Suppose that political ideologies can be represented on a 5-point scale. The sca

Question

Suppose that political ideologies can be represented on a 5-point scale. The scale identifies five clearly distinguishable ideologies on a spectrum from the far left (1) to the far right (5). Assume that your electorate is evenly spread over the whole political spectrum with 20% identifying with each of the five ideologies. The only two political parties that are about to form (the country is a new democracy) can choose their ideologies anywhere on the 5-point scale. We assume that voters will vote for a party that is closest to their ideology and if both parties are in equal distance the voters will split evenly between them. Model this situation as a game. Represent payoffs as percent of votes a party gets (the other party gets the remaining percentage of votes). Solve the game through iterated dominance. To get the solution in (a) do you need to assume cardinal payoffs or ordinal payoffs would be sufficient? In other words, to arrive at the equilibrium in (a) do you merely need to assume that parties prefer more votes over less or that they also have to prefer 40% over 20% twice as strongly as 30% over 20%?

Explanation / Answer

Political ideologies range in 1 to 5 scale.

Since voter is uniformly distributed among these ideologies each ideology population scale has probability = 20/100 =1/5

If Party 1 gets x% of votes and party 2 gets (100-x) % votes

x/100 = (20/100)* scale occupied i.e scale occupied by first is (x/20)

and scale by second is = 5 -(x/20) = (100-x)/20

For the first party to win majority scale occupied by it must be >=2.5 and vote percentage >=50% s

b) For the given example

Case 1: 40% of given scale = (40/100) *20 = 8% of total votes.

compared to twice 20%

Case 2 - a)If 20 % first and 20 % second are from different scales

then 20/100*(20) = 4% of total votes from scale X and (20/100)*(20) = 4% of total from scale Y

which is also 8% of total.

Case 2 b) If 20 % first and 20 % second are from same scale then

(20/100)*(20) = 4% of total

now from scond it is out of remaining (80% of that scale) = (20/100)*(80/100) * 20 = 3.2%

Adding for case 2 a) we get 7.2%

since 7.2%<8% so

so for one scale difference is (8-7.2) =0.8%

over 5 scales difference is 5*0.8% = 4%

Case 1 is better over case case 2 b)

but Case 1 is equivalent to case 2 a)

40% over a scale is better than 20% twice on same scale but

40% over one scale is equivalent to 20% twice if both 20% are from different scales.

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