Bluffing promotes overconfidence on social networks

Abstract

The overconfidence, a well-established bias, in fact leads to unrealistic expectations or faulty assessment. So it remains puzzling why such psychology of self-deception is stabilized in human society. To investigate this problem, we draw lessons from evolutionary game theory which provides a theoretical framework to address the subtleties of cooperation among selfish individuals. Here we propose a spatial resource competition model showing that, counter-intuitively, moderate values rather than large values of resource-to-cost ratio boost overconfidence level most effectively. In contrast to theoretical results in infinite well-mixed populations, network plays a role both as a "catalyst" and a "depressant" in the spreading of overconfidence, especially when resource-to-cost ratio is in a certain range. Moreover, when bluffing is taken into consideration, overconfidence evolves to a higher level to counteract its detrimental effect, which may well explain the prosperity of this "erroneous" psychology.

Figure 1

(a) Stable fraction of overconfident individuals f_O in structured population and (b) equilibrium frequency of overconfident individuals x* in infinite well-mixed population as a function of resource-to-cost ratio r/c for different values of α and β. Data presented in Panel (a) are obtained by means of Monte Carlo simulations, while in Panel (b) are obtained by means of extended replicator dynamic (see Methods for details). Hollow symbols in Panel (a) and dash line in Panel (b) correspond to cases where bluffing does not exist (β = 0). Other parameters: f_IO = 0.5, f_B = 0.5.

Figure 2. The 3-D plots showing the stationary fraction of overconfident individuals f_O in dependence on both α and β for (a) r/c = 1.5 and (b) r/c = 3.

The minimum f_O is about 0.77 in Panel (a) showing that overconfidence is always favored under moderate values of r/c. When r/c is high, f_O becomes more strongly dependent on the combination of α and β. The maximum f_O is reached when both α and β are high, indicating that higher values of overconfidence is evolved in response to higher bluffing activities.

Figure 3. Influence of initial overconfidence level f_IO and initial bluffing level f_B on stationary fraction of overconfident individuals f_O in the contour form.

Parameters are: α = 0.4, β = 0.3, and r/c = 2.5. Similar results can be obtained whenever α > β and r/c > 2.0. Overall, larger f_IO corresponds to higher f_O. For any given value of f_IO, increasing f_B further promotes f_O, showing that overconfidence is favored when bluffing exists.

Figure 4. Typical snapshots of spatial patterns formed by different types of individuals under initial overconfidence levels f_IO = 0.1 ((a)–(c)) and f_IO = 0.3 ((d)–(f)) at different time steps.

Overconfident ones are colored blue (high real ability) and green (middle and low real ability). Unbiased ones are colored yellow (high real ability) and red (middle and low real ability). The stationary fractions of overconfident ones are 42.5% for f_IO = 0.1 and 62.6% for f_IO = 0.3. In structured population, high real ability but unbiased individuals form clusters to resist the invasion by overconfident ones, explaining the survival of unbiased ones even with high r/c. The size of the square lattices is 100 × 100. Other parameters: α = 0.4, β = 0.3, and r/c = 3.

Figure 5. Comparison of the payoffs of individuals with different real abilities.

(a) The 3-D plot showing the normalized average accumulated payoffs of individuals with high (H, crimson), middle (M, light green), and low(L, blue) real abilities respectively, as a function of f_IO and f_B. H ranges from 0.9350 to 1; M ranges from 0.4859 to 0.5521; L ranges from 0.0443 to 0.1065. (b) Average payoff ratio between the individuals who bluff and those who do not (R = Payof f_bluff/Payof f_non–bluff) as a function of f_IO and f_B, when their real abilities are low (R^L), middle (R^M), and high(R^H) respectively. R^L ranges from 1.03 to 8.62; R^M ranges from 1.01 to 1.634; R^H ranges from 1.002 to 1.322. Data point is obtained by averaging over a period of 10000 time steps, and we have checked that longer time period does not qualitatively change the results. Other parameters: α = 0.4, β = 0.3, and r/c = 3.

Figure 6. Stable fractions of overconfident individuals f_O and bluffing individuals f_B as a function of r/c for different values of α and β, when players can learn both the overconfidence state and the bluffing state of others.

In Panel (a), α = 0.4 and β = 0.3. Bluffing prevails once r/c exceeds the critical value 2.0. Overconfidence does not drop to lower level with increasing r/c and remains above 0.89. In Panel (b), α = 0.7 and β = 0.6. Bluffing almost dominate the whole range of r/c. Overconfidence stays above 0.98 for r/c > 0.5. Other parameters: f_IO = 0.5, and f_IB = 0.5.