SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We claim that player  $i_j$  has an improving deviation by keeping its bid vector  $\mathbf{b}_{i_j}$  and stopping all payments to other players. Since the other players have neutral reactions to the deviations, we obtain an outcome  $(\widetilde{\mathbf{b}},\widetilde{\mathbf{t}})$  such that  $\widetilde{\mathbf{b}} = \mathbf{b}$ ,  $\widetilde{t}_{i_j,k} = 0$  for all  $k\in [n]$ , and  $\widetilde{t}_{k,i_j}\leq t_{k,i_j}$  for all  $k\neq i_j$ . The gain of player  $i_j$  from the deviation can be bounded by:
$$
\begin{array}{l} u _ {i _ {j}} (\widetilde {\mathbf {b}}, \widetilde {\mathbf {t}}) - u _ {i _ {j}} (\mathbf {b}, \mathbf {t}) = u _ {i _ {j}} (\mathbf {b}, \widetilde {\mathbf {t}}) - u _ {i _ {j}} (\mathbf {b}, \mathbf {t}) \quad \left(\text {S i n c e} \widetilde {\mathbf {b}} = \mathbf {b}.\right) \\ = \left[ m _ {i _ {j}} (\widetilde {\mathbf {t}}) + \left(\sum_ {k = 1} ^ {x _ {i _ {j}} (\mathbf {b})} v _ {i _ {j}, k}\right) - p (\mathbf {b}) \cdot x _ {i _ {j}} (\mathbf {b}) \right] - \left[ m _ {i _ {j}} (\mathbf {t}) + \left(\sum_ {k = 1} ^ {x _ {i _ {j}} (\mathbf {b})} v _ {i _ {j}, k}\right) - p (\mathbf {b}) \cdot x _ {i _ {j}} (\mathbf {b}) \right] \\ = m _ {i j} (\widetilde {\mathbf {t}}) - m _ {i j} (\mathbf {t}) \\ = - m _ {i _ {j}} (\mathbf {t}) \quad \left(\text {S i n c e} m _ {i _ {j}} (\widetilde {\mathbf {t}}) = 0.\right) \\ > 0. \quad (\text {S i n c e} m _ {i _ {j}} (t) <   0 \text {b y c h o i c e o f} i _ {j}.) \\ \end{array}
$$
Thus player  $i_j$  has a strictly improving deviation, which contradicts the choice of  $(\mathbf{b}, \mathbf{t})$  as core-stable with  $\mathbf{t} \neq \mathbf{0}$ . Thus the assumption must have been false. It follows that the only core stable profiles (if any) have  $\mathbf{t} = \mathbf{0}$ .
Step II: the social welfare is maximized. Next we show that if  $(\mathbf{b},\mathbf{t})$  is a core-stable outcome, then the allocation induced by  $\mathbf{b}$  is welfare maximizing. Assume towards a contradiction this is not the case. By Step I, we have  $\mathbf{t} = \mathbf{0}$ .
Let  $w_{1} \geq \ldots \geq w_{n \cdot K}$  be the bids sorted in decreasing order (breaking ties lexicographically) at the truth-telling bid profile  $\mathbf{v}$ . For each  $j \in [n \cdot K]$ , let  $\pi_{j}$  be the player that submitted bid  $w_{j}$  in this ordering.
Let  $\widetilde{w}_1 \geq \ldots \geq \widetilde{w}_{n \cdot K}$  be the bids sorted in decreasing order (breaking ties lexicographically) at the bid profile  $\mathbf{b}$ . Let  $\widetilde{\pi}_j$  be the player that submitted bid  $\widetilde{w}_j$  in this ordering.
Consider an undirected bipartite graph  $G = (L, R, E)$ , where  $L$  is the left part,  $R$  the right part, and  $E$  the set of edges. Define
-  $L = \{(i,j) \mid i \in [n], j \in [K]\}$ , and  $x_{i}(\mathbf{v}) \geq j\}$ . For example, if  $x_{i}(\mathbf{v}) = 2$ , then  $L$  has nodes  $(i,1)$  and  $(i,2)$ . If on the other hand  $x_{i}(\mathbf{v}) = 0$ , then  $L$  has no nodes  $(i,j)$ , for any  $j$ .
