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Roadmap to the appendix
Appendix A includes the proofs omitted from Section 3 (Offline Setting). Appendix B includes the proofs omitted from Section 4 (Online Setting). Appendix C includes the equilibrium analysis. A few results from prior work that we invoke are in Appendix D.
A Appendix: Offline Setting
In this section we include the proofs omitted from the main text for the offline setting.
Observation 1 (restated). Player  $i$  has an optimum bid vector  $\beta = (\beta_{1},\dots,\beta_{K})\in \mathbb{D}^{K}$  with  $\beta_{j}\in S_{i}$  for all  $j\in [K]$
Proof. Let  $\mathbf{c} = (c_1, \ldots, c_K)$  be an arbitrary optimum strategy for player  $i$ . Suppose  $\mathbf{c} \notin S_i^K$ .
We use  $\mathbf{c}$  to construct an optimum bid vector  $\beta \in S_i^K$  as follows. For each  $j\in [K]$ :
- If  $c_{j} \in S_{i}$ , then set  $\beta_{j} = c_{j}$ .
- Else, set  $\beta_{j} = \max \{y\in S_{i}\mid y\leq c_{j}\}$
Then in each round  $t$ , player  $i$  gets the same allocation when playing  $\beta$  as it does when playing  $\mathbf{c}$  and the others play  $\mathbf{b}_{-i}^{t}$  since the ordering of the owners of the bids is the same under  $(\mathbf{c}, \mathbf{b}_{-i}^{t})$  as it is under  $(\beta, \mathbf{b}_{-i}^{t})$ ; moreover, the price weakly decreases in each round  $t$ . Thus player  $i$ 's utility weakly improves. Since  $\mathbf{c}$  was an optimal strategy for player  $i$ , it follows that  $\beta$  is also an optimal strategy for player  $i$  and, moreover,  $\beta \in S_{i}^{K}$  as required.
Next we show how to find an optimal bid vector in polynomial time. The proof uses several lemmas, which are proved after the theorem.
Theorem 1 (restated, formal). Suppose we are given a number  $n$  of players, number  $K$  of units, valuation  $\mathbf{v}_i$  of player  $i$ , discretization level  $\varepsilon > 0$ , and bid history  $H_{-i} = (\mathbf{b}_{-i}^1, \dots, \mathbf{b}_{-i}^T)$  by players other than  $i$ . Then an optimum bid vector for player  $i$  can be computed in polynomial time in the input parameters.
Proof of Theorem 1. Compute the set  $S_{i}$  given by equation (1) and the graph  $G_{i}$  from Definition 1. The proof has several steps as follows.
$G_{i}$  is a DAG. All the edges in  $G_{i}$  flow from the source  $z_{-}$  to the nodes from layer 1, then from the nodes in layer  $j$  to those in layer  $j + 1$  (i.e. of the form  $(z_{r,j}, z_{s,j + 1})$  for all  $j \in [K - 1]$  and  $r, s \in S_{i}$  with  $r \geq s$ ), and finally from all the nodes in layer  $K$  to the sink  $z_{+}$ . A cycle would require at least one back edge, but such edges do not exist.
Bijective map between bid vectors and paths from source to sink in  $G_{i}$ . To each bid vector  $\beta = (\beta_{1},\ldots ,\beta_{K})\in S_{i}^{K}$ , associate the following path in  $G_{i}$ :  $P(\beta) = \left(z_{-},z_{\beta_{1},1},\dots ,z_{\beta_{K},K},z_{+}\right)$ .
We show next the map  $P$  is a bijection from the set  $S_{i}$  of candidate bid vectors for player  $i$  to the set of paths from source to sink in  $G_{i}$ . Consider arbitrary bid vector  $\beta = (\beta_{1},\dots,\beta_{K})\in S_{i}^{K}$ . By definition of a bid vector, we have  $\beta_{1}\geq \ldots \geq \beta_{K}$ . Then  $P(\beta) = (z_{-},z_{\beta_{1},1},\ldots ,z_{\beta_{K},K},z_{+})$  is a valid path in  $G_{i}$ .
