© Zyanya Santuario, MATH+
Author: Max Klimm (TU Berlin)
Project: AA3-18 Evolution Processes for Populations and Economic Agents
Challenge
Santa sports a sleigh drawn by two reindeer that are selected at random from a large population of reindeer at the North Pole. Every reindeer chosen for sleigh duty is of one of two types: it either pulls hard or it barely pulls. If both reindeer pull hard, the sleigh flies gracefully across the sky, everybody is content, and both reindeer receive 4 piles of hay each. If one reindeer pulls hard while the other barely pulls, the sleigh still flies well, but it is very exhausting for the hard-pulling reindeer, who is too tired to eat and, hence, receives no hay while the other reindeer, as a freerider, receives 5 piles of hay. If both reindeer barely pull, the sleigh moves very slowly, Santa is dissatisfied, and the reindeer receive only the minimum amount of 1 pile of hay each.
We are interested in understanding how the proportion of the hard-pulling reindeer in the population evolves using what is called replicator dynamics. Denoting this proportion at time t by x(t), we have that a reindeer randomly drawn at time t is hard-pulling with probability x(t), and barely pulling with probability 1-x(t). With this observation, it is easy to compute that a hard-pulling reindeer can expect
f_\text{h}(t) = 4 \cdot x(t) + 0 \cdot (1- x(t))
piles of hay when recruited for sleigh duty at time t. A similar equation can be obtained for the function f_\text{b}(t) describing the expected number of piles a barely pulling reindeer receives when recruited.
The central assumption in replicator dynamics is that the increase in the proportion of a type of reindeer is proportional to the excess in hay compared to an average reindeer from the population, if recruited for sleigh duty. More specifically, for the instantaneous rate of change x'(t) (cf. remark 2) of the proportion of hard-pulling reindeer at time t we have the equation
x'(t) = x(t) \biggl( f_\text{h}(t) - \bigl(x(t)f_\text{h}(t) + (1-x(t))f_\text{b}(t)\bigr)\biggr).
Which of the following statements is true? (The two remarks are not considered as statements.)
Remark 1: For a given point in time t_0 and a value c there is exactly one function x(t) satisfying x(t_0) = c as well as the above equation.
Remark 2: The instantaneous rate of change (also called derivative) can be interpreted as follows:
- If x'(t)=0 at all times t in a certain interval, then x(t) is constant in this interval.
- If x'(t) > 0, there exists a time interval around t where x(t) is increasing.
- Similarly, if x'(t) < 0, there exists a time interval around t where x(t) is decreasing.
For a linear function described by f(t)=at+b for real numbers a, b, at all times t, the instantaneous rate of change f'(t) is equal to a, i.e. equal to the slope of the function.
Possible Answers
- The formula for the expected number of piles of hay a barely pulling reindeer receives at time t when recruited is
f_{\text{b}}(t) = 5 \cdot (1-x(t)) + 1\cdot x(t).
- When at time 0, all reindeer are hard-pulling, at some time t, not all reindeer will be hard-pulling.
- When at time 0, all reindeer are barely pulling, it will stay like this forever.
- When at time 0, there is an even split of hard-pulling and barely pulling reindeer, it will stay like this forever.
- The proportion of hard-pulling reindeer at time 1 is always the same, no matter what the initial proportion of hard-pulling reindeer at time 0 is.
- For some proportion of hard-pulling reindeer at time 0, the proportion of hard-pulling reindeer is increasing within a certain period of time.
- Santa can choose an initial proportion x(0) of hard-pulling reindeer time 0 with 0\leq x(0)\leq 1/2, so that the proportion of hard-pulling reindeer stays above 1/4 for all times.
- Suppose Santa changes his feeding practice so that in every scenario every reindeer receives twice as much hay as before. This does not influence the dynamic of the proportion, no matter the initial proportion at time 0.
- Suppose Santa changes his feeding practice so that every reindeer receives one additional pile of hay. This influences the dynamic of the proportion for some initial proportion.
- Suppose Santa changes his feeding practice so that every reindeer always obtains 1 pile of hay no matter the sleighing performance. Then, for all intial proportions at time 0, after some time, all reindeer will be barely pulling.
Project Reference
The replicator dynamics is an example of a question studied in the field of evolutionary game theory. This area studies dynamical systems that describe both the dynamics of evolutionary forces that describe the development of different genotypes or phenotypes of a species, and the dynamics of different behaviors by economic actors. The main objects of study of the project AA3-18 are mathematical processes that describe both how different genotypes or different economic behavioral patterns emerge over time. This research area, where similar mathematical models are used to describe seemingly unrelated processes in two disconnected application domains in biology and economics, is an interesting example of the power of mathematics, aiming for a general understanding of fundamental processes that may have completely different interpretations, depending on the application in question.