Game Theory

By John Parus


The purpose of this article is to discuss solo gaming. The points I will address in this article include:

  • Reasons why to game solo.
  • Different approaches to take in solo gaming.
  • Mechanisms that can be used to assist in solo gaming.
  • Tips on mathematical modeling for use in solo (or any other) gaming.
  • Source material for additional information on solo gaming.

By way of background, I have been primarily a solo gamer for over 20 years. Although I greatly enjoy the camaraderie of face-to-face gaming when possible, my solo gaming is driven by necessity. Without going into the reasons why I've gamed mostly solo, there are a number of good reasons why you might try solo gaming. These include:

  1. Lack of a face to face opponent.
  2. Preparation of a scenario.
  3. Finishing an engagement started in a multi-player game, often where the results will carry over to the next m-p game.
  4. Conducting minor engagements "off-stage" that impact upon a greater battle or campaign, or for the enjoyment of the mini-game itself.
  5. Testing new tactics, or rule sets.
  6. Replaying situations to determine "...what if...?"
  7. The ability to involve yourself in much greater detail with a battle or campaign than you might be able to do in a face-to-face game due to time, availability or interest of fellow gamers.

The Solo Approach

Let's take each of these in turn and look at them further.

1. Lack of a face-to-face opponent. This is the most common reason for solo gaming. Whether it is a question of no one available to game with at all, or no one in your group who shares your interest in your period, the end result is the same. The lack of an "opponent" can cause great problems, particularly with the beginning soloist, until some tricks of the trade are learned for causing your "opponent" to use different strategies and to limit your control of the battlefield, or the information available to you as the game progresses.

2. Preparation of a scenario. Whether you are designing one from scratch from a rule set that you know in great detail, or are trying to do one using one of the "build your own" schemes included in some commercial rule sets, or if you are going to do a complete development of a scenario to be used with the rules you've written, this is a very useful tool for wringing out the bugs that may not have been obvious to you when putting the scenario together (I've detailed a good example in a later paragraph!). Running through your scenario once it is written can help identify flaws in rule mechanisms, deployment of assets, timing of events, and allocation of forces.

In those cases where you are gaming a purposefully unbalanced scenario (such as a delaying action), this approach will give you a better idea of how to handicap the results to allow the weaker side to "win" strategically, even if they die to the man in the process (hold the pass at Thermopolae for ## days, or delay the pursuing force for ## turns until your reinforcements can arrive, or consider it a victory if you can destroy a particular unit, or leader, or piece of equipment).

3. Finishing an event already started. In face-to-face gaming, I've been in situations where a major battle is taking place and the results will establish the basis and starting point for the next battle that the group will play. The game takes far longer than expected, & out of courtesy to the host, we wrap up at an unreasonably late hour (rather than staying until finished at a totally unreasonable hour, or breakfast, whichever comes later). The battle is unfinished, but not enough of it remains to fill up the next gaming opportunity. By solo gaming the results based on general battle plans drawn up by the other players, the next session can be a fresh start at a new battle, rather than trying to restart from what you left behind the last session. Honesty is essential, but if you can't trust your fellow gamer, maybe you should have been playing solo the first place! This usually isn't a problem though, since the others all get their chance to finish off the event when the game is held at their house.

4. Conducting minor engagements "off-stage" that impact upon a greater battle or campaign. Whether in preparation for another event or as a side event on its own, you can fill-up an evening or two conducting "one-off" situations that would not be enough for a group game, but can be interesting just the same. Rather than dicing on an events table to determine that 3 of the ships carrying reinforcements made it through the opposition's ships and no indication of what damage was done to those blockading (info of great use to the landing force in case they have to leave in a hurry, or are expecting additional forces), try gaming it out. This is a good chance get introduced to other rule sets (e.g. naval rules when you normally only game land battles or skirmish rules when you are normally at the brigade level, etc.).

