Quantum Auctions

TL;DR

This paper introduces a quantum auction protocol that uses superpositions and entanglement to preserve bid privacy and prevent bid revealing, applicable to various auction types and involving game-theoretical analysis.

Contribution

It presents a novel quantum auction protocol leveraging superpositions and entanglement, with analysis of strategic behavior and incentive-driven quantum algorithm design.

Findings

Bid privacy is maintained through quantum measurement.

The protocol applies to multiple auction formats, including combinatorial auctions.

Adiabatic search reduces strategic manipulation by bidders.

Abstract

We present a quantum auction protocol using superpositions to represent bids and distributed search to identify the winner(s). Measuring the final quantum state gives the auction outcome while simultaneously destroying the superposition. Thus non-winning bids are never revealed. Participants can use entanglement to arrange for correlations among their bids, with the assurance that this entanglement is not observable by others. The protocol is useful for information hiding applications, such as partnership bidding with allocative externality or concerns about revealing bidding preferences. The protocol applies to a variety of auction types, e.g., first or second price, and to auctions involving either a single item or arbitrary bundles of items (i.e., combinatorial auctions). We analyze the game-theoretical behavior of the quantum protocol for the simple case of a sealed-bid quantum, and show how a suitably designed adiabatic search reduces the possibilities for bidders to game the auction. This design illustrates how incentive rather that computational constraints affect quantum algorithm choices.

Figures

Tables

Table 1. Table 1: Notation for the quantum auction.

number of bidders	$n$
number of items in auction	$m$
number of qubits per bidder	$p$
state of qubits for bidder $j$	${\psi }_j$
state of all qubits	$\Psi ={\psi }_1\otimes \dots \otimes {\psi }_n$

Equations

$$U=U_1\otimes U_2\otimes \dots \otimes U_n$$

$$U_j=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& -1& 0\\ 0& 1& 0& -1\end{array}\right)$$

$${\Psi }_0=U{\Psi }_{\mathrm{init}}$$

$${\Psi }_s=UD(f)U^{†}P(f){\Psi }_{s-1}$$

$$P_{xx}(f)=\exp \left(-ifc(x)\Delta \right)$$

$$D_{xx}(f)=\exp \left(-i(1-f)d(x)\Delta \right)$$

$$\frac{1}{2}(|\varnothing ,\varnothing ⟩+|b_A,\varnothing ⟩+|\varnothing ,b_B⟩+|b_A,b_B⟩)$$

$$\frac{1}{\sqrt{2}}(|\varnothing ,b_B⟩+|b_A,b_B⟩)$$

$${\psi }_1=\sum _{i=0}^{2^p-1}{\alpha }_i|i⟩$$

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)(2^p-1{)}^k$$

$$2^{np}-n(2^p-1)\ge n(2^p-1)$$

$$2^{np}-n(2^p-1)\ge \sum _{k=1}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{k}\right)(2^p-1)=2^{np}-1-(2^p-1{)}^n$$

$$d(x)=-r(x)(\mathrm{mod}n+1)$$

$$\alpha |0⟩+\beta |a⟩$$

$$\left(\begin{array}{ccccc}1& -1& 0& 0& \\ 1& 1& 0& 0& \\ 0& 0& 1& -1& \\ 0& 0& 1& 1& \end{array}\right)\left(\begin{array}{ccccc}1& 0& -1& 0& \\ 0& 1& 0& -1& \\ 1& 0& 1& 0& \\ 0& 1& 0& 1& \end{array}\right)\left(\begin{array}{ccccc}1& 0& 0& -1& \\ 0& 1& 1& 0& \\ 0& -1& 1& 0& \\ 1& 0& 0& 1& \end{array}\right)$$

$$V=\left(\begin{array}{ccccc}1& -1& -1& 1& \\ 1& 1& -1& -1& \\ 1& -1& 1& -1& \\ 1& 1& 1& 1& \end{array}\right)$$

$$U_{x,y}=\prod _{i=1}^n(U_i{)}_{x_i,y_i}$$

$$\frac{1}{\sqrt{3}}(|\varnothing ,0⟩+|X,1⟩+|(X,Y),2⟩)$$

$$\frac{1}{\sqrt{4}}(|\varnothing ,0⟩+|X,1⟩+|X,3⟩+|(X,Y),4⟩)$$

$$2^{np}-((n+1{)}^m-1)2^{n{p}_{\mathrm{price}}}\ge n(2^p-1)$$

$$2^{np}-((n+1{)}^m-1)2^{n{p}_{\mathrm{price}}}\ge 2^{np}-1-(2^p-1{)}^n$$

$$(2^p-1{)}^n$$

$$\iff 2^{p_{\mathrm{price}}n}(2^m-2^{-p_{\mathrm{price}}}{)}^n$$

$$\iff (2^m-2^{-p_{\mathrm{price}}}{)}^n$$

$$(2^m-1{)}^n\ge (n+1{)}^m-1$$

$$(2^m-1{)}^{\frac{1}{m}}\ge (n+1{)}^{\frac{1}{n}}$$

$$\frac{(2^m-1{)}^{\frac{1}{m}}}{m}\left(\frac{2^m\ln (2)}{2^m-1}-\frac{\ln (2^m-1)}{m}\right)$$

$$\frac{2^m}{2^m-1}\frac{m}{{\log }_2(2^m-1)}>1$$

$$\frac{(n+1{)}^{\frac{1}{n}}}{n}\left(\frac{1}{1+n}-\frac{\ln (1+n)}{n}\right)$$

$$\ln (1+n)\frac{1+n}{n}>1$$

$$\frac{1}{\sqrt{3}}(|\varnothing ,0,\varnothing ,0⟩+|I_A,b_A,I_B,b_B⟩+|I_C,b_C,I_D,b_D⟩)$$

Full text

1 Introduction

Quantum information processing [23] offers potential improvements in a variety of applications. Computational advantages [26, 14] of quantum computers with many qubits have received the most attention but are difficult to implement physically. On the other hand, technology for manipulating and communicating just a few qubits could be sufficient to create new economic mechanisms by altering the information security and strategic incentives of the underlying game.

Examples of quantum mechanisms include the prisoner’s dilemma [10, 11, 7, 8], coordination [17, 21] and public goods provisioning [3]. In particular, a quantum mechanism can significantly reduce the free-rider problem without a third-party enforcer or repeated interactions, both in theory and practice [2].

In this paper, we examine quantum mechanisms for another economic scenario: resource allocation by auction [28]. While traditional auction mechanisms can efficiently allocate resources in many cases, quantum auction protocols offer improvements in preserving privacy of the losing bids and dealing with scenarios in which bidders care about what other bidders win when multiple items are auctioned. Specifically, using quantum superpositions to represent bids prevents the auctioneer and other bidders from viewing the bids during the auction without disrupting the auction process. Furthermore, the auction result reveals nothing but the winning bid and allocation.

The first part of the paper introduces a general quantum auction protocol for various pricing and allocation rules, multiple unit auctions, combinatorial auctions and partnership bids. For simplicity, we focus on the sealed-bid first-price auction. In this auction, each bidder has one opportunity to submit a bid. The winner is the highest bidder, who pays the amount bid for the item. This auction has been well studied both theoretically [28] and experimentally [5, 4], and contrasts with iterative auctions in which bidders can incrementally increase their bids depending on how others bid.

If the auction is not well-matched to the bidders preferences, it can introduce perverse incentives and result in poor outcomes, such as lost revenue for the seller or economically inefficient allocations where items are not allocated to those who value them most. Thus it is important to examine incentives introduced with a proposed auction design. In particular, our auction protocol involves quantum search, which introduces incentive issues beyond those examined in prior quantum games [11].

A full analysis of incentive issues is complicated, even for classical auctions. In this paper we focus on two incentive issues arising from the quantum auction protocol. The first incentive issue arises from the possibility of manipulating the search outcome by altering amplitudes associated with different bids. We show how to revise an adiabatic search method to correct this incentive problem, thereby preserving the classical Nash equilibrium. From a quantum algorithm perspective, this construction of the search illustrates how incentive issues affect algorithm design, in contrast to the more common concern with computational efficiency in quantum information processing.

Second, the quantum search for the highest bid is probabilistic, i.e., does not always return the highest bid. While the probability of finding the correct answer can be made as high as one wishes by using more iterations of the search, the small residue probability of awarding the item to someone other than the highest bidder may change bidding behavior. As a step toward addressing the effect of probabilistic outcomes, we show that, with sufficient steps in the quantum search, altering choices from those of the corresponding deterministic auction gives at most a small improvement for that bidder.

The paper is organized as follows. Sec. 2 describes the quantum auction and the bidding language encoding bids in quantum states. Sec. 3 describes the quantum search method to find the maximum bid. After these sections describing the auction protocol, in Sec. 4 we turn to strategic issues raised by the quantum nature of the auction beyond those in the corresponding classical auctions. Then, in Sec. 5 we give a game theory analysis of some of these strategic possibilities and describe how simple modifications of the quantum search improves the auction outcome, in theory. Sec. 6 generalizes the results to auctions of multiple items, including combinatorial auctions. Sec. 7 describes scenarios for which the quantum protocol offers likely economic advantages in terms of information security and ability to compactly express complex dependencies among items and bidders. Finally, Sec. 8 summarizes the quantum auction protocol and highlights a number of remaining economic questions.

