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What’s the Score?
November 7, 2009

This past Wednesday was a day full of sports for me. In the afternoon I had my first opportunity to learn and play the sport of curling. Then, later that evening at home, I caught the closing moments of Game 6 of the 2009 World Series. All of this exposure to sports got me thinking about scoring systems, something that underlies both sports and board games. Linked closely with this concept is that of two of the most important mechanics in any type of game: the ending condition and the winning condition.

To avoid confusion, let’s define those terms. (Sorry to any scholars who have already done so; I might not get these precisely correct, but they will be accurate in the ways that I use them in this discussion.) An ending condition is the set of criteria by which a game ceases to be played by its players and no further adjustment to scores may take place. A separate but related concept is that of something like the sports world’s overtime in which the ending condition is replaced by another so that further scoring my take place. A winning condition is the set of criteria by which the winning player of a game is identified. Again, there is a related concept, the tie-breaker, in which secondary scoring criteria are examined beyond the primary scoring criterion due to a tie in those primary scores.

Now, let’s take a survey of some games and these conditions:

Baseball

  • Ending: play of nine innings completed
  • Winning: more runs scored than the opposing team

Tennis

  • Ending: a player had satisfied the winning condition
  • Winning: take the majority of sets to be played (either 2 in best-of-3 or 3 in best-of-5)

Basketball

  • Ending: 60 minutes elapsed on the play clock
  • Winning: more points scored than the opposing team

Curling

  • Ending: play of eight ends completed
  • Winning: more points scored than the opposing team

Portal

  • Ending: play through all levels
  • Winning: satisfy the ending condition

Settlers of Catan

  • Ending: a player has satisfied the winning condition
  • Winning: accumulation of 10 victory points at one time

Power Grid

  • Ending: a player builds his/her (X)th city, where X varies by the number of players
  • Winning: ability to power the most cities

Imperial

  • Ending: a Great Power attains at least 25 Power Points, reaching the x5 region of the Power Track
  • Winning: highest sum of multiplied bond value and cash-on-hand

Puerto Rico

  • Ending: cannot refill the colonist ship OR run out of victory point chips OR build into 12th city space
  • Winning: highest total of victory points

Monopoly

  • Ending: a critical mass gets tired of playing
  • Winning: who knows?

Okay, so the last one isn’t quite right, but I’ve never played any other type of Monopoly game and I didn’t feel like sifting through the rules.

There’s many more I could name, probably some with interesting mechanics, but these will do for now. The first thing to note is that the ending condition and the winning condition are often closely linked. This has a number of advantages. Take tennis and baseball as the two opposites in this case. In tennis, the ending condition goes hand-in-hand with the winning conditions. This has the neat property that a player hasn’t won until he/she guarantees that his/her opponent cannot win. In other words, a US Open finalist engaged in the championship match can still hope to win even when he’s down two sets and trailing 5-0 in the third set. Don’t get me wrong, this kind of turnaround is highly unlikely, but there is still space for this player to outscore his opponent, and so the thinking goes that the game should continue until it is literally impossible to win. This is a nice property that doesn’t seem valuable at first glance, but you sorely miss when you’re playing a game without it.

This brings us to baseball, which is, in my opinion, a scoring mechanics disaster. (I pick on baseball, but most sports are guilty on this count: replace baseball and its terminology with the same pertaining to basketball, golf, football, soccer, and, yes, even curling.) The disaster comes from the frequency of the so-called “runaway leader problem” that you so often observe. This is when one player has acquired such a large leading score so as for it to be insurmountable by the opponents within the remainder of the game. Now, you complain (or should!) that many games have such a problem. The tennis example above comes into play again. While the described situation still allows for the trailing player to win, the actual possibility of this happening is about as close to zero as the baseball team facing an 18-0 run deficit in the first inning. Where tennis scores some brownie points, then, is from its ability to hasten the end of the game. The tennis match continues only as long as the losing player is able to mount some sort of comeback; meanwhile, the baseball spectators are stuck waiting around for eight more innings for whatever might happen. (I’ll admit that there is a re-ingnition of interest in these lop-sided scoring cases, a sort of “how high can it go?” interest.) But we see that if a game’s ending condition and winning condition are “in tune”, we can develop a better game-playing experience.

