Risk(tm) is also a board game of world domination by the Parker Brothers. Two to six players play on a world map divided into territories and continents with coloured tokens representing their armies.

Success in Risk requires not only sound strategy and luck but also good diplomacy in ... ahem ... "encouraging" your opponents to take each other out.

more information on Risk (the board game):
http://www.maths.nott.ac.uk/personal/odl/riskfaq.html


In the financial sense, risk is the amount of danger involved in a particular financial decision. This is usually reflected in the amount of capital that may be lost if the deal/contract/decision did not turn out as intended.

Most people misunderstand financial risk. While it is true that some financial instruments inherently carry more risk than others (to take an extreme example, options on futures carry a far higher risk than, say, property), a lot of people misjudge the amount of risk they exercise in their investing activities.

Understanding what actually constitutes risk in investing is one of the most important lessons in learning to become a good investor. With knowledge, wisdom and experience, the seasoned investor sees a wholly different level of financial risk than the novice investor even when they are looking at the exact same possible investments.

My father has owned a Risk (board game by the Parker brothers) set for 30 years, and on the box there's a picture of a family which was clearly taken in the 50's or 60's. I've never played Risk on anything other than this set.

And I've played a lot.

When I think of the game Risk, I think of the badly proportioned blobs which represent countries, with lovely shades of purple and yellow and green, depending on which continent the country is in. I think of the little blocks of colored wood that represent armies, all different sizes and some with corners missing and such. I think of the frayed cards and the box which needs to be held from the bottom on account of it's condition.

And now my father has bought a new risk set, with 3D illustrations of the coastlines, and instead of non-identical blocks of wood representing the armies, it comes with mass-produced detailed plastic soldiers, on which you can see their weapon and their facial expression. There are little plastic cannons and cavalry. I have to admit, it's pretty cool.

Now the old set is useless in comparision. It has been thrown out.

God-knows how many years of history has been thrown out with it. The time my brother and I actually beat my father at the ages of 6 and 8. The time we convinced my grandmother to play with her then whipped her ass silly. Every time my mother actually won for a change. It's gone.

That's my sad story for the day.

Risk is Megadeth's 8th album. Released in 1999 to mixed reviews by fans and non-fans alike.

Risk's sound was an huge change from their earlier works. Especially works like "Killing is my business... And business is good!" and "Rust in Peace".

Risk was produced by country guitarist Dann Huff, and co-produced by Dave Mustaine, the Megaman himself.

Tracklisting:

  1. Insomnia
  2. Prince of Darkness
  3. Enter the Arena
  4. Crush 'em
  5. Breadline
  6. The Doctor is Calling
  7. I'll be There
  8. Wanderlust
  9. Ecstasy
  10. Seven
  11. Time: The Beginning
  12. Time: The End

Risk even features, in his own stuck-on-a-mousetrap way, Vic Rattlehead. While he's not on the cover, he at least makes an appearance.

My copy of Risk came with a "Limited Edition Bonus 6 track CD". Featuring one song from each of their previous albums, save for "Killing" because of contractual reasons.

Risk
The condition in which there is a possibility of an adverse deviation from a desired outcome that is expected or hoped for. 1
I was told a story once by a professor:
So my wife and I are coming back from a Hawkeye game, and we stop to get gas. We'd left early to beat the traffic, and in the car she kept saying "You know, the Hawkeyes were doing pretty well, I think they'll win." So I'm inside paying for the gas, and I hear on the radio that the Hawkeyes lost. So I go back out to the car and offered my wife a wager. I gave her really good odds. Of course, when she found out...

So was there a risk in the situation there?
We all said yes, thinking that she didn't know, so she was taking a risk. And he asked us, "Well, was there risk in what I was doing? Then how is it any different for her?" There was no risk; the outcome was certain. His wife was a victim of uncertainty, doubt caused by lack of knowledge about the future, and not risk. The lesson there is that risk occurs whether we know about it or not.

