Optimal Nash Equilibria for Bandwidth Allocation

04/06/2019 ∙ by Benjamin Plaut, et al. ∙ Stanford University 0

In bandwidth allocation, competing agents wish to transmit data along paths of links in a network, and each agent's utility is equal to the minimum bandwidth she receives among all links in her desired path. Recent market mechanisms for this problem have either focused on only Nash welfare branzei_nash_2017, or ignored strategic behavior goel_beyond_2018. We propose a nonlinear variant of the classic trading post mechanism, and show that for almost the entire family of CES welfare functions (which includes maxmin welfare, Nash welfare, and utilitarian welfare), every Nash equilibrium of our mechanism is optimal. Next, we prove that fully strategyproof mechanisms for this problem are impossible in general, with the exception of maxmin welfare. Finally, we show how some of our results can be directly imported to the setting of public decision-making via a reduction due to garg_markets_2018.

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1 Introduction

Bandwidth allocation is a classic resource allocation problem where competing agents wish to transmit data across paths in a network. Each link has a fixed capacity, and each agent’s utility is equal to the minimum bandwidth she receives among all links in her desired path, i.e., the rate at which she is able to transmit data. We follow the standard model of Kelly et al. [23], where there are no monetary payments, and each agent’s path is fixed in advance.

Although one could consider a model where bandwidth allocation and routing are handled simultaneously (i.e., by allowing agents to choose their paths), that would be less accurate in terms of how the internet actually works: routing (which is handled by IP) and bandwidth allocation (which is handled by TCP) are generally separate problems111See Section 1.3 for a discussion of routing games.. This paper is about bandwidth allocation, where pricing-based schemes (like trading post) naturally correspond to signaling mechanisms that indicate which links are congested, and an end-point protocol like TCP [10] can be thought of as agent responses. One of the foundational works in the area of bandwidth allocation is Kelly et al. [23], whose pricing scheme results in the allocation maximizing Nash welfare (the product of utilities).

In this paper, we take the role of a social planner, whose goal is to design a mechanism that leads to a “desirable” outcome (for some definition of “desirable”). We study this through the lens of implementation theory. A mechanism is said to Nash-implement a social choice rule (for example, could denote Nash welfare maximization) if every problem instance has least one Nash equilibrium, and every Nash equilibrium outcome is optimal with respect to . This is similar to saying that the price of anarchy – the ratio of the optimum and the “worst” Nash equilibrium – of the mechanism is 1.222The price of anarchy [24] concept applies only when can be written as the maximization of some cardinal function. This is true when denotes Nash welfare maximization, but is not true in general. In this paper, we focus on pure Nash equilibria, i.e., we do not consider randomized strategies.

The result of Kelly et al. [23] assumes that agents are not strategic, and thus the Nash equilibria of their mechanism may be poor. In contrast, our augmented trading post mechanism will lead to optimal Nash equilibria, not just for Nash welfare, but for an entire family of welfare functions.

1.1 Trading post

Our main tool will be an augmented version of the trading post mechanism. In the standard trading post mechanism, each agent submits a bid on each good , with the constraint that for each agent . Let be the fraction of good that agent receives: then trading post’s allocation rule is . In words, each agent receives a share of the good proportional to her share of the aggregate bid on that good. The bids consist of “fake money”: agents have no value for leftover money.

Trading post has the desirable property that the information requirements are quite light. Each agent’s best response only depends on the aggregate bid of the other agents (i.e., ), not on their individual bids. Furthermore, the allocation rule is decentralized in the sense that there is no centralized price computation, and each link only needs to know the bids .

However, the vanilla version of trading post also has limitations. First of all, it is not even guaranteed to have a Nash equilibrium for every problem instance.333This happens when there is a good that has large enough supply that is not the “rate limiting factor” for any agent; see Sections 1.3 and 2.2.1 for additional discussion. A partial solution to this was proposed by [8]. For every , they gave a modified version of trading post (parameterized by ) that always has a Nash equilibrium, and where every Nash equilibrium attains at least of the maximum possible Nash welfare.444They study Leontief utilities, which is a generalization of bandwidth allocation to the setting where agents may desire goods in different proportions. In the language of implementation theory, this mechanism Nash-implements a approximation of Nash welfare. In the course of our main result, we will strengthen this to full Nash implementation. It is important to note that their mechanism still uses the linear constraint of ; their modification has to do with a minimum allowable bid (see Section 1.3 for additional discussion).

In this paper, we augment the trading post mechanism by allowing nonlinear bid constraints: instead of , we require for each agent , where each is a nondecreasing function chosen by us ahead of time. Importantly, all agents are still subject to the same bid constraint, and we use the same allocation rule of . This novel augmentation allows us to Nash-implement a wide range welfare functions, as opposed to just Nash welfare. Specifically, we will Nash-implement almost the entire family of CES welfare functions (see Section 1.4 for more details). This is our main result.

