Proceedings of the Thirtieth International Florida Artificial Intelligence Research Society Conference

Optimizing Expected and Stability in Role Based Hedonic Games

Matthew Spradling University of Michigan - Flint mjspra@umflint.edu

Abstract Table 1: Ex. RBHG instance with |P | =4,R= {A, B} In the hedonic coalition formation game model Roles Based r, c u (r, c) u (r, c) u (r, c) u (r, c) Hedonic Games (RBHG), agents view teams as compositions p0 p1 p2 p3 of available roles. An agent’s utility for a partition is based A, AA 2200 upon which roles she and her teammates fulfill within the A, AB 0322 coalition. I show positive results for finding optimal or sta- B,AB 3033 ble role matchings given a partitioning into teams. In set- B,BB 1111 tings such as massively multiplayer online games, a central authority assigns agents to teams but not necessarily to roles within them. For such settings, I consider the problems of op- timizing expected utility and expected stability in RBHG. I As per (Spradling and Goldsmith 2015), a Role Based He- show that the related optimization problems for partitioning donic Game (RBHG) instance consists of: are NP-hard. I introduce a local search heuristic method for • P approximating such solutions. I validate the heuristic by com- : A population of players parison to existing partitioning approaches using real-world • R: A set of roles data scraped from League of Legends games. • C: A set of compositions, where a composition c ∈ C is a multiset (bag) of roles from R. Introduction • U : P × R × C → Z defines the utility function ui(r, c) In coalition formation games, agents from a population have for each player pi. Generally, I assume that for all pi ∈ P various for different partitions, where a partition is a and for all r ∈ R, ui(r, {r})=0. set of teams (subsets of agents). Hedonic games are coalition A solution to an RBHG instance is a partition π of each formation games in which an agent’s for a parti- agent into a team (set of agents) t and a matching M of tion is determined only by the coalition to which the agent agents to roles in R. Because RBHG is a hedonic game, util- is assigned (Sung and Dimitrov 2010). Hedonic games are ity of a player for a partition π and matching M is equal to used to model matching of agents to teams when agents only i i i that player’s utility for (π(pi),M(pi))=(r ,c ,t ). I will care about the utility of their own teams, not others. i i refer to a player pi’s utility for (π, M) as ui(r ,c ). Stability of an assignment is of primary importance when This work is motivated by team formation among au- agents are able to leave the team if they are not satisfied tonomous drones (Hock and Schoellig 2016) and by hu- (Spradling and Goldsmith 2015). While a utility-function man team formation in massively multiplayer online games maximizing partition is attractive, forming such a partition (MMOs). A drone may be specially suited to fulfilling a sub- is not helpful if the agents cannot be guaranteed to keep the set of roles needed by a local swarm. It must efficiently de- assignment. An optimal assignment may not be stable. It has termine which group to assist and in what capacity as new been observed that, for some stability measures, there may tasks arise. In the paradigm of MMOs, the top five online be multiple stable assignments with variable utility levels games in the world each incorporate some variation of role (Spradling et al. 2013). A stable assignment is not guaran- based team formation. Including Overwatch and League of teed to have an optimal utility among all stable assignments. Legends, these games accounted for 259.22 million hours of In this paper, I consider the variant Role Based Hedonic combined viewing over the Twitch streaming service world- Game (RBHGs) (Spradling and Goldsmith 2015). In this wide in January, 2017 alone (Gamoloco 2017). In both set- model, an agent’s utility for a partitions is based upon the tings, role assignments are chosen by the agents after teams role it fulfills on its team and the roles fulfilled by its team- or coordinates are determined by a central authority. An mates. The multiset of roles fulfilled by a team is termed its agent then chooses a role depending upon an internal utility composition. function. For this reason, I focus on the problems of finding Copyright c 2017, Association for the Advancement of Artificial partitions which have optimal expected stability and optimal Intelligence (www.aaai.org). All rights reserved. expected utility over the possible matchings.