There are two types of risk, and they are a world apart.
  • Speculative risk is a risk with a possibility of gain or loss. In other words, you either win or lose.
    • Example: I put a quarter in a slot machine (buy tech stock, whatever). I pull the handle. Now there is a risk; I either win some random amount of money or I lose my quarter.
  • Pure risk merely involves the possibility of a loss. You lose or you don't lose, you never win anything.
    • Example: You buy a house. There is now a risk that your house will burn, and that risk is permanent (never goes away). If your house doesn't burn, you don't win another house; you simply don't lose the house you bought.
The only risk which is insurable (or ought to be, anyway) is a pure risk. Purchasing insurance, then, transfers the risk to the insurer (it never goes away entirely). Speculative risks are simply investments or gambles.

A sensible question to ask, then, may be "Well, then if insurance companies are accepting other people's risks, how do they make money?" The answer is in the insurance company's ability to predict losses over a large number of exposures, or risks. If a company only insures 10 homes against total loss by fire, for example, and they predict 1 loss (10%), there's a good chance they will have exactly 1 loss. But what if no houses burn? Huge profits. What if they have 2? The company would probably take a loss. Now imagine that the company insures 1000 homes against total loss by fire. They still predict 10% loss, which now equals 100 homes. A deviation of one home, or even 10 homes, above or below the prediction means far less to the company than it did when they were only covering 10 exposures. The point to transferring your risk to a professional risk bearer is that they are better equipped to deal with your risk; they pool thousands, even millions of risks, minimizing impact of losses and even making money in the process.

In a corporation, a risk manager is a person either hired or contracted by a company to manage that company's risks. This person's function is to use any tools at his/her disposal to ensure that the company can continue to function as intended. They develop processes to either avoid risks altogether, minimize their chance of occurring, or minimize the financial impact of those losses that do occur (by purchasing insurance).
1Taken from Vaughan and Vaughan, Essentials of Risk Management, second ed.

Unfortunately, humans are no longer chased across the savannah by a enemy predator, normally a wild beast such as a tiger. This is mostly due to evolution, we have spread outside of Africa and are now living in cities which protect us from these kind of predatory risks.

This may sound like a good thing. But alas, people are still keen to have an element of risk in their lives, however, not by raging blood thirsty tigers, but through other means.

One of those means, is...Roller Coasters. Yes these revolutionary fear emulators create the same feelings/emotions inside our heads that tigers used to make. The adrenaline and fear created when thrown around an iron cage going at 100 mph, inside out, and upside down makes some people...'feel alive'.

Other people, apparently, who haven't got the means to get to a Roller Coaster (they may live in the countryside away from smokey civilisation) might find other ways to recreate the 'tiger fear'. These may include, shop lifting, bank robbery, sky diving, and/or walking into a road blindfolded.

Personally, I prefer stealing a policeman's helmet, the chase you get from a young police officer is akin to the tiger in the savannah.

The best strategy for winning at Risk is to only engage in battles you are sure to win with minimal casualties - ironic, eh? There are a number of ways to do this; however, this is purely a heuristic, and it is not the only way to win. Risk is part formula, part thought, and you must adapt to the circumstances. This ability can be learned only through practice.


1. Initial placement. Do not put yourself in a position where you can be attacked from many directions. This means avoiding Asia, Europe (easily the worst continent on the board due to its relatively low worth compared to the number of places it can be attacked from), and perhaps Africa. Try to take North America, because it has few places to be attacked from, and gives a fairly large amount of men per turn. However, do not put up too much of a fight for North America in the beginning of the game, because once again, you want to get in as few battles as possible. In fact, you do not want to put yourself in any position in the beginning of the game in which you are fighting for control of a continent.

2. Fall back. If you do not want the enemy to take a territory, take a single territory of theirs in said country and pull your men back at the end of the turn. This will ensure that they will not get the bonus men that the country grants, and will also ensure that you do not get into a fight where you will lose a lot of men.

