# WARP implies completeness, transitivity and thus rationalizability. What is wrong with the statement?

Let $$A$$ be a menu and $$R$$ be a complete and transitive binary relation. Define choice correspondence generated by $$R$$:

$$c_R(A)=\{x\in A|| xRy \ \forall y\in A\}.$$

Theorem (from Kreps 1988): for any correspondence $$c$$, the following conditions are equivalent:

1. $$c$$ satisfies WARP
2. There exists a transitive and complete $$R$$ such that $$c=c_R$$.

Given $$A$$ is finite and $$c=c_R$$, it is obvious that $$c$$ can be represent by a utility function. Then why do we need SARP/GARP to rationalize a choice correspondence with a utility function? What was wrong with my logics?

• Is Kreps (1988) the book titled Notes on the Theory of Choice? I don't think the book mentions WARP as such. Can you give a precise statement of the theorem in Kreps (1988)? – Herr K. May 14 '19 at 21:35

SARP basically posits conditions that explicitly rule out intransitive cycles in revealed preferences with a general $$n$$-good environment.