134 I provide polynomial time algorithms for finding optimal vide a model in which stable partitions of several forms can and stable matchings M given a partitioning π of agents be found when the agents seek to help or avoid harming al- into teams. I consider the complexity of recognizing parti- lies in the network (Nguyen et al. 2016). tions with optimal expected stability and optimal expected Past work considers a scenario where all partitioning is utility, both of which I prove to be NP-complete for Max- performed by the central authority. In massively multiplayer Sum and MaxMin optimization. I introduce a greedy local online games, agents may be partitioned to teams by the search heuristic which runs in polynomial time. I validate server but not necessarily the roles within them. Once parti- the heuristic by comparison to “what if analysis” and a pre- tioned, the matching is determined by the agents while the viously described greedy voting heuristic (Spradling et al. game is in progress. Even if agents are assigned to a par- 2013) using real world matchmaking data from League of tition which contains an optimal and stable matching, it is Legends games. The experiments show improvements to ex- not guaranteed that the agents will stabilize on that partic- pected utility, expected stability, and to the percentage of ular matching. The match may end, or someone may quit teams with a stable matching. In particular, greedy local playing, before agents discover how they can collaborate. search identifies stable role and team recommendations for For human agents, I consider the problem of finding par- over 50% of players in the test population. These results in- titions in which players are most likely to stabilize upon a dicate that greedy local search can be applied to real world matching. I call this the optimal expected stability partition. RBHG matchmaking scenarios. Similarly, I consider the problem of finding the optimal ex- pected utility partition in which the average optimal compo- Related Work: Stability and Optimization in sition utility for each team is maximized. Coalition Formation Games Expected Utility and Stability In , hedonic coalition formation games model sit- Given a partition of agents to a particular team t, the utilities uations where agents join teams to collaboratively produce of the agents within t over the roles within some composi- goods for themselves (Dreze` and Greenberg 1980). Hedonic tion c can be stored in a |t| x |t| matrix. One such matrix for game models have been applied wide ranging fields, includ- each of C compositions represents the utilities of the agents ing distributed task allocation in wireless agents (Saad et al. of t for each composition. 2011a), communications networks (Saad et al. 2010), vehic- ular networks (Saad et al. 2011b; Xu et al. 2015), adaptive Definition 1 Given a partition π of an instance B of RBHG, smart grid management (Kim 2014), federation formation a team t ∈ π, and a composition c ∈ C, the optimal utility among cloud providers (Guazzone, Anglano, and Sereno of t given c is the matching of agents to roles within c which 2014), and measuring impact on perceived learning out- maximizes the sum of the utilities for all agents on that team. o comes (Alexiou and Doerga 2015). An optimal composition of t is a composition ct ∈ C for For general hedonic games, finding a stable coalition which the optimal utility of t for all c ∈ C is maximized. structure is NP-complete (Ballester 2004). Farsighted stabil- Theorem 1 When agents are matched into roles by a cen- ity in general hedonic games considers a case where agents t avoid defecting from a team if the defection would induce a tral authority, an optimal or stable composition for a team worse change by agents on their new or former team (Dia- can be identified in polynomial time. mantoudi and Xue 2003). It has been shown that core stable Proof 1 The optimal utility of a team t given a composi- and Nash stable solutions are farsighted stable while individ- tion c can be computed in time O(|t|3) by application of ually stable and contractually individually stable solutions the Kuhn Munkres algorithm for optimizing square matrices do not guarantee this (Diamantoudi and Xue 2003). (Edmonds and Karp 1972). o Work in role based hedonic games first considered the The optimal composition ct ∈ C of t can be identified in roles and teams hedonic game (RTHG) model, a special case time O(|C|·|t|3) by computing optimal utility for each c ∈ of RBHG in which all compositions are of equal size. Even C. When agents have 1-0 “accept/-reject” utilities, a stable under this constraint, finding a perfect, maxsum, or maxmin composition will have optimal utility 1 for each pi ∈ t.  optimal partition is NP-hard (Spradling et al. 2013). While Without a central authority to assign agents to roles within optimization would be valuable, agents in online games may a team t, agents in t will either stabilize upon a composition choose not to play and reject an unstable partition. c or one or more agents will defect from the team t. There- For general RBHG several stable partitioning problems fore, I consider evaluating the quality of a team t by either are NP-complete or coNP-complete (Spradling and Gold- its expected utility or expected stability across compositions. smith 2015). Similar results have been found for stability a and optimization problems related to additively separable Definition 2 An acceptable composition ct for a team t is a hedonic games (Sung and Dimitrov 2010; Aziz, Brandt, and one for which up(r, ct ) > 0 for at least one agent in p ∈ t. Seedig 2010; 2011), hedonic games represented by an in- For a 0-1 instance of RBHG where agents either accept or dividually rational list of coalitions (Ballester 2004), anony- reject each role and composition pair, a stable composition s s mous hedonic games (Banerjee, Konishi, and Sonmez¨ 2001) ct for a team t is one for which up(r, ct )=1for each the Group Activity Selection Problem (GASP) (Darmann et agent p ∈ t given an optimal utility matching of roles. al. 2012), and hedonic games with dichotomous preferences Observe that any stable composition is also an acceptable among others (Peters 2016). Altruistic hedonic games pro- composition but that the reverse is not necessarily the case.

