SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Social regret: The social regret scales as  $O((MN - (K^{*})^{2})\log (T) / \Delta)$ . When participant buyers are replaced by non-participant buyers the valuation decreases which leads to  $O(K^{*}(N - K^{*})\log (T) / \Delta)$  regret. Similarly, participant seller getting replaced by non-participant seller introduces  $O(K^{*}(M - K^{*})\log (T) / \Delta)$  social regret. Finally, as goods move from non-participant buyers to non-participant sellers we obtain  $O((M - K^{*})(N - K^{*})\log (T) / \Delta)$  regret.
Non-participant regret: The individual regret scales as  $O(\sqrt{M}\log (T) / \Delta)$  for a non-participant buyer, whereas it scales as  $O(\sqrt{N}\log (T) / \Delta)$  for a non-participant seller. For any non-participant buyer once it has  $O(\log (T) / \Delta^2)$  samples it no longer falls in top  $K^{*}$  buyer. However, it can keep participating until each non-participant seller collects enough samples, i.e.  $O(\log (T) / \Delta^2)$  samples. Finally, regret is shown to be  $O(\sqrt{\#\text{participation}\log(T)})$  which leads to the  $O(\sqrt{M}\log (T) / \Delta)$ .
Participants: The leading  $O(\sqrt{T\log(T)})$  term in the regret for each optimal participating buyer and seller do not scale with the size of the system. This leading term depends mainly on the random fluctuation of the bid of the lowest bidding participating buyer and the highest bidding participating seller. The  $O\left(\frac{MN}{\Delta}\log (T)\right)$  regret for the participating buyer and seller comes because each time a non-participating buyer or seller ends up participating the price deviates.
Special Case  $K^{*} \approx N \approx M$ : We see that when the system the number of participants is very high, i.e.  $(N - K^{*}), (M - K^{*}) = O(1)$ , then the  $O(\log(T)/\Delta)$  component, hence the regret, per buyer/seller does not scale with the system size. Furthermore, the social regret also scales as  $O((MN - (K^{*})^{2})\log(T)/\Delta) \approx O(\log(T)/\Delta)$ . This indicates that there is rapid learning, as all the participants are discovering her own price almost every round.
4.1 Proof Outline of Regret Upper Bounds
We now present an outline to the proof of Theorem 2. The full proof is given in Appendix A. A salient challenge in our proof comes from two-sided uncertainty. In Double auction, the outcomes of the buyers' and sellers' sides are inherently coupled. Hence error in buyers' side propagates to the sellers' side, and vice versa, under two-sided uncertainty. For example, if the buyers are bidding lower than their true valuation then the optimal non-participating sellers, even with perfect knowledge of their own valuation, ends up participating. We show that optimal participants (both buyers and sellers) get decoupled, whereas in optimal non-participants regret leads to new information in both buyer and seller side. This breaks the two-sided uncertainty obstacle.
Monotonicity and Information Flow: Our proof leverages monotonicity of the average mechanism, which means if the bid of each buyer is equal or higher, and simultaneously bid of each seller is equal or lower, then the number of participants increases (Proposition 7). Additionally, UCB ensures each buyer w.h.p. bids higher than its true value, and LCB ensures each seller w.h.p. bids lower than its true value. Therefore, we have domination of the information flow:  $K(t) \geq K^{*}$  w.h.p. for all  $t \geq 1$ .
Social Regret Decomposition: For any sample path with  $K(t) \geq K^{*}$  the social regret, namely  $r_{SW}(T)$ , under average mechanism can be bounded as
$$
\begin{array}{l} r _ {S W} (T) \leq \sum_ {i \leq K ^ {*} <   i ^ {\prime}} \left(B _ {i} - B _ {i ^ {\prime}}\right) \sum_ {t = 1} ^ {T} \mathbb {1} \left(i \notin \mathcal {P} _ {b} (t), i ^ {\prime} \in \mathcal {P} _ {b} (t)\right) \\ + \sum_ {j \leq K ^ {*} <   j ^ {\prime}} \left(S _ {j ^ {\prime}} - S _ {j}\right) \sum_ {t = 1} ^ {T} \mathbb {1} \left(j \notin \mathcal {P} _ {s} (t), j ^ {\prime} \in \mathcal {P} _ {s} (t)\right) + \sum_ {i ^ {\prime}, j ^ {\prime} > K ^ {*}} \left(S _ {j ^ {\prime}} - B _ {i ^ {\prime}}\right) \sum_ {t = 1} ^ {T} \mathbb {1} \left(j ^ {\prime} \in \mathcal {P} _ {s} (t), i ^ {\prime} \in \mathcal {P} _ {b} (t)\right). \\ \end{array}
$$
The first term corresponds to a non-participant buyer replacing a participant buyer. The second term is the same for sellers, whereas the final term corresponds to two non-participant buyers and seller getting matched.
