集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织 - 用户贡献 [zh-cn]

多稳态

2021-01-03T15:48:14Z

舒寒：

此词条暂由彩云小译翻译，翻译字数共232，未经人工整理和审校，带来阅读不便，请见谅。
此词条由舒寒初步翻译。

In a [[dynamical system]], '''multistability''' is the property of having multiple [[Stability theory|stable equilibrium points]] in the [[vector space]] spanned by the states in the system. By mathematical necessity, there must also be unstable equilibrium points between the stable points. Points that are stable in some dimensions and unstable in others are termed unstable, as is the case with the first three [[Lagrangian points]].

在动力学系统中，多重稳定性是在矢量空间中具有多个稳定的平衡点的特性，该平衡点由系统中的状态跨越。根据数学上的必然性，在稳定点之间也一定存在不稳定的平衡点。与前三个拉格朗日点一样，在某些维度上稳定而在其他维度上不稳定的点被称为不稳定点。

== Bistability ==

{{main|Bistability}}

Bistability is the special case with two stable equilibrium points. It is the simplest form of multistability, and can occur in systems with only one state variable, as it only takes a one-dimensional space to separate two points.

双稳态是具有两个稳定平衡点的特殊情形。这是多重稳定性的最简单形式，可以发生在只有一个状态变量的系统中，因为它只需要一个一维空间来分隔两个点。

== Initial instability ==

{{see also|Path dependence}}

Near an unstable equilibrium, any system will be sensitive to noise, initial conditions and system parameters,

在不稳定平衡点附近，任何系统对噪声、初始条件和系统参数都很敏感,

which can cause it to develop in one of multiple divergent directions.

这导致它向多个不同的方向中的一个发展。

In economics and social sciences, path dependence gives rise to divergent directions of development.

在经济学和社会科学中，路径依赖引起了发展方向的分歧。

Some path dependent processes are adequately described by multistability,

多重稳定性充分描述了一些路径的依赖过程,

by being initially sensitive to input, before reaching a stagnant state –

在进入停滞状态之前，最初对输入很敏感

for example [[market share]] instability, which can develop into a stable [[monopoly]] for one of multiple possible vendors.

例如，市场份额的不稳定，可能会演变成多个可能的供应商之一的稳定垄断。

== Multistable perception ==

[[Image:Multistability.svg|right|thumb|260px|Ambiguous images]]

Ambiguous images

模棱两可的图像

{{main|Multistable perception}}

In vision science, [[multistable perception]] characterizes the wavering percepts that can be brought about by certain visually ambiguous pattern such as the [[Necker cube]], [[monocular rivalry]] or [[binocular rivalry]].

在视觉科学中，多稳态知觉描述了由于某些视觉上模糊的模式，如内克魔方、单眼竞争或双眼竞争，而引起的摇摆感知。

Through lateral inhibition, a pattern in which one image, when stimulated, inhibit the activity of neighboring images.

通过横向抑制，其中一个图像在受到刺激时会抑制邻近图像的活动。

==External links==

*[https://web.archive.org/web/20070928121528/http://www.icbm.de/komplsyst/9905.html Def of multistability from icbm.de]

[[Category:Optical illusions]]

Category:Optical illusions

分类: 视觉错觉

{{sci-stub}}

<noinclude>

This page was moved from [[wikipedia:en:Multistability]]. Its edit history can be viewed at [[多重稳定性/edithistory]]</noinclude>

[[Category:待整理页面]]

周期点

2021-01-03T15:30:07Z

舒寒：

此词条暂由彩云小译翻译，翻译字数共488，未经人工整理和审校，带来阅读不便，请见谅。
此词条由舒寒初步翻译。

In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.

在数学中，特别是在迭代函数和动力系统的研究领域中，函数的周期点是系统在一定次数的函数迭代或一定时间后返回的点。

== Iterated functions ==

Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,

给定一个从集合X到自身的映射f,

:<math>f: X \to X,</math>

a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that

X中的点x称为周期点，如果存在一个n使

:<math>\ f_n(x) = x</math>

where <math>f_n</math> is the ''n''th [[iterated function|iterate]] of ''f''. The smallest positive [[integer]] ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a [[periodic function]]).

其中f_n为f的第n次迭代。满足上述条件的最小正整数n称为点x的素数周期或最小周期。如果X中的每一个点都是周期为n的周期点，那么 f被称为周期点，周期为n(这不能和周期函数的概念混淆)。

If there exist distinct ''n'' and ''m'' such that <math>f_n(x) = f_m(x)</math>

如果存在不同的n和m使:<math>f_n(x) = f_m(x)</math>

then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.

那么x称为前周期点。所有周期点都是预周期点。

If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>f_n^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if

:<math>|f_n^\prime|\ne 1,</math>

如果f是微分流形的微分同胚，则定义了导数<math>f_n^\prime</math>，如果:<math>|f_n^\prime|\ne 1,</math>，那么f是双曲周期点，

that it is ''[[Attractor|attractive]]'' if :<math>|f_n^\prime|< 1,</math>

如果:<math>|f_n^\prime|< 1,</math>，则称周期点f为吸引子，

and it is ''repelling'' if:<math>|f_n^\prime|> 1.</math>

如果:<math>|f_n^\prime|> 1.</math>，则称周期点f为排斥子。

If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]].

如果周期点或不动点的稳定流形的维数为零，则称其为源点；如果不稳定流形的维数为零，则称其为汇点；如果稳定流形和不稳定流形都有非零维数，则称其为鞍点。

=== Examples ===

A period-one point is called a [[fixed point (mathematics)|fixed point]].