3. Buffer zones. Tempting as it may seem to place large amounts of men at the edge of your territories, this is a bad idea -unless you are desperate for the extra men a given continent will give you. You will most likely end up in a fight where you lose large amounts of men for no reason. Rather, after taking a continent, take a few territories in the adjacent continent around the entry point to your continent. Then pull back, leaving only one troop in each of those territories outside of your continent. This way, if an enemy attacks you, he will have to wade through a buffer zone, losing troops along the way until he stops or is weakened to a manageable size.

4. Let other people fight your battles. In the beginning of the game, and through much of the middle of it, there will be many fights as people try to gain continents. For the most part, you want to avoid this. Do not get stuck in the middle of a fight for a piece of territory, and gain alliances with the people near you - and try to make sure your allies do the fighting for you.

That's all for now. One last piece of advice: study the other players. Try to predict what they will do next. These strategies can often go down the drain if you are facing a suicidal player who will sacrifice all his men if he can get to you. Also, if you do not want the other players to gang up on you, try not to gain too much power if you don't think you can take them all on. Being the first to take a continent early in the game is a surefire way to end up at the top of everyone's hit list. Similarly, if you seriously hurt a player, make sure you can finish him off. If you hurt them, they will most likely carry a grudge against you.

Risk strategy

Risk is game of strategy, without a good strategy you are bound to lose the game, however well you perform on individual turns. But when it comes down to it Risk is also a game of numbers, a game of statistics. When the player is looking at the board, deciding his next move, what he is trying to work out is whether or not he will be able to win each battle and, even if he does win, if he will be left weak and vulnerable afterwards.

Thus I present to you

The Statistics of Risk and its Implications on Strategy

Before we start it is important for me to briefly clarify a few definitions that I will use. An attack is one roll of the dice. A battle is a series of attacks that continue until the defending army is completely destroyed or the attacker decides to stop, for example if he were to be worn down to only having one army left. A win, when applied to a battle, is when the defender loses all of his pieces, when applied to an attack it is merely when the defender loses pieces and the attacker does not. A loss, when applied to a battle, is when the attacker stops attacking (if, for example, he were to only have one army left and thus can not attack) yet the defender still has at least one army in his territory. When applied to an attack it is when the attacker loses pieces and the defender does not. There is also the special case of a draw, this can only apply in attacks where the attacker is using 2 or 3 dice and the defender is using 2 dice, if both the attacker and the defender lose one army each then they have drawn.

I will assume that you know how attacks in Risk are conducted, but for those of you who have not played, or who are a bit fuzzy about it then here you go. The attacker can only attack from a territory which has more than one army in it. The attacker can roll up to three dice to attack, but the number of dice rolled must be less than the number of armies in the attacking territory. So if the attacker only has two armies in that territory then he can only roll one dice (if he has three armies then two dice, if four or more then all three die may be rolled). The defender may roll one or two dice. If the defender has one or two armies then only one die may be rolled, if he has three or more armies then he may defend with two dice.

Once the number of dice that may be rolled by both the attacker and the defender has been decided then both sides roll their dice. Once the dice have been rolled then the attackers highest rolled dice is matched with the defenders highest rolled dice, then the second highest with the second highest (assuming that either side attacked using more than one die). The values are then compared. If the attacker's dice has a higher value than the defender's then the defender loses an army. If the values are equal, or if the attacker's dice has a lesser value than the defender's then the attacker loses an army. This is done for each pair of dice. So it can be seen that, providing both attacker and defender roll at least two dice each, there are three possible outcomes: the defender loses two armies, the attacker loses two armies, or both attacker and defender lose an army. The last case happens when the attacker loses one of the pairs and wins the other pair, this case is a draw.

