Stable Structures of Coalitions in Competitive and Altruistic Military Teams. M. Aurangzeb, D. Mikulski, G, Hudas, F.L. Lewis, and Edward Gu

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1 Stable Structures of Coaltons n Compettve and Altrustc Mltary Teams M. Aurangzeb, D. Mkulsk,, Hudas, F.L. Lews, and Edward u M. Aurangzeb, The Unversty of Texas at Arlngton, Research Insttute, USA, aurangze@uta.edu D. Mkulsk, U.S. Army Tank Automotve Research, Development and Engneerng Ctr. Unted States darusz.mkulsk@us.army.ml. Hudas, U.S. Army Tank Automotve Research, Development and Engneerng Ctr. Unted States greg.hudas@us.army.ml F.L. Lews Unversty of Texas n Arlngton Unted States lews@uta.edu Edward u, Oakland Unversty Unted States guy@oakland.edu For submsson to the Specal Sesson on Intellgent Behavors Chars. Hudas, D. Mkulsk, and F.L. Lews Abstract In heterogeneous battlefeld teams, the balance between team and ndvdual objectves forms the bass for the nternal topologcal structure of teams. The team structure s studed by presentng a graphcal coaltonal game (C) wth Postonal Advantage (PA). PA s Shapley value strengthened by the Axoms of value. The noton of team and ndvdual objectves s studed by defnng altrustc and compettve contrbuton made by an ndvdual; altrustc and compettve contrbutons made by an agent are components of ts total or margnal contrbuton. Moreover, the paper examnes the dynamc team effects, by defnng three onlne sequental decson games. These sequental decson games are based on margnal, compettve and altrustc contrbutons of the ndvduals towards team. The stable graphs under these sequental decson games are studed and found to be any connected, complete, or tree respectvely. Keywords: raphcal Coaltonal ame, Postonal Advantage, Margnal Contrbuton, Compettve Contrbuton, Altrustc Contrbuton, Sequental Decson ames, Stable Structures 1. Introducton Battlefeld teams are generally heterogeneous coaltons consstng of nteractng humans, ground sensors, and unmanned arborne (UAV) or ground vehcles (UV). In these teams relatons are based on drect local nteractons, yet nformaton s conveyed along a chan of command. The balance between coaltonal and ndvdual objectves forms the bass for the nternal topologcal structure of coaltons. In ths paper, the coaltonal structure s studed by usng cooperatve game theory. A novel graphcal coaltonal game (C), wth Postonal Advantage (PA), s presented. PA s based on Shapley value strengthened by the Axoms of value. Axoms of valve are based on ts connectvty propertes wthn the coalton. Under the Axoms of Value the C s convex, far, cohesve, and fully cooperatve. Three measures of the contrbutons of agents to a coalton are ntroduced: margnal contrbuton (MC), compettve contrbuton (CC), and altrustc contrbuton (AC). ame theory, ntroduced as a formal dscplne of mathematcs by J. von Neumann [13], [14], s now an establshed mathematcal dscplne that deals wth ssues and strateges nvolvng compettons and cooperaton between several enttes [15]. In the scope of mathematcal game theory these enttes are called players or agents [5], [15], [17], [20]. ame theory s used n 1

2 Report Documentaton Page Form Approved OMB No Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 1204, Arlngton VA Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE 17 MAR REPORT TYPE Journal Artcle 3. DATES COVERED to TITLE AND SUBTITLE Stable Structures of Coaltons n Compettve and Altrustc Mltary Teams 6. AUTHOR(S) M Aurangzeb; D Mkulsk; reg Hudas; F Lews; Edward u 5a. CONTRACT NUMBER 5b. RANT NUMBER 5c. PRORAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMIN ORANIZATION NAME(S) AND ADDRESS(ES) Research Insttute,The Unversty of Texas at Arlngton,701 S. Nedderman Dr,Arlngton,TX, SPONSORIN/MONITORIN AENCY NAME(S) AND ADDRESS(ES) U.S. Army TARDEC, 6501 East Eleven Mle Rd, Warren, M, PERFORMIN ORANIZATION REPORT NUMBER ; # SPONSOR/MONITOR S ACRONYM(S) TARDEC 11. SPONSOR/MONITOR S REPORT