Your Turn!

by Kirill Chernomaz and Yen-Sheng Chiang

Overview—

Using the method of genetic algorithm, we attempt to model how individuals volunteer to give presentations in a sequence emerging from the consideration of two aspects: Individuals would like to present as close as possible to their exogenously given preferred positions while trying not to, in order to avoid an unfavorable contrast, present right after a talk whose quality is deemed as better than theirs. We are interested in how individuals actually choose the order of presentation as the strategies evolve over generations.

The Model—

Consider a group of ten students who are going to present their papers. The order of presentation is based on a volunteering basis. For each position, the instructor calls for a volunteer. If there are many hands up, the tie is broken by randomly choosing one from the volunteers. Similarly, if nobody volunteers the instructor chooses a student at random from those who have not yet presented. Each student has a preferred spot of the order, which is exogenously given by the discreet uniform distribution between 1 and 10. The quality of students’ presentations is also predetermined by a continuous uniform distribution: U(0,1). Moreover, students do not want to present if they think the quality of the preceeding presentation is better than theirs. This is captured by the specific payoff function given below.

U(s, x, q, q_(t-1))= -|s-x|*1_(q>q(t-1)) - a|s-x|*1_(q<q(t-1))

where: s is the preferred spot

x is the actual spot

q is the quality of the student’s presentation

q_(t-1) is the quality of the last presentation

a is a punishment factor (if present after a work with better quality)

1 is an index variable

Such a payoff structure inevitably constitutes a trade off—students might pass on their preferred spot because of a better presentation coming before it, or they would choose to present earlier than their preferred spot because of witnessing an inferior presentation and a chance that the following presentation will be better than theirs.

Each student possesses a strategy which is a pair of functions (u(s), v(s)), mapping from his/her preferred spot s. If a student’s preferred spot is s, then he/she will volunteer to present at the u(s)^th spot if the (u(s)-1)^th presentation is worse than his/hers and volunteer to present at the v(s)^th spot if the (v(s)-1)^th presentation is better. We focus on cut-off strategies: if a student is ready to present at the x^th spot after a worse/better presentation and is not chosen due to a tie, he/she will continue to raise his/her hand later on if the presentations continue to be worse/better than his/hers. As an example, consider a strategy that prescribes the following: u(5)=3 and v(5)=8

The example shows that the preferred spot is 5, but the student will jump in if the 2^nd presentation is worse than his/hers. If there is no inferior work till the 7^th spot, the actor will jump at the 8^th spot even if the 7^th presentation is superior.

The evolution of strategies is modeled by a genetic algorithm. Students are agents who participate in 50 sessions of presentations and get an average payoff from them. To produce a new generation of strategies, agents go through a standard process of genetic algorithm: selection, mutation and cross-over. All the genetic operators are applied to the 2 populations of functions: those that map a preferred spot after a better presentation and those that map a preferred spot after a worse one. Each function is represented by a binary string of length 100. Cross-over probability is set to be 0.4, mutation probability is 0.1, and pair-wise tournament is chosen as the selection mechanism. Sequences of 10,000 generations are considered.

Results—

The following reports are based on the averages over the last 100 generations across 10 independent trials. First, we examine how students actually chose the spots in contrast to their preferred ones. The plots below recorded the chosen spots (vertical axis) against the preferred spots (horizontal axis).

Figure 1 Figure 2 Figure 3

If there is no punishment (a=1), it is no different to present before a superior or inferior work. The actual chosen spots will be close to the preferred spots. Figure 1 supports this conjecture. Deviations from the 45-degree-angle line can be attributed to random selections when ties occur. Once punishment is imposed on presenting after a better work, one would expect that students may decide to jump in earlier having observed an inferior work and will not postpone presentation much from their preferred spots after a superior presentation. That is exactly what we deduce from the simulation data. Having observed a superior presentation agents raise their hand around their preferred spots. It is not surprising since the next presentation is likely to be even better and the punishment for deviation is too severe to raise one’s hand before the preferred spot. Instead, it is the spot of presentation after an inferior work that moves further to the front end capturing the effect of punishment. This can be observed in Figure 2 and 3, where as punishment becomes severe the red line (actual chosen spot after a worse presentation) falls down, especially for those students with larger preferred spots (>5). We can obtain the same conclusion by observing the following plots that record the difference between the preferred spots and the chosen spots after an inferior and a superior presentation respectively. From Figure 4, we see increasing differences when punishment cost goes up. However, the differences are much smaller in Figure 5.

Figure 4 Figure 5

Finally, we found for a given preferred spot the distances between the chosen spots after an inferior and a superior presentation are increasing as punishment cost rises. This is especially apparent for cases of larger preferred spots (see Figure 6).

Figure 6

Conclusion and Discussion—

By assumption, each student has their own preferred spots of presentation order. But since the quality of presentation varies and students try not to embarrass themselves by presenting right after a seemingly good work done by others, the preferred spots will not always be chosen. The evolution of strategies, operated by genetic algorithm, shows that students actually chose spots somewhere near or earlier than their preferred spots, depending on the quality of the precious presentation. More interestingly, as the motivation of not presenting after a superior work becomes stronger, students tend to present earlier than their preferred spots if conditions permit (seeing an inferior work).

There are some directions for future exploration. First, in the present model the distributions of preferences of spots and quality of work are both uniform. Future work can test if other distributions or degrees of variation will lead to different results. In addition, in the model students are restricted to take into account the direction of quality comparison only (whether one’s work is better of worse than the last observed presentation). However, absolute levels of quality matter for the probabilities of observing a specific quality of presentation later in a session. For example, given a preferred spot of 5 and quality 0.9 (out of 1) an agent may make different decisions after observing presentations in the 3^rd spot with qualities 0.1 or 0.89. He/she may want to wait in the former case (expecting next presentation to have quality <0.9) but raise the hand in the latter case (expecting next presentation to have quality >0.9). Future work could extend the analysis along this direction. The difficulty is that strategy space becomes increasingly multi-dimensional.

There are many examples of ordering of sequences, such as airplanes waiting to take off or the release of movies. Our model particularly considers the variation of quality and the psychological mechanism where people avoid following a better work. Our simulation shows volunteers will congest in the middle range of spots of the presentation order. One could take this into account to design a more effective mechanism to designate the presenter rather than by a random selection when the peak of volunteers happens. The particular design can point to specific purpose such as increasing an overall better quality of presentation or the maintenance of fairness.

Final Remarks—

The simulation is implemented in JAVA. The code can be obtained upon request from the authors. Please email to: chernomaz.1@osu.edu