Sharks and Suckers: Overview
Sharks and Suckers
In order to test the robustness of Simpson's Effect, we experimented with some computer games. The first game is a variation on John Conway's "Game of Life". The computer screen is divided into a 30 by 30 array of cells. Each cell can be empty or filled with either a Shark or a Sucker. Suckers are shown in red, Sharks in blue. The mouse can be used to set up an initial field of Suckers and Sharks or the program can provide a random initial field.
Given such an initial field, the program determines the survival and spread of Sharks and Suckers for generation after generation. For each generation, the program applies rules which determine whether each cell will be a Shark, a Sucker or empty in the next generation. For each cell, A, the rules are as follows:
Cell A is empty: "Birth or barrenness"
Rule 1: "Too crowded":
If cell A is empty, and three or more adjoining cells are full, then it stays empty.
Rule 2: "Too lonely"
If cell A is empty, and fewer than two adjoining cells are full, then it stays empty.
Rule 3: "Sharks rule":
If cell A is empty, and just two adjoining cells, B and C, are full, then:
If both B and C are Suckers, A becomes a Sucker.
If just one of B and C is a Shark, A becomes a Shark.
If both of B and C are Sharks, A stays empty.
Cell A is full: "Death or survival"
Rule 4: "Too crowded":
If cell A is full, and three or more adjoining cells are full, then it becomes empty.
Rule 5: "Too lonely":
If cell A is full, and no adjoining cells are full, then it becomes empty.
Rule 6: "Status quo":
If cell A is full, and just one adjoining cell is full, then:
If A is a Sucker, A stays a Sucker.
If A is a Shark, A stays a Shark.
Rule 7: "Sharks rule":
If cell A is full and just two adjoining cells B and C are full, then:
If A is a Sucker and both B and C are Suckers, then A stays a Sucker.
If A is a Sucker and either B or C (or both) are Sharks, then A becomes empty.
If A is a Shark, and both B and C are Sharks, then A becomes empty.
If A is a Shark, and either B or C (or both) are Suckers, then A stays a Shark.
Notice that the "Too crowded", "Too lonely" and "Status quo" rules apply in exactly the same way to both Sharks and Suckers. Rules 3 and 7 on the other hand, give Sharks a local advantage over Suckers. An empty cell can become a Sucker only if it has exactly two neighbours which are both Suckers. A Sucker will be killed if one or more of its neighbours are Sharks, but a Shark can only be killed by two neighbouring Sharks. In every case, a given cell would never be worse off if it were a Shark and never better off if it were a Sucker. Despite this, we discovered a robust tendency for Suckers to survive and even prosper.
Suppose we begin with an initial randomized field consisting of equal numbers of Sharks and Suckers and set the program running. What happens?
Try
it out for yourself. Click on the 'Step' button a few times
to see the first few generations. Click on 'Start' to set
the program running through the generations. Click on 'Stop'
to halt the program. Click on 'Random' to generate a new
random field and start again.
The figures show the proportion
of Sharks and Suckers that have survived from the initial population.
After the first generation, the number of Sharks and Suckers drops quite dramatically, due to overcrowding. The number of Suckers then begins to increase, while the number of Sharks starts to level off. We end up with more Suckers than Sharks in the population and a greater percentage of the initial population of Suckers surviving than Sharks.
Suckers do not always do better than Sharks. If we have few Sharks and lots of Suckers in the initial population, then Shark survival rates will be higher. Sharks do best in an environment where there are few competing Sharks to contend with and lots of Suckers.
Try it out for yourself. Click on the 'Step' button to a few times to see the first few generations and then click on 'Start' to set the program running. Click on 'Stop' to halt the program. Click on 'Random' to generate a new random field and start again. This grid is set up to produce random fields consisting of 10% Sharks and 90% Suckers.
Sharks also do better than Suckers when there are lots of Sharks and few Suckers in the initial population, because such an initial ratio tends to drive the small population of Suckers extinct very quickly. But since life is so hard for everybody in such an environment, sometimes all of the Sharks will become extinct too.
Try it out for yourself. Click on the 'Step' button a few times to see the first few generations. You will most likely arrive at a static state after just four of five steps. Click on 'Random' to generate a new random field and start again. This grid is set up to produce random fields consisting of 90% Sharks and 10% Suckers.
Figure 1 shows how survival rates for both Sharks and Suckers decreases as the percentage of Sharks in the population increases. The more Sharks there are, the worse everybody does. These figures were obtained by generating random fields consisting of each proportion of Sharks from 5% to 100%. Survival rates for Sharks and Suckers after one generation were then computed. The final figures are the average of 50 such random trials for each proportion of Sharks.
Figure
1
In the long-run however, survival rates for Suckers are often better than that for Sharks. Figure 2 shows average survival rates for Sharks and Suckers after 1000 generations, for different initial ratios of Sharks to Suckers (100 trials for each initial ratio). The overall average survival rate for Sharks after 1000 generations was 29%. For Suckers the average was 38%. Notice that Sharks do best in the extreme cases where we have lots of Sharks and few Suckers or lots of Suckers and few Sharks. But in most cases, Suckers do better in the long run than Sharks.
Figure 2
The success of Suckers is due to a clustering effect. Suppose we start with an initial randomized field of 50% Sharks and 50% Suckers. Sharks and Suckers are mixed together fairly evenly at first. But after the first generation, clusters of Sharks with Sharks and Suckers with Suckers begin to appear.
Once these clusters have appeared, Suckers begin to gain ground. The clusters of Sharks remain stable, while the clusters of Suckers start to expand. Sucker survival rates are now consistently higher than Shark survival rates, even after many generations.
You can see this clustering effect for yourself on the grid below. Click on 'Random' to create an initial random field of 50% Sharks and 50% Suckers. Now click on the "Step' button a few times and you should see clusters of Sharks and Suckers begin to appear. Continue to click on 'Step' and the Sucker clusters will then start to expand, while the Shark clusters remain fairly stable.
Why does this clustering occur? Once a Sucker cluster has formed, it will remain a Sucker cluster. Likewise, once a Shark cluster has formed it will remain a Shark cluster. On the other hand, clusters of Sharks with Suckers are unstable and will either become extinct or turn into either a Shark or Sucker cluster.
Try this out for yourself on the grid below. Click once in a cell to turn it into a Sucker. Click again to turn it into a Shark. Clicking a third time will return the cell to empty. Try setting up a cluster of two or three Suckers next to each other, and then click on "Step' to see what happens. Repeat the experiment for a cluster of two or three Sharks. Then see what happens with clusters of cells containing a mixture of Sharks and Suckers - two Sharks and two Suckers, or one Shark and three Suckers, for example.
So, Shark clusters and Sucker clusters are stable, but mixtures of Sharks and Suckers are unstable - either they become extinct or turn into Shark-Shark or Sucker-Sucker clusters. Because of this robust clustering effect, most of the Sharks in a population will be next to other Sharks and most of the Suckers will be next to other Suckers.
This means that most of the Sharks will be in areas where survival rates are low and most of the Suckers will be in areas where survival rates are higher, since survival rates decrease the more Sharks there are in a population and increase the more Suckers there are. (See Figure 1).
Sharks may do better than Suckers locally, but because most of the Sharks are next to other Sharks and most of the Suckers are next to other Suckers, overall survival rates are higher for Suckers. Thus, the local advantages of Sharks over Suckers does not add up to a global advantage because of the operation of a robust Simpson's effect.