135 s Definition 3 Let qt be the quantity of stable compositions instance A = P, R, C, U of PERFECT BINARY RBHG, a     for t and qt be the quantity of acceptable compositions for I construct an instance f(A)=P ,R,C ,U ,k of a t. The expected stability of t is 0 if qt =0or the ratio MAXMIN ES RBHG such that A ∈ PERFECT BINARY s a a qt /qt for qt > 0. RBHG iff f(A) ∈ MAXMIN ES RBHG. a a Let ut be the sum of optimal utilities for all qt ac- Let B = P, R, C, U be an instance of PER- ceptable compositions for t. The expected utility of t is 0 FECT BINARY RBHG. I construct an instance f(B)= a a a a     if qt =0or the ratio ut /qt for qt > 0. P ,R,C ,U ,k of MAXMIN ES RBHG as follows. Let P  = P , R = R, C = C, U  = U, and let k =1/|C|. The goal it this setting is to select a partition π which I claim that there is a π of f(B) s.t. each t ∈ π has optimize expected utility, expected stability, or the percent expected stability ≥ k iff there is a (π, M) of B s.t. B ∈ of teams for which there is a stable composition. PERFECT BINARY RBHG. Suppose there is a solution π of f(B) such that f(B) ∈ Complexity of Optimization MAXMIN ES RBHG. Then I construct a solution (π, M) of In this section, I define the decision problems related to find- B as follows. Let π = π. For each t ∈ π, for each c ∈ C, ing optimal expected utility and optimal expected stability find the optimal role assignment m for (c, t) in time O(|t|3) partitions in RBHG. I prove each to be NP-complete by re- and add m to M. Given the assumption that each t ∈ π duction from PERFECT RBHG. has expected stability ≥ 1/|C|, there must be at least one of the |C| compositions which has a perfect assignment for t. Definition 4 An instance of Binary RBHG is an instance of (π, M) B f(B) ∈ RBHG such that for each agent p ∈ P , for each c ∈ C, for Therefore is a perfect solution for . Thus MAXMIN ES RBHG implies that B ∈ PERFECT BINARY each r ∈ c, up(r, c) →{0, 1}. A perfect solution (π, M) of an instance of Binary RBHG RBHG. p p Suppose there is a solution (π, M) of B such that B ∈ has the property that, for each p ∈ P , up(r ,c )=1. PERFECT BINARY RBHG. Then I construct a solution π of Definition 5 The language PERFECT RBHG consists of f(B) as follows. Let π = π. Given the assumption that M those instances of RHBG for which a perfect solution ex- is a perfect matching, it must be the case that each t ∈ π has ists. PERFECT BINARY RBHG consists of those instances a perfect assignment for as least one of the |C| composi- of Binary RBHG for which a perfect solution exists. tions. Therefore π is a partition of f(B) s.t. each t ∈ π has ≥ 1/|C| B ∈ Observation 1 An instance A = P, R, C, U, m of RTHG expected stability . Thus PERFECT BINARY f(B) ∈ is an instance B = f(A)=P, R, C, U of RBHG where RBHG implies that MAXMIN ES RBHG.  the set C of compositions consists of all multisets of m roles Therefore MAXMIN ES RBHG is NP-hard. within R. Therefore, an instance A of RTHG has a perfect Definition 7 The language MAXSUM ES RBHG consists solution iff B = f(A) has the same perfect solution. There- of pairs (B,k), where B is an instance of RBHG, k is an P fore PERFECT RTHG ≤m PERFECT RBHG. integer, and there exists a partition π of B such that the sum total expected stability of all t ∈ π is ≥ k. Observation 2 PERFECT RTHG is NP-complete even for special cases where utilities are binary (Spradling et al. Theorem 3 MAXSUM EU RBHG is NP-complete. 2013). Therefore PERFECT RBHG and PERFECT BINARY Proof Sketch To show that MAXSUM ES RBHG is in NP, RBHG are both NP-hard. consider the following NP algorithm. Given an instance of Definition 6 The language MAXMIN ES RBHG consists MAXSUM ES RBHG, guess a partition π and evaluate the of pairs (B,k), where B is an instance of RBHG, k is an sum total expected stability over all teams in π. This can be 2 integer, and there exists a partition π of B such that for each computed in time O(|P |·|C|·MAX(|t|) ). Stop and reject if t ∈ π expected stability is ≥ k. the average expected utility value is

