1 Introduction and Overview of Results
1.1 Evolutionary Game Theory
1.2 Adaptive Dynamics Singular Strategy Types
Defining | Nondegen | Codim | Normal form \(h\)
|
---|---|---|---|
– |
\(f_{yy}\)
|
\(0\)
|
\(\varepsilon (x+\delta (y-x))(y-x)\)
|
\(f_{yy}-f_{xx}\)
|
\(\varepsilon =\mathrm {sgn}(f_{yy}-f_{xx}) \quad \delta =\varepsilon \mathrm {sgn}(f_{yy})\)
| ||
\(f_{yy}\)
|
\(f_{yy}-f_{xx}\)
| 1 |
\(\varepsilon x(y-x)\)
|
\(\varepsilon =-\mathrm {sgn}(f_{xx})\)
| |||
\(f_{yy}-f_{xx}\)
|
\(f_{yy}\)
| 1 |
\(\varepsilon (y-x+x^2)(y-x)\)
|
\(E_1\)
|
\(\varepsilon =\mathrm {sgn}(f_{yy})\)
| ||
\(f_{yy}-f_{xx}\)
|
\(f_{yy}\)
| 2 |
\(\varepsilon (y-x+ \delta x^3)(y-x)\)
|
\(E_1\)
|
\(E_2\)
|
\(\varepsilon =\mathrm {sgn}(f_{yy}) \quad \delta =\varepsilon \; \mathrm {sgn}(E_2)\)
| |
\(f_{xx}\)
|
\(E_1\)
| 3 |
\(((x+\sigma (y-x))^2+\delta (y-x)^2)(y-x)\)
|
\(f_{yy}\)
|
\(E_3\)
|
\(\delta =\mathrm {sgn}(E_3) \quad \sigma = (f_{xyy}+f_{yyy})/\sqrt{4|E_3|}\)
|
1.3 Equivalence of Strategy Functions
1.4 Singularity theory
1.4.1 Determinacy Theory for Strategy Functions
1.4.2 Classification for Strategy Functions
1.4.3 Unfolding Theory and Codimension for Strategy Functions
Universal unfolding H of normal form h | Necessary conditions for \(F\) to be an unfolding of \(f \simeq h\)
|
---|---|
\(\varepsilon (y-x)(x+ay)\)
|
\(\det \begin{pmatrix} f_{yy}-f_{xx} &{}\quad f_{xyy}+f_{yyy}\\ 2F_{a y} &{}\quad F_{a yy} \end{pmatrix} \ne 0\)
|
\(\varepsilon (y-x)(y-x+x^2+a)\)
|
\(F_{ay} \ne 0\)
|
\(\varepsilon (y-x)(y-x+\delta x^3+a+bx)\)
|
\(\det \begin{pmatrix} F_{ay} &{}\quad F_{ayy}-F_{axx}\\ F_{by} &{}\quad F_{byy}-F_{bxx} \end{pmatrix} \ne 0\)
|
\((y-x)((x+\sigma (y-x))^2+\delta (y-x)^2+a+b(y-x))\)
|
\(\det \begin{pmatrix} 0 &{}\quad f_{xxx}+f_{xxy} &{}\quad f_{xyy}+f_{yyy} \\ F_{ay} &{}\quad F_{axx} &{}\quad F_{ayy} \\ F_{by} &{}\quad F_{bxx} &{}\quad F_{byy} \end{pmatrix} \ne 0\)
|
1.4.4 Determinacy Theory for Universal Unfoldings
1.5 Structure of the Paper
2 Geometry of Unfolding Space
2.1 Transition Varieties
Universal unfolding \(G'(\cdot ,\alpha )\)
|
\(\fancyscript{B}\)
|
\(\fancyscript{E}\)
|
\(\fancyscript{C}\)
|
---|---|---|---|
\(\varepsilon (u+\delta v)\)
| – | – | – |
\(\varepsilon (u+av) \)
| – |
\(a=0\)
| – |
\(\varepsilon (v + u^2+a)\)
| – | – |
\(a=0 \)
|
\(\varepsilon (v +\delta u^3+a+bu) \)
| – | – |
\(a^2-\frac{4}{27} \delta b^3=0\)
|
\((u+\sigma v)^2 + \delta v^2+a+bv \)
|
\(b^2-4\delta a=0\)
|
\(b^2+4\sigma ^2a=0\)
|
\(a=0\)
|
2.2 Persistent Pairwise Invasibility Plots
Codimension 0 |
\(\mathrm {CvSS}_+\mathrm {ESS}_+\)
| |
\(\mathrm {CvSS}_-\mathrm {ESS}_+\)
| ||
\(\mathrm {CvSS}_+\mathrm {ESS}_-\)
| ||
\(\mathrm {CvSS}_-\mathrm {ESS}_-\)
| ||
Codimension 1 |
\(\mathrm {CvSS}_+\mathrm {ESS}_0\)
| |
\(\mathrm {CvSS}_-\mathrm {ESS}_0\)
| ||
\(\mathrm {CvSS}_0\mathrm {ESS}_+\)
| ||
\(\mathrm {CvSS}_0\mathrm {ESS}_-\)
| ||
Codimension 2 |
\(\mathrm {CvSS}_0\mathrm {ESS}_0\)
|
3 The Dieckmann–Metz Example
Player B Hawk | Player B Dove | |
---|---|---|
Player A Hawk |
\(\frac{1}{2}(V-C)\)
|
\(V\)
|
Player A Dove |
\(0\)
|
\(\frac{1}{2}V\)
