---
title: "Farkas' Lemma"
source: https://www.jemoka.com/posts/kbhfarkas_lemma/
---

\begin{equation} Ax \leq 0, c^{T}x &lt; 0 \end{equation}
and
\begin{equation} A^{T}y + c = 0, y \geq 0 \end{equation}
are strong alteratives. This is through duality.
investment arbitrage invest \(x_{j}\) in each of \(n\) assets \(1 &hellip; n\) with prices \(p_1 &hellip; p_{n}\) suppose \(V_{ij}\) is the payoff of asset \(j\) and outcome \(i\) risk-free (cash): \(p_{1} = 1\), \(V_{i1} = 1\) for all \(i\) as our first investment Arbitrage means there exists some \(x\) with \(p^{T}x &lt; 0, V x \succeq 0\) (first thing is you borrow money?, and then second thing is you get more money back).
By Farkas&rsquo; Lemma, there being no arbicharnge implies there exists some \(y\) such that \(V^{T}y = p\). Recall our first column is \(1\) (from risk-free cash) making \(1^{T}y = 1\).
