alexander skopalik - heinz nixdorf institut · sperner’s lemma lemma (sperner’s lemma) every...
Post on 06-Aug-2020
20 Views
Preview:
TRANSCRIPT
Algorithmic Game Theory
Alexander Skopalik
Today
1. Strategic games (Normal form games) – Complexity
2. Zero Sum Games – Existence & Complexity
Outlook, next week:
• Network Creation Games Guest lecture by Andreas Cord-Landwehr
Nash‘s Theorem
Proof of Nash‘s Theorem
• Idea: Construct a function 𝑓: 𝑋 → 𝑋 (with 𝑋 being set of mixed states) that has a fixpoint 𝑓 𝑥∗ =𝑥∗ if and anly if 𝑥∗is a mixed Nash equilibrium.
• Show: 𝑋 and 𝑓 satisfies the conditions for Brower’s Theorem.
• Proof: a fixpoint 𝑥∗is a mixed Nash equilibrium and vice-versa.
• Then a fixpoint exists and, thus, a mixed equilibrium.
Complexity of NASH
Theorem: Finding a Nash equilibrium is PPAD-complete. 1. Nash is in PPAD.
It is not harder than for example End-of-a-line. Show that you reduce the problem NASH to End-of-a-line (and then you could use an algorithm for end of the line).
2. Nash is PPAD-hard. It is at least as hard as End-of-a-line. Reduce End-of-a-line to NASH, so you could use an algorithm for NASH to solve End-of-a-line.
We only show 1. The proof of 2. is very involved.
Nash is in PPAD
Lemma: Finding an (approximate) Nash equilibrum is in PPAD.
Outline of the proof:
1. Reduction: Finding fixed points
2. Subdivide the space into finite number of smaller areas and apply some fancy coloring
3. Use End-of-a-line to find an area close to a fixed point.
End-of-a-line
Step 2: Subdivision - Simplifying assumptions.
• For simplicity, we only consider only 𝐷 = ℝ2 and functions 𝑓: 0,1 2 → 0,1 2
• Furthermore, we transform 0,1 2 to a triangle 𝑇:
and consider the corresponding problem on 𝑇.
Step2: Subdivision
• Divide T into smaller triangles.
• Draw an arrow at the vertices into the direction of f.
These pictures and a nice introduction to PPAD can be found on http://cgi.csc.liv.ac.uk/~pwg/PPADintro/PPADintro.html Thanks to Paul Goldberg.
Step2: Subdivision
• Give the 3 outermost (extremal) vertices of the triangle the 3 colors red, green and blue
• Color the vertices according to the direction of the arrows.
• Each vertex with an arrow gets colored with a color of an extreme vertex that it is moving away from.
• Note: vertices on the edge between the outermost red and green vertices will be either red or green. Similarly for vertices on edges between other vertices.
• Intuitively, fixpoints of 𝑓 lie in vicinity of small triangles whose 3 vertices get 3 different colors.
• This is because the points are being dragged in 3 different directions
• By continuity, if we triangulate at a sufficiently fine resolution, we converge to a fixpoint.
• Here: 3 small triangles with vertices of all colors, marked with black spots
Sperner’s Lemma
Definition: A Sperner coloring of the vertices of a subdivided triangle satisfies: • Each extremal vertex gets a
different color. • A vertex on a side of the
largest triangle gets a color of one of the corresponding endpoints.
• Other vertices are colored arbitrarily.
Lemma (Sperner’s Lemma) Every Sperner coloring of a subdivided triangle contains a trichromatic triangle.
Sperner’s Lemma
Lemma (Sperner’s Lemma)
Every Sperner coloring of a subdivided triangle contains a trichromatic triangle.
• If we can find a trichromatic triangle, we find an (approximate) Nash equilibrium.
• The proof of Sperner‘s Lemma tells us how to find one using End-of-a-line.
Sperner’s Lemma
• We start with a valid Sperner Coloring.
Sperner’s Lemma
• We start with a valid Sperner Coloring
• We extend the triangulation by connecting some extremal vertex to all vertices along one of its incident edges.
Sperner’s Lemma
• We start with a valid Sperner Coloring
• We extend the triangulation by connecting some extremal vertex to all vertices along one of its incident edges.
• Treat each red-green edge as having a gateway through it.
Sperner’s Lemma
• We start with a valid Sperner Coloring
• We extend the triangulation by connecting some extremal vertex to all vertices along one of its incident edges.
• Treat each red-green edge as having a gateway through it.
Sperner’s Lemma
• We can do the same thing with the red-blue edges
Sperner’s Lemma
• We can do the same thing with the red-blue edges
• And again we find a trichromatic triangle by following the path through the gateways.
Sperner’s Lemma
We can construct a directed graph:
• vertices are the tiny triangles
• edges are pairs of adjacent triangles connected by a red-blue edge.
Sperner’s Lemma
• The graph has one known source: The outer plane.
• Each vertex has in indegree and outdegree at most one.
• We can construct simple algorithms/circuits that compute successor and predecessor.
• Now we have an instance of End-of-a-line
Summary
We have shown that finding an (approximate) Nash equilibrium is in PPAD.
In fact finding a Nash equilibrium is PPAD-complete – even for games with two players.
What does this mean?
top related