1.2 CES welfare functions

A welfare function [4, 36] assigns a real number to each possible outcome, with higher numbers (i.e, higher welfare) indicating outcomes that are more desirable to the social planner. Different welfare functions represent different priorities: in particular the tradeoff of overall efficiency and individual fairness. For any constant , the constant elasticity of substitution (CES) welfare function is defined by

where is agent ’s utility, and is the elasticity parameter. When , this is the utilitarian welfare, i.e., sum of utilities. Taking limits as goes to and 0 yields maxmin welfare (the minimum utility) [34, 37, 38] and Nash welfare (the product of utilities) [22, 29], respectively. This class of welfare functions was first proposed by Atkinson [3] (although under a different name), and further developed by [5]. See [28] for a modern introduction to this class of welfare functions.

The closer gets to , the more the social planner cares about individual equality (maxmin welfare being the extreme case of this), and the closer gets to 1, the more the social planner cares about overall societal good (utilitarian welfare being the extreme case of this). The CES welfare function (as opposed to the CES agent utility function) has received almost no attention in the computational economics community, despite being well-studied in the traditional economics literature [3, 5].

These welfare functions also admit an axiomatic characterization [28]:

  1. Monotonicity: if one agent’s utility increases while all others are unchanged, the welfare function should prefer the new allocation.

  2. Symmetry: the welfare function should treat all agents the same.

  3. Continuity: the welfare function should be continuous.

  4. Independence of common scale: scaling all agent utilities by the same factor should not affect which allocations have better welfare than others.

  5. Independence of unconcerned agents: when comparing the welfare of two allocations, the comparison should not depend on agents who have the same utility in both allocations.

  6. The Pigou-Dalton principle: all things being equal, the welfare function should prefer more equitable allocations [12, 33].

Ignoring monotonic transformations of the welfare function (which of course do not affect which allocations have better welfare than others), the set of welfare functions satisfying these axioms is exactly the set of CES welfare functions with 555Without the Pigou-Dalton principle, is also allowed. This can result in unnatural cases where it is optimal to give one agent everything and the rest none, even when this does not maximize the sum of utilities., including Nash welfare [28].666This actually does not include maxmin welfare, which obeys weak monotonicity but not strict monotonicity. This axiomatic characterization shows that we are not just focusing on an arbitrary class of welfare functions: CES welfare functions are arguably the most reasonable welfare functions.

Recently, [20] showed that for any CES welfare function, nonlinear pricing can be used to obtain market equilibria with optimal CES welfare. However, their equilibrium notion – price curve equilibrium – assumes that agents are not strategic.

1.3 Related work

Trading post and market games.

The trading post mechanism – first proposed by Shapley and Shubik [39], and sometimes called the “Shapley-Shubik game”777A plethora of other names have been applied to this mechanism as well, including the proportional share mechanism [17], the Chinese auction [27], and the Tullock contest in rent seeking [9]. – is an example of a strategic market game (for an overview of strategic market games, see [19]). The study of markets has a long history in the economics literature [2, 6, 40, 42]888Recently, this topic has garnered significant attention in the computer science community as well (see [41] for an algorithmic exposition)., but most of this work assumes that agents are price-taking, meaning that they treat the market prices are fixed, and do not behave strategically to affect these prices.999There is some work treating price-taking market models as strategic games; see e.g., [1, 7, 8]. A market game, however, treats the agents as strategic players who wish to selfishly maximize their own utility. Trading post does not have explicit prices set by a centralized authority: instead, prices arise implicitly from agents’ strategic behavior. In particular, – the aggregate bid on good – functions as the implicit price of good . Although the trading post mechanism is well-defined for any utility functions, the Nash equilibria are not guaranteed to have many nice properties in general, except in the limit as the number of agents goes to infinity [14] (in this case, the trading post Nash equilibria converge to the price-taking market equilibria).

The paper most relevant to ours is [8], which analyzed the performance of trading post (with a linear bid constraint) with respect to Nash welfare. They showed that for Leontief utilities (which generalize bandwidth allocation), a modified trading post mechanism approximates the Nash welfare arbitrarily well. Specifically, for any , they gave a mechanism (parameterized by ) which achieves a Nash welfare approximation: there is at least one Nash equilibrium, and every Nash equilibrium has Nash welfare at least times the optimal Nash welfare. Thus the price of anarchy is at most ; equivalently, this mechanism Nash-implements a approximation of Nash welfare. The reason that they were unable to perfectly implement Nash welfare is because when there is a good with supply much larger than other goods101010Specifically, this occurs when a good has price zero. Having a much larger supply than other goods is sufficient but not necessary for this., vanilla trading post may not even have a Nash equilibrium. To fix this, they added a minimum allowable bid, and showed that for any , there is a minimum bid that gives them a Nash implementation. Instead of having a minimum allowable bid, we will add a special bid , which will allow us to strengthen this to full Nash implementation (see Section 2.2.1).