136 similar to those utilized for MAXSUM ES RBHG and Algorithm 1 GreedyLocalSearch(RBHG instance B, empty MAXMIN ES RBHG. Guess a partition then verify the ex- partition π, MAX(|t|)=m) pected utility of each team, which can be computed in time for |P |/m teams do polynomial in the size of the input. select a pivotal agent p To show NP-hardness, I show that MAXSUM ES RBHG m − 1 P for positions do ≤m MAXSUM EU RBHG and that MAXMIN EU RBHG i P set max index MAX to null ≤m MAXMIN ES RBHG.  set max score sMAX to MIN(up(r, c)) ·|P | for |P |/m remaining agents do   Heuristic Matchmaking Methods calculate expected utility sp for t ∪ p (in time In this section I outline three matchmaking methods for O(|C|·|m|3))    RBHG: “What if” analysis, greedy voting (Spradling et al. if sp >sMAX, set iMAX = p and sMAX = sp 2013), and greedy local search. end for set t = t ∪ p “What if” Analysis end for “What if” analysis generates a partition uniformly at random set π = π ∪ t and computes the value of expected stability, expected util- end for ity, or some other optimization criteria. If there is improve- ment over the previous best the new partition is kept. This procedure is repeated with a new partition until all |P |! par- Observation 3 The time complexity of greedy local search titions have been enumerated, until some threshold value δ is O(|P |2/m · f), where f is the running time of the opti- is met for the optimization criteria, or until some maximum mization function. In the case of optimizing expected utility number of partitions Δ have been evaluated. or expected stability, f = O(|C|·m3) and the total run time Each iteration of “what if” analysis requires O(|P |/m) is O(|P |2 ·|C|·m2). computations of the optimization metric being considered, m For a special case of RBHG which I call soul mates where is the maximum team size. Given that comput- RBHG, greedy local search always returns a perfect parti- ing expected utility or stability for a team has a cost of O(|C|·m3) tion. For each agent in such an instance, there is a unique set , the total cost of considering one partition for of agents perfectly complimenting its preferences. this optimization goal is O(|P |·|C|·m2).IfΔ is left un- capped, this algorithm is guaranteed to return a partition Definition 9 An instance B of soul mates RBHG is a 0-1 in- nearest to δ. In the worst case, this requires O(|P |!) itera- stance of RBHG which has the property that, for each agent tions to enumerate all partitions of P . p ∈ P , there is a unique set of agents P  ⊆ P such that each acceptable composition is also a stable composition. I call a Greedy Voting set t = p ∪ P  having this property soul mates. Greedy voting considers agent utilities as scoring rule Theorem 5 For any instance B of soul mates RBHG, style votes in an election over the available compositions Greedy Local Search always returns a perfect partition. (Conitzer and Sandholm 2005). Agent utilities for each com- positions, over the roles within that composition, are added Proof Sketch Consider the first iteration of greedy local B together. The composition c ∈ C with the highest total score search heuristic on an instance of soul mates RBHG. Se- p is selected along with the |c| agents that most preferred c. lect some agent as the pivotal agent. By definition, there P  ⊆ P This team is added to the partition and their scores for all is a unique set of agents which are soul mates with p p compositions are subtracted from the totals. The procedure . When selecting the first agent who locally optimizes tp = p ∪ p repeats with a new vote over the compositions and a new expected stability of , the agent must be some p ∈ P  team being formed until all players have been matched. The . O(|P |2/m) Observe that, by the distributive property, p ∪ P  =(p ∪ greedy voting heuristic has a running time of ,      or O(|P |·|C|·m) if |P | < |C|·m2 (Spradling et al. 2013). p ) ∪ (P ∩ p ) given that p ∈ P . Therefore, the soul mates property is preserved between p ∪ p and P  ∩ p. In sub-   Greedy Local Search sequent iterations, the remaining members of P ∩ p will be iteratively selected and added to the team tp until it is I compare to the