# Fair cost allocations under conflicts a game-theoretic point of view

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1 Fair cost allocations under conflicts a game-theoretic point of view Yoshio Okamoto (ETH Zürich) 14th Annual International Symposium on Algorithms and Computation (ISAAC 2003) Rihga Royal Hotel Kyoto, Kyoto, Japan December 15 17, 2003 Supported by the Berlin-Zürich Joint Graduate Program

2 p 1. Framework: What is all about? Several people are willing to work together... They want to have a largest possible benefit optimization problem They want to allocate the benefit in a fair way game-theoretic problem. Cooperative Game Theory 1

3 p 1. Framework: What is all about? Several people are willing to work together... They want to have a largest possible benefit optimization problem They want to allocate the benefit in a fair way game-theoretic problem. Game Theory? Noncooperative Game Theory Cooperative Game Theory 1

4 p 2. Def.: Cooperative games A cooperative game (or a game) is a pair (N, γ) of a finite set N (set of players) a function γ : 2 N IR with γ( ) = 0 (characteristic function)... Goal: To allocate γ(n) to each player in a fair way.. This work: study on minimum coloring games. 2

5 p 2. Def.: Cooperative games A cooperative game (or a game) is a pair (N, γ) of a finite set N (set of players) a function γ : 2 N IR with γ( ) = 0 (characteristic function).. Interpretation: For S N, { the max. benefit gained by S γ(s) represents the min. cost owed by S when the players in S work in cooperation. }. Goal: To allocate γ(n) to each player in a fair way.. This work: study on minimum coloring games. 2

6 p 2. Def.: Cooperative games A cooperative game (or a game) is a pair (N, γ) of a finite set N (set of players) a function γ : 2 N IR with γ( ) = 0 (characteristic function).. Interpretation: For S N, { the max. benefit gained by S γ(s) represents the min. cost owed by S when the players in S work in cooperation. }. Goal: To allocate γ(n) to each player in a fair way.. This work: study on minimum coloring games. 2

7 p 3 G = (V, E) an undirected graph Definition: minimum coloring game A proper k-coloring of G is a surjective map c : V {1,..., k} s.t. if {u, v} E, then c(u) c(v). The chromatic number χ(g) of G = min{ k : a proper k-coloring of G exists }. The minimum coloring game on G is a cooperative game (V, χ G ). χ G : 2 V IN is defined as χ G (S) = χ(g[s]), where G[S] is the subgraph induced by S V. 3

8 p 4 χ G (S) = χ(g[s]) for S V. Example: minimum coloring game S χ G S χ G S χ G S χ G Goal: To allocate χ(g) to each vertex in a fair way. 4

9 p 5 Conflict graph: a model of conflict the vertices = the agents, the principals... the edges = between two in conflict. Background: model of conflicts min. coloring game: a simplest model of the fair cost allocation problem in conflict situations 5

10 p 6 We study minimum coloring games, and investigate the following kinds of fairness concepts: Objective of our work Core (Gillies 53) Nucleolus (Schmeidler 69) τ-value (Tijs 81) Shapley value (Shapley 53) Past works on minimum coloring games: Deng, Ibaraki & Nagamochi 99 Deng, Ibaraki, Nagamochi & Zang 00 Okamoto Why minimum coloring games?? 6

11 p 6 We study minimum coloring games, and investigate the following kinds of fairness concepts: Objective of our work Core (Gillies 53) Nucleolus (Schmeidler 69) τ-value (Tijs 81) Shapley value (Shapley 53) Past works on minimum coloring games: Deng, Ibaraki & Nagamochi 99 Deng, Ibaraki, Nagamochi & Zang 00 Okamoto Why fair cost allocation problems?? 6

12 p 7 Background: Operations Research Fair cost allocation problems are studied in OR community from the game-theoretic viewpoint. Megiddo 87 First noticed the computational issue in fair cost allocation problems. So far, a lot of results have appeared in Mathematics of Operations Research, Mathematical Programming, Mathematical Methods of Operations Research, Discrete Applied Mathematics, International Journal of Game Theory, Games and Economic Behavior, etc. They assume practical applications. 7

13 p 8 Background: Algorithmic game theory There are many kinds of fairness concepts (called solutions ) in cooperative game theory.. Q. Which fairness concept should be used better??. Thesis: Bounded Rationality Decisions by realistic economic agents cannot involve unbounded resources for reasoning.. Thesis: (Deng & Papadimitriou 94) [Economic concept] [Algorithmic concept] = Algorithmic study of cooperative game theory 8

