On the Efficiency of the Proportional Allocation Mechanism for Divisible Resources

We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction proportional to her bids. We quantify the inefficiency of Nash equilibria by studying the Price of Anarchy (PoA) of the induced game under complete and incomplete information. When agents' valuations are concave, we show that the Bayesian Nash equilibria can be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure equilibria. Next, we upper bound the PoA over Bayesian equilibria by 2 when agents' valuations are subadditive, generalizing and strengthening previous bounds on lattice submodular valuations. Furthermore, we show that this bound is tight and cannot be improved by any simple or scale-free mechanism. Then we switch to settings with budget constraints, and we show an improved upper bound on the PoA over coarse-correlated equilibria. Finally, we prove that the PoA is exactly 2 for pure equilibria in the polyhedral environment.


Introduction
Allocating network resources, like bandwidth, among agents is a canonical problem in the network optimization literature. A traditional model for this problem was proposed by Kelly [14], where allocating these infinitely divisible resources is treated as a market with prices. More precisely, agents in the system submit bids on resources to express their willingness to pay. After soliciting the bids, the system manager prices each resource with an amount equal to the sum of bids on it. Then the agents buy portions of resources proportional to their bids by paying the corresponding prices. This mechanism is known as the proportional allocation mechanism or Kelly's mechanism in the literature.
The proportional allocation mechanism is widely used in network pricing and has been implemented for allocating computing resources in several distributed systems [5]. In practice, each agent has different interests for different subsets and fractions of the resources. This can be expressed via a valuation function of the resource allocation vector, that is typically private knowledge to each agent. Thus, agents may bid strategically to maximize their own utilities, i.e., the difference between their valuations and payments. Johari and Tsitsiklis [12] observed that this strategic bidding in the proportional allocation mechanism leads to inefficient allocations, that do not maximize social welfare. On the other hand, they showed that this efficiency loss is bounded when agents' valuations are concave. More specifically, they proved that the proportional allocation game admits a unique pure equilibrium with Price of Anarchy (PoA) [15] at most 4/3.
An essential assumption used by Johari and Tsitsiklis [12] is that agents have complete information of each other's valuations. However, in many realistic scenarios, the agents are only partially
Remark 2. Lattice submodular functions used in [20] are subadditive (see Section 4). In the case of a single variable (single resource), any concave function is subadditive; more precisely, concave functions are equivalent to lattice submodular functions in this case. However, concave functions of many variables may not be subadditive [18].
In the Bayesian setting, the valuation of each agent i is drawn from a set of possible valuations V i , according to some known probability distribution D i . We assume that D i 's are independent, but not necessarily identical over the agents.
A mechanism can be represented by a tuple (x, q), where x specifies the allocation of resources and q specifies the agents' payments. In the mechanism, every agent i submits a non-negative bid b ij for each resource j. The proportional allocation mechanism determines the allocation x i = (x ij ) j and payment q i , for each agent i, as follows: When all agents bid 0, the allocation can be defined arbitrarily, but consistently.
Nash Equilibrium. We denote by b = (b 1 , . . . , b n ) the strategy profile of all agents, where b i = (b i1 , . . . , b im ) denotes the pure bids of agent i for the m resources.
we denote the strategies of all agents except for i. Any mixed, correlated, coarse correlated or Bayesian strategy B i of agent i is a probability distribution over b i . For any strategy profile b, x(b) denotes the allocation and q(b) the payments under the strategy profile b. The utility u i of agent i is defined as the difference between her valuation for the received allocation and her payment: Definition 3. A bidding profile B forms the following equilibrium if for every agent i and all bids b ′ i : Pure Nash equilibrium: The first four classes of equilibria are in increasing order of inclusion. Moreover, any mixed Nash equilibrium is also a Bayesian Nash equilibrium.
Price of Anarchy (PoA). Our global objective is to maximize the sum of the agents' valuations for their received allocations, i.e., to maximize the social welfare SW(x) = i∈[n] v i (x i ). Given the valuations, v, of all agents, there exists an optimal allocation . . , o im ) we denote the optimal allocation to agent i. For simplicity, we use SW(b) and v i (b) instead of SW(x(b)) and v i (x i (b)), whenever the allocation rule x is clear from the context. We also use shorter notation for expectations, e.g. we use where E(I) is the set of pure Nash, mixed Nash, correlated, coarse correlated or Bayesian Nash equilibria for the specific instance I ∈ I, respectively 3 .
Budget Constraints. We also consider the setting where agents are budget-constrained. That is, the payment of each agent i cannot be higher than c i , where c i is a non-negative value denoting agent i's budget. Following [2,20], we use Effective Welfare as the benchmark: EW(x) = i min{v i (x i ), c i }. In addition, for any randomized allocation x, the expected effective welfare is defined as:

Concave Valuations
In this section, we show that for concave valuations on multiple resources, Bayesian equilibria can be arbitrarily inefficient. More precisely, we prove that the Bayesian PoA is Ω( √ m) in contrast to the constant bound for pure equilibria [12]. Therefore, there is a big gap between complete and incomplete information settings. We state our main theorem in this section as follows.
Theorem 5. When valuations are concave, the PoA of the proportional allocation mechanism for Bayesian equilibria is at least √ m 2 . Proof. We consider an instance with m resources and 2 agents with the following concave valuations. v 1 (x) = min j {x j } and v 2 (x) is drawn from a distribution D 2 , such that some resource j ∈ [m] is chosen uniformly at random and then v 2 ( Under this bidding profile, agent 1 bids the same value for all resources, and agent 2 only bids positive value for a single resource associated with her valuation. Suppose that agent 2 has positive valuation for resource j, i.e., v 2 (x) = x j / √ m. Then the rest m − 1 resources are allocated to agent 1 and agents are competing for resource j. Bidder 2 has no reason to bid positively for any other resource. If she bids any value b ′ 2j for resource j, her utility would be . For b 1j = δ/m − δ, the utility of agent 2 is maximized for b ′ 2j = 1/( √ m + 1) 2 = δ by simple calculations.
Since v 1 (x) equals the minimum of x's components, agent 1's valuation is completely determined by the allocation of resource j. So the expected utility of agent 1 under 3 The expectation over v is only needed for the definition of Bayesian PoA.

The inequality comes from the fact that
But the optimal social welfare is 1 by allocating to agent 1 all resources. So, PoA ≥ √ m 2 .

Subadditive Valuations
In this section, we focus on agents with subadditive valuations. We prove that the proportional allocation mechanism is at least 50% efficient for coarse correlated equilibria and Bayesian Nash equilibria, i.e., PoA ≤ 2. We further show that this bound is tight and cannot be improved by any simple or scale-free mechanism. Before proving our PoA bounds, we show that the class of subadditive functions is a superclass of lattice submodular functions. Proof. It has been shown in [20] that for any lattice submodular function The second equality is due to the definition of partial derivative and the inequalities is due to the monotonicity of ∂v ∂x j (x).

Upper bound
A common approach to prove PoA upper bounds is to find a deviation with proper utility bounds and then use the definition of Nash equilibrium to bound agents' utilities at equilibrium. The bidding strategy described in the following lemma is for this purpose.

Lemma 7.
Let v be any subadditive valuation profile and B be some randomized bidding profile. For any agent i, there exists a randomized bidding strategy a i (v, B −i ) such that: Proof. Let p ij be the sum of the bids of all agents except i on resource j, i.e., p ij = k =i b kj . Note that p ij is a random variable that depends on b −i ∼ B −i . Let P i be the propability distribution of p i = (p ij ) j . Inspired by [9], we consider the bidding strategy a i (v, The first inequality follows by swapping p ij and b ′ ij and using the subadditivity of v i . The second inequality comes from the fact that o v ij ≤ 1. The lemma follows by summing up over all agents and the fact that i∈[n] o v ij = 1.
Theorem 8. The coarse correlated PoA of the proportional allocation mechanism with subadditive agents is at most 2.
Proof. Let B be any coarse correlated equilibrium (note that v is fixed). By Lemma 7 and the definition of the coarse correlated equilibrium, we have Theorem 9. The Bayesian PoA of the proportional allocation mechanism with subadditive agents is at most 2.
Proof. Let B be any Bayesian Nash Equilibrium and let v i ∼ D i be the valuation of each agent i drawn independently from D i . We denote by C = (C 1 , C 2 , . . . , C n ) the bidding distribution in B which includes the randomness of both the bidding strategy b and of the valuations v. The utility of agent i with valuation v i can be expressed by By the definition of the Bayesian Nash equilibrium, we obtain By taking expectation over v i and summing up over all agents,

