Online Market Equilibrium with Application to Fair Division

1 Introduction . 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 [ 17 , 20 , 28 , 31 , 38 , 43 ] . 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 , 39 , 47 ] . 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 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 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 [ 12 , 46 ] . 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 , 29 , 40 ] , 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 . Main contributions . We consider the problem of allocating and pricing sequentially arriving items to n buyers . This setting is termed as an online Fisher market . Given a sequence of item arrivals , we define an online market equilibrium as the items ’ allocations and prices that , in hindsight , ensure buyer optimality and market clearance . We propose the PACE dynamics , which can be viewed as a nontrivial instantiation of the dual averaging algorithm on a reformulation of the dual of the Eisenberg-Gale convex program . Leveraging the convergence theory of dual averaging , we show that , when item arrivals are drawn from an ( unknown ) underlying distribution s , possibly over an infinite/continuous item space , PACE ensures the following . • The pacing multipliers generated by PACE converge to the static equilibrium utility prices . Here , “ static ” means w.r.t . to an underlying static Fisher market . • Buyers ’ time-averaged utilities converge to the static equilibrium utilities . • Buyers ’ time-averaged expenditures converge to their respective budgets . These convergences are all in mean square with rates O ( ( log t ) /t ) , O ( ( log t ) /t ) and O ( ( log t ) 2/t ) , respectively , where the constants in these rates involve moderate polynomials of n. In this way , PACE generates allocations and prices that constitute an online market equilibrium in the limit . In particular , the allocations and prices ensure that the allocation is Pareto optimal , and buyers have no regret , no envy , and get at least their proportional share asymptotically . We also extend PACE to the case of quasilinear buyer utilities , which yields the first online algorithm for computing first-price pacing equilibria . Finally , numerical experiments suggest that PACE converges much faster than its theoretical rates in terms of pacing multipliers , utilities and expenditures . 2 Static and Online Fisher Markets . Static Fisher markets and equilibria . We first introduce static Fisher markets and their equilibria . Following the recent work [ 24 , §2 ] , we consider a measurable ( possibly continuous ) item space . Below are the technical preliminaries for the subsequent online setting . They can be skimmed through and referred back to as needed . From now on , we define [ k ] : = { 1 , . . . , k } for any k ∈ N : = { 0 , 1 , 2 , . . . } and R+ ( R++ , resp . ) as the set of nonnegative ( positive , resp . ) real numbers . Let I { A } ∈ { 0 , 1 } denote the indicator function of an event A . ( a ) There are n buyers ( individuals ) , each having a budget Bi > 0 . ( b ) The item space is a finite measure space ( Θ , M , µ ) with 0 < µ ( Θ ) < ∞ . From now on , Lp ( and Lp+ , resp . ) denote the set of ( nonnegative , resp . ) L p functions on Θ for any p ∈ [ 1 , ∞ ] ( including p = ∞ ) . Below are some concrete special cases for illustration . ( i ) Finite : Θ = [ m ] , M = 2 [ m ] = { A : A ⊆ [ m ] } and µ ( A ) = ∑ a∈A µ ( a ) ( all 2 m subsets are measurable and the measure is given by a point mass on each item ) . ( ii ) Lebesgue-measurable : µ is the Lebesgue measure on Rd , M is the Lebesgue σ-algebra and Θ is a ( Lebesgue- ) measurable subset of Rd with positive finite measure . For example , Θ can be a compact subset of Rd with a nonempty interior . ( iii ) Countably infinite : Θ = N and µ ( A ) = ∑ a∈A µ ( a ) for any A ⊆ N , where µ ( N ) < 0 . For example , µ ( a ) can be the probability mass of a Poisson distribution , in which case ( N , M , µ ) is a probability space . ( c ) The supplies of items is s ∈ L∞+ , i.e. , item θ ∈ Θ has supply s ( θ ) . Since Θ is compact , it