The first thing we must consider as we embark on this enterprise are the dice. The game is made or broke on the rolls of these simple cubes. There are three possible outcomes of any roll of the dice, the roll can be Won, Drawn or Lost. There are also six possible combinations of die rolls. So we should start of by looking at the probabilities of winning, losing or drawing any dice roll.

                               DEFENCE
WIN                      DRAW                  LOSE
     |   1   |   2   |     |   1   |   2   |     |   1   |   2   |
A  --+-------+-------+   --+-------+-------+   --+-------+-------+
T  1 | 0.417 | 0.174 |   1 | 0.000 | 0.000 |   1 | 0.583 | 0.826 |
T  --+-------+-------+   --+-------+-------+   --+-------+-------+
A  2 | 0.660 | 0.228 |   2 | 0.000 | 0.324 |   2 | 0.340 | 0.448 |
C  --+-------+-------+   --+-------+-------+   --+-------+-------+
K  3 | 0.802 | 0.372 |   3 | 0.000 | 0.336 |   3 | 0.198 | 0.293 |
   --+-------+-------+   --+-------+-------+   --+-------+-------+

In these tables the rows are the number of dice that the attacker is rolling (from one to three) and the columns are the number of dice that the defender is rolling (one or two). So, for example, we can see that if the attacker is rolling one dice and the defender is rolling one dice then the probability of the attacker winning is 0.417 and the probability of the attacker losing is 0.583. Remember of course that the defender has a slight advantage since the defender wins the roll if the numbers are equal. Obviously in any case where either side is rolling less than two dice then a draw can not occur, thus having a probability of zero.

From these tables we should note several things. Firstly we should observe that the defenders ability to roll more than one die has a major effect on the attacker's chance of winning, in all cases more than halving the attacker's chance of winning. It should also be noted that, in all cases where the defender rolls two dice, the odds are not stacked in particular favor of the attacker. In all cases where the defender rolls two dice there is an at least 0.629 chance of the attacker losing at least one army, rising to 0.826 if the attacker was to attack with one army against the two.

These figures also demonstrate that a common strategy does not work. Many people, while defending their territory, will choose to defend with just one die (against three) under the preconception that it will reduce their loses. If we compare the probable outcomes of one attack, where the attack uses three dice and the defender uses two, to a series of two attacks in which the attacker uses three dice and the defender uses only one, we can see the following. In the first case of three against two the chances of the defender losing two armies is 0.372 and the chance of losing just one army is 0.336. In the second case of two attacks of three armies against just one army we can work out that the chances of the defender losing two armies (one in each battle) actually increases to 0.643. The chance of losing just one army does however drop to 0.318. If we combine these numbers then we can see that the chances of the defender losing at least one army is just 0.629 in the first case compared to an enormous 0.961 in the second case. This clearly shows that, in all cases, the defence should maximize their ability to defend by using both dice whenever they can, irregardless of the myth of "slowing down your loses".

These numbers clearly show us that in any attack losses are to be expected. Later we will return to this and look at the most likely number of losses that can be expected when fighting a battle. We will also look at the likely number of losses that a defender can be expected to sustain should the attacker lose the battle. But before we move on to that we shall first look at the probabilities of a battle being won or lost.

To generate the following figures I did the following. I started off by using the probabilities from the tables above. I made the assumption that, in all cases, both the attacker and defender will use the maximum number of dice that they can roll.

In any attack (where both attacker and defender are using at least two dice each) there are three possible outcomes: the attacker can win, lose or draw. Thus we need to look at the chances of any one of these three outcomes occurring. So we can work out the chance of the attacker winning a battle with x armies against the y armies of the defender as follows:

To start we must first define three functions, p_ow(x, y), p_od(x, y), p_ol(x, y). These give the probabilities of wining, drawing or losing an attack. Basicly they are just the contents of the dice tables given above. For the sake of completeness I will repeat them here.

