Social Choice with Non Quasi-linear Utilities

04/06/2018 ∙ by Hongyao Ma, et al. ∙ Technion Harvard University 0

Without monetary payments, the Gibbard-Satterthwaite theorem proves that under mild requirements all truthful social choice mechanisms must be dictatorships. When payments are allowed, the Vickrey-Clarke-Groves (VCG) mechanism implements the value-maximizing choice, and has many other good properties: it is strategy-proof, onto, deterministic, individually rational, and does not make positive transfers to the agents. By Roberts' theorem, with three or more alternatives, the weighted VCG mechanisms are essentially unique for domains with quasi-linear utilities. The goal of this paper is to characterize domains of non-quasi-linear utilities where "reasonable" mechanisms (with VCG-like properties) exist. Our main result is a tight characterization of the maximal non quasi-linear utility domain, which we call the largest parallel domain. We extend Roberts' theorem to parallel domains, and use the generalized theorem to prove two impossibility results. First, any reasonable mechanism must be dictatorial when the utility domain is quasi-linear together with any single non-parallel type. Second, for richer utility domains that still differ very slightly from quasi-linearity, every strategy-proof, onto and deterministic mechanism must be a dictatorship.

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1 Introduction

We study social choice mechanisms that aggregate individual preferences and select one among a finite set of alternatives. Our interest is in the existence of strategy-proof social choice mechanisms under general, non quasi-linear utility functions. In the classical voting problem without money, the seminal Gibbard-Satterthwaite theorem (Gibbard, 1973; Satterthwaite, 1975) states that if agents’ preferences can be any ordering over the alternatives, the only deterministic, onto (i.e. every alternative can be selected) and strategy-proof mechanisms for three or more alternatives are dictatorial. On the other hand, with the introduction of monetary transfers and quasi-linear utilities, the Vickrey-Clarke-Groves (VCG) mechanism (Vickrey, 1961; Clarke, 1971; Groves, 1973) maximizes social welfare in dominant strategies, and can be generalized to implement any affine maximizer of agents’ values (Roberts, 1979; Krishna and Perry, 1998).

However, quasi-linearity is a strong assumption, violated for example in domains with budget constraints and problems with lack of liquidity (Cramton, 1997) or with wealth effect and risk-aversion (Pratt, 1992). Non quasi-linearity can also arise as a result of the timing of payments coupled with temporal preferences (Frederick et al., 2002), when payments are contingent on agents’ actions and the presence of payments affects decisions and thus the likelihood of contingencies (Ma et al., 2016b, c), and in the context of illiquid currencies such as points-based allocation schemes (Kash et al., 2007).

To the best of our knowledge, few papers considered social choice mechanisms with monetary transfers and non quasi-linear utilities.111A preliminary, short version of this paper (Ma et al., 2016a) proved a special case of the impossibility result for two agents and the richest non quasi-linear utility domain, with additional assumptions of unanimity and neutrality. The proof techniques are different, and this preliminary version does not make use of the generalized Roberts theorem. The main result of this paper is a tight characterization of the maximal utility domain, which we name the largest parallel domain, where there exist non-dictatorial mechanisms that are strategy-proof, onto, deterministic, individually rational and satisfy no subsidy (i.e. no positive transfers from the mechanism to the agents). These properties are those of the VCG mechanism in quasi-linear utility domains. As a special case, we prove that for utility domains that contain all quasi-linear types but do not reside within the largest parallel domain, the only mechanisms satisfying the above conditions must be dictatorial. The proofs make use of a generalized Roberts’ theorem, which we extended to non quasi-linear utility domains with suitable properties. We also provide a negative result for a broader class of mechanisms: by allowing richer utility domains that still differ very slightly from quasi-linearity, we establish the impossibility of non-dictatorial mechanisms even without requiring individual rationality or no subsidy.

