Abstract
An interior convergent path in an evolutionary dynamic can be seen as a sequence of perturbed quasi-equilibria to refine an equilibrium. In this note, we investigate a variety of conditions to establish the connection between interior convergence in regular payoff monotone selections and versions of proper equilibrium and use the connection for equilibrium selection.
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- 1.
Hendon et al. (1996) prove that the limit state under a fictitious play is a sequential equilibrium in an extensive form game. They consider two kinds of fictitious play in a sequential-move game and the theorem applies to both. Local fictitious play is the one played by “agents” in an agent-normal form, while sequential fictitious play is the one played in a normal form of the sequential-move game.
- 2.
The assumption of unit mass is made just for notational simplicity. We could easily extend the model and the results to general cases where different populations have different masses.
- 3.
For a finite set \(\mathcal{Z} =\{ 1,\ldots,Z\}\), we define \(\varDelta \mathcal{Z}\) as \(\varDelta \mathcal{Z}:=\{ (\rho _{1},\ldots,\rho _{Z}) \in {[0, 1]}^{Z}\vert \sum _{z\in \mathcal{Z}}\rho _{z} = 1\}\), i.e. the set of all probability distributions on \(\mathcal{Z}\).
- 4.
A bold letter represents a column vector. Precisely x is a column vector \((x_{1}^{1},\ldots,\) \(x_{{S}^{1}}^{1},x_{1}^{2},\ldots,\) \(x_{{S}^{2}}^{2},\ldots,\) \(x_{1}^{P},\ldots,x_{{S}^{P}}^{P})\).
- 5.
Notice that a symmetric game \(\boldsymbol{\varPi }\in {\mathbb{R}}^{S\times S}\) with the pure strategy set \(\mathcal{S}\) is expressed as a general two-player game as \(U_{s\hat{s}}^{1}:=\varPi _{s\hat{s}}\) and \(U_{s\hat{s}}^{2} =\varPi _{\hat{s}s}\) with \({\mathcal{S}}^{1} = {\mathcal{S}}^{2}:= \mathcal{S}\). That is, it is a bimatrix game with \(({\mathbf{U}}^{1},{\mathbf{U}}^{2}) = (\boldsymbol{\varPi },\boldsymbol{\varPi }^{\prime})\).
- 6.
Cressman (2003, Definition 2.3.2) defines uniform monotonicity by imposing ∃K ≥ 1 such that \(K\vert F_{s}^{p}(\mathbf{x}(t)) - F_{\hat{s}}^{p}(\mathbf{x}(t))\vert \geq \vert \dot{x}_{s}^{p}(t)/x_{s}^{p}(t) -\dot{ x}_{\hat{s}}^{p}(t)/x_{\hat{s}}^{p}(t)\vert \geq {K}^{-1}\vert F_{s}^{p}(\mathbf{x}(t)) - F_{\hat{s}}^{p}(\mathbf{x}(t))\vert \). This implies our regularity (PM3).
- 7.
\(\mathbf{e}_{a} = (e_{ab})_{b=1}^{n} \in {\mathbb{R}}^{n}\) is a basis vector in \({\mathbb{R}}^{n}\), with e aa = 1 and e ab = 0 for any b ≠ a.
- 8.
For the same reason, the tBRD does not satisfy Cressman’s uniformity.
- 9.
For example, see Cressman (2003, p. 11).
- 10.
In Example 6.5, interior Nash equlibria in N 0 do not satisfy condition (ii).
- 11.
For epistemological foundation of properness, see Blume et al. (1991).
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Acknowledgements
I would like to thank Ross Cressman, Dimitrios Diamantaras, Josef Hofbauer, Bill Sandholm, and the participants at ISDG meeting, as well as Vlastimil Kr̆ivan and an anonymous referee, for their useful comments.
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Zusai, D. (2013). Interior Convergence Under Payoff Monotone Selections and Proper Equilibrium: Application to Equilibrium Selection. In: Křivan, V., Zaccour, G. (eds) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol 13. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-02690-9_6
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