It is worth noting that [8] also considers a broader class of valuations than Leontief, but for this broader class, only a approximation is achieved. Another recent paper gave a strategyproof mechanism achieving a approximation of the optimal Nash welfare [11]. Their approximation guarantee is weaker than the 1/2 guarantee of [8] (and the guarantee for Leontief), but strategyproofness is sometimes more desirable that Nash implementation. Unfortunately, strategyproofness in the bandwidth allocation setting is generally impossible (Theorem 5.1).

Price-taking markets.

The simplest mathematical model of a price-taking market is a Fisher market, due to Irving Fisher [6]. In a Fisher market, there is a set of goods for sale, and each buyer enters the market with a budget she wishes to spend. Each good has a price, and each buyer purchases her favorite bundle among those that are affordable under her budget constraint. Prices are linear, meaning that the cost of a good is proportional to the quantity purchased, and buyers are assumed to have no value for leftover money, so they will always exhaust their entire budgets. A market equilibrium assigns a price to each good so that the demand exactly equals the supply. For a wide class of agent utilities, including bandwidth allocation utilities, an equilibrium is guaranteed to exist [2].111111Specifically, an equilibrium is guaranteed to exist as long agent utilities are continuous, quasi-concave, and non-satiated. The full Arrow-Debreu model also allows for agents to enter to market with goods themselves and not only money; the necessary conditions on utilities are slightly more complex in that setting. The seminal work of Eisenberg and Gale showed that for linear prices and a large class of agent utilities (including bandwidth allocation), the market equilibria correspond exactly to the allocations maximizing Nash welfare [15, 16].121212The conditions for the correspondence between Fisher market equilibria and Nash welfare are slightly stricter than those for market equilibrium existence, but are still quite general. Sufficient criteria were given in [15] and generalized slightly by [21]. Furthermore, the prices are equal to the optimal Lagrange multipliers in the convex program for maximizing Nash welfare (the Eisenberg-Gale convex program).

Recently, [20] extended this model to allow nonlinear prices, where the cost of a good may be any nondecreasing function of the quantity purchased. These functions are called price curves. They showed that for bandwidth allocation, for any , there exist price curves that make every maximum CES welfare allocation a market equilibrium. Furthermore, these prices take a natural form: the cost of purchasing of good is , for some nonnegative constants . Interestingly, for – which denotes Nash welfare – this function form reduces to a linear price , and we know that linear pricing maximizes Nash welfare. Furthermore, are the optimal Lagrange multipliers in the convex program for maximizing CES welfare.

Trading post with linear bid constraints () can be thought of as a market game equivalent of the Fisher market model: it implements Nash welfare ([8] proved a approximation, but we will strengthen this to exact implementation), and the implicit trading post prices (the aggregate bids) are equal to the Fisher market equilibrium prices. Our augmented trading post, with bid constraint , can be thought of as a market game equivalent of the price curves model. The augmented trading post mechanism we use to implement CES welfare will use for each good , further strengthening this analogy.

Bandwidth allocation.

Bandwidth allocation has been studied both with and without monetary payments; we focus on the later setting, following the model of Kelly et al. [23]. Although it has been known that different marking schemes (such as RED and CHOKe [18, 32]) and versions of TCP lead to different objective functions (eg. [31]), a market-based understanding was developed only for Nash Welfare, starting with the pioneering work of Kelly et al. [23]. Furthermore, the market scheme of Kelly et al. is in the price-taking setting; the only strategic market analysis of bandwidth allocation that we are aware of is the approximation of Nash welfare due to [8].

Routing games.

A related topic is that of routing games. In a routing game, each agent has a fixed source and destination in the network, but chooses which path she uses to get there. Each agent incurs a cost for each link she travels over, and the cost each agent pays is typically nondecreasing function of the total traffic over that link. Each agent wishes to minimize the total cost she incurs by strategically choosing which path to follow. In the standard bandwidth allocation model, each agent has a fixed path, and her goal is to maximize the total amount of flow she is able to send from her source to her destination (which is equal to the minimum bandwidth she receives among links in her path). Instead of choosing which path to follow, each agent’s strategy is how she bids (or more generally, how she interacts with the allocation mechanism). For an overview of routing games, see [35].

Implementation theory.

Implementation theory is the study of designing mechanisms whose outcomes coincide with some desirable social choice rule. A social choice rule could be the maximization of a cardinal function, such as a CES welfare function, or something else, such as the set of Pareto optimal allocations. A full survey is outside the scope of this paper; we direct the interested reader to [26].