5. Testing new tactics, or rule sets. Most of you have already done this with board games you purchased before you play them competitively. It also works well to try different approaches for use by your miniature armies, particularly if the army is one you aren't familiar with. How you best implement the historical Scythian tactics you've researched if the only army you've ever gamed with is your Early Hoplite Greeks? How does the tactic of having your riflemen fall prone just prior to the advancing infantry coming into volley range affect the outcome of the battle? Do the rules support your use of tactics that may be uncommon or distinct to a particular group that you are playing? This is a good way to find out before you run up against an actual opponent.

6. Replaying situations to determine "...what if...?" Taking #5 above one step further, you take your Scythian army to the gaming shop after test driving it against your Hoplites. Your opponent is using a Later Assyrian army, as he has done in convention play for the past 4 years.
End results:
Assyria:1
Scythia:0

Now, what if you had tried an off-board flanking move to get your cavalry behind his chariots? What if your army included the optional large number of cheap infantry and a wagon laager defense instead of all cavalry? What if...

7. The ability to involve yourself in greater detail with a battle or campaign than you might be able to do in a face-to-face game due to time availability or interest of fellow gamers. This is perhaps the most compelling of the voluntary reasons to game solo. The experience of setting up and running a full campaign can be the most rewarding, and, at times, it can be the most frustrating you'll find in gaming. This approach allows the inclusion of such elements as commerce, industry, population, politics, diplomacy, espionage, characterization, and a host of other side issues. The most common mistake one can make in this regard is trying to do too much. The bookkeeping can quickly overwhelm you. But once you've done a bit of solo gaming and tried an initial campaign or two, you will have a much better feel for what you want and need in your campaign, and what is "chrome" that really adds nothing of significance, but costs a great deal of time and effort.

More Approaches

The next topic I wanted to cover is the different approaches to take in solo gaming. These range from consciously deciding to play one of the two solo sides (even though this a battle between union and confederate forces, I'm playing the confederate side) which introduces the possibility for some bias in the outcome (I can't believe my sharpshooters did that badly... maybe if I re-rolled the dice ... ). This approach also works best where you have a mechanism developed to provide an intelligent opponent, using pre-programmed responses.

The other extreme is where you play what I call the "Grand Clockmaker". In this case, you set up a scenario and all of the rules (hopefully) to cover expected situations, set the two sides in motion, and watch how the whole thing turns out without taking a position favoring either side. Of course, the outcome is largely dependent on how well you set the problem up, and how well your rules cover the situations.

The approach you decide upon will in some respects determine how much you must have established ahead of time as opposed to how much you will 'ad-lib'. You may also find it convenient to play a campaign in one mode (clockmaker) to set the stage for the battles, then step in and play one side or the other. In any approach, intelligence in the play of the other side can be provided by a variety of means. These include:

