# Improving Computational Efficiency in Crowded Task Allocation Games with Coupled Constraints
## 5 Results and Discussion
### Number of Confirmations  
In this section, the effects of variation in number of confirmations on the global score and computation times are examined. Simulation of an operation setting similar to the example in Table 5 but with varying number of confirmations at 10, 50, 100, 500, and 1000, respectively, were considered and the results are shown in Figure 4 with the means and standard deviations for the various aspects of the simulation presented in Tables 7-9. The observed global scores are generally similar at the selected levels of confirmations and the standard deviation generally decreases with increasing number of confirmations, providing a tighter bound to the range of scores. Intuitively, the decrease in standard deviation is in relation to the increasing probabilities of the solution being a Nash equilibrium (refer to Appendix A). With only 10 confirmations, the mean global score is recognizably lower, and the range is also obviously greater as compared to the global scores for games with higher levels of confirmation. This lower levels of performance can be attributed to the extremity of the constraints placed on the agents' action set. With a sample size of 20 and 10 confirmations, the agents are expected to be exposed to a maximum of 191 unique actions which barely covers approximately 5% of all possible actions. The exposure of an agent refers to the number of actions that it has seen from the moment when it last changed its action. As such, it is plausible that the agents do not have sufficient understanding of the full range of possible actions, leading to agreements that are less efficient. However, it should be noted that such inefficiencies lead to a global score which is merely 3% lower as seen in Figure 5. Indeed, it can be argued that agents do not need to understand the entirety of their action sets to arrive at reasonable solutions. With 50 confirmations, the maximum exposure is at 951 unique actions, or approximately 26% of all possible actions. Yet, mean global scores comparable to those at higher levels of exposure are observed. With 1000 confirmations, it is highly likely that every agent has seen its entire possible action set since there are only 3610 possible actions while it is exposed to 19,001 non-unique actions. Interestingly, the maximum scores achieved for each category are similar with a value of 259.862 except for simulations with only 10 confirmations having a slightly lower maximum of 259.791. This phenomenon will be amplified in dense task allocation settings since most tasks will be closely grouped together and most action sequences will provide similar levels of reward. As such, most Nash equilibria will provide near-optimal solutions and it is, therefore, unnecessary for agents to have a complete understanding of their possible actions since emphasis is not placed on achieving the solution with an optimal global score or the Nash equilibrium with the maximum global score. When constraints are provided at every iteration, agents have lesser numbers of choices to make at each iteration, allowing them to arrive at a general agreement more quickly and easily as evident in the short computation times for games with only 10 confirmations. Improvements to the agreement are then made as the agents continue to explore the possible action space for each iteration. However, it is necessary to note that increasing the confirmations do not necessarily lead to increase in global scores for every play. Recall that in DSA, progression is stochastic and hence the Nash equilibrium obtained for every play may differ and therefore, there is a non-zero probability of reaching a Nash equilibrium that has the lowest possible score despite considering a high number of confirmations in one play while having a Nash solution with a higher score when considering lower number of confirmations. Another observation made, from Figure 6, is that the computation times increase with the number of confirmations as expected.