In some sovereign states there is a high degree of heterogeneity within the population across different territories. A great source to empirically observe this phenomenon are the election results. In this article we analyse the Spanish general elections results for 2011 to detect the different communities within the Spanish state. We continue describing a model in which we measure the global happiness of the population with the results of the elections. The final section of the article shows an algorithm that is able to optimize the global happiness by more than 11% by creating a few new borders.
We define the happiness of each voter as the representation (or share of votes) that the party he voted for got. Its range is from 0 to 1, higher meaning more happy. For example, if party A gets 70% of the votes, a person who voted for party A will be 0.7 happy.
We then define the happiness of a whole country as the sum of the happiness of each individual. We can calculate this number for the country of Spain given the elections of 2011 and the result is 7,570,311.
We can express this mathematically. The respresentation of party p is.
Where Vij is the number of votes of party i and region j.
The we can calculate the total happiness of a country as.
It's easy to see that with the previous definition, a country with more consensus will be happier than a heterogeneous one. It is possible however, to create borders to split a heterogeneous state into more than one homogeneous smaller states. In the next section we present an algorithm that will let us do it efficiently.
To calculate the happiness of several states, we just calculate the happiness of each state and we add them up.
We use the method of Extremal optimization for this task. We keep a vector S with as many elements as regions. The i-th element of the vector contains the state the region i belongs to. At the beginning of the algorithm, all the elements of the vector are set to 0, wich means that all the regions belong to the same state. Then we search for the region that would yield to the highest happiness if it belonged to a different state (by setting the correspoding element of S to 1). Next, we repeat the process to find a second region that will join the first into the newly created state that will also optimize the happiness. Then we try with a 3rd region. The process goes on until there is no improvement possible for the happiness measure.
This is the pseudo code of the algorithm.
S = [0,0,...,0] h0 = happiness(V,S) maxi = 0 while maxi != -1 maxi = -1 maxhappiness = happiness(V,S) for i = 0 to numregions S[i] = 1 h1 = happiness(V,S) if h1 > maxhappiness maxi = i S[i] = 0
We can rerun the algorithm to generate another state. In this case we find that the previously created state has been divided into two, each one containing the Autonomous Communities of Catalonia and Basque Country [Figure 2]. Now the happiness is 8,412,420 which is more than 11% increase over the single state.
If we create yet another state. We will separate the regions of Sevilla, Asturias, Jaén, Córdoba, Huelva, Granada, Badajoz, Cádiz, Zaragoza, Cáceres and Huesca [Figure 3]. This new state would generate a happiness of 8,483,369 wich is an increase of 12% wich is not very different from the previous result. Besides, this new state is disconnected with 3 components.
We have proposed a simple model to calculate happiness in a country based on eletoral results. Next, we designed a strategy to increase this happiness by splitting a heterogeneous country into several homogeneous smaller countries. We apply our approach using 2011 Spanish electoral data, and we find that creating two new connected states increases the happiness of the population by more than 11%.
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