-  $R = \{(i,j) \mid i \in [n], j \in [K], \text{ and } x_i(\mathbf{b}) \geq j\}$ .
-  $E = (s_1, s_2)$ , for all  $s_1 \in L$  and  $s_2 \in R$ .
Since both allocations  $\mathbf{x}(\mathbf{v})$  and  $\mathbf{x}(\mathbf{b})$  allocate exactly  $K$  units, we have  $|L| = |R| = K$ . Consider now a graph  $G_{1} = (L_{1}, R_{1}, E_{1})$  obtained from  $G$  as follows. Set  $G = G_{1}$ . Then for each node  $(i,j) \in L$ : if the node also appears in  $R$ , then delete both copies of the node, together with any edges containing them.
Thus in  $G_{1}$ , the left side  $L_{1}$  consists of nodes  $(i,j)$  with  $x_{i}(\mathbf{v}) \geq j$  but  $x_{i}(\mathbf{b}) < j$ . For each such node  $(i,j)$ , let  $k$  be the rank of the valuation  $v_{i,j}$  in the ordering  $\pi$ . By definition of  $G_{1}$ , we have that  $R_{1}$  does not have any node of the form  $(i,s)$  for any  $s$ . To see this, observe that
- nodes  $(i, s)$  with  $s < j$  were deleted when constructing  $G_{1}$  from  $G$ , and
- nodes  $(i, s)$  with  $s \geq j$  do not exist even in  $G$  (if they did, then  $(i, j)$  would exist in both  $L$  and  $R$  and so would have been deleted when constructing  $G_1$ ).
Let  $d(i,j)$  be the player that displaces player  $i$ 's bid for the  $j$ -th unit at the bid profile  $\mathbf{b}$ . Formally,  $d(i,j)$  is the owner of bid  $\widetilde{w}_k$  when considering the bids  $\mathbf{b}$  in descending order. Let  $\omega(i,j) \in \mathbb{N}$  be such that player  $d(i,j)$  obtains an  $\omega(i,j)$ -th unit in their bundle at  $\mathbf{b}$ .
Since  $x_{i}(\mathbf{b}) < j$ , we have  $i \neq d(i,j)$ . Since at the truth-telling profile player  $i$  gets a  $j$ -th unit but player  $d(i,j)$  does not get an  $\omega(i,j)$ -th unit, we have  $v_{i,j} \geq v_{d(i,j),\omega(i,j)}$ . Moreover, since the allocation  $\mathbf{x}(\mathbf{v})$  maximizes welfare but  $\mathbf{x}(\mathbf{b})$  does not, the inequality is strict for some  $(i,j) \in L_1$ .
We claim that  $[n]$  is a blocking coalition. To show this, we will argue there is a bid profile  $\mathbf{b}^*$  and vector of transfers  $\mathbf{t}^*$  such that at  $(\mathbf{b}^*,\mathbf{t}^*)$  the utility of each player  $i\in [n]$  is weakly improved and the improvement is strict for at least one player. For each  $i\in [n],k\in [K]$ , let
$$
b _ {i, j} ^ {*} = \left\{ \begin{array}{l l} v _ {i, j} & \text {i f} x _ {i} (\mathbf {v}) \geq j \\ 0 & \text {o t h e r w i s e .} \end{array} \right.
$$
Then  $\mathbf{x}(\mathbf{b}^{*}) = \mathbf{x}(\mathbf{v})$  and  $p(\mathbf{b}^{*}) = 0$ . Also define monetary transfers  $\mathbf{t}^{*}$  as follows:
- Initialize  $\mathbf{t}^* = 0$ . Let  $\varepsilon = \min_{(i,j) \in L_1} (v_{i,j} - v_{d(i,j),\omega(i,j)}) / 2$ . Then  $v_{i,j} \geq v_{d(i,j),\omega(i,j)} + \varepsilon$  for each  $(i,j) \in L_1$ .
- For each  $(i,j) \in L_1$ , let  $t_{i,d(i,j)}^* := t_{i,d(i,j)}^* + v_{d(i,j),\omega(i,j)} + \varepsilon$ .
Thus each player  $i$  that got a  $j$ -th unit in their bundle at the truth-telling profile did so because their bid for the  $j$ -th unit had rank  $k \leq K$ . Since  $i$  does not get the  $j$ -th unit at bid profile  $\mathbf{b}$ , there is a player  $d(i, j)$  whose bid for the  $j$ -th unit had rank  $k$  and who received this way a  $\omega(i, j)$ -th unit in their bundle.
