COMPLEXITY OF HEDONIC GAMES

Transcription

1 COMPLEXITY OF HEDONIC GAMES WITH DICHOTOMOUS PREFERENCES Dominik Peters University of Oxford advised by Edith Elkind This work will be presented at AAAI 2016.

2 Hedonic Games [Drèze and Greenberg, 1980; Banerjee et al., 2001; Bogomolnaia and Jackson, 2002] Finite set N of agents, each agent having preferences < i over groups of agents: i 2 N {1, 2} 1 {1, 2, 3} 1 {1} 1 {1, 3} Outcome: a partition of the agents.

3 Hedonic Games [Drèze and Greenberg, 1980; Banerjee et al., 2001; Bogomolnaia and Jackson, 2002] Finite set N of agents, each agent having preferences < i over groups of agents: i 2 N {1, 2} 1 {1, 2, 3} 1 {1} 1 {1, 3} Outcome: a partition of the agents.

4 Single-Agent Stability A partition is Nash stable (NS) if no agent wants to join a different coalition. A partition is individually stable (IS) if no agent can join a different coalition, all members of which welcome the new agent.

5 Group Stability A partition is core stable (CR) if no group S of agents all strictly prefer that group to where they are in the partition.

6 Preference Representation Assign utilities to agents, not coalitions: v i : N! R. Additively Separable Games: S < i T () X j2s v i (j) > X j2t v i (j). We can also take the average (Fractional Hedonic Games), the minimum (W-hedonic games), the median, and others.

7 SNS SCR CORE NASH IS Sum ( Additively Sep. ) Average ( FHGs ) Minimum (no ties) ( W-preferences ) Minimum ( W-preferences ) NP-hard NP-hard [Sung, Dimitrov 10] NP-hard [Sung, Dimitrov 10] NP-hard NP-hard NP-hard [Brandl, Brandt, Strobel 15] NP-hard [Sung, Dimitrov 10] [Brandl, Brandt, Strobel 15] (poly-time) (poly-time) [Aziz, Harrenstein, Pygra 12] NP-hard [Cechlárová, Hajduková 04] [Aziz, Harrenstein, Pygra 12] [Sung, Dimitrov 10] [Brandl, Brandt, Strobel 15] [Aziz, Harrenstein, Pygra 12] WB-preferences NP-hard NP-hard Median NP-hard NP-hard List all Acceptable ( IRCL ) NP-hard NP-hard NP-hard [Ballester 04] [Ballester 04] [Ballester 04] Weighted Formulas ( HC-nets ) NP-hard NP-hard NP-hard [Elkind, Wooldridge 09] Social FHGs NP-hard NP-hard (poly-time) (poly-time) Midrange (½B + ½W) NP-hard NP-hard 4-Approval NP-hard NP-hard NP-hard

8 Theorem. (Informal Statement) Instance: a (concisely represented) hedonic game Question: does the game admit stable outcome? This problem is NP-hard whenever agents may order coalitions {i, j} arbitrarily and agents may have enemies such that they dislike any coalition containing enemies. from Peters and Elkind (2015, IJCAI): Simple Causes of Complexity in Hedonic Games

9 DICHOTOMOUS PREFERENCES {1, 2} 1 {1, 3} 1 {1, 2, 3} 1 {1}, {2, 3} 2 {2, 1} 2 {1, 2, 3} 2 {2}, {3, 1} 3 {3, 2} 3 {1, 2, 3} 3 {3}. {1, 2} 1 {1, 3} 1 {1, 2, 3} 1 {1}, {2, 3} 2 {2, 1} 2 {1, 2, 3} 2 {2}, {3, 1} 3 {3, 2} 3 {1, 2, 3} 3 {3}.

10 Theorem: Every hedonic game with dichotomous preferences admits an outcome that is simultaneously core stable (resistant to group deviations) individually stable (resistant to single-agent deviations).

11 Theorem: Every hedonic game with dichotomous preferences admits an outcome that is simultaneously core stable (resistant to group deviations) individually stable (resistant to single-agent deviations). Proof: Repeatedly find a maximal (w.r.t. set inclusion) coalition that is approved by everyone in it. Once there are no such coalitions anymore, put all the remaining players (the losers) into one coalition.

12 BOOLEAN HEDONIC GAMES Dichotomous Preferences can be specified with propositional logic: every agent has a goal. Example: Agent i approves a coalition S N if and only if Alice Bob (Charlie (David Eve)) This is the model studied by Aziz, Lang, Harrenstein, Wooldridge (2014), Boolean Hedonic Games

13 COMPLEXITY RESULTS SW PF PO NS IS CR SCR Boolean NP-c. NP-c. NP-h. NP-c. P FNP-h. p 2 -c. 1-lists NP-c. P P P P P NP-c. 2-lists NP-c. P P P P P NP-c. 3-lists NP-c. NP-c. NP-h.? P P NP-c. 4-lists NP-c. NP-c. NP-h. NP-c. P P NP-c. Anonymous NP-c. NP-c. NP-h. NP-c. P P NP-c. Intervals P P P? P P? Roommates P P P NP-c. P P P Majority? P? P P P P

14 MAJORITY PREFERENCES Suppose the agents form the vertices of a graph, and any two agents who have similar opinions are connected by an edge. An agent i approves those coalitions in which i is connected to a majority of agents in the coalition, so that majority decisions in the group tend to favour agent i. Formally, a coalition S N is approved by i if d G[S] (i) > S /2 In hedonic games with these preferences, an outcome that is both core- and Nash-stable always exists, and can be found in polynomial time.

15 COMPLEXITY OF HEDONIC GAMES WITH DICHOTOMOUS PREFERENCES Dominik Peters University of Oxford advised by Edith Elkind This work will be presented at AAAI 2016.