14 p 8 Background: Algorithmic game theory There are many kinds of fairness concepts (called solutions ) in cooperative game theory.. Q. Which fairness concept should be used better??. Thesis: Bounded Rationality (Simon 70s) Decisions by realistic economic agents cannot involve unbounded resources for reasoning.. Thesis: (Deng & Papadimitriou 94) [Economic concept] [Algorithmic concept] = Algorithmic study of cooperative game theory 8

15 p 8 Background: Algorithmic game theory There are many kinds of fairness concepts (called solutions ) in cooperative game theory.. Q. Which fairness concept should be used better??. Thesis: Bounded Rationality (Simon 70s) Decisions by realistic economic agents cannot involve unbounded resources for reasoning.. Thesis: (Deng & Papadimitriou 94) [Economic concept] [Algorithmic concept] A solution follows Computation can be Bounded Rationality done in poly. time = Algorithmic study of cooperative game theory 8

16 p 9 Background: Combinatorial optimization games Combinatorial Optimization Games = cooperative games arising from combinatorial optimization problems Assignment games (Shapley & Shubik 71) Minimum-cost spanning tree games (Bird 76) Steiner tree games (Megiddo 78) Simple flow games (Kalai & Zemel 82) Travaling salesman games (Potters, Curiel & Tijs 92) Chinese postman games (Granot, Hamers & Tijs 99) Facility location games (Goemans & Skutella 00) Minimum coloring games (Deng, Nagamochi & Ibaraki 99)... 9

17 10 p Background: Set functions E a finite set. Def.:. Appearance: A set function on E is a function f : 2 E IR. Cooperative game theory Combinatorial optimization Pseudo-boolean functions Nonadditive measure theory (fuzzy measure theory)... They study different aspects of set functions. 10

18 11 p Scope of today s talk Focus on cores and nucleoli of minimum coloring games Def.: cost allocation Def.: core Def.: nucleolus Characterization: the nucleolus for a chordal graph Open problems 11

19 12 p Cost allocation. Def.: A cost allocation for a game (N, γ) is a vector z IR N such that {z[ i ] : i N} = γ(n).. Interpretation: z[ i ] = the amount of cost the player i must pay when all players in N work together Q. What kinds of cost allocations are considered fair??... Core, Nucleolus, τ-value, Shapley value, etc. 12

20 12 p Cost allocation. Def.: A cost allocation for a game (N, γ) is a vector z IR N such that {z[ i ] : i N} = γ(n).. Interpretation: z[ i ] = the amount of cost the player i must pay when all players in N work together Q. What kinds of cost allocations are considered fair??... Core, Nucleolus, τ-value, Shapley value, etc. 12

21 13 p Core. Def.: A cost allocation z IR N for (N, γ) is a core allocation if {z[ i ] : i S} γ(s) for all S N The core of (N, γ) is the set of all core allocations.. Interpretation: i S γ(s) : z[ i ] : Each subset S N is satisfied with z cost owed by S when people in N work together cost owed by S when people in S work together. 13

22 14 p Example: Core z[ 3 ] 3 V = {1, 2, 3} z[ 1 ] Core = z IR3 : z[ 2 ] 1 2 χ G ( ) 0 χ G ({1}) 1 χ G ({2}) 1 χ G ({3}) 1 χ G ({1, 2}) 1 χ G ({1, 3}) 2 χ G ({2, 3}) 2 χ G ({1, 2, 3}) 2 z[ 1 ] 1, z[ 2 ] 1, z[ 3 ] 1, z[ 1 ] + z[ 2 ] 1, z[ 1 ] + z[ 3 ] 2, z[ 2 ] + z[ 3 ] 2, z[ 1 ] + z[ 2 ] + z[ 3 ] = 2 14

23 14 p Example: Core z[ 3 ] 3 V = {1, 2, 3} z[ 1 ] z[ 2 ] 1 Core = conv {( ) 2, ( χ G ( ) 0 χ G ({1}) 1 χ G ({2}) 1 χ G ({3}) 1 χ G ({1, 2}) 1 χ G ({1, 3}) 2 χ G ({2, 3}) 2 χ G ({1, 2, 3}) 2 )} 14

24 15 p Problems We want to know: (1) When is the core empty?? (2) If the core is nonempty, (a) What are the extreme points of the core?? (b) How can we compute a core allocation?? Thm. (Deng, Ibaraki & Nagamochi 99) It is NP-complete to decide the min coloring game of a given graph has a nonempty core or not.. Thm. (Deng, Ibaraki & Nagamochi 99) For a bipartite graph G, core = conv(the char. vectors of edges of G). 15