is measurable with a finite measure . For the finite case Θ = [ m ] , we have s = ( s1 , . . . , sm ) ∈ Rm+ . ( d ) The valuation of each buyer i on all items is vi ∈ L1+ , i.e. , buyer i has valuation vi ( θ ) on item θ ∈ Θ . For the finite case Θ = [ m ] , we have vi = ( vi1 , . . . , vim ) ∈ Rm+ . ( e ) For buyer i , an allocation of items xi ∈ L∞+ gives a utility of ui ( xi ) : = ⟨vi , xi⟩ : = ∫ Θ vi ( θ ) xi ( θ ) dθ , where the angle brackets are based on the notation of applying a bounded linear functional xi to a vector vi in the Banach space L1 and the integral is the usual Lebesgue integral . For the finite case Θ = [ m ] , we have xi = ( xi1 , . . . , xim ) ∈ Rm+ and the utility is ui ( xi ) = ⟨vi , xi⟩ = ∑ j vijxij , the usual Euclidean vector inner product . We will use x ∈ ( L∞+ ) n to denote the aggregate allocation of items to all buyers , i.e. , the concatenation of all buyers ’ allocations . ( f ) The prices of items are modeled as p ∈ L1+ ; in other words , the price of item θ ∈ Θ is p ( θ ) . For the finite case Θ = [ m ] , we have p = ( p1 , . . . , pm ) ∈ Rm+ . ( g ) For a measurable item subset A ⊆ Θ , let vi ( A ) : = ∫ A vi ( θ ) dθ ( and similarly for p and s ) , the vi-induced measure of A . For the finite case Θ = [ m ] , for any item subset A ⊂ [ m ] , vi ( A ) = ∑ j∈A vij ( and similarly for p ( A ) and s ( A ) ) . ( h ) Without loss of generality , we assume a unit total budget ∥B∥1 = 1 , a unit total supply s ( Θ ) = 1 and normalized buyer valuations ⟨vi , s⟩ = 1 . In other words , all items have a total value of 1 for every buyer . Definition 1 . Given item prices p ∈ L1+ , the demand of buyer i is its set of utility-maximizing allocations given the prices and budget : Di ( p ) : = argmax { ⟨vi , xi⟩ : xi ∈ L∞+ , ⟨p , xi⟩ ≤ Bi } . The associated utility level Ûi ( p ) is defined as the value of ⟨vi , xi⟩ for any xi ∈ Di ( p ) . Definition 2 . A market equilibrium ( ME ) is an allocation-price pair ( x∗ , p∗ ) ∈ ( L∞+ ) n × L1+ such that the following holds . ( i ) Supply feasibility : ∑ i x ∗ i ≤ s. ( ii ) Buyer optimality : x∗i ∈ Di ( p∗ ) for all i . ( iii ) Market clearance : ⟨p∗ , s− ∑ i x ∗ i ⟩ = 0 ( any item with a positive price is fully allocated ) . In the above definition and subsequently , all equations involving measurable functions are understood as “ holding almost everywhere. ” For example , ∑ i xi ≤ s means the ( measurable ) set { θ ∈ Θ : ∑ i xi ( θ ) ≤ s ( θ ) } has the same measure as Θ . Given a ME ( x∗ , p∗ ) , we often denote the ( unique ) equilibrium utilities as u∗i = ⟨vi , x∗i ⟩ . For a finite-dimensional linear Fisher market , it is well known that a ME can be computed via solving the EG convex program . Recently , [ 24 ] generalized this framework to handle the case of an infinite item space . More specifically , consider the following ( possibly infinite-dimensional ) convex programs . sup x∈ ( L∞+ ) n ∑ i Bi log⟨vi , xi⟩ s.t . ∑ i xi ≤ s. ( PEG ) inf p∈L1+ , β∈Rn+ ( ⟨p , s⟩ − ∑ i Bi log βi ) s.t . p ≥ βivi , ∀ i . ( DEG ) The following theorem summarizes the results in [ 24 , §3 ] regarding the above convex programs capturing market equilibria . As shown in that work , the above convex programs satisfy strong duality and their optimal solutions ( which correspond to ME ) can be characterized by the KKT optimality conditions . We slightly generalize the assumptions of [ 24 ] by allowing non-uniform item supplies s instead of s ( θ ) = 1 for all θ ∈ Θ . For completeness , a proof , which is mainly based on the proofs of the results in [ 24 , §3 ] , can be found in the Appendix . Theorem 1 . The following hold regarding ( PEG ) and ( DEG ) . • Both suprema are attained . • Given x∗ feasible to ( PEG ) and ( p∗ , β∗ ) feasible to ( DEG ) , they are both optimal if and only if the following holds : ( i ) ⟨p∗ , s− ∑ i x ∗ i ⟩ = 0 ( market clearance ) , ( ii ) ⟨p∗−β∗i vi , x∗i ⟩ = 0 ( buyer i only receives items within its ‘ winning set ’ { p∗ = β∗i vi } ) ( ii ) and ⟨vi , x∗i ⟩ = u∗i : = Bi/β∗i ( buyer i gets its maximum possible utility from x∗i ) . In this case , ( x ∗ , p∗ ) is a ME . • Conversely , for a ME ( x∗ , p∗ ) , it holds