Behavioral and Emotional Dynamics of Two People Struggling to Reach Consensus about a Topic on Which They Disagree

We studied the behavioral and emotional dynamics displayed by two people trying to resolve a conflict. 59 groups of two people were asked to talk for 20 minutes to try to reach a consensus about a topic on which they disagreed. The topics were abortion, affirmative action, death penalty, and euthanasia. Behavior data were determined from audio recordings where each second of the conversation was assessed as proself, neutral, or prosocial. We determined the probability density function of the durations of time spent in each behavioral state. These durations were well fit by a stretched exponential distribution, with an exponent, , of approximately 0.3. This indicates that the switching between behavioral states is not a random Markov process, but one where the probability to switch behavioral states decreases with the time already spent in that behavioral state. The degree of this “memory” was stronger in those groups who did not reach a consensus and where the conflict grew more destructive than in those that did. Emotion data were measured by having each person listen to the audio recording and moving a computer mouse to recall their negative or positive emotional valence at each moment in the conversation. We used the Hurst rescaled range analysis and power spectrum to determine the correlations in the fluctuations of the emotional valence. The emotional valence was well described by a random walk whose increments were uncorrelated. Thus, the behavior data demonstrated a “memory” of the duration already spent in a behavioral state while the emotion data fluctuated as a random walk whose steps did not have a “memory” of previous steps. This work demonstrates that statistical analysis, more commonly used to analyze physical phenomena, can also shed interesting light on the dynamics of processes in social psychology and conflict management.


APPENDIX S1 Effective Kinetic Rate Constant
The probability of switching between behavioral states can also be described by the effective kinetic rate constant, k ef f , which is the probability to switch states given that the participant has already remained in that state for a time t. This parameter has proved valuable in analyzing a number of other different systems [1]. The effective kinetic rate constant has also been used in renewal theory, epidemiology and actuarial sciences. For example, k ef f in renewal theory, called age-specific failure rate, is the probability that a light bulb fails to produce light given that it has already produced light for t amount of time.
In actuarial sciences, k ef f is called survival rate and used for life insurance policies. Therefore, we now derive that k ef f from the stretched exponential form used in the previous sections to show how our results here can be related to previous results found in other fields.
The escape probability k ef f is defined as [2] where P (t) is the cumulative probability with the initial conditions of P (t = 0) = 1 and P (t = ∞) = 0.
Once we carry out the derivative, we obtain The probability density function (PDF) and P (t) are related via As we have observed in fitting different forms of the PDFs to the experimental behavior data, the PDF for our data is well represented by the stretched exponential form Integrating equation (6) from t = 0 to t = t leads to The integration on the right hand side of the (7) is not trivial. Therefore, we solve the integral with the help of Maple software and find the analytical solution of (7) as where M(p, q, τ ) is the Whittaker M function, which is related to the confluent hypergeometric function and arises as one of two linearly independent solutions to the Whittaker differential equation [3]. Since the solution is not a simple solution, we first check that the initial conditions for the cummulative probability (8) are satisfied, i.e., P (t = 0) = 1 and P (t = ∞) = 0. The constant B in (7) or (8) is not an arbitrary constant, and can be found using the relation Once we insert (4) in (13) we obtain where Γ(1/α) is the gamma function and defined as [4] Γ Finally, we can summarize our results and put the effective kinetic rate constant in its final form. From equations (2) and (3), k ef f is given by To show why behaviors depend on the memory, let's first plot ln(k ef f ) vs ln(t) for both extreme groups.
The values of A and α in calculating (8) Figure S2. Plots of the effective rate constant for the tractable dyads. The logarithmic graphs of the effective rate constant versus time for all the tractable dyads for A) proself (state 1), B) neutral (state 2), C) prosocial (state 3) and D) the combined data from all three states (state 123). The plots indicate that k ef f is not a constant, as would be expected for a Markov process, but that it decreases with the time t already spent in a state.
Checking the Result for the Case α = 0.5 To justify the general result we have obtained in the previous section, we will re-derive the k ef f (17) for the special case for α = 0.5. We first get B using (15) as Once we take the integral, the cummulative probability will be One can easily check that P (t = 0) = 1 and P (t = ∞) = 0. This form of P (t) looks much simpler than the one given in (8). To see the equivalence of (8) and (27) for the value of α = 0.5, we compute and plot the cummulative probability using (8) and (27)