Most well-studied problems in computational social choice [Brandt et al., 2016] deal with combining individual preferences over alternatives – often expressed as rankings – into a collective choice [Procaccia et al., 2012, Skowron et al., 2016, Caragiannis et al., 2017]. More often than not, the mechanisms employed for this aggregation task are ordinal, i.e., they do not use the intensities of the preferences of the individuals, and non-truthful, which is justified by several impossibility theorems [Gibbard, 1973, Satterthwaite, 1975, Gibbard, 1977]. On the other hand, the class of truthful cardinal mechanisms has been shown to be much richer [Freixas, 1984, Barbera et al., 1998, Feige and Tennenholtz, 2010] and the additional information provided by the numerical values can notably increase the well-being of society [Guo and Conitzer, 2010, Filos-Ratsikas and Miltersen, 2014, Cheng, 2016].
At the same time, truthful mechanisms with money are pretty well-understood and the welfare-maximizing mechanisms for a wide class of problems are known [Nisan et al., 2007]. However, in a rich set of hybrid social choice problems, where monetary transfers are possible for some individuals and not for others, designing truthful, cardinal mechanisms is more challenging – one needs to combine elements of mechanism design with money and social choice.
We provide a few examples of such hybrid social choice scenarios. Government agencies routinely sell public assets such as spectrum, land, or government securities, by transferring their ownership (or usage rights). As such transfers may have huge impact to citizens, the decision about the new ownership is not simply the outcome of some competitive process among the potential buyers (e.g., through an auction), but it usually also involves experts from the citizen community who provide advice regarding the societal impact of each potential ownership transfer [Janssen, 2004, PTI, 2018]. In contrast to each potential buyer who faces a value-for-money trade-off, the experts care only about societal value; their compensation is unrelated to the ownership decision and instead depends on their reputation and experience only. The government needs both parties for a successful transfer of the public assets and a reasonable goal would be to maximize the social welfare, which aggregates the values of buyers and experts for the ownership transfer.
A very similar situation occurs for private ownership transfers. Mergers and acquisitions play a central role in the competition among private players in a market, and the rules or the policies that dictate the mergers are often up for debate.111EU data show that more than 6500 mergers have taken place in the EU since 1990, and strict rules are in effect for mergers [European Commission, 2018]. There is ample evidence to support the fact that the transfer of ownership of an organization has a significant impact on the economy of the employees and the customers [Auerbach, 2008, Hitt et al., 2001]. The current owner or the administration can employ industry experts for their opinion on the transfer and ask the potential buyers to quote their values. Similarly to the previous example, the administration takes into account the input of both parties and social welfare maximization among them is a reasonable goal. Furthermore, in the organization of sporting events, the bids of the potential hosts are taken into consideration along with the recommendations of a respective sports’ administrative body (e.g., IOC for Olympic Games, FIFA for the World Cup, FIA for Formula One, etc.).
Motivated by these examples, we consider a setting where the bidders offer monetary compensations (e.g. they “buy-into” a new company or a government asset), but the experts (e.g., the citizen representatives or the administrative body) do not. The objective is to achieve the decision that maximizes the social welfare, which includes the cardinal values of both the expert and the bidders. This is a hybrid social choice setting, which blends together classical social choice and classical mechanism design with money, but is distinct from both of them, thereby rendering celebrated solutions like the VCG mechanism [Vickrey, 1961, Clarke, 1971, Groves, 1973] insufficient.
1.1 Our contribution and techniques
More concretely, we study the very simple but fundamental setting of two competing bidders and and a single expert with cardinal preferences over the three options of selling to bidder , selling to bidder , or not selling at all (in which case the ownership transfer does not take place). A mechanism takes as input the bids and the expert’s preferences and decides one of the three options as outcome. In general, mechanisms are randomized
. For a given input, they select the outcome using a probability distribution (orlottery) over the three options.
We consider mechanisms that can be “truthfully implemented” as follows. First, the outcome of the mechanism is complemented with payments that are imposed to the bidders. Then, the lottery and the payments should be such that the expert is incentivized to report her true preferences in order to maximize her (expected) value for the outcome and the bidders are incentivized to report their true values as bids in order to maximize their utility, i.e., their expected value for the outcome minus their payment to the mechanism. In the following, we refer to mechanisms with such implementations as truthful mechanisms.