2 Quantum Auction Protocol

In our auction protocol, each bidder selects an operator that produces the desired bid from a prespecified initial state. The auctioneer repeatedly asks the bidders to apply their individual operators in a distributed implementation of a quantum search to find the winning bid. More specifically, the quantum auction protocol for sealed-bid auctions involves the following steps:

Auctioneer announces conventional aspects of the auction: type of auction (e.g., first or second price and any reservation prices), the good(s) for sale, the allowed price granularity (e.g., if bids can specify values to the penny, or only to the dollar), and the criterion used to determine the winner(s), e.g., maximizing revenue for the seller 2. 2.

Auctioneer announces how quantum states will be interpreted, i.e., as specifying a price if only one good is for sale, or a combination of price and a set of goods if combinations are for sale; and also announces the initial quantum state. This state uses $p$ qubits for each bidder. Auctioneer announces the quantum search procedure. 3. 3.

Each bidder selects an operator on $p$ qubits. Bidders keep their choice of operator private. 4. 4.

Auctioneer produces a set of particles implementing $p$ qubits for each bidder, initializing the set to the announced initial state. 5. 5.

Auctioneer and bidders perform a distributed search for the winner

Fig. 1 illustrates this procedure for two bidders and repeating the steps of the search twice. Realistic search involves a larger number of steps. In contrast with other quantum games, e.g., public goods, that involve just one round of interaction, the search required to identify the winners involves multiple rounds of interaction among the participants. The required number of iterations depends on the search method. In practice, the auctioneer could pick the number of iterations based on prior experience with similar auctions, or from simulating several test cases using valuations randomly drawn from a plausible distribution of values for the auction items. Alternatively, the auctioneer could repeat the procedure several times (possibly with steps from each repetition interleaved in a random order) and use the best result from these repetitions.

This auction protocol uses a distributed search so bidders’ operator choices remain private. Specifically, the search operation requiring input from the bidders is applied locally by each bidder, giving the overall operator

[TABLE]

where $n$ is the number of bidders and $U_{i}$ the operator of bidder $i$.

3 Quantum Auction Implementation

A quantum auction requires finding the winning bid and corresponding bidder. This procedure has two components: the interpretation of the qubits as bids, and the search procedure to find the winner. The following two subsections discuss these components in the context of a single-item auction. Sec. 6 generalizes this discussion to multiple items.

3.1 Creation and interpretation of quantum bids

We define a bid as the amount a bidder indicates he is willing to pay for the item. An allocation is a list of bids, one from each bidder. The quantum auction protocol manipulates superpositions of allocations. We use an allocation rule to indicate how allocations specify a winner and amount paid.

Example 1**.**

Consider an auction of one item with three bidders, willing to pay $1,$3 and $10 for the item, respectively. We represent these bids as$\left|$1\right\rangle$,$\left|$3\right\rangle$and$\left|$10\right\rangle$, and the corresponding allocation as the product of these states, i.e.,$\left|$1,$3,$10\right\rangle$with the ordering in the allocation understood to correspond to the bidders. A simple allocation rule selects the highest bidder as the winner, who pays the high bid. In this example, this rule results in the third bidder winning, and paying$10 for the item.

Each bidder gets $p$ qubits and can only operate on those bits. Thus each bidder has $2^{p}$ possible bid values, and can create superpositions of these values. A superposition of bids specifies set of distinct bids, with at most one allowed to win. The amplitudes of the superposition affect the likelihood of various outcomes for the auction. For a single-item auction, a bidder will typically have only one bid. As discussed below, more complicated superpositions are useful for information hiding. Specifically, bidder $j$ selects an operator $U_{j}$ on $p$ qubits to apply to the initial state for that bidder’s qubits ${\psi_{\rm init}}$ specified by the auctioneer. The resulting state, $\psi_{j}=U_{j}{\psi_{\rm init}}$, is a superposition of bids, each of the form $\left|b_{i}^{(j)}\right\rangle$ where $b_{i}^{(j)}$ is bidder $j$’s bid for the item. The subscript $i$ indicates one of the possible bids that can be specified with $p$ qubits according to the announced interpretation of the bits.

We define the subspace used by bidder $j$ as the set of states spanned by the basis eigenvectors in $\psi_{j}$. Only these basis vectors appear in allocations relevant for the search. As bidders apply their operators during the search, the superposition of allocations remains within the subspace of each bidder. In this case, where each bidder applies an operator only to their own qubits, the superposition of allocations is always a factored form, i.e., $\Psi=\psi_{1}\otimes\ldots\otimes\psi_{n}$. More generally, groups of bidders could operate jointly on their qubits, entangling their bids in the allocations as discussed in Sec. 7.

To exploit information hiding properties of superpositions, the state revealed at the end of the search should specify only the bidder who wins the item and the corresponding bid. To achieve this, instead of a direct representation of bids, we interpret bids formed from the $p$ qubits available to a bidder as containing a special null value, $\varnothing$, indicating a bid for nothing. This null bid has additional benefits in multiple item settings, as discussed in Sec. 6 and Sec. 7.

Example 2**.**

Consider bidder $j$ with two qubits and the initial state ${\psi_{\rm init}}=\left|00\right\rangle$ corresponding to the vector $(1,0,0,0)$, which is interpreted as the null bid. The other bid states are $\left|01\right\rangle$, $\left|10\right\rangle$ and $\left|11\right\rangle$ corresponding to vectors (0,1,0,0), (0,0,1,0) and (0,0,0,1). These three states are interpreted as three bid values in some preannounced way, e.g., $1,$2 and $3, respectively.

The operator

[TABLE]

gives the initial state $\psi_{j}=U_{j}{\psi_{\rm init}}$ as $(\left|00\right\rangle+\left|10\right\rangle)/\sqrt{2}$ and specifies the search subspace whose basis is the first and third columns of $U_{j}$ in this example. Thus the possible allocations involve only $\left|00\right\rangle$ and $\left|10\right\rangle$ for this bidder, corresponding to the null bid and a bid of $2, respectively.

In the presence of a null bid, we consider an allocation to be a feasible if it contains exactly one bid not equal to $\varnothing$. The corresponding allocation rule assigns no winner to infeasible allocations and, for feasible allocations, the winner is the single bidder in the allocation whose bid is not $\varnothing$, and he pays the amount bid. This allocation rule corresponds to a first-price single-item auction, except there can be no winner, analogous to the situation in auctions with a reservation price when no bidder exceeds that price.

3.2 Distributed Search

The auctioneer must find the best state according to an announced criterion, e.g., maximum revenue. Specifically, the auctioneer has a evaluation function $F$ assigning a quality value to each allocation. The function $F$ assigns a lower value to infeasible allocations than to any feasible one. An example is $F$ equal to the revenue produced by the allocation (if feasible) and otherwise is $-1$.

The auctioneer uses quantum search to find the allocation in the subspace selected by the bidders giving the maximum value for $F$ (e.g., a feasible allocation giving the most revenue to the auctioneer). This could be done via repeated uses of a decision-problem quantum search [14, 1] as a subroutine within a search for the minimum threshold value of $F$ giving a solution to the decision problem, e.g., with a classical binary search on threshold values or using results of prior iterations of the decision problem [9]. Alternatively, we could use a method giving the maximum value directly (e.g., adiabatic [12] if run for a sufficiently long time or heuristic methods [15, 16] based on some prior knowledge of the distribution of bidders values). For definiteness, we focus on the adiabatic method.

The adiabatic search is conventionally described as searching for the minimum cost state. We use this convention by defining a state’s cost to be the negative of the evaluation function $F$. The adiabatic search procedure, if run sufficiently slowly, changes the initial superposition into a final superposition in such a way that the amplitude in each initial eigenstate maps to the same amplitude in the corresponding final eigenstate, up to a phase factor (for nondegenerate eigenstates). We refer to this mapping of initial to final eigenstates as a perfect search. In practice, with a finite time for the search, there will be some transfer of amplitude among the eigenstates so the search will not be perfect in the sense defined here. Instead the auction outcome is probabilistic: the auction will not always produce the best outcome when starting from the ground state. For example, an auction intending to find the highest bid could sometimes produce the second highest bid instead. Conventionally, the search operations are chosen so the uniform superposition is the lowest cost initial eigenstate. In our case, bidders are free to choose their operators and need not create uniform superpositions.

A discrete implementation of adiabatic search consists of the following steps:

• •

The auctioneer selects a number of search steps $S$ and parameter $\Delta$. These need not be announced to the bidders.

• •

The auctioneer initializes the state of all $np$ qubits to ${\Psi_{\rm init}}={\psi_{\rm init}}\otimes\ldots\otimes{\psi_{\rm init}}=\left|0,\ldots,0\right\rangle$, with $n$ factors of ${\psi_{\rm init}}$ in the product, and ${\psi_{\rm init}}=\left|0\right\rangle$ is the initial state for the $p$ qubits for a single bidder.

• •

The auctioneer sends these initialized qubits to the bidders who use their individual operators and then return the qubits to the auctioneer, jointly creating the state

[TABLE]

• •

For $s=1,\ldots,S$, the auctioneer and bidders update the state to

[TABLE]

with $f=s/S$ the fraction of steps completed. The bid operator $U$ and its adjoint $U^{\dagger}$ are performed by sending bits to the bidders as described in Sec. 2. The diagonal matrices $D(f)$ and $P(f)$ are described below.

