Markets Beyond Nash Welfare for Leontief Utilities
We study the allocation of divisible goods to competing agents via a market mechanism, focusing on agents with Leontief utilities. The majority of the economics and mechanism design literature has focused on linear prices, meaning that the cost of a good is proportional to the quantity purchased. Equilibria for linear prices are known to be exactly the maximum Nash welfare allocations.
Price curves allow the cost of a good to be any (increasing) function of the quantity purchased. We show that price curve equilibria are not limited to maximum Nash welfare allocations with two main results. First, we show that an allocation can be supported by strictly increasing price curves if and only if it is group-domination-free. A similarly characterization holds for weakly increasing price curves. We use this to show that given any allocation, we can compute strictly (or weakly) increasing price curves that support it (or show that none exist) in polynomial time. These results involve a connection to the agent-order matrix of an allocation, which may have other applications. Second, we use duality to show that in the bandwidth allocation setting, any allocation maximizing a CES welfare function can be supported by price curves.
In a market, buyers and sellers exchange goods according to some sort of pricing system. One of the most important concepts in the study of markets is market equilibrium, which describes when the supply provided by the sellers and the demands of the buyers exactly match. Market equilibrium theory dates back to Walras’s seminal work in 1874 . In 1954, Arrow and Debreu finally showed that a market equilibrium is guaranteed to exist for a wide class of utility functions . This includes Leontief utilities, which will be our focus.
The simplest mathematical model of a market is a Fisher market, first proposed in 1891 by Irving Fisher (see  for a modern introduction). A Fisher market consists of a set of goods available for sale, and a set of agents, each with a fixed amount of money to spend. It is usually assumed that agents have no value for leftover money. In Fisher markets, each good has a single real-number price , and the cost of buying some quantity of good is . We refer to such prices as linear, meaning that the cost is proportional to the quantity purchased. A market equilibrium assigns a price to each good such that when each agent purchases her favorite bundle of goods that is affordable under these prices, the demand exactly matches the supply.
There are three motivations behind this work. First, in real market economies, prices are often not linear, and depend on the quantity purchased111One consequence of this is that there can be an incentive for agents to “team up”, which is not the case in linear pricing. For example, it could be cheaper for one person to purchase the resource in bulk and then distribute it, as opposed to each person buying her own: imagine ordering pizza for a party. We do not consider strategic behavior in this paper; see Section 2.1 for additional discussion.. We refer to prices of this form as price curves. For example, “buying in bulk” may allow agents to purchase twice as much of some resource for less than twice the price. In this case, the marginal price decreases as more of the good is purchased. On the other hand, for a scarce resource, a central authority may choose to impose increasing marginal costs to ensure that no single individual can monopolize the resource. Israel’s pricing policy for a water is a good example of this, where each additional unit of water costs more than the previous one . A tremendous amount of work has been devoted to understanding the nature of linear prices, despite the pervasiveness of price curves in the real world. This paper attempts to ask the same fundamental questions of price curves that have been answered for linear prices.
Second, imagine a social planner or mechanism designer who wishes to design a pricing scheme that will maximize some objective function. The objective function of a social planner is typically referred to as welfare. There are many different social welfare functions, the most well studied being utilitarian welfare (the sum of agent utilities), Nash welfare (the product of agent utilities) [19, 16], and max-min welfare (the minimum agent utility [24, 27, 28])222The utilitarian welfare is also known as the Benthamite welfare, after Jeremy Bentham. The max-min welfare is also known as the Rawlsian welfare, after John Rawls, or the egalitarian welfare.. Max-min welfare can be seen as caring only about equality across individuals. The utilitarian welfare measures overall good across the entire population, possibly at the expense of certain individuals. The Nash welfare is something of a compromise between these two extremes.
Eisenberg and Gale famously showed that for linear prices, the market equilibria correspond exactly to the allocations maximizing Nash welfare [10, 11]. This result is powerful, but also limiting: what if the social planner wishes to maximize a different welfare function? Is it possible that using price curves instead of linear prices allows a wider set of allocations to be equilibria? In particular, are there welfare functions other than Nash welfare such that welfare-maximizing allocations can always be supported by price curves? (We say that an allocation can be supported by price curves if there exist prices curves that make that allocation an equilibrium.) Our paper answers these questions in the affirmative.
The third motivation involves a more conceptual connection between markets and welfare functions, both of which have been extensively studied in the economics literature. We know that linear-pricing equilibria correspond to maximizing Nash welfare, but does this connection go deeper? Our work hints at an affirmative answer to this question as well.
1.1 CES welfare functions
For any constant , the constant elasticity of substitution (CES) welfare function is given by
where is agent ’s utility, and is the elasticity parameter. Setting yields utilitarian welfare, and taking limits as goes to and 0 yields max-min welfare and Nash welfare, respectively. This class of welfare functions was first proposed by Atkinson , although he did not call it by the same name. The smaller is, the more the social planner cares about individual equality (max-min welfare being the extreme case of this), and the larger is, 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 very little attention in the computational economics community, despite being well-studied in the traditional economics literature [2, 5].