Interestingly, the theory of mechanism design allows us to abstract away from payments and view truthful mechanisms simply as lotteries. Well-known characterizations for single-parameter mechanism design with money from the literature, as well as new characterizations that we prove here for lotteries that guarantee truthfulness from the expert’s side, are the main tools we use in order to constrain the design space of truthful mechanisms in our setting.
Additional informational restrictions can further divide truthful mechanisms into the following classes: (i) ordinal mechanisms, which ignore the exact bids and the expert’s preference values and instead take into account only their relative order, (ii) bid-independent mechanisms, which ignore the bids and base their decision solely on the expert’s cardinal preferences, (iii) expert-independent mechanisms, which ignore the expert’s preferences and base their decision solely on the bids, and (iv) general truthful mechanisms, which may take both the bids and the expert’s preference values into account.
We measure the quality of truthful mechanisms in terms of the social welfare, i.e., the aggregate value of the bidders and the expert for the outcome. Unfortunately, our setting does not allow for a truthful implementation of the social welfare-maximizing outcome. So, we resort to near-optimal truthful mechanisms and use the notion of the approximation ratio to measure their quality.
For the classes of ordinal, bid-independent, and expert-independent mechanisms, we prove lower bounds on the approximation ratio of truthful mechanisms in the class and identify the best possible among them, with approximation ratios of , , and , respectively. Furthermore, by slightly enhancing expert-independent mechanisms and allowing them to utilize a single bit of information about the expert’s preferences, we define a template for the design of new truthful mechanisms. The template defines “always-sell” mechanisms that select either bidder A or bidder B as the outcome. We present two mechanisms that follow our template, one deterministic and one randomized, with approximation ratios and , respectively. The former is best-possible among all deterministic truthful mechanisms. The latter is best-possible among all always-sell truthful mechanisms. We also present an unconditional lower bound of on the approximation ratio of any truthful mechanism. These results are summarized in Table 1.
|class of mechanisms||approximation ratio||reference|
|ordinal||mechanisms EOM, BOM (Theorem 3)|
|best possible (Theorem 4)|
|bid-independent||mechanism BIM (Theorem 7)|
|best possible (Theorem 8)|
|expert-independent||mechanism EIM (Theorem 11)|
|best possible (Theorem 11)|
|our template||randomized mechanism (Theorem 14)|
|best possible, always-sell (Theorem 15)|
|deterministic mechanism (Theorem 14)|
|best possible, deterministic (Theorem 17)|
|all mechanisms||unconditional lower bound (Theorem 16)|
Both our positive and negative results have been possible by narrowing the design space using the truthfulness characterizations, the particular structure in each class of mechanisms, as well as the goal of low approximation ratio. In most cases, the design of new mechanisms turns out to be as simple as drawing a curve in a restricted area of a -dimensional plot (e.g., see Figures 2 and 3).
1.2 Related work
Our setting can be viewed as an instance of approximate mechanism design, with [Nisan and Ronen, 2001] and without money [Procaccia and Tennenholtz, 2013], which was proposed for problems where the goal is to optimize an objective under the strict truthfulness requirement. A result that will be very useful to our analysis is Myerson’s characterization [Myerson, 1981] for single-parameter domains, which provides necessary and sufficient conditions for (deterministic or randomized) mechanisms (with money) to be truthful. On one hand, it allows us to abstract away from the payment functions (which are uniquely determined given the selection probabilities) on the bidders’ side. On the other hand, similar arguments based on the same characterization enable us to reason about the structure of truthful mechanisms (without money) on the expert’s side as well.
For settings with money, the well-known Vickrey-Clarke-Groves (VCG) mechanism [Vickrey, 1961, Clarke, 1971, Groves, 1973] is deterministic, truthful, and maximizes the social welfare in many settings of interest. However, as we pointed out in the discussion above, in our hybrid mechanism design setting, one needs to take the values of the expert into account as well, and therefore VCG is no longer truthful nor optimal. On the expert’s side, truthful mechanisms can be though of as truthful voting rules; any positive results for deterministic such rules are impaired by the celebrated Gibbard-Satterthwaitte impossibility theorem [Gibbard, 1973, Satterthwaite, 1975] which limits this class to only dictatorial mechanisms.