• •

The auctioneer measures the state $\Psi_{S}$, resulting in specific values for all the bits, from which the winner and prices are determined by the allocation rule described in Sec. 3.1.

The diagonal matrix $P(f)$ adjusts the phases of the amplitudes according to the cost associated with each allocation. In particular, using the cost $c(x)=-F(x)$ for allocation $\left|x\right\rangle$, we have

[TABLE]

Similarly, the diagonal matrix $D(f)$ adjusts amplitude phases as defined by a function $d(x)$:

[TABLE]

The key property of $d(x)$ is assigning the smallest value, e.g., 0, to $\left|0\right\rangle$, thereby making the first column of $U$ the ground state eigenvector. Aside from this key property, the choice of $d(x)$ is somewhat arbitrary. The conventional choice in the adiabatic method uses the Hamming weight of the state, i.e., $d(x)$ equal to the number of 1 bits in the binary representation of $x$. However, as described in Sec. 5, other choices for $d(x)$ can improve the incentive properties of the auction.

The discrete-step implementation of the continuous adiabatic method [12] involves the limits $\Delta\rightarrow 0$ and $S\Delta\rightarrow\infty$, in which case the final state $\psi_{S}$ has high probability to be the lowest cost state. In practice, this outcome can often be achieved with considerably fewer steps using a fixed value of $\Delta$, corresponding to a discrete version of the adiabatic method [16].

4 Strategies with Quantum Operators

Ideally, an auction achieves the economic objective of its design (e.g. maximum revenue for the seller). In practice, an auction design may not provide incentives for participants to behave so as to achieve this objective. Usually auction designs are examined under the assumption of self-interested rational participants. In conventional auctions, strategic issues include misrepresentation of the true value, collusion among bidders and false name bidding (where a single bidder submits bids under several aliases). Some of these issues can be addressed with suitable auction rules, e.g., second price auctions encourage truthful reporting of values. Developing suitable designs of classical auctions in a wide range of economic contexts remains a challenging problem [28].

Quantum auctions raise strategic issues beyond those of classical auctions. In our case, every step of the adiabatic search requires each bidder to perform an operation on their qubits. Ideally, the bidder should use the same operator $U$ for creating ${\psi_{\rm init}}$ as in every step of the search in Eq. (4). In addition, bidders should include the null bid in their subspaces. In the classical first-price sealed-bid auction, the bidder makes one choice: the amount to bid. In our quantum setting, this choice amounts to selecting the subspace to use with the quantum search. The remaining freedom to select $U$, and possibly a different $U$ for each step in the search, are additional choices provided by the quantum auction.

Bidders may be tempted to exploit the flexibility of choosing operators in several general ways. First, they could use a subspace not including the null bid. Second, they could use a different operator for creating ${\psi_{\rm init}}$ than they use in the rest of the search, thereby producing an altered initial amplitude that is not the ground state eigenvector. Third they could change operators during the search. If any such changes give significant probability for low bids to win, bidders would be tempted to make such changes and include a low bid in their subspace, hoping to profit significantly by winning the auction with a low bid.

The remainder of this section describes some strategic issues unique to quantum auctions and possible solutions. We further discuss a game theory analysis of some of these issues in Sec. 5.

4.1 Selecting the Subspace

The use of the null bid in our protocol raises the strategic issue illustrated in the following example:

Example 3**.**

Consider an auction of a single item with two bidders Alice and Bob. Using operators producing uniform amplitudes for the sake of illustration, they ought to apply operators that create

$\frac{1}{\sqrt{2}}(\left|\varnothing\right\rangle+\left|b_{A}\right\rangle)$* and $\frac{1}{\sqrt{2}}(\left|\varnothing\right\rangle+\left|b_{B}\right\rangle)$*

respectively, where $b_{A}$ and $b_{B}$ are their desired bids. The initial superposition for all the qubits is the product of these individual superpositions, i.e., $\Psi_{0}$ is

[TABLE]

If bidders use these same operators during the search, the search algorithm finds the highest revenue allocation, i.e., giving the item to the highest bidder. Suppose instead Bob picks an operator with a one-dimensional subspace, producing an initial state $\left|b_{B}\right\rangle$ rather than including $\varnothing$. The product superposition is then

[TABLE]

Since the search remains in this subspace and the second allocation is infeasible, the search will return $\left|\varnothing,b_{B}\right\rangle$ no matter what Alice bids. Thus Bob always wins the item, and can win using the lowest possible bid.

This example shows bidders have an incentive to exclude the null set from their subspace. If all bidders make this choice, there will be no feasible allocations in the joint subspace and the auction will always give no winner. For auctions with more than two bidders, selecting subspaces excluding $\varnothing$ is a weak Nash Equilibrium for the quantum auction because any other choice by a single bidder still results in no feasible allocations.

4.2 Altering Initial Amplitudes

Strategic choices for bidders also arise from the search procedure itself, even when using the correct subspace consisting of $\varnothing$ and the desired bid. In particular, the probabilistic outcome of the search means the optimal bid according to the auction criterion (e.g., highest revenue) will not always win. For the adiabatic search method, bidders could try to arrange for especially tiny eigenvalue gaps between the state corresponding to the best outcome and another state allowing them to win with a low bid. A sufficiently small gap could make the number of steps the auctioneer selects insufficient to give the optimal state with high probability and instead give a significant chance of producing the more favorable outcome. However, because the eigenvalues are a complicated function of the operators of all bidders, and individual bidders do not know the choices made by others, it will be difficult for a bidder to determine how to make such especially small gaps and do so in a way that gives a favorable outcome. Nevertheless, even fairly small probabilities for not finding the optimal state could alter the strategic behavior of the bidders.

A more direct way a bidder can arrange for a low bid to win is by altering the initial state of the adiabatic search to start not in the ground state but in an eigenvector corresponding to one of the first few eigenvalues above the ground state. The adiabatic search takes such eigenvectors, with high probability, to an outcome in which a bid lower than the highest wins. While a single bidder cannot create an arbitrary initial condition, one bidder can ensure that it is not the ground state. For example, a bidder could chose an operator that gives a nonuniform amplitude for the initial state, in particular $(\left|\varnothing\right\rangle-\left|b_{A}\right\rangle)/\sqrt{2}$, while using the uniform state $(\left|\varnothing\right\rangle+\left|b_{A}\right\rangle)/\sqrt{2}$ as the ground state through the remainder of the search in Eq. (4). This can result in significant probability for a low bid to win, and so a bidder is tempted to deviate from the nominal operator choice.

Fig. 2 illustrates this behavior. Instead of starting in the ground state, the bidder’s choice gives the initial state as a linear combination of the ground state and the single-deviation state for that bidder, denoted as $\left|01\right\rangle$ or $\left|10\right\rangle$ for the two bidders in Fig. 2. Here a “single deviation” state is one that a single bidder can create, i.e., by operating on just the qubits available to that bidder. The adiabatic search splits the degeneracy, thereby giving some probability for the lowest bid to win and some probability for an infeasible allocation.

More generally, bidder $i$ uses this strategy by selecting two different operators $U_{i}^{\rm init}$ and $U_{i}$ to use for forming the initial state and during the search, respectively. These choices result in different joint operators, in Eq. (1), used in Eq. (3) and (4).

As with selecting a subspace without $\varnothing$, if many or all bidders make this choice, the initial state will have significant amplitude in eigenvectors corresponding to large eigenvalues, which produce infeasible outcomes and hence a high probability for no winner. Thus with standard adiabatic search, if everyone uses the same operator for both initialization and search, then each bidder is tempted to use a different initialization operator and bid low, gaining a chance to win with a low bid. However, if multiple bidders attempt this, the outcome will most likely be an infeasible state, with no winner.

We can address this problem by reordering the eigenvalues given by the $d(x)$ function in Eq. (6) so that any change in initial operator by a single bidder increases probability of infeasible allocation but not the probability of any feasible allocation with a bid lower than the highest bid. This is possible because bidders only have access to their own bits, so can only form initial superpositions from a limited set of basis vectors. Fig. 3 illustrates the resulting situation. We give an analysis of this approach in Sec. 5.2.

4.3 Changing Operator During Search

The distributed search of Eq. (4) has each bidder using the same operator for every step of the search. Thus bidders may gain some advantage by altering their operator during the steps of the search. Gradually changing the operator during the search amounts to a different path from initial to final Hamiltonian during the adiabatic search. Thus, provided the auctioneer uses enough steps, such changes will have at most a minor effect on the outcome probabilities unless the bidder can arrange for particularly small eigenvalue gaps among favorable states. Such arrangement is difficult, particularly since the bidder does not know the choices of other bidders and the auctioneer could treat the bits from the bidders in an arbitrary, unannounced order.

More significant changes in outcome is possible with sudden, large changes in the operator during search. Since the use of bidders operators gradually decreases during the search (i.e., $D_{xx}(f)$ given in Eq. (6) approaches the identity operator as $f$ approaches 1), the most problematic situation is for an abrupt change in operator at the beginning of the search. After such a change, the adiabatic search continues its gradual change of states, but now instead of starting in the ground state, it will instead have a linear combination of various states obtained by mapping the original basis onto the basis after the change.

