攻防博弈的不同版本和纳什均衡策略

mathe · 发表于 7 天前

KeyTo9_Fans 发表于 2025-4-30 22:05
回贴里给出的解是【“轰对轰”可以相互抵消，双方各扣1点能量后，游戏继续（而不是同归于尽，平局）】的版 ...

Singular的数值解:

[6]:
[1]:
1
[2]:
0
[3]:
0
[4]:
0.5
[5]:
0.5322164753917473785730130135837516764153
[6]:
0
[7]:
0.4677835246082526214269869864162483235847
[8]:
0.2338917623041263107134934932081241617924
[9]:
0.5435599392114846833945392592447825794875
[10]:
0
[11]:
0.4564400607885153166054607407552174205125
[12]:
0.1067575702040283885142929385064802401963
[13]:
0.3333333333333333333333333333333333333333
[14]:
0
[15]:
0.6666666666666666666666666666666666666667
[16]:
0
[17]:
0.6767027602640444037698408890552034157298
[18]:
0.3232972397359555962301591109447965842702
[19]:
0
[20]:
0.7661082376958736892865065067918758382076
[21]:
0.2890991910655504452694973388386245833886
[22]:
0.1677023090143674924812794066684739089692
[23]:
0.5431984999200820622492232544929015076422
[24]:
0.5
[25]:
0.229723376779308992354268786751867182406
[26]:
0.1484504857764892184708816136855450780036
[27]:
0.6218261374442017891748495995625877395904
[28]:
0.345634414455267594632411852533035565922
[29]:
0.2960166353232403562260597893149840700427
[30]:
0.1964044119198634755202773300619960957683
[31]:
0.507578952756896168253662880623019834189
[32]:
0.1964044119198634755202773300619960957683
[33]:
0.5435599392114846833945392592447825794875
[34]:
0.4564400607885153166054607407552174205125
[35]:
0
[36]:
0.8932424297959716114857070614935197598037
[37]:
0.235080991685955290728811793040237656087
[38]:
0.2272600907872339989668723236668473384232
[39]:
0.5376589175268107103043158832929150054898
[40]:
0.654365585544732405367588147466964434078
[41]:
0.1865110237728683122925916452336539105152
[42]:
0.2093678467839996568771268907180845244511
[43]:
0.6041211294431320308302814640482615650337
[44]:
0.5
[45]:
0.2456982845434307717922939735167945759736
[46]:
0.3267169971550315044857450841469151865197
[47]:
0.4275847183015377237219609423362902375067
[48]:
0.3267169971550315044857450841469151865197
[49]:
1
[50]:
0
[51]:
0
[52]:
1
[53]:
0.172208189184588117717598023423322049212
[54]:
0.2266464353678437119922203717975587385607
[55]:
0.6011453754475681702901816047791192122273
[56]:
0.8035955880801365244797226699380039042317
[57]:
0.1201079962998774388409560385447331389395
[58]:
0.2264580093900595521875537931614364880211
[59]:
0.6534339943100630089714901682938303730394
[60]:
0.6732830028449684955142549158530848134803
[61]:
0.1476907126624975141627392922523725624813
[62]:
0.4261546436687512429186303538738137187593
[63]:
0.4261546436687512429186303538738137187593
[64]:
0.5

