The issue in solving social problems is how to respect minority opinions , which are often ignored in general majority rules. To build consensus on pluralistic values and make social choices in consideration of minority opinions, we propose aggregation methods that give weighting to the minority's positionality on cardinal cumulative voting. Based on quadratic and linear voting, we formulated three weighted aggregation methods that differ in the ratio of votes to cumulative points and the weighting of the minority to all members, and calculated the frequency distributions of the aggregation results, assuming that the distributions of votes follow normal distributions . From these calculation results, we found that minority opinions are  likely to be reflected as weighting increases proportionally in two of the above three methods. This means that Sen and Gotoh's idea of considering the social position of unfortunate people on ordinal ranking , that welfare economicsconsiders under an axiomatic approach, was shown by weighting the minority's positionality on cardinal voting. In addition, we can know the contents such as the number and positionality of the minority from the analysis of the aggregation results. It will be useful for promoting mutual understanding between the majority and minority by  visualizing the contents of the proposed aggregation methods interactively in the consensus-building process. With the further development of information technology, the consensus building on cardinal choices based on big data will be necessary. We would like to use the proposed aggregation methods for making social choices for pluralistic values such as social, environmental, and economic.
Respecting minority opinions is vital in solving social problems . However, minority opinions are often ignored in general majority rules. To build consensus on pluralistic values and make social choices that consider minority opinions, we propose aggregation methods that give weighting to the minority's positionality on cardinal cumulative voting. Based on quadratic and linear voting, we formulated three weighted aggregation methods that differ in the ratio of votes to cumulative points and the weighting of the minority to all members, and assuming that the distributions of votes follow normal distributions , we calculated the frequency distributions of the aggregation results. We found that minority opinions are more likely to be reflected proportionately to the average of the distribution in two of the above three methods. This implies that Sen and Gotoh's idea of considering the social position of unfortunate people on ordinal ranking in the welfare economics, was illustrated by weighting the minority's positionality on cardinal voting. In addition, it is possible to visualize the number and positionality of the minority from the analysis of the aggregation results. These results will be useful to promote mutual understanding between the majority and minority by interactively visualizing the contents of the proposed aggregation methods in the consensus-building process. With the further development of information technology, the consensus building based on big data will be necessary. We recommend the use of our proposed aggregation methods to make social choices for pluralistic values such as social, environmental, and economic.
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