SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Uniform pricing represents a type of fair pricing, since when everyone gets charged the same price per unit, no bidder envies another bidder's allocation. Fairness is an important property that is at odds with maximizing revenue, since for the purpose of improving revenue the auctioneer is better off charging higher prices to the buyers that show more interest in the goods. However, there is evidence that buyers are unhappy with discriminatory prices (see, e.g., [AS08]), which led to a body of literature focused on designing fair auction mechanisms  $\left[\mathrm{GHK}^{+}05\right.$ , FGL14, FFLS12, CAEFF16, Sek17, AVGN17, BFMZ17, FMTV20,  $\mathrm{DGJ}^{+}22]$ . Collusion in auctions was also studied both experimentally and theoretically in various models (e.g., [GN92, vdFH93, GNR15]).
3 Offline Setting
In the offline setting, we are given a player  $i$  with valuation  $\mathbf{v}_i$  and a history  $H_{-i} = (\mathbf{b}_{-i}^1,\dots ,\mathbf{b}_{-i}^T)$  with the bids of other players in rounds 1 through  $T$ . We assume the bids are restricted to a discrete domain  $\mathbb{D} = \{k\cdot \varepsilon \mid k\in \mathbb{N}\}$ , for some  $\varepsilon >0$ . The goal is to find an optimum fixed bid vector in hindsight:  $\mathbf{b}_i^* = \arg \max_{\boldsymbol {\beta}\in \mathbb{D}^K}\sum_{t = 1}^{T}u_i(\boldsymbol {\beta},\mathbf{b}_{-i}^t)$ , where we recall that  $u_{i}(\boldsymbol {\beta},\mathbf{b}_{-i}^{t})$  is the utility of player  $i$  at the bid profile  $(\beta ,\mathbf{b}_{-i}^{t})$ . We will see the maximum in the definition of  $\mathbf{b}_i^*$  is well defined since there is an optimum bid vector  $\beta$  with entries at most  $\varepsilon$  above the maximum historical bid.
For any  $\varepsilon > 0$ , the optimal solution on the discrete domain  $\mathbb{D}$  also yields an  $(\varepsilon \cdot K)$ -optimal strategy on the continuous domain. The offline problem we study here has some resemblance to the problems studied in [RW16, DGPL19, DPS21]. There, the goal is to optimize reserve prices in VCG auctions while having access to a dataset of historical bids.<sup>4</sup>
To design a polynomial time algorithm for solving the offline problem, we define a set  $S_{i}$  of "candidate" bids for player  $i$ , which has polynomial size in the history, player  $i$ 's valuation, and  $1 / \varepsilon$ :
$$
\mathcal {S} _ {i} = \left\{0 \right\} \cup \left\{b _ {j, k} ^ {t} \mid j \in [ n ] \backslash \{i \}, k \in [ K ] \right\} \cup \left\{b _ {j, k} ^ {t} + \varepsilon \mid j \in [ n ] \backslash \{i \}, k \in [ K ] \right\}. \tag {1}
$$
Observation 1. 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  $j\in [K]$ , where  $S_{i}$  is defined in equation (1).
The proof of Observation 1 is simple and deferred to Appendix A. Next we construct a directed acyclic graph (DAG)  $G_{i}$  that will be used to compute an optimum bid vector for player  $i$ .
Definition 1 (The graph  $G_{i}$ ). Given valuation  $\mathbf{v}_{i}$  of a player  $i$ , history  $H_{-i} = (\mathbf{b}_{-i}^{1},\dots,\mathbf{b}_{-i}^{T})$ ,  $\varepsilon >0$ , and set  $S_{i}$  from (1), define a graph  $G_{i}$  with the following features.
[RW16] also presents an online version of their offline algorithms under full information feedback; see also [NGW+21] for an online version of [RW16]'s algorithm for both full information and bandit settings using Blackwell approachability [Bla56].
Vertices. Create a vertex  $z_{s,j}$  for each  $s \in S_i$  and  $j \in [K]$ . We say vertex  $z_{s,j}$  is in layer  $j$ . Add source  $z_-$  and sink  $z_+$ .
