) is solved over a nearest-neighbor candidate pool, so it is optimal against a high-quality field rather than proven best across all partitions for the largest leagues.

And it prices only miles. Conferences carry rivalry, television markets, history, and competitive balance that no distance matrix can see. A high premium is not a verdict that a league is wrong, only a measure of what its line costs in travel.

Methods and Sources

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Coordinates for every team, including the two presumptive NBA expansion clubs, live in one shared file, and every distance is the haversine great-circle mile computed once into a matrix. Each optimal alignment is the exact minimum of a set-partition integer program (CBC, via OR-Tools or PuLP) where the enumeration is tractable, and the same program over a nearest-neighbor candidate set where it is not. The schedule weighting uses each league’s real division, conference, and cross-conference game ratios under the frozen-cross model, which the identity above proves leaves the partition untouched.

The centroid towns are a flourish, not analysis: each is the nearest populated place to the average position of a real division, found by reverse-geocoding the centroid, and the populations are 2020 US Census counts where a count exists. The maps follow the page’s light or dark mode, and every premium, recovery figure, and panel reads from the one coordinate file and the one solve.

def centroid(teams_in_division):
    lat = sum(t.lat for t in teams_in_division) / len(teams_in_division)
    lon = sum(t.lon for t in teams_in_division) / len(teams_in_division)
    return lat, lon

Sources and data. Team coordinates are arena locations compiled by hand; populations are 2020 United States Census counts where a count exists. Distances are haversine great-circle miles, and each optimal alignment is the exact minimum of a set-partition integer program solved with CBC through PuLP. The league histories are cited in the notes below.

The reasoning is the deliverable: the premium only means something once you have seen the maps, the distance metric, the integer program that picks the divisions, and the schedule argument that makes the comparison fair.