5 Quantum Auction Design

In this section, we focus on mechanism design to reduce incentive issues arising from the quantum aspects of the auction. We analyze incentive issues with the Nash equilibrium (NE) concept commonly used to evaluate auctions [28]. A given set of behaviors for the bidders is an equilibrium if no single bidder can gain an advantage (i.e., higher expected payoff) by switching to another behavior. Specifically, Sec. 5.1 describes an approach to encouraging bidders to include the null set in their bids. In Sec. 5.2 we show that using the ground state eigenvector is a NE provided bidders do not change the operators during the search. Sec. 5.3 then discusses how the auctioneer can discourage bidders from changing operators. Sec. 5.4 describes how the auction can be made symmetric across the different bidders. We focus on single-item auctions in this section, but the ideas extend to quantum combinatorial auctions, as described in Sec. 6.

5.1 Checking for the Null Set

One approach to the incentive to exclude the null set, described in Sec. 4.1, is for the auctioneer to perform a second search: for the allocation with the most $\varnothing$ values. This search uses the same distributed protocol of Eq. (4) but with separate qubits and a different cost function to define $P(f)$, i.e., setting $c(x)$ to the number of non-$\varnothing$ values in the allocation $x$. Interleaving the additional search in a random order within the steps of the search for the winning bid prevents bidders from knowing which search a given step belongs to. So bidders could not consistently select different operators for the two searches.

If all bidders include $\varnothing$ in their selected subspace, this additional search returns $\left|\varnothing,\varnothing,\ldots\right\rangle$. Any bidder found not to have included $\varnothing$ could be excluded from winning the auction. At this point the auctioneer could either announce there is no winner, or restart the auction for the remaining bidders without announcing this restart. The adiabatic search has a small but nonzero probability of returning the wrong result, which would then incorrectly conclude some bidder did not include $\varnothing$. As long as the probability of such errors is smaller than the error probability of the search for the winner, these errors should not greatly affect the incentive structure of the mechanism. Alternatively, the auctioneer could use a search completing with probability one in a finite number of steps, i.e., with different choices of $D$ and $P$ in Eq. (4), the auctioneer could implement Grover’s algorithm [14] to search for the allocation $\left|\varnothing,\varnothing,\ldots\right\rangle$ in the joint subspace of the bidders. Since the auctioneer does not know the size of the subspaces selected by the bidders, the auctioneer would need to try various numbers of steps [1] before concluding $\left|\varnothing,\varnothing,\ldots\right\rangle$ is not in the selected subspaces. Unlike the adiabatic search, failure would only indicate some bidder had not included $\varnothing$, but not which one. Thus the auctioneer’s only alternative in this case is to announce the auction has no winner.

While this approach removes the immediate benefit of not including the null bid, its affect on broader strategic issues in the full auction is an open question.

5.2 The First-Price Sealed-Bid

Auction

In this section we examine the incentive structure of the auction with permuted eigenstates described in Sec. 4.2. We first review how game theory applies to auctions. We then consider the quantum auction when the search runs long enough to give successful completion almost always (“perfect search”). Finally, we consider the more realistic case of search with small, but not negligible, probability for non-optimal outcomes.

5.2.1 A Game Theory Approach to Auctions

Game theory is a common approach to evaluating auctions [20, 28]. Consider $n$ people bidding for an item, with person $i$ having value $v_{i}$ for the item. Unlike discrete choice games, such as the prisoner’s dilemma, a strategy for a private value auction involves a bidding function $b(v)$, mapping a bidder’s value to a corresponding bid. Theoretical analysis of auctions usually involves identifying a NE strategy, if any. This is a strategy for all players such that no bidder gains by changing this strategy given everyone else is using it. This focus on possible changes by a single bidder assumes bidders do not collude.

A primary issue for auction behavior is how much participants know about other bidders’ values. Such knowledge can affect the choice of bid. The most popular model of such knowledge is independent private values, where the $v_{i}$ are independently drawn from the same distribution. Each bidder knows his own value, but not the values of other bidders. However, the distribution from which values come is common knowledge, i.e., known to all bidders, each bidder knows the others know this fact, and so on. A final ingredient for the analysis is an assumption of bidders’ goals. For illustration, we use the common assumption that bidders are risk neutral expected utility maximizers, and within the context of the auction, utility is proportional to profit.

We illustrate this approach for a first-price sealed-bid auction, in which each bidder submits a single bid without seeing any of the other bids. This corresponds to the auction scenario considered in this paper. The bidder with the highest bid gets the item and pays the amount of his bid. Thus if bidder $i$ bids $b_{i}$, his profit is $v_{i}-b_{i}$ if he wins the auction and zero otherwise. To avoid possibly losing money, bidders should ensure $b_{i}\leq v_{i}$, and bids are required to be nonnegative.

In the symmetric case where bidders’ values all come from the same distribution, a NE is a bidding function $b(v)$. A bidder’s expected payoff is $(v-b(v))P(b)$ where $v$ is his value, $b$ is his bidding function and $P(b)$ is the probability of winning if he is using $b(v)$ (which is also the function others use in equilibrium). Let $F$ be the cumulative distribution of values, i.e., probability a value is at most $v$, and $n$ be the number of bidders. The equilibrium condition leads to a differential equation satisfied by $b(v)$ [28]. As a simple example, when $v$ is uniformly distributed between 0 and 1, $F(v)=v$ and the NE is $b(v)=(n-1)v/n$. Thus, in the equilibrium strategy, a bidder bids somewhat less than his value and the bid gets closer to the value when there is more competition, i.e., larger $n$.

If bidders have differing value distributions, a NE involves a set of bidding functions, $\{b_{i}(v)\}$. An auction may have multiple equilibria.

5.2.2 Behavior with Perfect Search

With perfect search and non-colluding bidders, if bidders use the same operators for every step of the search, including initialization, and pick a subspace with the null bid then the adiabatic search described in Sec. 3.2 finds the highest revenue state. We now show that the auctioneer can choose eigenvalues for the search so that bidders have no incentive to create an initial state different from the ground state. This choice corresponds to the auctioneer selecting an appropriate function $d(x)$ in Eq. (6).

Suppose bidder $i$ uses operator $U_{i}$, giving the overall operator $U$ with Eq. (1). Suppose all bidders except bidder 1 use the same operator to create the initial state as they use for the subsequent search. But bidder 1 uses two operators: $U_{1}^{\rm init}$ to form the initial state and $U_{1}$ for the search. Thus the initial state produced by bidder 1, $\psi_{1}=U_{1}^{\rm init}{\psi_{\rm init}}$, i.e., the first column of $U_{1}^{\rm init}$, is not necessarily equal to the first column of $U_{1}$ that bidder 1 uses for the subsequent search. Instead, $\psi_{1}$ may have contributions from all columns of $U_{1}$, i.e.,

[TABLE]

where $\left|i\right\rangle$ corresponds to column $i$, ranging from [math] to $2^{p}-1$, of $U_{1}$. Combining with the initial state of all other bidders, Eq. (3) gives $\Psi_{0}=\sum_{i}\alpha_{i}\left|i,0,\ldots,0\right\rangle$, instead of the initial ground state $\left|0,0,\ldots,0\right\rangle$.

Significantly, because a bidder can only operate on the $p$ qubits from the auctioneer and not on any of the qubits sent to other bidders, a single bidder can only create a limited set of “single-deviation” initial states. In the case of bidder 1, these states all have the form $\left|i,0,\ldots,0\right\rangle$. Similarly, if bidder $j$ is the one using different initial and search operators, the states all have the form $\left|\ldots,0,i,0,\ldots\right\rangle$, where only the $j^{th}$ position can be nonzero. Thus, among the $2^{np}$ basis states in the full search space, aside from the correct ground state, only $n(2^{p}-1)$ are possible states some single bidder can create when all other bidders use the same operator for initialization and search.

More generally, $k$ bidders can create superpositions of $(2^{p}-1)^{k}$ basis states in which none of them use the ground state initially, by selecting different operators for initialization and search. Thus there are

[TABLE]

$k$-deviation states that some set of $k$ bidders can create, while the other $n-k$ bidders use the ground state.

Our formulation has $n(2^{p}-1)$ feasible allocations, i.e., situations in which exactly one of the bidders has a non-$\varnothing$ bid while all other bidders have $\varnothing$. To see this, each of the $n$ bidders could have the non-$\varnothing$ bid, and this bid could have any of $2^{p}-1$ values (since the remaining value for the bidder’s bits represents $\varnothing$). The remaining $n-1$ bidders have only one choice each, i.e., $\varnothing$.

Suppose the auctioneer selects $d(x)$ such that $d(\left|0,\ldots,0\right\rangle)=0$ is the lowest eigenvalue and $d(x)$ for all single-deviation states $x$ is the largest value, with intermediate values for all other states. Provided the number of infeasible allocations is at least equal to the number of single-deviation states, a perfect search will then map every single-deviation state to an infeasible allocation, resulting in no winner for the auction. This condition amounts to

[TABLE]

The following claim shows that Eq. (9) always is true in an auction scenario.

Claim 1**.**

Eq. (9) is true for all integers $n,p\geq 1$

Proof.