Edges. For each index  $j \in [K - 1]$  and pair of bids  $r, s \in S_i$  with  $r \geq s$ , create a directed edge from vertex  $z_{r,j}$  to vertex  $z_{s,j + 1}$ . Moreover, add edges from source  $z_{-}$  to each node in layer 1 and from each node in layer  $K$  to the sink  $z_{+}$ .
Edge weights. For each edge  $e = (z_{r,j}, z_{s,j+1})$  or  $e = (z_{r,K}, z_+)$ , let  $\beta = (\beta_1, \ldots, \beta_K) \in S_i^K$  be a bid vector with  $\beta_j = r$  and  $\beta_{j+1} = s$  (we define  $s = 0$  if  $j = K$ ). For each  $t \in [T]$ , let  $\mathbf{h}^t = (\boldsymbol{\beta}, \mathbf{b}_{-i}^t)$ . Define the weight of edge  $e$  as
$$
w _ {e} = \sum_ {t = 1} ^ {T} \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) \geq j \right\}} \left(v _ {i, j} - r\right) + j \left[ \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) > j \right\}} (r - s) + \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) = j \right\}} \left(r - p \left(\mathbf {h} ^ {t}\right)\right) \right], \tag {2}
$$
recalling that  $p(\mathbf{h}^t)$  and  $\mathbf{x}(\mathbf{h}^t)$  are the price and allocation at  $\mathbf{h}^t$ , respectively<sup>5</sup>. The edges incoming from  $z_{-}$  have weight zero.
A pictorial illustration of the graph  $G_{i}$  is displayed in Figure 1.
Figure 1: Example of DAG as in Definition 1. Here, we have  $K = 4$  units and the set of candidate bids for player  $i$  is  $S_{i} = \{0,1,2\}$ . The graph has a source  $z_{-}$ , a sink  $z_{+}$ , and nodes  $z_{r,j}$  for all  $r \in S_{i}$  and  $j \in [4]$ . Edges are of the form  $(z_{r,j},z_{s,j + 1})$  for all  $j < 4$  and  $r,s \in S_{i}$  with  $r \geq s$ . There are also edges from the source  $z_{-}$  to nodes  $z_{j,1}$  and from nodes  $z_{j,4}$  to the sink  $z_{+} \forall j \in S_{i}$ .
At a first glance, it may seem as though the weight  $w_{e}$  of an edge  $e$  depends on the whole bid vector  $\beta$  since we use the value of  $x_{i}(\mathbf{h}^{t})$  in its definition. However, this is not the case. Meaning that, this formula gives us the same value for any choice of  $\beta$  that satisfies  $\beta_{j} = r$  and  $\beta_{j + 1} = s$ . This is due to the fact that the values of variables  $\mathbb{1}_{\{x_i(\mathbf{h}^t)\geq \}}$ ,  $\mathbb{1}_{\{x_i(\mathbf{h}^t) > j\}}$ , and  $\mathbb{1}_{\{x_i(\mathbf{h}^t) = j\}}$  can be determined only knowing the values of  $\beta_{j + 1}$  and  $\beta_{j}$  (without knowing the rest of the vector  $\beta$ ).
We will show that computing an optimum strategy for the player is equivalent to finding a maximum-weight path in the DAG  $G_{i}$  from Definition 1.
Theorem 1 (formal version). 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  $H_{-i} = (\mathbf{b}_{-i}^1, \ldots, \mathbf{b}_{-i}^T)$  by players other than  $i$ . Then an optimum bid vector for player  $i$  can be computed in polynomial time in the input parameters.
The proof of Theorem 1 is in Appendix A, together with the other omitted proofs of this section.
At a high level, for any bid vector  $\beta$ , we first write the cumulative utility of player  $i$  when playing  $\beta$  while the others play according to  $\mathbf{b}_{-i}^{t}$ . Then observe the cumulative utility can be rewritten as a sum of  $K$  terms, so that each term only depends on  $j$ ,  $\beta_{j}$ , and  $\beta_{j+1}$ . Create a layered DAG as in Definition 1, where the weight of edge  $(\beta_{j}, j)$  to  $(\beta_{j+1}, j+1)$  is the  $j$ -th term of the sum above, with special handling for edges involving the source or the sink. The entire graph can be computed in polynomial time.