p_ow(2, {1|2})  = 0.417   p_od(2, {1|2})  = 0.000   p_ol(2, {1|2})  = 0.583
p_ow(2, >2)     = 0.174   p_od(2, >2)     = 0.000   p_ol(2, >2)     = 0.826 
p_ow(3, {1|2})  = 0.660   p_od(3, {1|2})  = 0.000   p_ol(3, {1|2})  = 0.340
p_ow(3, >2)     = 0.228   p_od(3, >2)     = 0.324   p_ol(3, >2)     = 0.448
p_ow(>3, {1|2}) = 0.802   p_od(>3, {1|2}) = 0.000   p_ol(>3, {1|2}) = 0.198
p_ow(>3, >2)    = 0.372   p_od(>3, >2)    = 0.336   p_ol(>3, >2)    = 0.293

To make life simpler and easier to understand: p_ow means "probability of outcome win", p_od and p_ol are "outcome draw" and "outcome lose" respectively.

We can now define our function p_win(x, y). For the sake of ease of reading, p_win stands for probability win.

p_win(x > 0, 0) = 1
p_win(1, y > 0) = 0

while({x,y}>2)
	p_win(x, y) = p_ow(x, y)p_win(x, y - 2) + p_od(x, y)p_win(x - 1, y - 1) + p_ol(x, y)p_win(x - 2, y)
else
	p_win(x, y) = p_ow(x, y)p_win(x, y - 1) + p_ol(x, y)p_win(x - 1, y)

As you can see, this creates a nice recursive function which, conveniently enough, is nice and easy to code. Basiclly all the algorithm does is to create a probability tree and then sum the probabilities of each branch of the tree where the attacker wins. By plugging these algorithms into a nice C program I generated the following table of the probability of any attack winning with up to twenty armies against up to twenty armies. The hardcore risk players might want to print this out to refer to mid-game...


                                                           D E F E N C E

      |  01 |  02 |  03 |  04 |  05 |  06 |  07 |  08 |  09 |  10 |  11 |  12 |  13 |  14 |  15 |  16 |  17 |  18 |  19 |  20 |
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   01 |0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   02 |0.417|0.174|0.030|0.005|0.001|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   03 |0.802|0.588|0.239|0.144|0.056|0.033|0.013|0.008|0.003|0.002|0.001|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   04 |0.961|0.887|0.564|0.412|0.258|0.172|0.107|0.068|0.042|0.026|0.016|0.010|0.006|0.004|0.002|0.001|0.001|0.001|0.000|0.000|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   05 |1.000|0.971|0.737|0.593|0.429|0.317|0.221|0.156|0.106|0.073|0.049|0.033|0.021|0.014|0.009|0.006|0.004|0.003|0.002|0.001|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   06 |1.000|1.000|0.863|0.738|0.596|0.469|0.360|0.269|0.199|0.143|0.103|0.073|0.051|0.035|0.025|0.017|0.011|0.008|0.005|0.003|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
A  07 |1.000|1.000|0.922|0.835|0.716|0.604|0.489|0.391|0.303|0.234|0.175|0.131|0.096|0.070|0.050|0.036|0.025|0.018|0.013|0.009|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
T  08 |1.000|1.000|0.960|0.898|0.812|0.712|0.611|0.508|0.417|0.333|0.264|0.204|0.157|0.118|0.089|0.066|0.049|0.035|0.026|0.018|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
T  09 |1.000|1.000|0.978|0.939|0.875|0.799|0.708|0.617|0.523|0.438|0.358|0.290|0.230|0.181|0.140|0.108|0.082|0.062|0.046|0.034|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
A  10 |1.000|1.000|0.989|0.964|0.922|0.861|0.790|0.707|0.623|0.536|0.456|0.379|0.313|0.253|0.204|0.160|0.126|0.097|0.075|0.057|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
C  11 |1.000|1.000|1.000|0.980|0.950|0.908|0.850|0.784|0.707|0.630|0.548|0.473|0.399|0.334|0.274|0.224|0.180|0.144|0.113|0.089|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
K  12 |1.000|1.000|1.000|0.988|0.970|0.939|0.898|0.842|0.780|0.708|0.636|0.559|0.487|0.416|0.353|0.294|0.244|0.198|0.161|0.128|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   13 |1.000|1.000|1.000|1.000|0.982|0.962|0.930|0.889|0.836|0.777|0.710|0.641|0.568|0.500|0.432|0.370|0.312|0.262|0.215|0.177|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   14 |1.000|1.000|1.000|1.000|1.000|0.976|0.955|0.923|0.883|0.832|0.776|0.711|0.646|0.577|0.512|0.445|0.387|0.329|0.278|0.231|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   15 |1.000|1.000|1.000|1.000|1.000|0.987|0.971|0.948|0.916|0.877|0.828|0.775|0.713|0.652|0.586|0.522|0.457|0.398|0.339|0.286|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   16 |1.000|1.000|1.000|1.000|1.000|1.000|0.983|0.966|0.943|0.911|0.873|0.825|0.774|0.716|0.656|0.589|0.523|0.461|0.394|0.340|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   17 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.979|0.961|0.938|0.906|0.868|0.823|0.774|0.714|0.645|0.586|0.516|0.445|0.387|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   18 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.987|0.975|0.957|0.933|0.900|0.860|0.817|0.760|0.695|0.632|0.564|0.495|0.446|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   19 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.985|0.971|0.952|0.924|0.891|0.848|0.801|0.737|0.664|0.606|0.549|0.459|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
   20 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.982|0.967|0.945|0.913|0.872|0.832|0.769|0.698|0.615|0.552|0.466|
  ----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