NUMBER(S) # DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release; dstrbuton unlmted 13. SUPPLEMENTARY NOTES For submsson to the Specal Sesson on Intellgent Behavors 14. ABSTRACT In heterogeneous battlefeld teams, the balance between team and ndvdual objectves forms the bass for the nternal topologcal structure of teams. The team structure s studed by presentng a graphcal coaltonal game (C) wth Postonal Advantage (PA). PA s Shapley value strengthened by the Axoms of value. The noton of team and ndvdual objectves s studed by defnng altrustc and compettve contrbuton made by an ndvdual; altrustc and compettve contrbutons made by an agent are components of ts total or margnal contrbuton. Moreover, the paper examnes the dynamc team effects, by defnng three onlne sequental decson games. These sequental decson games are based on margnal, compettve and altrustc contrbutons of the ndvduals towards team. The stable graphs under these sequental decson games are studed and found to be any connected, complete, or tree respectvely. 15. SUBJECT TERMS raphcal Coaltonal ame, Postonal Advantage, Margnal Contrbuton, Compettve Contrbuton, Altrustc Contrbuton, Sequental Decson ames, Stable Structures 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAE unclassfed ABSTRACT Publc Release 18. NUMBER OF PAES 12 19a. NAME OF RESPONSIBLE PERSON

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4 many walks of lfe [7], [21] nvolvng stuatons of competton and cooperaton. Owng to the occurrences of such stuatons n engneerng, game theory has found ts way n engneerng applcatons [2], [4], [5], [9], [10], [11], [16], [17], [18], [22], [23]. ame theory s prmarly dvded nto two areas: noncooperatve game theory [5], and cooperatve game theory [15], [20]. Cooperatve games can be dvded nto three classes [5]: Canoncal Coaltonal ames, Coalton Formaton ames, and Coaltonal raph ames [4], [17]. Closely related to the coaltonal graph games are onlne or sequental-n-tme decson games. These are games where agents make moves through tme sequentally to maxmze ther prescrbed objectve functons [20]. Myerson [12] used the Shapley value [19] as the payoff to agents n a coalton defned by a graph. Jackson and Wolnsky [9] studed coalton graph games where the payoff depends on the connectvty of an agent and a cost was mposed for mantanng graph edges. Ths paper defnes a graphcal coaltonal game wth novel propertes. The frst pont of mpact of the paper s based on a Value Functon that s requred to satsfy four formal axoms. Owng to these axoms mposed on the Value Functon, t s possble to perform a rgorous study of the nternal structure of teams. The Shapley value wth the Value Functon satsfyng the four Axoms s nterpreted as the worth of an agent n a team, and when strengthened by the Axoms of Value, s called the Postonal Advantage (PA). PA formalzes the noton of wellconnectedness n a team. The second pont of mpact of the paper s to study three types of contrbutons of ndvduals wthn a team- the margnal (MC), compettve (CC), and altrustc (AC) [1]. The PA, whch ncludes the formal Axoms of Value, allows to rgorously developng certan propertes of these three contrbutons, ncludng ther dependence on the nternal team structure and changes n the team structure. In ths work the nternal team structure s represented as a smple graph [6]. The thrd pont of mpact of the paper s the study of the dynamc team effects such as the formaton of teams by addng agents, the destructon of teams by defecton of agents, and other ssues nvolvng sequental decsons by agents to jon, leave, or form teams. To study the dynamc team effects, three onlne sequental decson games are defned. These sequental decson games are based on the three contrbutons MC, CC, and AC. It s shown that the stable graphs under the objectve of maxmzng the MC are any connected graph. The stable graphs under the objectve of maxmzng the CC are the complete graph. The stable graphs under the objectve of maxmzng