A key observation is that the critical property of agents’ utilities that enables non-dictatorial mechanisms is not the linear dependency on payments. We say the utility functions of an agent is of parallel type if for any two alternatives and , within the range of interest, no matter how much the agent is charged for , to achieve the same utility, the additional amount she is willing to pay for stays the same. Quasi-linear utility functions have this property, but there can also be non quasi-linear parallel types. Intuitively, a type being parallel requires that regardless of which alternative is selected, the agent’s marginal cost for money is the same as long as she has the same utility level— the trade-off with money depends on how happy the agent is, not how much she is paying. A domain where all types are parallel is called a parallel domain, and the largest parallel domain is the set of all parallel types.

The rest of the paper is organized as follows. After a brief discussion of related work, we provide in Section 2 a formal definition of parallel domains. We prove a positive result in Section 3, that within the parallel domains, the family of generalized weighted VCG mechanisms are strategy-proof, onto, deterministic, individually rational (IR), and satisfy no subsidy. In Section 4, we generalize Roberts’ Theorem (Roberts, 1979) to parallel domains, and prove that when the differences in agents’ willingness to pay for different alternatives are unrestricted, maximizers of affine functions of the willingness to pay are the only implementable choice rules amongst mechanisms with these properties. With this characterization, we prove in Section 5 our main result— that when agents have types outside of the parallel domain, the only mechanisms that are strategy-proof, onto, deterministic, individually rational, and satisfy no subsidy must be fixed-price dictatorships, i.e. there exists a dictator who chooses her favorite alternative given fixed prices associated with each alternative. We also develop a negative result for a broader class of mechanisms: by allowing utility domains that are slightly richer in their non quasi-linearity, we show that individual rationality and no subsidy can be relaxed, while the impossibility of non-dictatorial mechanisms can still be established. We give in Section 5.1, for example, an impossibility result for a two-slopes domain, when the utility function of each agent for each alternative is linear, with the slope taking one of two possible values. Proofs omitted from the body of the paper are provided in Appendix A.

1.1 Related Work

In social choice without monetary payments, the classical Gibbard-Satterthwaite theorem has been extended to more restricted preference domains, including the saturated domains (Kalai et al., 1979), linked domains (Aswal et al., 2003), circular domains (Sato, 2010), and weakly connected domains (Pramanik, 2015). Domains for which there are positive results have also been extensively studied, see Black’s majority rule (Black, 1948), Moulin’s median voting schemes and generalizations (Moulin, 1980; Nehring and Puppe, 2007) and results on graphs with metric spaces (Barbera, 2001; Schummer and Vohra, 2002; Dokow et al., 2012).

For social choice with payments and quasi-linear utilities, Roberts (1979) showed that with three or more alternatives, when the values can take any real numbers, positive association of differences is necessary and sufficient for strategy-proof implementation, and that the only implementable choice rules are affine maximizers of agent values. Such choice rules can be implemented by weighted VCG mechanisms (Krishna and Perry, 1998). Characterizations of strategy-proof implementations have also been developed for mechanism design problems in specific domains (Myerson, 1981; Rochet, 1987; Lavi et al., 2003; Bikhchandani et al., 2006; Saks and Yu, 2005).

One approach to mechanism design without quasi-linearity is to assume that the functional form of agent utility functions is known to the designer, for example auctions public budget constraints (Dobzinski et al., 2012), or auctions with known risk preferences (Maskin and Riley, 1984). In contrast, we assume that the functional form of agent utility functions are private. For private non quasi-linear utilities, under suitable richness of type space, Kazumura et al. (2017) prove a “taxation principle” style characterization and a “revenue uniqueness” result of truthful mechanisms, and show various applications to problems other than social choice, e.g. single item allocation. The existence of truthful and non-dictatorial social choice mechanisms is not discussed.