The “outcome” of a mechanism is not really well-defined; we need to specify a solution concept. The solution concept that we focus on for most of this paper is Nash equilibrium. Possibly the most crucial result regarding implementation in Nash equilibrium (Nash implementation, for short) is due to Maskin [25], who identified a necessary condition for Nash implementation, and a partial converse. He showed that in a very general environment (much broader than bandwidth allocation), any Nash-implementable social choice rule must satisfy what he calls monotonicity. Monotonicity, in combination with a property called no veto power, is sufficient for Nash implementation. In Section 4.2.1, we show that CES welfare functions do not satisfy no veto power, and so cannot be Nash-implemented by Maskin’s approach.

1.4 Our results

    ,

 

Nash-implementable?     (Thm. 5.3) (Thm. 4.1) ?
DSE-implementable?     (Thm. 5.2) (Thm. 5.1) (Thm. 5.1)
Table 1: A summary of our main implementation results. Here denotes maxmin welfare, includes Nash welfare as , and denotes utilitarian welfare. DSE stands for “dominant strategy equilibrium”. “✓” indicates that the type of implementation specified by the row is possible for the social choice rule specified by the column, while “✗” indicates that we give a counterexample, and “?” indicates an open question.

Our results fall into two categories, both summarized by Table 1.

Nash-implementing CES welfare functions.

We view the Nash implementation of CES welfare functions by trading post as our main result (Theorem 4.1). For each , we define an augmented trading post mechanism with a nonlinear bid constraint of for each agent .131313The reader may notice that for – which corresponds to Nash welfare – this constraint reduces to the standard linear constraint of , which is what we should expect: we know from [8] that trading post with the linear constraint leads to good Nash welfare. We denote this mechanism by . We show that has at least one Nash equilibrium, and that all of its Nash equilibria maximize CES welfare.

Our result improves that of [20] by strengthening their price curve equilibrium (which assumes agents are not strategic) to a strategic equilibrium, and improves that of [8] by generalizing from just Nash welfare to all CES welfare functions (except ) and strengthening their approximation to exact implementation.141414It is worth noting that the result of [8] holds for Leontief utilities, a generalization of bandwidth allocation utilities. Furthermore, because the price curve equilibria can be computed in polynomial time [20], our Nash equilibria can also be computed in polynomial time.

Our proof makes use of the following results (stated informally):

  1. Theorem 3.1: Any Nash equilibrium of can be converted into an “equivalent” price curve equilibrium.

  2. Theorem 3.2: Any price curve equilibrium can be converted into an “equivalent” Nash equilibrium of .

  3. Lemma 4.3 [20]: If is a maximum CES welfare allocation, then there exist price curves of the form such that is a price curve equilibrium.

  4. Lemma 4.4 [20]: If is a price curve equilibrium and each has the form , then is a maximum CES welfare allocation.

Lemmas 4.3 and 4.4 together imply that is a maximum CES welfare allocation if and only if it is a price curve equilibrium with respect to some price curves of the form (where are nonnegative constants). Theorems 3.1 and 3.2 allow us to convert between price curve equilibria and Nash equilibria of , and thus enable us to apply Lemmas 4.3 and 4.4 to the Nash equilibria of . Specifically, Theorem 3.1 in combination with Lemma 4.4 will show that any Nash equilibrium of maximizes CES welfare, and Theorem 3.2 in combination with Lemma 4.3 will show that has at least one Nash equilibrium.

Section 3 is devoted to proving our reduction between price curve equilibrium and Nash equilibria of trading post: Theorems 3.1 and 3.2. This reduction is the main tool we use to Nash-implement CES welfare maximization. Section 4 then uses this reduction, in combination with Lemmas 4.3 and 4.4, to prove our main theorem: Theorem 4.1.

Our trading post approach breaks down for and . We are able to Nash-implement by a different mechanism (see below), but we were not able to resolve whether is Nash-implementable. We leave this as an open question.

Results for dominant strategy implementation and maxmin welfare.

A natural question is whether these results can be improved from Nash implementation to implementation in dominant strategy equilibrium (DSE). Section 5 shows that the answer is mostly no: for any , there is no mechanism which DSE-implements CES welfare maximization (Theorem 5.1). We do this by showing that there is no strategyproof mechanism for this problem: the revelation principle tells us that DSE-implementability implies strategyproofness, so impossibility of strategyproofness implies impossibility of DSE implementation.

On the positive side, we show that maxmin welfare () can in fact be DSE-implemented by a simple revelation mechanism (Theorem 5.2). This is actually stronger than strategyproofness: strategyproofness requires truth-telling to be a DSE, but does not rule out the possibility of additional dominant strategy equilibria that are not optimal. In contrast, DSE implementation requires every DSE to be optimal.