the AC are any tree. Ths paper s organzed as follows. In Secton 2, a graphcal coaltonal game (C) havng novel propertes, s defned, wth ts axoms on the value. The Postonal Advantage (PA) of an agent wthn a graph topology s also defned on the grounds establshed by Myerson [12] and Shapley [19]. Secton 3 defnes three types of contrbutons of agents n a coalton based on the communcaton graph structure. Margnal, compettve, and altrustc contrbutons are defned. Results about these three contrbutons of the agents based on topologcal graph propertes are establshed. In Secton 4, three onlne sequental decson games are developed on top of the C. The stable graph structures under each of these three games are studed. Smulaton examples of onlne sequental decson games are presented n Secton Postonal Advantage n raphcal Coaltonal ames The noton of sequental decson games for coalton formaton s detaled n Secton 4; these games rest on the dea of graphcal coalton game (C) ntroduced n ths secton. In ths secton a C satsfyng certan axoms s defned. These axoms make t possble to establsh the noton of Postonal Advantage (PA) of agents wthn a coalton based on ther postons wthn the graph. Ths secton starts wth a few essental notatons of graph theory [6]. 2

5 2.1 raph Defntons Consder a smple undrected graph ( V, E) wth V a fnte nonempty set of agents or vertces and E s a set of edges. Two agents are nterpreted to have an edge between them f and only f they drectly communcate wth each other. The number of elements n V s called the order or sze of the graph and s denoted as, also denoted as N. The graph theoretc concepts ncludng neghborhood, path, connectvty, ds-connectvty, components, cycle, tree, cut vertex, cut edge, subgraph, and nduced subgraph are vtal for the understandng of ths paper and can be seen n [3], [6]. The followng notatons are used n ths work. In ths paper S, denotes that S s an nduced subgraph of. Whereas, an nduced subgraph of obtaned by excludng a vertex from V s denoted as \ { }. If an edge e s deleted from a graph then the new graph obtaned s denoted as e, and f an edge e s added n then the new graph s denoted as e. 2.2 raphcal Coaltonal ame In ths secton a graphcal coaltonal game ( v, ) wth transferable utlty s proposed. The game s based on the communcaton structure of the agents wthn a coalton. ven a graph, wth agents as nodes, and there exst edges between the nodes f and only f the correspondng agents drectly communcate wth each other. The Value Functon v s formally defned as v : 2 wth v( ) v 0 (1) where 2 s the collecton of all the nduced subgraphs of and s the empty set. The Value Functon satsfes the followng Axoms of Value. In these axoms S 2 s an nduced subgraph of. Axoms of the Value 1. If S s a connected component wth S m then ( ) 0 0 v S v m 2. If S s havng k connected components S : 1, 2,... k wth S m then v( S) v : v 0 m 3. If N m n 0 then n. v. m m m v n 4. If N 1 m n 0 then. v v v v m 1 m n 1 n It s to be mentoned here that the Axom 2 s accordng to allocaton rules the coaltonal graph game defned by Myerson n Secton 3 of [12], where the coaltons are restrcted by the underlyng communcaton graph. A condton on v 1 and v stronger than Axom 3 s 2 v 2v (2) Ths condton s used to establsh some refnements and strengthenng of results. The next defnton provdes a fundamental noton used n ths paper. Defnton 1: raphcal Coaltonal ame. ven a graph, the graphcal coaltonal game s defned as the game ( v, ) where the Value Functon v satsfes Axom 1-4. Remark 1: It can be establshed that Axom 1 s mpled by the Axom 2 and Axom 3 s mpled by the Axom 4. Nevertheless, Axoms 1 and 3 are retaned as axoms because of the ease of ther use n establshng the game propertes [3]. 3