  1. Objectively playing each side (Okay, I see the flanking move by his cavalry... My best counter to this is to take cover in the heavy woods and let my superior firepower do its stuff against the mounted troops ... O.K., now I'm the cavalry guy... since he has taken cover in the woods where my mobility is greatly restricted, I'll stay in position just out of range, and wait for my ground forces to close. By the way, just how dry is the forest and which way is the wind blowing?).
  2. Playing one side against a straight-forward foe. U. S. Marines vs. a Jap Banzai , or French Foreign Legionnaires vs. a band of tribal desert raiders (not noted for subtlety or diversionary tactics), forces storming a fixed defensive position, forces fighting a delaying action vs. a greatly superior number of forces that will try to cut off the delaying troops and deal with them quickly while pushing on with the remainder of the pursuit forces are examples that work fairly well in this manner.
  3. Playing against a programmed opponent. There are a couple of excellent books that give the basic tactics, randomly chosen, for each side. Without too much effort you can draw up a logic tree for your opposition to follow (see example at and of article).
  4. Drawing up a series of possibilities for each side, and randomly choosing from them when required, i.e. with a force of 1 HI, 2 LI, and 4 MC, in this sort of terrain, against a numerically superior foe, the four most likely dispositions I would take are a, b, c, and d, with a 40% chance of a, and 20% each of b, c, and d. The roll gives me c. In a (choice c) disposition, I might either try to hold the terrain feature with the HI and delay with the light troops (50%) or fight a falling back defense while waiting for an off-board flanking move by my cavalry (50%). Similarly, with the same force against a numerically inferior foe, the choices would be e, f, and g with percentages of 50% e, and 25% for each of the others. Alternately, the decision could be keyed to terrain type rather than the relative numerical advantage of one side over the other.
  5. Setting out the general scenario for a compatriot who can't play and let them provide you with their battle plan. A long past article in Lone Warrior was just this sort of request, with the result being a number of very different proposed responses, each of which would have been interesting to game out. For most of us, a phone call or a letter with a simple diagram to a friend is the normal way to go and a lot quicker. At a previous duty station, I (who have no interest in gaming WW II land battles) would provide inputs to a friend, who in turn would do the same for me (since I couldn't even bribe him into playing ancients!). You can even get this type of help from non-gamers, not totally unrealistic when you consider the backgrounds of some of the military leaders the world has seen, and some of the decisions (strange in hindsight) made by supposedly professional soldiers.
  6. Incorporating personality of the key figures in the determination of what course of action is taken and how successful it is. It is similar to the leadership traits used in WRG for sub-generals, but expanded to cover a greater number of possibilities. This is covered in greater detail below, but you basically determine personality traits for all of your key figures and base their decisions, tactics etc. accordingly.

The next topic has to do with mechanisms that can be used to assist in solo gaming. There are mechanisms to set scenarios, determine the actions of characters and the results of character participation, and to take away the full control you have over the battlefield. The last of these are the same as are used in face-to-face gaming... variable bounds, varying movement distances, random terrain, the use of chance event cards and the like, and as such, will not be covered in this article.

For the others I've referred to, your mechanism will do what you design it to do. If you've designed it well, this will also be what you want it to do. If not, the design will still do something - it just may not be anything like what you had wanted. Poor design, lack of thought as to what you are trying to model, lack of understanding of how something would work in the real world, or not understanding the way probability distribution function (using 1D100 or 10D10 will give quite different distributions for almost the same range of values) for the item you are modeling can all cause problems.

Bank Robbery

I remember well the first time I tried to run a scenario in the "clockmaker' mode. I had set up a map of a western town, purchased a few sets of AIRFIX HO cowboy figures, and wanted to game a bank robbery in the James Gang style. I had set up a table of attributes for all of my characters (weapon speed, weapon accuracy, ability to take damage, hand-to-hand skill, throwing, etc.) and I even had a few special characters with abilities and pasts not known to the community. These were such as a Texas Ranger passing through the town on the trail of other outlaws, an ex-gunslinger trying to hide his past, the lightning fast loud-mouthed kid (who couldn't hit a target on a bet, he just had a reputation of being too fast with a gun to mess with), and other similar stereotypical people as you would find in any Western movie.

So as not to bias the outcome either for or against my bank robbers, I randomly determined where the Town Marshal and all other key characters would be when the robbery was disclosed. Then, I determined the starting locations of all of the townspeople, and set them into motion using a die roll for direction (hex map 1-6) and distance. The result was a great demonstration of Brownian motion with a lot of "molecular particles" running into each other, but having absolutely no resemblance to the real (or gaming) world. The basics of the scenario were fine, but the model for character motion kept me from ever getting to the bank robbery stage.

This result (failure of the model to deliver the desired outcome) closely paralleled an incident that a fellow gamer had in a naval engagement he was doing as an "off-stage" event. The parallel was not with the mechanics involved, but with the end result of the event being far from what was desired. His weather table, which determined the tracks of his ships, was flawed in that it allowed one to get stuck in one type of weather (gale) with an unrealistically small chance of getting out of it. The result was that a planned naval engagement and amphibious assault that was anticipated to take 12 days actually took 34. The tracks of the opposing warships never crossed and the amphibious landing that was to trigger a general invasion plan was 22 days late!

