Abstract
We study a class of asymmetric games with compact Polish strategy sets and provide sufficient conditions for the stability and convergence of profiles under the infinite-dimensional replicator dynamics on such games. We apply these results to analyze the dynamic behavior of the Cournot duopoly with different pricing mechanisms, the rope-pulling game, and a game with a Nash equilibrium profile consisting of uniform distributions. Further, we prove that the set of all Gaussian profiles remains invariant under the replicator dynamics on a large class of quadratic games. Moreover, we study the dynamics restricted to the set of Gaussian profiles, both analytically and numerically.
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All numerical results in the paper are obtained via simulations using parameters mentioned in the paper. No part of the paper uses any external data set.
Notes
A compact invariant set with no proper attractors. For details, see Prop. 5.3 in [5].
A population state which is stable with respect to perturbations caused by a small proportion of invading mutants.
The readers who are not familiar with the notion of a Polish space may find it beneficial to visualize an Euclidean space, without losing the essence of the main results in the paper.
By strong solution, we mean a solution \(\Phi _t\) taking values in the Banach space \(\mathbb {M}(S_1)\times \mathbb {M}(S_2)\) equipped with the total variation norm.
For example, the Prohorov metric; see Shiryaev [49].
The support supp(P) is the complement of the largest open set having P-measure zero.
Basar and Olsder discuss this game in the context of differential games. We restrict the strategy sets to constant functions of time and treat it as a game on \(S^1\).
That is, \(U_{[0,1]}(A)=\int \limits _A 1\textrm{d}x,\;A\in \mathcal {B}([0,1])\).
Equivalence of the two dynamical systems is as described in Lemma 5.
By \(P\sim N(m,v)\), we mean that \(P(A)=\int \nolimits _A \frac{1}{\sqrt{2\pi v}}e^{-\frac{(x-m)^2}{2v}}\textrm{d}x,\;A\in \mathcal {B}(\mathbb {R})\).
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The authors would like to express their gratitude to the editor and two anonymous referees for their constructive feedback that helped improve the manuscript. The first author would like to thank Chinmay Ajay Tamhankar for the helpful discussions.
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Appendix A Proof of Theorem 10: Bounding \(k_1\) by an Integrable Function
Appendix A Proof of Theorem 10: Bounding \(k_1\) by an Integrable Function
We now bound \(k_1\) by a \(P_t\)-integrable function. Using Eq. (27), we obtain
Substituting this in Eq. (26), we get
We now define
Observe that
From Eqs. (A2) and (A4), we get
Note that, for any \(C^2\)-function \(f:\mathbb {R\xrightarrow []{}\mathbb {R}}\) and \(h>0\), we have
Since \(i_1\) is infinitely differentiable in the variable h, from Eqs. (A5) and (A6), we get
Since \(0\le u\le h\), we obtain
We now bound the integrand appearing in the inequality (A7). Note that
Since \(m_i\) and \(v_i\), \(i=1,2\) are \(C^1\)-functions, these functions and their derivatives are bounded on \([t,t+h]\). Hence, we have the following bounds:
for some \(A_1>0\). From Eq. (13), it follows that \(\sigma _1(\cdot ,P_t,Q_t)\) is a quadratic function in the variable x and consequently \(\sigma _1^2(\cdot ,P_{t+u},Q_{t+u})\) is a fourth-degree polynomial in the variable x. Using the fact that the trajectories are \(C^1\), \(\sigma _1^2\) can be bounded as follows:
for some \(A_2>0\). From Eq. (A8), we get
where \(A=A_1+A_2\). We next bound \(i_1\) in the expression (A9). From Eqs. (13), (A3) and the fact that \(a_1<0\), we obtain
Using the fact that \(m_i\) and \(v_i\), \(i=1,2\), being continuous, are bounded on the compact interval \([t,t+h]\) , we get
for some B, \(C>0\). Combining Eqs. (A9) and (A10), we obtain
From Eqs. (A7) and (A11), we get
This yields the necessary \(P_t\)-integrable upper bound for \(k_1\).
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Lewis, K.D., Shaiju, A.J. Asymmetric Replicator Dynamics on Polish Spaces: Invariance, Stability, and Convergence. Dyn Games Appl (2023). https://doi.org/10.1007/s13235-023-00546-3
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DOI: https://doi.org/10.1007/s13235-023-00546-3
Keywords
- Evolutionary game theory
- Replicator dynamics
- Asymmetric games with continuous strategy sets
- Evolutionary stability
- Dynamic stability