Risk Fighting

Have you ever wondered what the chance of each outcome in a Risk battle is?  Here you go.  I wrote R code to sample as many battles as you want with alternate dice combinations like those in Risk Factions or Risk 2210 to be included.  This only applies to battles where each side can lose two guys.

riskSim=function(att,def,samp){

attLostFreq=numeric(3);

for(i in 1:samp){

attRolls=numeric(length(att));

for(j in 1:length(att)){

attRolls[j]=sample(att[j],1)}

defRolls=numeric(length(def));

for(j in 1:length(def)){

defRolls[j]=sample(def[j],1)}

attWins=sum((sort(attRolls,decreasing=T)[1:2])>(sort(defRolls,decreasing=T)[1:2]));

attLostFreq[3-attWins]=attLostFreq[3-attWins]+1}

finalVec=c(attLostFreq/samp,(attLostFreq[1]+attLostFreq[2]/2)/samp);

return(finalVec)}

You enter the attack dice for att and the defense dice as def.  Here's the syntax - c(8,6,6) means you have one eight-sided die and two six-sided dice.  Samp is the number of trials.  

I did 10000 trials for every combination in risk, risk factions, and risk 2210.  Note that even though that seems like a large sample, there can still be some error.  

Here is an example of the standard matchup:

> riskSim(c(6,6,6),c(6,6),10000)

[1] 0.3712 0.3380 0.2908 0.5402

What this means is that the attack has three six-sided dice, the defense has two six-sided dice, and there were 10000 trials.  The results say that in those trials, 37.12% of the time, the defense lost 2, 29.08% of the time, each side lost 1, and 29.08% of the time, the attack lost 2.  In total, the attack had a 54.02 winning percentage.   

Risk 2210:

> riskSim(c(6,6,6),c(6,6),10000)

[1] 0.3716 0.3316 0.2968 0.5374

> riskSim(c(8,6,6),c(6,6),10000)

[1] 0.4745 0.3086 0.2169 0.6288

> riskSim(c(8,8,6),c(6,6),10000)

[1] 0.55440 0.27590 0.16970 0.69235

> riskSim(c(8,8,8),c(6,6),10000)

[1] 0.63380 0.24050 0.12570 0.75405

> riskSim(c(6,6,6),c(8,6),10000)

[1] 0.2707 0.3610 0.3683 0.4512

> riskSim(c(8,6,6),c(8,6),10000)

[1] 0.35520 0.34730 0.29750 0.52885

> riskSim(c(8,8,6),c(8,6),10000)

[1] 0.4525 0.3210 0.2265 0.6130

> riskSim(c(8,8,8),c(8,6),10000)

[1] 0.5112 0.2948 0.1940 0.6586

> riskSim(c(6,6,6),c(8,8),10000)

[1] 0.21300 0.34070 0.44630 0.38335

> riskSim(c(8,6,6),c(8,8),10000)

[1] 0.27880 0.33810 0.38310 0.44785

> riskSim(c(8,8,6),c(8,8),10000)

[1] 0.3481 0.3346 0.3173 0.5154

> riskSim(c(8,8,8),c(8,8),10000)

[1] 0.4066 0.3258 0.2676 0.5695

Risk Factions:

> riskSim(c(6,6,6),c(6,6),10000)

[1] 0.37650 0.33770 0.28580 0.54535

> riskSim(c(6,6,6,6),c(6,6),10000)

[1] 0.4620 0.3288 0.2092 0.6264

> riskSim(c(6,6,6),c(6,6,6),10000)

[1] 0.23220 0.29890 0.46890 0.38165

> riskSim(c(6,6,6,6),c(6,6,6),10000)

[1] 0.3136 0.3062 0.3802 0.4667

Here are some things to note.  Improving dice helps defense more, especially in Factions.  Also, look at how likely the 2-0 and 0-2 sweep are.  Consider the standard Risk Factions battle.  Attack won 54.535% of the time.  For two independent battles with attack winning 54.535% of the time, the distribution of series results are: 

2-0

> .54535*.54535

[1] 0.2974066

1-1

> .54535*(1-.54535)*2

[1] 0.4958868

0-2

> (1-.54535)*(1-.54535)

[1] 0.2067066

That's 49.58868% for independent battles compared to 33.77% in Risk.