TIFR CSE 2013 | Part A | Question: 2

Question

Consider the following two types of elections to determine which of two parties $A$ and $B$ forms the next government in the 2014 Indian elections. Assume for simplicity an Indian population of size $545545 (=545 * 1001)$. There are only two parties $A$ and $B$ and every citizen votes.

TYPE C: The country is divided into $545$ constituencies and each constituency has $1001$ voters. Elections are held for each constituency and a party is said to win a constituency if it receive a majority of the vote in that constituency. The party that wins the most constituencies forms the next government.

TYPE P: There are no constituencies in this model. Elections are held throughout the country and the party that wins the most votes (among $545545$ voters forms the government.

Which of the following is true?

• A. If the party forms the govt. by election TYPE C winning at least two-third of the constituencies, then it will also forms the govt. by election TYPE P.
• B. If a party forms govt. by election TYPE C, then it will also form the govt. by election TYPE P.
• C. If a party forms govt. by election TYPE P, then it will also form the govt. by election TYPE C.
• D. All of the above
• E. None of the above

Answer 1

Minimum condition for Type C winning: if any party wins $273$ constituencies out of $545$ and with vote $501$ out of $1001$ for each.
Minimum condition for Type P winning: if any party wins $272773$ votes out of $545545$

Option A:

• Type C : Let party $A$ wins $2/3$ of  constituencies i.e., $364$ wins by $1$ vote and $181$ loss by all vote $364\times 501 + 181\times 0 = 182364$
• Type P : If $A$ wins it should have more than half of vote i.e., $272773$

$\qquad$ So, it is FALSE since $A$ got only $182364$ votes

Option B : Similar to option A

Option C:

• Type P: Let $A$ win by $272773$ votes 
• Type C: If $A$ wins $272$ constituencies with $1001$ votes and loss $272$ with $1001$ and $1$ constituency with $1000$ votes then, total received votes $=272\times 1001+272\times 0+1\times 1= 272773$. But $A$ lose the  election since $A$ won only $272$ constituencies out of $545$ 

So option E must be correct.  

Comments

There is artihmetic error in last point:

272*1001 + 272*0 + 1*1 = 272273 (not 272773 as written in answer)

Type C: A wins 272 constituencies with 1001 votes making vote tally for A to be 272272, 501 votes remain( 272773 - 272272 = 501 ). Distribute these 501 votes among remaining 273 constituencies in any way possible except one way ( 272*0+1*501 as this would make A win in 273 constituencies overall). Other than this every other distribution of 501 votes in 273 constituencies would make A lose.