In contrast, the class of randomized truthful voting rules is much richer and includes many reasonable truthful rules that are not dictatorial. In fact, Gibbard  characterized the class of all such ordinal rules; a general characterization of all cardinal voting rules is still elusive. To this end, a notable amount of work in the classical economics literature as well as in computer science has been devoted towards designing such rules and proving structural properties for restricted classes. Gibbard  provided a similar characterization to his 1977 result, which however only holds for discrete strategy spaces, and later Hylland 222Quite remarkably, this paper is unpublished - the result was revisited in Dutta et al. . proved that the class of truthful rules that are Pareto-efficient reduces to random dictatorships. Freixas , used the differential approach to mechanism design proposed by Laffont and Maskin  to design a class of truthful mechanisms which actually characterizes the class of twice differentiable mechanisms over subintervals of the valuation space; the best-possible truthful bid-independent mechanism that we propose in this paper can be seen as a mechanism in this class. Barbera et al.  showed that there are many interesting truthful mechanisms that do not fall into the classes considered by Freixas . In the computer science literature, Feige and Tennenholtz  designed a class of one-voter cardinal truthful mechanisms, where the election probabilities are given by certain polynomials.
Social welfare maximization without payments has been studied in a plethora of related papers in the computer science literature, in general social choice settings [Filos-Ratsikas and Miltersen, 2014, Bhaskar et al., 2018], as well as in restricted domains, such as matching and allocation problems [Filos-Ratsikas et al., 2014, Cheng, 2016, Guo and Conitzer, 2010]. Similarly to what we do here, Filos-Ratsikas and Miltersen  make use of one-voter cardinal truthful mechanisms to achieve improved welfare guarantees. However, the presence of the bidders significantly differentiates our setting from theirs (as well as the other related works), since we have to consider both sides in the design and analysis of mechanisms. Another relevant notion is that of the distortion of (non-truthful) mechanisms which operate under limited information (typically ordinal mechanisms) [Caragiannis and Procaccia, 2011, Boutilier et al., 2015, Anshelevich et al., 2015, Caragiannis et al., 2017, Caragiannis et al., 2016]. While the lack of information has also been a restrictive factor for some of our results (in conjunction with truthfulness), we are mainly interested in cardinal mechanisms for which truthfulness is the limiting constraint.
The rest of the paper is structured as follows. We begin with preliminary definitions, notation and examples in Section 2. Then, Sections 3, 4, and 5 are devoted to ordinal, bid-independent and expert-independent mechanisms, respectively. Our template and the corresponding best possible deterministic and randomized mechanisms are presented in Section 6, while our unconditional lower bounds are presented in Section 7. We conclude with a discussion of possible extensions and open problems in Section 8.
Our setting consists of two agents and who compete for an item (to be thought of as an abstraction of a merger or acquisition) and an expert . The agents have valuations and denoting the amount of money that they would be willing to spend for the item, and the expert has a valuation function over the following three options: agent is selected to get the item, or agent is selected, or no agent is selected to get the item. We use to denote this last option; hence, . We use to denote an agent profile and let be the set of all such profiles. Similarly, we use to denote an expert profile and let be the set of all such profiles. The domain of our setting is . From now on, we use the term profile to refer to elements of .
A mechanism takes as input a profile and decides, according to a probability distribution (or lottery) the pair consisting of an option
and a vectorindicating the payments that are imposed to the agents. The execution of the mechanism yields a utility to the expert and the agents. Given an outcome of the mechanism, the utility of the expert is ; the utility of agent is if and otherwise.
The expert and the agents are asked to submit an expert’s report and bids to the mechanism and may have incentives to misreport their true values in order to maximize their utility. We are interested in mechanisms that do not allow for such strategic manipulations. We say that a mechanism is truthful for agent if for any agent value and any profile ,
where the expectation is taken with respect to the lottery . This means that bidding her true value is a utility-maximizing strategy for the agent, no matter what the other agent and the expert’s report are. Mechanism is truthful for the expert if for any expert profile and any profile ,
Again, this means that reporting her true valuation profile is a utility-maximizing strategy for the expert, no matter what the agents bid. A mechanism is truthful if it is truthful for the agents and truthful for the expert.
Our goal is to design truthful mechanisms that achieve high social welfare, which is the total value of the agents and the expert for the outcome.333Note that this is the usual definition, as we assume the collected payments from the agents to be transferred to the company after the merger. That is, the seller here is not the expert, but rather the company that the agents “buy into” and the expert is assumed to be an independent entity that is not concerned with the price of the sale. For a meaningful definition of the social welfare that weighs equally the expert’s and the agents’ valuations, we adopt a canonical representation of profiles. The expert has normalized von Neumann-Morgenstern valuations, i.e., she has valuations of and for two of the options and a value in for the third one. The agent values are normalized in the definition of the social welfare, which is defined as
We measure the quality of a truthful mechanism by its approximation ratio, which (by abusing notation a bit and interpreting as the option decided by the mechanism) is defined as
Of course, low values of , as close as possible to , are most desirable.
