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In my Games Theory 101 Article on "The Bomb", I explain why the majorities scoring used in games such as Acquire, El Grande, and Union Pacific is used so widely. Briefly, it enables explosive scoring opportunities in which the payoff of an additional share or cube can be very disproportionate to the investment. This creates tension as the need to get that one or two additional shares (or whatever) becomes very great.
Scoring for majorities is one of the most popular mechanics in Eurogames. I scanned my own collection and found that out of 100 games, I identified 35 of them that use scoring for majorities in some form. But there are other ways that designers have come up with to grant disproportionate, typically escalating, scoring in their games. It's pretty darned hard to come up with something new, but fortunately there are a lot of creative designers out there. I thought it would be fun to see what people are using, and to see what different effects they can have on game play.
We're going to talk a lot about "set collecting" games here, and for our purposes I'm going to use a very broad definition. A set collecting game for this discussion is any game in which players can collect assets which come in different varieties and whose collective value is not proportionate to the quantity of each variety of asset. The idea is that the game is either encouraging you to concentrate your collecting efforts (Coloretto, Civilization, diversify them (Tigris & Euphrates), or offer some conflicting alternative (the monuments in "Ra", and any majority game such as "Union Pacific"). Having different varieties just means that you can collect cards of different colors, or shares of different companies or whatever. If there's just one type, then the game isn't directing players to collect sets of anything, it's just encouraging them to build up assets. If value is just proportional to the quantity of an item you collect, then there is no set collecting element either. Suppose you get 1 VP for every yellow card, 5 VP for every red card, and 10 VP for every blue card. In such a case cards are just a way of storing VP's, like money in different denominations. But 1f you get 5 VP's for every color of card in which you have the majority - but 0 VP's for all other cards - you have the elements of a set collecting game. In this case the incremental value of a red, blue, or yellow card all depends on the situation. It might be worth 5 VP's and it might be worth none. There's now the rudiments of a "game" here, depending on how players acquire the cards.
Moreover, for this purpose, "area control" games such as "El Grande" and "Web of Power/China" are really set collecting games in disguise. The difference between an "area control" game and a "set collection" game is just the difference between "put" and "take". Want to convert "El Grande" into a set collecting game? Create different colored cards, one for every region on the board. Now, instead of placing two cubes of your own color into "Baskenland", you take two green "Baskenland" cards. When scoring, each player sees how many cards of each type they have to determine majorities. Of course, "Web of Power" also gives bonuses for connecting chains - but to that extent it is neither a set collecting game nor an area control game. Want to convert "Union Pacific" into an area control game? Create a board showing all 10 companies. Now every time you'd normally take a share card, just discard it and put your colored cube into that company's area. Voila; "Union Pacific " is now an area control game. In fact, that's how Goldbrau operates. Players play shares in beer halls and breweries, but mark their control with their colored cubes.Now there are good reasons to choose one form versus the other. The theme may involve collecting shares - and that is best represented directly. Your game play may involve a map, and there may be particular rules concerning areas that are adjacent to each other so that putting cubes into a central board makes play easiest to follow. But when it comes to scoring, both forms are equivalent. Our goal here is to discuss how designers address the issue of scoring these sets, and what the implications of their choices are.
A resolution: I intend, in this blog, to avoid algebraic notation, even when that may be the most efficient way to express things. I have two reasons for this. One is that I don't want to tune anybody out. I'm pretty good at math, and even my own eyes glaze over when I start seeing formulas. I just want to express everything in plain English. The other reason has to do with black holes. In the introduction to his "A Brief History of Time", author Stephen Hawking says that his editor required that he write his entire book without using a single mathematical formula. Well, I figure that if Hawking can explain the big bang, charmed quarks, and black holes without using algebra, then I sure as hell better be able to explain board games that way.
