Outstanding conjectures and open problems in (algorithmic) game theory?
Here are several open problems:

1-Major open problem is the problem of computing approximate Nash equilibria.

2- Existence of efficient algorithm for computing pure Nash equilibria in congestion
games?

3-finding equilibria with minimum inefficiency?

4-Tim Roughgarden, in Communications of the ACM , posed the following open problem:


  to what extent is “incentive-compatible” efficient computation fundamentally less powerful
  than “classical” efficient computation?


Algorithmic game theory,
Communications of the ACM, Volume 53 , Issue 7,  (July 2010)

Also, these references contains some open problems:
Nisan, Roughgarden, Tardos, and Vazirani, editors. Algorithmic Game Theory. Cambridge
University Press, 2007.

T. Roughgarden. Algorithmic game theory: Some greatest hits and future directions. In TCS ’08, p. 21–42.