I just started working out of Game Theory by Fudenberg and I'm having trouble understanding the formula for calculating ''player i's payoff to profile $\sigma$, the formula is $u_i(\sigma)\equiv \sum_{s\in S }(\prod_{j=1}^{I}\sigma _{j}({s_j}))u_i(S)$
For example:
This image is the game matrix for a example in the book
where the payoff for player 1 to profiles $\sigma_1=(1/3,1/3,1/3)$ and $\sigma_2=(0,1/2,1/2)$ are $u_1(\sigma_1,\sigma_2)=(1/3)*(0*4+1/2*5+1/2*6) +(1/3)*(0*2+1/2*8+1/2*3) + (1/3)*(0*2+1/2*9+1/2*2)$
Im genuinely confused how the formula above relates to this calculation.
Here is a link to the textbook the material i'm concerned with is on pages 4 and 5 of chapter 1.1
It seems the explicit calculation has parenthesis for clarity but it's achieved the opposite.
In the example, for the strategy profile $s=(U,L)$, we have that $\sigma_1(s_1)=\frac{1}{3},\sigma_2(s_2)=0$, and $u_1((U,L))=4$. So then $\sigma_1(s_1)\sigma_2(s_2)u_1((U,L))$ should equal $\frac{1}{3}(0)(4)$.