Most of the modern wargames have some "luck." The classical Chess and Go, of course, do not. Even with no "luck" involved, the results of two chess games between two grandmasters are not always the same. Yet the best wargames players act to cut luck down as far as possible, because they profit from the game being more controlled by skill.

First I want to say what I mean by that emotional word "luck." I want to be able to compare sets of rules for one game, not one game with another. When a game's outcome is stated on a scale like the victory points of France 1940, then consider this "thought experiment." Imagine two infinitely skillful and balanced players. They will play a large number of games. Then luck is some measure of the range (such as standard deviation) of the results. When the game's outcome is an on-off condition such as taking a city or not, I haven't a good definition. The trouble is comparing small differences in skill with differences in luck. But the basic idea is this: Given players of different skill, there is less luck the more often the more skillful can expect to win.

The main way that luck comes into wargames is in the combat rules. In many cases, luck is as clear as the rolling of a die. Quite simply, this varies the results. I don't think game designers usually know how much to vary the results (what the variance should be). They can only wring their hands and wish they knew.

For instance, here is a typical miniatures combat rule (from Hardtack). Roll a die, and take as many casualties as spots show. Now, I have seen evidence for arguing whether the average result, 3 1/2 figures destroyed, is too high or too low. But I have not seen any evidence about the various fractions of the time that various results were obtained. We know that results are not certain (zero variance). Nor are they completely uncertain (infinite variance). But we just don't know how uncertain it should be.

Thus, in general, a game with lowered luck may wel I be as accurate a model as one with a large amount. In this vein, I will discuss ways to lower luck in combat rules for existing or planned games.

Certainty Equivalent

This is a brute force technique for reducing luck in existing games. Suppose a player finds that a group of his pieces is to be destroyed, or not, by a die roll. Then allow him to choose to destroy at least as many factors as he could expect to lose. Otherwise he can still roll the die. For instance, in Stalingrad, you have a 3-1 attack against a doubled 5-7-4. Then you choose between rolling the die or removing at least five factors. In games where some pieces are more valuable than others add the following rule. The factors removed will be of the most valuable type among the pieces being considered for removal. In the example the five factors would have to be armor, unless 14 factors of infantry were available. Add replacement factors if the game hasn't got them already.

Then give the player the difference between factors removed and factors expected to be lost in replacement factors. The player with the 5-7-4, if retreat were available, could remove his piece and receive 1/3 x 7 equals 2 replacement points. But games become dull if this rule is always chosen. Rolling a die is exciting. I've noticed also that without the random losses wargames seem to have little variety (compared to Chess or Go, for instance). Perhaps this is because of the relatively small number of moves, each one being rather complicated.

Multiple Decisions

What makes luck important in a game's result is having the result depend heavily on just a few die rolls. The reduction of Tobruk in Afrika Korps is an example. The half dozen river breaking moves of the Germans in Stalingrad is another. Luck can be cut down, without being cut out, by rolling the die three times, say, instead of once. Just take the average of the 3 rolls to get the losses. It is still possible in this version to lose all 14 factors, or none, in that 3-1 river breaking attempt. It just isn't so likely. Games such as Spirit of '76 which use a tactical board to decide strategic board results have pushed this idea to its limit. Step reduction is another way to make more small decisions. Unless the situations in which a subgame will be used are both clearly defined and few, the game becomes slow. However it does retain the pleasure of variability while keeping it in bounds.

Simultaneous Choice

Here the combat result depends on choices by the attacker and the defender at the same time. Kreigspiel and Diplomacy use it. It appears to cut out luck, because most players don't use a die. But the appearance is deceiving. For a rational player would use a die according to game theory.

