Online Addendum for “Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency” Feryal Erhun • Pınar Keskinocak • Sridhar Tayur Department of Management Science and Engineering, Stanford University, Stanford, CA 94305-4026, USA School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA ferhun@stanford.edu • pinar@isye.gatech.edu • stayur@andrew.cmu.edu A.1. Limited Capacity Case In this section, we consider the case where the supplier has a capacity of K units that he can sell throughout N -periods. Let Q N correspond to the total production quantity of an N -period uncapacitated game. For the N -period limited capacity case, when the capacity is “tight,” i.e., C S ≤ a 4b , or when the capacity is “abundant,” i.e., C S ≥ Q N , the results are straightforward and intuitive. In the first case, as the capacity is tight, the supplier does not change his price through the game, so the N -period game is equivalent to a single period game. When C S ≥ Q N , the problem is equivalent to an unlimited capacity game (Proposition 1). What happens in between these extremes is more interesting. Our main result is as follows. Proposition A.1 The SPNE for the N -period capacitated model for C S <Q N is as follows: Let N * ∈{1, 2, ··· ,N } be such that Q N * -1 <C S ≤ Q N * . For the first N − N * periods, the supplier and the buyer do not play the game. In the last N * -periods, they play the following N * -period game. For i =1, 2, ··· ,N * − 2: q N -N * +1 = C S − N  j =N -N * +2 q j , q N = a 2b − C S , q N -i =  2i 2i +1  q N -i+1 , w N -N * +1 = N * -1  i=1  2i +1 2i  w N , w N =2 ( a 2 − bC S ) , w N -i =  2i +1 2i  w N -i+1 . A-1