These welfare functions also admit an axiomatic characterization:
Monotonicity: if one agent’s utility increases while all others are unchanged, the welfare function should prefer the new allocation.
Symmetry: the welfare function should treat all agents the same.
Continuity: the welfare function should be continuous.
Independence of common scale: scaling all agent utilities by the same factor should not affect which allocations have better welfare than others.
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.
Up to monotonic transformations of the welfare function (which of course do not affect which allocations have better welfare than others), the set of welfare functions that satisfy these axioms is exactly the set of CES welfare functions with , including Nash welfare 333This actually does not include max-min 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 really are the welfare functions.
2 Results and prior work
We assume throughout the paper that agents have Leontief utility functions. An agent with a Leontief utility function desires the goods in fixed proportions, e.g., one unit of CPU for every two units of RAM. We can express agent ’s utility as
where is the amount of good agent receives, and is agent ’s (nonnegative) weight for good . Although this definition may seem limiting, it subsumes network bandwidth allocation, where agents have Leontief utilities with for all and . Bandwidth allocation is a well-studied area in its own right (for example, the work of Kelly  on proportional fairness.).
Leontief utilities exhibit certain convenient properties that other utility functions do not. In particular, such an agent will always purchase her goods exactly in the same proportions, and all that changes is how much she purchases. We also assume that each agent has the same amount of money to spend. However, most of our results do extend to the case of unequal budgets, as noted later on.
We now describe our two main results.
A necessary and sufficient condition for the existence of price curves.
Section 4 presents our first main result, which concerns the first motivation described above: trying to understand fundamental properties of price curve equilibria. In particular, this section answers the following question: given some allocation, is there a way to tell whether there exist price curves that make this allocation an equilibrium? Furthermore, can such price curves be efficiently computed?
The answer boils down to a property we call group-domination-freeness. Roughly, a set of agents group-dominates a set of agents if these sets are the same weight, but for every good and every threshold , the number of agents in receiving at least of of good is at least as large as the number of agents in receiving at least of good . The formal definition of group domination is given in Section 4. An allocation is group-domination-free (GDF) if no group dominates any other group. We show that an allocation can be supported by strictly increasing price curves if and only if the allocation is GDF (Theorem 4.1)444This result extends to the setting of unequal budgets if one instead considers “budget-weighted group-domination-freeness”. We elaborate on this in Section 4.. This characterization results in a polynomial time algorithm to compute the underlying price curves or show that none exist (Theorem 4.2). Appendix A gives an analogous characterization theorem and polynomial time algorithm for weakly increasing price curves (Theorems A.1 and A.2).
Although the definition of group domination may seem slightly technical, we also demonstrate its relation to the concept of stochastic dominance, and argue that it may in fact be interpreted as a fairness notion. The stochastic dominance interpretation will also suggest that group domination is quite a strong property, and hence group-domination-freeness is a quite a weak assumption.
The proof of these characterization theorems involves the construction of a special matrix we call the agent-order matrix , which is a function of the allocation. We show that existence of strictly increasing price curves is captured by strongly positive solutions (every entry of the solution vector is positive) to . We relate group-domination-freeness to a property of this matrix, and then invoke a duality theorem equivalent to Farkas’s Lemma  to complete the proof. The algorithm for computing price curves is a linear program involving the agent-order matrix.
Maximum CES welfare allocations can be supported in bandwidth allocation.
Our second main result concerns the second and third motivations: a social planner who wishes to maximize a welfare function other than Nash welfare, and understanding the connection between markets and welfare functions. We know that the maximum Nash welfare allocations can be supported by linear prices. If we allow price curves, are there other welfare functions whose maxima can be supported?
First, will need some assumption on the agents’ weights. To see this, consider just two agents and one good. Since the agents have the same budget, they must receive equal amounts of the good no matter the price curve. However, if one agent derives less utility per unit of the good, this allocation doesn’t maximize any CES welfare function except for Nash welfare555This example actually holds for a much wider class of utilities, not just Leontief. This is because for a single good, all anyone can do is buy as much of that good as they can..
One natural assumption would be to normalize the agents’ weight vectors under some norm – we discuss such results in the following section. However, we begin with the simpler assumption that for every agent and good . This means that each agent has some set of goods she desires, and she desires all of those goods in the same proportion. This corresponds to the well-studied bandwidth allocation setting, where each good corresponds to a link in a network. Each agent wishes to transmit some data along a fixed path of links, and thus desires bandwidth for exactly those links in equal proportion.
In this setting, price curves correspond naturally to a signaling mechanism that provides congestion signals (eg. in the form of a packet mark or drop) and an end-point protocol such as TCP  corresponds naturally to agent responses. While it was understood that different marking schemes (such as RED and CHOKe [13, 21] and variants of TCP lead to different social welfare objectives (eg. ), a market-based understanding was developed only for Nash Welfare, starting with the seminal work of Kelly et al. .
Our second main result is that in the bandwidth allocation setting, the welfare-maximizing allocations for any fixed CES welfare function with can be supported by price curves (Theorem 6). We prove this by writing a convex program to maximize CES welfare, and using duality to construct explicit price curves. Furthermore, these price curves take on a natural form: the cost of buying of good is , where is a constant derived from the dual666These results extend to agents with unequal budgets if we instead consider the “budget-weighted CES welfare”, i.e., , where is agent ’s budget. We discuss this in Section 6.1. The price curves will take the exact same form..