following greedy local search heuristic. Se- completely formed and added to the partition π. lect an agent p ∈ P as the pivotal agent and locally opti- mize the chosen metric for the team t including p and some new agent p ∈ P . The local search continues to add agents Scraping League of Legends Game Data to the team t until no local improvement is available or all For my experiments, I consider a population of League of positions have been filled, at which point t is added to the Legends players where |P | = 1081. I considered a set final partition. A new pivotal agent is then selected, form- |R| =5of popular champion roles, Jungler, AD Carry, ing a new team around the pivot. This procedure is repeated Tank, Support, and Middle. These roles were identified as the for each of |P |/MAX(|t|) teams. In the following implemen- most common roles of champions (game avatars) selected by tation (Algorithm 1) I optimize expected utility, though an- the players. I scraped data from the most recent 20 matches other utility function f may be substituted in place. for each player p ∈ P from www.lolking.net, storing the

137 roles, compositions, and win/loss records for each match. I 50 selected players uniformly at random by generating account Table 2: Algorithm mean results over trials numbers from 20,000,000 to 60,000,000. This gives a range Method RT %S EU¯ EU˜ ES¯ ES˜ of accounts having been created roughly between the years W (%S) 207.4 22.4% 2.09 2.08 0.03 0 2012 and 2014. Active accounts having played 10 matches W(EU¯ ) 207.8 16.0 2.13 2.13 0.02 0 within the last 7 days were kept for further scraping, with W(ES¯ ) 207.8 22.3 2.09 2.09 0.03 0 new matches identified by the match date. V 3.8 34.3 2.10 2.13 0.05 0 I applied frequent item set mining over team compositions L (rand) 236.6 55.5 2.75 2.78 0.10 0.13 to identify those compositions which were used in at least L (max) 245.9 57.4 2.75 2.8 0.10 0.13 3% of matches, and rejected other compositions. This left a L (min) 230.5 50.5 2.78 2.82 0.09 0.13 set |C| =8of compositions used in more than 60% of all scraped matches combined. To compute a player’s utilities for role and composition Table 3: Algorithm std deviations of results over 50 trials pairs I used the following wins and losses utility function. Method RT %S EU¯ EU˜ ES¯ ES˜ Definition 10 Wins and losses utility function: For each W (%S) 4.3 0.9% 0.01 0.06 0.00 0 agent p ∈ P , for each c ∈ C set up(r, c)=1if the agent ¯ (r, c) u (r, c)=−1 W(EU) 4.6 2.3 0.01 0.01 0.00 0 won more matches with than they lost, p ES¯ if the agent lost more matches with (r, c) than they won, and W( ) 5.0 0.8 0.01 0.06 0.00 0 V 0.1 0.0 0.00 0 0.00 0 up(r, c)=0otherwise. L (rnd) 4.6 1.8 0.02 0.04 0.00 0.01 I consider an agent to accept a pair (r, c) if up(r, c)=1 and otherwise to reject it. L (max) 4.8 0.0 0.00 0.00 0.00 0 L (min) 4.8 0.0 0.00 0.00 0.00 0 Testing and Results Mean results of the experiments are presented in Table 2 ing showed improved optimization results for the heuristic while standard deviations are presented in Table 3. I per- versus random partitioning as the population size increased. formed matchmaking using “what if” analysis (W), greedy My experiments tested several different formulas for how voting (V) (Spradling et al. 2013), and greedy local search to randomize the utilities of agents. It became clear that the (L). I compare the algorithms in terms of run time (in sec- π π availability of high quality results depended on how the util- onds) to form the partition (RT), the percent of teams in ity matrix was formed. I saw that the algorithm generally with at least one stable matching (%S), mean expected util- found higher utility solutions on the scraped instances, per- ity (EU¯ ), median expected utility (EU˜ ), mean expected sta- ¯ ˜ haps because it leveraged the underlying structure of real bility (ES), and median expected stability (ES). Expected human preferences. utility ranges from −5 to 5 while expected stability ranges from 0 to 1. I consider 50 of the 52 trials performed, drop- Rate of growth for “what if” analysis ping the two