A Dynamic Version of the Hawk-Dove Game in Nature
https://link.springer.com/content/pdf/10.1007/BF02071588.pdf
We discussed in class how the hawk-dove game consists of two players which can either play a cooperative (dove) strategy or aggressive (hawk) strategy – if both play dove, they each get a small payoff; if one plays dove and one plays hawk, the aggressive player gets a payoff much greater than the cooperative players; if both play hawk, both players get a payoff of 0. The hawk-dove game can play out in nature through an animal encountering a piece of food, but also encountering an adversary for that food – they could share, reducing the payoff they would’ve gotten if they were aggressive and the other passive – or they could both fight, which can be severely unfavorable to both. The study linked has noted that previous models simulating complete hawk behavior in a predator’s population end in extinction, because fights are very energy-costly and the tradeoffs between risking winning a fight and being severely injured/killed are insufficient over time. This is consistent with the observation in class that hawk-hawk is not an ESS as a hawk in a hawk-majority population will often encounter other hawks, which it does extremely poorly against.
This study aims to examine how the hawk-dove game can be dynamically altered to better fit the complex and constantly changing variables that an animal faces in a natural environment. More specifically, the study investigates how an animal’s behavior changes when two factors are introduced: time and hunger. For simplified reasons, we assume an animal’s ultimate goal is to avoid dying, which can happen from starvation or with a given probability every time they choose to fight another animal. An animal looks for food over the course of a day and loses a set amount of energy overnight; if they reach 0 energy, they die. The researchers calculated for different situations an energy threshold c(t); it is an animal’s best strategy to always play dove until their energy depletes to a point below this threshold, at which point they will play hawk. This also makes sense in the real world – an animal has little reason to risk dying in a fight until they are almost starving, in which case it is better to take that risk instead of continuing to starve to death. With this setup, the researchers started changing other variables in their model to see how that could impact an organism’s likeliness to play hawk:
In this graph, what is changed is the amount of energy an organism tends to lose overnight. One can see that when this quantity is greater as in curve iii, the proportion of hawk-players is also greater because there is a bigger chance of starvation. Accordingly, the proportion of hawk-players in each curve grows slightly over the course of the day as the stakes of going without enough food for the night grow. One could apply this to species with a higher metabolic rate as possibly tending to be more aggressive in some circumstances.
In this graph, what is changed is the probability of a food item being contested by an opponent when the animal encounters it. The animal will always want to play hawk when there are few hawks in the population (left side of the graph) as the probability of a fight is lower and therefore the risk of dying from injury is lower. Other factors can complicate the right side of the graph, as animals will also weigh their personal state and the time of day in addition to the likeliness of a fight.
Therefore, this study demonstrates how introducing new, dynamically changing factors to an environment where behaviors are modeled by game theory can cause changes in an individual’s perceived payoffs and therefore changes in expected behavior probabilities over time.