Then the proof shows there is a bijection between the set of bid vectors of the player and the set of paths from the source to the sink in the graph. Thus the solution to the offline problem is obtained by computing a maximum weight path in the graph and returning the corresponding bid vector.
The algorithm for the offline problem is summarized in the next figure.
That is,  $x_{k}(\mathbf{h}^{t})$  is the number of units received by player  $k$  at allocation  $\mathbf{x}(\mathbf{h}^t)$ .
Input: Player  $i$  with valuation  $\mathbf{v}_i = (v_{i,1},\dots,v_{i,K})$ , a history  $H_{-i} = (\mathbf{b}_{-i}^{1},\dots,\mathbf{b}_{-i}^{T})$  of bids submitted by players other than  $i$ , and a grid  $\mathbb{D} = \{k\cdot \varepsilon \mid k\in \mathbb{N}\}$  with allowed bid values.
Output:  $\mathbf{b}_i^* = \arg \max_{\boldsymbol{\beta} \in \mathbb{D}^K} \sum_{t=1}^{T} u_i(\boldsymbol{\beta}, \mathbf{b}_{-i}^t)$ .
- Compute set  $S_{i}$  as defined in equation (1). By Observation 1, player  $i$  has an optimum bid vector  $\mathbf{b}_i^*$  with  $b_{i,j}^{*}\in S_{i}$  for each  $j\in [K]$
- Construct directed acyclic graph  $G_{i}$  as in Definition 1. By Theorem 1, there is a bijective map between paths from the source to the sink in  $G_{i}$  and bid vectors  $\beta \in S_{i}$  of player  $i$ .
- Compute a maximum-weight path in graph  $G_{i}$  and output the corresponding bid vector.
algorithm 1: Optimum Strategy for the Offline Problem.
4 Online Setting
In the online setting, we consider the repeated auction where in each round  $t$  the auctioneer announces  $K$  units for sale. Then the players privately submit the bids  $\mathbf{b}^t$  and the auctioneer computes the outcome. At the end of round  $t$ , the auctioneer gives full information or bandit feedback. The information available to player  $i$  at the beginning of each round  $t$  is the history:
$$
H _ {i} ^ {t - 1} = \left(\mathbf {b} _ {i} ^ {1}, \dots , \mathbf {b} _ {i} ^ {t - 1}\right) \cup H _ {- i} ^ {t - 1}, \text {w h e r e} ^ {6}
$$
$$
H _ {- i} ^ {t - 1} = \left\{ \begin{array}{l l} \left(\mathbf {b} _ {- i} ^ {1}, \dots , \mathbf {b} _ {- i} ^ {t - 1}\right) & \text {i n t h e f u l l i n f o r m a t i o n s e t t i n g}, \\ \left(p (\mathbf {b} ^ {1}), \dots , p (\mathbf {b} ^ {t - 1})\right) & \text {i n t h e b a n d i t s e t t i n g}. \end{array} \right.
$$
A pure strategy for player  $i$  at time  $t$  is a function  $\pi_i^t: H_i^{t-1} \to \mathbb{R}^K$ , where  $H_i^{t-1}$  is the history available to player  $i$  at the beginning of round  $t$ . Thus the pure strategy tells player  $i$  what bid vector to submit next given the information observed by the player about the previous rounds. A mixed strategy is a probability distribution over the set of pure strategies. Consequently, the bid vector  $\mathbf{b}_i^t$  submitted by player  $i$  at time  $t$  is the realization of the mixed strategy  $\pi_i^t(H_i^{t-1})$ . We denote by  $\pi_i = (\pi_i^1, \dots, \pi_i^T)$  the overall strategy of player  $i$  over the entire time horizon.
