On a daily basis, we face problems and situations that should be evaluated and solved, and we are challenged to understand different perspectives to think about these situations. Most of us are building our cognitive thinking based on previous similar situations or experiences. However, this may not guarantee a better solution for a problem, as our decision may be affected by emotions, non-prioritized facts, or other external influences that reflect on the final decision. Therefore, critical thinking tends to build a rational, open-mined process that depends on information and empirical evidence.
The National Council for Excellence in Critical Thinking defines critical thinking as an “intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action.” The process tends to help us judge and evaluate situations based on understanding the related data, analyze it, build a clear understanding of the problem, choose the proper solution, and take actions based on the established solution.
The critical thinking process prevents our minds from jumping directly to conclusions. Instead, it guides the mind through logical steps that tend to widen the range of perspectives, accept findings, put aside personal biases, and consider reasonable possibilities. This can be achieved through six steps: knowledge, comprehension, application, analyze, synthesis, and take action. Below is a brief description of each step and how to implement them.
Step 1: Knowledge
For every problem, clear vision puts us on the right path to solve it. This step identifies the argument or the problem that needs to be solved. Questions should be asked to acquire a deep understanding about the problem. In some cases, there is no actual problem, thus no need to move forward with other steps in the critical thinking model. The questions in this stage should be open-ended to allow the chance to discuss and explore main reasons. At this stage, two main questions need to be addressed: What is the problem? And why do we need to solve it?
Step 2: Comprehension
Once the problem is identified, the next step is to understand the situation and the facts aligned with it. The data is collected about the problem using any of the research methods that can be adopted depending on the problem, the type of the data available, and the deadline required to solve it.
Step 3: Application
This step continues the previous one to complete the understanding of different facts and resources required to solve the problem by building a linkage between the information and resources. Mind maps can be used to analyze the situation, build a relation between it and the core problem, and determine the best way to move forward.
Step 4: Analyze
Once the information is collected and linkages are built between it the main problems, the situation is analyzed in order to identify the situation, the strong points, the weak points, and the challenges faced while solving the problem. The priorities are set for the main causes and determine how they can be addressed in the solution. One of the commonly used tools that can be deployed to analyze the problem and the circumstances around it is the cause effect diagram, which divides the problem from its causes and aims to identify the different causes and categorize them based on their type and impact on the problem.
Step 5: Synthesis
In this stage, once the problem is fully analyzed and all the related information is considered, a decision should be formed about how to solve the problem and the initial routes to follow to take this decision into action. If there are number of solutions, they should be evaluated and prioritized in order to find the most advantageous solution. One of the tools that contribute choosing the problem solution is the SWOT analysis that tends to identify the solution’s strength, weakness, opportunity, and threats.
Step 6: Take Action
The final step is to build an evaluation about the problem that can be put into action. The result of critical thinking should be transferred into action steps. If the decision involves a specific project or team, a plan of action could be implemented to ensure that the solution is adopted and executed as planned.
The critical thinking method can be adopted to replace emotions and perusal biases when trying to think about a situation or a problem. The time for adopting critical thinking varies based on the problem; it may take few minutes to number of days. The advantage of deploying critical thinking is that it contributes to widening our perspectives about situations and broadening our thinking possibilities. However, these steps should be translated into a plan of action that ensures that the decided resolution is well achieved and integrated between all the involved bodies.
For each simulation done for fixed S and R, we store the following quantities: , the mean of player’s collaboration probabilities pi in one group, , the mean players’ fitness πi in one group, and ∑max, the capacity ∑ reached by a given group after 2000 rounds (i.e., the final value of ∑). Each quantity is stored after averaging over 20 games. Our first result, as expected by evolutionary nature of our algorithm, is that for any fixed R, the final results for all groups of given size S are the same. So for example, all n = 4 groups with S = 25 players ultimately reach the same , and ∑max for any given R. This means it suffices to consider only one group of size S. Therefore, in what follows we represent our findings via final values of , and ∑max as function of R and S.
We begin by examining , the mean collaboration probability in a group of size S which is undertaking the task of simplicity R. The results are shown in Fig. 1 as a surfaceplot. For very hard problems (small R), players have no preference on collaborating, so regardless of S. In contrast, for very easy tasks, players in small groups strongly tend to collaborate, while players in big groups tend to work individually.
On the other hand, for small groups the tendency to collaborate increases with increasing task simplicity, while for big groups this tendency actually decreases. In between, for groups of size roughly S = 50, the collaboration probability is basically constant with R, and hence independent on the task that such group is facing.
To examine these trends in more detail, we show in Fig. 2 the sequence of profiles of for constant R (left) and for constant S (right).
