Online Market Equilibrium with Application to Fair Division

A market is said to be in equilibrium when supply is equal to demand. Computing prices and allocations which constitute a market equilibrium (ME) has long been a topic of interest [43, 28, 38, 20, 17, 31]. Most existing work focuses on the case of static markets. However, in this paper we consider the case of online markets where items arrive sequentially. We consider the extension of market equilibrium to this setting and provide market dynamics which quickly converge to an equilibrium in the case of online Fisher markets.

In static Fisher markets there is a fixed supply of each item, individual preferences are linear, additive, and items are divisible (or equivalently, randomization is allowed so individuals can purchase not just items but lotteries over items). In general, finding market equilibria is a hard problem [14, 47, 39]. However, in static linear Fisher markets, equilibrium prices and allocations can be computed via solving the Eisenberg-Gale (EG) convex program [22, 37].

We consider an online extension of Fisher markets where buyers are constantly present but items arrive one-at-a-time. Buyers’ budgets are per-period and represent their respective ‘bidding powers’ instead of being binding constraints. We extend the definition of market equilibrium to the online setting: online equilibrium allocations and prices are time-indexed and, when averaged across time, form an equilibrium in a corresponding static Fisher market where item supplies are proportional to item arrival probabilities. Due to the stochastic nature of online Fisher markets, any online algorithm can only attain an online market equilibrium asymptotically, that is, the allocations and prices approximately satisfy the equilibrium conditions after running the algorithm for a long time.

We propose market dynamics that find these equilibria in an online fashion based on the dual averaging algorithm applied to a reformulation of the dual of the EG convex program. We refer to this mechanism as PACE (Pace According to Current Estimated utility). In PACE, each buyer is assigned a utility pacing multiplier at time $0$. When an item arrives, the individual with the highest adjusted utility (its valuation times the multiplier) receives that item and pays a price equal to its adjusted utility. The pacing multipliers of all individuals are then adjusted according to a closed-form rule which is given by the time average of the subgradient of the dual of the EG program. Intuitively, the pacing multipliers of those that did not receive the item go up while the receiver’s typically (but not always) goes down. We show that PACE yields item allocations and prices that satisfy various equilibrium properties asymptotically, for example no-regret and envy-freeness.

One important application of market equilibrium is fair allocation using the competitive equilibrium from equal incomes (CEEI) mechanism [46, 12]. In CEEI, each individual is given an endowment of faux currency and reports her valuations for items; then, a market equilibrium is computed and the items are allocated accordingly. However, many fair division problems are online rather than static. These include the allocation of impressions to content in certain recommender systems [34], workers to shifts, donations to food banks [2], scarce compute time to requestors [25, 40, 29], or blood donations to blood banks [32]. Similarly, online advertising can also be thought of as the allocation of impressions to advertisers via a market though with a budget of real money rather than faux currency. In the static CEEI case with linear additive preferences, the resulting equilibrium outcomes (i.e. results of the EG program) have been described as “perfect justice” [3]. In the online case, PACE achieves the same fair allocations as CEEI asymptotically. See Appendix A for more related work in the areas of (static and online) equilibrium computation and fair division.

We evaluate PACE experimentally in several market datasets. Convergence to good outcomes happens quickly in experiments. Taken together our results, we conclude that PACE is an attractive algorithm for both computing online market equilibria and online fair division.

In this section, we introduce the PACE (Pacing According to Current Estimated utility) dynamics that prices and allocates sequentially arriving items via (i) maintaining a pacing multiplier for each buyer and (ii) simple, distributed updates.³³3Pacing and pacing multipliers are terminology in budget management in large-scale ad auctions [18, 19]. In §5, we will show that PACE is an instantiation of dual averaging [48], a stochastic first-order method for regularized optimization, applied to a reformulation of ($\mathcal{D}_{\rm EG}$).

