Hasbro’s Risk and Strategy Analysis
http://web.mit.edu/sp.268/www/risk.pdf
Hasbro’s game of Risk is a strategy game in which two to six players fend and expand their territories in the ultimate war for world domination. Troops are strategically placed in a territory in which a player is expected to play passively or aggressively — defend or attack. This is closely related to Nash equilibrium in which a player weighs out an opponent’s best response and strategically acts on what he or she believes the other player will make. However, a player is given the option to attack in multiple ways: rolling a single die, rolling two dice, or rolling three dice. In a similar manner, defending players are given the option to defend with a single die or two dice. In this game, the number of dice rolled corresponds to the number of troops expended to battle. If an attacker’s die rolls a higher number than his or her opponent’s then he or she wins the battle while the defender must remove a number of troops corresponding the number of die. The same applies to a defender’s success in battle. Faced with these numerous strategies, how can one possibly tell which action to take in a game that revolves around probability?
Fortunately, a research article from MIT illustrates the probabilities of players’ success in each strategy from a sample of over 100,000 battles as shown below:
From the table above one can see that the attacker’s dominant strategy is to always roll three dice if he or she has expendable troops while the defender’s dominant strategy is to roll two dice. One of the most important concepts of the game is knowing when to attack and when to defend. As the article mentions, attacking consistently can not only leave a player’s forces spread out over many territories but also leave a territory more vulnerable than before his or her turn. On the other hand, defending seems like the most viable option between the two because a large concentration of troops in a territory with few entry points for attack can “act as a deterrent for large attacking forces” as stated in the article. Ultimately choosing to defend, albeit not as exciting as its counterpart, is the key to success in this game for world domination.