Although every DSE is also a Nash equilibrium, DSE-implementability does not imply Nash-implementability [13]. A DSE implementation requires every DSE to be optimal, but there could be Nash equilibria (which are not dominant strategy equilibria) that are not optimal. This means that Theorem 5.2 does not imply Nash-implementability of maxmin welfare. In fact, our revelation mechanism which DSE-implements maxmin welfare is not a Nash implementation: there exist Nash equilibria which are not optimal (see Section 5.2.1 for an example). Our last result of Section 5 is that there is a different mechanism which does Nash-implement maxmin welfare (Theorem 5.3).151515The mechanism for Theorem 5.3 is unrelated to trading post: our trading post approach breaks down for both maxmin welfare and utilitarian welfare. This is because is not a valid price curve when or when .

The rest of the paper is structured as follows. Section 2 formally defines the models of bandwidth allocation, price curves, trading post, and implementation theory. Section 3 presents our reduction between price curves and our augmented trading post mechanism. In Section 4, we use this reduction to Nash-implement CES welfare maximization for . Finally, Section 5 handles DSE-implementation and maxmin welfare.

2 Model

Let be a set of agents, and let be a set of divisible goods, each representing a link in a network. Throughout the paper, we use and to refer to agents and and to refer to goods. Let denote the available supply of good . The social planner needs to determine an allocation , where is the bundle of agent , and is the quantity of good allocated to agent . An allocation cannot allocate more than the available supply: is a valid allocation if and only if for all .

Agent ’s utility for a bundle is denoted by . We assume agents have bandwidth allocation utilities, which take the form:

where is the set of links that agent requires. We assume that for all , i.e., each agent desires at least one good. It will sometimes be useful to define weights where if , and 0 otherwise.

Just as agents have utilities over the bundles they receive, we can imagine a social planner who wishes to design a mechanism to maximize some societal welfare function . One can think of as the social planner’s utility function, which takes as input the agent utilities, instead of a bundle of goods. The most well-studied welfare functions are the maxmin welfare , the Nash welfare , and the utilitarian welfare . As discussed in Section 1, these three welfare functions can be generalized by a CES welfare function:

where is a constant in . The limits as and yield maxmin welfare and Nash welfare, respectively. Throughout the paper, we will use and to denote maxmin welfare and Nash welfare (e.g., “ This theorem holds for ” would include Nash welfare but not maxmin welfare).

For , this function is strictly concave in , so every optimal allocation

has the same utility vector.

161616There could be multiple optimal allocations, however. For example, consider one agent who desires two goods with supply and . The agent’s optimal utility will be , but we can either allocate the rest of the second good anyway, or leave some unallocated; the utility is unaffected.

2.1 Price curves

Price curves, introduced by [20], generalize the well-studied Fisher market model. In a Fisher market [6], each good is available for sale and each agent enters the market with a fixed budget she wishes to spend. Each good has a price , and the cost of purchasing of good is . Price curves allow the cost of a good to be any nondecreasing function of the quantity purchased. When each price curve is defined by , this reduces to the Fisher market model. We do not consider strategic behavior in the price curves model; instead, we use this model as a tool for analyzing the Nash equilibria of our augmented trading post.

Like the Fisher market model, the price curves model assumes that agents have no value for leftover money; this will imply that each agent always spends her entire budget. Throughout the paper, we will assume that all agents have the same budget, and normalize all budgets to 1 without loss of generality. In this paper, we will assume that price curves are either strictly increasing, or identically zero (denoted )171717This assumption is not made in [20], but is helpful for our purposes.. We also assume that price curves are normalized ) and continuous.

Informally, a price curve equilibrium assigns a price curve to each good so that the agents’ demand equals supply. Formally, for price curves , the cost of a bundle is

and the demand set is the set of agent ’s favorite affordable bundles:

If is strictly increasing for all , an agent with bandwidth allocation utility will only purchase goods in her set , and will purchase the exact same quantity of each. When , any agent can add more of that good at no additional cost: this cannot improve her utility, but it also cannot hurt it. This complication with zero-price goods is discussed more in Section 2.2.1.

A price curve equilibrium (PCE) is an allocation and price curves such that

  1. Each agent receives a bundle in her demand set: .

  2. The market clears: for all , , and whenever .181818The definition given in [20] omits this condition, because that paper is also interested in equilibria where the supply is not exhausted. Such equilibria cannot be optimal for our purposes, so we disallow them in our definition.

The second condition states that the demand never exceeds the supply, and that any good whose supply is not completely exhausted must have a price of zero. This implies that no agent has utility for the leftover goods: otherwise she would simply buy more at no additional cost.

