Here are the Penn-Columbia tiebreakers for the 2017 Ivy League men’s basketball tournament

Updated Tuesday, Feb. 28, with a scenario I hadn't thought of that matters.

After Penn's loss Saturday at Columbia, Quakers coach Steve Donahue said that from what he knew, his team controlled its own destiny in the race for fourth place in the Ivy League.

Donahue wasn't the only person who believed that. Many observers across the league interpreted the tiebreaker rules to mean that the first tiebreaker after head-to-head results - results against other teams in the league - only extended as far as results against the other teams that qualify for the tournament.

That belief was incorrect.

On Sunday, the Ivy League issued a clarification that the second tiebreaker extends all the way down the standings - starting with first place (Princeton) and moving to last place.

(This was not a late rule change, to be clear. The rule was this way all along, but the language in the league's initial explanation wasn't clear enough.)

Only after going all the way through the standings does the third tiebreaker come into play: an average of teams' rankings in the NCAA RPI, Sagarin, Ken Pomeroy and ESPN College Basketball Percentage Index standings.

With that in mind, here is a look at potential scenarios that could unfold on the final weekend of conference play. Penn hosts Dartmouth on Friday and Harvard on Saturday, while Columbia visits Brown on Friday and Yale on Saturday.

One major disclaimer: You will see scenarios in here in which the Harvard-Princeton result does not matter. That only applies to breaking the tie for fourth place. It does not apply to potentially having to break a tie for first place, which is for another post that would likely take up far too much of everyone's time. 

If Penn and Columbia sweep the weekend and finish 7-7, and Yale is clear in third place

There are a few scenarios, as follows. Brown-Cornell is the game to focus on.

Scenario 1: Brown beats Cornell

Penn beats Dartmouth
Columbia beats Brown
Yale beats Cornell
Harvard beats Princeton

Penn beats Harvard
Columbia beats Yale
Brown beats Cornell
Dartmouth beats Princeton

The standings would be:

Princeton 12-1
Harvard 11-3
Yale 8-6
Penn/Columbia 7-7
Brown/Dartmouth 4-10
Cornell 3-11

Penn and Columbia have both been swept by Princeton and split with Harvard and Yale. The next tiebreaker would be Penn and Columbia's respective records against Brown and Dartmouth combined. Both teams would be 2-2.

At that point, the ratings index average tiebreaker would come into play. Penn has an advantage in that heading into the weekend.

Scenario 2: Cornell beats Brown

The standings would be:

Princeton 12-1
Harvard 11-3
Yale 8-6
Penn/Columbia 7-7
Cornell/Dartmouth 4-10
Brown 3-11

Penn's record against Cornell and Dartmouth combined would be 3-1. Columbia would be 2-2. Penn gets fourth place.

If Penn and Columbia sweep the weekend and Yale loses twice at home, resulting in a three-way tie at 7-7

Scenario 1

Results

Penn beats Dartmouth
Columbia beats Brown
Cornell beats Yale
Princeton-Harvard doesn't matter

Penn beats Harvard
Columbia beats Yale
Brown beats Cornell
Dartmouth beats Princeton

Yale, currently in third place at 7-5, would be out. Penn and Columbia would have splits with Yale. Princeton swept all three teams, so the tiebreaker would be results against Harvard. The Crimson swept the Bulldogs, split with the Lions, and would have a split with the Quakers. 

The order of bids would be settled by each team's combined results against the next place team or teams if there is a tie.

So...

Standings

Princeton/Harvard in some order
Penn/Columbia
Yale 7-7
Brown/Cornell/Dartmouth 4-10

As with the above scenario of a three-way tie at 4-10, the ratings tiebreaker would be used.

Scenario 2

Results

Cornell beats Brown
Dartmouth beats Princeton

Standings

Penn/Columbia 7-7
Yale 7-7
Cornell/Dartmouth 4-10
Brown 3-11

Penn's combined record against Cornell and Dartmouth is 3-1, and Columbia's is 2-2. Penn gets the No. 3 seed and Columbia gets the No. 4 seed.

Scenario 3

Results

Cornell beats Brown
Princeton beats Dartmouth

Standings

Cornell 4-10
Brown/Dartmouth 3-11

Penn was 2-0 against Cornell and Columbia was 1-1. Penn takes third and Columbia takes fourth.

Scenario 4

Brown beats Cornell
Princeton beats Dartmouth

Brown 4-10
Cornell/Dartmouth 3-11

Columbia swept Brown, and Penn split with Brown. So Columbia would be the No. 3 seed and Penn would be the No. 4 seed.