Conversely, since  $G_{i}$  has an edge  $(z_{r,j}, z_{s,j+1})$  if and only if  $r \geq s$ , each path from source to sink in  $G_{i}$  has the form  $Q = (z_{-}, z_{\beta_{1},1}, \ldots, z_{\beta_{K},K}, z_{+})$  for some numbers  $\beta_{1}, \ldots, \beta_{K} \in S_{i}$ , and so it can be mapped to bid vector  $\beta = (\beta_{1}, \ldots, \beta_{K})$ . Thus  $Q = P(\beta)$ , and so  $P$  is a bijective map.
Utility of player  $i$  and weight of a path of length  $K$  in  $G_{i}$ . Let  $\beta = (\beta_{1},\dots,\beta_{K})\in S_{i}^{K}$ . Additionally, let  $\beta_{K + 1} = 0$ . The total utility of player  $i$  when bidding  $\beta$  in each round  $t$  while the others bid  $\mathbf{b}_{-i}^{t}$  is  $U_{i}(\boldsymbol {\beta}) = \sum_{t = 1}^{T}u_{i}(\mathbf{h}^{t})$ , where  $\mathbf{h}^t = (\boldsymbol {\beta},\mathbf{b}_{-i}^t)\forall t\in [T]$ .
Let  $P(\beta) = (z_{-}, z_{\beta_1,1}, \ldots, z_{\beta_K, K}, z_{+})$  be the path in  $G_i$  corresponding to  $\beta$ . The edges of the path  $P(\beta)$  are  $(z_{-}, z_{\beta_1,1}), (z_{\beta_1,1}, z_{\beta_2,2}), \ldots, (z_{\beta_{K-1}, K-1}, z_{\beta_K, K}), (z_{\beta_K, K}, z_{+})$ , while the weight of the path is equal to the sum of its edges. Summing equation (2), which gives the weight of an edge, across all edges of  $P(\beta)$  implies that the weight of path  $P(\beta)$  is equal to
$$
w (P (\boldsymbol {\beta})) = \sum_ {j = 1} ^ {K} \sum_ {t = 1} ^ {T} \left[ \mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) \geq j \}} \left(v _ {i, j} - \beta_ {j}\right) + j \left(\mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) > j \}} \left(\beta_ {j} - \beta_ {j + 1}\right) + \mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) = j \}} \left(\beta_ {j} - p \left(\mathbf {h} ^ {t}\right)\right)\right) \right]. \tag {7}
$$
We claim that  $U_{i}(\boldsymbol {\beta}) = w(P(\boldsymbol {\beta}))$ . The high level idea is to rewrite the utility to "spread it" across the edges of the path corresponding to bid profile  $\beta$ . Towards this end, recall the utility of player  $i$  at strategy profile  $\mathbf{h}^t = (\boldsymbol {\beta},\mathbf{b}_{-i}^t)$  is
$$
u _ {i} \left(\mathbf {h} ^ {t}\right) = \sum_ {j = 1} ^ {K} \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) \geq j \right\}} \left(v _ {i, j} - p \left(\mathbf {h} ^ {t}\right)\right). \tag {8}
$$
By Lemma 1, equation (8) is equivalent to
$$
u _ {i} \left(\mathbf {h} ^ {t}\right) = \sum_ {j = 1} ^ {K} \left[ \mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) \geq j \}} \left(v _ {i, j} - \beta_ {j}\right) + j \left(\mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) > j \}} \left(\beta_ {j} - \beta_ {j + 1}\right) + \mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) = j \}} \left(\beta_ {j} - p \left(\mathbf {h} ^ {t}\right)\right)\right) \right]. \tag {9}
$$
Summing  $u_{i}(\mathbf{h}^{t})$  over all rounds  $t$  gives
$$
\begin{array}{l} U _ {i} (\boldsymbol {\beta}) = \sum_ {t = 1} ^ {T} u _ {i} (\mathbf {h} ^ {t}) \\ = \sum_ {t = 1} ^ {T} \sum_ {j = 1} ^ {K} \left[ \mathbb {1} _ {\{x _ {i} (\mathbf {h} ^ {t}) \geq j \}} (v _ {i, j} - \beta_ {j}) + j \left(\mathbb {1} _ {\{x _ {i} (\mathbf {h} ^ {t}) > j \}} (\beta_ {j} - \beta_ {j + 1}) + \mathbb {1} _ {\{x _ {i} (\mathbf {h} ^ {t}) = j \}} (\beta_ {j} - p (\mathbf {h} ^ {t}))\right) \right] \tag {By equation (9)} \\ = \sum_ {j = 1} ^ {K} \sum_ {t = 1} ^ {T} \left[ \mathbb {1} _ {\{x _ {i} (\mathbf {h} ^ {t}) \geq j \}} (v _ {i, j} - \beta_ {j}) + j \left(\mathbb {1} _ {\{x _ {i} (\mathbf {h} ^ {t}) > j \}} (\beta_ {j} - \beta_ {j + 1}) + \mathbb {1} _ {\{x _ {i} (\mathbf {h} ^ {t}) = j \}} (\beta_ {j} - p (\mathbf {h} ^ {t}))\right) \right] \\ \end{array}
$$
(Swapping the order of summation)
$$
= w (P (\beta)). \quad \text {(B y e q u a t i o n (7))}
$$
Thus the weight of the path  $P(\beta)$  is equal to the utility of player  $i$  from bidding  $\beta$ . The implication is that to find an optimum bid vector, it suffices to compute a maximum weight path in  $G_{i}$ .