So what can we learn from this?

Firstly we must note that this table merely gives the probability of winning, it tells us nothing about how many armies we are likely to have left afterwards, although we will be looking into this later. We should also note that it is very simple to work out the probability of winning if you also wish to have a certain number of armies left at the end, all you have to do is take as your attacking force the number of armies you have minus the number of armies you wish to have left (so if I have 20 armies and want to have 5 left at the end, then I take the probability of attacking with 15 armies). Lastly, the top row (attacking with one army) is supposed to be all zeros - you can't attack if you only have one army!.

One of the most obvious questions to ask of this data is "How many armies do I need to have good chance of winning?". To answer this using the data provided we must first define what is meant by a "good" chance of winning. For the sake of the reality of the game I am going to define a "good" chance as being two thirds. In my opinion any attack that has a two thirds chance of winning is "good". So let us look at the numbers provided above. The first thing that we can see (with the aid of a nice graphing program...) is that the ratio of attackers to defenders to ensure a chance of winning that is at least 0.66 follows an inverse exponential law. The ratio goes down from needing 3 attackers to defeat 1 defender, a ratio of 3, to a ratio of just 1.125 with 18 attackers against 16 defenders. We can also see a slightly surprising fact here. In order to have at least a 0.66 (two thirds) chance of beating a defender, the attacker needs to have merely two armies more than the defender has. Entirely counter-intuitive, but this result is entirely borne out by the numbers.

To take this further we must now look at a table of the most likely number of armies remaining after an attack. To generate these next two tables I used the same algorithm as previously but, instead of looking at the probabilities, I looked at the number of armies left at the end of each possible attack outcome. I then just worked out what the most likely number of armies was from this range.

The first table is the most likely number of armies the attacker will have left assuming that the attacking wins the battle. The second table is most likely number of armies that the defender will have remaining assuming that the attack fails.