We argue that all the players weakly improve their utility at  $(\mathbf{b}^{*},\mathbf{t}^{*})$ , and the improvement is strict for at least one of them.
- For each pair  $(i,j) \in L_1$ , under the bid profile  $\mathbf{b}^*$  player  $i$  receives the  $j$ -th unit at a cost of zero and transfers an amount of  $v_{d(i,j),\omega(i,j)} + \varepsilon$  to player  $d(i,j)$ .
The component of the utility that player  $i$  gets from unit  $j$ , counting the value, price, and transfer related to unit  $j$ , is  $v_{i,j} - 0 - (v_{d(i,j),\omega(i,j)} + \varepsilon) \geq 0$ , where the inequality holds by choice of  $\varepsilon$ . This is a weak improvement compared to the utility that player  $i$  gets from unit  $j$  at profile  $(\mathbf{b},\mathbf{0})$ , which is zero. Moreover, the improvement is strict for at least one pair  $(i,j) \in L_1$ , since the bid profile  $\mathbf{b}$  does not induce a welfare maximizing allocation.
- For each pair  $(i,j) \in L \setminus L_1$ , at the profile  $(\mathbf{b}, \mathbf{0})$  player  $i$  gets utility  $v_{i,j} - p(\mathbf{b})$  from unit  $j$ , since it makes no transfers. At profile  $\mathbf{b}^*$ , player  $i$  gets unit  $j$  at a price of zero and makes no transfers (towards other players) related to unit  $j$ . Thus the component of the utility related to unit  $j$  is  $v_{i,j} - 0 - 0 = v_{i,j}$ . Since  $p(\mathbf{b}) \geq 0$ , we have  $v_{i,j} - p(\mathbf{b}) \leq v_{i,j}$ , a weak improvement for player  $i$  with respect to unit  $j$ .
- For each pair  $(i,j) \notin L$ : if  $(i,j) \notin R$ , then player  $i$  does not get a  $j$ -th unit under either  $\mathbf{b}$  or  $\mathbf{b}^*$ , so its utility from unit  $j$  is zero at both profiles. If on the other hand  $(i,j) \in R$ , since  $(i,j) \notin L$ , it must be that  $(i,j) \in R_1$ . At profile  $(\mathbf{b},\mathbf{t})$  the utility of player  $i$  from unit  $j$  is  $v_{i,j} - p(\mathbf{b})$  since it gets the unit and receives no transfers. At profile  $(\mathbf{b}^*,\mathbf{t}^*)$  player  $i$  does not receive the unit but receives a transfer of  $v_{i,j} + \varepsilon$  from the player that gets the unit instead. This is again a weak improvement.
Thus all players weakly improve their utility at  $(\mathbf{b}^{*},\mathbf{t}^{*})$  and the improvement is strict for at least one player, so the profile  $(\mathbf{b},\mathbf{0})$  is not stable. This is a contradiction, thus the assumption that  $\mathbf{x}(\mathbf{b})$  is not welfare maximizing must have been false.
Step III: the price is zero. Next we show that if profile  $(\mathbf{b},\mathbf{0})$  is core-stable, then  $p(\mathbf{b}) = 0$ . By Step II, the bid profile  $\mathbf{b}$  induces a welfare maximizing allocation.
Suppose towards a contradiction that  $p(\mathbf{b}) > 0$ . Then we show there exists a blocking coalition. Let  $w_{1} \geq \ldots \geq w_{n \cdot K}$  be the bids sorted in decreasing order (breaking ties lexicographically) at bid profile  $\mathbf{b}$ . For each  $i \in [n \cdot K]$ , let  $\pi_{i}$  be the owner of bid  $w_{i}$ .
