Intra-party decision making, party formation, and moderation in multiparty systems: Examples

Transcription

1 Intra-party decision making, party formation, and moderation in multiparty systems: Examples GeraldPech October 8, Detailsofexample1inthemaintext Example1 ConsideragamewithN ={1,2,3,4,5}andu i (x)= x x i, withthedistributionofblisspoints x 1 =0.1, x 2 =0.2, x 3 =0.5, x 4 =0.55, x 5 =1andx sq =1. Agentsarepessimistic. Partition π 1 = {{1},{2,3,4},{5}} results in M = {2,3,4} and gives the payoffvector(.3167,.2167,.1167,.1333,.5833), partition π 2 = {{1,2},{3,4},{5}}resultsinM={{1,2},{3,4}}andgives(.2375,.1875,.1875,.2125,.6625), partition π 3 = {{1,2},{3,4,5}} results in M = {3, 4, 5} and gives (.1667,.1333,.2333,.2833,.7333) as does partitionπ 4 ={{1},{2},{3,4,5}}andpartitionπ 5 ={{1},{2},{3,4},{5}}gives (.3167,.2167,.2333,.2833,.7333). These partitions dominate every otherpartitioninπ,i.e. thereisadeviationbyacoalitionwhichresultsin π 1,π 2,π 3,π 4 orπ 5. Yetnoneofthesepartitionsisstable: Coalition{1,2} wantstodeviatefromπ 1 andπ 5 andinducesπ 2,coalition{3,4,5}wantsto deviatefromπ 2 andinducesπ 3. Coalition{2,3,4}wantstodeviatefromπ 3 andπ 4,inducingπ 1. KIMEP, gp@geraldpech.net 1

2 2 Detailsofexample2inthemaintext Example two of the main text demonstrates that the following remark is true: Remark 1 Inastablepartystructure,x Z M andx m x sq ispossible. Example2 Consideragamewithx sq =0.6,n=11andu i (x)= x x i with the distribution of bliss points x 1 = 0.35, x 2 = 0.35+ε, x 3 = 0.4, x 4 = 0.454, x 5 = 0.499, x 6 = m=0.5, x 7 = , x 8 = ε and x i =0.6+(i 9)εfori=9,10,11,ε 0. Agentsarepessimistic. We want to show stability of partition π = {{1},S,{8},{9},..} with S={2,3,4,5,6,7}. G(π )={S}isasingletonandinS,agent5ispivotal. If agent2proposes0.398,agents3,4and5supporttheproposal. Beforeshowingthatstabilityofπ weshowstabilityofπ 1 ={{1,2,3,4,5},{6,7},{8},{9}, {10},{11}} and π 6 = {{1,2},{3,4,5,6,7,8},{9},{10},{11}}. The indices refer to the table. Stability of π 1 : π and π 6 are preferred by all agents i 6. Yet deviations from π 1 to intermediate partitions such as π = π 12 or π 13 are deterredbecauseπ 1 Ξ(π)viaπ 14 andπ 2. Stabilityof π 6 : 1,2,3and4preferπ. {1,2,3}maydeviatetoπ=π 15 or{1,2,3,4}toπ=π 16,yetforeachπ,π 6 Ξ(π)viaadeviationof{5,6} toπ 17 (π 18 ). Nowwecanshowstablitityofπ. First,considerdeviationsbyi m. π 1 isstable(whichrulesoutstability ofpartitionsπ 2 andπ 3 )andpreferredbyi 7. However,6cannotbeforced intoπ 1 : 7maybringaboutπ 4,butπ Ξ(π 4 )(viaπ 5 andπ 2 ). {5,6,7,8}wouldpreferstableπ 6 overπ. Deviationsof{5,6,7}toπ 8 or of{5,6}toπ 9 orof{6}toπ 9 ={{1},{2,3,4,5},{6},...}-resultinginthe samepayoffasπ 9 -aredeterredbecauseforeachπ,π Ξ(π). Adeviation of{5,6,7,8}toπ 7 isdeterredbecauseπ Ξ(π 7 )viaπ 9. A deviation of {4,5,6,7} to π 10 by is deterred because π Ξ(π 10 ). Inclusion in government of agents further on the right than 8 is objected evenby5and6. Furthermore,{1,2,3}or{1,2}donotwanttodeviatebecausepartitions whichcouldbereachedsuchasπ 11 areunattractiveforthedeviators. 3 Tables 2

3 Partition government U 1 U2 U 2 U 3 U 4 U 5 Deviation π 1 = * π 2 = π 2 = ,34* π 3 = , , π 3 = * π 1 = π 4 = * π 1 = , , π 5 = ,34* π 2 = , , ,2, ,2, π= min π 5 = ,23 (M1) ,4 (M2) ,5<M ,23,4 (M3) ,4,5 <M π= min π 3 = ,3 (M1) ,4 < M ,5 <M ,4,5 <M ,3,4 (M2) π= ,2,3* π 2 = i,45;i=1,2, π= ,3* π 2 = , , π= ,23* π 5 = , , π= * π 2 = π= * π 2 = , , π= * π 3 = π= * π 4 = π= * π 1 = π= min π 1 = ,2,3 (M1) ,2,4 <M i,j,5 for i,j< ,3,4 (M2) ,3,4 (M3) ,2,3,4 (M4) Table 1: Details of example 1. Pay offs U i for selected partitions. signifies a party boundary, M i a potential government, signifies uniqueness and < M j domination by M j via. Parties in M i are separated by commas. min signifies taking the minimum over pay offs for different M i. The final column notes possible deviations to other partitions. Please select Sheet 2" for computations and definitions

4 Partition Government U 1 =U 2 U2 U 3 U 4 U 5 U 6 =U m U 7 =U 8 U 9 =U 10 =U 11 π*= * , , π 1 = ,67* π 2 = min ,2345,67 (M1) ,67 (M2) π 3 = min ,2345,678* ,678* π 4 = ,23456* , ,23456, π 5 = ,23456* , ,23456, π 6 = * , π 7 = min , ,234, π 8 = min ,234,567 (M1) ,567,8 (M2) ,234,567,8 (M3) π 9 = min ,234,56 (M1) ,234,56,78 (M2) ,56,78 (M3) π 10 = min ,23,4567 (M1) ,4567,8 (M2) ,4567 (M3) π 11 = min ,456 (M1) ,456,7 (M2) ,456,7,8 (M3) π 12 = min ,6 (M1) ,6,7 (M2) π 13 = ,456* ,456, π 14 = ,678* π 15 = ,45678* π 16 = ,5678* π 17 = min ,4,56 (M1) ,4,56,78 (M2) ,56,78 (M3) ,56 < M π 18 = min ,56 (M1) ,56,78 (M2) Table 2: Details of example 2. Pay offs U i for selected partitions. signifies a party boundary, M i a potential government, signifies uniqueness and < M j domination by M j via. Parties in M i are separated by commas. min signifies taking the minimum over pay offs for different M i. please select "Sheet 2" for computations, definitions and further partitions

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