One may wonder if Theorem 6 could be extended to , i.e., utilitarian welfare. Example 1 shows that the answer is no, unfortunately. One may also wonder if Theorem 6 would generalize if we relax our constraint from to . The answer is again no; this counterexample is more involved and is given by Theorem C.1 in Section C.
|agent 1||agent 2||agent 3|
We prove several additional results. We consider max-min welfare in Section 5, and show that as long as agents’ weights are reasonably normalized, allocations with optimal max-min welfare can be supported. In Appendix B, we use the characterization theorem for weakly increasing price curves (Theorem A.1) to prove several results about CES welfare functions. First, we provide an alternative proof of the bandwidth setting result (although this proof only holds for ). We then show that for two agents whose weights are normalized in any reasonable way, any maximum CES welfare allocation can be supported. We also show that when agents’ weights are normalized with respect to itself, any maximum CES welfare allocation can be supported. These arguments are quite technically involved, with L’Hôpital’s rule making a surprising cameo appearance.
2.1 Prior work
The study of markets has a long histories in the economics literature [32, 30, 1, 6]. Recently, this topic has received significant attention in the computer science community as well (see  for an algorithmic exposition). The vast majority of the literature has focused on linear prices, but there are some papers that consider more complex pricing schemes. One such paper is , which showed that for linear but fully personalized prices (i.e., we can independently assign different prices to different agents for the same good), one can support any Pareto optimal allocation. One drawback of fully personalized prices is that we lose any claim to fairness, since agents may be subjected to totally different prices for the same resource. Price curves are personalized in a much weaker sense: the cost of a good depends only on how much you buy of that good, not on who you are or how much you buy of another good.
The concept of a social welfare function – a function which encapsulates a societal value system – was first proposed in 1938 , and further developed by . For brevity, we will just refer to them as welfare functions. Since then, several different welfare functions have been proposed, the most well-studied being utilitarian welfare, Nash welfare [19, 16], and max-min welfare (the minimum agent utility) [24, 27, 28]. The class of CES welfare functions was first proposed by  and further developed by , although not under the same name. See  for a modern discussion of welfare functions.
We briefly mention an important property in mechanism design: strategy-proofness. A mechanism is strategy-proof if agents can never improve their utility by lying about their preferences. Unfortunately, even in simple settings, the only mechanism for resource allocation that can simultaneously guarantee strategy-proofness and Pareto optimality is dictatorial, meaning that one agent receives all of the resources . This is clearly unacceptable, so we sacrifice strategy-proofness in favor of Pareto optimality. Specifically, we assume throughout the paper that agents always truthfully report their preferences.
The remainder of the paper is organized as follows. Section 3 formally defines the model. Section 4 presents our first main result: that for strictly increasing price curves, an allocation can be supported if and only if it is group-domination-free. In Section 5, we use this characterization to show that allocations with optimal max-min welfare can be supported by price curves in a wide range of settings. Section 6 gives our second main result: that in the bandwidth allocation setting, every maximum CES welfare allocation can be supported by price curves. At this point we conclude the main paper, and move on to supplementary results. In Appendix A, we generalize our characterization theorem from Section 4 to account for weakly increasing price curves. We use this theorem to prove several more results about CES welfare in Appendix B. Appendix C provides counterexamples to various claims that one might have hoped to prove. We also discuss in that section why certain other classes of utilities – in particular, linear utilities – are much more difficult to analyze. Finally, Appendix D provides some proofs that are omitted from earlier sections.
Let be the set of agents, and let be a set of divisible goods, where . Throughout the paper, we use and to refer to agents and and to refer to goods. The social planner needs to determine an allocation , where is the bundle of agent , and is the fraction of good allocated to agent . An allocation cannot allocate more than the available supply: is a valid allocation if and only if for all .777Unless otherwise stated, we assume without loss of generality that the supply of each good is normalized to 1.
Agent ’s utility for a bundle is denoted by . The literature studies many subclasses of utility functions. For example, Leontief utilities have the form
where represents the weight agent has for good . For brevity, we will usually write and leave the condition implicit. The same holds for other contexts where we are dividing by a value that may be zero. We assume that agents have Leontief utilities throughout the paper, and we assume that each agent has nonzero weight for at least one good.
Just as agents have utilities over the bundles they receive, we can imagine a social planner who wishes to design a pricing 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 max-min welfare , the Nash welfare , and the utilitarian welfare .888One could also imagine the (arguably less natural) case of a social planner who cares about some agents’ utilities more than others, which would manifest as weights in her utility function. We briefly consider this case in the setting of CES welfare with unequal budgets (Section 6.1). These three welfare functions can be generalized by a CES welfare function:
where is a constant in .
3.1 Fisher markets
The simplest market model is a Fisher market . In a Fisher market, each good is available for sale and each agent enters the market with a fixed budget she wishes to spend. It is assumed that agents have no value for leftover money; this will imply that each agent always spends her entire budget. Unless otherwise stated, we will assume that all agents have the same budget, and normalize all budgets to 1 without loss of generality.
