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