25 15 p Problems We want to know: (1) When is the core empty?? (2) If the core is nonempty, (a) What are the extreme points of the core?? (b) How can we compute a core allocation?? (1). Thm. (Deng, Ibaraki & Nagamochi 99) It is NP-complete to decide the min coloring game of a given graph has a nonempty core or not. (2). Thm. (Deng, Ibaraki & Nagamochi 99) For a bipartite graph G, core = conv(the char. vectors of edges of G). We will generalize it to... 15

26 16 p Perfect graphs and min coloring games. Def.: G is perfect if for every induced subgraph H G χ(h) = max size of the cliques in H. (A clique is a vertex subset which induces a complete graph.). Thm. (Deng, Ibaraki, Nagamochi & Zang 00) A min coloring game (V, χ G ) is totally balanced G is perfect.. Rem.: = 16

27 16 p Perfect graphs and min coloring games. Def.: G is perfect if for every induced subgraph H G χ(h) = max size of the cliques in H. (A clique is a vertex subset which induces a complete graph.). Thm. (Deng, Ibaraki, Nagamochi & Zang 00) A min coloring game (V, χ G ) is totally balanced G is perfect.. Rem.: A game is totally balanced = it has a nonempty core. 16

28 17 p Characterization of cores for perfect graphs. Thm. G a perfect graph core = conv(the char. vectors of the maximum cliques of G). z[ 3 ] ) )} Core = conv {( 1 0 1, ( z[ 2 ] z[ 1 ] 17

29 18 p Corollaries. Cor. We can do the following in poly. time. (1) Find a core allocation for a perfect graph. ( Find a maximum clique in poly. time.) (2) Decide whether a given vector belongs to the core or not for a perfect graph. ( The membership problem for the clique polytope.) They are the consequences of the previous theorem and a result by Grötschel, Lovász & Schrijver ( 83). 18

30 19 p We saw the core... What is good/bad for the core??. Good :) Easy to investigate. Much is known.. Bad :( Might be empty. Even if not empty, there might be many allocations in the core. (We need another criterion to choose one of them.) The nucleolus is another fairness concept, which uniquely exists for every min coloring game. 19

31 20 p Scope of today s talk Focus on cores and nucleoli of minimum coloring games Def.: cost allocation Def.: core Def.: nucleolus Characterization: the nucleolus for a chordal graph Open problems 20

32 21 p Excess Let (N, γ) a game z IR N a cost allocation S N (often called a coalition). Def.: An excess e(s, z) is defined as e(s, z) := i S z[ i ] γ(s).. Interpretation: (1) z core e(s, z) 0. (2) The smaller e(s, z), the happier S with z. (3) e(s, z) how much z satisfies the condition of the core for S. 21

33 22 p Enumerating the excesses... Let (N, γ) be a game, z IR N a cost allocation Consider the following procedure. Enumerate e(s, z) for all S 2 N \ {, N}. Arrange these excesses in non-increasing order to obtain θ z IR 2 N 2. (θ z [ i ] θ z [ j ] if i j.). Example: z = θ z = ( 1, 1 2, 1 ) 2 ( ) 1 2, 0, 1 2, 1 2, 1 2, 1 S γ(s) e(s, z) 0 (0) {1} 1 0 {2} 1 1/2 {3} 1 1/2 {1, 2} 1 1/2 {1, 3} 2 1/2 {2, 3} 2 1 {1, 2, 3} 2 (0) 22

34 22 p Enumerating the excesses... Let (N, γ) be a game, z IR N a cost allocation Consider the following procedure. Enumerate e(s, z) for all S 2 N \ {, N}. Arrange these excesses in non-increasing order to obtain θ z IR 2 N 2. (θ z [ i ] θ z [ j ] if i j.). Example: z = θ z = ( 1, 1 2, 1 ) 2 ( ) 1 2, 0, 1 2, 1 2, 1 2, 1 S γ(s) e(s, z) 0 (0) {1} 1 0 {2} 1 1/2 {3} 1 1/2 {1, 2} 1 1/2 {1, 3} 2 1/2 {2, 3} 2 1 {1, 2, 3} 2 (0) 22

35 23 p Nucleolus. Def.: ν(n, γ) = The nucleolus of (N, γ) is defined as { z IR N : z lex-mins θ z over all cost alloc s y s.t. y[ i ] γ({i}) i N }.. Interpretation: The smaller e(s, z), the happier S with z. Want an allocation which minimizes max excess.. Thm. (Schmeidler 69) The nucleolus consists of a single vector. So we usually say ν(n, γ) = z instead of ν(n, γ) = {z}. 23