In the PACE dynamics, each buyer maintains a pacing multiplier $\beta^{t}_{i}$ (with an initial value $\beta^{0}_{i}=(B_{i}+1)/2$, or any value in $[B_{i},1]$). At time step $t$, the following events take place.

1. (a)

   An item $\theta_{t}$ appears and each buyer $i$ sees a value $v_{i}(\theta_{t})$ for the item.

2. (b)

   Each buyer $i$ bids their paced value $\beta^{t}_{i}v_{i}(\theta_{t})$ for the item.

3. (c)

   The item is allocated to the highest bidder (the winner at $t$): $i_{t}=\operatorname*{arg\,max}_{i}\beta^{t}_{i}v_{i}(\theta_{t})$, with ties broken arbitrarily. For concreteness, we always choose the lowest winning index, i.e.,

   $$i_{t}=\min\operatorname*{arg\,max}_{i}\beta^{t}_{i}v_{i}(\theta_{t}).$$

   Then, the price of $\theta_{t}$ is set by the first-price rule

   $$p^{t}(\theta_{t})=\max_{i}\beta^{t}_{i}v_{i}(\theta_{t})=\beta^{t}_{i_{t}}v_{i}(\theta_{t})$$

   and the winner $i_{t}$ pays this price $p^{t}(\theta_{t})$ for the item $\theta_{t}$.

4. (d)

   Each buyer $i$ gets a utility

   $$u^{t}_{i}=v_{i}(\theta_{t})\mathbb{I}\{i=i_{t}\}.$$

   In other words, the winner $i_{t}$ gets $v_{i_{t}}(\theta_{t})$ and other buyers get zero.

5. (e)

   Each buyer $i$ updates its cumulative average utility $\bar{u}^{t}_{i}$:

   $$\bar{u}^{t}_{i}=\frac{1}{t}\sum_{\tau=1}^{t}u^{\tau}_{i}=\frac{t-1}{t}\bar{u}^{t-1}_{i}+\frac{1}{t}u^{t}_{i}.$$

6. (f)

   Each buyer $i$ updates their pacing multiplier $\beta^{t+1}_{i}$ as follows:

   $$\beta^{t+1}_{i}=\Pi_{[l_{i},h_{i}]}(B_{i}/\bar{u}^{t}_{i}):=\min\{\max\{l_{i},B_{i}/\bar{u}^{t}_{i}\},h_{i}\}.$$

   where $l_{i}=B_{i}/(1+\delta_{0})$ and $h_{i}=1+\delta_{0}$ for some fixed $\delta_{0}>0$ (e.g., $\delta_{0}=0.05$).

As will be seen in §5, buyer $i$’s equilibrium pacing multiplier (utility price) satisfies $l_{i}<\beta^{*}_{i}<h_{i}$ and her per-period utility $u^{t}_{i}$ corresponds to the $i$th component of a stochastic subgradient of a function on $\beta$ in a reformulation of the convex program ($\mathcal{D}_{\rm EG}$), on which we run dual averaging. Furthermore, the update rule for $\beta_{i}^{t+1}$ is such that, if the realized utilities $\bar{u}^{t}_{i}$ were the true static equilibrium utility for buyer $i$, then $\beta_{i}^{t+1}$ would be the equilibrium multiplier. Note that PACE does not randomize (any randomness can only come from the market environment from which item arrivals are drawn) and assigns every item to a single buyer without splitting it.

The simplicity and distributed nature of PACE makes it desirable for large-scale practical use.

• •

  It can be run on arbitrary sequential item arrivals and only requires buyers’ valuations $v_{i}(\theta_{t})$ on the arrived items (rather than all valuations $v_{i}$ over the potentially large item space). No parameter tuning is needed (in particular, no stepsize tuning as in many first-order optimization methods).