Informally, a Fisher market equilibrium assigns a price to each good so that the agents’ demand equals supply. Formally, for a price vector , the cost of bundle is . Bundle is affordable for agent if . Agent ’s demand set is the set of her favorite affordable bundles, i.e.,
If for all , an agent with Leontief utility will always purchase in exact proportion to her weights: since agent ’s utility is determined by , violating these proportions cannot increase her utility. Thus when discussing an arbitrary allocation , we assume that each bundle is in proportion to agent ’s weights: otherwise there is no hope of supporting such an allocation. For brevity, we leave this assumption implicit through the paper, rather than always stating “for an arbitrary allocation where each bundle is in proportion to agent ’s weights”.
The careful reader may note that we are glossing over a detail: if for some good , agent can add more of good to her bundle at no additional cost. This does not affect agent utilities at all, but is technically possible. In order to avoid handling this uninteresting and sometimes messy edge case, we assume throughout the paper that for agents with Leontief utilities, demand sets and arbitrary allocations are always in exact proportion to agent weights.
Formally, a Fisher market equilibrium is an allocation and price vector such that
Each agent receives a bundle in her demand set: .
The market clears: for all , . Also, if , then .
When all agents have the same budget, this is also called the competitive equilibrium from equal incomes .
For a wide class of agent utilities, including Leontief utilities, an equilibrium is guaranteed to exist 999Specifically, 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.. Furthermore, the equilibrium allocations are the exactly the allocations which maximize Nash welfare101010The 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  and generalized slightly by .. This is made explicit by the celebrated Eisenberg-Gale convex program [10, 11], and combinatorial approaches to computing market equilibria [9, 15].
3.2 Price curves
Our paper considers an expanded model where instead of assigning a single price to each good, we assign each good a price curve . The function expresses the cost of good as a function of the quantity purchased. When we say “price curve”, we mean a function that is weakly increasing (buying more of a good cannot cost less), normalized (), and continuous. Setting for all and all yields the Fisher market setting.
Given a vector of price curves , the cost of a bundle is now . Although the functions may not be linear, the cost of a bundle is still additive across goods. Each agent’s demand set is defined identically to the Fisher market setting: .
The demand set is intuitively the same as in the Fisher market setting: each agent purchases exactly in proportion to her weights, and buys as much as she can without exceeding her budget. A price curve equilibrium as an allocation and vector of price curves such that
Each agent receives a bundle in her demand set: .
The demand does not exceed supply: for all 111111For Fisher market equilibria, the second condition also stipulated that whenever , . Without this additional condition, cranking up all prices to infinity would result in trivial equilibria where all agents purchase almost nothing and so would certainly not maximize Nash welfare. Such trivial price curve equilibria do exist under our definition, but since we are not going to make any claims of the form “all price curve equilibria maximize a certain function”, there is no issue with allowing these trivial equilibria to exist..
We say that price curves support an allocation if is a price curve equilibrium. The fundamental question we address in this paper is: what allocations can be supported?
4 Group domination
Recall that we require price curves to be continuous and weakly increasing. We wish to theoretically characterize which allocations can be supported by price curves so that we can (1) apply this characterization in our subsequent proofs, and (2) construct an algorithm which can calculate price curves in polynomial time.
The true necessary and sufficient condition for an allocation to be supported by price curves – and an algorithm to compute them – is given in Appendix A. However, this condition (“locked-agent-freeness”) is somewhat unwieldy. Although weakly increasing price curves are sometimes necessary121212Consider an instance with two goods and two agents, whose weights are given by and . Nash welfare is maximized by splitting good 1 evenly between the two agents, and allowing agent 1 to purchase an equal quantity of good 2. This only possible if the price of good 2 is zero: otherwise, agent 1 is paying more than agent 2. Recall that the Fisher market equilibrium prices are the dual variables of the convex program for maximizing Nash welfare: thus the price of good 2 being zero corresponds to the fact that the supply constraint for good 2 is not tight in this instance., for now we will consider only strictly increasing price curves. The corresponding necessary and sufficient condition is the cleaner notion of group-domination-freeness.
4.1 Group domination
We have termed the necessary and sufficient condition for the existence of strictly increasing price curves “group-domination-freeness” (GDF). To gain intuition for this condition, consider an allocation and agents . We will say that agent dominates agent if and there exists for which this inequality is strict. Observe that this would prevent the existence of strictly increasing price curves supporting allocation – both agents must spend their full budget (otherwise they could buy more of every good, since price curves are continuous), but agent buys strictly more than agent . A similar scenario arises when considering two weighted groups of agents with the same total weight. If for every possible quantity of any good , considering only agents purchasing at least of good , the weight of the agents in is greater than or equal to the weight of agents in , then can never be made to pay more than . Essentially, for each additional of any good, as many agents from must purchase that as agents from , so no matter how we price these increments, never pays more. If this difference in weights is strict for any ( pair, that implies some increment must cost 0 (so that the total expenditure of and is equal), violating the requirement that price curves be strictly increasing.
