# Risk

• Bronze
Joined: 27.05.2010
Hi all, was playing risk with my cousins yesterday and was asked what the optimal strategy was when defending. Unfortunately my passion for games and (severely overestimated) knowledge of game theory mean i often get asked questions like this and more often than not have no idea. However it seemed a simple probability problem so today i decided to solve it. Thought i'd spread some knowledge here for anyone too lazy to do the maths themselves (obviously i'm not sharing the results with cousins though )

Solved only in the case of big armies (armies greater than or equal to 3)

The basic problem is this: The aggressor rolls three dice and the best two of these are his 'attack scores'. The incumbent then has the decision to roll one dice or two dice. If our 'defence scores' are greater than or equal to his 'attack scores' we win. If we roll one dice it must equal his highest scoring dice in order for us to win and kill one of his armies. If we roll two dice our highest scoring dice is up against his highest scoring dice and our lowest scoring dice is against his second highest scoring dice. Depending on what he rolls, should we roll one dice or two dice?

The loss of each army gives a payoff of -1 and the defeat of our opponent's army a payoff of 1. If we decide to roll both dice we therefore either win both rolls (payoff 2), win one roll and lose one roll (payoff 0), or lose both rolls (payoff -2). I simply calculated the probabilities of defeating certain rolls and the EV given these probabilites and payoffs. Results below:

 code: ```2 Dice Opp |win|lose| P(win) | P(los) | EV 66x | 2 | -2 | 0.0278 | 0.6944 | -1.3333 65x | 2 | -2 | 0.0833 | 0.6667 | -1.1667 64x | 2 | -2 | 0.1389 | 0.5833 | -0.8889 63x | 2 | -2 | 0.1944 | 0.4444 | -0.5000 55x | 2 | -2 | 0.1111 | 0.4444 | -0.6667 54x | 2 | -2 | 0.2222 | 0.4167 | -0.3889 53x | 2 | -2 | 0.3333 | 0.3077 | 0.0513 44x | 2 | -2 | 0.2500 | 0.2500 | 0.00000 1 Dice Opp |win|lose| P(win) | P(los) | EV 6xx | 1 | -1 | 0.1667 | 0.8333 | -0.6667 5xx | 1 | -1 | 0.3333 | 0.6667 | -0.3333 4xx | 1 | -1 | 0.5000 | 0.5000 | 0.00000```

Please note with 2 dice P(win) + P(lose) doesn't equal 1 because of the probability of a draw. Always choose the roll with the highest EV (or lowest negative EV). For smaller rolls EV rises (obviously). Therefore roll with two dice unless your opponent rolls 66x, 65x, 64x, 55x or 54x. In these special cases roll with one dice.
Next task is to work out how big an advantage (if any) a defender gets, to decide what army size i need in order to attack.

LE: Also note i did google this before working it all out myself and couldn't find it anywhere.
• 4 replies
• Black
Joined: 03.07.2008
Don't play Risk, there are so many good board games out there that rolling dice for 3 hours seems such a waste :/
• Bronze
Joined: 27.05.2010
Originally posted by Anssi
Don't play Risk, there are so many good board games out there that rolling dice for 3 hours seems such a waste :/
...
• Black
Joined: 03.07.2008