TGP calculation help

Events
22

So, do you get it? Because I don’t. Here’s the issues I’m having:

There’s quite obviously something wrong with the 4pl, 1ko column. The numbers don’t even match the numbers in the TGP Guide. @pinwizj any idea what happened there? Just look at the first row: 4 people play a single game, that’s it. How is this 3 EV*? My guess is that there was a mistake in the table, which has been fixed at some point, so an updated version should exist somewhere which the TGP Guide is actually based on.

The TGP Guide says:

3 player groups, 1 player eliminated per round until final 4 players remain (assumes 1 game for final 4).

Looking at the table, I don’t think that’s correct, and it should actually be until final 3 players remain (assumes 1 game for final 3). Same goes for 2-player groups.

The reason is this: Look at the first row of 3pl, 1ko, it says Max 2, meaning the winner of the tournament could have played a maximum of 2 games. That’s not possible if the final game is supposed to be played with 4 players, and there’s only 4 players total in the tournament.

Even for the columns other than the 4pl, 1ko one, and assuming the correct number of players in the final game, the numbers don’t exactly match up with mine. @dbs are you still around and can maybe shed some light on how you did it? Let’s look at the first row of 3pl, 1ko for example. Here’s how I would calculate the EV+:

  • There’s 4 players total, and each one contributes to the expected number of games played by the winner based on this: If they win the tournament, how many games will they have played? And what’s the probability that they do actually win the tournament?

  • One player enters the tournament in the final game. There’s 3 people in that game, so for the probability to win it (and thus the tournament) we can assume 1/3. That player will only have played 1 game in the end, so they contribute 1/3 * 1 to the EV. Since it’s a 3-player game, there’s a 1.5x multiplier, so they contribute 1/3 * 1 * 1.5 to the EV+, which is 0.5.

  • For the other 3 players it’s like this: In order to win the tournament, they have to advance a round (don’t come in last in a 3-player game, so a probability of 2/3) and then win the last game (1/3, as before), so a total probability of 2/3 * 1/3. If any of them does that, they will have played 2 games in the end, so that’s a contribution of 2/3 * 1/3 * 2 to the EV. This applies to 3 players total, so in total those 3 players contribute 2/3 * 1/3 * 2 * 3 to the EV. 3-player multiplier again, so combined they contribute 2/3 * 1/3 * 2 * 3 * 1.5 to the EV+, which is 2.

  • So, the sum of all players’ contributions to the EV+ is 2.5, meaning the EV+ is 2.5.

  • The table says 2.3, though—how do you get that? I mean, in order to calculate the EV, a probability for each player to win / not lose a game has to be assumed. I gave all players an even probability to win each game, which seems reasonable, and it’s the same assumption the tables for the bye playoff structures seem to make, so I guess you’re doing the same? What’s the difference then?

The table seems inconsistent with the Common Playoff Structures table.

Some formats are actually identical, but they are listed with different EV values in different tables. E.g. row 5 (8 players) in the 4p, 2ko ladder table plays identical to row 3 (8 players, 2 x 2 byes, 2 x 1 bye) in the 4-player common playoff structures table, but the first lists an EV* of 3.6, while the second lists an EVR of 1.75, so an EV* of 3.5 (rounded up to 4 in the table).

Here’s that playoff table for comparison:

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My previous post had a mistake in my math, so I took it down. But, no, I have not figured out a way to reverse engineer the chart that @pinwizj posted.

I agree with everything you’ve posted here, @umbilico. In particular, Ladder Np8, 4pl, 2ko should be exactly equivalent to Elimination Match Play Np8, 4pl, 2br2, 1br2 and the fact that the Ladder Chart gives EV* 3.6 rather than 3.5 tells me something is wrong with these charts. I also assume it’s wrong in the Ladder chart because the EVR from the Elimination Match Play chart is easy to reverse engineer in a way that is consistent with the other values of the chart and matches my understanding of what expected value means.

If Dave has the program or spreadsheet formulas that were used to generate the ladder chart that’d go a long way towards helping us uncover the reason for the difference and from there we can either explain it or correct it.

Another possibility: @pinwizj is Ladder intentionally less valuable than Elimination Match Play? I ask because Flip Frenzy, for example, is intentionally less valuable than the similar head-to-head formats. But in the case of Flip Frenzy the mechanism and rationale are clearly explained in the WPPR Rules and I don’t see anything like that for Ladder.