My motion model provided random movement of the characters, but their motion lacked "purpose" or a logical endpoint. It did not model the motion of people going to or from places. The weather model allowed for the random development of weather, but the table had a flaw which had change, but no rationale for a readjustment of the weather down from a higher state.

Back to Break the Bank

These two incidents clearly illustrate one of the opening points of this section of the article: If you want to be satisfied with the results of your event, you must first decide what it is you are trying to do. In my case, I didn't want a model of citizens moving around, I wanted a bank robbery that would take place with some uncertainty for the bandits as to where the law men and key townsfolk would be.

I went back to the beginning, determined a set of likely locations for all of the key figures, randomly distributed the remaining townsfolk (assumed stationary until the robbery commenced, or at time = 0 in the game), started the robbery, and then rolled the dice to pick from the set of locations where the key figures actually were (e.g. was the marshal in his office, at one of three places in the street making his rounds, at the corral checking on his horse, or over at the saloon having a drink. Was the Texas Ranger at the telegraph office, asleep in his room, at the corral getting his horse, or already on his way out of town, etc.) and ran the scenario from there.

In the case of the possible naval battle, the question to ask was whether what was desired was the very realistic and likely result of no naval engagement ever taking place (weather not withstanding) leaving the possibility of the attacked countries able to have their ships function as sea raiders operating behind the lines and tying up a much larger naval force, or, if the desired result was to have the naval engagement take place.

To examine the options for establishing a scenario, let's look at the naval engagement question. Assuming the decision was that a naval battle would take place, the specifics of the scenario could be decided by:

  1. An 'act of god.' Decide that the combined naval forces of the attacking countries would meet up with the defenders on (this) day at (this) time with (this) weather. The skills of the respective crews and captains would be (this) respectively, (this) side would have surprise, and (these) random occurrences might occur. The battle would continue until (this) outcome for the attacker or (this) outcome for the defender occurs. Or,
  2. Randomly determine all of the above. Or,
  3. "Know" some of this information from previous gaming (the crew of the good ship Moor Hound are only fair ship handlers, but very good gunners, while the Painted Lady was the fastest ship around (and a good thing since she is lightly armed by comparison to her foes) and Captain Terrement is head and shoulders above all of his contemporaries in all respects (he gets more performance out of his ship than anyone else could, and his crew gains +2 in any morale check due to their confidence in his ability to get them out of even the toughest scrapes). Or,
  4. Some other method of the gamer's choice.

The mechanisms used to resolve items can vary greatly. The one chosen will depend on time available, interest of the gamer, and level of detail desired. If rushed for time, or not wanting to game the naval battle out, one could take an approach much like Mr. Tony Bath did in a campaign game he was running. When faced with the issue of how successful one gamer would be in his attempt to blockade an opponent's harbor by scuttling some older ships in the mouth of the harbor. Mr. Bath assigned a range of 2-12 to the event with 2 being a total failure and 12 a total success. With the roll of two dice, the outcome was determined.

In another situation, it might have been more desirable to set the scenario up and actually game the event out. I am an inveterate tinkerer with rules systems and mathematical models, both commercially produced and those I've written myself. In general, I wouldn't be happy with a simple resolution by the 2-12 dice roll. I complicate things for myself by trying to set up a better representation of the range of results. I may start with a rule set and then bias the sets of dice rolls to account for variables (like the +2 for Terrement's crew) or actually drawing up a mathematical model to cover the event in question.

I Spy

For example, I recently saw a gaming article where the results of your efforts in spying on an opponent as resolved with a six-sided dice roll. By comparison, when I was faced with the same issue, I set up a series of tests to determine the availability and accuracy of results given the number and skill of spies at work, the amount of time they spent, the amount of money spent on bribes, the difficulty in obtaining the info they were after, the number and skill of the equivalent to today's secret police working against them, and a luck roll for each side.