We match the players as follows. Create a bipartite graph with left part  $L = (\pi_1, \dots, \pi_K)$  (allowing repetitions) and right part  $R = (\pi_{K+1}, \dots, \pi_{2K})$  (allowing repetitions). For each  $i \in [K]$ , create edge  $(\pi_i, \pi_{K+i})$ . Consider the profile  $(\mathbf{b}^*, \mathbf{t}^*)$ , where
$$
b _ {i, j} ^ {*} = \left\{ \begin{array}{l l} v _ {i, j} & \text {i f} x _ {i} \geq j \\ 0 & \text {o t h e r w i s e .} \end{array} \right.
$$
Define the transfers as follows:
- Initialize  $\mathbf{t}^{*} = \mathbf{0}$ . Set  $\varepsilon = p / (2K)$ . For each  $i \in [K]$ , let player  $\pi_{i}$  pay an additional amount of  $\varepsilon$  to player  $\pi_{K + i}$ :  $t_{\pi_i, \pi_{K + i}}^* = t_{\pi_i, \pi_{K + i}}^* + \varepsilon$ .
Then each player  $i$  with  $x_{i}(\mathbf{b}) > 0$  gets the same allocation at  $\mathbf{b}^*$  as at  $\mathbf{b}$ , but pays a price of zero for the units and makes a transfer of at most  $p / 2$  to other players, resulting in improved utility compared to the utility at profile  $(\mathbf{b},\mathbf{0})$ .
On the other hand, each player  $i$  with  $x_{i}(\mathbf{b}) = 0$  on the other hand gets the same allocation at  $\mathbf{b}^*$  as at  $\mathbf{b}$  (i.e. no units), but receives a non-negative amount of money from other players, and the net amount of money received is strictly positive for all the players  $\pi_{K + 1},\dots ,\pi_{2K}$ .
Thus there is an improving deviation, which contradicts the choice of  $(\mathbf{b},\mathbf{t})$  as core-stable. Thus the assumption must have been false, and  $p(\mathbf{b}) = 0$ .
C.2 Core without transfers
A strategy profile in this setting is described by a bid profile  $\mathbf{b}$ , where  $\mathbf{b}_i = (b_{i,1},\dots,b_{i,K})$  is the bid vector of player  $i$ . In the setting without transfers, given a profile  $\mathbf{b}$  of bids, a blocking coalition  $S$  needs to agree on simultaneously changing their bids, such that when the players outside  $S$  still bid according to  $\mathbf{b}$ , the players in  $S$  weakly improve their utility and the improvement is strict for at least one player in  $S$ .
The core stable profiles are those that have no such blocking coalitions. Formally, we have:
Definition 5 (Blocking coalition, without transfers). Let  $\mathbf{b}$  be a bid profile. A group  $S \subseteq [n]$  of players is a blocking coalition if there exists a bid profile  $\widetilde{\mathbf{b}}$ , at which each player  $i \in S$  weakly improves their utility, the improvement is strict for at least one player in  $S$ , and  $\widetilde{b}_{i,j} = b_{i,j}$  for all  $i \in [n] \setminus S, j \in [K]$ .
Definition 6 (The core with transfers). The core with transfers consists of bid profiles  $\mathbf{b}$  at which there are no blocking coalitions. Such profiles are core-stable.
The non-transferable utility core can be characterized as follows.
Theorem 5 (Core without transfers; restated). Consider  $K$  units and  $n > K$  hungry players. The core without transfers of the  $(K + 1)$ -st auction can be characterized as follows:
- every bid profile  $\mathbf{b}$  that is core stable has price zero (i.e.  $p(\mathbf{b}) = 0$ );
- each allocation  $\mathbf{z}$  where all the units are allocated can be supported in a core stable bid profile  $\mathbf{b}$  with price zero (i.e.  $\mathbf{x}(\mathbf{b}) = \mathbf{z}$  and  $p(\mathbf{b}) = 0$ ).
Proof. The proof is in two parts.
Part I: the price in the core is zero. Consider a bid profile  $\mathbf{b}$  that is core-stable. Suppose towards a contradiction that  $p(\mathbf{b}) > 0$ .
Let  $M = \sum_{\ell_1 = 1}^n\sum_{\ell_2 = 1}^K v_{\ell_1,\ell_2}$ . Define a bid profile  $\widetilde{\mathbf{b}}$  such that for all  $i\in [n],j\in [K]$ :
$$
\widetilde {b} _ {i, j} = \left\{ \begin{array}{l l} M & \text {i f} j \leq x _ {i} (\mathbf {b}) \\ \varepsilon & \text {o t h e r w i s e .} \end{array} \right. \tag {34}
$$
At  $\widetilde{\mathbf{b}}$ , the players only submit bids equal to  $M > 0$  for the units they are supposed to get at allocation  $\mathbf{x}(\mathbf{b})$ , and moreover, there are exactly  $K$  strictly positive bids. Thus  $\mathbf{x}(\widetilde{\mathbf{b}}) = \mathbf{x}(\mathbf{b})$ . Moreover,  $p(\mathbf{b}) = 0$  since the  $(K + 1)$ -st highest bid is 0.
