The History of EOR

March 1997, by Bruce Biskup

Designing a game for distribution over the Internet presents unique difficulties for the game designer to overcome. For example, the designer must decide just how large the game files may become since large files are more time consuming for customers to download. Game counters and pieces are also a major concern since the designer must guess what type of print capability exists in the customer base before designing the game. In this month’s Rambling I will be telling the tale of how BoneGames (really Joshua Howard) solved the problem of designing a randomization scheme that would work for a game distributed across the Internet.

Most games played today use dice to generate random events. Dice are a tried and true randomization system and lend themselves well for mathematical analysis. This allows the game designer to mathematically simulate the “odds” and play out scenarios prior to the first playtest. Unfortunately for Joshua and me, no matter how hard we tried, we could not figure out how to send dice down the telephone lines. They just don’t fit. Those readers familiar with our first game know how we overcame this problem with our first game, The Barons of Fyn. In BOF, we employed a system that used the playing cards themselves to generate the required random numbers. This system works rather well and is a familiar feature for several in several commercial games. Avalon Hill’s card game Up Front uses a similar system.

For an Internet game designer like ourselves, using cards as randomization pieces is a two edged sword. The cards really need to be tightly integrated into the game design and not just an add-on. To do otherwise makes the game rather large since separate cards are needed for the randomization scheme. This problem was overcome in BOF by using the cards as regular playing pieces in addition to their use as randomization pieces. Another problem is consistency. A trait of a good game is a game whose parts when taken together make sense. The rules and pieces must add something to the game and not just become a gimmick. We have all seen and played games that have been hyped as new or innovative only to find that all that is really new is a new gimmick. The problem with the BOF randomization system was that when used with other designs it looked more like a gimmick rather than a well thought out and integrated game component. We really needed something better.

Enter EOR. Like many of our more creative ideas, EOR came from the fertile mind of Joshua. It was late 1993 early 1994 and I was looking for a real job after finishing my graduate studies. Joshua already had a real job, complete with real wages and some free time for idle thought. I was mailing my resume to everyone and traveling around the country in search of employment. I really don’t know how Joshua came up with EOR but I remember him calling me and telling me that he had a real neat randomization system that worked with any dice. Like normal, I had a little trouble understanding what he was getting at after hearing his description and gave him my standard line: Sounds great, I can’t wait until I see it in person. Luckily, I had some job leads in Dallas and since that was where Joshua was, I could see him in person the coming weekend. So off to Dallas I went.

EOR is a dice pool scheme. In a dice pool, you roll several dice whose total outcome determines the final result. At the time I had never heard of a dice pool and I have never seen one really used in a game. Joshua’s EOR is different in that it can use any dice since results in EOR are based on the sum of two mutually exclusive events and not the numeric result of the dice roll. If I have lost you let me explain. In most dice pool schemes the results are like this: the # of 6 rolled with 8 6-sided dice. In EOR, the results are read as the # of “odd” results out of # dice. Since all dice have an even number of sides then any type of die can be used with EOR. Coins work well too. Even a standard deck of playing cards can be used but this is slow since the deck needs to be reshuffled every time. In fact, anything that has two mutually exclusive events can be used. Several different types of dice work just as well as a uniform set of dice. We made the assumption that most of our customers would have access to coins, dice, or cards and hence an Internet friendly randomization scheme was created.

I had a very bad time looking for a job in Dallas that weekend. I was rudely told to leave at one establishment and never got beyond the door at the other. I was depressed and very annoyed. So what did I do, did I go back to Joshua’s, look up other prospects and go out once more to look for that regular paying job? I did not. In my dejection, I went back and designed a game, using EOR.

I would like to say it was a good game. It was not. In fact, very few people have even seen the draft of the rules I created. The game was a miniatures rules system designed to be used with 1/72 scale figures. I was building a 15 mm Napoleonic Corps (the V to be exact) and was interested in miniature wargaming. In the game, the odds for a successful outcome were expressed as a fraction such as 3/8. This means out of 8 “dice” you must roll 3 hits for the result to be successful. The result of my experimentation demonstrated to Joshua and me that EOR could be used in a game design.

