Question

Question 1. The international olympic committee consists of 55 members and needs to decide on the host city of the next summer games. There are five candidate cities: Berlin (B), Doha (D), Paris (P), Rome (R), and Tokyo (T). The individual preferences are: 18 members have ranking RPDBT, 12 mem- bers have ranking TDPBR, 10 members have ranking BTDPR, 9 members have ranking PBDTR, 4 members have ranking DTPBR, and 2 members have ranking DBPTR. (i) Is there a Condorcet winner? Justify your answer. (ii) What is the social preference according to Borda count rule?

Answer 1

Condorcet Winner is the one who beats all of its opponent in a one to one clash.

But even if any one gets beaten by any one of the other opponent then he can't be a Condorcet Winner.

I lets have the preferences as per question 18 mak.bg : - R >PЬ >>Т T > > P > B >R 12 members :- 10 members :- B > T > DOPR 9 members :- P > B> D > T R ^ >T 1 D 4 members :- DST > P. . 2 members :- D> B > P > I > R ) For Condorcet winnes. tale check for one-to-one match ups . • Rive li Es R>P :- 18 L P R - 12 + 10+9+4+2 = 24 80, TP >Ras 37 >18.

& R can't be a condorcet minner as it lost to p • - p us D . POD :- 18+9 = 27 5 PKD: 184 12 +10 +4+2 = 28 & IPCD as 20 27 228 . So minner as it p can't be a lost to condoreet D . . = 37 Dus R. DR - 12+10 +9+4+2 L OOR >D :- 18 ko, D >r as 37 > 18 .. us b. 5 0 > B 4 B > D :- 18+12+4+2 = 36 : 10+9 = 19 ko, PD 38 as 36>19 • busT. Ost :- 18+9+4+2 - 35 4 TD in 12+10 = 22

6, D>t as 33 > 22. sine, o beats every one else se. P, R, B, T. Hence, soha (D) is the condorcet minner ii) for borda count me need to give pointe as per the rankings As there are o cities so me will give points from I to 5 With 1 point ginen to least preferred and 5 point to the most preferred .. points 5 points - 4 forints - y points - 18 R. P P 12 T D 이 ㅇ red 10 9 4 2 B D D T B T B' D P p ola Ana 2 pointe - Tint - TR RR R R so lets iaculate the points for each candidate R :- (1945) + (12*2) + (1042) + (9x1) + (941) + (2x1)

= 90 + 12 + 10 + 9 + 4+2 [R = 127 points for p :- (1984) + (1243) + (10x2) + (9x5) + (4х3) + (2x3) E 72 + 36 + 20 + 45 + 12 +6. Tp = 101 points FOR D :- (1843) + (1284) + (10x3) + (9x3) + (4x5) + (2x5) < 54 + 48 + 30 + 27 + 20 + 10 B = 189 points FR B :- (18x2) + (12x2) + (10x5) + (2x4) 1 + (4x2) + (2x4) 36 24 + 50 + 36 + 8 + 8 . 1b = 162 foints

Foc T : (18x1) + (12 x 5) + (1024) + ax (9x 2) + (464) + (2x 2) = 18 + 60 + 40 + 18 + 16 + 4 .. [I = 156 points 7 T So, we can summarize all the points R 127 pointe - 5th P → 191 points - 1st D 189 hoints - B 162 points - 344 T 156 points - 4th do, social Preference as per Borda count - p> D > B > I > R . 1 or PDBTR
Summary

i) So, There is a Condorcet Winner that is D (Doha)

ii) Social Preference according to Borda count is : PDBTR