Then the grand coalition  $C = [n]$  is blocking with the profile  $\tilde{\mathbf{b}}$ , in contradiction with  $\mathbf{b}$  being core stable. Thus the assumption must have been false, so  $p(\mathbf{b}) = 0$ .
Part II: every allocation can be implemented at a core-stable bid profile. Let  $\mathbf{z}$  be an arbitrary allocation at which all the units are allocated.
Define  $\mathbf{b}$  such that for all  $i\in [n],j\in [K]$ , we have  $b_{i,j} = M$  if  $j\leq z_{i}$  and  $b_{i,j} = 0$  otherwise. Then  $\mathbf{x}(\mathbf{b}) = \mathbf{z}$  and  $p(\mathbf{b}) = 0$ . Let  $W = \{i\in [n]\mid z_i > 0\}$  be the set of "winners" at  $\mathbf{b}$ .
Assume towards a contradiction that  $\mathbf{b}$  is not stable. Then there is a blocking coalition  $C = \{i_1, \ldots, i_k\} \subseteq [n]$  with alternative bid profile  $\mathbf{d} = (\mathbf{d}_{i_1}, \ldots, \mathbf{d}_{i_k})$ . Denote  $\widetilde{\mathbf{b}} = (\mathbf{d}, \mathbf{b}_{-C})$  the profile where each player  $i \in C$  bids  $\widetilde{\mathbf{b}}_i = \mathbf{d}_i$  and each player  $i \notin C$  bids  $\widetilde{\mathbf{b}}_i = \mathbf{b}_i$ . We must have  $u_i(\widetilde{\mathbf{b}}) \geq u_i(\mathbf{b})$  for all  $i \in C$ , with strict inequality for some player  $i \in C$ .
If  $C \cap W = \emptyset$ , then the only way for at least one of the players in  $C$  to change their allocation is to make some of their bids at least  $H$ . But this increases the price from zero to at least  $H$ , which yields negative utility for everyone. Thus  $C \cap W \neq \emptyset$ . We consider two cases:
1. Case  $p(\widetilde{\mathbf{b}}) > 0$ . Each player  $i \in C \cap W$  requires strictly more units at  $\widetilde{\mathbf{b}}$  than at  $\mathbf{b}$ , to compensate for the higher price at  $\widetilde{\mathbf{b}}$ . Thus  $\mathbf{x}_i(\widetilde{\mathbf{b}}) > z_i$  for all  $i \in C \cap W (\dagger)$ . If  $x_i(\widetilde{\mathbf{b}}_i) < z_i$  for some player  $i \in W \setminus C$ , there would have to exist at least  $K + 1$  bids with value at least  $H$ , and so  $p(\widetilde{\mathbf{b}}) \geq H$ , which would give negative utility to all the players including the deviators. Thus  $x_i(\widetilde{\mathbf{b}}_i) \geq z_i$  for all  $i \in W \setminus C (\ddagger)$ .
Combining  $(\dagger)$  and  $(\ddagger)$  gives a contradiction:
$$
\sum_ {i \in C \cup (W \backslash C)} x _ {i} (\widetilde {\mathbf {b}}) > \sum_ {i \in C \cup (W \backslash C)} z _ {i} = K.
$$
Thus  $p(\widetilde{\mathbf{b}}) > 0$  cannot hold.
2. Case  $p(\mathbf{b}) = 0$ . Then each player  $i \in C \cap W$  requires  $u_i(\widetilde{\mathbf{b}}) \geq z_i$ . Since  $p(\widetilde{\mathbf{b}}) = 0$ , the top  $K$  bids are strictly positive and the remaining bids are zero. Then each player  $i \in C \cap W$  submits exactly  $z_i$  strictly positive bids. It follows that  $\mathbf{x}_i(\widetilde{\mathbf{b}}) = \mathbf{x}_i(\mathbf{b})$  and  $p(\widetilde{\mathbf{b}}) = p(\mathbf{b})$  for each  $i \in [n]$ , which means no player in  $C$  strictly improves. Thus  $C$  cannot be blocking.