When $p=1$, Eq. (9) reduces to $2^{n-1}\geq n$, which is true for all $n\geq 1$.

We prove a stronger condition for $p\geq 2$, namely there are enough infeasible states to handle up to $n-1$ bidders deviating. Using Eq. (8), this stronger condition is

[TABLE]

with the $k=1$ term in the sum corresponding to the right-hand side of Eq. (9). Writing $x\equiv 2^{p}-1$, Eq. (10) becomes $f(x,n)\equiv x^{n}-nx+1\geq 0$.

Since $p\geq 2$, we have $x\geq 3$. For this range of $x$ and for $n\geq 1$, $f(x,n)$ is monotonically increasing in both arguments. To see $f$ is monotonic for $x$, the derivative of $f(x,n)$ with respect to $x$ is $n(x^{n-1}-1)$ which is nonnegative since $n\geq 1$ and $x>1$. Similarly, the derivative with respect to $n$ is $x(x^{n-1}\ln(x)-1)$ which is at least $3(\ln(3)-1)>0$ since $n\geq 1$ and $x\geq 3$. Thus for the relevant range of $n$ and $x$, $f(x,n)\geq f(3,1)=1$ so Eq. (10) is true for all $n\geq 1$ and $p\geq 2$.

Combining these cases for $p=1$ and $p\geq 2$ establishes the claim. ∎

Using this claim, we demonstrate the permuted eigenvalue choices remove incentives to alter the initial amplitudes:

Theorem 1**.**

If (a) auctioneer chooses eigenvalues as described above, (b) $\{b^{*}_{i}(v)\}_{i=1}^{n}$ is an equilibrium for the first-price classical auction, and (c) bidders include the null set as part of their bids and use the same operator in each step in the search except, possibly, for the initial state, then the strategy of using bidding functions $\{b^{*}_{i}(v)\}_{i=1}^{n}$ and the same operator for their initial state as they use in the search is a NE for corresponding quantum auction.

Proof.

Without loss of generality, suppose only bidder 1 deviates and all the other bidders use $\{b^{*}_{i}(v)\}_{i=2}^{n}$ and the same operator for initialization and search. Then, as described above, the initial state $\Psi_{0}$ is $\sum_{i}\alpha_{i}\left|i,0,\ldots,0\right\rangle$ for some choice of amplitudes $\alpha_{i}$, with $i$ ranging from [math] to $2^{p}-1$.

A perfect adiabatic search maps each of these states to a corresponding allocation. In particular, with $d(\left|0,\ldots,0\right\rangle)$ having the smallest value of the function $d(x)$, the lowest cost allocation is produced with probability $|\alpha_{0}|^{2}$. This allocation corresponds to the highest bid winning. Moreover, each $\left|i,0,\ldots,0\right\rangle$ with $i\neq 0$ has the largest value of $d(x)$, and so, because of Eq. (9), maps to an infeasible allocation, giving no winner and hence no value to bidder 1.

Hence the expected value for bidder 1 is $|\alpha_{0}|^{2}V$ where $V$ is the value of the expected profit of the corresponding classical auction to bidder 1. Since $|\alpha_{0}|^{2}V\leq V$, bidder 1 cannot gain from such a deviation.

Furthermore, there is no gain from deviating from the bidding function $b^{*}_{1}(v)$ since it will only decrease $V$, because, by assumption, $\{b^{*}_{i}(v)\}_{i=1}^{n}$ is a NE for the corresponding classical auction.

Because of Eq. (9), this discussion applies to deviations by any single bidder, not just bidder 1. Thus, using bidding function $\{b^{*}_{i}(v)\}_{i=1}^{n}$ and using the same operator for their initial state as they use in the search is a NE.

∎

The stronger condition, Eq. (10), shows that the number of infeasible states is enough to give no winner for any choice of initial amplitudes that up to $n-1$ bidders can produce, provided $p\geq 2$. Thus if an auctioneer implements a collusion-proof classical auction with the quantum protocol and assigns infeasible states as described then the resulting quantum auction is collusion-proof up to $n-1$ bidders for initial amplitude deviations.

The choice for $d(x)$ satisfying the above requirements is not unique. As one example, let $x$ be the state index in the full search space, running from 0 to $2^{np}-1$. Consider $x$ as written as a series of $n$ base-$2^{p}$ numbers, $\left|x_{1},x_{2},\ldots,x_{n}\right\rangle$. Define

[TABLE]

where $r(x)$ is number of nonzero values among $x_{1},x_{2},...,x_{n}$. The mod operation gives all $d(x)$ values in the range 0 to $n$. For the initial ground state, $x=\left|0,\ldots,0\right\rangle$, $r(x)=0$ so $d(x)=0$, and this is the smallest possible value. Single-deviation states have exactly one of the $x_{i}$ nonzero, giving $r(x)=1$ and $d(x)=n$, the largest possible value. More generally, all $k$-deviation states have $r(x)=k$ so $d(x)=n+1-k$. This function definition gives values directly from the representation of the state $x$, so, in particular, the auctioneer can implement it without any knowledge of the subspaces selected by the bidders.

The assumption of perfect search is a sufficient but not necessary condition for the proof of Theorem 1. The necessary conditions are more complicated because we only need that every single bidder deviation maps to a linear combination of infeasible states. Thus mixing among different single-deviation states during search (e.g., due to small eigenvalue gaps among those states), or among states corresponding to two or more bidders deviating, does not affect the proof.

5.2.3 Bounded Number of Search Steps

Theorem 1 shows the quantum auction has the same NE as the classical first price auction if the search is perfect and each bidder uses the same operator for every search step of Eq. (4). Since adiabatic search, run for a finite number of steps, is not perfect we examine the effect on the NE of an imperfect search. We show that the NE for perfect search, i.e., bidding as in a classical first price auction and using the same operator initially and during the search, is an $\epsilon$-equilibrium for the auction with imperfect search. Furthermore, $\epsilon$ converges to zero as the number of search steps goes to infinity. A strategic profile is an $\epsilon$-equilibrium [24] if for every player, the gains of unilateral defecting to another strategy is at most $\epsilon$. This weaker equilibrium concept is useful in our case because determining how to exploit imperfect search is computationally difficult. Specifically, with the small eigenvalue gaps and degeneracy it is hard to know whether imperfect search benefits a particular bidder. Thus computational cost will likely outweigh the small possible gain. In this situation, an $\epsilon$-equilibrium is a useful generalization of NE.

We must prove that for any $\epsilon$ there exists an $N$ so that if the search process uses at least $N$ steps, the equilibrium of the game with a perfect search is also an $\epsilon$-equilibrium when using the actual search. To do so, we bound the possible gain from deviation based on prior knowledge of the range of possible bidder values. That is, we assume the distribution of values has a finite upper bound $\bar{v}$. In our context, one such bound is the maximum bid value expressible by the announced interpretation of each bidders qubits.

Theorem 2**.**

If the conditions of Theorem 1 are met, and assuming the possible bidder values are bounded by $\bar{v}$, for any $\epsilon>0$, there exists an $N$ so that the NE in the quantum auction with a perfect search, shown in Theorem 1, is also an $\epsilon$-equilibrium of the same auction with an imperfect search using $N$ search steps.

Proof.

Let $p_{h}$ be the probability of the highest bid wins. Let $p_{\rm inf}$ be the probability of reaching an infeasible state. Then $p_{o}=1-p_{h}-p_{\rm inf}$ is the probability of a bid other than the highest bid wins.

With the adiabatic search, with nonzero eigenvalue gaps, the probability of correctly mapping the initial to final states converges to one as the number of search steps increases. Thus for any $\delta>0$, there always exists a $N$ where $p_{o}$ is at most $\delta$.

We define an equilibrium expected payoff function for bidder $i$ with value $v$ as $\pi^{*}_{i}(v)$, when all bidders use their equilibrium bidding functions.

Without loss of generality, from the perspective of bidder $i$ with value $v$, the probability of achieving the equilibrium payoff, $\pi^{*}_{i}(v)$, if that bidder does not deviate is $1-\delta$. Thus the expected payoff of deviating is at most $\pi^{\rm deviate}_{i}(v)\leq(1-\delta)\pi^{*}_{i}(v)+\delta\bar{v}$ because (a) the most any bidder can gain is bounded by $\bar{v}$, and (b) with probability $1-\delta$ the auction either produces no profit ($p_{\rm inf}$) or is identical to a classical auction ($p_{h}$).

The expected gain $g$ from deviating is the expected payoff from deviating minus the expected payoff with no deviation, i.e., $g=\pi^{\rm deviate}_{i}(v)-\pi^{*}_{i}(v)\leq\delta(\bar{v}-\pi^{*}_{i}(v))$, which in turn is at most $\delta\bar{v}$. Thus for any choice of $\delta$, there always exists an $N$ where the maximum deviation benefit is at most $\delta\bar{v}$.

For any $\epsilon>0$, using $\delta=\epsilon/\bar{v}$ in the above discussion shows there always exists an $N$ where the deviation is at most $\epsilon$. ∎

5.3 Testing for Changed Operators During

Search

One approach to the incentive issue of changing operators during search, described in Sec. 4.3, is for the auctioneer to test the bidders by randomly inserting additional probe steps in the search.

