Thursday, June 22, 2017

WBC: Expected Laurel Case Studies

During this week's Agricola stream we had a bit of a side discussion going about laurels at WBC based off of my last post. Of particular concern was that I may not be properly considering the time invested in making it to the semis which dovetailed into the idea that even if we can use the butt-hour formula (which determines prize levels) to approximate available laurels that it may not hold across different tournament formats. By this I mean that maybe a single elimination tournament is just more efficient than a heats into semifinal tournament. Or, as Randy suspected, the opposite may be true. Twilight Struggle was the game that was brought up as a particular example. My gut feeling is that Twilight Struggle is an excellent game to play, if you are good at it, because it is a skill intensive game with a ton of laurels on the line. Randy thought that the amount of time you need to invest in a day long tournament (it's 5 3-hour rounds all in one day with a final afterwards) would be a huge problem.

I suspect it would probably be a problem because losing an entire day probably kills off a bunch of other tournaments. But my gut feeling is that that's only a problem for someone who plays other games and would be looking to add another and not something intrinsic to the single elimination format. So it's probably a terrible game for Randy or I to pick up in a quest for Consul, but that someone could build a Consul plan around it. But I don't want to just go on gut feelings, I want to crunch some numbers!

Another game that came up was Advanced Civ. It was brought up as being way too much time for the potential payout and my response was that it was likely true, but only because the formula caps at 6 and that Advanced Civ was probably worth about a 10 because of how many hours get invested and therefore it's a bad play because of formula inefficiency. So I wanted to check into that... Turns out I definitely have egg on my face here because it only has a 5 prize level! Having to sink 16 or 24 hours into a game is a really big investment, especially compared to Stone Age only needing 10 hours. But actually, maybe your odds of earning laurels could be a lot higher? (Keven Youells has earned laurels in it for 14 straight years...)

So I want to crunch some numbers for a few games to see how things line up with a couple assumptions. After that I'll decide if I care enough to go through all of the games or maybe if I'll learn some shortcuts that can be used to make assumptions about the rest of the games? Who knows!

Twilight Struggle

This game is run swiss style, but they play until they have 2 undefeated people and then only those two play in the finals. So it's basically single elimination when it comes to 1st or 2nd, but for 3rd-6th you can keep playing after a loss. I'm going to assume you enjoy the game enough to keep playing with a single loss but will drop out with 2 losses. (Actually, the say they use strength of schedule to determine 3rd-6th, so probably I should assume a loss in one of the first 2 rounds is a drop.) The last couple years have seen attendance swing up barely above the magical 64 number so I'm actually surprised they've been able to finish in only 6 rounds. From the recap they had only 3 undefeated players after 4 rounds last year which really doesn't make sense. That implies only 48 people were really playing but they had 70 sign up. There were also only 2 draws in the whole event, so it isn't like that was eliminating people either. So there must have been quite a few people who showed up, won a round, and dropped. So I'm going to assume there are only 48 people in the tournament even if 70 show up, which will inflate the laurel numbers a little because in reality you could be the person who loses to someone who drops.

(Alternatively it could have gone 70->35->17->8->3 if one of the draws was between undefeated people in round 3. I'm not sure which is more likely to be honest. I should hedge a little and assume more like 54 people show up.)

Here are your potential outcomes, assuming a 50% chance to win each game.

50% - drop after 3 hours (0-1)
25% - drop after 6 hours (1-1)
6.25% - drop after 12 hours (2-2)
6.25% - drop after 15 hours (3-2)
3.125% - make finals
9.375% - make top 7

Twilight Struggle has 5 prizes, so you're looking at...

50% - 3 hours for 0 laurels
25% - 6 hours for 0 laurels
6.25% - 12 hours for 0 laurels
6.25% - 15 hours for 0 laurels
1.5625% - 18 hours for 50 laurels
1.5625% - 18 hours for 30 laurels
1.3393% - 15 hours for 20 laurels1.3393% - 15 hours for 15 laurels
1.3393% - 15 hours for 10 laurels
1.3393% - 15 hours for 5 laurels
1.3393% - 15 hours for 0 laurels

For a total EV of 2.1875 laurels earned for 6.65625 hours invested. Or .32 laurels per hour.

Your odds of winning are not going to be 50%, though. This is where a little bit of art needs to seep into our science. If we're looking at someone who is actively good at the game what are there odds of winning a game? Those odds would need to get worse as you got later in the tournament as the worse players would get removed from the pool. Looking at the laurel list the top player has a massive 443 laurels with second place having 161. There are many people with a significant number of laurels which makes me think this is a very high skill game. I think I want to start our mythical great player off with a 90% chance of winning in round 1 and linearly trend that down to 60% in the finals. That changes the above numbers to:

10% - 3 hours for 0 laurels
14.4% - 6 hours for 0 laurels
4.66% - 12 hours for 0 laurels
9.69% - 15 hours for 0 laurels
16.8% - 18 hours for 50 laurels
11.2% - 18 hours for 30 laurels
6.65% - 15 hours for 20 laurels6.65% - 15 hours for 15 laurels
6.65% - 15 hours for 10 laurels
6.65% - 15 hours for 5 laurels
6.65% - 15 hours for 0 laurels

For a total EV of 15.09 laurels earned for 13.2 hours invested. Or 1.14 laurels per hour. Better, but that actually doesn't feel very good...