2.1 An alternative view of profiles
In order to simplify the exposition in the following sections, we devote some space here to introduce two alternative ways of representing profiles, the expert’s view and the agents’ view. Without essentially restricting the space of mechanisms that can achieve good approximation ratios according to our definition of the social welfare, we focus on mechanisms that base their decisions on the normalized bid values, i.e., on the quantities and . It will be convenient to use the following two alternative ways
to represent the profiles, which is a different way of representing a profile . These representations are the expert’s and agents’ view of the profile, respectively. Each column corresponds to an option. According to the expert’s view at the left, the columns are ordered in terms of the expert’s values, which appear in the first row. The quantities , , and hold the normalized agent bids for the corresponding option and for option . Essentially, is the value that the expert’s favourite option has, which can be equal to if it corresponds to the value of the agent with the highest value (high-bidder), equal to some value if it corresponds to the value of the agent with the lowest value (low-bidder), or if it corresponds to the no-sale option . Similarly, and are the values that expert’s second and third favourite options have, respectively. According to the agents’ view at the right, the columns are ordered in terms of the bids, which appear in the second row. The quantities , , and now hold the expert valuations for the corresponding options. Essentially, according to this view, is the value that the expert has for the high-bidder, is the value of the expert for the low-bidder, and is the value that the expert has for the no-sale option. All of them can take values in the interval .
These representations yield a crisper way to argue about truthfulness for the expert and the agents in our main results. Specifically, in Section 4, we will study bid-independent mechanisms, and therefore it makes sense to use the expert’s view of profiles, whereas in Section 5, it will be easier to argue about our expert-independent mechanisms based on the agents’ view instead. The agents’ view will also be used in Section 6, where, the mechanisms we present use the expert’s opinion only to appropriately partition the input profiles into categories, and it is therefore easier to argue about their properties using the agent’s view.
Similarly to the expert’s and the agents’ view described above, we use two different representations of the lottery , depending on whether we represent profiles according to the expert’s or the agents’ view. From the expert’s viewpoint, is represented by three functions , , and , which correspond to the probability of selecting the first, second, and third favourite option of the expert, respectively. Similarly, from the agents’ viewpoint, is represented by three functions , , and , which correspond to the probability of selecting the high-bidder, the low-bidder, or option .
Let us present an example. Consider a profile with expert valuations for option , for option , and for option and normalized bids of and from bidders and , respectively. Consider a lottery which, for the particular profile, uses probabilities , , and for options , , and , respectively. The expert’s and agents’ views of the profile are
respectively. The functions , and are defined over the -tuple of arguments which compactly represents the expert’s view and take values , , and , respectively. Similarly, functions , , and are defined over the -tuple of arguments which compactly represents the agents’ view and take values , , and , respectively.
In order to handle situations of equal values (e.g., equal bids), we adopt the convention to resolve ties using the fixed priority in order to identify the high- and low-bidder as well as the highest and/or lowest expert valuation. For example, if the expert has valuations of for options and , we interpret this as option being her most favourite one. Similarly, agent is always the high-bidder and agent is the low-bidder when their bids are equal. This is used in the definition of our mechanisms only; lower bound arguments do not depend on such assumptions in order to be as general as possible.
2.2 Reasoning about truthfulness
Let us now explain the truthfulness requirements having these profile representations in mind. There are two different kinds of possible misreports by the expert.
First, she can attempt to make a “level change in the reported valuation” (or ECh, for short) by changing her second highest valuation without affecting the order of her valuations for the options.
Second, she can attempt to make a “reported valuation swap” (or ESw, for short), i.e., change the order of her valuations for the options as well as the particular values.
For example, the profile
is the result of a reported valuation swap by the expert who changes her valuations from (that she has in Example 1) to for the three options .
Similarly, there are also two different kinds of possible misreports by each agent. In particular, the agent can attempt to make
a “level change in the reported bid” (BCh) by changing her bid without affecting the order of bids or
a “bid swap” (BSw) by changing both the bid order and the corresponding values.
For example, the profile
is the result of a bid swap deviation by the low-bidder, who increases her bid in the profile of Example 1 to a new bid that is four times the bid of the other agent.