Regardless of the results of the event, I then computed the probability of the spies being discovered given the level of their involvement, the amount of money they spent, the level of effort and quality of the secret police, etc. If discovered, I then calculated the probability of capture given a similar set of variables, and if either not captured, or not discovered, calculated the probability that the spy would get the info to his chain of command given the distance involved, level of pursuit, hostility of the area, etc. (Nothing quite like having your best spy pull off a major coup in intelligence, have him get detected, escape the pursuit, and have him killed by a highwayman just short of delivering the information!!!).

I recognize that not everyone enjoys this sort of thing, and in the end, going back to the harbor blockage problem, a result of "60% harbor blockage, with 3 months required to clear is the same whether it comes from an arbitrary decision by the solo gamer to make the scenario more interesting, from a 2-12 roll against a simple results table, from a much more detailed system, or actually gaming it out on a wargames table. It's all a function of what you want to do.

For one of my most useful mechanisms to determine what will transpire, I borrowed an idea from Mr. Donald Featherstone. He suggests setting up a campaign diary, and keeping a record of the successes and failures of your units and leaders, the idea being that the color added to your game is worth the extra work. Few will remember that on a dice roll of a 9 followed by a 12 a gate was taken, but many more will remember the event if it is cast in terms of "...in spite of his serious wound (the 9), Sgt. Faversham bravely crawled forward under heavy enemy fire to re-light the fuse on the charge that blew apart the main gate to the rebel stronghold (the 12)."

This also suggests the possibility of introducing characterization to assist in determining who does what and why. For example, why did Sgt. Faversham risk death (Duty? Desire for glory? Despondent over sweet Nelly's "Dear John" letter to him?), while Sgt. Blackadder found a nice shaded rock to take cover behind (Wounded? A desire to be sipping on the gin in his canteen?). Drawing on this approach, I developed a method of characterization that could be used to determine how my key characters will proceed, and hence, to some extent, how the battle will be fought. The mechanics will vary according to your tastes and level of detail you want to include. Here's some approaches you can try:

  1. From personal desire, previous play, or as in the above example, you "know" the individuals and play them in character as the situation dictates. Sgt. Faversham is a born gambler who believes his luck cannot fail. He will always take the risky road if there is a chance for substantial gain. Sgt. Blackadder believes life is to be lived, and he intends to do so for as long as possible. He will comply with the letter of the orders given, but avoid the spirit of them as much as possible, or find the reasons why they can't be complied with.
  2. You randomize the decisions, but by weighting the choices made or through the use of "filters" you bias the results in the direction that you would expect them to go.
  3. You establish numerical values that define your character's abilities and limitations and must satisfy the choice with a dice roll that corresponds to the attribute in question.

An example of the first type is what I would do in answer to the questions posed by a fellow gamer as to how to proceed with a situation in his campaign. Using the "reasonable man" approach, I looked at the characters in question, made some suppositions as to their abilities and motivations and said "if I were he, what would I think it would be reasonable to do. All things being equal, a leader who tends to be a gambler, aggressive in his approach, will likely see a different set of opportunities and challenges than will a less experienced, more tentative and defensive minded leader when both are confronted with the same situation.

The second case would be where you know, for example, that the most likely course of action for the CINC is to attack in a given manner, and draw up a battle plan accordingly. Then, by coin flip, dice roll, or whatever, determine if the plan will be used. If a 'no' was received, a different offensive plan would be proposed. A second (or however many you think is right for this character) failure to get a 'yes' to the proposal would then cause consideration of a defensive proposal, and so forth, until a decision was made. This is a case where the 'filter' of forcing a certain number of one type of decision before looking at the alternate approach causes the character, over the long haul, to behave in the manner expected most of the time, but does not preclude an occasionally different, unexpected result.

The same result could be attained by drawing up a full range of options and assigning varying probabilities to their occurrence based on the likelihood of their choice by the character. A single random draw would decide the course taken. One way the weighting of choices could be done is by saying that a given proposition has a basic 50/50 chance, but you add/subtract modifiers to reflect the individual or situation (Capt. Terrement has a +20% modifier for any decision against his sworn enemy, Capt. Beheler. Faced with a choice of whether to engage or not, assuming all else to be equal, the 50/50 choice would become 70/30).