For the assignment problem with unit demand, the minimum Walrasian-equilibrium mechanism is known to be truthful (Demange and Gale, 1985; Alaei et al., 2010; Aggarwal et al., 2009; Dütting et al., 2009; Morimoto and Serizawa, 2015), even for any general non-increasing utility function in payment. On the other hand, truthfulness cannot be achieved together with Pareto-efficiency for allocation problems in which agents may demand more than one unit of good, or when agents have multi-dimensional type spaces (Kazumura and Serizawa, 2016; Dobzinski et al., 2012; Baisa, 2016). We do not impose Pareto-efficiency (PE) in proving our impossibility results.222With PE, however, we prove a similar dictatorship result, which in comparison to our main results, requires weaker assumptions on agents utility domains. See Theorem 5 in Appendix B. Randomized mechanisms for bilateral trade (Garratt and Pycia, 2014) and revenue-optimal auctions in very simple settings (Baisa, 2017; Che et al., 2012; Pai and Vohra, 2014) have also been studied in the context of private budget constraints. We focus here on social choice rather than assignment or allocation problems, settings for which there is more structure on agents’ preferences and also indifference toward outcomes where an agent’s own assignments are the same.

2 Preliminaries

Denote as the set of agents and as the set of alternatives. A social choice mechanism accepts reports from agents as input, selects a single alternative , and may also determine payments. A mechanism is onto if for any alternative , there exists a preference profile for which is selected. A mechanism is dominant-strategy incentive compatible (DSIC) if no agent can gain by reporting false preferences.

We allow monetary transfers, and the utility of an agent may depend both on the selected alternative and her assigned payment. Denote as the utility of agent if alternative is selected and she needs to pay . determines agent ’s type and is her private information. Denote as a type profile, and as the type profile of agents except for agent .

Denote the utility of alternative to agent at zero payment as , which we call the value of alternative to agent . Let and be a most and a least preferred alternative at zero payment. A utility profile is quasi-linear if for all , and . In this case, the values fully determine the type profile. Let the quasi-linear domain be the set of all quasi-linear types of a single agent where the ’s can take any value in , and let be the set of all quasi-linear type profiles.

We consider non quasi-linear utilities such that for all and all ,

  1. is continuous and strictly decreasing in ,

  2. .

Property (S1) guarantees that agents strictly prefer to make smaller payments. Property (S2) means that every agent prefers the worst alternative at zero payment to any alternative at some very large payment. Denote the general non quasi-linear utility domain as the set of all types of an agent satisfying (S1) and (S2), and let be the general non quasi-linear utility domain for a set of agents.

A social choice mechanism on type domain is composed of a choice rule and a payment rule . Thus if the reported type profile is , the choice made is , and the utility of agent is . A mechanism is DSIC if and only if, for any agent , any type of agent , and any reported profile from other agents , agent cannot gain by misreporting any type :

(1)

A mechanism is individually rational (IR) if and only if, by truthfully participating in the mechanism, regardless of the reports made by the other agents, no agent can be worse off than having their worst alternative at zero payment selected and not making any payment.333For a mechanism where IR is violated, an agent may benefit from not participating. See Section 5 for more discussions on voluntary participation. Assuming for all and , may take any non-negative value, for DSIC mechanisms, this definition of IR is equivalent to requiring that the utility of any truthful agent is non-negative. That is, ,

(2)

We are interested in mechanisms with the following set of properties.

  1. Dominant-strategy incentive compatible

  2. Deterministic

  3. Onto

  4. Individually rational

  5. No subsidy

Ontoness only requires that all alternatives will be selected given some type profile, but does not require all payment schedules are achievable. No subsidy requires that the mechanism does not make positive transfers to the agents.

Before continuing, we review a well-known characterization of deterministic DSIC mechanisms. We say that a mechanism is agent-independent if an agent’s payment is independent of her report, conditioned on a particular alternative being selected; i.e. fixing the type profile of the rest of the agents , , . Given an agent-independent mechanism and any , if there exists s.t. , let the agent-independent price be the payment pays when is selected: , which depends only on . Otherwise, if for all , let . An agent-independent mechanism is also agent-maximizing if given the agent-independent prices , the alternative selected by the mechanism maximizes the utilities of all agents simultaneously, i.e. , s.t. for all , and . The properties of agent-independence and agent-maximization are necessary and sufficient for deterministic DSIC mechanisms with quasi-linear utilities (Vohra, 2011), and this equivalence can be easily generalized to general utilities that strictly decrease with payment.