一个周期——一个点叫做不动点。

The [[logistic map]]

后勤地图

:<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>

exhibits periodicity for various values of the parameter ''r''. For ''r'' between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which [[attractor|attracts]] all orbits). For ''r'' between 1 and 3, the value 0 is still periodic but is not attracting, while the value {{nowrap|(''r'' − 1) / ''r''}} is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + {{radic|6}}, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and {{nowrap|(''r'' − 1) / ''r''}}. As the value of parameter ''r'' rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of ''r'' one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

参数r的各种值呈现周期性。对于介于0到1之间的r，0是唯一的周期点，周期为1（给出了吸引所有轨道的序列0,0,0，... ）。对于介于1到3之间的r，值0仍然是周期性的，但不是吸引点，而该值是周期1的吸引周期点。当r大于3但小于1 + 时，存在一对周期2的点，它们共同构成一个吸引序列，非吸引周期1点为0。当参数r的值上升到4时，会出现周期为正的一组周期点；对于 r 的某些值，这些重复序列中的一个被吸引，而对于其他值，则没有一个被吸引(几乎所有的轨道都是混乱的)。

== Dynamical system ==

Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],

给定一个实整体动力系统(R，X，Φ) ，其中X为相空间，Φ为演化函数，

<math>\Phi: \mathbb{R} \times X \to X</math>

a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' > 0 so that

如果存在 t > & thinsp; 0，则X中的点x称为周期为t的周期。因此

<math>\Phi(t, x) = x\,</math>

The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''.

具有此性质的最小正t称为点x的素数周期。

=== Properties ===

* Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R'''

* 给定一个周期为“p”的周期点“x”，则对于“R”中所有“t”的Phi(t,x) = \Phi(t+p,x)

* Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.

* 给定周期点“x”，则在轨道 <math>\gamma_x</math>上的所有点都具有相同的素数周期

==See also==

* [[Limit cycle]]

* [[限制周期]]

* [[Limit set]]

* [[限量套]]

* [[Stable manifold|Stable set]]

* [[稳定歧管|稳定集]]

* [[Sharkovsky's theorem]]

* [[Sharkovsky定理]]

* [[Stationary point]]

* [[固定点]]

* [[Periodic points of complex quadratic mappings]]

* [[复杂二次映射的周期点]]

{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}

[[Category:Limit sets]]

Category:Limit sets

类别: 极限集

<noinclude>

This page was moved from [[wikipedia:en:Periodic point]]. Its edit history can be viewed at [[周期点/edithistory]]</noinclude>

[[Category:待整理页面]]

周期点

2021-01-03T15:24:34Z

舒寒：

此词条暂由彩云小译翻译，翻译字数共488，未经人工整理和审校，带来阅读不便，请见谅。

In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.

在数学中，特别是在迭代函数和动力系统的研究领域中，函数的周期点是系统在一定次数的函数迭代或一定时间后返回的点。

== Iterated functions ==

Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,

给定一个从集合X到自身的映射f,

:<math>f: X \to X,</math>

a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that

X中的点x称为周期点，如果存在一个n使

:<math>\ f_n(x) = x</math>

where <math>f_n</math> is the ''n''th [[iterated function|iterate]] of ''f''. The smallest positive [[integer]] ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a [[periodic function]]).

其中f_n为f的第n次迭代。满足上述条件的最小正整数n称为点x的素数周期或最小周期。如果X中的每一个点都是周期为n的周期点，那么 f被称为周期点，周期为n(这不能和周期函数的概念混淆)。

If there exist distinct ''n'' and ''m'' such that

如果存在不同的n和m使

:<math>f_n(x) = f_m(x)</math>

then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.

那么x称为前周期点。所有周期点都是预周期点。

If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>f_n^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if

If f is a diffeomorphism of a differentiable manifold, so that the derivative <math>f_n^\prime</math> is defined, then one says that a periodic point is hyperbolic if

:<math>|f_n^\prime|\ne 1,</math>

如果f是微分流形的微分同胚，则定义了导数f_n，如果f_n^prime不等于1，那么f是双曲周期点，

that it is ''[[Attractor|attractive]]'' if

:<math>|f_n^\prime|< 1,</math>

如果f_n^prime < 1，则称周期点f为吸引子，

and it is ''repelling'' if

:<math>|f_n^\prime|> 1.</math>

如果f_n^prime > 1，则称周期点f为排斥子。

If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]].

如果周期点或不动点的稳定流形的维数为零，则称其为源点；如果不稳定流形的维数为零，则称其为汇点；如果稳定流形和不稳定流形都有非零维数，则称其为鞍点。

=== Examples ===

A period-one point is called a [[fixed point (mathematics)|fixed point]].

一个周期——一个点叫做不动点。

The [[logistic map]]

The logistic map

后勤地图

:<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>

<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>

x_{t+1}=rx_t(1-x_t)，qquad 0 leq x _ t leq 1，qquad 0 leq r leq 4

exhibits periodicity for various values of the parameter ''r''. For ''r'' between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which [[attractor|attracts]] all orbits). For ''r'' between 1 and 3, the value 0 is still periodic but is not attracting, while the value {{nowrap|(''r'' − 1) / ''r''}} is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + {{radic|6}}, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and {{nowrap|(''r'' − 1) / ''r''}}. As the value of parameter ''r'' rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of ''r'' one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

参数r的各种值呈现周期性。对于介于0到1之间的r，0是唯一的周期点，周期为1（给出了吸引所有轨道的序列0,0,0，... ）。对于介于1到3之间的r，值0仍然是周期性的，但不是吸引点，而该值是周期1的吸引周期点。当r大于3但小于1 + 时，存在一对周期2的点，它们共同构成一个吸引序列，非吸引周期1点为0。当参数r的值上升到4时，会出现周期为正的一组周期点；对于 r 的某些值，这些重复序列中的一个被吸引，而对于其他值，则没有一个被吸引(几乎所有的轨道都是混乱的)。

== Dynamical system ==

Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],

给定一个实整体动力系统(R，X，Φ) ，其中X为相空间，Φ为演化函数，

:<math>\Phi: \mathbb{R} \times X \to X</math>

<math>\Phi: \mathbb{R} \times X \to X</math>

Phi:实数集{R}乘 X → X

a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' > 0 so that

a point x in X is called periodic with period t if there exists a t > 0 so that

如果存在 t > & thinsp; 0，则X中的点x称为周期为t的周期。因此

:<math>\Phi(t, x) = x\,</math>

Phi (t，x) = x，

The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''.