The third case is one that I use quite a bit, to make sure that my personal bias for certain characters / armies / nations that I have in my campaign doesn't enter into the decisions or the results. There are a number of ways to set up the mechanics for doing this. Mr. Tony Bath outlines several in his book on setting up campaigns, and quite a few of the Role Play games on the market have ideas you can use as well.

My current method is a distillation of them. I (a) lay out opposing personality traits, (b) randomly determine the values of the attribute, (c) use the attribute to determine the course of action taken in an appropriate circumstance, and (d) look for possible adjustment to the scale after a key decision or action takes place. The random roll can give you a wide spread in values and more likely extremes by using a straight 1-100 roll, a middle weighted bell curve by starting each value at 50% and +/- 3 ten sided dice, or a nice smooth bell by rolling 10D10. Let's look at a partial example. Using scores of 80 for bravery, 65 for headstrong, and 95 for trustworthiness, you have:

Cowardice	|...20|........80...|        Bravery
Reticent	|....35|......65....|        Headstrong
Dishonest	|.5.|........95.....|        Trustworthy

(etc. for as many attributes you need for your campaign or game, and can apply to nations or armies as well as individuals)

The Drunken Brawl

Our character has the values established as shown above. After several months on the job, Marshall John Odom finds himself in a situation where his duty requires him to enter the local saloon and break up a large drunken brawl. Since his deputy is out of town, he must do so alone. This situation calls for a check against his bravery score. The two choices are to (A) take the brave approach, boldly going in to (ruin the party) restore order, or, (B) decide that there are too many cowboys that are too drunk to try to do so alone and opt for the cowardly approach.

Let's say we start with case (A). Roll a percentage dice. On a 01-20, he fails the roll and backs down at the last minute. If 21-100, he does the brave approach. If he succeeds in his choice, he has a 100-current level chance of improvement in that attribute (100 - 80 = 20% chance of becoming more brave as a result of this incident). I use a 1D6 for the amount of change. If you thought that in this circumstance he would take the cowardly (more rational?) approach, he would do so on a roll of 01-20, with the corresponding chance (100 - 20 = 80% chance of becoming 1D6 more cowardly). I only check for changes when the choice I am testing for succeeded.

For explaining the results in gaming terms (such as you would for an entry in your campaign diary), in the first instance, case (A) John Odom's failure could be attributed to "good judgment", and in case (B), his entry at the last minute was prompted by a sudden burst of civic responsibility. The checks for improvement of attribute scores will provide changes readily for those scores near the middle, but the stronger an attribute becomes, the harder it will be to improve it further. This same type of mechanism can be used to determine if your sub-general Subotai will follow the battle plan or attack his hereditary foe, instead, or, if the country Maalinast will take advantage of internal unrest in neighboring Hohollga to extend its borders a bit.

Character involvement in a solo game can take on many forms. You can be a player-character, from emperor of your nations, to one of the 1/20th of a foot soldier represented by the second lead soldier on the third stand from the right. Most of us traditionally play the role of whatever the highest level is in the game we are playing. But, you don't have to. Two of the methods available to you to handle this are:

  1. (As suggested in an article in a past issue of Lone Warrior), you game out the entire battle as you normally would, except that when your figure would be going into a battle, you shift scales in both time and space, and game out that portion of the much bigger battle as a skirmish game, 1 figure = 1 man.
  2. (As provided in several of the Chivalry & Sorcery rules sets) there is a mathematical model to handle complete battles on paper, with the results you receive as a knight, for example, a function of how the overall battle went, how aggressively you fought (can't get glory by hanging back), and based on these and other factors what glory you achieved (and what wounds, including fatal ones, you may have received... ahhhh, the price of glory!). As you can see, these two approaches cover both extremes; there are doubtlessly other methods somewhere in the middle.

Game Theory: Part 2 in LW 117


Back to Table of Contents -- Lone Warrior 116
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