If Penn and Columbia each go 1-1 on the weekend and finish tied at 6-8

Scenario 1: Penn beats Dartmouth and loses to Harvard

Columbia would get the berth regardless of which game it wins, because it has one win over Harvard and Penn does not.

Scenario 2: It gets more complicated

Dartmouth beats Penn
Columbia beats Brown

Penn beats Harvard
Yale beats Columbia

Penn would get the bid. Each team would have a win over Harvard, and Columbia would have two losses to Yale.

Scenario 3: It gets really complicated

Dartmouth beats Penn
Brown beats Columbia

Penn beats Harvard
Columbia beats Yale

Penn and Columbia would both have splits with Harvard and Yale. So the bid would again come down to results against the teams ranked below the Quakers and Lions in the standings.

Scenario 3A

Results

Dartmouth beats Penn
Brown beats Columbia
Yale-Cornell doesn't matter
Princeton-Harvard doesn't matter

Penn beats Harvard
Columbia beats Yale
Brown beats Cornell
Dartmouth beats Princeton

Standings

Princeton
Harvard (can't catch Princeton because of the loss to Penn
Yale 8-6 or 7-7
Penn/Columbia 6-8
Brown/Dartmouth 5-9
Cornell 4-10 or 3-11

The tiebreaker becomes records against Brown and Dartmouth. Penn and Columbia are both 2-2. The next tiebreaker is records against Cornell, and Penn wins that.

Scenario 3B

Results

Dartmouth beats Penn
Columbia beats Brown
Yale-Cornell doesn't matter
Princeton-Harvard doesn't matter

Penn beats Harvard
Yale beats Columbia
Brown beats Cornell
Dartmouth beats Princeton

Standings

Princeton
Harvard
Yale 9-5 or 8-6
Penn/Columbia 6-8
Dartmouth 5-9
Brown 4-10
Cornell 4-10 or 3-11

The first tiebreaker is Penn and Columbia's respective records against Yale. Penn takes it with a win over Yale that Columbia would not have.

Scenario 3C

Results

Dartmouth beats Penn
Brown beats Columbia
Cornell beats Yale
Princeton-Harvard doesn't matter

Penn beats Harvard
Columbia beats Yale
Cornell beats Brown
Dartmouth beats Princeton

Standings

Princeton
Harvard
Yale 7-7
Penn/Columbia 6-8
Cornell/Dartmouth 5-9
Brown 3-11

Penn and Columbia would each be 1-1 against Yale and 2-2 against Cornell and Dartmouth. The ratings tiebreaker would come into play.

If Penn and Columbia both get swept and one of Brown, Cornell or Dartmouth joins them at 5-9:

If the third team is Brown, all three teams would be 1-1 against each other. Columbia would get the bid via its win over Harvard.

If the third team is Cornell, Penn would get the bid via a 3-1 record against the other two.

If the third team is Dartmouth, the Big Green would get the bid via a 3-1 record against the other two.

If there is a four-way tie with Penn, Columbia, Dartmouth and Brown or Cornell at 5-9:

Penn's wins would be over Columbia, Cornell, Brown, Yale, and Cornell again. Their round-robin record in a four-way tie would be 2-4 if Brown is the fourth team and 3-3 if Cornell is the fourth team.

Columbia's wins would be over Cornell, Dartmouth, Harvard, Brown and Penn. Their round-robin record in a four-way tie would be 3-3 whether Brown or Cornell is the fourth team.

Dartmouth's wins would be over Penn, Brown, Columbia, Penn again and Princeton. Their round-robin record in a four-way tie would be 4-2 if Brown is the fourth team and 3-3 if Cornell is the fourth team.

Brown's wins would be over Penn, Cornell, Dartmouth, Columbia and Cornell again. Their round-robin record in a four-way tie would be 3-3.

Cornell's wins would be over Columbia, Dartmouth, Dartmouth again, Yale and Brown. Their record in the round-robin would be 3-3.

So...

If Brown is the fourth team

Dartmouth 4-2
Columbia 3-3
Brown 3-3
Penn 2-4

If Cornell is the fourth team

Dartmouth 3-3 (with a win over Princeton)
Columbia 3-3
Penn 3-3
Cornell 3-3

Which means Dartmouth gets the bid in a four-way tie.

Thanks to longtime Ivy League basketball stats wonk Mike James, and Meghan Moore and Matt Panto at the Ivy League office, for help in figuring out the many scenarios listed here.