4-player vs. 3-player groups. Are there disadvantages/advantages?

I attended a tournament over the weekend and overheard a player complaining about being placed in a three player group. This tournament used Pinburgh style scoring (3-2-1-0, 3-1.5-0), and the player was trying to make the case that this scoring scheme puts players in a three player group at a disadvantage. The player made an additional argument that 7-5-3-1, 7-4-1 scoring does not have this problem.

From my perspective, this problem can be solved straightforwardly by computing the expected value of each scoring system. Using pinburgh style scoring, the expected value between a four player group and three player group is exactly the same (1.5). The same is true of 7-5-3-1 scoring (expected value = 4). Thus, from an expected value perspective, there really is no disadvantage or advantage for being in either a three player or four player group.

This got me thinking, do other players have similar feelings about three and four player groups? Do you have a preference for one group size over the other? If so, why? Am I failing to consider something that isn’t covered by expected value?

For the record, some scorings schemes do have expected value imbalances between three and four player groups. Here are the expected values for each of the main popular scoring schemes out there.

Equal value schemes

  • E(3-2-1-0) = 1.5 vs. E(3-1.5-0) = 1.5
  • E(7-5-3-1) = 4 vs. E(7-4-1) = 4
  • E(8-6-4-2) = 5 vs. E(8-5-2) = 5

Unbalanced schemes

  • E(4-2-1-0) = 1.75 vs. E(4-2-1) = 2.33
  • E(5-3-2-1) = 2.75 vs. E(5-3-1) = 3
  • E(7-4-2-0) = 3.25 vs. E(7-3-0) = 3.33
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The main advantage of 7/5/3/1 (“IFPA”) scoring is that players in three-player groups complain less. I think it’s the half-point that somehow throws people off and think they’re losing out.


These are literally the same scoring system in every meaningful way, unless a player completely skips a round. The point scoring in 7-5-3-1 is the same scoring as Pinburgh, with double the points, and a meaningless +1 participation point added to everyone.

Ask this player how they feel about 6-4-2-0 scoring!


That’s +1 meaningless participation WINS to you. Nobody calls them points :wink:


Except somewhere on the internet I’ve read that this leads to less ties, but I haven’t tried to verify that mathematically. (Not relevant in Pinburgh, but relevant if you’re having top two advance).

Everyone I’ve ever known who played in a three player group in Pinburgh has complained about it, including me. Maybe it’s all in our heads, but it feels much harder to get points. Someone in your group blows up a game? Best you can get is 1.5. It’s also less likely that someone will have one of those terrible games, so your okay scores that would have earned a 1 by beating someone, are now much more likely to take a zero.

I’ve been in a 3 player group with Bob Matthews. So now a top player in the world accounts for 50% of my opponents this round. I’d have to sweep that other player just to squeeze out a 6.

The NWPAS bonus point scoring is also a balanced scoring for 4/3/2 players. See “Game Scoring” here: http://nwpas.wapinball.net/main.html. It’s a variation of the 7-5-3-1 scoring with an average of 4 points per player regardless of number of players, but provides more target score levels that generally results in less ties, and especially for novice to intermediate players keeps them closer to the qualifying line than some other scoring formats. The scoring is based on what we used to do at FSPA (and I think they still do), but with an adjustment that ensures 1st always scores more points than 2nd, and not end up with situation where both get same number of points.

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Since you assume Bob will win all games regardless, go ahead and remove him from both evaluations… now you have 3P: 2 players chasing 6 points, or 4P: 3 players chasing 9 points. Both have 3 points available per player. Former scenario is basically first/last, whereas the 4P game gives you the ability to finish in the middle… people like having second place as a safety :slight_smile:


It warms my heart to hear people talking about expected values. :slight_smile: That’s exactly what you need to keep in mind when creating a scoring system. I work with lots of leagues to develop/refine their formats, and anytime a league proposes a scoring scheme that doesn’t have the same avg points/player/game, I try to talk them out of it. Even something that seems minor (say, 2.5 vs 2.25) can have a noticeable impact across a few dozen games in a league season or Pinburgh-style tournament.

If the scoring system is fair for all group sizes, I personally prefer, and often hear others prefer, 3-player groups. It seems to be a good balance between socializing with your opponents, and not having an interminable wait between balls.