The smallest positive t with this property is called prime period of the point x.

具有此性质的最小正t称为点x的素数周期。

=== Properties ===

* Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R'''
* 给定一个周期为“p”的周期点“x”，则对于“R”中所有“t”的Phi(t,x) = \Phi(t+p,x)

* Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.

* 给定周期点“x”，则在轨道gamma_x上的所有点都具有相同的素数周期

==See also==

* [[Limit cycle]]

* [[限制周期]]

* [[Limit set]]

* [[限量套]]

* [[Stable manifold|Stable set]]

* [[稳定歧管|稳定集]]

* [[Sharkovsky's theorem]]

* [[Sharkovsky定理]]

* [[Stationary point]]

* [[固定点]]

* [[Periodic points of complex quadratic mappings]]

* [[复杂二次映射的周期点]]

{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}

[[Category:Limit sets]]

Category:Limit sets

类别: 极限集

<noinclude>

This page was moved from [[wikipedia:en:Periodic point]]. Its edit history can be viewed at [[周期点/edithistory]]</noinclude>

[[Category:待整理页面]]

周期点

2021-01-03T15:21:51Z

舒寒：

此词条暂由彩云小译翻译，翻译字数共488，未经人工整理和审校，带来阅读不便，请见谅。

In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.

在数学中，特别是在迭代函数和动力系统的研究领域中，函数的周期点是系统在一定次数的函数迭代或一定时间后返回的点。

== Iterated functions ==

Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,

给定一个从集合X到自身的映射f,

:<math>f: X \to X,</math>

f：X→X，

a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that

X中的点x称为周期点，如果存在一个n使

:<math>\ f_n(x) = x</math>

f_n(x) = x

where <math>f_n</math> is the ''n''th [[iterated function|iterate]] of ''f''. The smallest positive [[integer]] ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a [[periodic function]]).

其中f_n为f的第n次迭代。满足上述条件的最小正整数n称为点x的素数周期或最小周期。如果X中的每一个点都是周期为n的周期点，那么 f被称为周期点，周期为n(这不能和周期函数的概念混淆)。

If there exist distinct ''n'' and ''m'' such that

If there exist distinct n and m such that

如果存在不同的n和m使

:<math>f_n(x) = f_m(x)</math>

f_n(x) = f_m(x)

then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.

那么x称为前周期点。所有周期点都是预周期点。

If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>f_n^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if

If f is a diffeomorphism of a differentiable manifold, so that the derivative <math>f_n^\prime</math> is defined, then one says that a periodic point is hyperbolic if

:<math>|f_n^\prime|\ne 1,</math>

如果f是微分流形的微分同胚，则定义了导数f_n，如果f_n^prime不等于1，那么f是双曲周期点，

that it is ''[[Attractor|attractive]]'' if

:<math>|f_n^\prime|< 1,</math>

如果f_n^prime < 1，则称周期点f为吸引子，

and it is ''repelling'' if

:<math>|f_n^\prime|> 1.</math>

如果f_n^prime > 1，则称周期点f为排斥子。

If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]].

如果周期点或不动点的稳定流形的维数为零，则称其为源点；如果不稳定流形的维数为零，则称其为汇点；如果稳定流形和不稳定流形都有非零维数，则称其为鞍点。

=== Examples ===

A period-one point is called a [[fixed point (mathematics)|fixed point]].

一个周期——一个点叫做不动点。

The [[logistic map]]

The logistic map

后勤地图

:<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>

<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>

x_{t+1}=rx_t(1-x_t)，qquad 0 leq x _ t leq 1，qquad 0 leq r leq 4

exhibits periodicity for various values of the parameter ''r''. For ''r'' between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which [[attractor|attracts]] all orbits). For ''r'' between 1 and 3, the value 0 is still periodic but is not attracting, while the value {{nowrap|(''r'' − 1) / ''r''}} is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + {{radic|6}}, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and {{nowrap|(''r'' − 1) / ''r''}}. As the value of parameter ''r'' rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of ''r'' one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

参数r的各种值呈现周期性。对于介于0到1之间的r，0是唯一的周期点，周期为1（给出了吸引所有轨道的序列0,0,0，... ）。对于介于1到3之间的r，值0仍然是周期性的，但不是吸引点，而该值是周期1的吸引周期点。当r大于3但小于1 + 时，存在一对周期2的点，它们共同构成一个吸引序列，非吸引周期1点为0。当参数r的值上升到4时，会出现周期为正的一组周期点；对于 r 的某些值，这些重复序列中的一个被吸引，而对于其他值，则没有一个被吸引(几乎所有的轨道都是混乱的)。

== Dynamical system ==

Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],

给定一个实整体动力系统(R，X，Φ) ，其中X为相空间，Φ为演化函数，

:<math>\Phi: \mathbb{R} \times X \to X</math>

<math>\Phi: \mathbb{R} \times X \to X</math>

Phi:实数集{R}乘 X → X

a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' > 0 so that

a point x in X is called periodic with period t if there exists a t > 0 so that

如果存在 t > & thinsp; 0，则X中的点x称为周期为t的周期。因此

:<math>\Phi(t, x) = x\,</math>

Phi (t，x) = x，

The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''.

The smallest positive t with this property is called prime period of the point x.

具有此性质的最小正t称为点x的素数周期。

=== Properties ===

* Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R'''
* 给定一个周期为“p”的周期点“x”，则对于“R”中所有“t”的Phi(t,x) = \Phi(t+p,x)

* Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.