Another way to gain intuition for group domination is by analogy to stochastic dominance. Distribution stochastically dominates distribution if for every possible payoff , the odds of getting at least from are at least as high as the odds of getting at least from . One consequence of stochastic dominance is that any rational agent should prefer to – there are no trade-offs, is simply better than (dominant over) . In fact, we can directly consider weighted groups of agents as probability distributions. The total weight of each group must be the same – without loss of generality, equal to 1. Consider sampling the allocations for any good with probability equal to the weight of each agent. The probability distribution stochastically dominating is exactly equivalent to the weighted group group-dominating . Thus not only does group domination create problems for pricing, it can arguably be considered unfair, as is in some sense objectively better-off than 131313See  for an introduction to stochastic dominance..
The formal definition of this condition is below.
Definition 4.1 (Group-domination-free (GDF)).
Let and be vectors in that assign a (possibly zero) weight to each agent, such that . Then group-dominates in an allocation (denoted ) if for all and for any threshold ,
and there exists a () pair where the inequality is strict. is group-domination-free if there do not exist vectors such that in .141414The “-free/-freeness” suffix may remind some readers of the popular fairness notion envy-freeness; this connection is intended. If one agent does envy another, this constitutes an instance of group domination in the allocation, so GDF implies envy-freeness. However, the reverse is not true: for an agent to envy agent , must receive strictly more of every good cares about; for group domination, the difference need only be strict on one good.
We will also assume without loss of generality that for all , at most one of and is nonzero: were this not the case, we could define and by and , and we would have as well.
Theorem 4.1 will show that an allocation can be supported with strictly increasing price curves if and only if it is GDF.
4.2 Characterization of allocations supported by strictly increasing price curves
In order to relate the existence of price curves to GDF, first observe that, for agents with Leontief utilities, the conditions for a price curve equilibrium take on a relatively simple form. Recall that by assumption, the allocation to be considered doesn’t violate supply, and each agent purchases goods in exact proportion to her weights (see Section 3.1). Then the condition that for all can be captured by Lemma 4.2, whose proof appears in Appendix D. Intuitively, agent fills up her bundle in proportion to her weights until (1) she reaches her budget and (2) there exists a good where buying more would cost more.
lemmalemLeontiefDemand Given price curves , if and only if both of the following hold: (1) , and (2) there exists such that for all , .
Given these requirements, we now rely on three key observations to relate price curves to GDF: (1) Only the points on the price curves corresponding to agent allocations matter. (2) Only the order of the agents, not their absolute allocations, matters. (3) The order of the agents can be captured in an agent-order matrix such that the column sums and row sums represent agent costs and group dominations, respectively.
We will illustrate these observations with the example allocation shown in Figure 0(a). First we deal with observation 4.2. Consider the possible price curves shown in Figure 0(c). Given the price that each agent pays for each good, these are the only points that matter, in the sense that (a) each agent’s total cost, which must equal 1, depends only on these points, and (b) an agent must be able to purchase more of a good if the next fixed point on that curve has the same price, and otherwise need not be able to do so. Thus when considering whether price curves are possible, we need only consider the set of prices corresponding to agent allocations.
A similar argument addresses observation 4.2. As long as we fix the order of points along a price curve, we can change the allocations arbitrarily (assuming they still obey the supply and proportional-purchase assumptions) without changing the prices. Obviously, every agent will still incur a cost of 1, and it will not change whether an agent can purchase more of a good (whether the next point along the curve has the same price).
Finally, we come to the more complicated observation 4.2. We will first lay out how the agent-order matrix is constructed, then illustrate its connection to both prices and group domination. The matrix will have rows, one for each agent, and a sub-block for each good, as shown in Figure 0(b). Within a sub-block, each column will correspond to a non-zero agent allocation (i.e., the non-zero points shown in Figure 0(c)). The entry corresponding to agent , good , and allocation threshold will equal 1 if and 0 otherwise. Essentially, this will indicate which agent pays the cost of the first, second, etc. section of each price curve. Additionally, we append a column of ’s to the end of the matrix. To see the connection to prices, consider a vector such that . For instance, Figure 1(a) exhibits such a vector for the matrix shown in Figure 0(b). will represent prices, so we require all the entries to be non-negative, denoted ; for strictly increasing price curves, we require to be strongly positive151515Recall that a strongly positive vector has every entry greater than 0., denoted . Specifically, within each block (corresponding to a good ), the first entry represents the cost of increasing from 0 of good to the first non-zero point on the price curve, the second entry represents the cost of increasing from the first point to the second point, and so on. The last entry in , which we can assume equals 1 without loss of generality, represents the total cost expended by each agent. Thus ensures that each agent spends exactly 1 unit of money. (Ensuring that condition 4.2 of Lemma 4.2 be met is slightly more complicated. However, for strictly increasing price curves, it is trivially satisfied.)
Thus we can see that column sums of the agent-order matrix correspond to agent expenditures. Row sums, however, correspond to group domination. To see the connection, consider a vector such that is strictly positive161616Recall that a strictly positive vector has entries in with at least one entry non-zero., denoted . For instance, Figure 1(b) exhibits such a vector for the matrix shown in Figure 0(b). Given , the positive entries correspond to the weights of agents in a dominating group , while the (absolute value of the) negative entries are the weights of agents in group . Since the last entry of must be nonnegative, the total weight of is at least as large as that of . And since , all the entries are non-negative and at least one other entry must be positive. This means that at every point on a price curve (any ), the weight of group purchasing at least of good is at least as much as the weight of group purchasing , and for some () this is strict. Clearly this is equivalent to .