One of the many aspects of EOR that we like is that it provides the designer with two avenues for modifying the final result. One method is to add additional dice to the roll. This would allow a player to use more or less dice to achieve the same result. For example a player might be allowed to use 9 “dice” to achieve the required 3 “odds” instead of the normal 8. The other method one can use to modify the “odds” is to add or subtract automatic results. For example, a –1 modifier would mean that to achieve a 3/8 result the player would need to achieve 4 “odd” results out of 8 since one would be lost. The second method has a more severe impact on the odds so most of our designs do not use that methodology modify rolls. Perhaps the best aspect of this system is that no matter how what the odds, there is always a chance of failure.

To date, EOR has been incorporated into the Sovereign Seas Lite game and LNL uses a more simplified version of the EOR system. Currently, none of our new game designs incorporate EOR. So have we abandoned the system? No, we just don’t want it to become a gimmick. As new game designs get built that work well with EOR, we will use EOR; until then we would rather not “force” it into every game we build.

This is it for this month. Next month I hope to preview some of our currently in-work projects.

Note from Joshua

There must be a math person out there who can tell us the specific formula one can use to generate the probability of any given EOR event. So here is a challenge – write a formula to solve for the EOR problem, that is, what is the probability of rolling at least X hits with Y dice, and exactly X hits with Y dice? Send your submissions to me at

Note from Joshua: Part Two

We had two solutions sent in, both essentially the same. Our thanks go out to Giles Duffin from the UK and Anthony Kam at MIT.

EOR Tables

Here are two tables that lists the percentages for various EOR events. The most common use of EOR is – essentially – getting at least X hits from Y dice. The below chart puts X and Y on the axis, and should be easily read. The other EOR event – exactly X hits from Y dice – is also presented.

Number of Strikes (At Least X hits with Y dice)
0 1 2 3 4 5 6 7 8 9 10 11 12
100.00% 50.00%
100.00% 75.00% 25.00%
100.00% 87.50% 50.00% 12.50%
100.00% 93.75% 68.75% 31.25% 6.25%
100.00% 96.88% 81.25% 50.00% 18.75% 3.13%
100.00% 98.44% 89.06% 65.63% 34.38% 10.94% 1.56%
100.00% 99.22% 93.75% 77.34% 50.00% 22.66% 6.25% 0.78%
100.00% 99.61% 96.48% 85.55% 63.67% 36.33% 14.45% 3.52% 0.39%
100.00% 99.80% 98.05% 91.02% 74.61% 50.00% 25.39% 8.98% 1.95% 0.20%
100.00% 99.90% 98.93% 94.53% 82.81% 62.30% 37.70% 17.19% 5.47% 1.07% 0.10%
100.00% 99.95% 99.41% 96.73% 88.67% 72.56% 50.00% 27.44% 11.33% 3.27% 0.59% 0.05%
100.00% 99.98% 99.68% 98.07% 92.70% 80.62% 61.28% 38.72% 19.38% 7.30% 1.93% 0.32% 0.02%
Number of Strikes (Exactly X hits with Y dice)
0 1 2 3 4 5 6 7 8 9 10 11 12
50.00% 50.00%
25.00% 50.00% 25.00%
12.50% 37.50% 37.50% 12.50%
6.25% 25.00% 37.50% 25.00% 6.25%
3.13% 15.63% 31.25% 31.25% 15.63% 3.13%
1.56% 9.38% 23.44% 31.25% 23.44% 9.38% 1.56%
0.78% 5.47% 16.41% 27.34% 27.34% 16.41% 5.47% 0.78%
0.39% 3.13% 10.94% 21.88% 27.34% 21.88% 10.94% 3.13% 0.39%
0.20% 1.76% 7.03% 16.41% 24.61% 24.61% 16.41% 7.03% 1.76% 0.20%
0.10% 0.98% 4.39% 11.72% 20.51% 24.61% 20.51% 11.72% 4.39% 0.98% 0.10%
0.05% 0.54% 2.69% 8.06% 16.11% 22.56% 22.56% 16.11% 8.06% 2.69% 0.54% 0.05%
0.02% 0.29% 1.61% 5.37% 12.08% 19.34% 22.56% 19.34% 12.08% 5.37% 1.61% 0.29% 0.02%