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 )