A truthful mechanism never incentivizes (i.e., it is incentive compatible with respect to) such misreports. We use the terms ECh-IC, ESw-IC, BCh-IC, and BSw-IC to refer to incentive compatibility with respect to the misreporting attempts mentioned above. A truthful mechanism, therefore, satisfies all these IC conditions. Before we proceed, we provide a few examples of truthful mechanisms.
Example 2 (A bid-independent ordinal mechanism).
Consider the following mechanism that ignores the bids reported by the agents. With probability , output the expert’s most preferred option and with probability , output the expert’s second most preferred option. Adopting the expert’s view and the corresponding representation of the lottery , the mechanism can be written as:
The mechanism can be seen to be truthful by the fact that (a) ignores the bids of the agents and (b) it always assigns higher probability to the most-preferred outcome for the expert and probability to the least-preferred outcome. Note that using the terminology above, any ordinal mechanism is ECh by construction, since changing the level in the reported valuation does not change the outcome.
Example 3 (A bid-independent mechanism which is not ordinal).
Consider the following mechanism that ignores the bids reported by the agents. Consider the expert’s view and the corresponding representation of the lottery and note that is the value of the expert for her second most-preferred outcome. Let be given by:
Note that the mechanism uses the cardinal information of the expert’s report and therefore it is not ordinal. This mechanism has been referred to in the literature as the quadratic lottery and has been proven to be truthful (Feige and Tennenholtz , Freixas ).
Example 4 (A deterministic expert-independent mechanism).
Consider the following mechanism that ignores the expert’s values for the different outcomes. Among the two agents, output the agent with the highest bid (breaking ties arbitrarily) and charge this agent a payment equal to the bid of the other agent. Charge the other agent a payment of . In terms of the agents’ view, the outcome of the mechanism can be written as:
This mechanism is the well-known second-price auction (Vickrey ), which is known (and easily seen) to be truthful.
It is not hard to see that none of the mechanisms presented in Examples 2, 3 and 4 can achieve very strong approximation ratios. As we will see in Section 3, the mechanism of Example 2 is actually the best possible among the restricted class of ordinal mechanisms; later on, the use of cardinal information will allow us to decisively outperform it. We also note that while the second-price auction in Example 4 is welfare-optimal for the agents, which is a well-known fact, it can only provide a -approximation when it comes to our objective of the combined welfare of the agents and the expert.
We continue with important conditions that are necessary and sufficient for BCh-IC and ECh-IC. The next lemma is essentially the well-known characterization of [Myerson, 1981] for single-parameter domains.
Lemma 1 (Myerson, 1981).
A mechanism is BCh-IC if and only if the functions and are non-increasing and non-decreasing in terms of their first argument, respectively.
The correct interpretation of the lemma is that, as long as the output of a mechanism satisfies the monotonicity condition above, one can always find payments for the agents that will make the mechanism BCh-IC. In fact, when the mechanisms are required to charge a payment of zero to an agent with a zero bid, then these payments are uniquely defined, and are given by the following formula
where is the probability that agent will be selected as the outcome, is the payment function, is the bid of agent and is the bid of the other agent. Therefore, we can avoid referring to the payment function when designing our mechanisms, as we can choose the above payment function, provided that the outcome probabilities satisfy the monotonicity conditions of Lemma 1. On the other hand, our lower bounds apply to all mechanisms, regardless of the payment function, as they only use the monotonicity condition.
Next, we provide a similar proof to that of Myerson  for characterizing ECh-IC in our setting.
A mechanism is ECh-IC if and only if the function is non-decreasing in terms of its first argument and the function satisfies
for every -tuple representing a profile as seen by the expert.
As a corollary, functions and are non-increasing in terms of the first argument.
To shorten notation, we use as an abbreviation of the information in the second row of a profile in expert’s view and as an abbreviation of . Also, we drop from notation (hence, is used instead of ) since it is clear from context. Due to ECh-IC, the expert has no incentive to attempt a level change of her valuation for her second favourite option from to . This means that
Similarly, she has no incentive to attempt a level change of her valuation for her second favourite option from to . This means that
which implies that is non-decreasing in terms of its first argument.
This means that function is convex with respect to its first argument and has as its subgradient [Rockafellar, 2015]. Hence, from the standard results of convex analysis we get
which is equivalent to (1).