|
4 The Restricted Tangent Space
4.1 Strategy Equivalence of Payoff Functions
4.2 Restricted Tangent Space of a Payoff Function
4.2.1 Maximal Ideals and Finite Codimension
4.2.2 Computation of the Restricted Tangent Space \(\mathrm {RT}(g)\)
4.3 Modified Tangent Space Constant Theorem
5 Recognition of Low Codimension Singularities
5.1 A Change of Coordinates
5.2 Recognition of Low Codimension Singularities
\((u+\sigma v)^3\)
|
\((u+\sigma v)^2v\)
|
\((u+\sigma v)v^2\)
|
\(v^3\)
| |
---|---|---|---|---|
\((u+\sigma v) p\)
|
\(1\)
|
\(0\)
|
\(\delta \)
| 0 |
\(v p\)
|
\(0\)
|
\(1\)
|
\(0\)
|
\(\delta \)
|
\(v^2p_v\)
|
\(0\)
|
\(0\)
|
\(\sigma \)
|
\(\delta \)
|
\(v^2p_u\)
|
\(0\)
|
\(0\)
|
\(1\)
|
\(0\)
|
5.3 Translation of Results from Payoff to Strategy Functions
\(g_{x} = g_u-g_v\)
|
\(g_{y} = g_v\)
| |
\(g_{xx} \!=\! g_{uu}-2g_{uv}+g_{vv}\)
|
\( g_{xy} \!=\! g_{uv}\!-\!g_{vv}\)
|
\(g_{yy} = g_{vv}\)
|
\(g_{xxx} \!=\! g_{uuu}-3g_{uvv}+3g_{uvv}-g_{vvv}\)
|
\(g_{xxy} \!=\! g_{uuv}\!-\!2g_{uvv}\!+\!g_{vvv}\)
| |
\(g_{xyy} = g_{uvv}-g_{vvv}\)
|
\(g_{yyy} = g_{vvv}\)
|
\(f=0\)
|
\(f_{xx}=-2g_x\)
|
\(f_{x}=-g\)
|
\(f_{xy}=g_x-g_y\)
|
\(f_{y}=g\)
|
\(f_{yy}=2g_{y}\)
|
\(f_{xxx} = -3g_{xx}\)
|
\(f_{xxxx} = -4g_{xxx}\)
|
\(f_{xxy} = g_{xx}-2g_{xy}\)
|
\(f_{xxxy} = g_{xxx}-3g_{xxy}\)
|
\(f_{xyy} = 2g_{xy}-g_{yy}\)
|
\(f_{xxyy} = 2g_{xxy}-2g_{xyy}\)
|
\(f_{yyy} = 3g_{yy}\)
|
\(f_{xyyy} = 3g_{xxy}-g_{yyy}\)
|
\(f_{yyyy} = 4g_{xxx}\)
|
6 The Universal Unfolding Theorem
6.1 Tangent Spaces and Unfolding Theorems
6.2 Universal Unfoldings of Low Codimension Singularities
\(h\)
| Codim |
\(\mathcal {I}(h)\)
|
\(T(h)\)
|
\(V_h\)
|
---|---|---|---|---|
\(\varepsilon u+\delta v\)
|
\(0\)
|
\(\mathcal {M}\)
|
\(\mathcal {E}\)
| – |
\(\varepsilon u\)
|
\(1\)
|
\(\langle u, v^2\rangle \)
|
\(\langle u,v^2\rangle \oplus \mathbf {R}\{1\}\)
|
\(\mathbf {R}\{v\}\)
|
\(\varepsilon (v+u^2)\)
|
\(1\)
|
\(\langle u^2,v\rangle \)
|
\(\mathcal {M}\)
|
\(\mathbf {R}\{1\}\)
|
\(\varepsilon (v+\delta u^3)\)
|
\(2\)
|
\(\langle u^3, v \rangle \)
|
\(\langle u^2, v \rangle \)
|
\(\mathbf {R}\{1,u\}\)
|
\((u+\sigma v)^2+\delta v^2\)
|
\(3\)
|
\(\mathcal {M}^3\oplus \mathbf {R}\{h,vh_v\}\)
|
\(\mathcal {M}^3\oplus \mathbf {R}\{h_u, h,vh_v\}\)
|
\(\mathbf {R}\{1,v,v(u+\sigma v)\}\)
|
6.3 Recognition Problem for Universal Unfoldings
\(h\)
| Codim |
\(T(g)\)
| Necessary condition |
---|---|---|---|
\(\varepsilon u\)
|
\(1\)
|
\(\mathcal {M}^2 \oplus \mathbf {R}\{1,u\}\)
|
\(\det \begin{pmatrix} g_u &{}g_{uv}\\ G_{\alpha }&{} G_{\alpha v} \end{pmatrix}\ne 0\)
|
\(\varepsilon (v+u^2)\)
|
\(1\)
|
\(\mathcal {M}\)
|
\(G_\alpha \ne 0\)
|
\(\varepsilon (v+\delta u^3)\)
|
\(2\)
|
\(\mathcal {M}^2+\langle v \rangle \)
|
\(\det \begin{pmatrix} G_{\alpha }&{} G_{\alpha u} \\ G_{\beta } &{} G_{\beta u} \end{pmatrix} \ne 0\)
|
\((u+\sigma v)^2+\delta v^2\)
|
\(3\)
|
\(\mathcal {M}^3\oplus \mathbf {R}\{g_u,ug_u,vg_v\}\)
|
\(\det \begin{pmatrix} 0 &{} g_{uu} &{} g_{uv} \\ G_{\alpha } &{} G_{\alpha u} &{}G_{\alpha v} \\ G_{\beta } &{} G_{\beta u} &{} G_{\beta v} \\ \end{pmatrix} \ne 0\)
|