Regret. From now on we fix an arbitrary player  $i$ . Given a bidding strategy  $\pi_i$  of player  $i$ , the regret of the player is defined with respect to a history  $H_{-i}^{T}$  of bids by other players:
$$
\begin{array}{l} \operatorname {R e g} _ {i} \left(\pi_ {i}, H _ {- i} ^ {T}\right) = \max  _ {\boldsymbol {\beta} \in \mathcal {S} _ {i} ^ {K}} \sum_ {t = 1} ^ {T} \sum_ {j = 1} ^ {K} \left(v _ {i, j} - p (\boldsymbol {\beta}, \mathbf {b} _ {- i} ^ {t})\right) \cdot \mathbb {1} _ {\left\{x _ {i} \left(\boldsymbol {\beta}, \mathbf {b} _ {- i} ^ {t}\right) \geq j \right\}} \\ - \sum_ {t = 1} ^ {T} \mathbb {E} _ {\mathbf {b} _ {i} ^ {t} \sim \pi_ {i} ^ {t} \left(H _ {i} ^ {t - 1}\right)} \left[ \sum_ {j = 1} ^ {K} \left(v _ {i, j} - p (\mathbf {b} ^ {t})\right) \cdot \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {b} ^ {t}\right) \geq j \right\}} \right]. \tag {3} \\ \end{array}
$$
For the purpose of equipping player  $i$  with a bidding algorithm, we will think of the other players as adversarial, and aim to achieve small regret regardless of  $H_{-i}^{T}$ . A way to interpret the regret is that player  $i$  is competing against an oracle that (a) has the perfect knowledge of the history of bids, but (b) is restricted to submit a time-invariant bid  $\beta$ . The regret quantifies how much worse player  $i$  does compared to the oracle.
Before studying the feedback models in greater detail, we replace the set  $S_{i}$  of candidate bids for player  $i$  from equation (1) of the offline section by a coarser set  $S_{\varepsilon} = \{\varepsilon, 2\varepsilon, \dots, \lceil v_{i,1} / \varepsilon \rceil \varepsilon \}$ .
In the full information scenario, at the end of each round  $t$  the auctioneer announces the bids  $\mathbf{b}^t$  submitted by the players in round  $t$ . We define a graph  $G^t$  that forms the basis of the learning
<sup>6</sup>We omit the allocation information in the history, for it is not informative when other players are adversarial: from the price information, player  $i$  knows her own allocation; treating all other players as a giant adversarial player, that player gets the rest of the allocation.
algorithm for the player. The graph  $G^{t}$  is the same as in the offline setting (Definition 1), except for the next differences:
- The vertex  $z_{s,j}$  is indexed by  $s \in S_{\varepsilon}$ , instead of  $S_{i}$ ;
- The edge weights are based only on the current round  $t$ , rather than the sum over all rounds in the offline setting. More specifically, for edge  $e = (z_{r,j}, z_{s,j+1})$  or  $e = (z_{r,K}, z_{+})$ , set
$$
w ^ {t} (e) = \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) \geq j \right\}} \left(v _ {i, j} - r\right) + j \left[ \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) > j \right\}} (r - s) + \mathbb {1} _ {\left\{x _ {i} \left(\mathbf {h} ^ {t}\right) = j \right\}} \left(r - p \left(\mathbf {h} ^ {t}\right)\right) \right]. \tag {4}
$$
The formal definition is deferred to Definition 2 in Appendix B. This DAG structure enables us to formulate the learning problem as an online maximum weight path problem, and therefore treat each path as an expert and apply online learning algorithms to compete with the best expert. Theorem 2 shows such an algorithm (Algorithm 2) works for both  $K$ -th and  $(K + 1)$ -st price auctions.
Input: time horizon  $T$ , valuation  $\mathbf{v}_i = (v_{i,1},\dots ,v_{i,K})$  of the player  $i$
Output: bid vectors  $(\mathbf{b}_i^1,\dots ,\mathbf{b}_i^T)$
Set the resolution parameter  $\varepsilon = v_{i,1}\sqrt{K / T}$ , and learning rate  $\eta = \sqrt{\log T} / (v_{i,1}\sqrt{KT})$ . For every  $t = 1,\dots,T$ :
- Construct a graph  $G^{t}$  as in Definition 2, initially without weights on the edges.