Individual Regret Decomposition: We now turn to the individual regrets. Regret of a non-participant buyer  $i$ , can be bounded by
$$
\sum_ {t: i \in \mathcal {P} _ {b} (t)} (p (t) - B _ {i}) \leq \sum_ {t: i \in \mathcal {P} _ {b} (t)} (b _ {i} (t) - B _ {i}) \lesssim \sum_ {t: i \in \mathcal {P} _ {b} (t)} \sqrt {\frac {\alpha_ {b , i} \log (t)}{n _ {b , i} (t)}} \lesssim \sqrt {\alpha_ {b , i} n _ {b , i} (T) \log (T)}.
$$
We have  $p(t)$  lesser than  $b_{i}(t)$  because  $i$ -th buyer participates in round  $t$ . By UCB property w.h.p.  $b_{i}(t)$  is at most  $\sqrt{\frac{\alpha_{b,i}\log(t)}{n_{b,i}(t)}}$  away from  $B_{i}$ . A similar argument shows for a non-participating seller  $j$  the regret is roughly  $\sqrt{\alpha_{s,j}n_{s,j}(T)\log(T)}$ .
For a participating buyer  $i$ , we have the regret bounded as
$$
\sum_ {t: i \notin \mathcal {P} _ {b} (t)} \left(B _ {i} - p ^ {*}\right) + \sum_ {t: i \in \mathcal {P} _ {b} (t)} \left(p (t) - p ^ {*}\right) = \left(T - n _ {b, i} (T)\right) \left(B _ {i} - p ^ {*}\right) + \sum_ {t: i \in \mathcal {P} _ {b} (t)} \left(p (t) - p ^ {*}\right).
$$
Similarly, a participating seller  $j$  has regret bound  $(T - n_{s,j}(T))(p^{*} - S_{j}) + \sum_{t:j\in \mathcal{P}_{s}(t)}(p^{*} - p(t))$ .
Decoupling Participants: In Lemma 10 in Appendix A, we show that learning is decoupled for the optimal participating buyers, and optimal participating sellers. It lower bounds the number of participation for optimal participating buyers and sellers.
Lemma 3 (Informal statement of Lemma 10). For  $\alpha_{\mathrm{min}} > 4$ , w.h.p., for every  $i, j \in [K^*]$ , and  $i', j' > K^*$
$$
\begin{array}{l} \left(T - n _ {b, i} (T)\right) \lesssim \sum_ {i ^ {\prime \prime} \geq K ^ {*} + 1} \frac {\alpha_ {b , i ^ {\prime \prime}} \log (T)}{\left(B _ {i} - B _ {i ^ {\prime \prime}}\right) ^ {2}}, \left(T - n _ {s, j} (T)\right) \lesssim \sum_ {j ^ {\prime \prime} \geq K ^ {*} + 1} \frac {\alpha_ {s , j ^ {\prime \prime}} \log (T)}{\left(S _ {j ^ {\prime \prime}} - S _ {j}\right) ^ {2}}, \\ \sum_ {t = 1} ^ {T} \mathbb {1} (i \notin \mathcal {P} _ {b} (t), i ^ {\prime} \in \mathcal {P} _ {b} (t)) \lesssim \frac {\alpha_ {b , i ^ {\prime}} \log (T)}{(B _ {i} - B _ {i ^ {\prime}}) ^ {2}}, a n d \sum_ {t = 1} ^ {T} \mathbb {1} (j \notin \mathcal {P} _ {s} (t), j ^ {\prime} \in \mathcal {P} _ {s} (t)) \lesssim \frac {\alpha_ {b , j ^ {\prime}} \log (T)}{(S _ {j ^ {\prime}} - S _ {j}) ^ {2}}. \\ \end{array}
$$
It argues after a optimal non-participant  $i''$  gets  $O\left(\frac{\log(T)}{(B_i - B_{i''})^2}\right)$  samples it can not participate while  $i$  does not participate as the bid of  $i$  is higher than the bid of  $i''$  with high probability. Similarly seller side result follow.