• •

  When run as a centralized allocation mechanism, PACE only needs to maintain $O(n)$ scalars, namely, $\beta^{t}_{i}$, $B_{i}$ and $\bar{u}^{t}_{i}$ for all $i$. At time $t$, it observes buyers’ valuations $v_{i}(\theta_{t})$ of the item $\theta_{t}$, compute bids $\beta^{t}_{i}v_{i}(\theta_{t})$, finds the winner $i_{t}$, set the price as the maximal bid $\beta^{t}_{i_{t}}v_{i_{t}}(\theta_{t})$ and allocates the item to the winner; finally, it updates $\bar{u}^{t}$ and $\beta^{t+1}$ as in f, which takes $O(n)$ time.

• •

  PACE can also be run among the buyers in a decentralized manner, in which case each buyer only maintains two scalar values: the pacing multiplier $\beta^{t}_{i}$ and time-averaged utility $\bar{u}^{t}_{i}$. When a new item arrives, each buyer only performs a few simple arithmetic operations to create a bid $\beta^{t}_{i}v_{i}(\theta_{t})$, receives her utility (if she wins) and subsequently updates $\bar{u}^{t}_{i}$ and $\beta^{t+1}_{i}$.

These make PACE suitable for Internet-scale online fair division and online Fisher market applications. In particular, it is very reminiscent of how Internet advertising auctions are run. There, a similar auction-based system is used, with the pacing multiplier ensuring that each advertiser smooths out their budget expenditure across the many auctions. The primary difference between this and our setting is that (i) the auction can be first-price or second-price and (ii) buyers usually have quasilinear utilities, that is, utility of the item minus the expenditure (price paid) [18, 8, 6, 7]. In §C, we extend PACE to quasilinear utilities, which provides a novel online algorithm for first-price pacing equilibrium computation [19].

In this section, we briefly recap the setup and general convergence results of dual averaging [48, 35], which will be used in the analysis of PACE. First, we introduce some notation for this and subsequent sections. Let $\mathbf{e}^{(i)}$ denote the $i$’th unit basis vector in $\mathbb{R}^{n}$ and $\mathbf{1}\in\mathbb{R}^{n}$ denote the vector of $1$’s. For $x,y\in\mathbb{R}^{n}$, $[x,y]$ denotes the Cartesian product of intervals $\prod_{i=1}^{n}[x_{k},y_{k}]\subseteq\mathbb{R}^{n}$. All norms $\|\cdot\|$ without a subscript are Euclidean $2$-norms, unless otherwise stated.

Let $\Psi$ be a closed convex function with domain ${\rm dom}\,\Psi:=\{w\in\mathbb{R}^{n}:\Psi(w)<\infty\}$. Let $Z\subseteq\mathbb{R}^{d}$ be an arbitrary sample space. For each $z\in Z$, let $f_{z}$ be a convex and subdifferentiable function on ${\rm dom}\,\Psi$. Considers the following regularized convex optimization problem [48, §1.1]:

where the expectation is taken over a probability distribution $\mathcal{D}$ on $Z$. A more general online optimization setting, as described in [48, §1.2] is as follows. At each time $t=1,2,3,\dots$, we must choose an action $w^{t}$ before a new, unknown convex loss function $f_{t}$ arrives, which incurs a loss $f_{t}(w^{t})$ (a special case is i.i.d. sampled functions, i.e., $f_{t}=f_{z_{t}}$, where $z_{t}$ are i.i.d. samples drawn from $\mathcal{D}$). The goal is to minimize regret when comparing our sequence of actions $w^{1},w^{2},\ldots$ to any fixed action $w$. Here, the regret against $w$ is defined as

and the overall (maximal) regret up to time $t$ is $R_{t}=\max_{w}R_{t}(w)$. We assume access to an oracle that, given any $f_{t}$ and $w\in{\rm dom}\,\Psi$, returns a subgradient $g^{t}\in\partial f_{t}(w)$. The dual averaging algorithm (DA) [48, Algorithm 1] is as follows. First, set $w_{1}\in{\rm dom}\,\Psi$ and $\bar{g}^{0}=0$. Then, for each $t=1,2,\dots$, DA performs the following steps:

1. (1)

   Observe $f_{t}$ and compute $g^{t}\in\partial f_{t}(w^{t})$.