To verify that the conditions of the lemma are also sufficient for a mechanism to be ECh-IC, assume that is non-decreasing, satisfies equation (1), and the expert has incentive to make a level change of her valuation for her second favourite option from to . This means that
which, by replacing , is equivalent to
and the assumption that is non-decreasing. Hence, the expert must not have any incentive to make such level changes. ∎
Before we conclude the section, we remark here that while Lemma 2 will be fundamental for our proofs, it does not provide a characterization of all truthful one-voter mechanisms in the unrestricted social choice setting (such mechanisms are referred to as unilateral in the literature). The reason is that (a) it applies only to changes in the intensity of the preferences and not swaps in the ordering of alternatives and (b) it only provides conditions for three alternatives, as opposed to many alternatives in the general setting.
3 Ordinal mechanisms
We will consider several classes of truthful mechanisms depending on the level of information that they use. Let us warm up with some easy results on ordinal mechanisms, i.e. mechanisms which do not use the exact values of the expert’s report and the bids but only their relative order. It turns out that the best possible approximation ratio of such mechanisms is and is achieved by two symmetric mechanisms, one depending only on the ordinal information provided by the expert (expert-ordinal), while the other depends only on the relation between the bids (bid-ordinal).
The expert-ordinal mechanism EOM selects the expert’s favourite and second best option with probabilities and , respectively. Symmetrically, the bid-ordinal mechanism BOM selects the high- and low-bidder with probabilities and , respectively.
Mechanisms EOM and BOM are truthful mechanisms that have approximation ratio at most .
EOM is clearly truthful for the agents since it ignores the bids. It is also clearly truthful for the expert since the probabilities of selecting the options follow the order of the expert’s valuations for them. BOM is clearly truthful for the expert (since her input is ignored); truthfulness for the bidders follows by observing that the probability of selecting an agent is non-decreasing in terms of her bid.
We prove the approximation ratio for mechanism BOM only; the proof for EOM is completely symmetric. Consider the profile in agents’ view. We distinguish between two cases. If , the optimal welfare is and the approximation ratio is
since . If , the optimal welfare is and the approximation ratio is
since . ∎
We conclude the section by showing that both EOM and BOM are optimal among all ordinal mechanisms.
The approximation ratio of any ordinal mechanism is at least .
Let and consider the following two profiles:
Since the order of the expert utilities and the bids is the same in both profiles, an ordinal mechanism behaves identically in all these profiles for every . Assume that such a mechanism selects the middle option with probability . Then, the approximation ratio of this mechanism is at least the maximum between its approximation ratio for these two profiles. Considering all profiles for , we get an approximation ratio of at least
This is minimized to for . ∎
4 Bid-independent mechanisms
In this section, we consider cardinal mechanisms but restrict our attention to ones that ignore the bids and base their decisions only on the expert’s report. It is convenient to use the expert’s view of profiles . Then, a bid-independent mechanism can be thought of as using univariate functions , , and which indicate the probability of selecting the expert’s first, second, and third favourite option when she has value for the second favourite option. We drop from notation since the mechanism will be clear from context. The next lemma provides sufficient and necessary conditions for bid-independent mechanisms with good approximation ratio.
Let be a bid-independent mechanism that uses functions , and . Then has approximation ratio at most if and only if the inequalities
hold for every .
Consider the application of on the profile . If the optimal welfare is and the approximation ratio is
The first inequality follows since and the second inequality follows since the expression at the RHS is non-increasing in and non-decreasing in . Then, the first inequality of the statement follows as a sufficient condition so that has approximation ratio at most . To see why it is also necessary, observe that the inequalities in the derivation above are tight for , , and .
If the optimal welfare is and the approximation ratio is
The first inequality follows since and the second inequality follows since the expression at the RHS is non-increasing in and non-decreasing in . Then, the second inequality of the statement follows as a sufficient condition so that has approximation ratio at most . To see why it is also necessary, observe that the two inequalities in the derivation above are tight for , , and . ∎
Truthfulness of bid-independent mechanisms in terms of the agents follows trivially (since the bids are ignored). In order to guarantee truthfulness from the expert’s side, we will use the characterization of ECh-IC from Lemma 2 together with additional conditions that will guarantee ESw-IC. These are provided by the next lemma.
An ECh-IC bid-independent mechanism is truthful if and only if the functions , , and it uses satisfy and for every pair .
