For convenience, we will as- Proof. Consequently, the algorithmic normal vector is nonnegative. Similarly a face of any di- problem of finding Nash equilibria can be tackled by con- mension is called useful if it is contained in some useful sidering problems relating to the number of points on the facet.
Our interest in useful facets comes from the fact that convex hull of a set of n random points in d dimensions. In only nonnegative vectors induce feasible mixed strategies. Section 3 we extend analysis of such random polytopes for In particular, we obtain from Lemma 1 the following corol- our purpose. This allows us to examine the quality of our lary.
In particular, consider the convex Corollary 2. Then hull of n random points in d dimensions. Then, our main theorem can be stated as fol- lows. Let Theorem 1. Since the distributions we consider and so we get our algorithmic the entries of A and B are chosen at random, the corre- result as a corollary. In particular, we need to study the faces of random polytopes. Finally, we remark that extending our work to al- points N1 is number vertices, Nd number of facets. Again, low matrix entries to have arbitrary means would give a since the points are in general position any such face is in- polynomial-time randomized algorithm for finding approx- duced by exactly i points.
We have the following simple imate Nash equilibria in arbitrary games. We begin by showing how Nash equilibria are closely re- lated to random polytopes.
Take a vertex x and consider any d facets containing x. We will view the elements of sion s that contain x. The union of these sets may contain at B S as points in Rd , and denote by conv B S the polytope most d copies of any face.
Moreover, as all the points are in corresponding to the convex hull of these points. Thus, summing over all vertices, we obtain the result. Lemma 1. We will use the following notation. We write Theorem 4. We will need the simple only on d. Then there are convex sets C1 ,. First we need to develop some understanding of the behaviour of Ni. There has already been a large amount of work studying Ni for various distributions; for a rather comprehensive survey see [39].
Theorem 3. This is points from a distribution with density function f. Let F be only the covering near a single vertex, the origin, of the unit the set of subsets of points that induces facets of conv P , cube, which is extended to a complete covering using the and let Y1 and Y2 be two subsets of X. What is the probabil- symmetries of the cube in the obvious way. One can define ity that both Y1 and Y2 induce facets, i. Such a covering exists understood for many important distributions, including the for the Normal distribution if the complement of a suitably Gaussian and the cube.
We prove the following refinements large ball is deleted. We will use cap coverings to prove the following gener- alization of Lemma 4.
We focus now on the case when K is Lemma 3. Suppose f is the Normal density or the uniform the unit cube, but the proof applies whenever a cap covering density over a cube. If Y1 , Y2 are disjoint subsets of P, then exists.
Lemma 4. Assume x1 ,. For a cube, however, things Proof. It follows from are more complicated and we will use an economic cap- the main theorem in [5] that the random polytope conv P covering [6].
For a fixed Ch , it is not hard to see that N f, g. We replace this restricted integral by a double sum in the following way. Let Mf1 be the set of these elements of Mf. For smooth convex bodies N f, g is a constant, and the computation is simpler. We prove the lemma directly for the Normal density.
Recall that the algorithm exhaustively checks for Nash equilibria with supports of size 1, then for Nash equi- We now estimate this when f is the standard Normal libria with supports of size 2, etc. There are nd pairs of supports of size d and de- tance of H Y1 from the origin. So, it will be convenient to termining whether a pair of supports induce a Nash equilib- parametrize in terms of hyperplanes, and positions on them.
This is achieved by the Blaschke-Petkantschin formula. For Thus, provided that our game has a Nash equilibrium in- a spherically symmetric density function f we have, duced by supports of constant size, we obtain a polynomial time algorithm. More generally, we consider the probability that there is Z g x1 ,. Let S and x1 , Note that S and T induce a Nash equilibrium, H from the origin. The integrand on the RHS of 1 the payoff matrices for Alice and Bob, the probability that depends only on the distance of H Y1 from the origin.
So libria. Summing up over all pairs, S, T , Claim 1. We will use the notation Proof. The result then follows from Lemma 3. Similarly, applying Lemma 4, or Lemma 5, we obtain Lemma 7. We remark that only the cardinalities of these in- tersections will be of consequence. Theorem 1 now follows from 2 and Lemma 7.
If a pair of strategies. The normals to these facets also give the probability distri- However, butions on the strategy supports at the Nash equilibrium. Our result is unaffected by.
The probability of hitting any of those points exactly is the sum of hitting each of those points. But since each point has a probability measure of 0, the sum is also 0. Now if a dart were thrown and landed exactly at the origin, that is an event that is possible. If you think about the set of finite games as the dartboard, then the games that have an even or infinite number of solutions are like the collection of single points technical point: a line in a dartboard also has measure zero.
Games with an even number of Nash equilibria certainly exist, and the set can even be a collection of an infinite number of items. However, these games are a set of measure zero relative to the entire set of games.
However, some professors are purposefully tricky and will present games that have an even number of solutions. The flip side is you should not rely on games having an odd number. Do not fall for the Oddness Theorem fallacy. I run the MindYourDecisions channel on YouTube , which has over 1 million subscribers and million views. As you might expect, the links for my books go to their listings on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.
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