2.2 The trading post mechanism

In the standard trading post mechanism, each agent places a bid , where is the amount bids on good . Each agent must obey the constraint . We use to represent the matrix of all bids.

Each agent receives a fraction of the good in proportion to the fraction of the total bid on that good. Formally,

As in the Fisher market model, we assume that agents have no value for leftover money. The aggregate bid on good is , and can be thought of as the “price” of good : in fact, this analogy will be crucial in our proofs.

We augment the standard trading post mechanism in two ways. The first is necessary in order to ensure the existence of equilibrium when goods have price zero, and the second is to extend this mechanism to implement CES welfare functions beyond Nash welfare.

2.2.1 Handling goods with price zero

In Fisher markets, it is possible for some goods to have price zero. This occurs when that good is not the “rate-limiting factor”, i.e., there is enough of that good for everyone and the supply constraint is not tight. This is a problem for standard trading post: in order to receive any amount of good , agent must bid . But if the supply constraint is not tight in the Fisher market setting, there will be at least one agent receiving more of the good than they need. Such an agent will decrease their bid so that she is only receiving what she needs. However, this process will continue infinitely, with agents repeatedly decreasing their bids on this good, but never reaching bid 0.

To handle this, we present the following modified allocation rule. We allow an additional special bid of so that . Conceptually, a bid of 0 indicates that the agent actually does not want the good; bidding indicates that the agent desires the good, but is hoping to get it for free, so to speak. We treat as zero in arithmetic, for example, in the constraint . Similarly, we interpret to mean .

Our modified allocation rule follows this series of steps:

  1. If at least one agent bids a positive (i.e., neither 0 nor ) amount on good , we follow the standard trading post rule: .

  2. However, if all agents bid 0 either or on good , then we allow each agent to have as much good as they want. Specifically, for any agent with , let be an arbitrary good with . Then we allocate . For completeness, if there is no good with (although this will never happen at equilibrium), we set . For agents bidding 0 on good , we set .

  3. After following the above steps, for any good where (violating the supply constraint), for all bidding on good , we set for all as a penalty. In words, if so many agents try to get good for free that the supply constraint is violated, they are all penalized by receiving nothing. Not to worry: this will never happen at equilibrium.

This modification will allow us to simulate a good having price zero.

It is important that we allow separate bids of 0 and . Consider a good where for all . Suppose some agent does not need good , and bidding would cause the supply constraint to be violated and the Step 3 penalty to be invoked. Such an agent can bid 0 on good , which allows her to still spend no money on this good, without the possibility of invoking the Step 3 penalty.

2.2.2 Allowing nonlinear constraints

It will turn out that trading post with the standard constraint of implements Nash welfare. To implement other CES welfare functions, let be nondecreasing functions from to . We call the constraint curves. Like price curves, we assume that each is continuous and normalized. Unlike price curves, we require each to be strictly increasing: is not allowed. Throughout the paper, we will use to denote constraint curves and to denote price curves.

We define the mechanism as follows. Given bids , allocates each good according to the three-step allocation rule described in the previous section. However, each agent’s bid constraint is now

We can define like we defined for price curves and a bundle . Specifically, . Thus each agent’s constraint is in the price curves model, and is in the trading post model.

The most natural case will be when are all the same function. In particular, let be the mechanism where for all . In general, we will use to denote the allocation produced by the mechanism when agents bid .

2.3 Implementation theory

This section covers only the basic concepts of implementation theory; we direct the reader to [26] for a broad overview of this area.

A social choice rule takes as input a utility profile and returns a set of “optimal” outcomes. In our case, will represent maximizing a CES welfare function. Define by

In general, a social choice rule need not express the maximization of any cardinal function.

Let be a solution concept (e.g., Nash equilibrium), be a mechanism (sometimes called a “game form”), and be the induced game for utility profile .191919In general, the difference between a game and a mechanism is that the game definition includes the agent utilities, whereas a mechanism does not. Let be the set of strategy profiles202020A strategy profile is a list of strategies , where is the strategy played by agent . For trading post, a strategy is , and a strategy profile is . satisfying for that game. For example, if denotes Nash equilibrium, then would be the set of Nash equilibria of the game . To distinguish between equilibrium strategies (e.g., what agents bid) and equilibrium outcomes (e.g., the resulting allocation), we use to denote the set of outcomes resulting from strategy profiles satisfying .

Definition 2.1.

A mechanism implements a social choice rule if for any utility profile ,

Using the running example of Nash equilibrium, Nash-implements if for any utility profile , there is at least one Nash equilibrium, and every Nash equilibrium of results in an outcome that is optimal under . We denote the set of Nash equilibria of by , and the set of outcomes resulting from some Nash equilibrium by . When only a single utility profile is under consideration, we will frequently leave implicit and write .

