Algorithmic Game Theory
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In the last few years game theory has had a substantial impact on computer science, especially on Internet- and e-commerce-related issues. More than 40 of the top researchers in this field have written chapters that go from the foundations to the state of the art. Basic chapters on algorithmic methods for equilibria, mechanism design and combinatorial auctions are followed by chapters on incentives and pricing, cost sharing, information markets and cryptography and security. Students, researchers and practitioners alike need to learn more about these fascinating theoretical developments and their widespread practical application.
by seeking supports. Unfortunately, none of them are known to be efficient – to always succeed after only a polynomial number of steps. 2.2 Is the NASH Equilibrium Problem NP-Complete? Computer scientists have developed over the years notions of complexity, chief among them NP-completeness (Garey and Johnson, 1979), to characterize computational problems which, just like Nash and satisfiability,2 seem to resist efficient solution. Should we then try to apply this theory and prove that Nash is
and Papadimitriou, 2005) thus proving the first part of this conjecture. The proof in Daskalakis and Papadimitriou (2005) was local, and worked by modifying the gadgets so that the conflict graph became three-colorable; this approach had therefore reached its limit, because for the graphical game to work the conflict graph must contain triangles. It was again conjectured in Daskalakis and Papadimitriou (2005) that the two-player case can be solved in polynomial time. In contrast, the proof in
Players can then randomize locally with behavior strategies. This classic theorem (Corollary 3.12) is turned into an algorithm with the sequence form (Sections 3.10 and 3.11) which is a strategic description that has the same size as the game tree. We give in this chapter an exposition of the main ideas, not of all earliest or latest developments of the subject. Section 3.12 summarizes the main references. Further research is outlined in Section 3.13. 3.2 Bimatrix Games and the Best Response
(3.17) this means that not only the m nonbasic variables are zero, but also at least one basic variable. Such a degenerate basic feasible solution results from a pivoting step where the leaving variable (representing the label that is picked up) is not unique. As an example, consider the 3 × 2 game ⎡ ⎤ ⎡ ⎤ 3 3 3 3 A = ⎣2 5⎦ , B = ⎣2 6⎦ , (3.18) 0 6 3 1 which agrees with (3.3) except that b15 = 3. The polytope Q for this game is the same as before, shown on the right in Figure 3.2. The polytope P
describe NashProp informally is to say that each V does this computation once for each of its neighbors, each time “pretending” that this neighbor plays the role of the downstream neighbor W in ApproximateTreeNash, and the remaining neighbors play the roles of the upstream Ui . If all discretized table entries are initialized to the value of 1,1 it easy to show that the only possible effect of these local computations is to change table values from 1 to 0, which in effect refutes conditional NE