Game’s Degeneracy Running Proportional to Polytope’s Redundancy.

For a given set of vertices V ⊆ RK a Polytope P can be defined as the following set of points:

P = {∑i=1|V|λivi ∈ RK | ∑i=1|V|λi = 1; λi ≥ 0; vi ∈ V}

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Polytope is an intersection of boundaries that separate the space into two distinct areas. If a polytope is to be defined as an intersection of half spaces, then for a matrix M ∈ Rmxn, and a vector b ∈ Rm, polytope P is defined as a set of points

P = {x ∈ Rn | Mx ≤ b}

Switching over to a two-player game, (A, B) ∈ Rmxn2>0, the row/column best response polytope P/Q is defined by:

P = {x ∈ Rm | x ≥ 0; xB ≤ 1}

Q = {y ∈ Rn | Ay ≤ 1; y ≥ 0}

The polytope P, corresponds to the set of points with an upper bound on the utility of those points when considered as row strategies against which the column player plays.

An affine combination of points z1,….zk in some Euclidean space is of the form ∑i=1kλizi, where λ1, …, λk are reals with ∑i=1kλi= 1. It is called a convex combination, if λ≥ 0 ∀ i. A set of points is convex if it is closed under forming convex combinations. Given points are affinely independent if none of these points are an affine combination of the others. A convex set has dimension d iff it has d + 1, but no more, affinely independent points.

A polyhedron P in Rd is a set {z ∈ Rd | Cz ≤ q} for some matrix C and vector q. It is called full-dimensional if it has dimension d. It is called a polytope if it is bounded. A face of P is a set {z ∈ P | cz = q0} for some c ∈ Rd, q0 ∈ R, such that the inequality cz ≤ q0 holds for all z in P. A vertex of P is the unique element of a zero-dimensional face of P. An edge of P is a one-dimensional face of P. A facet of a d-dimensional polyhedron P is a face of dimension d − 1. It can be shown that any nonempty face F of P can be obtained by turning some of the inequalities defining P into equalities, which are then called binding inequalities. That is, F = {z ∈ P | ciz = qi, i ∈ I}, where ciz ≤ qi for i ∈ I are some of the rows in Cz ≤ q. A facet is characterized by a single binding inequality which is irredundant; i.e., the inequality cannot be omitted without changing the polyhedron. A d-dimensional polyhedron P is called simple if no point belongs to more than d facets of P, which is true if there are no special dependencies between the facet-defining inequalities. The “best response polyhedron” of a player is the set of that player’s mixed strategies together with the “upper envelope” of expected payoffs (and any larger payoffs) to the other player.

Nondegeneracy of a bimatrix game (A, B) can be stated in terms of the polytopes P and Q as no point in P has more than m labels, and no point in Q has more than n labels. (If x ∈ P and x has support of size k and L is the set of labels of x, then |L ∩ M| = m − k, so |L| > m implies x has more than k best responses in L ∩ N. Then P and Q are simple polytopes, because a point of P, say, that is on more than m facets would have more than m labels. Even if P and Q are simple polytopes, the game can be degenerate if the description of a polytope is redundant in the sense that some inequality can be omitted, but nevertheless is sometimes binding. This occurs if a player has a pure strategy that is weakly dominated by or payoff equivalent to some other mixed strategy. Non-simple polytopes or redundant inequalities of this kind do not occur for “generic” payoffs; this illustrates the assumption of nondegeneracy from a geometric viewpoint. (A strictly dominated strategy may occur generically, but it defines a redundant inequality that is never binding, so this does not lead to a degenerate game.) Because the game is nondegenerate, only vertices of P can have m labels, and only vertices of Q can have n labels. Otherwise, a point of P with m labels that is not a vertex would be on a higher dimensional face, and a vertex of that face, which is a vertex of P, would have additional labels. Consequently, only vertices of P and Q have to be inspected as possible equilibrium strategies. Algorithmically, if the input is a nondegenerate bimatrix game, and output is an Nash equilibria of the game, then the method employed for each vertex x of P − {0}, and each vertex y of Q − {0}, if (x, y) is completely labeled, the output then is the Nash equilibrium (x · 1/1x, y · 1/1y).

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Mailvox: The Origins of the Alt-Retard

Alt-Left-perspectives3

The reaction to degeneracy can sometimes happen within the spirit of degeneracy. Genocide is not the morally wholesome solution to whoredom. The Marxist-Lenninsts regard Fascism as form of bourgeois reaction. That is their frame, it is how they like to position their argument as it emphasises the difference between the two, but I think it is far better to think of Socialism as Left Modernism and Fascism as being Right Modernism. With Left and Right being dispositional/temperamental distinctions. They might be different teams but they’re both playing the same game.

A Generation X reader sent me this analysis of the Fake Right Clown Posse, which somehow manages to be both sympathetic of the plight being faced by the young men of today and contemptuous of what some of them have become in response. I think he is largely correct, and explains why their attempts to defend their race and their nations so often go awry.

We have no choice but to help them. The challenge is that the only answer to ignorance is information, and as we know, as we have witnessed, there are some who cannot be instructed by information.

Mailvox: The Origins of the Alt-Retard

Cultural Alchemy: Berlin Sin City of the 1920s

Berlin was a cesspit of degeneracy and vice power primarily by the demand of rich and middle class patrons who had their services supplied by the poor, who chose this option due to their desperate poverty.  Anyone wanting to get a measure of the moral squalor of Germany during those years should look at the magazine Simplicissimus which would make comment of the social problems of Germany at that time through the medium of art. Long term sufferers of this blog will know that I am a huge fan of the work of Otto Dix and George Grosz, men whom I wouldn’t of shared political affinity with but men who were nonetheless disgusted at the moral abyss which Germany had fallen into after the First World War.  Their work is profoundly disturbing, disgusting and degenerate until you realise that that was what they were trying deliberately get across in their work. Berlin, especially was a morally destitute city. It’s easy to see how the moral revulsion generated by the antics of Berlin would engender a lot of sympathy for a man like Hitler. Purging the filth, regardless of the details, becomes very appealing. Just saying Social Pathologist.