Algorithmic Game Theory. Alexander Skopalik

Transcription

1 Algorithmic Game Theory Alexander Skopalik

2 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

3 Nash s Theorem

4 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.

5 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.

6 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.

7 End-of-a-line

8 Step 2: Subdivision - Simplifying assumptions. For simplicity, we only consider only D = R 2 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.

9 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 Thanks to Paul Goldberg.

10 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.

11 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

12 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

13 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.

14 We start with a valid Sperner Coloring. Sperner s Lemma

15 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.

16 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.

17 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.

18 We can do the same thing with the red-blue edges Sperner s Lemma

19 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.

20 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.

21 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

22 Summary We have shown that finding an (approximate) Nash equilibrium is in PPAD. In fact finding a Nash equilibrium is PPADcomplete even for games with two players. What does this mean?