* 给定周期点“x”，则在轨道gamma_x上的所有点都具有相同的素数周期

==See also==

* [[Limit cycle]]

* [[Limit set]]

* [[Stable manifold|Stable set]]

* [[Sharkovsky's theorem]]

* [[Stationary point]]

* [[Periodic points of complex quadratic mappings]]

* [[限制周期]]

* [[限量套]]

* [[稳定歧管|稳定集]]

* [[Sharkovsky定理]]

* [[固定点]]

* [[复杂二次映射的周期点]]

{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}

[[Category:Limit sets]]

Category:Limit sets

类别: 极限集

<noinclude>

This page was moved from [[wikipedia:en:Periodic point]]. Its edit history can be viewed at [[周期点/edithistory]]</noinclude>

[[Category:待整理页面]]

负熵

2020-10-26T15:59:55Z

舒寒：

此词条由[[用户:Jxzhou|Jxzhou]]（[[用户讨论:Jxzhou|讨论]]）翻译
由CecileLi初步审校

{{Distinguish|Entropy and life#Negative entropy}}{{Redirect|Syntropy||Syntropy (software)}}

在信息论 information theory和统计学 statistics中，负熵 negentropy被用于度量与正态分布之间的距离。“负熵”这一概念是埃尔温·薛定谔Erwin Schrödinger在他1944年的科普著作《生命是什么？》 What is Life?中首次提出，<ref>Schrödinger, Erwin, ''What is Life – the Physical Aspect of the Living Cell'', Cambridge University Press, 1944</ref>后来莱昂·布里渊Léon Brillouin把这个短语缩写为“负熵”。<ref>Brillouin, Leon: (1953) "Negentropy Principle of Information", ''J. of Applied Physics'', v. '''24(9)''', pp. 1152–1163</ref><ref>Léon Brillouin, ''La science et la théorie de l'information'', Masson, 1959</ref>1974年，阿尔伯特·圣捷尔吉Albert Szent-Györgyi提出用“同向”这个可能源于20世纪40年代意大利数学家 Luigi fantappi的术语来代替“负熵”以建立一个生物学和物理学的统一理论。虽然巴克敏斯特·福乐 Buckminster Fuller试图推广这一思想，但是迄今为止负熵仍更为人们广泛使用。

在《生命是什么？》的一项注释中，薛定谔解释了他使用这个短语的原因。

如果我只为了迎合物理学家们，那么我就会让讨论转向“[[热力学自由能|自由能 free energy]]”。在这个语境中，自由能是物理学更熟悉的概念。但是，这个高度专业的术语在语言学上似乎太接近于能量 energy，以至于普通读者无法生动地看到两者之间的区别。

2009年，Mahulikar 和 Herwig 将动态有序子系统的负熵重新定义为有序子系统相对于周围混沌的特定熵赤字。<ref>Mahulikar, S.P. & Herwig, H.: (2009) "Exact thermodynamic principles for dynamic order existence and evolution in chaos", ''Chaos, Solitons & Fractals'', v. '''41(4)''', pp. 1939–1948</ref>因此，根据单位质量的熵定义负熵的 SI 单位为（J kg−1 K−1），其中(K−1)的定义基于单位能量的熵。这个定义表明: i)动态有序存在的尺度不变的热力学表示，ii)动态有序存在和演化的专门物理原理，iii)薛定谔关于负熵的数学解释。

==Information theory 信息论==

在信息论和统计学中，负熵可用来度量某个分布到正态分布的距离。<ref>Aapo Hyvärinen, [http://www.cis.hut.fi/aapo/papers/NCS99web/node32.html Survey on Independent Component Analysis, node32: Negentropy], Helsinki University of Technology Laboratory of Computer and Information Science</ref><ref>Aapo Hyvärinen and Erkki Oja, [http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb/node14.html Independent Component Analysis: A Tutorial, node14: Negentropy], Helsinki University of Technology Laboratory of Computer and Information Science</ref><ref>Ruye Wang, [http://fourier.eng.hmc.edu/e161/lectures/ica/node4.html Independent Component Analysis, node4: Measures of Non-Gaussianity]</ref> 在所有具有给定均值和方差的分布中，正态分布或高斯分布的熵最大。负熵也用来度量具有与它相同均值和方差的给定分布和正态分布之间熵的差距。因此，负熵总是非负的，在任何线性可逆的坐标变换下都是不变的，当且仅当分布表现为高斯分布 Gaussian distribution时才变为零。

负熵定义为

:<math>J(p_x) = S(\varphi_x) - S(p_x)\,</math>

其中<math>S(\varphi_x)</math>表示与<math>p_x</math>具有相同均值和方差的高斯密度的微分熵，<math>S(p_x)</math>表示<math>p_x</math>的微分熵：

:<math>S(p_x) = - \int p_x(u) \log p_x(u) \, du</math>

负熵通常用于统计和信号处理。它与网络熵有关，网络熵常用于进行独立成分分析 independent component analysis。<ref>P. Comon, Independent Component Analysis – a new concept?, ''Signal Processing'', '''36''' 287–314, 1994.</ref><ref>Didier G. Leibovici and Christian Beckmann, [http://www.fmrib.ox.ac.uk/analysis/techrep/tr01dl1/tr01dl1/tr01dl1.html An introduction to Multiway Methods for Multi-Subject fMRI experiment], FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.</ref>

一个分布的负熵等于 <math>p_x</math> 和具有与 <math>p_x</math> 相同均值和方差的正态分布的 Kullback-Leibler 散度(参见正态分布的微分熵 Differential entropy和最大化)。特别地，负熵总是非负的。

==Correlation between statistical negentropy and Gibbs' free energy 统计学负熵与吉布斯自由能的关联==

[ Willard Gibbs 在1873年展示的可用能量(自由能)图以一个垂直于 v (体积)轴和通过点A的平面为例，A点表示物体的初始状态。MN 是耗散能曲面的交线。Qε 和 Qη 分别是平面''η'' = 0 和 ''ε'' = 0的交线，因此分别与 ε (内能)和 η (熵)轴平行。AD 和 AE 分别代表物体初始状态的能量和熵，AB 和 AC 分别代表物体的有效能(吉布斯能)和熵的容量(在不改变物体能量或增加物体体积的情况下物体可以增加的熵的量)。