- If  $t = 1$ : for each edge  $e$ , initialize an edge probability as
$$
\phi^ {1} (e) = \left\{ \begin{array}{l l} 1 / | \mathcal {S} _ {\varepsilon} | & \text {i f} e = (z _ {-}, z _ {r, 1}); \\ 1 / | \{r ^ {\prime} \in \mathcal {S} _ {\varepsilon} \mid r ^ {\prime} \leq r \} | & \text {i f} e = (z _ {r, j}, z _ {s, j + 1}), r \geq s; \\ 1 & \text {i f} e = (z _ {r, K}, z _ {+}). \end{array} \right.
$$
If  $t\geq 2$
- starting from  $\Gamma^{t - 1}(z_{+}) = 1$ , recursively compute in the bottom-up order
$$
\Gamma^ {t - 1} (u) = \sum_ {v: e = (u, v) \in E} \phi^ {t - 1} (e) \cdot \exp (\eta \cdot w ^ {t - 1} (e)) \cdot \Gamma^ {t - 1} (v), \quad \forall u \in V (G ^ {t});
$$
- for each edge  $e$ , update the edge probability
$$
\phi^ {t} (e) = \phi^ {t - 1} (e) \cdot \exp (\eta \cdot w ^ {t - 1} (e)) \cdot \frac {\Gamma^ {t - 1} (v)}{\Gamma^ {t - 1} (u)}, \quad \text {f o r a l l} e = (u, v) \in E (G ^ {t}). \tag {5}
$$
- For  $j = 1,2,\dots ,K$  : sample  $b_{i,j}^{t}\sim \phi^{t}(z_{b_{i,j - 1,j - 1}^{t}},\cdot)$ , where  $z_{b_{i,0}^t,0} := z_-$ .
- The player observes  $\mathbf{b}_{-i}^{t}$ , and sets the edge weights  $w^{t}(e)$  in  $G^{t}$  according to (4).
algorithm 2: Selecting bids using the weight pushing algorithm in [TW03].
In Algorithm 2, inspired by the path kernel algorithm in [TW03], we define  $\phi^t(u, \cdot)$  as a probability distribution over the outneighbors of vertex  $u$ . To determine the bid vector in each round, we use a random walk starting from the source  $z_{-}$ , where the learner selects a random neighbor  $z_{1}$  of  $z_{-}$  using  $\phi^t(z_{-}, \cdot)$ , then selects a random neighbor  $z_{2}$  of  $z_{1}$  using  $\phi^t(z_{1}, \cdot)$ , and so on, until reaching the sink  $z_{+}$ . Remarkably, as we prove in Theorem 2, the distribution of the resulting path  $\mathfrak{p}^t$  is the same as that in the Multiplicative Weight Update (Hedge) algorithm [LW94].
Thus while the number of paths in the DAG (corresponding to experts in the Hedge algorithm) is exponentially large, we can run the Hedge algorithm with polynomial space and time complexity by leveraging the DAG structure. We do not need to keep track of the weights of every path, but only need to store and update the weights of every edge in the DAG using the variables  $\phi^t(e)$  and the recursive formula in equation (5). To accomplish this, we use the variables  $\Gamma^t(u)$ , for every vertex  $u$  in the DAG, which can be viewed as the weights of paths starting at vertex  $u$ .
We obtain the next theorem, the proof of which can be found in Appendix B.
Theorem 2. For each player  $i$  and time horizon  $T$ , under full-information feedback, Algorithm 2 runs in time  $O(T^2)$  and guarantees the player's regret is at most  $O(v_{i,1}\sqrt{TK^3\log T})$ .
4.2 Bandit feedback
The next theorem presents an upper bound on the regret under the bandit feedback. Similar to the full-information case, the algorithm we design works for both the  $K$ -th and  $(K + 1)$ -st price auctions.