2. (2)

   Update the average subgradient (the dual average) via $\bar{g}^{t}=\frac{t-1}{t}\bar{g}^{t-1}+\frac{1}{t}\bar{g}^{t}$.

3. (3)

   Compute the next iterate $w^{t+1}=\operatorname*{arg\,min}_{w}\{\langle\bar{g}^{t},w\rangle+\Psi(w)\}$.

Here, we do not employ any auxiliary regularizing function, since our problem has a natural source of strong convexity (i.e., a strongly convex $\Psi$) through the $-B_{i}\log\beta_{i}$ terms in ($\mathcal{D}_{\rm EG}$). The following convergence guarantee on DA is proved as part of the proof of Corollary 4 in [48].

When solving the stochastic optimization problem (2), in Theorem 2, we can set $G^{2}$ to be an upper bound on $\sup_{w\in{\rm dom}\,\Psi}\mathbf{E}\|g_{z}(w)\|^{2}$, where $g_{z}(w)$ is a subgradient oracle mapping each $(z,w)\in Z\times{\rm dom}\,\Psi$ to a subgradient and the expectation is over $z\sim\mathcal{D}$ and possible randomness of the subgradient oracle. We will shortly see that a reformulation of $\mathcal{D}_{\rm EG}$, when cast into the form (2), exhibits stochastic subgradients that are exactly buyers’ received utilities in each time step. Using Theorem 2, we can show that the sequence of pacing multipliers $\beta^{t}$ generated by PACE converges to the underlying (equilibrium) utility prices $\beta^{*}$ of the static Fisher market.

We will now show that PACE correspond to running DA on the vector $\beta^{t}$ of pacing multipliers for the buyers. To this end, we first reformulate ($\mathcal{D}_{\rm EG}$) into a (finite-dimensional) convex program in $\beta$ in the form of (2):

where $\delta_{0}>0$ is an arbitrarily small constant. The bounds on $\beta$ do not change the optimal solution, because $\beta^{*}_{i}\in(B_{i},1)$ for each $i$. Detailed steps of the reformulation are given in Appendix B.

In order to run DA, we need to compute a subgradient of $f_{\theta}:\beta\mapsto\max_{i}\beta_{i}v_{i}(\theta)$ at any $\theta\in\Theta$. Following [24, §5], since $f_{\theta}$ is a piecewise linear function, a subgradient is

where $i^{*}=\min\operatorname*{arg\,max}_{i}\beta_{i}v_{i}(\theta)$ is the winner (see, e.g., [10, Theorem 3.50]).

We can now show that the PACE dynamics corresponds to running DA on (3). First, choose an arbitrary $\beta^{0}\in(B/(1+\delta_{0}),(1+\delta_{0})\mathbf{1})$ and $\bar{g}^{0}=0$. At each time step $t=1,2,\dots$, given the current pacing multiplier $\beta^{t}$, DA applied to (3) unrolls the following steps.

• •

  An item $\theta_{t}$ arrives, having value $v_{i}(\theta_{t})$ for each buyer $i$. The function $f_{t}$ in DA is

  $$f_{\theta_{t}}:\beta\mapsto\max_{i}\beta_{i}v_{i}(\theta_{t}).$$

• •

  The winner is $i_{t}=\min\operatorname*{arg\,max}_{i}\beta^{t}_{i}v_{i}(\theta_{t})$ and a subgradient is

  $$g^{t}=v_{i_{t}j_{t}}\mathbf{e}^{(i_{t})}\in\partial f_{t}(\beta^{t}).$$

  Its $i$th entry is exactly the realized (single-period) utility of individual $i$ at time $t$ in PACE, that is, $g^{t}_{i}=v_{i}(\theta_{t})\mathbb{I}\{i=i_{t}\}=u^{t}_{i}$.