存在一个与自由能(自由焓 free enthalpy)密切相关的物理量，它具有熵的单位并且与我们所知的统计学和信息论中的负熵同构。1873年，威拉德·吉布斯创建了一个图表，说明了自由能对应于自由焓的概念。在图表上，我们可以看到称为熵的容量 capacity for entropy的物理量。这个量表示在不改变内能 internal energy体积的情况下可增加的熵值。<ref>Willard Gibbs, [http://www.ufn.ru/ufn39/ufn39_4/Russian/r394b.pdf A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces], ''Transactions of the Connecticut Academy'', 382–404 (1873)</ref> 换句话说，它是在假定条件下可能的最大熵与实际熵之间的差异。它恰好符合统计学和信息论中负熵的定义。1869年，Massieu 在等温过程 isothermal process<ref>Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. ''C. R. Acad. Sci.'' LXIX:858–862.</ref><ref>Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. ''C. R. Acad. Sci.'' LXIX:1057–1061.</ref><ref>Massieu, M. F. (1869), ''Compt. Rend.'' '''69''' (858): 1057.</ref>(两个量只有一个图形符号不同)中引入了一个类似的物理量，<ref>Planck, M. (1945). ''Treatise on Thermodynamics''. Dover, New York.</ref>后来 Planck 把这个概念引入到等温-等压 isothermal-isobaric process过程中。近期，Massieu–Planck 提出的热力学势，也被称为自由熵，<ref>Antoni Planes, Eduard Vives, [http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html Entropic Formulation of Statistical Mechanics], Entropic variables and Massieu–Planck functions 2000-10-24 Universitat de Barcelona</ref>已被证明在所谓的统计力学熵表述中发挥了重要作用，应用于分子生物学<ref>John A. Scheilman, [http://www.biophysj.org/cgi/reprint/73/6/2960.pdf Temperature, Stability, and the Hydrophobic Interaction], ''Biophysical Journal'' '''73''' (December 1997), 2960–2964, Institute of Molecular Biology, University of Oregon, Eugene, Oregon 97403 USA</ref> 和热力学非平衡过程。<ref>Z. Hens and X. de Hemptinne, [https://arxiv.org/pdf/chao-dyn/9604008 Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures], Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium</ref>

:: <math>J = S_\max - S = -\Phi = -k \ln Z\,</math>

其中：

<math>S</math> 是熵
<math>J</math> 是负熵(吉布斯“熵的容量”)
<math>\Phi</math> 是 Massieu 势
<math>Z</math> 是配分函数
<math>k</math> 是波兹曼常数

特别地，数学上的负熵(负熵函数，在物理学中解释为自由熵)是 LogSumExp 的凸共轭(在物理学中解释为自由能)。

==Brillouin's negentropy principle of information 布里渊信息负熵原理==

1953年，布里渊推导出一个一般性方程，证明了改变一个信息的比特值至少需要 kT ln (2)的能量。这与理想情况下 Leó Szilárd 的引擎所产生的能量相等。在他的书中，<ref>Leon Brillouin, ''Science and Information theory'', Dover, 1956</ref>他进一步探讨了这个问题并得出结论: 任何条件下一个比特的改变(测量、关于是/否问题的决定、擦除、显示等等)都需要消耗同等的能量。

==See also 参见==

* [[Exergy 㶲]]

* [[Extropy]]

--[[用户:CecileLi|CecileLi]]([[用户讨论:CecileLi|讨论]]) 【审校】如果手误与上下文衔接应为 entropy，熵；如果是Extropy，未查询到专门中文翻译，只有英译英：the prediction that human intelligence and technology will enable life to expand in an orderly way throughout the entire universe

* [[Free entropy 自由熵]]

* [[Entropy in thermodynamics and information theory 热力学和信息论中的熵]]

==Notes 注释==

{{Reflist|20em}}

{{Wiktionary}}

[[Category:Entropy and information]]

Category:Entropy and information

类别: 熵和信息

[[Category:Negative concepts]]

Category:Negative concepts

类别: 否定概念

[[Category:Statistical deviation and dispersion]]

Category:Statistical deviation and dispersion

类别: 统计偏差和离散度

[[Category:Thermodynamic entropy]]

Category:Thermodynamic entropy

类别: 熵

<noinclude>

This page was moved from [[wikipedia:en:Negentropy]]. Its edit history can be viewed at [[负熵/edithistory]]</noinclude>

[[Category:待整理页面]]

策略

2020-09-22T03:32:29Z

舒寒：

本词条由舒寒初步翻译。

In [[game theory]], a [[player (game)|player]]'s '''strategy''' is any of the options which he or she chooses in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others.<ref>[[Ben Polak]] ''Game Theory: Lecture 1 Transcript'' ECON 159, 5 September 2007, [[Open Yale Courses]].</ref> A player's strategy will determine the action which the player will take at any stage of the game.

In game theory, a player's strategy is any of the options which he or she chooses in a setting where the outcome depends not only on their own actions but on the actions of others. A player's strategy will determine the action which the player will take at any stage of the game.

在博弈论中，一个玩家的策略是他或她在一个不仅取决于他们自己的行动，而且也取决于其他人的行动的环境中所选择的任何一个选项。玩家的策略将决定玩家在游戏的任何阶段将采取的行动。

The '''strategy''' concept is sometimes (wrongly) confused with that of a '''move'''. A '''move''' is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A '''strategy''' on the other hand is a complete [[algorithm]] for playing the game, telling a player what to do for every possible situation throughout the game.

The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.

战略的概念有时(错误地)与行动的概念相混淆。行动是一个玩家在博弈过程中的某个时刻所采取的行动(例如，在国际象棋中，将 white 的 Bishop 移动到 b3)。另一方面，策略是博弈的完整算法，告诉玩家在整个游戏中的每个可能情况下应该做什么。

A '''strategy profile''' (sometimes called a '''strategy combination''') is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.