A dictatorship in social-choice without money identifies an agent as the dictator, and always selects her favorite alternative. We generalize this concept for social choice with money as follows:

Definition 1 (Fixed Price Dictatorship).

Under a fixed price dictatorship, there exists a dictator and fixed prices . Given any type profile , one of the dictator’s favorite alternatives under these prices is selected, i.e. , and the dictator pays .

2.1 Parallel Domains

Given any type of an agent , for each alternative , we define the willingness to pay as the payment for at which the agent is indifferent between getting at this payment, and getting her least preferred alternative at zero payment:

(3)

See Figure 1. is the maximum amount the agent can be charged if alternative is selected, without violating IR. always holds, and (S1)-(S2) imply that for all , exists, and .

Figure 1: An example type in the parallel domain. All horizontal sections of the shaded area (e.g. the dotted arrows) are of the same length, i.e. utility curves are horizontal translations of each other within the range but need not be straight lines.
Definition 2 (Parallel Domain).

A utility domain for an agent is a parallel domain if for all ,

(4)

See Figure 1. We call a a parallel type if (4) is satisfied. For a parallel type , for s.t. , for any utility level , we have

(5)

In words, the differences in the payments on and in order to achieve is a constant that does not depend on , i.e. the additional amount an agent is willing to pay for over does not depend on how much the agent is charged for .444There is no requirement on the shape of the utility functions below the utility level , or where the payments are negative, since utilities in these ranges are irreverent to the incentive properties of mechanisms that are IR and satisfy no subsidy. This also implies that the shape of the utility function of the least preferred alternative is irrelevant, and that when there are only two alternatives the general non quasi-linear domain is parallel.555Abstracting away the cardinal utility to an agent for an outcome, and modeling instead the prices at which one alternative is more preferable than another, a type being parallel then requires that in the range of interest, for any pair of alternatives , which one is more preferable depends only on the difference in the prices . For an agent with parallel type, her marginal cost for money is the same as long as she has the same utility level, regardless of which alternative is selected. Intuitively, the trade-off with money depends on how happy the agent is, not how much she is paying. For example, while paying for a better alternative, an agent with this kind of wealth effect would care less about money at the same payment amount than compared to a weaker alternative.

Denote as the largest parallel domain (i.e. the set of all parallel types), and . The quasi-linear domain is a parallel domain, where for all and for all . Another special case of the parallel domain is the linear parallel domain, where for every , there exists s.t. and all , (so the quasi-linear domain is a special case when ). For these two domains, the “vertical” distances between the utility curves also stay the same (which need not be the case for a general parallel domain), and the utility functions are horizontal translations of each other everywhere.

A utility domain for an agent is said to have unrestricted willingness to pay if for any

-dimensional non-negative vector with at least one zero entry, there exists

s.t. the willingness to pay according to is equal to this vector element-wise (at least one zero entry is required since an agent always has zero willingness to pay for ). Formally, for which s.t. , there exists s.t. for all .666There are multiple (actually, infinite number of) types in with the same vector of willingness to pay, and we only require at least one of them to be included. We call a parallel domain with unrestricted willingness to pay an unrestricted parallel domain. In particular, is an example of an unrestricted parallel domain. A utility domain is an unrestricted parallel utility domain if each of the is unrestricted and parallel.

We now prove two lemmas.

Lemma 1.

Let be a DSIC and deterministic social choice mechanism on a utility domain with unrestricted willingness to pay. The mechanism satisfies (P4) IR and (P5) No subsidy if and only if and , the agent-independent prices satisfy:

  1. for all alternatives ,

  2. there exists an alternative s.t. .