6 The allocaton of the net value of a coalton to ts ndvdual agents s a fundamental problem n coaltonal games. Allocaton s the share gven to each agent of the net value of the coalton efforts. It s establshed n [12] that for a Value Functon defned n games on graphs, the Shapley value [19] provdes a far allocaton. Therefore the next defnton s made. Defnton 2: Postonal Advantage of an Agent n the raphcal Coaltonal ame. ven the C ( v, ), defne the Postonal Advantage of agent as the Shapley value 1 ( ( { }) ( )) () (3) v S v S S \{ } S v, 1 Where the Value Functon v satsfes the Axoms of value, S \{ } means S s an nduced subgraph of the graph \ { }. Moreover, S {} denotes an nduced subgraph of, contanng all the agents n S and the agent. Fg. 1. Two smple graphs. (a) Example 1- Three agents n a chan (b). Example 2- Three agents n a complete graph. The followng examples explan the procedure to compute the PA of agents wthn a coalton and show the ratonale for Axoms 1-4 of the game as lad down. They reveal the mportance of PA n comparng the relatve mportance of agents n contrbutng to the communcaton structure of a coalton as represented by the graph. Example 1: Consder a chan of three agents {1, 2,3} as shown n Fg. 1(a). The PA of the agents s calculated by usng (3). For agent 1 (1) ( ) 1 v({1,2,3}) v({2,3}) ( v({1,2}) v({2})) ( v({1,3}) v({3})) v({1}) v( ) Usng Axoms 1-4 of the game ths s smplfed to (1) ( v v v ). It can be easly seen that (3) (1). The PA of the agent 2 s gven by (2) ( ) 1 v({1,2,3}) v({1,3}) ( v({1,2}) v({1})) ( v({2,3}) v({3})) v({2}) v( ) 3 1 Usng the axoms of the game ths s smplfed as 1 (2) ( v v 2 v ). 3 3 Usng the axoms of the game t can be readly seen that (2) (1), (3). Ths s accordng to the heurstcs for the gven communcaton structure, snce 2 s n the mddle of 1 and 3 and so logcally contrbutes more to the communcaton structure of the coalton. Example 2: Consderng a complete graph of three agents {1, 2, 3} as shown n Fg. 1(b), the PA of agent 1 s calculated by usng the defnton gven n (3). (1) ( ) 1 v({1,2,3}) v({2,3}) ( v({1,2}) v({2})) ( v({1,3}) v({3})) v({1}) v( ) 3 1 Usng the axoms of the game ths equaton s smplfed as 1 (1) v

7 It can be easly seen that both (2) and (3) have the same PA. Ths agan s accordng to ntuton, snce all the three nodes are symmetrcally dstrbuted n the graph and evenly contrbute to the communcatons wthn the coalton. In these two examples, two small graphs were taken to demonstrate the utlty of the game ( v, ) wth respect to the communcaton structures. In these examples, the nodes that are placed more advantageously and so contrbute more to the communcatons wthn a coalton have a greater PA as calculated through the payoff functon (3) of the game. Moreover, the nodes whch, accordng to the communcaton heurstcs should have same relatve mportance, actually do have the same PA as calculated through the payoff functon of the game. 3. Contrbuton of Agents wthn a Coalton A graphcal coaltonal game s ntroduced n the prevous secton. The game assgns a value to each coalton based on communcaton structure represented as a graph structure. The assgnment of value s based on the connectvty propertes of the graph structure. All the agents nvolved have some allocaton or share n the value assgned to the graph structure. Ths secton defnes the contrbuton made by an agent to the coalton and ts components. The contrbuton and ts components are further used to defne sequental decson games n the next secton. Defnton 5: Allocaton of a Set of Agents n an Induced Subgraph. Let A, and B, be nduced subgraphs of such that A B.The allocaton or payoff of the agents n coalton A when only the coalton B s consdered s denoted as ( B ) and defned as A ( B) ( ) (4) A A 3.1 Defntons of Contrbutons of Agents n a Coalton The total contrbuton of a group A of agents n a coalton s called the margnal contrbuton of A n and wrtten as m ( A ) [1]. The margnal contrbuton of a group of agents s dvded n two parts: one part s the contrbuton of the agents n the subset A for the sake of themselves, and the second part s the contrbuton of agents n A for the sake of the other agents n \ A. These two parts are termed the compettve contrbuton c ( A ) and the altrustc contrbuton a ( A ) respectvely. These contrbutons are formally defned next for a sngle agent. Defnton 6: Margnal Contrbuton of an Agent. The margnal contrbuton m () of an agent s defned as m ( ) ( ) ( \{ }) (5) \{ } Defnton 7: Compettve Contrbuton of an Agent. The compettve contrbuton c () of an agent s defned as c ( ) ( ) ( ) (6) \{ } a Defnton 8: Altrustc Contrbuton of an Agent. The altrustc contrbuton () of an agent s defned as a ( ) ( ) ( \{ }) (7) Accordng to these defntons Usng (4) n (5), (6), (7) gve \{ } \{ } B m ( ) c ( ) a ( ) (8) 5

8 (9) j (10) j \ j \ (11) j \ \ m ( ) ( j) ( j) j j \ \ c ( ) ( j) ( j) a ( ) ( j) ( j) If s connected, usng Axoms 1 and 2 of the graphcal coaltonal game and (4), equaton (9) can be wrtten as m ( ) v v : k 1 k j j j 1 j 1 p p (12) Here, p s the number of dsconnected components of obtaned by the deleton of the agent and k : j 1, 2,.., p are the szes of these components. j Lemma 1 [3]: The compettve contrbuton of an agent n a C s the same as ts Postonal Advantage. That s c ( ) ( ) (13) 3.2 Dependence of Contrbuton of Agents on raph Topology Changes n margnal, compettve, and altrustc contrbutons are mportant n the onlne sequental decson games detaled n Secton 4. Some results demonstratng the dependence of the contrbutons made by an agent and on graph topology and change n graph topology are presented next. Proofs are n [3]. Lemma 2: ven the C ( v, ) n Defnton 1, n any connected graph all the agents whch are not cut vertces of have the same margnal contrbuton. Moreover ther margnal contrbuton s the mnmum possble margnal contrbuton wthn the connected graph. Ths mnmum margnal contrbuton s ndependent of the connected graph and only depends upon. Remark 2: In a connected graph, f there s no cut vertex then the margnal contrbutons of all the agents are dentcal and ndependent of the graph structure. Lemma 3: In a connected graph of sze N, the maxmum possble margnal contrbuton an agent may have s of the center pont of a star The next results concern the altrustc contrbuton. Lemma 4: In a C, f an agent s solated then ts altrustc contrbuton s 0 The next result shows that under condton (2), ths result s also suffcent. Lemma 5: In a C wth a game havng v 2v, f the altrustc contrbuton of an agent s 0 then t s solated. Theorem 1: In a C, f a new edge s formed between two connected agents and j n a graph, then the margnal contrbuton of ts end vertces remans constant. Moreover the altrustc contrbutons of ts end vertces change by equal non-postve values whch are the negatves of the changes n the compettve contrbutons of ts end vertces. Proof: Let be the graph obtaned from by addng the edge {, j }. Wth the help of (9) The margnal contrbuton of the agent n the graph s (14) m ( ) ( j) ( j) j j \ \ 6

9 Snce the new edge s formed wthn the same component of, thus the frst term n the rght hand sde of the above equaton s same as the frst term n the rght hand sde of (9). Moreover, the value of the second term s ndependent of any edge ncdent at agent. Thus from (9) and (14) t s mpled that m ( ) m( ). Rest of the result follows from (8), Lemma 1, detals can be found n [3]. 4. Onlne Sequental Coalton Decson ames Based on the margnal contrbuton and ts components, compettve and altrustc contrbutons, as defned n Secton 3, three onlne coalton sequental decson games are defned n ths secton on top of the graphcal coaltonal game ( v, ) of Defnton 1, and the stable coalton graph topologes under these three games are presented. In a sequental decson game, agents take turns sequentally n tme to make vald or allowed moves A background on sequental decson games can be found n Chapter 5 of [20]. 4.1 Sequental Decson ames The propertes of sequental decson games depend on the allowed moves and the prescrbed objectve functon. In the onlne games defned here, agents are free to make coaltons by makng or breakng edges wth other agents. Allowed Moves. In these onlne games, at each move, an agent s selected at random. Ths agent s free to unlaterally break any edge ncdent at t or to blaterally make an edges provded the other agent ncdent on the edge agrees to make t, as detaled below. In a sngle step, an agent s allowed ether to make or break several edges. Objectve Functons. An objectve functon f (), for each agent n a coalton represented by graph s a real, nonnegatve functon. Edges are made or broken by a selected agent n order to maxmze f (). Based on the Allowed Moves and the Objectve Functon the sequental decson game s defned as follows. Defnton 9: Sequental Decson ames. In a sequental decson game a selected agent makes or breaks edges accordng to the rules: a) An agent forms an edge e {, j} f f ( ) f ( ) and f ( j) f ( j) b) An agent breaks an edge e {, j} f f ( ) f ( ) e e e Based on the margnal, compettve and altrustc contrbutons n Secton 3 the motves of agents for formng and breakng the edges are dfferent. Takng these contrbutons as objectve functons, three sequental decson games can be defned.. ame of Maxmal Margnal Contrbuton (MMC) In ths onlne game the objectve functon f ( ) m ( ).. ame of Maxmal Compettve Contrbuton (MCC) In ths onlne game the objectve functon f ( ) c ( ).. ame of Maxmal Altrustc Contrbuton (MAC) In ths onlne game the objectve functon f ( ) a ( ) In these three sequental decson games an agent s sad to have a motve to make an edge f the condton (a) of the game s satsfed and t s sad to have a motve to break an edge f the condton (b) of the game s satsfed. 7

10 4.2 Stablty of raph Topologes under Sequental Decson ames ( 1)/2 For a group of N agents there are 2 N N possble smple graphs. When agents are allowed to make vald moves, as they proceed, they may reach a graph where no agent has a motve to make any further moves. Such graphs are called stable graphs [9], [12]. The Structure of stable graphs s thus dependent on the allowed moves and the objectve functon of the sequental decson game. In [12] Myerson allowed only the breakage of an edge as a vald move. Under such allowance, for the game n Defnton 1 every graph s stable. In [9] the rules of makng and breakng edges are nearly the same as those n Secton 4.1. However, n [9] there are costs assocated wth makng edges. The balance between the value of beng connected and the cost of mantanng edges has a pvotal role n determnng the stable graph structures. Defnton 10: Stable raph. In any onlne sequental decson game, a graph s called stable when no agent has a motve ether to make an edge or to break an edge. Remark 3: It s establshed n [3] that the underlyng game ( v, ) n Defnton 1 s far. Therefore, n the onlne ame of Maxmal Compettve Contrbuton (MCC) no agent has a motve to break an edge. Smlarly by results n Secton 3 4.2, n the onlne ame of Maxmal Margnal Contrbuton (MMC) no agent has a motve to break an edge. Only n the onlne ame of Maxmal Altrustc Contrbuton, may an agent have a motve to break an edge. Theorem 2: In an onlne ame of Maxmal Margnal Contrbuton any connected graph s stable. Moreover, wth a game havng the strengthened condton v 2v, any stable graph s connected. Proof: In an onlne ame of Maxmal Margnal Contrbuton, f the graph s connected an agent whch s not a cut vertex wll never become a cut vertex no matter how many edges t makes. Thus accordng to Lemma 2 t wll contnue to have the same mnmal margnal contrbuton, thus t has no motve to make an edge. For an agent whch s a cut vertex, ts margnal contrbuton s gven by (9). In the rght hand sde of ths, the frst term s constant for a connected graph also no matter how many new edges agent makes the second term wll also reman unchanged. Thus agent has no motve to make any edge. The second term n the rght hand sde of (9) s ndependent of any edge ncdent at the agent, whle under the gven condton v 2v, the frst term s maxmum only when the graph s connected. Thus n an onlne ame of Maxmal Margnal Contrbuton a dsconnected graph cannot be stable. Remark 4: Let an onlne ame of Maxmal Margnal Contrbuton be started from a completely dsconnected graph, and let every agent be allowed to make as many edges as t desres. Then the agent who gets the frst move to make edges wll make edges wth all the other agents. Ths wll make a star, whch s also a tree, wth the frst agent at the center. Then there s no motve for any other player to make or break an edge. Theorem 3: In an onlne ame of Maxmal Compettve Contrbuton any complete graph s stable. Moreover, wth a game havng v 2v, any stable graph s complete. Proof: In an onlne ame of Maxmal Compettve Contrbuton a complete graph s stable snce there s no more edge to make and there s no motve to break an edge [3] and condton (b) of ame of Maxmal Compettve Contrbuton. It s establshed n Lemma 1 that the compettve contrbuton of an agent s the same as the Postonal Advantage of an agent. Whenever an edge s added n a graph, the PAs of both of ts 8

11 end vertces ncrease under the gven condton [3]. Thus there s always a motve to make an edge whenever t s possble. Theorem 4: In an onlne ame of Maxmal Altrustc Contrbuton any tree s a stable graph. Moreover, wth a game havng v 2v, f a graph s stable then t s a tree. Proof: The altrustc contrbuton of an agent s gven by (11) or (15) a ( ) ( j) ( j) j \ j \ \ (16) a ( ) ( j) ( ) ( j) j j \ \ By the axoms of the game n Defnton 1 and usng the known facts about the PA n (3), nherted from the Shapley value [19], all the connected graphs havng the same number of agents have the same value of ( j) v. Snce tree s a connected graph, formaton of a j new edge by any agent does not change the value of the frst term at the rght hand sde of (16). Moreover, for each new edge agent makes, ts PA ncreases or remans constant [3]. Moreover the last term at the rght hand sde of (16) s ndependent of agent. Thus the agent have no motve to make an edge. Snce a tree s mnmally connected graph, all of ts edges are cut edges and all the agents are reachable from any other vertex n the tree. Thus, breakage of any edge by the agent, ncdent at t wll not ncrease the frst term n the rght hand sde of (15) [3]. Moreover the second term at the rght hand sde of (15) s ndependent of any edge ncdent at the agent. The altrustc contrbuton a () of the agent thus reduces or remans constant upon the breakage of any edge ncdent at t. Thus agents have no motve to break any of the edge ncdents at them. The set of smple graphs can be parttoned nto three classes: dsconnected graphs, mnmally connected graphs and non-mnmally connected graphs. It s to be establshed that under gven condton v 2v, s nether dsconnected nor t s non-mnmally connected. If s dsconnected then there always exst at least two agents and j whch are not reachable from each other. From (3) and (15), under the gven condton, makng of the edge fulflls the condton (a) of ame of Maxmal Altrustc Contrbuton. A dsconnected graph s thus unstable. If s non-mnmally connected then there must exst an edge e {, j} such that remans connected even after ts removal. Removal of edge e {, j} by the agent thus does not change the frst term n the rght hand sde of (16), moreover the last term n the rght hand sde of (16) s ndependent of any edge ncdent at, and under gven condton, () decreases upon removal of edge e [3]. A non-mnmal connected graph s also unstable. Remark 5: It follows from the results establshed n ths secton that f v 2v, then n any of the three onlne sequental decson games defned n ths secton a stable graph s always connected. 5. Smulaton Examples of Onlne Sequental Decson ames Smulaton results for the three sequental decson games are presented here. In these smulatons the games are started from an ntal graph and the agents are free to make allowed moves as n Defnton 9. One agent s randomly selected to makes moves at each tme, and can make or break as many edges as t desres to mprove ts contrbuton objectve functon. The method establshed n [8] for fast computaton of Shapley value s used n the smulaton. The smulatons were run untl one of the stable graphs s reached. The smulaton results support the theory developed n Secton 4. These results show that any connected graph s stable n ame 9

12 of Maxmal Margnal Contrbuton (MMC), only a complete graph s stable n ame of Maxmal Compettve Contrbuton (MCC), and any tree s stable n ame of Maxmal Altrustc Contrbuton (MAC). Some of the smulaton results are presented n the followng Fgures 2-4. Fg. 2. Evoluton of graph n MMC when agents are allowed to make or break as many edges as desred. (a) Intal, completely dsconnected graph. (b) Stable Connected raph (a tree) results after one move. Fg. 3.Evoluton of graph n MCC when agents are allowed to make or break as many edges as desred. (a) Intal, completely dsconnected graph. (b)-(e) Transton states on sequental moves of randomly selected agents. (f) Stable, Complete raph. Fg. 4. Evoluton of graph n MAC when agents are allowed to make or break as many edges as desred. (a) Intal, random graph. (b)-(d) Transton states on sequental moves of randomly selected agents. (e) Stable graph, a Tree. 10

13 6. Concluson A raphcal Coaltonal ame that assgns a value to each ndvdual n a team, represented by a communcaton graph, based on ts connectvty s presented n ths paper. The game defnes the noton of Postonal Advantage of agents n a coalton from a connectvty aspect. The components of the contrbuton made by an agent to team efforts are defned and ther dependence on the connectvty n teams s establshed. The setup of C and sequental decson games provdes an nsght to the nternal structure of coaltons, provdes a gudelne for the formaton of coaltons, and serves to examne the compettveness and altrusm aspects of the coalton. REFERENCES [1] Arney, D.C., and Peterson, E., Cooperaton n Socal Networks: Communcaton Trust and Selflessness, Proceedngs of the Army Scence Conference, Orlando, FL [2] Arslan,., and Shamma, J. S., Dstrbuted convergence to Nash equlbra wth local utlty measurements, In 43rd IEEE Conference on Decson and Control, pp , [3] Aurangzeb, M., On Postonal Advantages and Costs Compettve and Altrustc Coaltonal raph ames, Doctoral Thess, Department of Electrcal Engneerng, Unversty of Texas at Arlngton. (In Preparaton) [4] Baras, J. S., Jang, T., and Purkayastha, P., Constraned Coaltonal ames and Networks of Autonomous Agents, Invted paper, 3rd Internatonal Symposum on Communcatons, Control and Sgnal Processng, Malta, March 12-14, [5] Başar, T., and Olsder,. J., Dynamc Noncooperatve ame Theory, 2nd ed., Phladelpha, PA: SIAM, [6] Destel, R., raph Theory, Sprnger, Berln, [7] Dmand, M. A., and Dmand, R.W., The Hstory of ame Theory, vol. 1: From begnnng to 1945, Routledge, London, [8] Dragan, I., Potters, J., and Tjs, S. (1989). Superaddtvty of solutons of coaltonal games. Lbertas Matematca, vol.ix, [9] Jackson, M. O., and Wolnsky, A., A Strategc Model of Socal and Economc Networks, Journal of Economc Theory, 71, 44-74, Artcle No. 0108, [10] Marden, J. R., and Werman, A., Overcomng Lmtatons of ame-theoretc Dstrbuted Control, IEEE Conference on Decson and Control CDC, held jontly wth 28th Chnese Control Conference, [11] Marden, J. R., Arslan,., and Shamma, J. S., Jont strategy fcttous play wth nerta for potental games, IEEE Transactons on Automatc Control, [12] Myerson, R. B., raphs and Cooperaton n ames, Mathematcs of Operatons Research, vol. 2, No. 3, August [13] Neumann, J. von., Zur Theore der esellschaftsspele, Mathematsche Annalen, 100(1), p p Englsh translaton: A. W. Tucker and R. D. Luce, ed. (1959), On the Theory of ames of Strategy, Contrbutons to the Theory of ames, v. 4, p p (1928) [14] Neumann, J. von and O. Morgenstern, Theory of ames and Economc Behavor, Prnceton: Prnceton Unversty Press, [15] Peters, H., ame Theory, A Mult-Leveled Approach, Sprnger-Verlag Berln Hedeberg. ISBN (2008). 11

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