Nuances of The Evolutionarily Stable Strategy in Vampire Bats
Pretend for a second that you are a vampire (bat) that needs to drink (usually, animal blood) every two nights to survive. Each night, only ⅔ of the population successfully feeds and ⅓ go hungry. After returning to your roost, you have the option to either share what you foraged with others or to withhold.
The following table displays a payoff matrix for the scenario. If both bats withhold, there is a neutral effect. If they both share, both are slightly negatively impacted. If one bat shares and the other bat withholds, the bat that shares is harmed while the bat that withholds benefits.
| BAT B | |||
| SHARE | WITHHOLD | ||
| BAT A | SHARE | -2, -2 | -5, 5 |
| WITHHOLD | 5, -5 | 0, 0 | |
First, I used the strategies discussed in class to find if there are any pure Nash equilibrium or dominant strategies. If A shares, B would withhold. If A withholds, B would withhold. If B shares, A would withhold. If B withholds, A would withhold. It’s evident that withholding is the dominant strategy for both Bat A and Bat B, and none would want to deviate from the strategy. Furthermore, the Nash equilibrium would be for Bat A and Bat B to withhold, for a payoff of 0 for each.
Now, determine whether the pure strategy Nash equilibrium is an Evolutionarily Stable Strategy. An ESS is defined as a strategy that cannot be invaded by another strategy if a small fraction x decides to deviate from the popular strategy. Suppose we are in a population that mainly withholds. That is, x represents the small fraction that would deviate to share and 1-x represents the rest of the population that withholds.
In a population that mainly withholds
If Bat A withholds,
(1-x)(5)+(x)(0)=5-5x
If Bat A shares,
(1-x)(-2)+(x)(-5)=-2+2x-5x=-2-3x
Since 5-5x>-2-3x for a small X, withholding is shown to theoretically be an evolutionarily stable strategy.
However, interestingly enough, in a 1984 study by G.S. Wilkinson, most vampire bats choose to share in an act of altruism. I would like to explore why they contradict the conclusion of the evolutionary stable strategy as well as the shortcomings of game theory when applied to real-life scenarios.
The vampire bat situation is unique due to repeat encounters, since bats tend to roost together for years. As a result, these bats that they would share/withhold with are not complete strangers. There is a high chance of them meeting again and then perhaps needing blood from them in the future to survive– so they are more inclined to be altruistic in hopes of reciprocation. Furthermore, there is a high risk of dying or harming the population by not sharing, thus, the stakes are a lot higher. It’s also been observed that hungry vampire bats are more likely to receive aid. The fact that not all vampire bats are able to feed each night raises the question whether game theory can even be applied to this situation, since it’s so complicated.
To experiment, I will set a different scenario with higher consequences for withholding that more accurately account for realistic long-term effects. That is, change the payoff for both withholding from 0 to -5 and adjust the others slightly as well.
| BAT B | |||
| SHARE | WITHHOLD | ||
| BAT A | SHARE | -2, -2 | -4, 6 |
| WITHHOLD | 6, -4 | -5, -5 | |
The pure strategy Nash equilibriums are (SHARE, WITHHOLD) and (WITHHOLD, SHARE). There is no dominant strategy for either player. There also exists a Nash equilibrium with mixed strategies, since the players would randomize between (6, -4) and (-4, 6), and neither can increase its expected payoff by playing an alternate strategy. The following is the computation of the payoff for a mixed strategy Nash equilibrium.
Payoff for Bat A
−2(q) − 4(1 − q) = 6q − 5(1 − q)
-2q-4+4q=6q-5+5q
2q-4=11q-5
1=9q
q=1/9
Payoff for Bat B
−2(p) − 6(1 − p) = −4p − 5(1 − p)
p=1/9
These are symmetrical since the table proposed is symmetrical, and demonstrates that both bats would cooperate with the other 1/9 of the time but choose to betray 8/9 of the time to have the highest possible personal payoff.
Despite the modification, it still appears that according to pure game theory statistics, these bats will choose to withhold the majority of the time. Perhaps there are more complicated approaches and methods beyond what was discussed in class, but this displays one of the many limitations of utilizing such a model to apply to real life situations.
Sources:
Wilkinson, Gerald S. “Reciprocal Food Sharing in the vampire bat.” Nature, vol. 308, no. 5955, 1984, pp. 181–184, https://doi.org/10.1038/308181a0.
Metzler, Dirk. Evolutionary Ecology – LMU, evol.bio.lmu.de/_statgen/EvolEcol/ws1718/ess_handout.pdf.
Srivastava, Saniya, and Heidi Zhang. Modeling Altruism in Evolutionary Biology Using Game Theory, math.mit.edu/research/highschool/primes/circle/documents/2021/Srivastava_Zhang.pdf.