As an example, suppose in a Diplomacy fall move there is a French A Sp, and English F MAO. The French and English are enemies. No other piece can move to Sp or Por. The Eng can play F MAO-Sp (I don't care which coast) or F MAO-Por. The Fr have the same two choices. Neither player can do better than flipping a coin to determine his move. Similar situations, and incredibly more complicated guessing games, occur in every game of Diplomacy. Simultaneous choice may well be used for other reasons, but it does hot avoid luck between opposed interests. In particular in two player games where ore loses whattheother gains, it cannot avoid luck.

Sequential Choice

I was always irked by the first generation wargames because they had the same odds table for Corps (Stalingrad), Divisions (D-Day), and Regiments (Afrika Korps). So once I made a series of trials. I wanted to see the results for the corps level when the regimental level actually followed the classical combat results table. The result was almost determined by the forces engaged. It seemed to depend on the defender's orders--to hold at all costs, or to fall back inflicting damage. But it did not depend on the attacker's orders--to gain territory, or inflict the most damage possible. The results could thus be expressed as a choice by the attacker, followed by a choice by the defender.

First the attacker specifies the forces. When the defender can choose to cut his losses and possibly give up the position. Or he can choose to try to keep the position but take heavier losses. This is the odds table that resulted:

Odds	Hold	Yield
1-3	---	(0,6,0,0)
1-2	---	(0,4,1,0)
2-3	---	(0,3,1,0)
1-1	---	(0,2,2,0)
3-2	(0,2,3,0)	(0,2,2,2)
2-1	(0,1,4,0)	(0,2,2,1)
3-1	(0,1,4,1)	(1,2,2,1)
4-1	(1,1,4,1)	---

The parentheses read (probability that the attacker advantages, the attacker's loss, the defender's loss, probability that the defender retreats). The losses are in units of one-sixth of the smal ler force involved. The total losses on the regimental scale were almost certain. So use the "certainty equivalent" method discussed above to take the losses on the corps level. The resulting games are more interesting tactically, but still seem to lack enough variability.

Giving the defender a decision does increase the effective number of moves in the game. I believe that some of the apparent variability of results in large scale conflicts is due to choices. Game designers should try to identify these choices.

Summary.

For an interesting wargame, luck is necessary. To cut it down, without cutting it out, make more die rolls for critical events. The designer can make subgames for the few most critical decisions. The subgame must not occur too often, else it becomes tedious.

SOME DEFINITIONS

Talking about luck without using these terms is hard. First I had them in, then I had them out. Finally I left them slightly in (mainly in parentheses). Then Don asked for definitions to one side. So here goes.

Suppose just one of N possible events can happen. Each event has a value associated with it. (Symbolically call these values v1, v2,.. vN.) For example if I can either lose a piece or keep it, N is 2. Suppose the piece has 5 combat factors. Then v1 is -5 , and v2 is 0.

Probabilities for the N events are any set of N numbers satisfying (1) each number is greater than or equal to zero, and (2) the sum of the numbers is one. In the example if I am to roll a die and lose the piece if I roll a 2 or a 5, then I set p(no loss) equal 2/6, and I set p(no loss) equal 4/6. (Symbolically call these numbers pl , p2,.. . pN.)

The expected value of the event is the sum of the probability of each possible outcome times its value. In the example, the expected value is 1/3 x (-5) plus 2/3 x 0 equals -1 2/3 combat factors. (Symbolically E(v) equals p1v1 plus p2v2plus ... plus pNvN.)

The Variance is a measure of the spread of results. Get set for this definition in English: Variance is the sum of the products of the probabilities of the possible outcomes times the square of the difference between the value for the possible event and the expected value. (Or symbolically Var equals P1(v1-E(v))2 plus p2(v2-E(v))2plus ... plus pN(vN-E(v))2). Roughly the further away from the expected value the individuals are, the larger the variance. In the example I'm doing, Var is 50/9 (combat factor)2. And then since nobody knows what a combat factor squared is, there is the...

...standard deviation, which is the square root of the variance. In the example the standard deviation is about 2.355 combat factors. There is only one way to have a standard deviation of zero--each outcome has the same value. Then the event is certain.