Thus, the agent-independent prices any agent faces under a mechanism satisfying (P1)-(P5) must be standard, i.e. the minimum price among all alternatives is zero. We leave the proof of the lemma to Appendix A.1.

Lemma 2.

For any parallel type and any standard prices :

  1. s.t. and ,

  2. .

Proof.

We first prove part (i). Assume w.l.o.g that . We know from the monotonicity of and the definition of parallel domain (4) that

For part (ii), observe that if the price for at least one of the alternatives is zero, the highest utility at the given prices among all alternatives is at least . Therefore, for any alternative s.t. , the alternative cannot be agent-maximizing. Among the alternatives s.t. , the agent-maximizing alternative(s) coincides with the maximizer(s) of , according to part (i). ∎

Thus, in a parallel domain, the agent-maximizing alternative given standard prices is the maximizer of the difference between the willingness to pay and the price: . As a result, the willingness to pay serves similar roles as values in the quasi-linear domain, and it is this connection that enables us to generalize Roberts’ theorem to unrestricted parallel domains.

3 The Generalized Weighted VCG Mechanism

We prove in this section a positive result, that in parallel domains, the generalized weighted VCG mechanisms implement in dominant strategy any affine maximizer of willingness to pay: for non-negative weights , and real constants .

Definition 3 (Generalized Weighted VCG).

The generalized weighted VCG mechanism, parametrized by non-negative weights and real constants , collects a type profile from agents, and computes the willingness to pay . It is defined as

  • Choice rule: , breaking ties arbitrarily, independent to agents’ reports.

  • Payment rule: for s.t. ; for s.t. :

    (6)

    where .

Theorem 1.

With type domain , any non-negative coefficients and any real constants , the generalized weighted VCG mechanism is DISC, IR and does not make positive transfers to the agents.

Proof.

We first consider an agent s.t. . Given , we can check that for any agent reports s.t. , agent ’s agent-independent payment would be

(7)

For agent s.t. , and coincide. No matter what agent reports, is always selected and she does not pay anything, thus

(8)

Since is the maximizer of , all agent-independent prices are non-negative. Moreover, has the minimum price among all alternatives, which is exactly zero: . By Lemma 1, we know that the prices are standard, and the mechanism satisfies (P4) IR and (P5) No subsidy if it is DSIC. What is left to prove is choosing at such agent-independent prices is agent-maximizing for all agents, which implies DSIC. From Lemma 2 we know that we only need to prove is the maximizer of for all agents. This is immediate for agents with . For an agent with , for any alternative , we have

thus indeed maximizes . ∎

Note that when , we have for all , , and maximizing an affine function of the willingness to pay is equivalent to maximizing the same affine function of the values. Thus, this mechanism coincides with the weighted VCG mechanisms when utilities are quasi-linear. Ontoness is satisfied if for at least one agent and when the utility domain is unrestricted.

4 Generalizing Roberts’ Theorem

With quasi-linear utilities, Roberts (1979) showed that with three or more alternatives, if each agent’s value for each alternative can be any real number, the choice rule of any social choice mechanism that is (P1) DISC, (P2) deterministic and (P3) onto must be a maximizer of some affine function of agents’ values. With two additional conditions, (P4) IR and (P5) No subsidy, we generalize Roberts’ theorem to the unrestricted parallel domains.

Theorem 2 (Roberts’ Theorem on Parallel Domains).

With three or more alternatives and an unrestricted parallel utility domain , for every social choice mechanism satisfying (P1)-(P5), there exist non-negative weights (not all equal to zero) and real constants such that for all ,

We prove in Lemma 3 that the weak monotonicity condition defined in terms of willingness to pay (which is eqivalent to the W-Mon condition in terms of values (Bikhchandani et al., 2006) when utilities are quasi-linear, see Definition 4) is a necessary condition of incentive compatibility. Following the same steps as in the first proof of Roberts’ theorem presented in Lavi et al. (2009) while treating willingness to pay as “values” in the proof, we conclude that affine maximizers of the willingness to pay are the only implementable choice rules (see Appendix A.2 for the details). The coefficients cannot be all zero in order to satisfy (P3) Ontoness.