with highest and lowest %S for each method. For “what if” analysis, I consider three different opti- As part of the “what if” analysis procedure I logged the num- mization goals; maximizing percent of teams with at least ber of partitions considered at each point that a new local one stable matching (%S), maximizing mean expected util- optima was discovered. I observed that the number of new ity (EU¯ ), and maximizing mean expected stability (ES¯ ). partitions which had to be considered to find an improve- ment increased by approximately O(2n) on these experi- For each optimization goal, “what if” analysis was given n a threshold of δ = |P | (perfect utility 1 for each agent) ments, where is the number of improvements found so far. and a cap of Δ=|P | partitions. Note that |P | iterations Each improvement costs exponentially more time. of “what if” analysis on these optimization functions will run in O(|P |2 ·|C|·m2). This matches the running time of Conclusions greedy local search on the same optimization functions. Over the League of Legends data set, greedy local search For greedy local search, I used expected utility as the op- out-performs both greedy voting and “what if” analysis on timization goal and considered three methods of selecting a all optimization measures I considered. As previously ob- pivotal agent: random pivoting where an agent is selected served, the cap of Δ=|P | partitions for “what if” analysis i.i.d. from P , max first pivoting which pivots around the gives it a similar running time to greedy local search with a agent maximizing qp = B up(r, c), and min first pivoting slightly smaller constant. In addition to improving expected which minimizes qp = B up(r, c). utility, greedy local search is the only one to achieve a non- Computations were run on a machine using 16 GB of zero median expected stability. In its favor, greedy voting is RAM and a 2.50 GHz Intel(R) Core(TM) i7-4710MQ CPU. significantly faster and still out-performs “what if” analysis All algorithms were implemented in Python 3.5. in terms of percent of teams with a stable matching (%S). Greedy voting could used as an alternative for split-second Testing on Randomized RBHG instances changes to a partition. I tested greedy local search on randomly generated RBHG On these experiments, greedy local search optimizes ex- instances for population sizes from 100 to 1000. The test- pected utility and expected stability at roughly the same

138 quality whether I used min first or max first pivoting. The Edmonds, J., and Karp, R. M. 1972. Theoretical improve- difference between the two is no more than the standard de- ments in algorithmic efficiency for network flow problems. viation of randomized pivoting. Pivot choice shows a more Journal of the ACM (JACM) 19(2):248–264. significant variance on the percent of teams with a stable Gamoloco. 2017. Leading gaming content on matching (%S). Here, max first pivoting improves by 1.9 twitch worldwide in january 2017, by number of hours percentage points over random pivoting and by 6.9 percent- viewed in millions. Statista - The Statistics Por- age points over min first pivoting. tal. https://www.statista.com/statistics/507786/leading- The percent of teams which can be stabilized is valuable game-content-twitch-by-number-hours-viewed/. to optimize when direct role assignment is possible, such Guazzone, M.; Anglano, C.; and Sereno, M. 2014. A game- as with drones being issued commands by home base. High theoretic approach to coalition formation in green cloud fed- expected utility and stability are valuable optimization goals erations. In Cluster, Cloud and Grid Computing (CCGrid), when a central authority cannot assume direct control of role 2014 14th IEEE/ACM International Symposium on, 618– matching, such as with human players in an online gaming 625. IEEE. environment. These experiments provide some positive re- sults for both settings. Even when direct role assignment is Hock, A., and Schoellig, A. P. 2016. Distributed iterative not possible or desirable, the optimal role assignments for a learning control for a team of quadrotors. In Decision and partition may be offered as recommendations to the agents Control (CDC), 2016 IEEE 55th Conference on, 4640–4646. once teams have been assigned. In the case of a League of IEEE. Legends game, this could take the form of suggesting par- Kim, S. 2014. An adaptive smart grid management ticular champions (avatars) the players have used well in the scheme based on the game model. ETRI Journal past and which compliment their team’s overall preferences. 