To better understand how Elimination Match Play works, I made a spreadsheet that reproduces the “4-Player Common Playoff Structures” chart. I’d describe it this way:

Effective Rounds (EVR) = SumForAllRounds((number of players actually playing this round) / (number of players that typically play this round))

So for Elimination Match Play Np8, 4pl, 2br2, 1br2 it’s:

EVR = 4/16 + 4/8 + 4/4 = 1.75
EV* = EVR x 2 = 3.5

I don’t know what math would lead to the EV* = 3.6 value in the Ladder Np8, 4pl, 2ko example.

Here’s another example for the last row in that chart Elimination Match Play Np40, 4pl, 2br4, 1br12:

EVR = 24/64 + 24/32 + 16/16 + 8/8 + 4/4 = 4.125 (rounded to 4.13 in the chart)
1GprEV* = EVR x 2 = 8.25 (rounded to 8 in the chart)
3GprEV* = EVR x 2 x 3 = 24.75 (rounded to 25 in the chart)
4GprEV* = EVR x 2 x 4 = 33
5GprEV* = EVR x 2 x 5 = 41.25 (rounded to 41 in the chart)

I don’t think it’d make sense but is ladder supposed to also work based on “full group match play bracket round size”? For example, Ladder Np12, 4pl, 2ko also has 5 rounds; is it computed like:

EVR = 4/64 + 4/32 + 4/16 + 4/8 + 4/4 = 1.9375
EV* = EVR x 2 = 3.875

It’s close to that, but the chart has 3.7 rather than 3.9. Even doing weird rounding, I can only get it to 3.8. :man_shrugging: I also don’t know why it should work that way even if I’m on the right track.

Another way to approach this… Forget the math, @pinwizj How should TGP% work for a Ladder finals in plain English? We can figure out the math from there and show our work; though it may turn out to be different that what’s in the TGP guide today.

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I also agree with (almost) everything you’ve posted here, @tommyv. A small remark in regards to your “shortcut” method for calculating the EV:

You gotta be careful there, because the method you’re laying out is based on the coincidence that the following two things are the same:

  • number of players that would typically play in a round if it were a full bracket
  • 1 / probability that a player playing that round goes on to win the tournament

Obviously, it’s the second one that’s relevant for calculating the EV of #rounds played by the tournament winner, not the first one. Those two coincide in formats where half of the players are eliminated each round (e.g. all of the bye playoff structures), but it breaks down in others (e.g. 1 player eliminated in 3 or 4 player group ladders).

… with the one caveat that it has to match the TGP of bye playoff structures, because those are basically the same format. And also it has to match the WPPR rule for finals components, “The TGP will be based on the expected number of meaningful games that the winner of a tournament will play”, unless there’s a very good reason to deviate from that. So, I don’t think there’s much wiggle room there.

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I’m not sure how @dbs did his math, but here’s how I would calculate it:

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This calculation shows expected number of games played for the winner (I think).

The excel sheet doesn’t show the 2X for 4-player matches, so simply double that number for the meaningful games played towards TGP stat.

This 10-person ladder that I did would have be 6.93 meaningful games played, which we would round up to 7 meaningful games played for TGP purposes (28%).

This assumes the tournament ends with that final 4 game as stated in the TGP Guide. If it was played out to the final winner, you would add 2.5 meaningful games played for the Round of 3 and the Round of 2.

Here’s a 20 person ladder based on the same formula:

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8 meaningful games played towards TGP assuming the last round is the final 4 round.

I believe the answer can be never more than 8 meaningful games played as the extended ladder does little to nothing to the EV for the winner. Same formula for 40 players was 3.9999046 games played.

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26

Great observation, thanks for the correction!

This is excellent, these are two descriptions of the same math as far as I can tell. That’s the good news.

The explanation from @umbilico describes each player’s contribution to EVR. The explanation from @pinwizj describes each round’s contribution to EVR. Either way, you get the same answers.

The bad news is that those answers are different than what’s in the chart and maybe the TGP guide. I think we’re all on the same page now and it should be possible to generate a new chart using either/both of these approaches.