Regarding the requirements on the utility domain: in the proof of Robert’s theorem (Lavi et al., 2009), the values can take any real numbers, whereas the willingness to pay for a non quasi-linear type takes non-negative values and one of them has to be exactly zero. This does not prevent us from generalizing the proof, since what is necessary is that the differences in the values for all can be any real numbers. We get this from the unrestricted parallel domain.

Definition 4 (Weak Monotonicity).

Let be a utility domain. A choice rule satisfies weak monotonicity (W-Mon) if for all and all ,

In words, W-Mon in a parallel domain means that if alternative is selected under and alternative is selected under , the additional willingness to pay for comparing with according to , i.e. , must be at least as big as the additional willingness to pay for comparing with according to : . This is a generalization of the W-Mon condition in terms of values as defined in (Bikhchandani et al., 2006), and the two are equivalent when utilities are quasi-linear, in which case holds for all .

Lemma 3.

With any parallel utility domain , every social choice mechanism satisfying (P1), (P2), (P4) and (P5) must satisfy W-Mon.

Proof.

Consider two types , and a social choice mechanism s.t. and . We know from agent-maximization and Lemma 2 that facing prices , alternative must be a maximizer of according to , and alternative must be a maximizer of according to . Thus, and must hold. Adding both sides of the two inequalities we get . ∎

5 Impossibility results

We now state the main result in this paper.

Theorem 3 (Dictatorship).

With at least three alternatives and a utility domain s.t.

  1. for each , contains an unrestricted parallel domain,

  2. for at least agents, ,

the only social choice mechanisms that satisfy (P1)-(P5) are fixed price dictatorships.

The quasi-linear domain is unrestricted and parallel, thus a special case of the theorem can be stated as: on any utility domain containing , if the utility domains of at least agents contain non-parallel types, the only mechanisms satisfying (P1)-(P5) are fixed price dictatorships. We provide here an outline of the proof and leave the full version to Appendix A.3.

Given any mechanism with (P1)-(P5) on a utility domain satisfying (C1) and (C2), we prove that the restriction of the mechanism on the parallel subspace must also satisfy (P1)-(P5). Theorem 2 then implies that on , the choice rule must be the maximizer of some affine function of agents’ willingness to pay. Fixing the choice rule, agent-independence and agent-maximization determine the agent-independent prices up to a constant (when there is no tie), and the requirement that prices being standard fully pins down as (7) and (8) for all and any s.t. is parallel for all . Theorem 2 requires that there exists at least one agent with a non-zero coefficient . If the number of agents for whom is more than one, for at least one of them. Assume w.l.o.g. that and . Fixing some parallel type profile of the rest of the agents , are not yet pinned down by the above mentioned characterization since is not parallel. We prove that no agent-independent price guarantees that an alternative that is agent-maximizing for all agents always exists for all economies where . This contradicts DSIC.

Now we know that there exists exactly one agent (say ) with a non-zero coefficient. This implies that when the reported profile is parallel (i.e. ), the outcome of the mechanism must be determined according to a fixed price dictatorship. By induction on the number of agents whose type is not parallel, we then prove that for any , the outcome must also be determined by the same fixed price dictatorship.

On the Tightness of the Negative Result

That the utility domain contains an unrestricted parallel domain is necessary for Theorem 2. If the number of agents that has a type outside of the parallel domain is smaller than , there are at least two agents whose types are always parallel. The generalized weighted VCG mechanism with only for these agents would satisfy all (P1)-(P5), and still would not be a dictatorship.

Regarding the properties (P1)-(P5), the mechanism being (P1) DSIC and (P2) deterministic are trivially necessary. (P3) ontoness is required, since if some alternatives are never selected, the number of alternatives can be effectively reduced to two, in which case all types satisfying (S1) and (S2) are parallel, thus even for the most general , any generalized weighted VCG mechanism with coefficients not all zero satisfies (P1)-(P5).