Looking at the left plot, for the most difficult tasks the is indeed constant with the group size, while for simpler tasks it exhibits a clear trend: smaller groups tend to collaborate more intensely, while players in the larger groups are predominantly individualists. Groups with approximately 50 players seem always to display regardless of R. This is also confirmed by the plots on right in Fig. 2: all curves with exception of S = 100 indicate that collaboration propensity increases with task simplicity. This increase becomes more and more moderate with the group size, so that finally, for S = 100 we find a clear negative trend: collaboration propensity decreases with the task simplicity. In between we find an almost constant curve for roughly S = 50.
We interpret these findings as follows. When the task is extremely difficult, it is hard to solve it collectively as well as individually. Thus, in such cases there is no particular preference for either approach, regardless of the group size. As the task becomes gradually easier, it is less beneficial for players in small groups to work individually, since this quickly reduces the size of the collaborating subgroup, ultimately making it so small that there is little or no fitness gained from it for anybody. This in turn means that by collaborating a player is more safe to gain at least some fitness. In contrast, easier problems encourage the players in large groups to defect collaboration, since the collaborating subgroup is always fairly large and contributes significant fitness. These circumstances allow for more “individually oriented” players to seek their luck by defecting, which they can do safely since at least some fitness is practically guaranteed. This feature captures the concept of free-riders discussed in the Introduction. These players contribute no knowledge of their own to the collective, but at the same time manage to benefit from the novelties reached collectively by the rest of the group. The equilibrium case of groups with approximately S = 50 players is the optimal system where both collaboration and individualism are equally beneficial for task of any difficulty. Such group has a good balance between its collectivist part which slowly but surely generates new knowledge, and its individualists who are still on average equally fit as collectivists.
To support these observations with the results for the mean fitness and the maximal capacity ∑max, we show in Fig. 3 the surfaceplots for these two quantities. Both surfaceplots display similar features: as expected, the biggest values are found for the simplest tasks when faced by the largest groups.
Smallest and ∑max correspond to the most difficult of tasks when approached by the smallest groups. The group with S = 50 players optimizes the gain in both quantities, and in fact performs better than the group with S = 100 players. This resonates with the earlier observation that this group has the best balance between collectivism and individualism. Indeed, by working excessively in isolation and looking too eagerly for free rides, the players in S = 100 group at the end achieve lesser total fitness, despite their larger size which could have made them all much better fit. It is easy to conclude that this trend would actually further increase with even larger groups, leaving S = 50 as the only optimal case. This confirms that the knowledge accumulation in too large groups is actually slower than in groups of certain moderate size, when the balanced tendencies for collective and individual approach. On the other hand, one could argue that in a real-life scenario this might in fact be in the long-run beneficial for the society. Namely, the individually gained unique knowledge, that could not have been gained via pure collectivism, might eventually be shared with the rest of the community, and hence available to everyone. By running the simulations with a number of players different from N = 100, we confirmed that the equilibrium group size is indeed robust to this, and is always approximately N/2.
We now show that the above numerical findings can be justified via analytical treatment, relying on standard methods from statistical physics. Let us consider the average fitness for a given player, depending on the player’s strategy and the strategy of the opponents. We call j the player’s group and αj the collectivists density in this group. The average fitness of a collectivist in j at time t is then
where is the capacity reached at time t in group j, and indicate averaging. On its turn, individualist (defector) average fitness is
where, by definition, is the mean value of the integer random number . Of course, within the same group j it is always , so that in absence of competition between the groups (i.e., S = N, n = 1) individualism is the best strategy, as confirmed by the simulations. On the other hand, for n ≥ 2, let us think about a group l made up by only collectivists, compared with a group j with just one individualist (we consider this scenario as it the best situation for a defector). In this case, l's collectivist have a better fitness than j's defector if it results
Here we distinguish two cases, the large group size , and the opposite regime where S ~ 1. We treat these two cases separately below.
In the large group size limit, the equation (3) reduces to
Equation (4) implies that in order to have the best fitness, collectivist in l must have reached at time t a capacity such that
This clearly implies that δ increases monotonically with R, going from 0 to 11/5. This means that, with increasing R cooperative groups have to reach a higher capacity to thrive, but this is balanced by the higher ease to solve the tasks: the combined effect is that, given the group size, the final cooperation level does not depend strongly on R, as shown in Fig. 1. On the other hand, increasing R enhances the global fitness, because it is easier to solve the tasks and, for a cooperative group, reach a higher capacity with respect to the others, as depicted in equation (5). This behavior is confirmed in Fig. 3, where an increase for increasing R is observable in the average agent fitness. The narrow tilt change for can be understood considering that when R becomes smaller than R* = 0.5, also δ gets smaller than , that is, there is practically no need to reach a higher capacity for cooperators to thrive, lowering the global fitness.
In the limit of S approaching to 1, equation (3) reduces to
relation automatically satisfied for , implying that as a group solves the second task, cooperation becomes the best strategy (in other words, free-riding is no longer convenient). This is also confirmed in Fig. 1, where it is clear that for small group size cooperation is in general enhanced, and increases more with increasing R. This analytical reasoning hence entirely confirms our numerical findings.