mathe · 发表于 6 天前

根据KeyTo9_Fans的结果给出的对应的Singular代码

LIB "primdec.lib";
LIB "solve.lib";
ring r=0,(v(0..63)),dp;
//VID(0,0,XI)+VID(0,0,HO)+VID(0,0,FA)=1
poly f0=v(0)+v(1)+v(2)-1;
//VID(0,1,XI)+VID(0,1,HO)+VID(0,1,FA)=1
poly f1=v(4)+v(5)+v(6)-1;
//VID(0,2,XI)+VID(0,2,HO)+VID(0,2,FA)=1
poly f2=v(8)+v(9)+v(10)-1;
//VID(0,3,XI)+VID(0,3,HO)+VID(0,3,FA)=1
poly f3=v(12)+v(13)+v(14)-1;
//VID(1,0,XI)+VID(1,0,HO)+VID(1,0,FA)=1
poly f4=v(16)+v(17)+v(18)-1;
//VID(1,1,XI)+VID(1,1,HO)+VID(1,1,FA)=1
poly f5=v(20)+v(21)+v(22)-1;
//VID(1,2,XI)+VID(1,2,HO)+VID(1,2,FA)=1
poly f6=v(24)+v(25)+v(26)-1;
//VID(1,3,XI)+VID(1,3,HO)+VID(1,3,FA)=1
poly f7=v(28)+v(29)+v(30)-1;
//VID(2,0,XI)+VID(2,0,HO)+VID(2,0,FA)=1
poly f8=v(32)+v(33)+v(34)-1;
//VID(2,1,XI)+VID(2,1,HO)+VID(2,1,FA)=1
poly f9=v(36)+v(37)+v(38)-1;
//VID(2,2,XI)+VID(2,2,HO)+VID(2,2,FA)=1
poly f10=v(40)+v(41)+v(42)-1;
//VID(2,3,XI)+VID(2,3,HO)+VID(2,3,FA)=1
poly f11=v(44)+v(45)+v(46)-1;
//VID(3,0,XI)+VID(3,0,HO)+VID(3,0,FA)=1
poly f12=v(48)+v(49)+v(50)-1;
//VID(3,1,XI)+VID(3,1,HO)+VID(3,1,FA)=1
poly f13=v(52)+v(53)+v(54)-1;
//VID(3,2,XI)+VID(3,2,HO)+VID(3,2,FA)=1
poly f14=v(56)+v(57)+v(58)-1;
//VID(3,3,XI)+VID(3,3,HO)+VID(3,3,FA)=1
poly f15=v(60)+v(61)+v(62)-1;
poly f16=2*v(3)-1;
poly f17=2*v(23)-1;
poly f18=2*v(43)-1;
poly f19=2*v(63)-1;
poly f20=v(19)+v(7)-1;
poly f21=v(35)+v(11)-1;
poly f22=v(39)+v(27)-1;
poly f23=v(51)+v(15)-1;
poly f24=v(55)+v(31)-1;
poly f25=v(59)+v(47)-1;
poly f26=v(2);
poly f27=v(18);
poly f28=v(34);
poly f29=v(50);
poly f30=v(1);
poly f31=v(5);
poly f32=v(9);
poly f33=v(13);
poly f34=v(0)-1;
poly f35=v(48)-1;
//VID(0,0,MT)
poly f36=v(0)*v(0)*v(23)+v(0)*v(2)*v(19)+v(2)*v(0)*v(7)+v(2)*v(2)*v(3)-v(3);
//VID(0,1,MT)
poly f37=v(4)*v(16)*v(27)+v(4)*v(18)*v(23)+v(6)*v(16)*v(11)+v(6)*v(17)*v(3)+v(6)*v(18)*v(7)-v(7);
//VID(0,2,MT)
poly f38=v(8)*v(32)*v(31)+v(8)*v(34)*v(27)+v(10)*v(32)*v(15)+v(10)*v(33)*v(7)+v(10)*v(34)*v(11)-v(11);
//VID(0,3,MT)
poly f39=0+v(12)*v(50)*v(31)+v(14)*v(49)*v(11)+v(14)*v(50)*v(15)-v(15);
//VID(1,0,MT)
poly f40=v(16)*v(4)*v(39)+v(16)*v(6)*v(35)+v(17)*v(4)+v(17)*v(6)*v(3)+v(18)*v(4)*v(23)+v(18)*v(6)*v(19)-v(19);
//VID(1,1,MT)
poly f41=v(20)*v(20)*v(43)+v(20)*v(22)*v(39)+v(21)*v(20)+v(21)*v(21)*v(3)+v(21)*v(22)*v(7)+v(22)*v(20)*v(27)+v(22)*v(21)*v(19)+v(22)*v(22)*v(23)-v(23);
//VID(1,2,MT)