• •

  Update the dual average (time-averaged utilities): for each $i$, compute $\bar{g}^{t}=\frac{t-1}{t}\bar{g}^{t-1}+\frac{1}{t}g^{t}$, i.e.,

  $$\bar{g}^{t}_{i}=\frac{t-1}{t}\bar{g}^{t-1}_{i}+\frac{1}{t}v_{i}(\theta_{t})\mathbb{I}\{i=i_{t}\}.$$

• •

  Update the pacing multipliers:

  $$\beta^{t+1}=\operatorname*{arg\,min}_{\beta\in[B/(1+\delta_{0}),(1+\delta_{0})\mathbf{1}]}\left\{\langle\bar{g}^{t},\beta\rangle-\sum_{i}B_{i}\log\beta_{i}\right\}.$$

  The minimization problem is separable in each $i$ and exhibits a simple and explicit solution which recovers step f in PACE (where $\bar{g}^{t}_{i}=\bar{u}^{t}_{i}$):

  $$\beta^{t+1}_{i}=\operatorname*{arg\,min}_{\beta_{i}\in[B/(1+\delta_{0}),1+\delta_{0}]}\left\{\bar{g}^{t}_{i}\beta_{i}-B_{i}\log\beta_{i}\right\}\ \Rightarrow\ \beta^{t+1}_{i}=\Pi_{[B_{i}/(1+\delta_{0}),1+\delta_{0}]}\left(\frac{B_{i}}{\bar{u}^{t}_{i}}\right).$$

As mentioned earlier, PACE does not require a stepsize parameter. This is because DA is stepsize-free given a strongly convex regularizer $\Psi$, which is indeed the case in our reformulation (3). In addition, in the above update step for $\beta^{t+1}_{i}$, the directions of change are as follows.

• •

  For a non-winner $i\neq i_{t}$, we have $u^{t}_{i}=0$ and hence $\bar{u}^{t}_{i}\leq\bar{u}^{t-1}_{i}$. This implies $\beta^{t+1}_{i}\geq\beta^{t}_{i}$. In words, a non-winner’s pacing multiplier weakly increases. The increase is strict if $\bar{u}_{i}^{t-1}>0$, i.e., buyer $i$ has already received a nonzero utility.

• •

  For the winner $i_{t}$, $\bar{u}^{t}_{i_{t}}$ may become greater than $\bar{g}^{t-1}_{i_{t}}$, in which case $\beta^{t+1}_{i_{t}}\leq\beta^{t}_{i_{t}}$. In words, the winner’s pacing multiplier may go up or down.

In order to analyze PACE, we assume $v_{i}(\Theta)=1$, $v_{i}\in L^{\infty}_{+}$ (normalized and a.e.-bounded valuations)⁴⁴4The a.e.-boundedness assumption is needed in subsequent convergence analysis. Since $\Theta$ has a finite measure, it holds that $L^{\infty}_{+}\subseteq L^{1}_{+}$. For a finite item space $\Theta=[m]$, both are equal to $\mathbb{R}^{m}_{+}$. and that there is an underlying item distribution $s\in L^{\infty}_{+}$ from which the item arrivals $\theta_{t}$, $t=1,2,\dots$ are drawn i.i.d.⁵⁵5The distributional assumption on item arrivals (i.e., they are drawn i.i.d. from an unknown distribution $s$) is needed to establish asymptotic equilibrium properties of PACE. See Appendix B for an example that any algorithm can yield arbitrarily suboptimal allocations without such a distributional assumption. Define the underlying static Fisher market as one having the same $n$ buyers (each with valuation $v_{i}$ and budget $B_{i}$) and item supplies $s$. Denote the equilibrium utilities and utility prices w.r.t. the underlying static market as $u^{*}$ and $\beta^{*}$, respectively. We further assume that the valuations are $v_{i}\in L^{\infty}_{+}$ (i.e., a.e.-bounded on the item space). This is not restrictive: since an individual item $\theta$ has value $v_{i}(\theta)$ for each buyer $i$, it should be a finite value.