                                                           D E F E N C E

      | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   01 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   04 | 04 | 04 | 04 | 04 | 03 | 04 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 03 | 04 | 03 | 04 | 03 | 04 | 03 | 04 | 03 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   06 | 06 | 06 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A  07 | 07 | 07 | 06 | 06 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T  08 | 08 | 08 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T  09 | 09 | 09 | 08 | 07 | 07 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A  10 | 10 | 10 | 09 | 08 | 07 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
C  11 | 11 | 11 | 10 | 09 | 08 | 08 | 07 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 05 | 05 | 04 | 04 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
K  12 | 12 | 12 | 11 | 10 | 09 | 09 | 08 | 07 | 07 | 06 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 05 | 04 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   13 | 13 | 13 | 12 | 11 | 10 | 10 | 09 | 08 | 08 | 07 | 07 | 06 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   14 | 14 | 14 | 13 | 12 | 11 | 11 | 10 | 09 | 09 | 08 | 07 | 07 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 05 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   15 | 15 | 15 | 14 | 13 | 12 | 11 | 11 | 10 | 09 | 09 | 08 | 08 | 07 | 07 | 06 | 06 | 06 | 06 | 06 | 05 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   16 | 16 | 16 | 15 | 14 | 13 | 12 | 12 | 11 | 10 | 10 | 09 | 08 | 08 | 07 | 07 | 07 | 07 | 06 | 06 | 06 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   17 | 17 | 17 | 16 | 15 | 14 | 13 | 13 | 12 | 11 | 10 | 10 | 09 | 09 | 08 | 08 | 07 | 07 | 07 | 07 | 06 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   18 | 18 | 18 | 17 | 16 | 15 | 14 | 14 | 13 | 12 | 11 | 11 | 10 | 09 | 09 | 08 | 08 | 08 | 07 | 07 | 07 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   19 | 19 | 19 | 18 | 17 | 16 | 15 | 15 | 14 | 13 | 12 | 12 | 11 | 10 | 10 | 09 | 09 | 08 | 08 | 08 | 08 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   20 | 20 | 20 | 19 | 18 | 17 | 16 | 16 | 15 | 14 | 13 | 12 | 12 | 11 | 10 | 10 | 10 | 09 | 09 | 09 | 09 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+

and now for the defence...


                                                           D E F E N C E

      | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   01 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   02 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   03 | 01 | 01 | 03 | 04 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   04 | 01 | 01 | 03 | 03 | 04 | 05 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   05 | 01 | 01 | 02 | 03 | 04 | 04 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   06 | 01 | 01 | 03 | 03 | 03 | 04 | 04 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A  07 | 01 | 01 | 02 | 03 | 03 | 04 | 04 | 05 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T  08 | 01 | 01 | 03 | 03 | 03 | 03 | 04 | 04 | 05 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T  09 | 01 | 01 | 02 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T  10 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 | 10 | 11 | 12 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A  11 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 | 10 | 11 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
C  12 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 | 10 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
K  13 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   14 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 07 | 08 | 08 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   15 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 07 | 08 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   16 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 06 | 07 | 07 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   17 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 07 | 07 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   18 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 06 | 07 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   19 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
   20 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 07 |
  ----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+

So what can we learn with the addition of these pieces of data?

Before we start to look more intensively at this new data I must make one point clear. The number given is the number of armies that the attacker (or defender) has at least a 50% chance of having remain. For example, if the attacker attacks with 20 armies against 6, then there is at least a 50% chance that the attacker will have 16 or more armies remaining after the attack. Conversely if the attacker were to lose then the defender has at least a 50% chance of having 3 or more armies remaining.

But what are all these figures useful for?

In risk there are normally two questions that are being asked "How far can I extend this attack" and "Can I hold this position". The best way to answer these questions are of course with an example. So let us look at some crude ASCII art of Africa, as it is depicted on a Risk board.

     +-----+-----+      NA - North Africa  5 Armies
     |NA   |E    |      E  - Egypt         5 Armies
  <--|     |  5  |      C  - Congo         1 Army
 to  |  5  |     |      EA - East Africa   1 Army
B    |     +-----|      SA - South Africa  1 Army
r    +-----+EA   |      M  - Madagascar    1 Army
a    |C    |  1  +--+
z    |  1  |     |M |
i    +-----+-----+  |
l    |SA         | 1|
     |     1     +--+
     +-----------+

Let us create the scenario where you are attacking Africa, as it stands above, from Brazil with 20 armies placed on Brazil. Is it possible to take all of Africa? We will use the tables above to analyze each step.