Having constructed the agent-order matrix and related its column and row sums to prices and group domination, respectively, the final step applies a previously-known duality result equivalent to Farkas’s Lemma , which establishes that valid prices (column sums) exist if and only if group domination (row sums) do not. Specifically, we make use of the following result originally due to Stiemke to prove Theorem 4.1.
Lemma 4.0 (1.6.4 in ).
For a commutative, ordered field , a matrix over , the following are equivalent.
has no solution
has a solution
Let be any allocation that obeys the supply constraints and gives at least one agent a nonempty bundle. Then be can supported by strictly increasing price curves if and only if is GDF.
Recall that an allocation can be supported if there exist price curves such that , and (i.e., obeys the supply constraints). The latter condition is satisfied by assumption, and by Lemma 4.2, for Leontief utilities and strictly increasing price curves, the former condition holds if and only if the cost .
Let be the set of distinct, non-zero amounts of good allocated to some agent under . Label the elements of as such that . Since , in some sense doesn’t matter – we only require that these “in-between” areas of the price curve don’t violate continuity and are strictly increasing. Thus there exist strictly increasing price curves supporting if and only if there exist functions such that and .
Now we are ready to set up the agent-order matrix to which we will apply Lemma 4.2. Since each column will represent an allocation point for a specific good (corresponding to its sub-block), we will write the column indices as , where indicates the sub-block and is the index within that sub-block.
Thus each row of represents an agent, and each column (except the last) represents one point of the functions . Since gives at least one agent a nonempty bundle by assumption, has at least two columns (one allocation point and the column of ’s). We know by Lemma 4.2 that such that if and only if such that . To complete the proof, we will show that the former condition is equivalent to the existence of strictly increasing price curves supporting , and the latter is equivalent to a group domination.
If such that , we may assume without loss of generality that the last entry in is 1. Furthermore, define (for convenience, define ). Clearly is equivalent to the requirement that . Additionally,
Thus is equivalent to the requirement that .
Finally, consider such that . This is equivalent to a group domination , where if , if , and all other entries are 0. Consider the product of the last column of with , which equals . Without loss of generality, we can assume , and thus . If this is not true, then would have greater weight than , and decreasing any weight in can only increase coordinates of or equivalently widen the gap between and in terms of group domination. Now observe that for any good and , is equal to the dot product of column of by , where is the largest value such that . This holds because is an indicator variable for , and by construction no agent can have an allocation amount between and . Therefore is equivalent to the requirement that for all () and that for some () this inequality is strict, i.e., is equivalent to . ∎
This characterization, in addition to allowing us to prove some of our subsequent results, implies that we can compute price curves (or show that they do not exist) for a particular instance in polynomial time. This is exhibited by the following linear program.
Given a set of agents , goods , and an allocation , let be the corresponding agent-order matrix. In the following linear program, the optimal objective value is strictly positive if and only if there exist strictly increasing price curves supporting , in which case defines such curves.
As per the proof of Theorem 4.1, there exist strictly increasing price curves supporting if and only if there is a solution to the system . To turn this into a valid linear program, instead of the strict inequality for each coordinate of , we write and attempt to maximize . Furthermore, we restrict the final entry of as , since otherwise can be scaled arbitrarily. If there is a solution with , then corresponds to price curves as before, with each entry representing the difference in price between adjacent allocation amounts. These points simply need to be connected, e.g., in a piecewise linear fashion, to constitute valid price curves. ∎
One may wonder if Theorem 4.1 generalizes to other classes of utility functions. Unfortunately, the answer in general is no. Example 3 gives an instance with linear utilities that is GDF, but cannot be supported by price curves.
In Section 5, we will show how the group-domination-freeness concept can be useful for proving that allocations of interest can be supported by price curves: specifically, allocations with optimal (or near optimal) max-min welfare. But first, a word about unequal budgets.
4.3 Unequal budgets
It turns out that the characterization theorem of the previous section easily generalizes to agents with unequal budgets. Since price curves are strictly increasing, the only additional requirement for an allocation to be supported is that each agent spends her entire budget . In the agent-order matrix, the last column of ’s corresponded to each agent’s expenditure, so we simply need to replace with for each row .
Following Lemma 4.2 with the modified agent-order matrix, the if-and-only-if characterization becomes “budget-weighted group-domination-freeness”. A budget-weighted group domination still requires that for all (),
and that there exists where the inequality is strict. The only difference is that instead of requiring both groups to have the same total weight, that weight is now scaled by each agent’s budget. That is, . Note that when for all , this recovers the definition of group domination.
5 Warm-up: max-min welfare
In this section, we show that under mild assumptions, price curves can support allocations with either optimal max-min welfare, or arbitrarily close to optimal max-min welfare. As before, we assume that agents have Leontief utility functions. Also, we refer to an allocation with optimal max-min welfare as a max-min allocation. The proof of this turns out to be quite simple, so we think of this section as a warm-up.