A strategy profile (sometimes called a strategy combination) is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.

战略配置文件(有时称为战略组合)是一套战略的所有球员，其中充分规定了这个博弈中的所有行动。一个战略档案必须包括一个和只有一个战略的每个球员。

==Strategy set==

A player's '''strategy set''' defines what strategies are available for them to play.

A player's strategy set defines what strategies are available for them to play.

玩家的策略集定义了他们可以使用的策略。

A player has a '''finite''' strategy set if they have a number of discrete strategies available to them. For instance, a game of [[rock paper scissors]] comprises a single move by each player—and each player's move is made without knowledge of the other's, not as a response—so each player has the finite strategy set {rock paper scissors}.

A player has a finite strategy set if they have a number of discrete strategies available to them. For instance, a game of rock paper scissors comprises a single move by each player—and each player's move is made without knowledge of the other's, not as a response—so each player has the finite strategy set {rock paper scissors}.

如果一个玩家有许多可用的离散策略，那么他的策略集是有限的。例如，石头剪刀布游戏包括每个玩家的一个动作---- 每个玩家的动作都是在不知道对方的动作的情况下进行的，而不是作为一种反应---- 所以每个玩家都有一个有限的策略集合{石头剪刀布}。

A strategy set is infinite otherwise. For instance the [[Fair division|cake cutting game]] has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}.

A strategy set is infinite otherwise. For instance the cake cutting game has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}.

否则策略集是无限的。例如，切蛋糕游戏在策略集{切蛋糕的0% 到100% 之间的任何地方}中有一个有限的连续统一体。

In a [[dynamic game]], the strategy set consists of the possible rules a player could give to a [[robot]] or [[Software agent|agent]] on how to play the game. For instance, in the [[ultimatum game]], the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject.

In a dynamic game, the strategy set consists of the possible rules a player could give to a robot or agent on how to play the game. For instance, in the ultimatum game, the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject.

在一个动态博弈中，策略集由玩家可能给机器人或代理人的游戏规则组成。例如，在最后通牒博弈中，为第二个参与者设置的策略将包括所有可能的规则，哪些提议可以接受，哪些可以拒绝。

In a [[Bayesian game]], the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information.

In a Bayesian game, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information.

在贝叶斯博弈中，策略集类似于动态博弈中的策略集。它包括对任何可能的私人信息采取何种行动的规则。

===Choosing a strategy set===

In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem to limit the strategy spaces, and ease the solution.

In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem to limit the strategy spaces, and ease the solution.

在应用博弈论中，策略集的定义是使博弈既可解又有意义的艺术的重要组成部分。博弈论者可以利用整体问题的知识来限制策略空间，并简化解决方案。

For instance, strictly speaking in the Ultimatum game a player can have strategies such as: ''Reject offers of ($1, $3, $5, ..., $19), accept offers of ($0, $2, $4, ..., $20)''. Including all such strategies makes for a very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤ ''x'', accept any offer > ''x''; for ''x'' in ($0, $1, $2, ..., $20)}.

For instance, strictly speaking in the Ultimatum game a player can have strategies such as: Reject offers of ($1, $3, $5, ..., $19), accept offers of ($0, $2, $4, ..., $20). Including all such strategies makes for a very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤ x, accept any offer > x; for x in ($0, $1, $2, ..., $20)}.

例如，严格地说，在最后通牒游戏中，玩家可以有以下策略: 拒绝($1，$3，$5，... ，$19) ，接受($0，$2，$4，... ，$20)。把所有这些战略都包括在内，会产生一个非常大的战略空间和一个有点难度的问题。博弈论者可能相信他们可以将策略集限制为: { Reject any offer ≤ x，accept any offer x; for x in ($0，$1，$2，... ，$20)}。

==Pure and mixed strategies==

A '''pure strategy''' provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation they could face. A player's '''strategy set''' is the set of pure strategies available to that player.

A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation they could face. A player's strategy set is the set of pure strategies available to that player.

纯策略为玩家如何玩游戏提供了一个完整的定义。特别是，它决定了球员在任何情况下都会采取的行动。一个玩家的策略集是该玩家可用的纯策略集。

A '''mixed strategy''' is an assignment of a [[probability]] to each pure strategy. This allows for a player to randomly select a pure strategy. (See the following section for an illustration.) Since probabilities are continuous, there are infinitely many mixed strategies available to a player.

A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. (See the following section for an illustration.) Since probabilities are continuous, there are infinitely many mixed strategies available to a player.

混合策略是给每个纯策略赋予一个概率。这允许玩家随机选择一个纯策略。(见以下部分的说明。)因为概率是连续的，所以一个参与人可以使用无限多的混合策略。

Of course, one can regard a pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is selected with probability ''1'' and every other strategy with probability ''0''.

Of course, one can regard a pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is selected with probability 1 and every other strategy with probability 0.

当然，我们可以把纯策略看作是混合策略的退化情形，在这种情形下，某个特定的纯策略的选择概率为1，而其他的策略的选择概率为0。

A '''totally mixed strategy''' is a mixed strategy in which the player assigns a strictly positive probability to every pure strategy. (Totally mixed strategies are important for [[equilibrium refinement]] such as [[trembling hand perfect equilibrium]].)

A totally mixed strategy is a mixed strategy in which the player assigns a strictly positive probability to every pure strategy. (Totally mixed strategies are important for equilibrium refinement such as trembling hand perfect equilibrium.)

