Regret theory is used for comparisons between lotteries. Define S s.t: * x"=>"y <=> S(x,y)=>0 * S linear in x,y * S(x,y)=-S(y,x) * S(x,y)=Sum(i=1,n,j=1,n,S(xi,xj)) Note that it is possible to violate transitivity: S(x,y)=>0 /\ S(y,z)=>0 /\ S(x,z)<=0