The conditions on the utility domain (C1) and (C2) in the statement of Theorem 3 require only a small deviation from quasi-linearity or the parallel domain. As an example, with and , the type domain and for any (e.g. as illustrated in Figure 2) satisfies both (C1) and (C2). Assuming only (C1) and (C2), truthful mechanisms that violate one or both of (P4) and (P5) may still exist. For the above described utility domain, the mechanism which always adds 1 to agent 2’s payment but otherwise functions exactly as a generalized weighted VCG mechanism satisfies (P1)-(P3) and (P5). More detailed discussions are provided in Appendix A.4, and we present an alternative impossibility result assuming only (P1)-(P3) in Section 5.1.

Figure 2: An example non-parallel type of agent 2.

Note that Theorem 5 does not rely on any efficiency assumption. Replacing ontoness by Pareto-efficiency, we can prove a similar dictatorship result which instead of (C2), requires that only a single agent has a single non-parallel type. See Theorem 5 in Appendix B.

On the Fixed Price Dictatorship

In a fixed price dictatorship, the dictator may still be asked to make some non-zero payments. It is clear that such a mechanism is DSIC, and when the prices that the dictator faces are standard, it is also IR and never pays the agents. In order to replace the fixed-price dictatorship in Theorem 3 with full dictatorship, i.e. the dictator chooses her favorite alternative free of charge, we can impose another condition, voluntary participation (VP), which means any agent can choose to walk away from the mechanism and accept the alternative decided by the rest of the agents without having to make any payment. If the dictator is charged a positive fixed price for some alternative , and is still the dictator’s favorite choice under the fixed prices, then when is selected by the sub-economy without the dictator (which should be the case for some given onto-ness), the dictator would have the incentive to walk away, in which case will be selected and the dictator pays . This contradicts VP.

VP is stronger than IR. To see this, note that IR requires that for every agent at least one of the prices is weakly below the agent’s willingness to pay. On the other hand, VP requires that the mechanism is well defined for any economy of agents and satisfy the same properties, moreover, each agent must face a zero-price for the alternative that would be selected in the sub-economy without her.

Regarding the payments from the rest of the agents— when the dictator strictly prefers a single alternative (i.e. , ), the rest of the agents do not make any payment: for all . In the degenerate case where , i.e. the dictator is indifferent toward multiple best alternatives, some or all the rest of the agents may be charged a non-zero payment to break ties among the dictator’s favorite alternatives (e.g., we could run a generalized VCG mechanism between two alternatives that are tied from the dictator’s perspective), and this may still satisfy (P1)-(P5). See Appendix A.3.

5.1 Relaxing IR and No Subsidy

We show in the rest of the section that with more richness in non quasi-linearity (e.g. with the linear domain with two slopes defined below), the dictatorship result remains given only (P1)-(P3).

Definition 5 (Linear Domain with Two Slopes).

is a linear domain with two slopes if there exists , s.t. for all , , either for all or for all .

We say a linear domain with two slopes is unrestricted if for each , the values can be any real numbers, and the slopes of utility functions for different alternatives can be any combination of and .

Theorem 4.

With at least three alternatives and a utility domain s.t. for each , contains an unrestricted linear domain with two slopes, a social choice mechanism satisfying (P1)-(P3) must be a fixed price dictatorship.

Intuitively, each contains as a sub-domain an unrestricted linear parallel domain (e.g. the set of s.t. for all and all ), which is a special case of a strictly parallel domain. For strictly parallel domains, with only (P1)-(P3), we can generalize Roberts’ Theorem, and still determine the agent-independent prices up to a constant. We then prove that if more than one agent has a positive coefficient in the choice rule, the induced agent-independent prices will result in contradictions when an agent’s utility domain consists of utility functions with mixed slopes. Then we can prove by induction that the only agent with a positive coefficient must be a fixed price dictator. See Appendix A.4.1 for the full proof.