Specifically, suppose at any step of the search the auctioneer, with some probability, decides to check a bidder by sending a new set of qubits in a known state $\left|\phi\right\rangle$, while storing the qubits for the search until a subsequent step. For the test step, the auctioneer sets $D$ or $P$ to the identity operator. The state returned by the bidder is then $U_{i}^{\prime}U_{i}^{\dagger}\left|\phi\right\rangle$ or $U_{i}^{\prime\dagger}U_{i}\left|\phi\right\rangle$, depending on which part of the search step in Eq. (4) the auctioneer is testing. Without loss of generality, we consider the former case.

Ideally, the bidder uses the same operator, so $U_{i}^{\prime}=U_{i}$ and $U_{i}^{\prime}U_{i}^{\dagger}$ is the identity. Suppose the test state is formed from some operator $V$, randomly selected by the auctioneer, $\left|\phi\right\rangle=V\left|0\right\rangle$. If $U_{i}^{\prime}U_{i}^{\dagger}$ is not the identity, the returned state has the form $\alpha\left|\phi\right\rangle+\beta\left|\phi_{\perp}\right\rangle$, where $\left|\phi_{\perp}\right\rangle$ is some state orthogonal to $\left|\phi\right\rangle$ and $|\alpha|^{2}+|\beta|^{2}=1$. The auctioneer then applies $V^{\dagger}$, giving

[TABLE]

for some value $a\neq 0$. The auctioneer then measures this state, getting [math] with probability $|\alpha|^{2}$, indicating the bidder passes the test. Otherwise, the auctioneer observes a different value, indicating the bidder changed the operator.

Hence the chance of getting caught depends on how often the auctioneer checks, and how big a change the bidder makes in the operator. Larger operator changes are more likely to be caught. This testing behavior is appropriate as small changes are not likely to have much affect on the search outcome, and instead simply act as an alternate adiabatic path from initial to final states. This technique is especially useful for risk averse bidders since then even a small chance to be caught might be enough to prevent bidders from wanting to change operators.

5.4 Assigning Eigenvalues to Subspaces

Quantum search acts on the full space of superpositions of the available qubits, i.e., in our case to all $2^{np}$ configurations of items and bids. In the auction context, bidders choose operators to restrict the search to a subspace of possible bids, namely the ones they wish to make. Conceptually, the search described above is then restricted to the subspace selected by the bidders.

The search can also be viewed as taking place in the full space of $2^{np}$ configurations. The operator $U$ appearing in the search algorithm is block diagonal (up to a permutation of the basis states), with only the block operating on the selected subspace relevant for the search outcome. This view of the search is that of the auctioneer, who has no prior knowledge of the subspace selected by each bidder. The operator $U$ is not known to any single individual: instead its implementation is distributed among the bidders, with each bidder implementing a part of the overall operator. The auctioneer chooses the eigenvalues for the initial Hamiltonian and the ordering for the qubits assigned to each bidder. These choices, which could change during the search, affect the incentive structure of the auction as described in Sec. 5.2.

This section describes how the auctioneer’s choice of $d(x)$ can give the same eigenvalues when restricted to the subspace actually selected for the search. For simplicity, we suppose each bidder uses a 2-dimensional subspace, consisting of $\varnothing$ and the desired bid for the single item. While not essential for the NE results discussed above, uniformity with respect to subspace choices means bidders are treated uniformly, so convergence of the search is independent of the order in which the auctioneer considers the bidders.

5.4.1 An Example

Consider $n=2$ bidders, each with $p=2$ bits, representing 4 values: $\varnothing$ and three bid values $1,2,3$. A set of 2-bit operators to form a uniform superposition of the form $(\left|\varnothing\right\rangle+\left|b\right\rangle)/\sqrt{2}$ where $b$ is the bid value, 1, 2 or 3, is $1/\sqrt{2}$ times

[TABLE]

which we can denote as $A_{1},A_{2},A_{3}$, respectively, with the first columns giving the uniform superposition of the three possible bid values. If the bidders select bids $b_{1},b_{2}$, respectively, the overall operator for the search is $U=A_{b_{1}}\otimes A_{b_{2}}$, used in Eq. (4) to perform each step of the search. Thus in this case there are 9 possible subspaces the two bidders can jointly select. Up to a permutation, $U$ is block diagonal with the block containing the nonzero entries of the first column, and hence all the nonzero amplitude during the search, equal to

[TABLE]

The search using $U$ in the full 4-bit space is thus equivalent to one taking place in the 2-bit subspace selected by the two bidders using this operator $V$.

The auctioneers’ choice of eigenvalues, i.e., the function $d(x)$ used in Eq. (6) should ensure the uniform superposition within the subspace defined by the two bidders has the lowest value, say 0, and all other eigenstates have larger values.

One possibility is the standard choice for the diagonal values $d(x)$ when searching in the full space of $2^{4}$ states defined by the $np=4$ bits, namely the Hamming weight of each state, i.e., the number of 1 bits in its binary representation, ranging from 0 to 4.

An alternative approach is picking $d(x)$ so eigenvalues for the four states appearing in $V$ have the same values as they would have with using the Hamming weight for a 2-bit search, ranging from 0 to 2. Doing so requires selecting the eigenvalues to match the corresponding Hamming weights for any choices the bidders make among $A_{1},A_{2},A_{3}$. In this example, each bidder has 2 qubits, so can represent 4 states, which we denote as $\left|0\right\rangle,\ldots,\left|3\right\rangle$. The states for both bidders are products of these individual states, $\left|0,0\right\rangle,\ldots,\left|3,3\right\rangle$. Examining the 9 possible cases for $U$, shows a consistent set of choices is $d(\left|x,y\right\rangle)$ equal to the number of nonzero values among $x,y$. With this $d(x)$, the adiabatic search in the subspace selected by the bidders is identical to the standard adiabatic search for two bits. This choice treats both bidders identically.

In this case we see the auctioneer can arrange the adiabatic search to operate symmetrically no matter what choice of subspace each bidder makes (i.e., no matter what value each bidder decides to bid). Thus from the point of view of the bidders, the search, in effect, takes place within the subspace of possible values defined by their bid selections.

5.4.2 General Case

For arbitrary numbers of bidders $n$ and bits $p$, we consider a single-item auction so each bidder would, ideally, pick an operator giving just two terms, with $b^{(j)}$ the bid of bidder $j$ for the single item and no bits needed to specify which item the bidder is interested in. The choice of $b^{(j)}$ corresponds to the bidder picking a 2-dimensional subspace of the $2^{p}$ possible states. The product of these subspaces gives a subspace $S$ of all $np$ qubits used in the auction. The subspace $S$ has dimension $2^{n}$ and its states $x_{S}$ can be viewed as strings of $n$ bits. More specifically, we suppose bidder $j$ implements the operator $U_{j}$ such that the rows and columns corresponding to $\varnothing$ and $b^{(j)}$ have nonzero values only for positions $\varnothing$ and $b^{(j)}$. That is, the elements of $U_{j}$ for these two values form a $2\times 2$ unitary matrix.

If the auctioneer knew the subspace $S$, the eigenvalue function $d(x)$ used in Eq. (6) could be selected to match any desired function $d_{S}(x_{S})$ of the states in $x_{S}\in S$. Without such knowledge, this is possible only for some choices for $d_{S}$.

Theorem 3**.**

Provided $d_{S}(x_{S})$ depends only on the Hamming weight of the states $x_{S}$, a single choice of $d(x)$ in the full space corresponds to $d_{S}(x_{S})$ in all possible subspaces the bidders could select that include the null set.

Proof.

Consider the full operator $U$ given by Eq. (1). For the element $U_{x,y}$, express the $np$ bits defining the states $x$ and $y$ as sequences of $p$-bit values, $x_{1},\ldots,x_{n}$ and $y_{1},\ldots,y_{n}$, respectively, with each $x_{i}$ and $y_{i}$ between 0 and $2^{p}-1$. From Eq. (1),

[TABLE]

The matrix $U$ is of size $2^{np}\times 2^{np}$ while each $U_{i}$ is of size $2^{p}\times 2^{p}$.

Consider the first column of $U$, i.e., $y=0$. $U_{x,0}$ is nonzero only for those $x$ such that all the $(U_{i})_{x_{i},0}$ are nonzero. For this to be the case, each $x_{i}$ is either 0 (corresponding to $\left|\varnothing\right\rangle$ for that bidder’s superposition) or $x_{i}=b^{(i)}$, i.e., the bid value. Similarly, for all columns with each $y_{i}$ equal to 0 or $b^{(i)}$. These values for $x,y$ are precisely the states in the selected subspace of the bidders, $S$. For these choices of $x_{i},y_{i}$, we can map 0 (i.e., $p$ bits all equal to zero) to the single bit 0, and each $b^{(i)}$ (specified by values for $p$ bits) to the single bit 1. This establishes a one-to-one mapping from states in the full space, of $np$ bits corresponding to the product of bidders’ superpositions, to states in the subspace treated as $n$-bit vectors. Thus a function $d_{S}(x_{S})$ applied to the subspace that depends on the Hamming weight, i.e., the number of 1 bits in $x_{S}$, is the same as a function on the full space depending on the number of nonzero $x_{i}$ values in $x=x_{1},\ldots,x_{n}$.