It is worth noting that some of the literature refers to Definition 2.1 as weak implementation, where full implementation requires that , i.e., every outcome that is optimal under should be a Nash equilibrium outcome of . We feel that this distinction is not important in our case, since the utility vector in is unique (with the exception of , which we do not Nash implement anyway): thus allocations differ only in what they do with leftover supply, i.e., supply that will not affect anyone’s utility. If one truly cared about this distinction, our augmented trading post mechanism could be further augmented by allowing each agent another special bid that indicated how much of the leftover supply they wanted. Since these special bids would not affect the utilities, the Nash equilibrium utilities would not be affected, and there would be a combination of leftover supply bids that achieves any maximum CES welfare allocation.212121We would also need to include another penalty step if the leftover supply bids lead to a supply constraint being violated.

We remind the reader of the following standard definitions:

  1. Nash equilibrium: a strategy profile where no agent can strictly improve her utility by unilaterally changing her strategy. We consider only pure Nash equilibria, i.e., we do not allow randomized strategies.

  2. Dominant strategy: a strategy that is optimal regardless of what other agents do.

  3. Dominant strategy equilibrium (DSE): a strategy profile where each agent plays a dominant strategy.

  4. Strategyproofness: A revelation mechanism (i.e., a mechanism that asks each agent to report her utility function) is strategyproof if telling the truth is a dominant strategy for every agent.

DSE-implementability implies strategyproofness via the revelation principle222222See Chapter 9 of [30] for an introduction to the revelation principle., but it is not generally true that any strategyproof social choice rule is DSE-implementable. Strategyproofness ensures that truth-telling is a dominant strategy equilibrium, but there could also be bad equilibria that are not consistent with .

By definition, every DSE is also a Nash equilibrium. However, it is not generally true that DSE-implementability implies Nash-implementability [13]. DSE-implementability requires that every DSE of the mechanism be optimal under , but the mechanism might have additional Nash equilibria (that are not dominant strategy equilibria) that are not consistent with . We will need to take both this and the previous paragraph into account when studying DSE implementation.

We now move on to our results, beginning with our reduction between price curves and . This reduction will be the main tool we use to show that Nash-implement CES welfare maximization.

3 Reduction between price curves and augmented trading post

In this section, we show that any equilibrium of our augmented trading post mechanism can be transformed into a price curve equilibrium, and vice versa. Section 4 will use this result (along with the existence of price curve equilibria maximizing CES welfare, due to [20]) to prove that the mechanism Nash-implements CES welfare maximization.

Section 3.1 describes the intuition behind the reduction. Section 3.2 presents some useful necessary and sufficient conditions for price curve equilibrium and trading post Nash equilibrium. Section 3.3 shows that any trading post Nash equilibrium can be transformed into a price curve equilibrium (Theorem 3.1), and Section 3.4 shows that any price curve equilibrium can be transformed into a trading post Nash equilibrium (Theorem 3.2).

3.1 Intuition behind the reduction

First, notice that augmented trading post and price curves have similar-looking constraints: and . If , these constraints become identical, so is a feasible bid if and only if is a feasible purchase subject to price curves . Suppose that is a price curve equilibrium. For now, assume each is strictly increasing (the formal proof will also handle the possibility of ). Let be the outcome of when agents bid (i.e., ), and suppose that for all : then

where the last equality uses the fact that when is a PCE and .

Thus the allocation resulting from under bids is in fact . Furthermore, since is a price curve equilibrium, each agent exhausts her price curve constraint: . Since and , this implies that for all . Furthermore, in any price curve equilibrium with all nonzero prices, each agent should be spending exclusively on goods in her set , and purchasing them in equal amounts. Thus in the trading post outcome , each agent also also spending exclusively on and acquiring them in equal amounts.

We claim that is a Nash equilibrium of . Suppose the opposite: then there must exist an agent and an alternate bid such that bidding instead of increases her utility. Thus under , she receives strictly more of all goods in . But this means that she must be bidding strictly more on each of these goods, which would violate her bid constraint, since is already tight. Therefore must be a Nash equilibrium of .

The above is an informal proof of one direction of the reduction: transforming price curve equilibria into trading post equilibria. Similarly, if we are given a Nash equilibrium of , we can let (actually, will be a scaled version of ) and , and use the same intuition to show that is a price curve equilibrium.

There are several additional complications. The largest of these is dealing with goods that have price zero in ; indeed, this is the issue that prevents vanilla trading post from implementing Nash welfare maximization [8]. Another difficulty is that in trading post, what you bid depends on others’ bids (whereas for price curves, it only depends on ). However, due to the nature of bandwidth allocation utilities, agents will always purchase in proportion to their weights , and the outcomes at equilibrium will correspond. We will end up with the following two theorems:

Theorem 3.1.