Step 1: Brazil into North Africa
Our tables tell us that 20 armies against 5 has a probability of winning of 1 (in acutallity its more like 0.999999... but rounding errors take their toll). Our table also tell us that we are likely to have 17 armies left. So we attack and we move all our armies (barring the one that must remain in Brazil) in to North Africa.
At the end of the 1st step we now have 16 armies in North Africa.

Step 2: North Africa into the Congo
15 vs. 1. Probability of winning: 1. Likely number of armies left: 15.
At the end of the 2nd step we have 14 armies in the Congo

Step 3: Congo into South Africa
14 vs. 1. Probability of winning: 1. Likely number of armies left: 14.
At the end of the 3rd step we have 13 armies in South Africa

Step 4: South Africa into Madagascar
13 vs. 1. Probability of winning: 1. Likely number of armies left: 13.
At the end of the 4th step we have 12 armies in Madagascar

Step 5: Madagascar into East Africa
12 vs. 1. Probability of winning: 1. Likely number of armies left: 12.
At the end of the 5th step we have 11 armies in East Africa

Step 6: East Africa into Egypt
11 vs. 5. Probability of winning: 0.950. Likely number of armies left: 8.
At the end of the 6th and last step we have 7 armies in the Egypt

So at the end of this we can see that the probability of taking the African continent as depicted above with 20 armies is (1 x 1 x 1 x 1 x 1 x 0.950) = 0.950 and you're likely to have 7 armies in Egypt at the end of that.

There is, however, a large flaw with the above calculation - it assumes that at each stage we will still have the most likely number of armies left. Any change in that number will have a large impact on all the following stages. It still does show that twenty armies will have a very good chance of conquering the Africa shown above.

Now let us move on to our final point about defence. Let us take once more our situation above. We shall assume that, after the attack, the attacker then moved (the fortification move) three armies from Egypt to North Africa, so there are now four armies in North Africa and four armies in Egypt. What will this defend us against? Handily this question has already been answered for us. Above we discussed the idea of a "good" chance at winning an attack and we noted that (if we go by 0.66 as a good chance of winning) the attacker must have two more armies than the defender in order to have a good chance. So we can easily see that four armies will defend us against up to six attacking armies against each location

I feel here that I have covered most of the basics of how the statistics of the die effects our strategies in Risk, but if you have any more questions, or if you would like me to look more closely at certain scenarios then please msg me. If there is sufficient demand I will happily put the tables up on the Internet in a prettier format.

Risk (?), n. [F. risque; cf. It. risco, risico, rischio, Pg. risco, Sp. riesgo, and also Sp. risco a steep rock; all probably fr. L. resceare to cut off; pref. re- re- + secare to cut; -- the word having been probably first used among sailors. See Section.]

1.

Hazard; danger; peril; exposure to loss, injury, or destruction.

The imminent and constant risk of assassination, a risk which has shaken very strong nerves. Macaulay.

2. Com.

Hazard of loss; liabillity to loss in property.

To run a risk, to incur hazard; to encounter danger.

Syn. -- Danger; hazard; peril; jeopardy; exposure. See Danger.

 

© Webster 1913.


Risk, v. t. [imp. & p. p. Risked (?); p. pr. & vb. n. Risking.] [CF. F. risquer. See Risk, n.]

1.

To expose to risk, hazard, or peril; to venture; as, to risk goods on board of a ship; to risk one's person in battle; to risk one's fame by a publication.

2.

To incur the risk or danger of; as, to risk a battle.

Syn. -- To hazard; peril; endanger; jeopard.

 

© Webster 1913.

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