Theorem 3. For each player  $i$  and time horizon  $T$ , there is an algorithm for bidding in the repeated auction with bandit feedback that runs in  $O(KT + K^{-5/4}T^{7/4})$  and guarantees the player's regret is bounded by  $O(v_{i,1} \cdot \min\{\sqrt[4]{T^3K^7\log T}, KT\})$ .
We describe the algorithm below, while the proof of the theorem can be found in Appendix B.
The bidding strategy and its efficient implementation are the same as those in the proof of Theorem 2, with the only difference as follows. Under bandit feedback, we are not able to compute  $w^t(e)$  for every edge  $e$  at the end of time  $t$ ; instead, we are able to compute  $w^t(e)$  for every edge on the chosen path  $\mathfrak{p}^t$  at time  $t$ . This observation motivates us to use the next unbiased estimator  $\widehat{w}^t(e)$ :
$$
\widehat {w} ^ {t} (e) = \bar {w} (e) - \frac {\bar {w} (e) - w ^ {t} (e)}{p ^ {t} (e)} \mathbb {1} _ {\{e \in \mathfrak {p} ^ {t} \}}, \quad \text {w i t h} p ^ {t} (e) = \sum_ {\mathfrak {p}: e \in \mathfrak {p}} P ^ {t} (\mathfrak {p}), \tag {6}
$$
where  $P^t$  is the player's action distribution over paths, and for each edge  $e$ :
$$
\overline {{w}} (e) = \left\{ \begin{array}{l l} v _ {i, 1} - r + j (r - s) & \text {i f} e = (z _ {r, j}, z _ {s, j + 1}); \\ v _ {i, 1} - r + K r & \text {i f} e = (z _ {r, K}, z _ {+}); \\ 0 & \text {i f} e = (z _ {-}, z _ {r, 1}). \end{array} \right.
$$
The resulting algorithm is exactly the same as Algorithm 2, with the only difference being that all  $w^t$  are replaced by  $\widehat{w}^t$ , with an additional step (6) for the computation of  $\widehat{w}^t$ .
The regret analysis will appear similar to that in [GLLO07], but a key difference is that we no longer have  $w^{t}(e) \leq v_{i,1}$  for every edge  $e$  in our setting. Instead, we apply an edge-dependent upper bound  $w^{t}(e) \leq \overline{w}(e)$ , and use the property that  $\sum_{e \in \mathfrak{p}} \overline{w}(e) \leq K v_{i,1}$  for every path  $\mathfrak{p}$  from source to sink.
4.3 Regret lower bounds
The following theorem establishes an  $\Omega(K\sqrt{T})$  lower bound on the minimax regret of a single bidder in the full information scenario. Clearly the same lower bound also holds for bandit feedback.
Theorem 4 (restated, formal). Let  $K \geq 2$ . For any policy  $\pi_i$  used by player  $i$ , there exists a bid sequence  $\{\mathbf{b}_{-i}^t\}_{t=1}^T$  for the other players such that the expected regret in (3) satisfies  $\mathbb{E}[\mathrm{Reg}_i(\pi_i, \{\mathbf{b}_{-i}^t\}_{t=1})] \geq cv_{i,1}K\sqrt{T}$ , where  $c > 0$  is an absolute constant.
The proof of Theorem 4 is in Appendix B. The idea is to employ the celebrated Le Cam's two-point method, but the construction of the hard instance is rather delicate to reflect the regret dependence on  $K$ . We also remark that this regret lower bound still exhibits a gap compared with the current upper bound in Theorem 2, while it seems hard to extend the current two-point construction to Fano-type lower bound arguments. It is an outstanding question to close this gap.
Several open questions arise from this work. For example, is the regret  $\Theta(K\sqrt{T})$  in both the full information and bandit setting? In the full information setting, the challenge in obtaining an algorithm with regret  $O(K\sqrt{T})$  is that the expert distribution is not a product distribution over the layers, but rather only close to a product. Furthermore, how does offline and online learning take place in repeated auctions when the units are not identical, the valuations do not necessarily exhibit diminishing returns, or the pricing rule is different (e.g. in the first price setting)?
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