Let be constraint curves where each is homogenous of degree for some . For bids , define nonnegative constants by . Define price curves by

Let . Then is a price curve equilibrium.

Theorem 3.2.

Let be any constraint curve. Let be a price curve equilibrium, and define and by

Then is a Nash equilibrium of .

3.2 Equilibrium conditions for price curves and trading post

Recall that if , and 0 otherwise. The following lemma for trading post states a useful necessary and sufficient condition for Nash equilibria of .

Lemma 3.1.

Let . Then if and only if all of the following hold:

  1. For all , for all where there exists with .

  2. For all , .

Proof.

Assume that the two conditions of the lemma are true. First, we claim that for all : agent only spends money on goods in . This is because for , but ensures that , so would be impossible. Therefore .

Now suppose that is not a Nash equilibrium: then there exists an agent and bid such that , where is the resulting allocation when agent bids and every agent still bids . Condition 1 implies that for all with (since for ). Since only when , we have whenever . Thus when . Since for all , we have when .

We next claim that whenever . If there exists with , then is necessary to ensure that . The only other possibility is that , but , and for all . But in this case, following Step 1 of ’s allocation rule, . Then . This is the highest utility agent could ever have, since for all . This contradicts . We conclude that whenever .

Therefore, since each is strictly increasing,

Since by assumption, we have . This means that violates the bid constraint, and so is not a valid bid. Therefore is a Nash equilibrium.

Suppose that is a Nash equilibrium of . If , violates the supply constraint, so cannot be a Nash equilibrium. If , agent can improve her utility by bidding slightly more on every good (and thus receiving slightly more of every good). Thus must hold.

Suppose for some where there exists with . By definition of , is impossible, so we must have . Consider a new bid where for all , but is such that (where is the resulting allocation when bids and each bids ). Thus .

By definition of , we have , but , since for all , and . Thus there must exist a bundle with for all , but , i.e., obeys the bid constraint. Furthermore, let be the resulting allocation when bids and each bids : then for all . Therefore . But this means cannot be a Nash equilibrium, which is a contradiction. ∎

Next, we give an analogous lemma for price curve equilibrium. Note that the last condition in Lemma 3.2 is simply one of the conditions in the definition of PCE.

Lemma 3.2.

An allocation and price curves are a PCE if and only if all of the following hold:

  1. For all , whenever .

  2. For all , .

  3. For all , , and whenever .

Proof.

The third condition is simply one of the two conditions in the definition of PCE. The other requirement for PCE is that for all , so it suffices to show that if and only if and whenever .

Suppose that , and whenever . We first claim that agent only spends money on goods in . This is because for (because for ), and spending money implies that and , which makes impossible. Thus .

Now suppose for sake of contradiction that there exists another bundle that is also affordable, and . For all with , we have (because for ), so for , . Therefore

Since, , we have . But this implies that is not affordable, which is a contradiction. Therefore .

Suppose . If , is not affordable, which is impossible. If , agent can improve her utility by purchasing slightly more of every good. Thus must hold.

Suppose for some where . By definition, is impossible, so we must have . Consider a bundle where for all , but . Then . Furthermore, , so . Consider another bundle where for all , but : this is always possible because each is continuous, and . Then is affordable, but . This contradicts . ∎

We are now ready to move on to the reduction itself.

3.3 Transforming trading post equilibria into price curve equilibria

This direction of the reduction will require an additional mild condition, involving the following definition.

Definition 3.1.

We say that a function is homogenous of degree if for any , .

Our main result of this section is the following theorem:

See 3.1

Before proving Theorem 3.1, we prove several helpful lemmas (Lemmas 3.33.5). Throughout Lemmas 3.33.5, we assume , and are defined as in Theorem 3.1. We also assume that . Let be the intermediate allocation after Step 2 of ’s allocation rule.

Our first lemma simply states that all agents end up with positive utility.

Lemma 3.3.

For all , .

Proof.

It is always possible for each agent to bid a nonzero amount on each good and obtain nonzero utility. Thus any Nash equilibrium must give each agent nonzero utility. ∎

The following lemma states that the intermediate allocation after Step 2 is in fact the final allocation.

Lemma 3.4.

We have .

Proof.

We need to show that Step 3 of ’s allocation rule is not invoked. Suppose it were invoked: then there is an agent who ends up with for all , and thus . But this contradicts Lemma 3.3. We conclude that . ∎

Lemma 3.5 states that under these constraint curves and bids, the bid constraint is equivalent to the price curves constraint.

Lemma 3.5.

For all , .

Proof.

By the allocation rule of , for all where there exists with , for all , we have

Lemma 3.4 implies that . Also, since we have , so

whenever there exists with . By the definition of , if and only if there exists with (since constraint curves are assumed to be strictly increasing). Therefore