I love linked campaigns. They’re probably my favorite way to play wargames – you build a little narrative in your head, you play some games, and at the end there’s a winner (and probably a planet reduced to rubble in the process). There’s a problem that’s always dogged them though – the runaway victory. When one side wins the first few games, and thus gets such a big lead, in whatever scoring or reward system your using (points per win, XP per character, etc.) that the other side might as well not play.

Lets face it, that’s less than fun.

The first time it happened to me, it was in a Mordheim campaign, and a Skaven player. Through a turn of good luck, he had so much gold he could drown anyone and anything in ratmen with slingshots. The goal of the campaign turned into “Escape games with the Skaven player having taken minimal casualties”. And it happens in the campaigns I play currently – there’s consistent dropout as it becomes clear that some point stand no chance of winning. As one of these people, I cope by coming up with alternate ways of winning – usually summed up as “hijinks” – but surely there’s a better way of doing this.

Enter…multiplication.

Note that the title of this post says “Non-competitive”. That is, campaigns and other series of games where there’s some call to have a “winner” (who captures the planet, etc.), but where the primary focus is for people to show up and have fun. This is expressly *not* for tournaments and other events where someone crushing all their opponents so handily that they might as well just skip the last round is, well, an achievement to be recognized and celebrated.

First, lets illustrate the problem with a simple example: A six-game campaign, where each time you win, you get a point. Lets also assume that the campaign is between two teams/people and is unbiased – that is, that everyone has a chance of winning. It might go something like this:

Team 2 (Green) wins the first and second games, Team 1 (Blue) wins the third, Team 2 wins the fourth and Team 1 wins the fifth. The red line denotes the difference in scores, which drops to one (3 to 2). The best Team 1 can hope for is a tie, but this is probably still worth going after. But if it had been 4 to 1…well…what would be the point? That last game wouldn’t matter. Simulating a thousand of these games yields this distribution:

About 600, or 60% of the games, are within one point. You can still force a tie. But that leaves ~ 400 games where the other team isn’t really in it at all. They might as well go play another game, grab a beer, or do laundry. Still, 60% isn’t so bad.

The problem is when there’s bias in the teams. It could be because logical divisions like “Order vs. Chaos” fell out with one side running really good armies and the other not, or one or two players who are just, well, a little better. Let’s look at a pretty light case of a biased campaign, where one team has a 60% chance of winning:

Still not so bad. The number of “still worth it” games drops from “a little above 600” to “a little below 600”, but we start to run into some problems with the really bad runaway victories, where the score difference is 5, and the other team not only can’t force a draw, but has just gotten so sufficiently blown out of the water that, well, what’s the point? This is made worse if you’re awarding extra units, character experience, benefits, etc. to the winning side. Now lets look at a campaign with pretty severe bias, where one team has a 75% chance of winning:

I think we can all agree that this campaign would be the definition of anti-fun. Only 35% of the time does the losing team have any hope of forcing a draw, and the campaign is now dominated by outcomes where the game was decided two or three games before the official ending. Carl at the Independent Characters talked about this with the campaign he was running – a Tyranid invasion of a planet. Essentially, the Tyranids were sufficiently outmatched by the Imperials that there wasn’t any point in going on in the campaign. What fun is that?

So is there a better way to handle this? A way that keeps things competitive, even in the face of a string of victories, or uneven odds?

Yep.

Break out your calculators.

In the old, additive system, your points after winning a game is as follows:

NewPoints = OldPoints + 1

Simple enough, right?

Now change it to this:

NewPoints = OldPoints + OpponentsPoints*Factor

Somewhat more complicated, but not massively so. Basically, instead of adding a flat number, you take your opponents points, and multiply it by some fraction to get your new points. For example, if you have 5 points, your opponent has 5 points, and the factor is 0.20…

NewPoints = 5 + (5*0.20) = 5 + 1 = 6

And if you had 5 points, your opponent had 6 points, and the factor is 0.20…

NewPoints = 5 + (6*0.20) = 5 + 1.2 = 6.2

So what we lose is clean, integer-based scores, but we gain something important: The more you’re losing by, the more a single win will bring up your score. The underdog is advantaged, and the current winners are at a disadvantage. Now it’s *always* better to win, but the degree of that isn’t flat.