Nice refinement, Dave. Another refinement that we’ve discussed (and I think a couple FSPA-derived leagues are using) is to have the bonus point be based on (P1 > (3 * P3)) instead of (P1 > (P2 + P3)) (and equivalent for 4-player groups). Argument in favor of this variation is: why should P2’s score affect this calculation, when they’re getting the same league points either way? Arguments against this variation: takes away the excitement of P3 cheering on P2 to “get close” to save their bonus point; also, the 3X is somewhat arbitrary.

[quote=“spraynard, post:1, topic:3172”]This tournament used Pinburgh style scoring (3-2-1-0, 3-1.5-0), and the player was trying to make the case that this scoring scheme puts players in a three player group at a disadvantage. The player made an additional argument that 7-5-3-1, 7-4-1 scoring does not have this problem.
What is this argument? These scoring systems are mathematically equivalent, as Bowen already pointed out, so I’d like to hear what the argument is :slight_smile:

Re: 3 vs 4 player groups. Probably most people want to play in the kind of group that most people are playing in: FSPA-style encourages 3-player groups, Pinburgh 4-…

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Anyone ever try 7-4-2-1?

10-5-1-0 or GTFO :slight_smile:


I played in a 3-player group at Pinburgh for round 6. In my head, I was pretty upset mostly because everyone was telling me that it was BS. I went into the round thinking it was BS, but after I played I didn’t really mind because the scoring worked out just fine. I believe there is more of a stigma around 3 player groups then anything.

New England Pinball league scores 10-6-4-2 10-5-2. For me, it’s a psychological problem rather than a math problem. I think in my head that I have less people to beat when I’m up against two players who I feel are better than me and I tank. I should really just concentrate and play better.

4 player groups feel like they have the potential for better results. I don’t care about your fancy wizard maths tricks because I know what’s right in my heart… :smirk:

I do suppose being in a 3 player group could suck in the very specific case of a matchplay tournament where the groups are drawn randomly (not compressed to tournament position) and so you’re on the bubble needing 2 or more points to have a chance at qualifying. Or am I still feeling wrongly?

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While mathematically the expectation might be same in the scoring systems, it sure seems like it assumes the probability of any result is equal (which it is not). The probability function is not known, but people certainly have a feel for it.

In other words, they know if they have a chance of succeeding against 1 player, 2 players, or 3 players. As the number of players in a group match is reduced, then as soon as a known top ranked/rated player is introduced into the mix, the lower ranked players already know (feel) what can be done.

In a 3 player match, they might feel that they now can only succeed against one player versus having 2 other lower ranked players of similar skills to potentially succeed against. Reduce a match to heads up play with a large disparity in rating/ranking, and for some people it would certainly beg the question of why even play the game out, other than to hope for a miracle.

Theoretical math is one thing, reality is another. :slight_smile:

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It’s totally the perspective of the player in these situations. If someone feels they have a disadvantage in a 3-player group . . . I would assume that another one of those players probably feels like it’s an advantage.

I know in our league we see this all the time. Personally I see 3-player groups as easier chances to grab 7 points (we do 7-5-3-1/7-4-1). I expect to win every game I play, so needing to defeat less players just makes that task easier.

However, I can sympathize with any players that feel like first place is off the table, and now their “MAX” is only 4 points per game instead of 5.

That stretches out even further as you go down our league standings. Lesser skilled players that may feel like they are playing for 3rd place in every game go from fighting for 3 points per game, to feeling like they shouldn’t even show up because they are destined for 1 point per game in a 3-player group.


I also played in a 3 player group in round 6 at pinburgh. I had no idea I was supposed to be upset about it. heh. I was actually excited for two reasons. It would make it easier to get a medallion. And I wanted to get a half point because I don’t like tie breakers. I ended the round with 10.5, just missed the medallion but I got one in round 7 against a full group!


Most people probably realize, but just pointing out that should be 7-4-1.

Fixed! Thanks

You must’ve misheard that part because he’s in a 3 player group again tonight and complaining as much as ever. I hear he already proved you’re all wrong because “if I get 2nd in 4 games, that’s 20 vs 16. Not the same.”

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