In both cases 1 and 2 we obtained a contradiction, so  $\mathbf{b}$  is core-stable,  $p(\mathbf{b}) = 0$ , and  $\mathbf{x}(\mathbf{b}) = \mathbf{z}$  as required.
D Theorems from prior work
In this section we include the theorem from [CBL06] that we use. In the problem of prediction under expert advice, there are  $N$  experts in total, and at each round  $t \in [T]$ :
- learner chooses a probability distribution  $p_t$  over  $[N]$ ;
- nature reveals the losses  $\{\ell_{t,i}\}_{i\in [N]}$  of all experts at time  $t$  where  $\ell_{t,i}\in [0,L]$ .
For a given sequence of probability distributions  $(p_1,\dots ,p_T)$ , the learner's regret is defined to be
$$
\operatorname {R e g} (T, \left(p _ {1}, \dots , p _ {T}\right)) = \sum_ {t = 1} ^ {T} \sum_ {i = 1} ^ {N} p _ {t} (i) \ell_ {t, i} - \min  _ {i ^ {\star} \in [ N ]} \sum_ {t = 1} ^ {T} \ell_ {t, i ^ {\star}}.
$$
For this problem, the exponentially weighted average forecaster, also known as the Hedge algorithm, is defined as follows. The algorithm initializes  $p_1 = \sigma$ , an arbitrary prior distribution over the experts  $[N]$  such that each expert  $i$  is selected with probability  $\sigma_i > 0$ . For  $t \geq 2$ , the forecaster updates
$$
p _ {t} (i) = \frac {p _ {t - 1} (i) \exp (- \eta \ell_ {t - 1 , i})}{\sum_ {j = 1} ^ {N} p _ {t - 1} (j) \exp (- \eta \ell_ {t - 1 , j})}, \quad \forall i \in [ N ],
$$
where  $\eta > 0$  is a learning rate.
Next we include the statement of Theorem 2.2 of [CBL06] in our notation.
Theorem 7 (Theorem 2.2 of [CBL06]). Consider the exponentially weighted average forecaster with  $N$  experts, learning rate  $\eta > 0$ , time horizon  $T$ , and rewards in  $[0,1]$ . Suppose the initial distribution  $\sigma$  on the experts is uniform, that is,  $\sigma = (1/N, \dots, 1/N)$ . The regret of the forecaster is
$$
\operatorname {R e g} (T, (p _ {1}, \dots , p _ {T})) \leq \frac {\log N}{\eta} + \frac {T \eta}{8}.
$$
The following corollary is a well known variant of the above theorem, to allow rewards in an interval  $[0,L]$  and an arbitrary initial distribution  $\sigma$  over the experts.
Corollary 1. Consider the exponentially weighted average forecaster with  $N$  experts, learning rate  $\eta > 0$ , time horizon  $T$ , and rewards in  $[0, L]$ . Suppose the initial distribution on the experts is  $\sigma$ . The regret of the forecaster is
$$
\operatorname {R e g} (T, (p _ {1}, \dots , p _ {T})) \leq \frac {1}{\eta} \max  _ {i \in [ N ]} \log \left(\frac {1}{\sigma_ {i}}\right) + \frac {T L ^ {2} \eta}{8}.
$$
The proof of Corollary 1 is identical to that of Theorem 7, except for the following two differences:
- Instead of  $W_{t} = \sum_{i=1}^{N} \exp(-\eta \sum_{s \leq t} \ell_{s,i})$ , we define  $W_{t} = \sum_{i=1}^{N} \sigma_{i} \exp(-\eta \sum_{s \leq t} \ell_{s,i})$ . In this way,
$$
\log \frac {W _ {T}}{W _ {0}} = \log W _ {T} \geq - \eta \min _ {i ^ {\star} \in [ N ]} \sum_ {t = 1} ^ {T} \ell_ {t, i ^ {\star}} - \max _ {i \in [ N ]} \log \frac {1}{\sigma_ {i}}.
$$
- When applying [CBL06, Lemma 2.2], we use the interval for the rewards as  $[0, L]$  instead of  $[0, 1]$ .