Simple mechanisms lower bound
Now, we show a lower bound that applies to all simple mechanisms, where the bidding space has size (at most) sub-doubly-exponential in m. More specifically, we apply the general framework of Roughgarden [19], for showing lower bounds on the price of anarchy for all simple mechanisms, via communication complexity reductions with respect to the underlying optimization problem. In our setting, the problem is to maximize the social welfare by allocating divisible resources to agents with subadditive valuations. We proceed by proving a communication lower bound for this problem in the following lemma.
Lemma 10. For any constant ε > 0, any (2 − ε)-approximation (non-deterministic) algorithm for maximizing social welfare in resource allocation problem with subadditive valuations, requires an exponential amount of communication.
Proof. We prove this lemma by reducing the communication lower bound for combinatorial auctions with general valuations (Theorem 3 of [17]) to our setting (see also [7] for a reduction to combinatorial auctions with subadditive agents). Nisan [17] used an instance with n players and m items, with n < m 1/2−ε . Each player i is associated with a set T i , with |T i | = t for some t > 0. At every instance of this problem, the players' valuations are determined by sets I i of bundles, where I i ⊆ T i for every i. Given I i , player i's valuation on some subset S of items is v i (S) = 1, if there exists some R ∈ I i such that R ⊆ S, otherwise v i (S) = 0. In [17], it was shown that distinguishing between instances with optimal social welfare of n and 1, requires t bits of communication. By choosing t exponential in m, their theorem follows.
We prove the lemma by associating any valuation v of the above combinatorial auction problem, to some appropriate subadditive valuation v ′ for our setting. For any player i and any fractional allocation It is easy to verify that v ′ i is subadditive. Notice that v ′ i (x) = 2 only if there exists R ∈ I i such that player i is allocated a fraction higher than 1/2 for every resource in R. The value 1/2 is chosen such that no two players are assigned more than that fraction from the same resource. This corresponds to the constraint of an allocation in the combinatorial auction where no item is allocated to two players. Therefore, in the divisible goods allocation problem, distinguishing between instances where the optimal social welfare is 2n and n + 1 is equivalent to distinguishing between instances where the optimal social welfare is n and 1 in the corresponding combinatorial auction and hence requires exponential, in m, number of communication bits.
The PoA lower bound follows the general reduction described in [19].
Theorem 11. The PoA of ǫ-mixed Nash equilibria 4 of every simple mechanism, when agents have subadditive valuations, is at least 2.
Remark 12. This result holds only for ǫ-mixed Nash equilibria. Considering exact Nash equilibria, we show a lower bound for all scale-free mechanisms in the following section.

Scale-free mechanisms lower bound
Here we prove a tight lower bound for all scale-free mechanisms including the proportional allocation mechanism. A mechanism (x, q) is said to be scale-free if a) for every agent i, resource j and constant c > 0, x i (c · b j ) = x i (b j ). Moreover, for a fixed b −i , x i (·) is non-decreasing and positive whenever b ij is positive. b) The payment for agent i depends only on her bids b i = (b ij ) j and equals to j∈[m] q i (b ij ) where q i (·) is non-decreasing, continuous, normalized (q i (0) = 0), and there always exists a bid b ij such that q i (b ij ) > 0.
Theorem 13. The mixed PoA of scale-free mechanisms when agents have subbaditive valuations, is at least 2.
Proof. Given a mechanism (x, q), we construct an instance with 2 agents and m resources. Let V be a positive value such that V /m is in the range of both q 1 and q 2 . This can be always done due to our assumptions on q i . Let T 1 and T 2 be the values such that q 1 (T 1 ) = q 2 (T 2 ) = V /m. W.l.o.g. we assume that T 1 ≥ T 2 . By monotonicity of q 1 , q 1 (T 2 ) ≤ V /m. Pick an arbitrary value a ∈ (0, 1), and let h 1 = x 1 (a, a) and h 2 = x 2 (a, a). By the assumption that We define the agents' valuations as: We claim that the following mixed strategy profile B is a Nash equilibrium. Agent 1 picks resource l uniformly at random and bids b 1l = y, and b 1k = 0, for k = l, where y is a random variable drawn by the cumulative distribution G(y) = mq 2 (y) Notice that G(·) and F (·) are valid CDFs, due to monotonicity of q i (·). Since G(T 2 ) = 1, F (T 2 ) = 1 and q i (·) is continuous, G(y) and F (y) are continuous in (0, ∞) and therefore both functions have no mass point in any y = 0. We assume that if both agents bid 0 for some resource, agent 2 takes the whole resource. We are ready to show that B is a Nash equilibrium. For the following arguments notice that G(T 2 ) = 1, F (T 2 ) = 1 and G(0) = 0. If agent 1 bids any y in the range (0, T 2 ] for a single resource j and zero for the rest, then she gets allocation of at least h 1 (that she values for 2v), only if y ≥ z, which happens with probability F (y). This holds due to monotonicity of x 1 (·) with respect to y. Otherwise her value is v. Therefore, her expected valuation is v + F (y)v. So, for every y ∈ (0, T 2 ] her expected utility is v + F (y)v − q 1 (y) = 2v − q 1 (T 2 ). If agent 1 picks y according to G(y), her utility is still 2v − q 1 (T 2 ), since she bids 0 with zero probability. Suppose agent 1 bids y = (y 1 , . . . , y m ), y j ∈ [0, T 2 ] for every j, with at least two positive bids, and w.l.o.g., assume y 1 = max j y j . If z > y 1 , agent 1 has value v for the allocation she receives. If z ≤ y 1 , agent 1 has value 2v, but she pays more than q 1 (y 1 ). So, this strategy is dominated by the strategy of bidding y 1 for the first resource and zero for the rest. Bidding greater than T 2 for any resource is dominated by the strategy of bidding exactly T 2 for that resource.
If agent 2 bids z ∈ [0, T 2 ] for all resources, she gets an allocation of at least h 2 for all the m resources with probability G(z) (due to monotonicity of x 2 (·) with respect to z and to the tie breaking rule). So, her expected utility is V + G(z)V − mq 2 (z) = V . Bidding greater than T 2 for any resource is dominated by bidding exactly T 2 for this resource. Suppose that agent 2 bids any z = (z 1 , . . . z m ), with z j ∈ [0, T 2 ] for every j, then, since agent 1 bids positively for any item with probability 1/m, agent's 2 expected utility is 1 Therefore, it is sufficient to bound the expected social welfare in B. Agent 1 bids 0 with zero probability. So, whenever agent 2 bids 0, she receives exactly m − 1 resources, which she values for V . Agent 2 bids 0 with probability On the other hand, the social welfare in the optimum allocation is 2(V + v) = 2V 1 + 1 √ m (agent 1 is allocated h 1 proportion from one resource and the rest is allocated to agent 2). We conclude that P oA ≥ 2 for large m, converges to 2.

