Computing Possible and Necessary Equilibrium Actions (and Bipartisan Set Winners)

Markus Brill, Rupert Freeman, Vincent Conitzer

Abstract:

In many multiagent environments, a designer has some, but
limited control over the game being played. In this paper, we
formalize this by considering incompletely specified games,
in which some entries of the payoff matrices can be chosen
from a specified set. We show that it is NP-hard for the
designer to make this choices optimally, even in zero-sum
games. In fact, it is already intractable to decide whether a
given action is (potentially or necessarily) played in equilibrium.
We also consider incompletely specified symmetric
games in which all completions are required to be symmetric.
Here, hardness holds even in weak tournament games
(symmetric zero-sum games whose entries are all −1, 0, or 1)
and in tournament games (symmetric zero-sum games whose
non-diagonal entries are all −1 or 1). The latter result settles
the complexity of the possible and necessary winner problems
for a social-choice-theoretic solution concept known as
the bipartisan set. We finally give a mixed-integer linear programming
formulation for weak tournament games and evaluate
it experimentally