一个完全混合策略是一个混合策略，其中玩家为每个纯策略赋予一个严格正的概率。(完全混合策略对于均衡精化非常重要，比如颤抖手完美均衡。)

== Mixed strategy ==

=== Illustration ===

{| align=right

{| align=right

{ | 右对齐

|-

|-

|-

|{{Payoff matrix | Name = Pure [[coordination game]]

|{{Payoff matrix | Name = Pure coordination game

| {支付矩阵 | 名称纯协调对策

| 2L = A | 2R = B |

| 2L = A | 2R = B |

2 l a | 2 r b |

1U = A | UL = 1, 1 | UR = 0, 0 |

1U = A | UL = 1, 1 | UR = 0, 0 |

1 u a | UL 1,1 | UR 0,0 |

1D = B | DL = 0, 0 | DR = 1, 1 |

1D = B | DL = 0, 0 | DR = 1, 1 |

0,0 | DR 1,1 |

Float = | Width = }}

Float = | Width = }}

浮动 | 宽度}

|}

|}

|}

Consider the [[payoff matrix]] pictured to the right (known as a [[coordination game]]). Here one player chooses the row and the other chooses a column. The row player receives the first payoff, the column player the second. If row opts to play '''A''' with probability 1 (i.e. play '''A''' for sure), then he is said to be playing a pure strategy. If column opts to flip a coin and play '''A''' if the coin lands heads and '''B''' if the coin lands tails, then he is said to be playing a mixed strategy, and not a pure strategy.

Consider the payoff matrix pictured to the right (known as a coordination game). Here one player chooses the row and the other chooses a column. The row player receives the first payoff, the column player the second. If row opts to play A with probability 1 (i.e. play A for sure), then he is said to be playing a pure strategy. If column opts to flip a coin and play A if the coin lands heads and B if the coin lands tails, then he is said to be playing a mixed strategy, and not a pure strategy.

考虑右边的收益矩阵(称为协调博弈)。在这里，一个玩家选择行，另一个玩家选择列。第一行玩家获得第一个回报，第二列玩家获得第二个回报。如果行选择以概率1(即。当然是 a) ，那么他就是在玩纯策略。如果一列选择抛出一枚硬币，如果硬币正面朝上，则选择 a，如果硬币正面朝上，则选择 b，那么他就是在玩一种混合策略，而不是单纯的策略。

=== Significance ===

In his famous paper, [[John Forbes Nash]] proved that there is an [[Nash equilibrium|equilibrium]] for every finite game. One can divide Nash equilibria into two types. ''Pure strategy Nash equilibria'' are Nash equilibria where all players are playing pure strategies. ''Mixed strategy Nash equilibria'' are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see [[Matching pennies]]. However, many games do have pure strategy Nash equilibria (e.g. the [[Coordination game]], the [[Prisoner's dilemma]], the [[Stag hunt]]). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2.

In his famous paper, John Forbes Nash proved that there is an equilibrium for every finite game. One can divide Nash equilibria into two types. Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see Matching pennies. However, many games do have pure strategy Nash equilibria (e.g. the Coordination game, the Prisoner's dilemma, the Stag hunt). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2.

在他著名的论文中，约翰·福布斯·纳什证明了每个有限博弈都存在均衡。我们可以把纳什均衡分为两种类型。纯策略纳什均衡是所有参与者都采用纯策略的纳什均衡。混合策略纳什均衡是至少有一个参与人采用混合策略的均衡。虽然 Nash 证明了每个有限博弈都有纳什均衡点，但并不是所有的博弈都有纯策略纳什均衡。有关纯策略中没有纳什均衡点的游戏的例子，请参阅匹配便士。然而，许多博弈确实存在纯策略纳什均衡(例如:。协调博弈，囚徒困境，猎鹿)。此外，博弈可以同时具有纯策略和混合策略均衡。一个简单的例子是纯协调博弈，其中除了纯策略(a，a)和(b，b)之外，还存在一个混合均衡，其中两个参与者都采用概率为1 / 2的策略。

=== A disputed meaning ===

During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic".<ref name="Aumann1985">{{cite book |authorlink=Robert Aumann |last=Aumann |first=R. |chapterurl=http://www.ma.huji.ac.il/raumann/pdf/what%20is%20game%20theory.pdf |chapter=What is Game Theory Trying to accomplish? |title=Frontiers of Economics |editor1-first=K. |editor1-last=Arrow |editor2-first=S. |editor2-last=Honkapohja |pages=909–924 |publisher=Basil Blackwell |location=Oxford |year=1985}}</ref> Randomization, central in mixed strategies, lacks behavioral support. Seldom do people make their choices following a lottery. This behavioral problem is compounded by the cognitive difficulty that people are unable to generate random outcomes without the aid of a [[Random number generator|random or pseudo-random generator]].<ref name="Aumann1985" />

During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic". Randomization, central in mixed strategies, lacks behavioral support. Seldom do people make their choices following a lottery. This behavioral problem is compounded by the cognitive difficulty that people are unable to generate random outcomes without the aid of a random or pseudo-random generator.

在20世纪80年代，混合策略的概念因“直觉上有问题”而受到猛烈抨击。以混合策略为核心的随机化缺乏行为支持。很少有人在买彩票后做出选择。这个行为问题是由认知困难，人们不能产生随机的结果没有随机或伪随机生成器的帮助。

In 1991,<ref name="Rubinstein1991">{{cite journal |authorlink=Ariel Rubinstein |last=Rubinstein |first=A. |title=Comments on the interpretation of Game Theory |journal=[[Econometrica]] |year=1991 |volume=59 |issue=4 |pages=909–924 |jstor=2938166|doi=10.2307/2938166 }}</ref> game theorist [[Ariel Rubinstein]] described alternative ways of understanding the concept. The first, due to Harsanyi (1973),<ref>{{cite journal |last1=Harsanyi |first1=John |author1-link=John Harsanyi |title=Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points |journal=Int. J. Game Theory |volume=2 |pages=1–23 |year=1973 |doi=10.1007/BF01737554}}</ref> is called ''[[purification theorem|purification]]'', and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors. However, it is unsatisfying to have results that hang on unspecified factors.<ref name="Rubinstein1991" />

In 1991, game theorist Ariel Rubinstein described alternative ways of understanding the concept. The first, due to Harsanyi (1973), is called purification, and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors. However, it is unsatisfying to have results that hang on unspecified factors.