Non-Participant regret leads to two-sided Learning: Unlike optimal participating agents, the effect of uncertainties on the optimal non-participating buyers and sellers cannot be decoupled directly. With at least one non-participating buyer with large estimation error in her valuation present, the non-participating sellers can keep participating even with perfect knowledge. However, this does not ensure directly that the estimation error of this non-participating buyer decreases. Next, in Lemma 12 in Appendix A, we upper bound the number of times a non-participant can participate. Informally we have
Lemma 4 (Informal). For  $\alpha_{\mathrm{min}} > 4$  with high probability for any  $i, j \geq (K^{*} + 1)$ ,
$$
n _ {b, i} (T) \lesssim \frac {\alpha_ {b , i} \log (T)}{(B _ {K ^ {*}} - B _ {i}) ^ {2}} + \sum_ {j ^ {\prime} \geq (K ^ {*} + 1)} \frac {\alpha_ {s , j ^ {\prime}} \log (T)}{(S _ {j ^ {\prime}} - B _ {i}) ^ {2}}, n _ {s, j} (T) \lesssim \frac {\alpha_ {s , j} \log (T)}{(S _ {j} - S _ {K ^ {*}}) ^ {2}} + \sum_ {i ^ {\prime} \geq (K ^ {*} + 1)} \frac {\alpha_ {b , i ^ {\prime}} \log (T)}{(S _ {j} - B _ {i ^ {\prime}}) ^ {2}},
$$
and  $\sum_{t=1}^{T} \mathbb{1}(j' \in \mathcal{P}_s(t), i' \in \mathcal{P}_b(t)) \lesssim \frac{\alpha_{\max} \log(T)}{(S_{j'} - B_{i'})^2}$ .
A non-participant buyer  $i$  after obtaining  $O\left(\frac{\log(T)}{(B_{K^*} - B_i)^2}\right)$  samples does not belong to top  $K^*$  w.h.p. Hence, she participates with non-negligible probability only if a seller  $j' \geq (K^* + 1)$  participates. However, this implies that with each new match of buyer  $i$ , additionally at least one non-participating seller is matched, decreasing both their uncertainties. For this buyer  $i$ , we argue such spurious participation happens a total of  $\sum_{j' \geq (K^* + 1)} O\left(\frac{\log(T)}{(S_{j'} - B_i)^2}\right)$  times. After that all the non-participating sellers  $j \geq (K^* + 1)$  will have enough samples so that their LCB bids will separate from the  $i$ -th buyer's UCB bid. Reversing sellers' and buyers' roles does the rest.
Bounding Price Difference: The final part of the proof establishes bound on the cumulative difference of price from the true price  $p^*$  (see, Lemma 15 in Appendix A).
Lemma 5. (Informal statement of Lemma 15) For  $\alpha_{\mathrm{min}} > 4$ , w.h.p.  $\sum_{t=1}^{T}|p(t) - p^*| \lesssim C\log(T) + \sqrt{\alpha_{\max}T\log(T)}$ , where  $C = O\left(\frac{(M - K^*)(N - K^*)}{\Delta}\right)$ .
Let us focus on the first upper bound, i.e. of the cumulative value of  $(p(t) - p^{*})$ . The proof breaks down the difference into two terms,  $2(p(t) - p^{*}) = (\min_{i\in \mathcal{P}_{b}(t)}b_{i}(t) - B_{K^{*}}) + (S_{K^{*}} - \max_{j\in \mathcal{P}_{s}(t)}s_{j}(t))$ . For rounds when the buyer  $K^{*}$  is present (which happens all but  $O\left(\frac{(N - K^{*})\log(T)}{(B_{i} - p^{*})^{2}}\right)$  rounds) we can replace  $\min_{i\in \mathcal{P}_b(t)}b_i(t)$  with  $b_{K^*}(t)$ . Finally noticing that  $\sum_{t}(b_{K^{*}}(t) - B_{K^{*}})\lesssim \sqrt{\alpha_{max}}\sqrt{T\log(T)}$  takes care of the first term. For the second term, we need to study the process  $\max_{j\in \mathcal{P}_s(t)}s_j(t)$ . First we bound the number of times sellers 1 to  $(K^{*} - 1)$  crosses the seller  $K^{*}$ . Next we eliminate all the rounds where at least one seller  $j\geq (K^{*} + 1)$  are participating. Such an elimination comes at a cost of  $O\left(\frac{(M - K^{*})}{\Delta}\right)$ . For any seller  $j\geq (K^{*} + 1)$  this happens  $O\left(\frac{(N - K^{*})\log(T)}{\Delta_j^2}\right)$  times for some appropriate  $\Delta_j$ , and gives  $\Delta_j$  regret in each round. This final step gives us the dominating  $O\left(\frac{(M - K^{*})(N - K^{*})}{\Delta}\right)$  term. The bound for the cumulative value of  $(p^{*} - p(t))$  follows analogously.
