The Game Theory of buying wards in Dota 2
http://dota2withdeemi.blogspot.com/2015/06/the-supports-dilemma-discussion.html
In this blog post, the blogger attempts to apply Game Theory to a particular decision making process at the start of most high level DotA 2 (Defense of the Ancients 2) games, which is whether or not to buy sentry wards to de-ward the enemy team’s observer wards. In a more simple sense, a team has to decide whether or not to spend additional resources (on the sentry wards) to eliminate the benefits the enemy team has gained through their own purchases (the observer wards). This means that buying sentry wards is useless unless the other team bought observer wards. The blogger likens this scenario to the Prisoner’s Dilemma covered in class, by comparing the four different scenarios that come about from both teams deciding what to do with the graph shown below. To come up with these numbers, the blogger assumes that both teams always buy observer wards, and additionally quantifies the effect of the observer wards in terms of gold, to come up with the number below (interestingly, the numbers below are actually incorrect based on the model he proposed in the blog, though it doesn’t change the decision making process). The blogger then argues that buying sentry wards always swings the net worth difference in said team’s favor, making purchasing sentry wards the dominant strategy for both teams as it provides a +100 gold net worth swing in the team’s favor no matter the other team’s decision.
After seeing this post, I wanted to see if I could expand this model to explore whether or not teams would even always want to purchase observer wards in the first place (according to the blogger’s model, not purchasing any wards while the other team purchases both observer and sentry wards causes your team to be up 50 gold in net worth simply because the sentries were useless). To do this, I effectively flattened a hyper-cube of decisions to get what is shown below, with the gold difference in Team A’s favor shown.
| Team A | ||||||
| Doesn’t Observer Wards | Buys Observer Wards | |||||
| Doesn’t buy Sentry Wards | Buys Sentry Wards | Doesn’t buy Sentry Wards | Buys Sentry Wards | |||
| Team B | Doesn’t buy Observer Wards | Doesn’t buy Sentry Wards | 3125 | 3125(+0) | 3125 | 2925(-200) | 2975 | 3125(+ 150) | 2975 | 2925(-50) |
| Buys Sentry Wards | 2925 | 3125(+200) | 2925 | 2925(+0) | 2925 | 2975(+50) | 2925 | 2775(-150) | ||
| Buys Observer Wards | Doesn’t buy Sentry Wards | 3125 | 2975(-150) | 2975 | 2925(-50) | 2975 | 2975(+0) | 2825 | 2925(+100) | |
| Buys Sentry Wards | 2925 | 2975(+50) | 2775 | 2925(+ 150) | 2925 | 2825(-100) | 2775 | 2775(+0) | ||
It’s rather hard to visualize, but I don’t believe there is even a Nash Equilibrium once the decisions are laid out like this (I could very well be wrong). This is because one should always buy sentries when the other has observers, but never when they don’t (this is pretty intuitive). However, if one were to take the approach of always minimizing the maximum loss, a team would actually opt to only buy observer wards, as then the most one could be down is 100 gold in net worth. Truth to be told, I believe this model is actually more accurate, as the common trend in more casual play is to only buy observer wards. In fact, in the high level games where sentry wards ARE bought, they are seldom used for contesting observer wards, but are rather used to contest other even more valuable resource (for those with DotA acumen, blocking camps).