Advanced Civilization

This game plays two heats and then advances the top 8 players to a final. Each game is 8 hours in length and you can't leave partway through. They get around 40 players total, so if every player played in both heats you'd be looking at somewhere between 10 and 12 winners. I don't know how likely that is to happen. The recap for last year says they only had 9 people play in both heats, with 28 people in the first heat and 16 in the second heat. So they only had 6 games total, with one guy winning in both heats. Two of the winners didn't even show up for the finals, so they advanced 5 people who hadn't won a game. By the sounds of it, showing up for the finals after playing a decent game advanced you. But two years ago they had 8 games in the 2 heats with one double winner with all winners showing up and a very tight battle for closest 2nd...

To be safe, I think we need to assert that you need a win or a very close second to advance. If that isn't true, and it turns out to be a 'soft' game, then enough of us will show up to make it become true for future years. It seems like games in the heats are often 7 players, but they could be anywhere from 6 to 8.

This means that it's likely that the breakdown for this game is going to be:

1/7 - 8 hours to make finals
6/49 - 16 hours to make finals
36*2/49/8 - 16 hours to advance as a close second (assuming you play both heats and that 2 of 8 2nd placers advance)
55% - 16 hours for 0 laurels

Then once you're in the finals you need to commit another 8 hours for a 1 in 8 chance at each possible result. It's a 5 prize event, so 50-30-20-15-10-5-0-0. The math churns out to be 7.29 laurels for 18.4 hours, or .395 laurels per hour. Better than Twilight Struggle when the games are coin flips!

But Advanced Civ games are _not_ coin flips. There is definitely some randomness, but since there's a guy who laureled 14 years in a row I think it's pretty safe to say that someone who is really good at the game is going to be really good at the game. But how good is really good? Are they going to be 50% to win a heat against 6 other players? More than that? What about their finals odds?

I think I want to give the good player 50% to win a heat, 25% to come a close second. Finals odds I want to be 20-20-20-20-5-5-5-5. Advanced Civ is a game that ends at quasi-random times, especially in a final where people can be playing for best position as opposed to a heat where I wouldn't anticipate a lot of playing for 3rd or 4th.

This puts the EV at 22.2 laurels in 19.5 hours for an overall laurels per hour of 1.14. I swear I didn't cook these numbers... They really do round to the same as Twilight Struggle.

Thurn & Taxis

This game is run with 3 heats of 2 hours each. Winning a heat is good enough to advance to the quarterfinals but if you do particularly well you can earn a bye into the semis. This leads to two different possible plans... You can try to win a single heat and then sit the rest out or you can play every heat in an attempt to earn that bye. If Thurn is the game you care about you definitely want to try to earn that bye but if you're trying to maximize total laurels it likely depends what you could be doing with those time slots.

Last year had 36 people play in 3 heats, 51 people play in 2 heats, and 61 people play in a single heat. That means something like 70 games were played. I believe 4 people got byes to the semis which means 2 wins is not good enough for a bye. I don't know how to track things forward to future years, but I suspect a decent assumption would be that 3 wins is worth a bye to the semis and everyone else has to play the quarters. So my player is going to play at least two heats but only commit to playing the third heat if they have 0 or 2 wins.

1/64 - spend 6 hours to make semis (WWW)
3/64 - spend 6 hours to make quarters (WWL)
3/16 - spend 4 hours to make quarters (WL)
3/16 - spend 4 hours to make quarters (LW)
9/64 - spend 6 hours to make quarters (LLW)
27/64 - spend 6 hours to cry (LLL)

From there it's a bunch of number crunching because of the different number of hours that can be spent on each branch, but my spreadsheet spits out that you expect to earn 1.27 laurels after spending 6.77 hours, for .188 laurels per hour. Which makes T&T a worse use of time than the previous two games when every game is a coin flip! I suspect the reason for this is that no-skill semis are actually a real bad use of time and no-skill quarters are even worse. 94% of people not earning any laurels at all is pretty rough! I guess that's the downside to 150ish player fields compared to 40 player fields!

