Cs294-1 Algorithmic Aspects of Game Theory 2.1 Overview 2.2 Sperner's Lemma

CS 294 - 1 Algorithmic Aspects of Game Theory Spring 2001 Lecture 2 : January 23

We begin by looking at a set of theorems from various disciplines and how they relate to one another. From combinatorics, we take Sperner’s Lemma which we can use to prove Brouwer’s Fixed Point Theorem from topology. Brouwer’s Fixed Point Theorem can be used to prove the Arrow-Debreu Theorem from economics which states that general equilibria exist, and can also be used to prove Kakutani’s Fixe...

CS294-1 Algorithmic Aspects of Game Theory Spring 2001

Cs294-1 Algorithmic Aspects of Game Theory 4.1 Overview 4.2 Social Choice Theory

• Dictatorship: ( 1, . . . , n) = 1 • Democracy: When |C| = 2, c1 ( 1, . . . , n) c2 iff |{i : c1 i c2}| ≥ |{i : c2 i c1}|. Theorem 4.3. When |C| = 2, majority voting is the only social welfare functional that is monotone (an agent’s preference change from c1 to c2 does not cause the global preference to change from c2 to c1) and symmetric with respect to agents (permuting the agents does not c...

PPAD-completeness of polyhedral versions of Sperner's Lemma

We show that certain polyhedral versions of Sperner’s Lemma, where the colouring is given explicitly as part of the input, are PPAD-complete. The proofs are based on two recent results on the complexity of computational problems in game theory: the PPAD-completeness of 2-player Nash, proved by Chen and Deng, and of Scarf’s Lemma, proved by Kintali. We show how colourings of polyhedra provide a ...

Algorithmic Game Theory Lecture 7 Lecturer :

That is, C operates on n coordinates x1, . . . , xn and outputs a color. We can see how having such a circuit is equivalent to a unique coloring of the cube: if we want to find the color at a vertex, just give the circuit the coordinates of the vertex, and use the output of the circuit. The Sperner problem then becomes the problem of finding a panchromatic simplex when given a coloring circuit ...