Budget Constraints
In this section, we switch to scenarios where agents have budget constraints. We use as a benchmark the effective welfare similarly to [2,20]. We compare the effective welfare of the allocation at equilibrium with the optimal effective welfare. We prove an upper bound of φ + 1 ≈ 2.618 for coarse correlated equilibria, where φ = √ 5+1 2 is the golden ratio. This improves the previously known 2.78 upper bound in [2] for a single resource and concave valuations.
To prove this upper bound, we use the fact that in the equilibrium there is no profitable unilateral deviation, and, in particular, the utility of agent i obtained by any pure deviating bid a i should be bounded by her budget c i , i.e., j∈[m] a ij ≤ c i . We define v c to be the valuation v suppressed by the budget c, i.e., v c (x) = min{v(x), c}. Note that v c is also subadditive since v is subadditive. For a fixed pair (v, c), let o = (o 1 , . . . , o n ) be the allocation that maximizes the effective welfare. For a fixed agent i and a vector of bids b −i , we define the vector p i as p i = k =i b k . We first show the existence of a proper deviation.

Lemma 14.
For any subadditive agent i, and any randomized bidding profile B, there exists a randomized bid a i (B −i ), such that for any λ ≥ 1, it is Moreover, for any pure strategyâ i in the support of a i (B −i ), jâ ij ≤ c i .
Proof. In order to find a i (B −i ), we define the truncated bid vectorb −i as follows. For any set S ⊆ [m] of resources, we denote by 1 S the indicator vector w.r.t. S, such that x j = 1 for j ∈ S and x j = 0 otherwise. For any vector p i and any λ > 0, let T := T (λ, p i ) be a maximal subset of Now consider the following bidding strategy a i (B −i ): sampling b ′ i ∼P i and bidding a ij = 1 o ij p ij , which contradicts the maximality of T . Next we show for any bid b i and λ > 0, Observe that The claim follows by rearranging terms and taking the expectation We are now ready to prove the statement of the lemma. (1)) For the second inequality, notice that the second term doesn't depend on b ′ i , so we apply Lemma 11 for every b ′ i . For the forth and fifth inequalities, o i ≤ 1 and We are ready to show the PoA bound by using the above lemma.
Theorem 15. The coarse correlated PoA for the proportional allocation mechanism when agents have budget constraints and subadditive valuations, is at most φ + 1 ≈ 2.618.
Proof. Suppose B is a coarse correlated equilibrium. Let A be the set of agents such that for every The latter inequality comes from that agents do not bid higher than their budgets. Let λ = φ. So 1 − 1/λ = 1/(1 + λ). By taking the linear combination and summing up over all agents not in A, we get For every i ∈ A, we consider the deviating bidding strategy a i (B −i ) that is described in Lemma 14, then By summing up over all i ∈ A and by combining with inequality (2) we get Therefore, the PoA with respect to the effective welfare is at most φ + 1. (recall that for Inequality (2) we set λ = φ) By applying Jensen's inequality for concave functions, our upper bound also holds for the Bayesian case with single-resource and concave functions. Proof. Suppose B is a Bayesian Nash equilibrium. Recall that in the Bayesian setting, agent i's type t i = (v i , c i ) are drawn from some know distribution independently. We use the notation C = (C 1 , C 2 , . . . , C n ) to denote the bidding distribution in B which includes the randomness of bidding strategy b and agents' types t, that is b i (t i ) ∼ C i . Then the utility of agent i with type t i is u i (B i (t i ), C −i ). Notice that C −i does not depend on any particular t −i .