Now that I’ve preached about how important it is to link the ending and winning conditions, I’m going to turn that on it’s head and claim that an even better game would de-couple them … to a point. Prime examples of this are some from the family of eurogames. Let’s take a 5-player Power Grid game for this analysis. The ending condition is the building of the 15th city, but the winner is the player who powers the most cities. In many games, these players might be one and the same, but the interesting part is that they need not be. This leads to all sorts of fun jockeying for position. Compare this with Settlers of Catan, which is an example of a game with closely linked ending and winning conditions. Yes, the game doesn’t drag on unnecessarily even in the runaway leader case, but its end is altogether less interesting.

So, what’s the secret recipe? As with a lot of things in life, the hybrid approach seems best: I would argue it’s having a correlation between the game ending condition and winning condition that gives us the best game experience.

When to Buy: the Math in Imperial
October 27, 2009

Imperial is a 2006 game from Mac Gerdts that rethinks the classic Diplomacy. A layman—in this case, a true board game geek—would have difficulty not mistaking the maps incorporated in both games. What I like about Imperial is that it takes the too-interactive, too-chesslike Diplomacy, adds a realism-contributing economic aspect, and comes out with something that requires even more social and mental depth.

A critical component of the Imperial economy are “bonds”. Players seek to acquire bonds because they grant both points and influence within the game. For each of the six powers in the game, eight distinct bonds are available. Bonds feature a face value and interest value. For instance, the lowest-valued bond is the 1-bond, so called because its interest value is $1million. What this means is that for an initial investment (i.e. buying the bond) for $2m, you receive $1m payments each time a payout is to be had. This occurs indefinitely, so you have the possibility for making back your initial investment and more as the game goes on. When you’re rich enough, you might be able to afford the highest-priced bond, the 8-bond. As you could assume, its interest payment is $8m on a face value of $25m. Notice that the one-time interest payments as a fraction of the initial investment become smaller, the higher the bond’s face value. Viz. $1m/$2m = 50% while $8m/$25 = 32%.

Bond interest payouts not only provide the income stream described above, but also the points needed to win the game. At the game’s end, you can multiply the interest value of a bond by its power multiplier, a multiplier determined by how well a nation did in the game. Thus, you would like to be holding all of your high-valued bonds in the nations that did well and low-valued or, better, no bonds in the nations that did poorly.

However, the opportunities you have to buy bonds are limited and the cash available at that instant limited even more, so the question of what the “right” bond to buy is one that will cross your mind several times throughout a game. In particular, because the power multipliers will not be resolved until the game is already over, it’s important to forecast a nation’s finishing position to decide how much to invest. Just as investing too little is problematic because you miss out on multiplying more of your investment by those power multipliers, investing too much is problematic as it will stunt your ability to “grow” your money as the game goes on; in other words, what you spend this turn isn’t available next turn.

With that groundwork laid, it’s time to get to some pictures. First, a chart to capture what I have been discussing and provide some math that we’ll need.

ImperialPoints

Net payoff of bond purchase for varying multipliers.

This chart shows the net payoff from investing in varying bonds (the vertical axis). Of course, the net payoff is conditional on the power multiplier at game end (the horizontal axis). The net payoff is defined as the multiplier points less the original bond cost (face value). Thus, we see that the 8-bond paying off with the x5 multiplier yields 8×5=40 gross points. From this, we subtract the initial amount paid to buy the bond, 25, to get a net payoff of 15, the highest offered in the game.