@pinwizj’s version looks like this:

EVR_RoundN = ( [number of players new per round] / [number of players per game] ) x [number of games per round] x [game size multiplier] + ( [number of players advancing per round] / [number of players per game] ) x ( EVR_RoundN-1 + [number of games per round] x [game size multiplier] )

EVR_Round0 = [number of games per round] x [game size multiplier]

Sum(EVR_RoundN) where N from 0 to ( ( [number of players] - [number of players per game] ) / [number of players new per round] )

@umbilico’s version looks like this:

EVR_PlayerN = ( ( [number of players advancing per round] / [number of players per game]) ^ ( [number of rounds for this player to win] - 1 ) x ( 1 / [number of players per game] ) x [number of rounds for this player to win] x [number of games per round] x [game size multiplier]

NB: the top seeded players who join in only for the last round, [number of rounds for this player to win] == 1 and this simplifies to:
EVR_PlayerN[for top [number of players new per round] seeded players]  = ( 1 / [number of players per game] ) x [number of rounds for this player to win] x [number of games per round] x [game size multiplier]

[number of rounds for this player to win]_PlayerN = ? tricky math that is easier to do with if statements?

Sum(EVR_PlayerN) where N from 1 to [number of players]

@Boise_D (and anyone else following along) my apologies for all of this nerdy TGP math talk. You do not need to understand any of this to be a good TD or to try different formats. Really just stick with this advice:

Ladder is always mediocre TGP, so do it if you think ladder is fun. After we finish all of this discussion, Ladder might be worth a tiny bit more. But as @pinwizj pointed out it inherently caps out pretty quickly.

My TD advice ignoring TGP% is that Ladder with more than 3 rounds is not very fun. The top seeded players get a lot of byes. While byes make it easier to win, no one really has fun waiting for more than 2 rounds before they get to play competitive pinball again. So cap it at a top 8 ladder with 4 player games and 2 players eliminated per round for (in my opinion) the upper threshold of what’s still fun. I speak from experience.

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This was done many years ago (back in 2016) so I can’t say I remember all of it. And I just looked back at my notes, and not surprisingly, not as detailed as I would have wanted to write myself.

In any case, those two formats are NOT identical, and that’s probably the source of the discrepancy.

That row 8 with 2 then 2 byes will have up to 3 games played for the winner:
5th thru 8th. Two winners play 3rd and 4th. Two winners play 1st and 2nd. That’s the last game, final position establishes 1st thru 4th.

The ladder 4p2ko will have up to 4 games played for the winner:
5th thru 8th. Two losers get KO. Bring on next two, two losers get KO. That leaves final four … two losers get KO. That leaves 2 players, assumption is they play one game to establish 1st and 2nd.

I believe how you handle the final 4 (or final 3 in 3pl1ko matches) is probably the difference in what you are computing, and what is in those charts.

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TGP Guide has been updated for Ladder Format.

It is now based on the ladder running to completion (final 2 players compete in one final head-to-head match). The previous version was really based on the PAPA Circuit final that stopped at the final 4.

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29

Thanks to everyone who has put time into this.

I don’t think you’re right about 4p2ko having 4 games max in your table, because the table clearly says Max 3 in the 8 player column. But even considering that possibility, I’m still not getting your numbers.

It probably doesn’t matter, though, we can just recalculate this stuff and update the guide.

Thanks for the update. I’m still getting different numbers for 3p and 2p groups, though, could you have another look at that? E.g., for 2p1ko with 2 players total, you’re getting 2 meaningful games. That can’t be right, it’s just a single 2-player game.

If it helps, I’ve whipped up a calculator at https://laddertgp.slapsave.com/ that should be able to handle all the different cases that might come up.

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I got caught “adding 1” at the end (for no reason whatsoever it seems) :slight_smile:

I’ll include your slapsave link to the guide as well. Thanks for making it.

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I think for 3-player groups the numbers in the TGP Guide are still off, and it doesn’t look like it’s just an off-by-1 error, cf. the calculator results for 3-player groups. I’m pretty confident my numbers are correct, but maybe @tommyv wants to check them.

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It doesn’t look like I saved the updated 3-player group calculation on my end. I just updated it based on the table below. Max is 5 meaningful games per TGP no matter the player count.

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Thanks, looking good now.

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Eyes glazed over… @pinwizj I thought we never round up for meaningful games played. So 14.59 games is reported as 14 games, correct?

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We’ve done natural rounding since WPPR v5.0 started. Anything 14.5 or higher would be reported as 15 meaningful games played.

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