We must show that a single choice of function $d(x)$ in the full space maps to the desired $d_{S}(x_{S})$ in any choice of bidder subspaces. To see this is the case, consider any state in the full space $x=x_{1},\ldots,x_{n}$. Among these $x_{i}$, suppose $h$ are nonzero, denoted by $x_{a_{1}},\ldots,x_{a_{h}}$. This state $x$ will appear in all selected subspaces in which bidder $a_{j}$ bids $b^{(a_{j})}=x_{a_{j}}$, for $j=1,\ldots,h$, and the remaining bidders have any choice of bid. That is, $x$ appears in $(2^{p}-1)^{n-h}$ possible subspaces $S$. Since $x$ has exactly $h$ nonzero values, in each of these possible subspaces it maps to a state $x_{S}$ with exactly $h$ bits equal to 1, i.e., it has the same Hamming weight, $h$, in all possible subspaces in which it appears. Thus any choice of function $d_{S}(x_{S})$ depending only on the Hamming weight of $x_{S}$ will have the same value in all these possible subspaces. This observation allows the auctioneer to select that common value as the value for $d(x)$, consistently giving the desired eigenvalue function for any possible subspace. Since this holds for all values of $h$, the auctioneer can operate in the full space with identical search behavior no matter what subspace the bidders select. ∎

For the auctioneer to operate without knowledge of the actual subspace selected by the bidders and treat bidders identically, we need $d(x)$ to map to the same function on any subspace selected. In this case, the search proceeds exactly as if the auctioneer did know the subspace choices made by the bidders. The theorem gives one type of function for in which this is the case. In particular, Eq. (11) is an example of a function satisfying this theorem.

6 Multiple Items and Combinatorial Auction

While the paper focuses on the single item first-price sealed-bid auction, the quantum protocol can apply to multiple items by changing the interpretation of the bids, i.e., the bidding language. Such changes affect the counting of deviation and feasible states, so we must check the validity of Theorem 1.

In the single item case, each bidder uses the $p$ qubits to specify the bid amount. With multiple items, the bid must specify both the items of interest and a bid amount for the items. Various bidding languages can encode this information.

For multiple items, we divide the $p$ qubits allocated to each bidder into two parts: ${p_{\rm item}}$ bits to denote a bundle of items and ${p_{\rm price}}$ bits to denote bid value (so $p={p_{\rm item}}+{p_{\rm price}}$). Since qubits are expensive, a succinct representation of items is best. Depending on the type of auction, we have various choices with different efficiency in using bits. For example, the ${p_{\rm item}}$ item bits could indicate the item in the bid, allowing ${p_{\rm item}}$ qubits to specify up to $2^{p_{\rm item}}$ different items. Another case is multiple units of a single item, so ${p_{\rm item}}$ could specify how many units a bidder wants (with the understanding the bid is for all those units not a partial amount) so the bits could specify $2^{p_{\rm item}}$ different numbers. In the general case, bids are on arbitrary sets of items or bundles, and we represent a bundle with $m$ bits, 1 if the corresponding item is a part of the bundle and 0 otherwise, i.e., $m={p_{\rm item}}$. We focus on this general case in the remainder of the section. Allowing bids on sets of items is called a combinatorial auction [6].

With multiple items, the bid operator $\psi_{j}=U_{j}{\psi_{\rm init}}$ gives a superposition of bids of the form $\left|I_{i}^{(j)},b_{i}^{(j)}\right\rangle$ where $b_{i}^{(j)}$ is bidder $j$’s bid for a bundle of items $I_{i}^{(j)}$. In this notation, the null bid is $\left|\varnothing,b\right\rangle$, and the specified amount $b$ is irrelevant so we take it to be zero in the examples. A superposition specifies a set of distinct bids, with at most one allowed to win.

Example 4**.**

Consider a combinatorial auction with two items $X$, $Y$ and integer prices ranging from 0 to 3. With $p=4$ bits for each bidder, using 2 bits each to specify item bundles and prices, is sufficient to specify the bids. The full space for a bidder has dimension $2^{p}=16$, consisting of $4$ possible item bundle choices and $4$ price choices. Suppose a bidder places a bid

[TABLE]

i.e., a bid of 1 for item X alone, and 2 for the bundle of both items. In this case, the bidder is not interested in item $Y$ by itself. The dimension of the subspace of this bid is 3. Another example is the bid

[TABLE]

The dimension of the subspace is 4. This superposition has multiple bids on the same item $X$.

This bidding language is both expressive and compact. For instance, a superposition of bundles of items readily expresses exclusive-or preferences, where a bidder wants at most one of the bundles. It is also compact because superpositions allow the bidder to use exactly the same qubits to place no bid (i.e., $\varnothing$) and to place all the exponential number of bundles in a combinatorial auction.

An allocation, as defined in Sec. 3.1, is a list of bids, one from each bidder. With multiple items, an allocation is feasible if the item sets are pairwise disjoint. As in the single item case, we consider the allocation when all item sets are empty as infeasible. The value of a feasible allocation is the sum total of the bid values of the different bids in the allocation. The number of feasible states is $((n+1)^{m}-1)2^{n{p_{\rm price}}}$. This is because we can assign $m$ items among $n$ bidders where all items need not be allocated in $(n+1)^{m}$ ways. The factor $n+1$ allows for some items to remain unallocated. Since the allocation when all bidders place the null bid is an infeasible state, we subtract 1. Each bidder can specify $2^{p_{\rm price}}$ different prices for the bundle giving $2^{n{p_{\rm price}}}$ possible choices for $n$ bidders. Note that the number of feasible states for a single item, $m=1$, is different from that in Sec. 5.2 because here we have changed the bidding language to represent items also.

The null bid in our protocol simplifies the evaluation of allocations for combinatorial auctions. To see this, consider a protocol without the null bid. In a single item case, $F(x)$ for any allocation vector $x$ would be maximum of the bids placed by the different bidders on the item, which is fairly easy to compute. But in the case of multiple items, there could be several allocations for a vector $x$. For example suppose Alice bids on the set $\{A,B\}$ and Bob bids on $\{B,C\}$. Without the null set then both bids appear in the same state and have to be evaluated by $F(x)$. The possible allocation to the bidders are

none to either 2. 2.

$\{A,B\}$ to Alice 3. 3.

$\{B,C\}$ to Bob, and 4. 4.

$\{A,B\}$ to Alice and $\{B,C\}$ to Bob (which is infeasible)

$F(x)$ will have to compute the maximum of the values in all these states. This is computationally complex when there are many items. By contrast, the bidding language with the null bid avoids this combinatorial evaluation within the search function $F(x)$.

As in the case of single item auctions, we restrict ourselves to a one-shot sealed bid classical combinatorial auction that we implement in a quantum setting. The total number of states is $2^{pn}$ and the total number of single bidder deviations states is $n(2^{p}-1)$. These expressions are the same as the single item case. The condition for all single-deviation states to be mapped to infeasible allocations, resulting in no winner, is

[TABLE]

This condition holds for cases relevant for auctions as seen in the following claim.

Claim 2**.**

Eq. (13) is true for all integers $m,{p_{\rm price}}\geq 1$ and $n\geq 2$.

Proof.

Recall $p=m+{p_{\rm price}}$. We prove a stronger condition for integers $n,m\geq 2$, i.e., there exists enough infeasible states to handle joint deviations up to $n-1$ bidders. The number of $k$-bidder deviation states is the same as the single-item case, i.e., Eq. (8). Thus this stronger condition, with the same right-hand side as Eq. (10), is

[TABLE]

Hence Eq. (14) is true if

[TABLE]

Since $2^{-{p_{\rm price}}}\leq 1$, Eq. (14) is true if

[TABLE]

which is true if

[TABLE]

Let $f(m)\equiv(2^{m}-1)^{\frac{1}{m}}$ and $g(n)\equiv(n+1)^{\frac{1}{n}}$. We establish the required inequality, $f(m)\geq g(n)$, by showing $f(m)$ is increasing in $m$ when $m\geq 2$, $g(n)$ is decreasing in $n$ when $n\geq 2$ and noting $f(2)=g(2)=\sqrt{3}$.

Taking the derivative of $f(m)$ with respect to $m$, we get,

[TABLE]

This is positive if and only if

[TABLE]

This is true because $\log_{2}(2^{m}-1)<\log_{2}(2^{m})=m$ and hence both fractions in the expression are greater than 1. Thus, $f(m)$ is increasing for all $m\geq 2$.

Taking derivative of $g(n)$ with respect to $n$, we get,

[TABLE]

This is negative if and only if

[TABLE]

This is true for $n\geq$ 2. Thus $g(n)$ is decreasing in $n$ for $n\geq 2$.

Thus we have shown that Eq. (13) is true for $n,m\geq 2$. It can be easily checked that Eq. (13), is not true for $n=1$ and true when $n=2$ and $m=1$. ∎

Thus, if a classical combinatorial auction has a NE then the corresponding quantum auction protocol also has a NE with respect to initial state deviations. Also there is an $\epsilon$-equilibrium of the same auction with an imperfect search using $N$ search steps. Moreover, the stronger condition of Eq. (14) shows that in auctions with at least two bidders ($n>1$), there are enough infeasible states to give no winner for any deviation of initial amplitudes that up to $n-1$ can produce. Thus no groups, up to size $n-1$, can collude to benefit from initial amplitude deviations in the quantum auction.