poly f42=v(24)*v(36)*v(47)+v(24)*v(38)*v(43)+v(25)*v(36)+v(25)*v(37)*v(7)+v(25)*v(38)*v(11)+v(26)*v(36)*v(31)+v(26)*v(37)*v(23)+v(26)*v(38)*v(27)-v(27);
//VID(1,3,MT)
poly f43=0+v(28)*v(54)*v(47)+v(29)*v(52)+v(29)*v(53)*v(11)+v(29)*v(54)*v(15)+v(30)*v(53)*v(27)+v(30)*v(54)*v(31)-v(31);
//VID(2,0,MT)
poly f44=v(32)*v(8)*v(55)+v(32)*v(10)*v(51)+v(33)*v(8)+v(33)*v(10)*v(19)+v(34)*v(8)*v(39)+v(34)*v(10)*v(35)-v(35);
//VID(2,1,MT)
poly f45=v(36)*v(24)*v(59)+v(36)*v(26)*v(55)+v(37)*v(24)+v(37)*v(25)*v(19)+v(37)*v(26)*v(23)+v(38)*v(24)*v(43)+v(38)*v(25)*v(35)+v(38)*v(26)*v(39)-v(39);
//VID(2,2,MT)
poly f46=v(40)*v(40)*v(63)+v(40)*v(42)*v(59)+v(41)*v(40)+v(41)*v(41)*v(23)+v(41)*v(42)*v(27)+v(42)*v(40)*v(47)+v(42)*v(41)*v(39)+v(42)*v(42)*v(43)-v(43);
//VID(2,3,MT)
poly f47=0+v(44)*v(58)*v(63)+v(45)*v(56)+v(45)*v(57)*v(27)+v(45)*v(58)*v(31)+v(46)*v(57)*v(43)+v(46)*v(58)*v(47)-v(47);
//VID(3,0,MT)
poly f48=v(48)*v(12)+v(48)*v(14)+v(49)*v(12)+v(49)*v(14)*v(35)+v(50)*v(12)*v(55)+v(50)*v(14)*v(51)-v(51);
//VID(3,1,MT)
poly f49=v(52)*v(28)+v(52)*v(30)+v(53)*v(28)+v(53)*v(29)*v(35)+v(53)*v(30)*v(39)+v(54)*v(28)*v(59)+v(54)*v(29)*v(51)+v(54)*v(30)*v(55)-v(55);
//VID(3,2,MT)
poly f50=v(56)*v(44)+v(56)*v(46)+v(57)*v(44)+v(57)*v(45)*v(39)+v(57)*v(46)*v(43)+v(58)*v(44)*v(63)+v(58)*v(45)*v(55)+v(58)*v(46)*v(59)-v(59);
//VID(3,3,MT)
poly f51=v(60)*v(60)*1/2+v(60)*v(62)+v(61)*v(60)+v(61)*v(61)*v(43)+v(61)*v(62)*v(47)+v(62)*v(61)*v(59)+v(62)*v(62)*v(63)-v(63);
//extra equation by sim
//XI[0,1]=FA[0,1]
poly f52=v(16)*v(27)+v(18)*v(23)-(v(16)*v(11)+v(18)*v(7)+v(17)*v(3));
//XI[0,2]=FA[0,2]
poly f53=v(32)*v(31)+v(34)*v(27)-(v(32)*v(15)+v(34)*v(11)+v(33)*v(7));
//XI[0,3]=FA[0,3]
poly f54=v(48)*0+v(50)*v(31)-(v(48)*0+v(50)*v(15)+v(49)*v(11));
//XI[0,3]=HO[0,3]
poly f55=v(48)*0+v(50)*v(31);
//FA[0,3]=HO[0,3]
poly f56=v(48)*0+v(50)*v(15)+v(49)*v(11);
//XI[1,0]=HO[1,0]
poly f57=v(4)*v(39)+v(6)*v(35)-(v(4)+v(6)*v(3));
//XI[1,1]=FA[1,1]
poly f58=v(20)*v(43)+v(22)*v(39)-(v(20)*v(27)+v(22)*v(23)+v(21)*v(19));
//XI[1,1]=HO[1,1]
poly f59=v(20)*v(43)+v(22)*v(39)-(v(20)+v(22)*v(7)+v(21)*v(3));
//FA[1,1]=HO[1,1]
poly f60=v(20)*v(27)+v(22)*v(23)+v(21)*v(19)-(v(20)+v(22)*v(7)+v(21)*v(3));
//XI[1,2]=FA[1,2]
poly f61=v(36)*v(47)+v(38)*v(43)-(v(36)*v(31)+v(38)*v(27)+v(37)*v(23));
//XI[1,2]=HO[1,2]
poly f62=v(36)*v(47)+v(38)*v(43)-(v(36)+v(38)*v(11)+v(37)*v(7));
//FA[1,2]=HO[1,2]
poly f63=v(36)*v(31)+v(38)*v(27)+v(37)*v(23)-(v(36)+v(38)*v(11)+v(37)*v(7));