Lets take a look at what that looks like:

This game started everyone at 5 points for showing up (instead of zero, because multiplying by zero is a bit of a problem…) and the factor I used was 0.20. As you can see, there’s a fair bit of bounding around in terms of whose in the lead. And importantly, going into Game 6, Team 2 can more than force a tie…they can actually win!

But how does this hold up to a bunch of simulations? Again, lets look at 1,000 unbiased games. This time, instead of looking at the difference, which is a tricky concept with this system, we’ll look at what we really care about, the percentage of games that are “pointless”, meaning the loser is going to lose regardless of the outcome of the last game.

Not bad – it’s about a 50/50 split for “Do the losers have a shot?”. Though maybe not as great as the 60/40 split for the additive system for “Can the losers force a tie?”. Never the less, still good. And there’s another key here – you can tweak the *factor*. The closer to 1 you go, the more you advantage the losing team (when you set the factor to 1, you literally make only the last game matter).

Lets look at the same campaign – unbiased, starting at 5 points, six games long, but this time, the factor = 0.40:

This…this looks good to me. 70% of the time, the last game matters. You have to be winning *hard* for their to be a real blowout victory, but being in the lead still matters. By awarding the victor of a game 40% of their opponents current score, rather than a fixed number, you definitely keep the losing side in the running more.

But unbiased campaigns weren’t really the problem were they? Lets look at that biased campaign again, the one where one of the teams has a 75% chance of winning, but using the multiplier-based campaign score:

*This* is what’s at the heart of the multiplier system. It’s entirely flipped the outcome of a biased campaign from only 35% of the time it’s worth the losing team showing up to only 35% of the time is it *not* worth it. The really pretty severe bias in favor of one team in the campaign is almost entirely negated by the scoring system itself, and things are only a little more hopeless for the unfavored team than they would be if the two teams were equally favored.

Essentially, using a multiplier both *keeps teams in the running longer* and *is robust to an unbalanced game*. Both of these are good things. They’re also easier to tweak – you can advantage the underdog more or less by increasing or decreasing the factor respectively, as your group wants. Yes, it is marginally more complicated to work out the scores, but it’s marginally more complicated in the “Does anyone have a smartphone?” sense of the word – this can still be done on the table in less than 5 minutes after a game.

As far as I can see it, there’s no reason, outside of the tournament scene, to ever use “Win a game and get N points” based scoring systems ever again.

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I love that idea! Would you combine that with a system where the winner gets a few bonuses for winning then? Seeing as it probably wouldn’t make as much of a difference.

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Yeah. I could see doing a Mordheim-esq XP system, or other bonuses for winning, because there’s less of a steep slope to climb to get back in the game. I’d still be careful with it, but even if that tilted things pretty heavily in one team’s favor (see the 75% bias scenario), there’s still a shot.

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Great post, and timely for me as I am mulling over scoring mechanisms for a league I’m running shortly.

One quick thought is that you can approximate these effects in a more intuitive scheme by multiplicative increases in the value of each match. So, say, the first round is worth 1 point, second 2, and third 5. A player losing the first two could still pull out the win. Again, just an approximation, but more straightforward to explain. While I completely agree that the calculations you’re talking about are very simple, I can easily see many players being confused by it. We use a mechanism like this to value the primary objectives in our Apocalypse games, and it does a good job of preventing runaway leads while being very understandable.

For those same reasons, I would also consider scaling everything and rounding in order to have integer points. I would guess many people in this context will feel just a little put off by having decimal scores.

Finally, one other way to deal with runaway leads is compartmentalize and isolate the different layers. In most of the events I run, the bonuses awarded for victories apply only at the campaign level, not in-game. That way, any two players are always fighting a fair battle. Even if you’re behind in the standings or losing the campaign, you could still win any given match and are not disadvantaged by what’s going on at the campaign level.

That doesn’t always make sense, or isn’t worth it. For example, in Kill Team campaigns we do usually grant in-game boons. It’s just too personalized of a game to not do so, and the stakes in terms of time spent per match too little to worry about it a ton. But for big game 40k, where you’re investing a lot of hours into a match and it’s less customized and personal, this is often a reasonable approach. That’s particularly the case in many of our events, where there isn’t a set group of players and instead the campaign incorporates whoever happens to show up for a particular day.

I talk about some of this in this post on narrative tournament design:

http://www.rocketshipgames.com/blogs/tjkopena/2015/05/narrative-tournament-design-in-tabletop-wargaming/

Thanks again for this post!