We first show that the first condition is necessary. Assume that the first condition is violated, i.e., for two points . If , by the monotonicity of we have and . Otherwise, by the monotonicity of , we have and . In any case, there must exist such that . Now consider the swap from expert valuation profile to the profile . The utility of the expert in the initial true profile is while her utility at the new profile becomes , which is strictly higher.
Now, we show that the second condition is necessary. Again, assuming that the second condition is violated, we obtain that there is a point such that . Now, the swap from expert’s valuation profile to the profile increases the utility of the expert from to , which is again strictly higher.
In order to show that the condition is sufficient for ECh-IC, we need to consider five possible attempts for valuation swap by the expert.
Case 1. Consider the swap from the valuation profile to the profile . The utility of the expert at the new profile is , where the inequality holds due to the fact that , for every . Observe that the RHS of the derivation is the expert’s utility at the initial true profile.
Case 2. Consider the swap from the valuation profile to the profile . The utility of the expert at the new profile is , which is her utility at the initial true profile. The first inequality follows by the condition of the lemma and the second one is due to the convexity of function . See also the proof of Lemma 2.
Case 3. Consider the swap from the valuation profile to the profile . The utility of the expert at the new profile is , which is at most due to the conditions of the lemma.
Case 4. Consider the swap from the valuation profile to the profile . The utility of the expert at the new profile is , which is her utility at the initial true profile.
Case 5. Consider the swap from the valuation profile to the profile . The utility of the expert at the new profile is and the proof proceeds as in Case 2 above. ∎
We are now ready to propose our mechanism BIM. Let , where is the Lambert function, i.e., is the solution of the equation . Mechanism BIM is defined as follows:
BIM is depicted in Figure 1. All functions are constant in and have (admittedly, counter-intuitive at first glance) exponential terms in . Interestingly, as we will show later in Theorem 8, this is the unique best possible bid-independent truthful mechanism (and, actually, we came up with it after proving Theorem 8). Its properties are proved in the next statement.
Mechanism BIM is truthful and has approximation ratio at most , where is the Lambert function.
Tedious calculations can verify that BIM is truthful. The function is non-decreasing in and is defined exactly as in equation (1); hence, ECh-IC follows by Lemma 2. ESw-IC follows since , , and satisfy the conditions of Lemma 6.
We now show that the above mechanism is optimal among all bid-independent truthful mechanisms. The proof exploits the characterization of ECh-IC mechanisms from Lemma 2, the characterization of ESw-IC bid-independent mechanisms from Lemma 6, and Lemma 5.
The approximation ratio of any truthful bid-independent mechanism is at least , where is the Lambert function.
Let be a bid-independent mechanism that uses functions , , and to define the probability of selecting the expert’s first, second, and third favourite option and has approximation ratio . By the necessary condition (1) for ECh-IC in Lemma 2, we know that
Let be any value in .
Due to the fact that , we have
Since, the mechanism is -approximate, Lemma 5 yields
(by applying inequality (5) with ) and
Using (7), this last inequality becomes
Now, let be a continuous function with in such that
Setting (clearly, is differentiable due to the continuity of in ), we get the differential equation
which, given that , has the solution
for . Hence,
5 Expert-independent mechanisms
In this section, we consider mechanisms that depend only on the bids. Now, it is convenient to use the agents’ view of profiles . Then, an expert-independent mechanism can be thought of as using univariate functions , , and which indicate the probability of selecting the high-bidder, the low-bidder, and the option in terms of the normalized low-bid . Again, we drop from notation. Following the same roadmap as in the previous section, the next lemma provides sufficient and necessary conditions for expert-independent mechanisms with good approximation ratio.
Let be an expert-independent mechanism that uses functions , , and with and for . If
for every , then has approximation ratio at most . Condition (12) is necessary for every -approximate expert-independent mechanism.
Consider the application of on the profile with agents’ view . We distinguish between two cases. If , assuming that condition (12) is true, the approximation ratio of is
The first inequality follows since when , while the second one is essentially the right inequality in condition (12).
Otherwise, if , the approximation ratio of is
The first inequality follows since when ; again, the second one is essentially the left inequality in condition (12).
To see that condition (12) is necessary for every mechanism, first consider a mechanism that uses functions , , and such that the function violates the left inequality in (12), i.e., for some . Then, using this inequality and the fact that , the approximation ratio of at profile is
Now, assume that function violates the right inequality in (12), i.e., . Then, using this inequality and the fact that , the approximation ratio of at profile is
as desired. ∎
Figure 2 shows the available space (grey area) for the definition of function , so that the corresponding mechanism has an approximation ratio of at most . It can be easily verified that this is the minimum value for which the LHS of condition (12) in Lemma 9 is smaller than or equal to the RHS so that a function satisfying (12) does exist.
