Team game adaptive dynamics

TL;DR

This work demonstrated the existence of solutions to the adaptive dynamics for the team game and determined their regularity, and identified all stationary solutions and proved that these are precisely the Nash equilibria of the team game.

Abstract

Adaptive dynamics describes a deterministic approximation of the evolution of scalar- and function-valued traits. Applying it to the team game developed by Menden-Deuer and Rowlett [Menden-Deuer & Rowlett 2019], we constructed an evolutionary process in the game. We also refined the adaptive dynamics framework itself to a new level of mathamatical rigor. In our analysis, we demonstrated the existence of solutions to the adaptive dynamics for the team game and determined their regularity. Moreover, we identified all stationary solutions and proved that these are precisely the Nash equilibria of the team game. Numerical examples are provided to highlight the main characteristics of the dynamics. The linearity of the team game results in unstable dynamics; non-stationary solutions oscillate and perturbations of the stationary solutions do not shrink. Instead, a linear type of branching may occur. We finally discuss how to experimentally validate these results. Due to the abstract nature of the team game, our results could be applied to derive implications and predictions in several fields including biology, sports, and finance.

Figures (13)

• Figure 1: The strategies of the team game are distributions of competitive ability. For the function-valued game, in the interval $(a,b)$ is the number (or percentage) of individuals with competitive ability between $a$ and $b$.
• Figure 2: Projection onto the set of functions constrained by $w(f)=0$. The selection gradient belongs to a subspace of $L^2[0,1]$ and the normal component, $P(\nabla E)$, is removed. Thus, $w((1-P)\nabla E)=0.$
• Figure 3: A function $f$ with $\mathop{\mathrm{MCA}}\nolimits(f)>\frac{1}{2}.$
• Figure 4: At the initial condition (\ref{['eq:impossible']}) with $r=3/4$, the adaptive dynamics is trying to move the function away from the space of strategies by breaking the non-negativity constraint.
• Figure 5: In this numerical example, $\mathop{\mathrm{MCA}}\nolimits(\bm{y}_0)<\frac{1}{2}$ and the sum of the components, $\sum_i y_i(t),$ is growing until $\mathop{\mathrm{MCA}}\nolimits(\bm{y})=\frac{1}{2}$.

Theorems & Definitions (47)

• Theorem 2.1: See Theorem 1 in menden2021biodiversity
• Theorem 2.2: See Theorem 1 in menden2021biodiversity
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof