Voting Games with Positive Weights and. Dummy Players: Facts and Theory

Transcription

1 Appled Mathematcal Scences, Vol 10, 2016, no 53, HIKARI Ltd, wwwm-hkarcom Votng Games wth Postve Weghts and Dummy Players: Facts and Theory Zdravko Dmtrov Slavov Varna Free Unversty, Varna, Bulgara Chrstna Slavova Evans The George Washngton Unversty, Washngton DC, USA Copyrght 2016 Zdravko Dmtrov Slavov and Chrstna Slavova Evans Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted Abstract In ths paper we consder a specal class of weghted votng games wth two basc characterstcs: each player has a postve weght and dchotomous rule - acceptance or reecton Frst, we present two real votng paradoxes where a player wth postve weght s a dummy Next, usng the knowledge of these paradoxes we generate some theoretcal statements Mathematcs Subect Classfcaton: 91A12 Keywords: votng game, postve weght, dummy, proper, decsve 1 Introducton Let N be a nonempty fnte set of players n a game G and every subset S N s referred to as a coalton The set N s called the grand coalton and N s called the empty coalton We denote the collecton of all coaltons by 2 and N the number of players of S 2 by S Let us label the players by 1,2, n, n N 2 The noton of a smple game was ntroduced by John von Neumann and Oscar Morgenstern n ther monumental book Theory of Games and Economc Behavor n 1944 [8] Ths game s a conflct n whch the only obectve s wnnng and the only rule s an algorthm to decde whch coaltons are wnnng

2 2638 Zdravko D Slavov and Chrstna S Evans Defnton 1 A smple game s a par ( N, v) where N { 1,2,, n} s a set N of players and v : 2 {0,1 } s the characterstc functon whch satsfes the followng three condtons: (1) v ( ) 0 (2) v ( N) 1 (3) s monotonc, e f S T N, then v( S) v( T) From the above defnton t follows that for each coalton S N we have ether v ( S) 0 or v ( S) 1 Defnton 2 For any coalton S N, S s wnnng f and only f v ( S) 1, S s losng f and only f v ( S) 0, and S s blockng f and only f S and N \ S are both losng The collectons of all wnnng, all losng and all blockng coaltons are denoted by W, L and B, respectvely By defnton, we obtan N W and L ; therefore, W and L are N nonempty, W L and W L 2 Observe that a coalton havng a wnnng sub-coalton s also wnnng, a sub-coalton of a losng coalton s also losng, the complement of a blockng coalton s also blockng, B can be ether empty or nonempty and B L For any player N, the collecton of all wnnng coaltons ncludng s denoted by W and the collecton of all wnnng coaltons excludng s denoted by nequalty W Clearly, f W W S W, then S { }W ; therefore, we obtan the Defnton 3 For any coalton S W, S s called a mnmal wnnng coalton f and only f S \{ } s a losng coalton for all S The collecton of all mnmal wnnng coaltons s denoted by MW It s easy to prove that MW and W are fnte sets, MW W and MW s nonempty Defnton 4 A player who does not belong to any mnmal wnnng coalton s called a dummy, e player N s a dummy f and only f S for all S MW A player who belongs to all mnmal wnnng coaltons s called a veto player, e player N has capacty to veto f and only f S for all S MW A player N s a dctator f and only f {} s a wnnng coalton Formally, for any player N, s a dctator s equvalent to { } MW, s a veto player s equvalent to S, s a dummy s equvalent to S MW S S MW 2 Basc Concepts and Defntons n Weghted Votng Games In ths secton we wll consder a specal class of smple games called weghted votng games wth dchotomous votng rule - acceptance ("yes") or reecton

3 Votng games wth postve weghts and dummy players 2639 ("no") These games have been found to be well-suted to model economc or poltcal bodes [5] The basc formal framework of ths study s as follows A weghted votng game ( N, v) s descrbed by G [ q; w1, w2,, wn ] where q and w 1, w2,, wn are postve nteger numbers such that q w k 1 k By conventon, we take w w f As a result we derve the followng propertes: (1) 1 q (2) n N 2 s the number of players (3) w 1 s the number of votes of player N (4) w1 w2 wn (5) q s the needed quota so that a coalton can wn (6) the symbol [ q ; w1, w2,, wn ] represents the smple game ( N, v) defned by 1, w q ks k v( S), where S N 0, w q ks k Of course, f N and w 0, then player s powerless, for more nformaton see [3] We wll consder three examples Example 1 The votng method of the Securty Councl of the Unted Natons, formed by 5 permanent (USA, UK, France, Russa and Chna) and 10 temporary members, s the game n whch each one of the permanent member has 7 votes and each one of the temporary member has only one vote, the establshed quota s 39 votes, and there are 45 total votes We observe that any coalton whch does not nclude all of the 5 permanent members has at most votes, whch s an nferor number to the fxed quota As a result ths coalton wll not be wnnng Hence, each one of the permanent members has the capacty to veto any proposal For more nformaton see [1] and [7] Example 2 The Bulgaran Parlament wth 240 seats uses two dfferent rules: a smple maorty by quota 121 (more than 1 ) and a qualfed maorty by 2 quota 161 (more than 2 ) The Fnsh Parlament wth 200 seats uses three 3 dfferent rules: a smple maorty by quota 101 (more than 1 ), a qualfed 2 maorty by quota 134 (more than 2 ), and n some specal cases by quota (more than 5 ) [1] 6 Defnton 5 A weghted votng game ( N, v) s called proper f and only f v ( S) v( N \ S) 1 for all S N Note that a weghted votng game s proper f and only f the complement of a wnnng coalton s not a wnnng coalton Ths means that n a proper game n

4 2640 Zdravko D Slavov and Chrstna S Evans the both coaltons S and N \ S cannot be wnnng In ths context, f S s wnnng, then N \ S s losng, but the converse s not true In ths paper, we wll only study proper games Defnton 6 A weghted votng game ( N, v) s called decsve f and only f v ( S) v( N \ S) 1 for all S N It s easy to prove that a smple game s decsve f and only f t has no blockng coalton For any coalton S, S s wnnng s equvalent to N \ S s losng If a smple game s decsve, then t has no blockng player, but the converse statement s not true If a smple game has a dctator, then t s decsve, but the converse statement s not true Let us consder a weghted votng game [ q ; w1, w2,, wn ], q qmn where mn 1 q 1 f s even and q mn f s odd It s easy to prove that: (1) q mn and 2q mn 1 for all postve nteger 2 (2) f s even, then 2q mn 1 In ths case ( q mn 1) ( qmn 1) ; therefore, t s possble to have a blockng coalton (3) f s odd, then 2q mn 1, e ( q mn 1) qmn In ths case, for q qmn we fnd that t s not possble to have a blockng coalton Hence, ths game s decsve Defnton 7 Consder two weghted votng games G1 [ q1; w1, w2,, wm ] and G2 [ q2; w1, w2,, wn ] Game G 1 s equvalent to game G 2 f and only f m n and MW ( G1 ) MW ( G2), e the set of all mnmal wnnng coaltons of game G 1 s equal to the set of all mnmal wnnng coaltons of game G 2 Note that the set of all wnnng coaltons and the set of all mnmal wnnng coaltons n the weghted votng game [ q ; w1, w2,, wn ] are the same as the set of all wnnng coaltons and the set of all mnmal wnnng coaltons n the weghted votng game [ q; w1, w2,, wn ] for every postve nteger number, respectvely As a result we obtan that the weghted votng game [ q ; w1, w2,, wn ] s equvalent to the game [ q; w1, w2,, wn ] Defnton 8 For any proper game ( N, v), a par of players, N s called symmetrc (or and are symmetrc) f and only f v( S { }) v( S { }) for all coaltons S N \{, } Of course, f w w, then players and are symmetrc, but the converse statement s not true For example, see game G [30;14,13,11,10] Consder a par of symmetrc players, N By defnton, f S N and S, then v( S) v( S { }\{ }) Now, t s easy to see that W W

5 Votng games wth postve weghts and dummy players Two Votng Paradoxes In ths secton we wll descrbe two paradoxes n real votng systems In prncple a player can be assgned weght zero, but n practce ths player would be slly, because t would be a dummy However, a player havng a postve weght can also be a dummy 31 The Votng Paradox of Luxembourg In ths subsecton we wll see that Luxembourg was a powerless country as a member of the European Unon Councl of Mnsters durng the perod Let us consder the Councl of Mnsters durng the above perod from a mathematcal pont of vew The decson rule s a weghted votng game L 58 [12; 4, 4, 4, 2, 2,1] In ths game the players are Germany, France, Italy, Belgum, Netherlands and Luxembourg, respectvely Note that the total sum of the weghts s 17 and the quota s 12, e n 6, 17 and q 12 It s easy to show that ths game s proper, see also [1] and [7] The number of wnnng coaltons s 14 and they are: {1,2,3,4,5,6}, {1,2,3,4,5}, {1,2,3,4,6}, {1,2,3,5,6}, {1,2,3,4}, {1,2,3,5}, {1,2,3,6}, {1,2,4,5,6}, {1,3,4,5,6}, {2,3,4,5,6}, {1,2,3}, {1,2,4,5}, {1,3,4,5} and {2,3,4,5} The number of mnmal wnnng coaltons s 4 and they are: {1,2,3}, {1,2,4,5}, {1,3,4,5} and {2,3,4,5} As a result we decde that player 6 s a dummy So player 6 or Luxembourg formally was never able to make any dfference n the votng process and was a dummy durng the perod [6] Obvously, game L 58 [12; 4, 4, 4, 2, 2,1] has no dctator, has no veto player and has a dummy player We also get that ths game has at least one blockng coalton Hence, ths game s not decsve For example, coalton { 1,4} s blockng and coalton { 4,5,6 } s losng but not blockng Clearly, players 1, 2 and 3 are symmetrc, and players 4 and 5 are symmetrc because they have equal weghts, respectvely 32 The Votng Paradox of Nassau Country In ths subsecton we wll study Nassau Country, New York, whch s a regon on Long Island The government of Nassau Country took the form of a Board of Supervsors n 1958 for the perod and n 1964 for the perod , one representatve for each of varous muncpaltes, who casts a block of votes [2] There are two specal weghted votng games used at varous tmes by Nassau Country We wll dscuss these two votng games 321 The Frst Votng Game The frst decson rule n 1958 was a weghted votng game N 58 [16; 9, 9, 7, 3,1,1] In ths game the players were Hempstead 1, Hempstead 2, North Hempstead, Oyster Bay, Long Beach and Glen Cove, respectvely In game N 58, the total sum of the weghts was 30 and the quota was 16, e n 6,

6 2642 Zdravko D Slavov and Chrstna S Evans 30 and q 16 As a result we fnd that game N 58 s proper It s easy to show that ths game has no blockng coalton; therefore, t s decsve The number of mnmal wnnng coaltons s 3 and they are: {1,2}, {1,3} and {2,3} As a result we see that Oyster Bay, Long Beach and Glen Cove (players 4, 5 and 6) are dummes Note that the number of all wnnng coaltons s , e W 32 and MW 3 In ths game players 1, 2 and 3 are symmetrc, but they do not have equal weghts Players 4, 5 and 6 do not have equal weghts and they are symmetrc too 322 The Second Votng Game The second decson rule n 1964 was a weghted votng game N 64 [58; 31, 31, 28, 21, 2, 2] In ths game the players were Hempstead 1, Hempstead 2, Oyster Bay, North Hempstead, Long Beach and Glen Cove, respectvely In game N 64, the total sum of the weghts was 115 and the quota was 58, e n 6, 115 and q 58 It s easy to show that ths game s also decsve The number of mnmal wnnng coaltons s also 3 and they are: {1,2}, {1,3} and {2,3} As a result we see that North Hempstead, Long Beach and Glen Cove (players 4, 5 and 6) are dummes The number of all wnnng coaltons s also 32 As n game N 58, we see n game N 64 that players 1, 2 and 3 are symmetrc, they do not have equal weghts, players 4, 5 and 6 are also symmetrc and they do not have equal weghts 323 On Decson Rules We have obtaned that both games N 58 [16; 9, 9, 7, 3,1,1 ] and N 64 [58; 31, 31, 28, 21, 2, 2] have no dctator, have no veto player and have no blockng coalton Obvously, Long Beach and Glen Cove are dummes n both games N 58 and N 64 Oyster Bay s a dummy player n game N 58, but North Hempstead s a dummy n game N 64 Note that game N 58 [16; 9, 9, 7, 3,1,1 ] s equvalent to game N 64 [58; 31, 31, 28, 21, 2, 2] We also see that the both games N 58 [16; 9, 9, 7, 3,1,1] and N 64 [58; 31, 31, 28, 21, 2, 2] are equvalent to game [ 8; 4, 4, 4,1,1,1 ] As a result, after 1964, the quota was rased to guarantee that no muncpalty was a dummy The next decson rule n 1970 was a weghted votng game N 70 [63; 31, 31, 28, 21, 2, 2], 115 and q 63 It s easy to see that {1,2,6} and {3,4,5,6} are two mnmal wnnng coalton, e ths game has no dummy player

7 Votng games wth postve weghts and dummy players Some Propertes of Weghted Votng Games In ths secton we wll consder a proper game [ q; w1, w2,, w n 1, wn ] when n 3 and each player has a postve weght Here, we generate some theoretcal statements Theorem 1 If player N s a dummy and S N, then: (a) S W mples S { } W and S \{ } W (b) S L mples S { } L and S \{ } L (c) v( S) v( S { }) ( S \{ }) (d) S B mples S { } B and S \{ } B Proof From Defnton 4 t follows that f S MW, then S (a) Let S W There are two cases: Case 1 If S, then S { } S W and S MW Player s a dummy, S W and S MW mply S \{ } W Case 2 If S, then S \{ } S W From S W t follows that S { } W Fnally, we obtan S { } W and S \{ } W (b) Let S L There are two cases: Case 1 If S, then S { } S L and S \{ } S L Case 2 If S, then S \{ } S L Let us assume that T S { } W Player s a dummy, S T, S and T mply T \{ } S W, see part (a) Ths leads to a contradcton; therefore, S { } L Fnally, we obtan S { } L and S \{ } L (c) Drectly from parts (a) and (b) (d) Let S B, e S L and N \ S L Accordng to part (b) for coaltons S and T N \ S we obtan S { } L, S \{ } L, T { } L and T \{ } L Ths means that S { } B and S \{ } B The theorem s proven Theorem 2 Consder a par of players, N The followng statements are true (a) If player s not a dummy and player s a dummy, then w w (b) If player s a dummy and w w, then player s also a dummy (c) If players and are symmetrc, then s a dummy s equvalent to s a dummy Proof (a) Let us denote MW k { S MW : k S} for k N It s easy to show that MW s not empty and MW s empty If S MW MW, then w q k S k For coalton T S \{ } t follows that w w q k T k and w q, e T s a losng coalton It s known that player s a dummy; k T k

8 2644 Zdravko D Slavov and Chrstna S Evans therefore, S and T Now let us consder a new coalton P T {} There are two cases: Case 1 Let P be a losng coalton In ths case we have w w q k T k Case 2 Let P be a wnnng coalton From the condton that s a dummy t follows that T P \{ } s a wnnng coalton too Ths leads to a contradcton Fnally, we obtan w w q k T k From the nequaltes w w q k T k and w w q k T k we get w w (b) Let us assume that player s not a dummy From player beng a dummy and part (a) t follows that w w Ths leads to a contradcton wth the condton w w ; therefore, player s a dummy (c) Let player be a dummy and assume that player s not a dummy Hence, there exsts a coalton S MW such that S Consder a new coalton T S \{ } Clearly, T L and, T ; therefore, v( T) v( T { }) v( T { }) v( S) because and are symmetrc, and s a dummy As a result we obtan v( T) v( S), v ( T) 0 and v ( S) 1 Ths s a contradcton; therefore, player s a dummy The theorem s proven Theorem 3 [4 - Theorem 7] Player N s a dummy f and only f the nequalty a w w 1 wn q holds where a max{ w : S {1,2,, 1}, S W} S Note that the two theorems above allow us to count the number of all dummy players and construct the set of all dummy players Now we wll focus our attenton on game L 58 On the basc of nformaton of ths game we generate the frst basc theorem n our paper Theorem 4 If n 4, k 2, 0 w n k wn 1, q 0 modulo k and w 0 modulo k for all { 1,2, n 1}, then player n s a dummy Proof Let p be a postve nteger number between q and, and denote W ( p) { S W : w } and MW ( p) MW W( p) It s easy to show that pq S p W W ( p) and MW MW ( p) pq Of course, f p [ q, ] and W ( p) s nonempty, then ether p wn modulo k or p 0 modulo k From W W ( p) s nonempty t follows that there exsts p [ q, ] pq such that W ( p) s nonempty too Here there are two cases Case 1 Let p w modulo k n

9 Votng games wth postve weghts and dummy players 2645 In ths case we have p q Let us assume that there exsts S W( p), e w p S and p q From w S wn modulo k t follows that n S and p q w n Consder a coalton T S \{ n} It s easy to see that T W( p wn ); therefore, S MW As a result we derve S MW ( p), e MW ( p) s empty Case 2 Let p 0 modulo k In ths case there exsts S W( p) such that w p From S S w 0 modulo k t follows that n S As a result we obtan S W( p) mples n S Let us denote A { p[ q, ]: ns W( p)} Obvously, n S for all S W( p) p A Fnally, we have that n S for all S p A W( p) and MW MW ( p) pq MW ( p) p A W( p) As a result we obtan n S pa for all S p A MW ( p) and MW p A W( p) It s easy to show that n S for all S MW, e player n s a dummy The theorem s proven Corollary 1 [6 Theorem 3] If n 3, w n1 2, 0 w n wn 1, q 0 modulo w n1 and w 0 modulo w n1 for all { 1,2,, n 2}, then player n s a dummy Proof Accordng to Theorem 4 for k w n 1 Lastly, let us analyze both games N 58 and N 64 Usng the knowledge of ths votng paradox we wrte the second basc theorem Theorem 5 Consder a proper game G [ q; w1, w2,, w n 1, wn ] when n 4 (a) If { 2,3} W, then MW {{1,2},{1,3},{2,3 }} In partcular, ths game has no dctator and player 4 s a dummy (b) If game G [ q; w1, w2,, wn ] s decsve, player 1 s not a dctator and player 4 s a dummy, then MW {{1,2},{1,3},{2,3 }} Proof (a) From { 2,3} W t follows that { 1,2},{1,3},{1,2,3} W Game [ q ; w1, w2,, wn ] s proper mples { 1},{2},{3} L So, we fnd that {{ 1,2},{1,3},{2,3}} MW, { 1,2,3} W \ MW and player 1 s not a dctator Now, from {{ 1,2},{1,3},{2,3}} MW and { 1,2,3} W t follows that { 4,, n},{1,4,, n},{2,4,, n},{3,4,, n} L Hence, f S MW and S 2, then S {{ 1,2},{1,3},{2,3 }} MW Let us assume that S MW and S 3 In ths case there exsts a subcoalton T S such that T {,, k} where, {1,2,3 } and k { 4,, n} We know that {, } MW ; therefore, T MW and ths mples S MW Ths leads to a contradcton Thus, we obtan MW {{1,2},{1,3},{2,3 }}

10 2646 Zdravko D Slavov and Chrstna S Evans (b) From player 1 s not a dctator and player 4 s a dummy t follows that { 1},{2},{3}L and { 1,2,3} W, see Theorems 1 and 2 Accordng to Theorem 1 t follows that { 1,4} L ; therefore, { 2,3} MW We also see that { 2,4},{3,4} L Thus, we also obtan { 1,2},{1,3} MW Player 4 s a dummy mples MW {{1,2},{1,3},{2,3}} The theorem s proven References [1] A Laruelle, F Valencano, Votng and Collectve Decson-Makng: Barganng and Power, Cambrdge Unversty Press, [2] D Lppman, Math n Socety, Creatve Commons BY-SA, 2012 [3] N Masser, Decson-Makng n Commttees: Game-Theoretc Analyss, Sprnger Berln Hedelberg, [4] T Matsu, Y Matsu, A Survey of Algorthms for Calculatng Power Indces of Weghted Maorty Games, Journal of the Operatons Research Socety of Japan, 43 (2000), no 1, [5] B Peleg, Chapter 8 Game-Theoretc Analyss of Votng n Commttees Chapter n Handbook of Socal Choce and Welfare, vol 1, Elsever, 2002, [6] Z Slavov, C Evans, On the Votng Paradox of Luxembourg and Decson Power Indces, Mathematcs and Educaton n Mathematcs, 43 (2014), [7] A Taylor, A Pacell, Mathematcs and Poltcs: Strategy, Votng, Power and Proof, Strnger -Verlag New York, [8] J Von Neumann, O Morgenstern, Theory of Game and Economc Behavor, Prnceton Unversty Press, 1944 Receved: July ; Publshed: August 31, 2016