The set collecting mechanic is a way of creating a bomb in a game. A bomb is a disproportionately high scoring opportunity that focuses players' actions and creates tension. If a game rewards a player with 5 points for every yellow card he collects, that's not a bomb, that's an incremental reward. If the game rewards him with with 1 point for the first yellow card, 2 for the second, 3 for the third, and so on - the designer has removed the proportionality and created something more explosive. Now the player has his eye on the long-term prize.
"Collecting a yellow card now is okay, but what I really want is to work toward collecting ten yellow cards. Those last cards will be worth a lot of points if I can come up with a plan to achieve them."
There are two generic ways that designers have achieved this: either with some sort of majority scoring, or with an escalating scoring system of the type I described above.
Sometimes, either one might work plausibly. But the two are very different and will create very different types of games.
The most glaring difference between the two is that a majority scoring is based on players' relative positions, and so it creates inherent player interaction, while escalating scoring is based on a player's position irrespective of his opponents, and so the game must get its interaction elsewhere.
Alan Moon's game, "Get the Goods" is an example of a majorities- scoring set collecting game stripped down to its essentials. You draft cards in any of ten colors, and you play them, and three times in the game you score points based on your majority position in each of the ten colors. You get 3 points for 1st place; 1 point for second place. There's very little else in the game. It's very similar to "Union Pacific" - but without the board. Instead of placing trains to determine what a company is worth, all are set at a base value for the duration of the game. (Get the Goods is reputed to be based on Freight Train, but Freight Train has a much more complicated system to draft and play cards. In its basic draft-vs.-play mechanic, Get the Goods is much more similar to Union Pacific.)With hardly any rules besides "pick up cards, play them, and score them based on majorities" Alan Moon created an entirely viable game whose player interaction is self evident. The player interaction comes, of course, in the competition to gain the majority when the scoring cards are revealed. There's always a bomb in a majorities game because no matter how many pieces of a set you've collected, if players are close, then two pieces can make the difference between a being the top dog and being a dead dog. Sometimes that difference can be too extreme, and so most frequently designers offer consolation prizes for second and later places.
The majorities mechanism can be fragile. If a game mechanism enables a player to easily secure a runaway lead, the competition peters out for everyone. Trailing players aren't motivated to catch the leader because the investment in resources is too great. Leading players similarly can go to sleep, knowing that they've outdistanced the competition. In games with limited shares available - such as Union Pacific or Acquire- players can get a permament lock on a majority. That can be a nice thing to get players to fight for - as long as it can't happen too early into the game.
Generally, one thing that makes a game fun is when there's no concrete way to evaluate a particular move. That's a built-in feature of majorities games. Another share might be useless - or it might help you takeover the lead. Of course, if runaway leaders can be created, the value of an additional share is easily known: it's worth nothing.
The primary alternative to majorities scoring is escalating scoring. This is where each incremental item added to a set is worth more than the last one. An early example of escalating scoring appears in Civilization, in which the value of any set of cards was based on the *square* of the quantity of cards. Five cards in a set isworth 5 x 5 = 25 (I'm not counting anything as an algebraic formula unless it includes a variable!) times its base value. The most common escalating scoring formula is probably the "triangle", in which the first card in a set is worth 1 point, the second is worth 2 points, and so on. To see why it's called a triangular progression, picture a pyramid with one stone on top, then two beneath it, and so on. You've got the shape of a triangle, and the number of stones in say a 5 level pyramid is equal to 1+2+3+4+5 = 15. This sort of scoring appears in the way that color sets are treated in "Coloretto", in the points scored for treasures in "Tikal", and, in a modified form, in the track laying bonuses in "Ticket to Ride" (1+1+2+3+4...). In each case, players have an incentive to collect unified sets with great intensity and little incentive to collect lots of unmatched pieces.
Escalating scoring is used to achieve two sorts of objectives, depending on the type of game. It can encourage players to concentrate in one flavor of set - as in Coloretto and Civilization - and it can just offer players incentive to concentrate on a particular strategy - as is done with aristocrats in St. Petersburg.Escalating scoring is used nowhere near as widely as majorities. In my sample of 100 games, I identified 10 that used escalating scoring, compared with the 35 games that used majorities (and at least one game, Tikal, that used both).