Computing a maximum weight path in  $G_{i}$ . The graph  $G_{i}$  is a DAG with a number of vertices of  $G_{i}$  that is polynomial in the input parameters. By Lemma 2, the edge weights of  $G_{i}$  can be computed in polynomial time.
A maximum weight path in a DAG can be computed in polynomial time by changing every weight to its negation to obtain a graph  $-G$ . Since  $G$  has no cycles, the graph  $-G$  has no negative cycles. Thus running a shortest path algorithm on  $-G$  will yield a longest (i.e. maximum weight) path on  $G$  in polynomial time, as required.
The unique bid vector corresponding to the maximum weight path found can then be recovered in time  $O(K)$  and it represents an optimum bid vector for player  $i$ .
Lemma 1. In the setting of Theorem 1, for each bid profile  $\beta \in S_i^K$  and round  $t \in [T]$ , define  $\mathbf{h}^t = (\beta, \mathbf{b}_{-i}^t)$ . Then we have
$$
u _ {i} \left(\mathbf {h} ^ {t}\right) = \sum_ {j = 1} ^ {K} \left[ \mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) \geq j \}} \left(v _ {i, j} - \beta_ {j}\right) + j \left(\mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) > j \}} \left(\beta_ {j} - \beta_ {j + 1}\right) + \mathbb {1} _ {\{x _ {i} \left(\mathbf {h} ^ {t}\right) = j \}} \left(\beta_ {j} - p \left(\mathbf {h} ^ {t}\right)\right)\right) \right]. \tag {10}
$$
Proof. Given bid profile  $\beta = (\beta_{1},\dots ,\beta_{K})$  , we also define  $\beta_{K + 1} = 0$
If  $x_{i}(\mathbf{h}^{t}) = 0$  then both sides of equation (10) are zero, so the statement holds. Thus it remains to prove equation (10) when  $x_{i}(\mathbf{h}^{t}) > 0$ . Let  $u_{i,j}(\mathbf{h}^t) = \mathbb{1}_{\{x_i(\mathbf{h}^t)\geq j\}}(v_{i,j} - p(\mathbf{h}^t))$  for each  $j\in [K]$ . The term  $u_{i,j}(\mathbf{h}^t)$  represents the amount of utility obtained from the  $j$ -th unit acquired by player  $i$  at price  $p(\mathbf{h}^t)$ , so  $u_{i}(\mathbf{h}^{t}) = \sum_{j = 1}^{K}u_{i,j}(\mathbf{h}^{t})$ .
We first show that for each  $j \in [K]$ :
$$
u _ {i, j} \left(\mathbf {h} ^ {t}\right) = \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) \geq j \right\}} \left(v _ {i, j} - \beta_ {j}\right) + \sum_ {k = j} ^ {K} \left[ \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) > k \right\}} \left(\beta_ {k} - \beta_ {k + 1}\right) + \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) = k \right\}} \left(\beta_ {k} - p \left(\mathbf {h} ^ {t}\right)\right) \right]. \tag {11}
$$
To prove (11), we will rewrite  $u_{i,j}(\mathbf{h}^t)$  by considering three cases and writing a unified expression for all of them.
1.  $x_{i}(\mathbf{h}^{t}) = j$ . Then