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Heading into the last day of the season there are still two meaningful games to be played: Phillies at Braves and Padres at Giants. Let's break down the various scenarios that are possible.

I took each team's third-order wins (not updated through today, although I don't expect that to make a significant difference), made a crude adjustment to account for home field advantage, and ran everyone through the log5 method to come up with expected win percentages for each game. A little cross multiplication and voila, odds. Without further ado, the likelihood of various outcomes:

Braves win, Giants win – 30%
The single most likely outcome leaves the Padres out in the cold, with the Giants and the Braves advancing to the playoffs. There's very little enjoyment to be had here, unless you happen to be rooting for the Giants or the Braves.
Braves lose, Padres win – 20%
The single least likely outcome means a tie, but one without any real entertainment value. Both teams—the Padres and Giants—advance to the playoffs, with the Padres winning the division on account of head-to-head record. Again, this is a real snoozer unless you happen to root for the Padres or the Giants
Braves lose, Giants win – 25%
This is where things start to get fun. In this scenario, the Giants win the division, but the Padres and Braves remain tied for the wild card, leading to a one game playoff. One game playoffs are the best thing ever. Except for…
Braves win, Padres win – 25%
This. This is what we're hoping for. A three-way tie for two playoff spots. It is something of a long shot (although it's not really any less likely than the other scenarios.) But it is so spectacularly enticing that I want to believe it's inevitable. As Joe Posnanski notes, this could be utterly crazy. Did I mention I really want this to happen?
The biggest spanner in the works for these odds is the Phillies, who don't need to win a game to advance to the playoffs. They'll probably make a show of being competitive, but their hearts won't be in it the same way the other three teams are. The Giants may be slightly less motivated than the other two teams, as there is no scenario that leads to their elimination tomorrow night, but that's not likely to be a major factor.

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anderson721
10/03
What happens in the AL if both the Yankees and Rays lose and the Twins win? Who gets home field?
cwyers
10/03
If the Yankees and Rays end the season tied, the Rays win the division (on account of having the better head-to-head record) and the Yankees win the wild card. I don't know what the tiebreaker is for LDS seeding yet, but I'm working on it.
lucasjthompson
10/03
Head-to-head record. And both the Yankees and Rays had the tie-breaker on the Twins, so the Yankees eliminated the Twins from the pursuit of the #1 seed with their win in the first game Saturday. In other words, now the best the Twins can do is tie for the best record, and that won't get the #1 seed for them.
Richie
10/03
The Giants will be every bit as motivated as the Pods and Braves. Or if not, 99.999% motivated to their 100%
cwyers
10/03
If the Braves don't come back (they're threatening), the Giants may not be as motivated.
drawbb
10/03
In typically Goldbergian fashion, Selig has an unnecessarily complicated solution set up for the potential three-way tie scenario in the NL. Two tiebreaker games is both extraneous and also not exactly correct.

If SF and SD tie for the West at 91-71 with Atlanta looming as a potential wild card at 91-71, MLB will force a first tiebreaker game between the Giants and Padres and then have the loser play a second tiebreaker game with the Braves. This inexplicably ignores the fact that the SF/SD loser would then be 91-72 and no longer tied with Atlanta, but instead 1/2 game behind the Braves.

I do not suggest ending the procedure with just the first tiebreaker game for the NL West, because doing so would leave us in the illogical position of the second-place Braves automatically being in the playoffs while the two teams tied for first place in the West would have only 50/50 odds of advancing.

However, a much more sensible solution exists that would require only one tiebreaker game and avoids the "half a game behind is still tied" paradox we now face.

My suggestion is simple: Since the intent of divisional play is to crown a division champion, settle the NL West tie not with a tiebreaker game but with records v. the NL West. In this case, SD would have a better record against the NL West than SF so the Padres would be the division winner. That would leave SF and Atlanta both remaining at 91-71 and THEY would play each other in a tiebreaker game Monday for the wild-card berth.
matuszek
10/04
I think you're going to have to more clearly describe what desirable characteristic your version has, and why it is more desirable than "No team in a tie makes the playoffs without winning another game, or misses the playoffs without losing another game."

The paradox you describe almost happened in 1999, when the 96-66 Mets and Reds had a one-game playoff to determine the wild card. If the 97-65 Astros had won one fewer game, under the rules at that time, Houston and Cincinnati would have played one game for the NL Central, making the Mets the wild card by one half game. Everyone agreed that a first-place team shouldn't be in a worse position than a second-place team.

Under the new system, the three teams all have to win a game to make the playoffs, but the better teams are rewarded, to wit:
The first-place team with the tiebreaker advantage has two chances, the first at home and the second on the road,
The first-place team without the tiebreaker advantage has two chances, both on the road,
And the second-place team gets just one chance.
drawbb
10/04
What's unclear about not wanting two tiebreaker games, the second of which would pit two teams who aren't even tied? That's an incongruity which should be easily understood.

Besides being tidier and more efficient, my solution also carries the additional benefit of not extending the schedule needlessly which appears to be a rising concern in the postseason.

Rather than your mission statement, I'd suggest the top priority should be keeping the schedule to one and only one tiebreaker game per league unless two are absolutely necessary to resolve a four-way tie.
rawagman
10/04
The two teams are tied through 162 games, which is the most important clarification. While the 163rd game counts in the ultimate standings and the statistics, as a way of pointing out why one team is in the playoffs and the other teams isn't, that is the game's only function. It does not change the two teams' places in respect to any other team in the league, whether for seeding or for draft picks.
Richie
10/04
My bad. I was assuming the Giants needed to win regardless to clinch a berth. Which is how it turned out in the end, so I guess I was more prophetic than wrong.