//XI[1,3]=FA[1,3]
poly f64=v(52)*0+v(54)*v(47)-(v(52)*0+v(54)*v(31)+v(53)*v(27));
//XI[1,3]=HO[1,3]
poly f65=v(52)*0+v(54)*v(47)-(v(52)+v(54)*v(15)+v(53)*v(11));
//FA[1,3]=HO[1,3]
poly f66=v(52)*0+v(54)*v(31)+v(53)*v(27)-(v(52)+v(54)*v(15)+v(53)*v(11));
//XI[2,0]=HO[2,0]
poly f67=v(8)*v(55)+v(10)*v(51)-(v(8)+v(10)*v(19));
//XI[2,1]=FA[2,1]
poly f68=v(24)*v(59)+v(26)*v(55)-(v(24)*v(43)+v(26)*v(39)+v(25)*v(35));
//XI[2,1]=HO[2,1]
poly f69=v(24)*v(59)+v(26)*v(55)-(v(24)+v(26)*v(23)+v(25)*v(19));
//FA[2,1]=HO[2,1]
poly f70=v(24)*v(43)+v(26)*v(39)+v(25)*v(35)-(v(24)+v(26)*v(23)+v(25)*v(19));
//XI[2,2]=FA[2,2]
poly f71=v(40)*v(63)+v(42)*v(59)-(v(40)*v(47)+v(42)*v(43)+v(41)*v(39));
//XI[2,2]=HO[2,2]
poly f72=v(40)*v(63)+v(42)*v(59)-(v(40)+v(42)*v(27)+v(41)*v(23));
//FA[2,2]=HO[2,2]
poly f73=v(40)*v(47)+v(42)*v(43)+v(41)*v(39)-(v(40)+v(42)*v(27)+v(41)*v(23));
//XI[2,3]=FA[2,3]
poly f74=v(56)*0+v(58)*v(63)-(v(56)*0+v(58)*v(47)+v(57)*v(43));
//XI[2,3]=HO[2,3]
poly f75=v(56)*0+v(58)*v(63)-(v(56)+v(58)*v(31)+v(57)*v(27));
//FA[2,3]=HO[2,3]
poly f76=v(56)*0+v(58)*v(47)+v(57)*v(43)-(v(56)+v(58)*v(31)+v(57)*v(27));
//XI[3,1]=FA[3,1]
poly f77=v(28)*1+v(30)*1-(v(28)*v(59)+v(30)*v(55)+v(29)*v(51));
//XI[3,1]=HO[3,1]
poly f78=v(28)*1+v(30)*1-(v(28)+v(30)*v(39)+v(29)*v(35));
//FA[3,1]=HO[3,1]
poly f79=v(28)*v(59)+v(30)*v(55)+v(29)*v(51)-(v(28)+v(30)*v(39)+v(29)*v(35));
//XI[3,2]=FA[3,2]
poly f80=v(44)*1+v(46)*1-(v(44)*v(63)+v(46)*v(59)+v(45)*v(55));
//XI[3,2]=HO[3,2]
poly f81=v(44)*1+v(46)*1-(v(44)+v(46)*v(43)+v(45)*v(39));
//FA[3,2]=HO[3,2]
poly f82=v(44)*v(63)+v(46)*v(59)+v(45)*v(55)-(v(44)+v(46)*v(43)+v(45)*v(39));
//XI[3,3]=FA[3,3]
poly f83=v(60)*1/2+v(62)*1-(v(60)*0+v(62)*v(63)+v(61)*v(59));
//XI[3,3]=HO[3,3]
poly f84=v(60)*1/2+v(62)*1-(v(60)+v(62)*v(47)+v(61)*v(43));
//FA[3,3]=HO[3,3]
poly f85=v(60)*0+v(62)*v(63)+v(61)*v(59)-(v(60)+v(62)*v(47)+v(61)*v(43));
poly f86=3*v(12)-1;
ideal ii=f0,f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20,f21,f22,f23,f24,f25,f26,f27,f28,f29,f30,f31,f32,f33,f34,f35,f36,f37,f38,f39,f40,f41,f42,f43,f44,f45,f46,f47,f48,f49,f50,f51,f52,f53,f54,f55,f56,f57,f58,f59,f60,f61,f62,f63,f64,f65,f66,f67,f68,f69,f70,f71,f72,f73,f74,f75,f76,f77,f78,f79,f80,f81,f82,f83,f84,f85,f86;
ideal i0=std(ii);
dim(i0);i0;
def AC=solve(i0,40,0);

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