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

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.

[1]  Noga Alon,et al.  Ranking Tournaments , 2006, SIAM J. Discret. Math..

[2]  Felix A. Fischer,et al.  Possible and necessary winners of partial tournaments , 2012, AAMAS.

[3]  Felix A. Fischer,et al.  On the Hardness and Existence of Quasi-Strict Equilibria , 2008, SAGT.

[4]  Claire Mathieu,et al.  How to rank with few errors: A PTAS for Weighted Feedback Arc Set on Tournaments , 2006, Electron. Colloquium Comput. Complex..

[5]  Vincent Conitzer,et al.  Computing Slater Rankings Using Similarities among Candidates , 2006, AAAI.

[6]  Vincent Conitzer,et al.  Approximation Algorithm for Security Games with Costly Resources , 2011, WINE.

[7]  Vincent Conitzer,et al.  Vote elicitation: complexity and strategy-proofness , 2002, AAAI/IAAI.

[8]  Sarit Kraus,et al.  Using Game Theory for Los Angeles Airport Security , 2009, AI Mag..

[9]  Claire Mathieu,et al.  Electronic Colloquium on Computational Complexity, Report No. 144 (2006) How to rank with few errors A PTAS for Weighted Feedback Arc Set on Tournaments , 2006 .

[10]  Anders Yeo,et al.  The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments , 2006, Combinatorics, Probability and Computing.

[11]  Yoav Shoham,et al.  Internal implementation , 2010, AAMAS.

[12]  Bhaskar Dutta,et al.  Comparison functions and choice correspondences , 1999 .

[13]  Moshe Tennenholtz,et al.  k-Implementation , 2003, EC '03.

[14]  Milind Tambe,et al.  Security and Game Theory: IRIS – A Tool for Strategic Security Allocation in Transportation Networks , 2011, AAMAS 2011.

[15]  Milind Tambe,et al.  TRUSTS: Scheduling Randomized Patrols for Fare Inspection in Transit Systems , 2012, IAAI.

[16]  Felix A. Fischer,et al.  Computing the minimal covering set , 2008, Math. Soc. Sci..

[17]  Bo An,et al.  PROTECT - A Deployed Game Theoretic System for Strategic Security Allocation for the United States Coast Guard , 2012, AI Mag..

[18]  M. Breton,et al.  The Bipartisan Set of a Tournament Game , 1993 .

[19]  Jérôme Lang,et al.  Voting procedures with incomplete preferences , 2005 .

[20]  Jörg Rothe,et al.  The Complexity of Computing Minimal Unidirectional Covering Sets , 2009, Theory of Computing Systems.

[21]  Jean-François Laslier,et al.  More on the tournament equilibrium set , 1993 .

[22]  David C. Fisher,et al.  Optimal strategies for a generalized “scissors, paper, and stone” game , 1992 .

[23]  J. Harsanyi Oddness of the number of equilibrium points: A new proof , 1973 .

[24]  Neil Burch,et al.  Heads-up limit hold’em poker is solved , 2015, Science.

[25]  Vincent Conitzer,et al.  Determining Possible and Necessary Winners under Common Voting Rules Given Partial Orders , 2008, AAAI.

[26]  Toby Walsh,et al.  Winner determination in voting trees with incomplete preferences and weighted votes , 2011, Autonomous Agents and Multi-Agent Systems.