The theorem still holds if , for all . Moreover, we would reach the same result if the ’s and ’s are known to the mechanism designer. Since the ’s and the ’s can be arbitrarily close, this theorem shows that very slight disturbances on the slopes of agents’ utility functions is sufficient to rule out the existence of truthful non-dictatorial mechanisms.

6 Conclusions

The existence of truthful and non-dictatorial social choice mechanisms strongly depends on whether monetary transfers are allowed. The seminal Gibbard-Satterthwaite theorem proves that without payments, the only truthful and onto mechanisms are dictatorial, whereas for the (rather restrictive) quasi-linear utility domain, any affine maximizer of agent values can be implemented if payments are allowed. We study social choice with payments and general utilities, distinguish types being parallel as the central property of quasi-linearity for DSIC mechanisms to exist, generalize (with additional conditions IR and No subsidy) Roberts’ theorem to parallel domains with unrestricted willingness to pay, and provide a tight characterization of the largest parallel domain. Within the largest parallel domain, the generalized weighted VCG mechanisms implement any affine maximizer of agents’ willingness to pay, and satisfy DSIC, onto, deterministic, IR and do not make payments to agents. Adding any non-parallel type to an unrestricted parallel domain, the only mechanisms with the above properties are dictatorial. We also discuss utility domains that are richer in their non quasi-linearity but still deviate very slightly from the quasi-linear domain, for which individual rationality and no subsidy can be relaxed and the dictatorship result remains.

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Appendix A Proofs

a.1 Proof of Lemma 1

See 1

Proof.

We first prove parts (i) and (ii) given (P4) and (P5). Assume part (i) does not hold, that there exists and s.t. . Consider s.t. . In any agent-maximizing mechanism, the agent is guaranteed utility at least , which is not possible without the mechanism making a positive payment to the agent, and this violates (P5). Now we only need to prove that it cannot be the case that for all . This is obvious, since if otherwise, there exists for whom for all thus , in which case (P4) is violated.

We now prove the other direction. Given part (i), it is obvious that no matter which alternative is selected, the transfer from any agent to the mechanism is non-negative thus (P5) holds. Given (ii), we know that the agent’s minimum possible utility under an agent-maximizing mechanism is , which is at least , thus (P4) holds. ∎

a.2 Proof of Theorem 2

See 2

We had proved in Lemma 3 that W-Mon is a necessary condition for (P1)-(P5) if the utility domain is unrestricted and parallel. We provide here a few more steps where the details differ slightly for values and willingness to pay, following the first proof presented in Lavi et al. (2009), however, the high level ideas are the same.

Definition 6 (Positive Association of Differences).

A social choice function on a parallel domain satisfies positive association of differences

(PAD) if: for all

and in , if and for all and all , then it must be the case that , as well.

Lemma 4 (Ic Pad).

A social choice mechanism on an unrestricted parallel domain with (P1), (P2), (P4) and (P5) satisfies PAD.

Proof.

By Lemma 3, must satisfy W-Mon. Let be type profiles such that and for all and all . We need to show that . Denote . We know . Assume for some , we show by contradiction that must also hold. By induction, this implies that .

Assume that there exists and s.t. but . Since all players except player have the same type in and in , we get by W-Mon that , which contradicts the PAD assumption on and . Thus, must hold. This completes the proof of the induction step. ∎

We now prove the result analogous to Claim 1 in Lavi et al. (2009).

Claim 1.

Assume a choice rule on an unrestricted parallel domain satisfies PAD. Fix type profiles s.t. . If holds for all for some , then .

Proof.

We follow the same proof as in Lavi et al. (2009), and prove this claim by contradiction. The construction differs slightly from the original proof since the willingness to pay is always normalized s.t. the an agent’s smallest willingness to pay among all alternatives is zero.

Suppose by contradiction that . For each , denote . We know that for all , and in addition, . Consider s.t. for all ,

and for s.t. ,

Then, let be the normalized version of , i.e.