Total order

Summary

Weak preferences as a total order

Weak preferences $≽$ is a total order on $R^2$ if for all bundles

• $\left( x_1,x_2 \right)$ $≽$ $\left( y_1,y_2 \right)$ or $\left( y_1,y_2 \right)$ $≽$ $\left( x_1,x_2 \right)$ ( totality )
• $\left( x_1,x_2 \right)$ $≽$ $\left( x_1,x_2 \right)$ ( reflexivity )
• if $\left( x_1,x_2 \right)$ $≽$ $\left( y_1,y_2 \right)$ and $\left( y_1,y_2 \right)$ $≽$ $\left( x_1,x_2 \right)$ , then $\left( x_1,x_2 \right)$ $∼$ $\left( y_1,y_2 \right)$ . ( antisymmetry )
• if $\left( x_1,x_2 \right)$ $≽$ $\left( y_1,y_2 \right)$ and $\left( y_1,y_2 \right)$ $≽$ $\left( z_1,z_2 \right)$ , then $\left( x_1,x_2 \right)$ $≽$ $\left( z_1,z_2 \right)$ ( transitivity )