Our aim now is to define an expert-independent truthful mechanism achieving the best possible approximation ratio of . Clearly, truthfulness for the expert follows trivially (since the expert’s report is ignored). We restrict our attention to the design of a mechanism that never selects option , i.e., it has for every . Lemmas 1 and 9 guide this design as follows. In order to be BCh-IC and -approximate, our mechanism should use a non-decreasing function in the space available by condition (12). Still, we need to guarantee BSw-IC; the next lemma gives us the additional sufficient (and necessary) condition.
A BCh-IC expert-independent mechanism is truthful if and only if .
Consider an attempted bid swap according to which the low-bidder increases her normalized bid of so that it becomes the high-bidder and the normalized bid of the other agent is . Essentially, this attempted bid swap modifies the initial profile to . The deviating agent corresponds to the middle column in the initial profile and has probability of being selected. In the new profile, she corresponds to the first column, and has probability of being selected. So, the necessary and sufficient condition so that BSw-IC is guaranteed is for every . Since, by Lemma 1, and are non-decreasing and non-increasing, respectively, this condition boils down to .
The case in which the high-bidder decreases her bid so that it gets a normalized value of is symmetric. ∎
We are ready to propose our mechanism EIM, which uses the following functions. For ,
and for .
Essentially, EIM uses the blue line in the upper right plot of Figure 2, which consists of the curve that upper bounds the grey area up to point and the curve that lower-bounds the grey area after that point. The properties of mechanism EIM are summarized in the next statement. It should be clear though that the statement holds for every mechanism that uses a non-decreasing function in the grey area that is below (together with the restriction , this is necessary and sufficient for BSw-IC). Given the discussion about the optimality of above, all these mechanisms are optimal within the class of expert-independent mechanisms.
Mechanism EIM is truthful and has approximation ratio at most . This ratio is optimal among all truthful expert-independent mechanisms.
6 Beyond expert-independent mechanisms
In this section, we present a template for the design of even better truthful mechanisms, compared to those presented in the previous sections. The template strengthens expert-independent mechanisms by exploiting a single additional bit of information that allows to distinguish between profiles that have the same (normalized) bid values.
We denote by the set of mechanisms that are produced according to our template. In order to define a mechanism , it is convenient to use the agents’ view of a profile as . We partition the profiles of into two categories. Category contains all profiles with or with such that the tie between the expert valuations and is resolved in favour of the low-bidder. All other profiles belong to category .
For each profile in category , mechanism selects the low-bidder with probability that is non-decreasing in and the high-bidder with probability . For each profile in category , mechanism selects the low-bidder with probability , and the high-bidder with probability . Different mechanisms following our template are defined using different functions . The mechanisms of the template ignore neither the bids nor the expert’s report; still, it is not hard to show that they are truthful.
Every mechanism is truthful.
We first show that is truthful for the agents. BCh-IC follows easily by Lemma 1, since and are non-decreasing in . To show BSw-IC, notice that a bid swap attempt from a profile of category creates a profile of category and vice versa. This involves either a high-bidder that decreases her bid and becomes the low-bidder in the new profile, or the low-bidder that increases her bid and becomes the high-bidder in the new profile. In both cases, the increase or decrease in the selection probability according to follows the increase or decrease of the deviating bid.
To show that is truthful for the expert, first observe that according to the expert’s view, the lottery uses constant functions , , and in terms of her valuation for her second favourite option. Hence, Lemma 2 implies ECh-IC. To show ESw-IC, observe again that an expert’s report swap attempt from a profile of category creates a profile of category and vice versa. The expected utility that yields to the expert in the initial profile is if it is of category and if it is of category . After the deviation, the utility of the expert becomes if the new profile is of category and if it is of category . Hence, such a swap attempt is never profitable for the expert. ∎
The next lemma is useful in proving bounds on the approximation ratio of mechanisms in .
Let be a mechanism of and be such that the function used by satisfies
Then, has approximation ratio at most .
Clearly, the approximation ratio of in profiles of category is always since the mechanism takes the optimal decision of selecting the high-bidder with probability .
Now, consider a profile of category , i.e., . We distinguish between two cases. If , then the approximation ratio of is