5 Lower Bounds
5.1  $\Omega (\sqrt{T})$  minimax lower bound on individual regret
We show a minimax regret lower bound of  $\Omega(\sqrt{T})$  in Lemma 21 by considering a simpler system that decouples learning and competition. In this system, the seller is assumed to  $(i)$  know her exact valuation, and  $(ii)$  always ask her true valuation as the selling price, i.e., is truthful in her asking price in all the rounds. Furthermore, the pricing at every round is fixed to the average  $p_t = \frac{B_t + S_t}{2}$  in the event that  $B_t \geq S$ . We show in Corollary 19 through a coupling argument that any algorithm for the classical two armed bandit problem can be converted to an algorithm for this special case. Then we use well known lower bounds for the bandit problem to give the  $\Omega(\sqrt{T})$  lower bound in Lemma 21. All technical details are in Appendix B.
5.2  $\Omega (\log (T))$  instance dependent lower bound on Social Welfare Regret
The key observation is that social-welfare regret in Equation (1) is independent of the pricing mechanism and only depends on the participating buyers  $\mathcal{P}_b(t)$  and sellers  $\mathcal{P}_s(t)$  at each time  $t$ . We will establish a lower bound on a centralized decision maker (DM), who at each time, observes all the rewards obtained by all agents thus far, and decides  $\mathcal{P}_b(t)$  and  $\mathcal{P}_s(t)$  for each  $t$ . In Appendix B.7, we show that the actions of the DM can be coupled to that of a combinatorial semi-bandit model [14], where the base set of arms are the set of all buyers and sellers, the valid subset of arms are those subsets having an equal number of buyers and sellers and the mean reward of any valid subset
$\mathcal{A} \subseteq 2^{\mathcal{D}}$  is the difference between the sum of all valuations of buyers in  $\mathcal{A}$  and of sellers in  $\mathcal{A}$ . In Appendix B we exploit this connection to semi-bandits to give a  $\Omega(\log(T))$  regret lower bound for the centralized DM. Thus our upper bound of  $O(\log(T))$  social welfare regret under the decentralized setting is order optimal since even a centralized system must incur  $\Omega(\log(T))$  regret.
6 Simulation Study
We perform synthetic studies to augment our theoretical guarantees. For a fixed system of  $N$  buyers,  $M$  sellers,  $K^{*}$  participants, and  $\Delta$  gap, the rewards are Bernoulli, with means themselves chosen uniformly at random. We vary the confidence width of the buyers,  $\alpha_{b}$ , and seller,  $\alpha_{s}$ , in  $[\alpha_{1},\alpha_{2}]$ . Next we simulate the performance of the UCB  $(\alpha_{b})$  and LCB  $(\alpha_{s})$  over 100 independent sample paths with  $T = 50k$ . We report the mean,  $25\%$  and  $75\%$  value of the trajectories. We plot the cumulative regret of the buyers,  $R_{b,i}(t)$ , and the sellers,  $R_{s,j}(t)$ , the number of matches in the system  $K(t)$ , and the price difference  $(p(t) - p^{*})$ . In Figure 2, we have a  $8\times 8$  system with  $K^{*} = 5$ . We see that  $K(t)$  converges to 5, where as  $(p(t) - p^{*})$  converges to 0. The social regret grows as  $\log (T)$ . The participant and non-participant individual regret of this instance is presented in the appendix in Figure 3. We defer the simulation of other systems used to study behavior of heterogeneous  $\alpha$ , varying gaps, and different system sizes to Appendix D.