The scoring system used for commodities in Civilization gives a player tremendous incentive to trade cards in an effort to specialize. Two sets of four cards would be equal to 32 points (2 x 4x4) but a single set of eight cards would be worth double that: (1 x 8x8) 64 points (times the base value). The "squaring" formula tends to be stronger than the "triangle" formula - especially with low numbers. If we were using triangle scoring, the trade would result in your score going up 80% (36/20) instead of 100% (64/32). This is meaningful if you're playing a game where collecting 8 of a set is viable. Say you're only realistically going to collect up to 4 of a set. Then what happens if you trade two sets of "2" for one set of "4"? With squared scoring, you go from 8 points (2x2x2) to 16 points (1x4x4)- again up 100%. With triangle scoring you go from 6 points (2x3) to 10 points - an increase now of only 67%.
In short, you get an unexpected result. The difference between squaring and using the triangle formula (1+2+3+...) to give a "concentration" bonus is actually most extreme with smaller numbers. So in a game like Coloretto, where players are typically collecting sets of 3, with 4 and 5 piece sets being really nice, if Michael Schacht had used squares for the scoring, the incentive to get big sets would have been even greater. Conversely, the penalty for collecting more than three colors (where you earn negative points) would have not been all that consequential, because players tend to get singles or doubles of those.
At first blush, squaring the number of units seems to escalate much faster than using the triangle approach - and for low numbers it does - but in fact the two formulas are closely related algebraically (but we won't go there!) However, just as the triangle numbers can be expressed as 1+2+3+4+... , square numbers can be expressed as 1+3+5+7+...Both majority scoring and escalating scoring achieve a basic common game function - providing explosive disproportionate benefits that provide tension and that direct players to focus on specific strategies. Their consequences in games are very different. Most obviously, majority scoring provides built in player interaction because the score is all based on your position relative to your opponents. A game with escalating scoring needs other mechanisms to supply the player interaction. In fact, without those other mechanisms, the scoring breaks down because tactics become so obvious - specialize and leave everyone else alone. Imagine a pure draft and play game like Get The Goods (or Union Pacific) where your score for a particular share was just based on the triangle pattern. Your approach would be trivial. You collect the yellows, let me collect the blues, and leave Greg alone to collect the purples. Any time I choose a yellow over a blue, I'm giving up lots of points for just one or two points. The cost to me probably at least equals what I'm costing you, and Greg ends up at an advantage (no wonder Greg always wins!) If everyone is free to pursue their best (and obvious) strategy unchallenged - there's no game.
With escalating scoring, collecting more of the same is always the best. It's not neccessarily that way with majority scoring. You take a single yellow - and now you're the leader. Your best next action might be to take another yellow, to help secure your lead, but in the short run, your best choice to take blue, which could give you the lead score now in two colors. Now for me, the best course might be to take purple - but if I take two yellows I can score in yellow *and* deprive you of that score. In fact, I think that's what I'll do ;-) . Even so, strategies in such a system can become trivial, which is why games of majority control will typically limit your choices in such a way that, perhaps you'll end up with a blue even if it's not your first choice - thereby prodding you into competition with me for control. Get the Goods and Union Pacific do this, for example, by limiting your selection to the available face up cards (and the face down card). El Grande uses the brilliant (and I don't use that term lightly) mechanism of the king, which forces players to all drop their cubes in regions adjacent to a "king" piece which typically moves each turn. Only a small subeset of all regions are available to the players at any given time, thereby forcing them into competion with each other. Another method of encouraging competition is rewarding second places. If you already have two yellows, it may not pay for me to invest in yellows, knowing that it will take at least three to overtake you. But if I can get a modest payoff for second place, I'll take a single yellow - and now you've become within easy shooting distance of me. Hey, maybe I'll challenge you for first place after all.