The first thing we observe is that when agent weights are unconstrained in magnitude, there is no hope to support any approximation of max-min welfare. Consider a single good and two agents with weights and on that good. In this case, each agent ’s utility is just , so the max-min welfare of an allocation is . Now imagine that is much larger than : agent 1 needs significantly more of the good to achieve the same utility as agent 2. Then any max-min allocation (or even any decent approximation) must give more of the good to agent 1 than agent 2. But since agents have the same budgets, any price curve equilibrium must result in each agent receiving half of the supply of good 1, which is a contradiction.
Thus in order to have any hope of even approximately supporting a max-min allocation, the agent weights must be normalized in some way. Theorem 5.1 states that under a mild normalization assumption, we can support a max-min allocation.
Suppose there exist strictly increasing functions such that for all , . Then there exists a max-min allocation that can be supported by price curves.
Since the max-min welfare of an allocation is determined by the minimum agent utility, the max-min welfare cannot be improved by making any agent’s utility higher than any other. Similarly, since each agent’s utility is determined by , the max-min welfare cannot be improved by allocating goods to an agent outside of her desired proportions. Thus there exists a max-min allocation where all agents have the same utility , and where for all and .
Since GDF is invariant to scaling by constants, this implies that is GDF if and only if the weight vectors themselves are GDF. That is, is GDF if and only if the allocation defined by is GDF. One realizes that the assumption of for all is literally assuming that there exist (strictly increasing) price curves that support the allocation . Thus is GDF by Theorem 4.1, so is GDF, which completes the proof. ∎
One natural corollary of Theorem 5.1 is the following:
Suppose there exists some so that for all . Then there exists a max-min allocation that can be supported by price curves.
Theorem 5.1 has an interesting conceptual implication. We can think of price curves themselves as a sort of “norm” on the allocation, and any allocation for which there is a “norm” which assigns the same value to each agent’s bundle is reasonable enough that it can be supported by price curves. The previous statement can be rephrased as “an allocation can be supported by price curves if and only if there exist price curves which assign the same cost to each agent’s bundle”, and so is functionally a tautology. Since there exists a max-min allocation which is a constant scaling of the agent weights, this near-tautology carries over.
One final observation is that there are some interesting norms, such as the norm, which cannot be written as the sum of increasing functions. In fact, there are cases where no max-min allocation can be supported when agent weights have the same norm.171717The norm is defined as . Furthermore, the following counterexample falls under the even simpler bandwidth allocation setting: for all .
There exist instances where for all and , but no max-min allocation can be supported.
Consider an instance with three agents and two goods, where the agent weights are given by the following table:
The unique max-min allocation is . Thus any price curves must satisfy , . But then , which is a contradiction. Thus no max-min allocation can be supported. ∎
The good news is that the norm can be approximated to arbitrary precision by norms, leading to the following theorem. We use to denote the max-min welfare of allocation .
thmmaxminInftyApx Suppose that for all . Then for every , there exists an allocation that can be supported by price curves where .
With this warm-up in hand, we now move on to our second main result, which concerns CES welfare functions.
6 CES welfare
In this section, we consider CES welfare functions:
This section contains our second main result: that in the bandwidth setting (i.e., agents have Leontief utilities where for all , ), for any , any maximum CES welfare allocation can be supported by price curves. Our proof uses the dual of the convex program for maximizing CES welfare to construct explicit price curves that support a maximum CES welfare allocation. The price curves take the very simple form of for constants that are derived from the dual.
theoremthmBandwidth If for all and , then for any , any maximum CES welfare allocation can be supported by price curves.
We begin by writing the following program to maximize CES welfare:
The objective is concave for any , so the resulting program is convex. For brevity, we will omit “where ” for the rest of the proof, and assume that any terms which could cause division by zero are simply omitted.
We can remove the exponent of from the objective without affecting the optimal point: the optimal value may be affected, but the optimal solution (i.e., the ) will not. When is negative, this changes the program to a minimization program, but this can be handled by adding a factor of to the objective.181818We add a factor of instead of because this will slightly simplify the analysis. Thus the new program is
Next, we write the Lagrangian of this program. Let be the Lagrange multiplier associated with the constraint and let be the Lagrange multiplier associated with the constraint . We will use and to denote the vectors of all such Lagrange multipliers. Then the Lagrangian is given by
Consider any maximum CES welfare allocation: this corresponds to a point which is optimal for the primal. We have strong duality by Slater’s condition, so there must exist and such that is optimal for .
Since the primal objective was concave, is concave in and , so the gradient of evaluated at must be zero. In particular,
for every . Since , we have , so . Similarly,
for every and , and so . This implies that . Then by the definition of Leontief utility, we have
We now use the Lagrange multipliers to construct explicit price curves. We define by . Since , we have , so these price curves are in fact increasing.