What should also grab your attention, though, is that you can accomplish the same thing with the 7-bond (7×5–20=15). This begs the question: why buy the 8-bond. The answer is invariably “because the 7 has already been bought”.* As mentioned previously, it’s better to hold on to your money if you can. (You can keep it to add to your score at game’s end, so it never goes to waste.) Thus, in the perfect world, we absolutely make sure we have our hands on the 7-bond in x5 countries, the 5-bond in x4 nations, the 3-bond in x3 nations, and no higher than a 3-bond in any nations that fall short of x3. Anything else is gravy as long as you’re not taking a loss (red boxes).

Of course, the world’s not perfect and you’re never going to have your money in all the right places. There’s too many other players interested in seeing that you don’t. But it can pay to come close and know what you’re shooting for, although this will require taking some probabilities into account.

During a game in progress, you’ll only be able to discern game-end multipliers to a statistical certainty. The relative probabilities of each outcome dictate which bonds to shoot for. If you think x4 is likely with an outside shot of x3, then it pays most to aim for the 5-bond or the 4-bond.

Payoff graph

The progression of net payoffs for each bond

Since the probabilities will always be changing and the math is complicated to do in-game in the first place, it helps to generalize which bonds provide the best bang for the buck. The chart above attempts to capture this. At the start of the game, it assumes the following probabilties of a nation finishing in a given multiplier:

x0   0.00%
x1   5.00%
x2  25.00%
x3  33.33%
x4  20.00%
x5  16.66%

This suffices for the beginning of the game, but as the game progresses, nations will reach higher multipliers thereby nullifying the probability that they will reach a previous level—they’ve already reached it! For this, we make a simplifying assumption that the probabilities of reaching higher levels redistribute proportionally among the remaining possible outcomes. An example will help illustrate this. Once we reach x2, the chance of being at x1 is now 0. Thus, we redistribute the original 5% chance of being at x1 among the remaining outcomes in proportion to their original probabilities. Thus, the x2 outcome gains an extra (25%/95%)*5% chance, x3 adds (33.33%/95%)*5%, x4 goes up (20%/95%)*5%, and x5 increases (16.66%/95%)*5%. For each bond, we multiply the payoff at each level by the probability of realizing that payoff, and we generate the expected payoff for each bond.

As you would expect, as we can become more certain of reaching a given level, the expected net payoff increases. Thus, in the chart above, the right edge of each color band show the expected payoff assuming a given multiplier position. The dark blue is the result at the start of the game. Here we find that the 5-bond provides the best value for money as its blue bar reaches furthest right. The 5-bond falls about even with the 6-bond once we’ve reached the green region and sit in the x3 multiplier. Once we’re at x4, the 7-bond becomes the best and it maintains this positions even as the 8-bond evens up with it at the x5 region. This was precisely what we had found earlier.

Thus, we can see that the 5-bond is a very safe early bond to own in any country. It provides an optimal value at the x3 region, the optimum at x4, and a good return even at x5. However, once a country has attained x3. It’s best to focus on the 7-bond if possible. We also see the losses that can be sustained by investing in the 8-bond when it falls short of the x5 multiplier.

__________

* There are other gameplay factors at work here such as trying to take control of a nation. These effects are ignored in this analysis.

Greetings!
October 26, 2009

My name is Rudy and I’ll also be contributing to this blog in conjunction with Dan. Like him, I graduated from Northwestern University in 2009 with a B.S. in computer engineering. My interests in gaming lie very much in the board game realm. The reason for this are two characteristics common to board games:

  • Direct, face-to-face social interaction; and
  • Varied, player-generated story development

What I most appreciate is that, like a good novel, movie, or other form of passive entertainment, a board game has a plot that places you at the center of the action. Now, this is not to say that I don’t enjoy other forms of gaming; I also have some favorite console and computer games, but in those media I find that the games I like most are, when you boil them down, an electronic implementation of something you could play at your kitchen table.

I’m excited to see where Dan and I (along with future contributors) can take this discussion on gaming. I expect my initial contributions to be very board game–focused and analytical, perhaps drawing on my background in economics to model game mechanics.