Recall that v c (x) = min{v(x), c}. It is easy to check v c is concave if v is concave. For any agents types t = (v, c), let o t = (o t 1 , ..., o t n ) be the allocation vector that maximizes the effective welfare. We define o t i i to be the expected allocation over t −i ∼ D −i to agent i, in the optimum solution with respect to effective welfare, when her type is The latter inequality comes from that agents do not bid above their budget. Let λ = φ. So 1 − 1/λ = 1/(1 + λ). By taking the linear combination, taking the expectation over all t i / ∈ A i and summing up over all agents not in A, we get For every t i ∈ A i , by Lemma 14, there exists a randomized bid a i (t i , B −i ) for agent i, such that, Therefore, the PoA is at most φ + 1.
Remark 17. Syrgkanis and Tardos [20], compared the social welfare in the equilibrium with the effective welfare in the optimum allocation. Caragiannis and Voudouris [2] also give an upper bound of 2 for this ratio in the single resource case. We can obtain the same upper bound by replacing λ with 1 in Lemma 14 and following the ideas of Theorems 8 and 9.

Polyhedral Environment
In this section, we study the efficiency of the proportional allocation mechanism in the polyhedral environment, that was previously studied by Nguyen and Tardos [16]. We show a tight price of anarchy bound of 2 for agents with subadditive valuations. Recall that, in this setting, the allocation to each agent i is now represented by a single parameter x i , and not by a vector (x i1 , . . . , x im ). In addition, any feasible allocation vector x = (x 1 , . . . , x n ) should satisfy a polyhedral constraint A · x ≤ 1, where A is a non-negative m × n matrix and each row of A corresponds to a different resource, and 1 is a vector with all ones. Each agent aims to maximize her utility where v i is a subadditive function representing the agent's valuation. The proportional allocation mechanism determines the following allocation and payments for each agent: where a ij is the (i, j)-th entry of matrix A. It is easy to verify that the above allocation satisfies the polyhedral constraints.
Theorem 18. If agents have subadditive valuations, the pure PoA of the proportional allocation mechanism in the polyhedral environment is exactly 2.
Proof. We first show that the PoA is at most 2. Let o = {o 1 , . . . , o n } be the optimal allocation, b be a pure Nash Equilibrium, and let p ij = k =i b ij . For each agent i, consider the deviating bid b ′ i such that b ′ ij = o i a ij p ij for all resources j. Since b is a Nash Equilibrium, The second inequality is true since A · x ≤ 1, for every allocation x, and therefore o i a ij < 1. The last inequality holds due to subadditivity of v i . By summing up over all agents, we get The last inequality holds due to the fact that p ij ≤ k∈[n] b kj and i∈[n] o i a ij ≤ 1. The fact that PoA ≤ 2 follows by rearranging the terms. For the lower bound, consider a game with only two agents and a single resource where the polyhedral constraint is given by x 1 + x 2 ≤ 1. The valuation of the first agent is v 1 (x) = 1 + ǫ · x, for some ǫ < 1 if x < 1 and v 1 (x) = 2 if x = 1. The valuation of the second agent is ǫ · x. One can verify that these two functions are subadditive and the optimal social welfare is 2. Consider the bidding strategies b 1 = b 2 = ǫ 4 . The utility of agent 1, when she bids x and agent 2 bids ǫ 4 , is given by 1 + ǫ · x x+ǫ/4 − x which is maximized for x = ǫ 4 . The utility of agent 2, when she bids x and agent 1 bids ǫ 4 , is ǫ · x x+ǫ/4 − x which is also maximized when x = ǫ 4 . So (b 1 , b 2 ) is a pure Nash Equilibrium with social welfare 1 + ǫ. Therefore, the PoA converges to 2 when ǫ goes to 0.