We claim that is a price curve equilibrium. To see this, we explicit compute the cost of agent ’s bundle :
The crucial use of is that . Therefore
Thus is affordable to agent . Furthermore, since these price curves are strictly increasing, no bundle with higher utility is affordable to agent , so is in agent ’s demand set. We also know that , since is a feasible solution to the primal. Therefore is a price curve equilibrium. ∎
The structure of the price curve themselves () is also interesting when we consider the interpretation of the parameter : the smaller is, the more we care about agents with small utility. Recall that taking of yields max-min welfare, where we only care about the minimum utility. When , we have utilitarian welfare, where we only care about overall efficiency. This roughly corresponds to caring more about agents with higher utility. The limit as corresponds to Nash welfare, which is a mix of caring about both agents with low utility and those with high utility.
We know that maximum Nash welfare allocations are supported by linear price curves, i.e., those with constant marginal prices. When , these marginal prices are increasing, making it easier for agents who are buying less of each good. Since , whenever , so the agents who are buying less are also the ones with lower utility. Thus price curves of this form for are benefiting the agents with low utility. Furthermore, the smaller is, the faster marginal prices grow, which corresponds to favoring agents with low utility even more. On the other hand, when , these marginal prices are decreasing. This favors agents with higher utility, which is consistent with the interpretation of the CES welfare function with .
6.1 Unequal budgets
Finally, we address the setting where agents may have different amounts of money to spend. Let be agent ’s budget. If we instead consider the budget-weighted CES welfare
then the above argument extends directly. Duality tells us that agent ’s utility must be
and by using the same price curves of , we get
so agent is indeed spending exactly her budget.
A social planner may prefer to give the same weight to each agent’s utility, even if the budgets are not the same. Unfortunately, allocations with optimal unweighted CES welfare cannot be supported (at least not exactly) when agents have different budgets. To see this, consider two agents with different budgets and a single good: whichever agent has more money must receive a larger portion of the good. But assuming the agents have the same weight for that good (which holds in the bandwidth allocation setting or when weights are normalized somehow), the unweighted CES welfare optimum would give each agent the same amount. This is analogous to the Fisher market setting: the Fisher market equilibria for unequal budgets are exactly the allocations which maximize the budget-weighted Nash welfare.
In this paper, we analyzed price curves in several different settings, focusing on agents with Leontief utilities. Our first main result was that for strictly increasing price curves, an allocation can be supported if and only if it is GDF. We proved this by defining the agent-order matrix, and using duality theorems to show the existence of a strongly positive solution to a particular system of linear equations. Our second main result was that in the bandwidth allocation setting, the maximum CES welfare allocation can be supported by price curves. These price curves took the simple form of . This is contrast to the standard linear pricing setting, where only maximum Nash welfare allocations can be equilibria.
There are many possible directions for future research. The first is the possibility of tâtonnement for price curves. A tâtonnement process iteratively computes an equilibrium allocation by asking agents for their demand given a set of prices, and then updates prices accordingly (typically by raising the price of goods whose demand exceeds supply and lowering the price of goods whose demand is less than the supply). These processes have been well-studied for linear prices. One approach makes use of the fact that the equilibrium prices are the dual variables in the convex program to maximize Nash welfare, and gives a tâtonnement process that is akin to gradient descent on the dual program. We think that this approach could also yield tâtonnement for price curves, in particular for maximum CES welfare allocations in bandwidth allocation. This is because are exactly the dual variables of the convex program for maximizing CES welfare, just as linear prices are the dual variables of the convex program for maximizing Nash welfare. We think this deserves further study.
A second possible direction is studying price curves for other classes of agent utilities, and in particular, linear utilities. We will discuss in Appendix C some of the challenges that linear utilities pose for analyzing price curves, but perhaps everything would fall into place with the right framework.
Third, future research could consider in more depth the setting of price curves with unequal budgets. Our results did extend to this setting in the sense of budget-weighted group-domination and budget-weighted CES welfare, but weighting agents’ utilities by the amount of money they have seems inappropriate for many contexts.
Last but not least, we are intrigued by the connection between GDF and the agent-order matrix and duality theorems, and we wonder if this connection could be useful for other resource allocation problems as well.
This research was supported in part by NSF grant CCF-1637418, ONR grant N00014-15-1-2786, and the NSF Graduate Research Fellowship under grant DGE-1656518.
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Appendix A Characterization of allocations supported by weakly increasing price curves
In Section 4, we showed that an allocation can be supported with strictly increasing price curves if and only it is GDF. In this section, we provide the analogous necessary and sufficient condition for the case where any (continuous, weakly increasing) price curves are permitted. This boils down to what we called locked-agent-freeness (LAF). LAF is not a particularly interesting condition on its own – though as with GDF it implies a polynomial time algorithm for finding price curves – but it is crucial in allowing us to prove that maximum CES welfare allocations can be supported.
For an allocation , we wish to determine whether there exist price curves such that is a price curve equilibrium. Assuming obeys the supply constraints, we just need to determine whether there exist price curves such that for all .
Recall that if for all and , , and there exists a pair such that the inequality is strict. As discussed in Section 4, this implies that the aggregate spending of is at least that of for any , i.e.,
for any price curves . Furthermore, we argued that for strictly increasing , the inequality is strict, so cannot be made to pay as much as . When we allow weakly increasing price curves, simply implies that, for any marginal price where