Anyway, how good can you be at T&T? This is a harder one for me to estimate because I simply don't grok the game at all. It has had repeat winners, I recognize the names of the winners as all being quite good at games, and the laurel list has some big numbers on top so there's definitely skill there. The TrueSkill list on Yucata makes me think it's more random than Stone Age, but still has a pretty high skill component. So I'm going to say our good player wins 45% of heats, 40% of QFs, 35% of SFs, and 30% of Fs.

Swapping in those win rates to my spreadsheet spits out 5.07 laurels in 8.27 hours, for .613 laurels per hour. Much worse than either of the last two games! Is that my being unfair to skill factors in the games, or is it just that the big Euro heat game is not a very good play for laurels? (Heats do get punished by the WBC butt-hour formula, for what it's worth.)


Innovation is a super short single elimination tournament. Heats are scheduled for an hour but it's pretty likely 4 rounds will get compressed into 3 hours. I think I need to keep assuming every round is a full hour though, because sometimes slow people play... At any rate, I'll be considering it to be a mulligan + 6 rounds, with everyone who makes it to the 4th round getting laurels. (The game historically has had 6 people make it that far.) I'm also going to assert that if you win the mulligan you don't show for round 1, but if you lose it then you do.

In coinflip land, this means:

1/4 - out after 2 hours (LL)
1/4 - out after 2 hours (W-L)
1/8 - out after 3 hours (LWL)
1/8 - out after 3 hours (W-WL)
1/16 - out after 4 hours (LWWL)
3/16 - top 6

From there it actually gets a little tricky because of issues with byes/eliminators and that potential extra hour from the mulligan and round 1. Eugh. I'm going to assume the eliminator always loses, which is not true historically so maybe you should bump the numbers up a bit. With that assumption, off to the spreadsheet... (Oh, and Innovation is a trial, so it's only worth 20 laurels for 1st place.)

It pans out to earning 1.5 laurels for an investment of 2.95 hours. This means .508 laurels per hour which is our best coinflip rate so far! I suspect this is because not enough people play so first place shouldn't be worth 20 in a perfect world. So the people who do play get extra value for doing so?

How about a skill factor? Well, one person (Pounder) has made the finals in each of the last 4 years. We've played quite a few times for fun over the years and he routinely smashes me. I beat him once that I can remember (in the finals in 2015, hah!) but other than that I'm not sure I've ever beaten him. There are 7 rounds, so we need 7 win percentages. Round 1 should be the highest number since all the mulligan winners are taking that round off. I feel like I want the finals odds for our great player to be 60, so we'll use a similar scaling backwards thing that we did in Twilight Struggle? With the mulligan round being the same as round 2? So 84%-90%-84%-78%-72%-66%-60%.

Doing that gives us 7.13 laurels in 4.39 hours, or 1.63 laurels per hour. Unsurprisingly the highest coinflip game thus far is also the highest skilled game thus far. Is it fair to say Innovation is as skill intensive as Twilight Struggle?

Vegas Showdown

I want to do at least one more Euro, so let's do one that I think is more random than T&T or Stone Age. The reason I think Vegas Showdown is more random is that you have to pick a strategy pretty early on in the game but the winning strategy can't be known without knowing the order of the card deck. There are certainly still edges that good players will eke out over the course of the game, Showdown isn't on the level of Can't Stop or anything, but I think even the best players are going to win less frequently at this than at some other Euros. (It probably doesn't help that the elimination games are 5 player games.)

There are 3 heats of Vegas Showdown cutting 25 players to the semifinals. The last 2 years have each had 39 games played across the heats so there are likely to be a couple people with a win who don't make the semis. Last year had 7 double winners, leaving 25 more single winners, so 7 winners didn't advance. As such I think you definitely need to play at least 2 heats, and should probably play the third unless you already have at least a first and a second. Heats tend to be 4 player games and this is a 4 prize event.

It ends up being one heck of a spreadsheet, but it churns out 1.89 laurels in 6.21 hours or .303 laurels per hour. Which puts it ahead of Thurn, but behind all of the other games looked at thus far. It feels like games where you need to do better than win a heat to advance are bad deals.

We need to pick some skill numbers for Showdown. I think it'll be fair to pick numbers a little lower than Thurn because Showdown feels more random to me. I'm thinking a 40% chance to win a heat, 30% chance to come second in a heat, 30% chance to win a semi and finals odds of 25%-25%-20%-15%-15% for the different places.

Plugging those numbers in gives us 4.84 laurels in 6.87 hours, or .705 laurels per hour. That doesn't change where it lands relative to the other games.

I am getting very tired, and it turns out to be a fair amount of effort to do individual games. I'm more than happy to discuss methodologies if people disagree with these numbers, but I don't feel like my mind has been changed by looking at these games. Needing to do better than a win in a heat feels really bad to me now. You're getting dinged in the butt-hour formula for having heats but you don't get to save any time by taking heats off. Trials do feel good though, since they're probably heavily overvalued by being worth 20 laurels for a win.

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