(a) Social Regret
Figure 2: Double Auction  $N = 8$ ,  $M = 8$ ,  $K^{*} = 5$ ,  $\Delta = 0.2$ ,  $\alpha_{1} = 4$ , and  $\alpha_{2} = 8$
(b) Convergence of match size  $K(t)$ , price difference  $(p(t) - p^{*})$
Classical mechanism design in double auctions: There is a large body of work on mechanism design for double auctions, following Myerson et al. [36]. The average-mechanism, which is the subject of focus in this paper achieves all the above desiderata except for being incentive compatible. The VCG mechanism was developed in a series of works [40], [12] and [20] achieves all desiderata except being budget balanced. This mechanism requires the auctioneer to subsidize the trade. More sophisticated trade mechanisms known as the McAfee mechanism [33], trade reduction, and the probabilistic mechanism [5] all trade-off some of the desiderata for others. However, the key assumption in all of these lines of work was that all participants know their own valuations, and do not need to learn through repeated interactions.
Bandit learning in matching markets: In recent times, online learning for the two-sided matching markets have been extensively studied in [29], [30], [39], [7], [16]. This line of work studies two-sided markets when one of the side does not know of their preferences apriori and learn it through interactions. However, unlike the price-discovery aspect of the present paper, the space of preferences that each participant has to learn is discrete and finite, while the valuations that agents need to learn form a continuum. This model was improved upon by [9] that added notions of price and transfer. The paper of [25] studied a contextual version of the two-sided markets where the agents preferences depend on the revealed context through an unknown function that is learnt through interactions.
Learning in auctions: Online learning in simple auctions has a rich history - [6], [13], [34], [8], [15], [41], [22], [17] to name a few, each of which study a separate angle towards learning from repeated samples in auctions. However, unlike the our setting, one side of the market knows of their true valuations apriori. The work of [26] is the closest to ours, where the participants apriori do not know their true valuations. However, they consider the VCG mechanism in the centralized setting, i.e., all participants can observe the utilities of all participants in every round.
Learning in bi-lateral trade: One of the first studies on learning in bilateral trade is [10]. The work of [10] considers the single buyer and seller model under a weaker ordinal feedback model, while in the present work we consider the multi-agent model under a stronger bandit feedback model. The feedback model in [10] is more restrictive since the gain from the trade cannot be estimated by the agents based on ordinal feedback while in our model, the gain from the trade can be estimated by each agent. However, our work considers the impact of multi-agent competition on regret minimization, which is not studied in [10]. In each round a buyer and a seller draw her own valuation i.i.d. from their respective distributions. Then an arbiter sets the price, with trade happening if the price is between the seller's and buyer's price. They show with full-information follow-the-leader type algorithm achieves  $O(\sqrt{T})$  regret, where as with realistic (bandit-like) feedback by learning the two distributions approximately  $O(T^{2/3})$  regret can be achieved.
Protocol Model: The 'protocol model' alludes to multiple agents following a similar protocol/algorithm with each agent executing protocol with private information (e.g. Platform-as-a-Service (PAAS) [21]). In the works on learning and markets mentioned here, although the decentralized systems can be modeled as games, the protocol model is studied as a tractable way to capture the essence of the problem [4, 3]. The technical basis for this assumption is that, in the limit when the number of participants are large, and the impact of any single participant is small, a protocol based algorithm is the limit in a precise sense of any equilibrium solution to the multi-agent game [19].
We study the Double auction with Average mechanism where the buyers and sellers need to know their own valuation from her own feedback. Using confidence based bounds - UCB for the buyers and LCB for the sellers, we show that it is possible to obtain  $O(\sqrt{T\log(T)})$  individual regret for the true participant buyers and sellers in  $T$  rounds. Whereas, the true non-participant buyers and sellers obtain a  $O(\log (T) / \Delta)$  individual regret where  $\Delta$  is the smallest gap. The social regret of the proposed algorithm also admits a  $O(\log (T) / \Delta)$  bound. We show that there are simpler systems where each buyer and seller must obtain a  $\Omega (\sqrt{T})$  individual regret in the minimax sense. Moreover, in our setting we show even a centralized controller obtains  $\Omega (\log (T))$  social regret. Obtaining a minimax matching a  $O(\sqrt{T})$  regret remains open. Another important future avenue is, developing a framework and bidding strategy with provable 'good' regret for general Double auction mechanisms. Alleviating drawbacks of protocol model, such as collusion among participants, and platform disintermediation are important directions to explore.
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