36 23 p Nucleolus. Def.: ν(n, γ) = The nucleolus of (N, γ) is defined as { z IR N : z lex-mins θ z over all cost alloc s y s.t. y[ i ] γ({i}) i N }.. Interpretation: The smaller e(s, z), the happier S with z. Want an allocation which minimizes max excess.. Thm. (Schmeidler 69) The nucleolus consists of a single vector. So we usually say ν(n, γ) = z instead of ν(n, γ) = {z}. 23

37 24 p Example: nucleolus z[ 3 ] 3 z[ 2 ] 1 2 ν = θ ν = ( 1 2, 1 ) 2, 1 is the nucleolus. ( 0, 0, 1 2, 1 2, 1 2 2), 1. z[ 1 ]. Fact: the core is nonempty the nucleolus the core. 24

38 25 p Computation of the nucleolus. Thm. (Kuipers 96, Faigle, Kern & Kuipers 01) The nucleolus can be computed in polynomial time for submodular games.. Thm. (Okamoto 03) χ G is submodular G is complete multipartite. 25

39 26 p Computation of the nucleolus. Thm. (Kuipers 96, Faigle, Kern & Kuipers 01) The nucleolus can be computed in polynomial time for submodular games.. Thm. (Okamoto 03). Cor. χ G is submodular G is complete multipartite. G complete multipartite the nucleolus of χ G computed in poly. time. 26

40 27 p Correspondence On the computation of the nucleolus of a min coloring game Graph Min col. game general NP-hard perfect??? complete multipartite Poly 27

41 27 p Correspondence On the computation of the nucleolus of a min coloring game Graph Min col. game general NP-hard perfect??? O-good complete multipartite characterization Poly 27

42 28 p O-good perfect graphs. Thm. Namely, The nucleolus for an O-good perfect graph G is the barycenter of the characteristic vectors of the maximum cliques of G. ν[ i ] = # of maximum cliques containing i # of maximum cliques. 28

43 29 p O-good perfect graphs. Thm. The nucleolus for an O-good perfect graph G is the barycenter of the characteristic vectors of the maximum cliques of G.. Remarks: (1) We omit the def. of O-good perfect graphs. (2) The class of O-good perfect graphs contains the graphs with unique maximum cliques the complete multipartite graphs the chordal graphs (especially the forests). (3) A graph is chordal if every induced cycle is of length 3. 28

44 29 p Outline of the proof (1) Use the LP formulation for the nucleolus computation. (Peleg) By solving a sequence of LP problems, we can obtain the nucleolus (not in poly time). (2) Identify the essential coalitions. (Huberman 80) The essential coalitions reduce the work load. S V is essential S is an independent set. (3) Analyze the LP. For O-good perfect graphs, we can precisely tell what are the optimal solutions in the LP problems with help of the characterization of the extreme points of the core. 29

45 30 p Corollary. Cor. G chordal the nucleolus of χ G computed in poly. time.. Proof Sketch (1) Chordal graphs are O-good perfect graphs. (2) In a chordal graph, # of maximal cliques # of vertices. We can enumerate them in poly time. (Fulkerson & Gross 65) 30

46 p 31 Summary: the nucleolus graphs zero integrality gap perfect complete multipartite chordal forests bipartite 31

47 32 p Example: a bipartite graph There is a bipartite graph for which the nucleolus is not the barycenter of the char. vectors of the maximum cliques. 2/3 1/3 1/3 1/3 1/3 32

48 33 p The state of the art Core graphs Nucleolus graphs zero integrality gap zero integrality gap perfect perfect complete multipartite chordal forests bipartite complete multipartite chordal forests bipartite τ-value graphs Shapley value graphs zero integrality gap zero integrality gap perfect perfect complete multipartite chordal forests bipartite complete multipartite chordal forests bipartite 33

49 34 p Open problems: minimum coloring games Computation of the nucleoli for perfect graphs?? bipartite graphs?? outerplanar graphs?? cographs?? 34

50 35 p Conclusion The list of what we discussed Def.: minimum coloring game Def.: cost allocation Def.: core Def.: nucleolus Characterization: the nucleolus for a chordal graph Open problems 35

51 36 p Research paradigm. Framework: Several people are willing to work together... They want to have a largest possible benefit. (optimization theory) They want to allocate the benefit in a fair way. (cooperative game theory). Status of algorithmic problems on cooperative games As many cooperative games as optimization problems!! Many algorithmic problems remain unsolved!! = Why not work on them?? [End of the talk] 36

52 36 p Research paradigm. Framework: Several people are willing to work together... They want to have a largest possible benefit. (optimization theory) They want to allocate the benefit in a fair way. (cooperative game theory). Status of algorithmic problems on cooperative games As many cooperative games as optimization problems!! Many algorithmic problems remain unsolved!! = Why not work on them?? [End of the talk] 36

53 Full-paper version in preparation: 37

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