1991年，博弈理论家阿里埃勒·鲁宾斯坦 · 马丁描述了理解这个概念的不同方式。首先，由于哈萨尼(1973) ，被称为净化，并假定混合策略解释仅仅反映了我们对参与者的信息和决策过程的知识的缺乏。显然，随机的选择被看作是未指定的、与回报无关的外生因素的结果。然而，结果依赖于未指明的因素是不令人满意的。

A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.

A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.

第二种解释假设游戏玩家代表一大群代理人。每个代理选择一个纯策略，回报取决于选择每个策略的代理所占的比例。因此，混合策略代表了每个种群所选择的纯策略的分布。然而，当球员是单个经纪人时，这并不能为这种情况提供任何理由。

Later, Aumann and Brandenburger (1995),<ref>{{cite journal |last1=Aumann |first1=Robert |author1-link=Robert Aumann |last2=Brandenburger |first2=Adam |author2-link=Adam Brandenburger |title=Epistemic Conditions for Nash Equilibrium |journal=Econometrica |volume=63|pages=1161–1180 |year=1995 |doi=10.2307/2171725 |jstor=2171725 |issue=5 |citeseerx=10.1.1.122.5816 }}</ref> re-interpreted Nash equilibrium as an equilibrium in ''beliefs'', rather than actions. For instance, in [[rock paper scissors]] an equilibrium in beliefs would have each player ''believing'' the other was equally likely to play each strategy. This interpretation weakens the predictive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to ''actually'' play a pure strategy of Rock.

Later, Aumann and Brandenburger (1995), re-interpreted Nash equilibrium as an equilibrium in beliefs, rather than actions. For instance, in rock paper scissors an equilibrium in beliefs would have each player believing the other was equally likely to play each strategy. This interpretation weakens the predictive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock.

后来，Aumann 和 Brandenburger (1995)重新解释了纳什均衡点，认为它是信仰的平衡，而不是行动的平衡。例如，在石头剪刀布游戏中，信念的均衡会让每个玩家相信另一个玩家同样可能使用每种策略。然而，这种解释削弱了纳什均衡点的预测能力，因为在这样的均衡中，每个玩家都有可能实际使用一种纯粹的摇滚战略。

Ever since, game theorists' attitude towards mixed strategies-based results have been ambivalent. Mixed strategies are still widely used for their capacity to provide Nash equilibria in games where no equilibrium in pure strategies exists, but the model does not specify why and how players randomize their decisions.

Ever since, game theorists' attitude towards mixed strategies-based results have been ambivalent. Mixed strategies are still widely used for their capacity to provide Nash equilibria in games where no equilibrium in pure strategies exists, but the model does not specify why and how players randomize their decisions.

从那时起，博弈论者对基于混合策略的结果的态度一直是矛盾的。在纯策略不存在均衡的博弈中，混合策略仍然被广泛用于提供纳什均衡，但是这个模型并没有说明为什么以及参与者如何随机化他们的决策。

===Behavior strategy===

While a mixed strategy assigns a probability distribution over pure strategies, a '''behavior strategy''' assigns at each information set a probability distribution over the set of possible actions. While the two concepts are very closely related in the context of normal form games, they have very different implications for extensive form games. Roughly, a mixed strategy randomly chooses a deterministic path through the game tree, while a behavior strategy can be seen as a stochastic path.

While a mixed strategy assigns a probability distribution over pure strategies, a behavior strategy assigns at each information set a probability distribution over the set of possible actions. While the two concepts are very closely related in the context of normal form games, they have very different implications for extensive form games. Roughly, a mixed strategy randomly chooses a deterministic path through the game tree, while a behavior strategy can be seen as a stochastic path.

当混合策略在纯策略之上分配一个概率分布 / 值时，行为策略在每个信息集上分配一个概率分布 / 值，在可能的行动集上分配一个值。尽管这两个概念在范式游戏的背景下是密切相关的，但是它们对于范式游戏有着非常不同的含义。粗略地说，混合策略通过博弈树随机选择一条确定性路径，而行为策略可以看作是一条随机路径。

The relationship between mixed and behavior strategies is the subject of [[Kuhn's theorem]]. The result establishes that in any finite extensive-form game with perfect recall, for any player and any mixed strategy, there exists a behavior strategy that, against all profiles of strategies (of other players), induces the same distribution over terminal nodes as the mixed strategy does. The converse is also true.

The relationship between mixed and behavior strategies is the subject of Kuhn's theorem. The result establishes that in any finite extensive-form game with perfect recall, for any player and any mixed strategy, there exists a behavior strategy that, against all profiles of strategies (of other players), induces the same distribution over terminal nodes as the mixed strategy does. The converse is also true.

混合策略和行为策略之间的关系是库恩定理的主题。结果表明，在任何具有完全回忆的有限扩展形式的博弈中，对于任何玩家和任何混合策略，都存在一种行为策略，这种策略针对所有策略(其他玩家的) ，在终端节点上诱导出与混合策略相同的分布。反之亦然。

A famous example of why perfect recall is required for the equivalence is given by Piccione and Rubinstein (1997){{full citation needed|date=September 2018}} with their ''Absent-Minded Driver'' game.

A famous example of why perfect recall is required for the equivalence is given by Piccione and Rubinstein (1997) with their Absent-Minded Driver game.

Piccione 和 Rubinstein (1997)在他们的心不在焉的 Driver 游戏中给出了一个著名的例子来说明为什么等价物需要完全回忆。

== See also ==

* [[Nash equilibrium]]

* [[Haven (graph theory)]]

*[[Evolutionarily stable strategy]]

==References==

{{Reflist}}

{{Game theory}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

<noinclude>

This page was moved from [[wikipedia:en:Strategy (game theory)]]. Its edit history can be viewed at [[策略/edithistory]]</noinclude>

[[Category:待整理页面]]