36(1):80–88. Software could say, for example, “Your team could use a Nguyen, N.-T.; Rey, A.; Rey, L.; Rothe, J.; and Schend, L. great Support like you. Why not play Blitzcrank this round?” 2016. Altruistic hedonic games. In Proceedings of the 2016 Future work will test live matchmaking scenarios where International Conference on Autonomous Agents & Multi- agents are assigned to teams and are alternatively matched agent Systems, 251–259. International Foundation for Au- to recommended roles or allowed to select roles freely. This tonomous Agents and Multiagent Systems. user study will test the acceptance of recommendations, the Peters, D. 2016. Complexity of hedonic games with di- quality of partitioning, and the accuracy of various utility chotomous preferences. In Proceedings of the 30th AAAI functions for computing role and composition preferences. Conference on Artificial Intelligence. AAAI Press. References Saad, W.; Han, Z.; Basar, T.; Hjorungnes, A.; and Song, J. B. 2010. Hedonic coalition formation games for sec- Alexiou, A., and Doerga, A. 2015. Sprites and stories: The ondary base station cooperation in cognitive radio networks. impact of hedonic game elements on perceived learning out- In Wireless Communications and Networking Conference comes. In Academy of Management Proceedings, volume (WCNC), 1–6. 2015, 17225. Academy of Management. Saad, W.; Han, Z.; Basar, T.; Debbah, M.; and Hjorungnes, Aziz, H.; Brandt, F.; and Seedig, H. G. 2010. Optimal parti- A. 2011a. Hedonic coalition formation for distributed task tions in additively separable hedonic games. arXiv preprint allocation among wireless agents. In Proc. IEEE Transac- arXiv:1005.4540. tions on Mobile Computing. Aziz, H.; Brandt, F.; and Seedig, H. G. 2011. Stable par- Saad, W.; Han, Z.; Hjørungnes, A.; Niyato, D.; and Hossain, titions in additively separable hedonic games. In Proc. AA- E. 2011b. Coalition formation games for distributed coop- MAS ’11, 183–190. eration among roadside units in vehicular networks. IEEE Ballester, C. 2004. Np-completeness in hedonic games. Journal on Selected Areas in Communications (JSAC), Spe- Games and Economic Behavior 49(1):1–30. cial issue on Vehicular Communications and Networks. Banerjee, S.; Konishi, H.; and Sonmez,¨ T. 2001. Core in a Spradling, M., and Goldsmith, J. 2015. Stability in role simple coalition formation game. Social Choice and Welfare based hedonic games. In Proc. FLAIRS-28. Springer. 18(1):135–153. Spradling, M.; Goldsmith, J.; Liu, X.; Dadi, C.; and Li, Z. Conitzer, V., and Sandholm, T. 2005. Communication com- 2013. Roles and teams hedonic game. In Proc. Algorithmic plexity of common voting rules. In Proceedings of the 6th Decision Theory. Springer. 351–362. ACM conference on Electronic commerce, EC ’05, 78–87. Sung, S.-C., and Dimitrov, D. 2010. Computational com- New York, NY, USA: ACM. plexity in additive hedonic games. European Journal of Op- Darmann, A.; Elkind, E.; Kurz, S.; Lang, J.; Schauer, J.; and erational Research 203(3):635–639. Woeginger, G. 2012. Group activity selection problem. In Xu, K.; Wang, K.-C.; Amin, R.; Martin, J.; and Izard, R. Internet and Network Economics. Springer. 156–169. 2015. A fast cloud-based network selection scheme us- Diamantoudi, E., and Xue, L. 2003. Farsighted stability in ing coalition formation games in vehicular networks. IEEE hedonic games. Social Choice and Welfare 21(1):39–61. Transactions on Vehicular Technology 64(11):5327–5339. Dreze,` J., and Greenberg, J. 1980. Hedonic coalitions: Op- timality and stability. Econometrica 48(4):987–1003.

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