# Are these two definitions of regret in RM algorithm equivalent?

(2.1c) on p.1130 of the original paper on regret matching states that $$D_t^i(j,k) = \frac{1}{t} \sum_{\tau=1}^t \big[ u^i(k,s_\tau^{-j}) - u^i(s_\tau) \big] \;.$$

At the end of page 5 of this tutorial from CMU seems to mean that $$D_t^i(j,k) = \frac{1}{t} \sum_{\tau=1}^t \big[ u^i(k,s_\tau^{-j}) - \sum_{n=1}^N x_\tau^n u^i(n,s_\tau^{-n}) \big] \;,$$ where $$X_\tau = (x_\tau^1,\cdots,x_\tau^N)$$ is the mixed strategy, i.e., a PMF at time $$\tau$$. The notation is modified to fit the original paper.

Or, in the notation of the tutorial, the paper has $$r^t = \sum_{\tau=1}^t \big( l^\tau - u(s_\tau) \vec{1} \big)\;,$$ where $$u(s_\tau)$$ is the reward received at time $$\tau$$, while the tutorial has $$r^t = \sum_{\tau=1}^t \big( l^\tau - \vec{1} \big) \;.$$

Where did I get wrong?

New contributor
Chp is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.