7 Applications of Quantum Auctions

Two properties of quantum information may provide benefits to auctioneers and bidders: the ability to compactly express complicated combinations of preferences via superpositions and entanglement and the destruction of the quantum state upon measurement. This section describes some economic scenarios that could benefit from these properties.

As one economic application, quantum auctions provide a natural way to solve the allocative externality problem [18, 25]. In this situation, a bidder’s value for an item depends on the items received by other bidders. For example, consider companies bidding on a big government project requiring multiple companies to work on different parts. Allocative externality refers to the issue that the costs for a company which wins a contract for one part depends on which other companies win other parts. So company A may be willing to bid more aggressively if it knows that company B will work on related parts. Multiple simultaneous auctions for separate parts will not handle these interdependencies and thus will be inefficient. One possible solution is to let companies form partnership bids. That is joint bids that are accepted together or not at all. Quantum information processing allows for a natural way of forming partnership bids via entanglement. With the protocol described in Sec. 6, multiple bids can be entangled so they will either all be accepted together or none will be. Furthermore, quantum auctions may provide more flexibility with respect to information privacy of partnership bids than classical methods.

Specifically, with multiple items, groups of bidders could select joint operators on their combined qubits, allowing them to express joint constraints (e.g., where they either all win their specified items or none of them do) without any of the other bidders or auctioneer knowing this choice. The bidders do so by creating an entangled state instead of the factored form for their qubits. Thus employing quantum entanglement provides bidders a natural way for expressing any allocative externality. This possibility shows bidding languages based on qubits are highly expressive and compact because bidders can use the same bits to express their individual bids and joint bids via entanglement.

Example 5**.**

Alice and Bob could jointly form the state

[TABLE]

to represent the bidders willing to pay $b_{A}$ and $b_{B}$ for items $I_{A}$ and $I_{B}$, or to pay $b_{C}$ and $b_{D}$ for items $I_{C}$ and $I_{D}$, but they are not willing to buy other combinations, such as $I_{A}$ for Alice and $I_{D}$ for Bob.

In this scenario, a direct representation of bids, i.e., without a null bid, would not guarantee the joint preferences are satisfied for all entangled bidders or none of them. That is, without null bids, the superposition could not express the joint preference through entanglement.

A group of $k$ bidders operating jointly on their qubits to form entangled bids could also produce initial amplitudes involving up to $k$-bidder deviation states. However the discussion with Eq. (14) on multiple item auctions shows our protocol can handle all deviation states a group of up to $n-1$ bidders can produce, i.e., by mapping them to infeasible outcomes. Thus the additional expressivity used for joint bids does not introduce additional opportunities for collusion to change the outcomes via initial amplitude selection.

A second economic application for quantum auctions arises from their privacy guarantee for losing bids. This property is economically useful when bidders have incentives to hide information. An example is a scenario in which companies are bidding for government contracts year after year. A company’s bid usually contains information about its cost structure. If there is reasonable expectation that the losing bids will be revealed, a company may want to bid less aggressively to reduce the amount of information passed to its competition for use in future auctions. This will lead to a less efficient auction than if bidders reveal their true values. In this situation, a privacy guarantee on the losing bids enables bidders to bid with less inhibition. More generally, this privacy issue is only relevant when there are additional interactions between these companies after the auction is concluded, such as future auctions or negotiations where participants may be at a disadvantage if their values are known to others.

This strong privacy property is unique to quantum information processing. Privacy can be enforced via cryptographic methods for multi-player computation [13], and in an auction can keep losing bids secret [22]. However, the information on the bids, and the key to decrypt them, remains after the auction completes. People who have access to the key may be legally compelled to reveal the information or choose to sell it. So while cryptography can be secured computationally, it cannot guarantee the integrity of the person(s) who have the means to decrypt the information. On the other hand, the quantum method destroys losing bids during the search for the winning one and it is physically impossible to reconstruct the bids after the auction process. Similarly, some of the other properties of quantum auctions, such as correlations for partnership bids, can be provided classically [19]. Moreover, quantum mechanisms are readily simulated classically [27] (as long as they involve at most 20 to 30 qubits). However, these classical approaches lack the information security of quantum states. More study is needed to determine scenarios where the privacy property of the quantum protocol is significant.

8 Discussion

This paper describes a quantum protocol for auctions, gives a game theory analysis of some strategic issues the protocol raises and suggests economic scenarios that could benefit from these auctions. These include the privacy of bids and the possibility of addressing allocative externalities. The search used in our protocol can use arbitrary criteria for evaluating allocations, thereby implementing other types of auctions with quantum states. Thus while we focus our attention on the first-price sealed-bid auction, the protocol is more general: it can implement other pricing and allocation rules, as well as multiple-unit-multiple-item auctions with combinatorial bids. For example we can use this protocol in a multiple stage, iterative auction. In fact, the protocol supports general bidding languages.

Encoding bids in quantum states raises new game theory issues because the bidders’ strategic choices include specifying amplitudes in the quantum states. The auction is not only probabilistic, but the winning probability is not just a function of the amount bid. Instead a bidder can change the probability of winning by altering the amplitudes of the quantum states encoding his bid. For example, in the context of the first-price sealed-bid auction, the auction does not guarantee the allocation of the item to the highest bidder.

We show that the correct design of the protocol can solve a specific version of this incentive problem. The salient design feature is an incentive compatible mechanism so that bidders do not want to cheat, as opposed to an algorithmic secure protocol that prevents bidders from cheating. Thus, our design is an example of a quantum algorithm, in this case adiabatic search, tuned to improve incentive issues rather than the usual focus in quantum information processing on computation or security properties of algorithms.

In addition, we show that the Nash equilibrium of the corresponding classical first-price sealed-bid auction is an $\epsilon$-equilibrium of the quantum auction and that $\epsilon$ converges to zero when the quantum search associated with the protocol uses an increasing number of steps, under the conditions listed in Theorem 1. This result is with respect to changes in the initial state of the search. It remains to be seen whether other bidder strategies give some unilateral benefit, requiring further adjustments to the auction design.

There are multiple directions for future work. First, we plan a series of human subject experiments on whether people can indeed bid effectively in the simple quantum auction scenario described in this paper. As with previous experiments with a quantum public goods mechanism [2], such experiments are useful tests of the applicability of game theory in practice, and also suggest useful training and decision support tools. In particular, people’s behavior in a quantum auction could differ from game theory predictions that people select a Nash equilibrium based on idealized assumptions of human rationality and full ability to evaluate consequences of strategic choices with uncertainty.

Second, we plan to extend studies of quantum auctions to more complicated economic scenarios, such as one with allocative externality. Our analysis considers a single auction. An interesting extension is to a series of auctions for similar items. If auctions are repeated, the game theory analysis is more complicated [28]. In particular, privacy concerns become more significant since information revealed by a bidder’s behavior in one auction may benefit other bidders in later auctions.

The quantum auction destroys all information about the losing bids. As a result, it is not possible to conduct after-the-fact audits to verify that the auction has been conducted correctly. Is there a way to modify the mechanism to enable audits while preserving some of the privacy guarantees? Security is another interesting issue. For example, there may be third parties, aside from the auctioneer and bidders who are interested in intercepting and changing bits in transit. Auctioneers may have incentives to detect a bidder’s bid or skew auction results. The question is whether we can build security around the protocol to prevent or at least detect these types of attacks.

Similarly, many economics issues surrounding the protocol remain to be resolved. For example, people behave as if they are risk averse in auction situations [5, 4] which can change the predictions of game theory. Another issue arises from the possibility of multiple Nash equilibria. We have only shown that the desirable outcome is an equilibrium. The quantum protocol can also have other equilibria. Since the Nash equilibrium concept alone does not indicate how people select one equilibrium over another, additional study is needed to determine when the desirable outcome is likely to occur.

Our protocol makes only limited use of quantum states, in particular encoding bids in the subspace selected by the bidders but not using the amplitudes separately. Thus it would be interesting to examine extensions to the protocol exploiting the wider range of options for bidders. For example, a protocol might use amplitudes of superpositions to indicate a bidder’s probabilistic preferences, say, as in constructing a portfolio of items with various expected values and risks. Such portfolios could be useful if bidders have some uncertainty in their values (e.g., in bidding for oil field exploration rights) rather than the standard private value framework considered in this paper, where bidders know their own values for the items. With uncertain values, probabilistic bids could allow bidders to match their risk preferences along with their value estimates within the auction process.

As a final note, the number of qubits necessary to conduct an auction is small compared to the requirement of complex computations such as factoring. For example, if each bidder uses 7 bits (corresponding to $2^{7}$ or about 100 bid values) and there are 3 bidders, about 25 qubits are needed, considerably less than thousands needed for factoring interesting-sized numbers. Thus with the advancement of quantum information processing technologies, economics mechanisms could be early feasible applications.

Acknowledgments

We have benefited from discussions with Raymond Beausoleil, Saikat Guha, Philip Kuekes, Andrew Landahl and Tim Spiller. This work was supported by DARPA funding via the Army Research Office contract #W911NF0530002 to Dr. Beausoleil. This paper does not necessarily reflect the position or the policy of the Government funding agencies, and no official endorsement of the views contained herein by the funding agencies should be inferred.