Since escalating scoring systems have potentially trivial strategies and no inherent player interaction, they are neccessarily dependent on other primary game mechanisms to provide what's missing. In Coloretto, one of the simplest games with escalating scoring, players are rewarded for collecting sets in any three colors, but penalized for collecting cards in any additional colors. The driving mechanism that Michael Schacht introduces is that players must take bundles of cards, which their opponents are trying to poison with cards of contrasting color. Creating and choosing the bundles is the heart of the game, and the scoring system just makes it work. One of the most beloved games with escalating scoring is Taj Mahal, which uses this method to score the elephant "commodity" tiles. (Arguably, escalating scoring is also used to score palaces, but there is so much else going on there that it's better to exclude them from the discussion.) Reiner Knizia does two very basic things to keep it interesting. The obvious one is that he requires players to bid for the tiles. On its own, this could still lead to trivial tactics as players specialized in their favorite flavors. The added element is that every tile has two commodities, each in a different combination. This naturally forces people into competition and forces a mixed strategy. Players can't implicitly divvy up the commodities and choose to go only for their one best choice. I have two tea, one spice and one rice, so I'd really rather go for tea... but that spice and gems tile coming up isn't too shabby. And I can be confident that there's at least one other player who is collecting either spices or gems. By putting the collectible items into bundles of mixed sets, Schacht and Knizia help to include greater player interaction and mixed, non-trivial strategies.
There's an interesting paradox in this scoring system. How much is one more item worth? The answer is not as obvious as it seems. Let's use Caylus as an example. In Caylus, a player can take his "King's Favor" on any one of four different tracks, and can mix it up during the course of the game. In the case of the Victory Point track, the first favor is worth 1 VP, the next is worth 2, and so on, up to a maximum of 5. So you take 6 King's Favors on this track, you get 1+2+3+4+5+5 = 20 points; the first one is "worth" one point, and so on. But that analysis misses something. Had you passed up any one of those scoring opportunities - even the first - it would have cost you five points. If you're embarking on a strategy of taking lots of King's Favors on the VP track, each missed opportunity will cost you 5 points, and has to be evaluated that way. Yet, when it's all over, your 6 king's favors are only worth 20 points, not 30. That stinks! Where did the other 10 points go? How do you reasonbly evaluate the value of the strategy?I can propose two answers. One is that this just highlights the intrigue and ambiguity of the scoring system. No scoring opportunity can be evaluated precisely. The other - which I think works fairly well in this case - is that the moment you commit yourself to taking lots of king's favors as VP's, you've put yourself ten points in the hole. That represents the shortfall between the incremental value of each favor (5 points), and the total you'll get from collecting five or more. So you need to evaluate each opportunity as though it's worth 5 points, but you need to evaluate the strategy as though it's always worth 10 points less than the sum of the parts. Of course, if you don't actually get at least 4 such scoring opportunities, everything changes. But if you really focus on the strategy and can plausibly attain it - that's the best way to evaluate it.
Which is also a way of saying: the escalating scoring system, in this case, really forces a commitment. It prompts you to evaluate each opportunity as having a high 5 point payoff, and then *penalizes* you by the difference between what you actually get and the 5 points you counted on if you chicken out or fail midway. You made a decision based on this move being worth 5 points, but it turned out to only be worth 3 points.
All our talk so far has covered 3 narrowly defined ways that designers have used to provide focused efforts with disproportionate rewards: majority bonuses, triangular scoring, squared scoring. What does a designer do when he wants to get a little creative? It's a tough problem because at first blush, there aren't a lot of choices out there, apart from just creating a table. But sometimes the above 3 ideas aren't quite what the designer wants - either because it doesn't provide the proper reward and incentives, or it's too cumbersome, or... he just wants to try something new. In the next installment of this article, we'll take a look at some of the alternatives that are out there.