that ( i ) x∗ is an optimal solution of ( PEG ) and ( ii ) ( p∗ , β∗ ) , where β∗i : = Bi/⟨vi , x∗i ⟩ , is an optimal solution of ( DEG ) . In the above theorem , β∗i is known as buyer i ’ s utility price , i.e. , price per unit utility at equilibrium . As is well known , in a ME ( x∗ , p∗ ) , the allocations x∗ are inline Pareto optimal , inline envy-free ( in a budget weighted sense , i.e. , ⟨vi , x∗i ⟩/Bi ≥ ⟨vi , x∗k⟩/Bk for all k ̸= i ) , inline proportional ( i.e. , ⟨vi , x∗i ⟩ ≥ ⟨vi , s⟩/n = 1/n ) ; see , e.g. , [ 24 , Theorem 3 ] . Online Fisher markets and equilibria . We now consider a simple online variant of the Fisher market setting , referred to as an online Fisher market ( OFM ) . There are n buyers , each with a valuation vi ∈ L1+ . Assume there are discrete time steps t = 1 , 2 , . . . . At each time step t , an item θt arrives and each buyer i sees a value vi ( θt ) . The item must be allocated irrevocably to one buyer . Each buyer i has a budget Bi > 0 representing her per-period expenditure rate . 1 Next , we introduce the notions of demand , utility level and online market equilibrium in an OFM . All of them are defined based on sequences of arrived items and their prices ; they do not require any distributional assumption on the item arrivals . Definition 3 . Let the arrived items be ( θτ ) τ∈ [ t ] . An allocation ( of arrived items ) is ( xτi ) ( τ , i ) ∈ [ t ] × [ n ] , where xτi ∈ [ 0 , 1 ] is the fraction of the item θτ allocated to buyer i.2 Let the prices of the arrived 1This assumption is similar to one made in the literature on budget management in auctions , where each buyers has a per-period expenditure rate and the overall budget equal to the rate times the number of time periods . If a hard budget cap across all time periods is desired , then PACE and similar mechanisms may deplete some buyers ’ budgets close to the end of the horizon [ 6–8 ] . 2We allow fractional allocations in the definition for more generality . As we will see , fractional allocation is not needed : PACE generates allocations and prices that satisfy the OME conditions asymptotically via assigning each arrived item to one buyer . items be ( pτ ( θτ ) ) τ∈ [ t ] . The demand of each buyer i ( in hindsight ) at time t is Dti = argmax ( zτi ) τ∈ [ t ] { 1 t t∑ τ=1 vi ( θτ ) z τ i : 0 ≤ zτi ≤ 1 , ∀ τ , 1 t t∑ τ=1 pτ ( θτ ) z τ i ≤ Bi } . ( 1 ) Let Û ti be the utility level associated with this demand , i.e. , the maximum value in ( 1 ) . An online market equilibrium ( OME ) is a pair of allocations ( xτi ) ( τ , i ) ∈ [ t ] × [ n ] and prices pτ ( θτ ) such that the following holds . ( i ) Total allocation does not exceed the unit amount of the item ∑ i x τ i ≤ 1 for all τ . ( ii ) Buyers ’ realized allocations are optimal in hindsight : ( xτi ) τ∈ [ t ] ∈ Dti for all i . ( iii ) Market clearance : ∑ i x τ i = 1 for τ such that p τ ( θτ ) > 0 . In words , Û ti is the maximum possible ( time-averaged ) utility buyer i could have attained via choosing from the arrived items ( θτ ) τ∈ [ t ] in hindsight , subject to their respective posted prices ( pτ ( θτ ) ) τ∈ [ t ] and her current total budget tBi , with Dti being the set of such utility-maximizing ( time-indexed ) allocations subject to per-period item availability constraints . An OME is a pair of allocations and prices that make buyers optimal in hindsight and market cleared . Given an OFM , we define the associated underlying static Fisher market as having the same n buyers and an item space Θ with supply s being the ( unknown ) distribution from which the arriving items θt are drawn . To clarify the concepts of OFM and OME , we consider some simple special cases . • Suppose all item arrivals θ1 , . . . , θt are known in advance . Then , the OFM is the same as a static n × t Fisher market with the same buyers and the t items , each having a unit supply . Here , buyer i ’ s valuation of item τ is viτ = vi ( θτ ) . To compute an OME , it suffices to solve the classical ( finite-dimensional ) Eisenberg-Gale convex program , that is , ( PEG ) with Θ = [ t ] , s = ( 1 , . . . , 1 ) ∈ Rt+ and x ∈ Rn×t+ . Let the static ME be ( x∗ , p∗ ) ∈ Rn×t+ ×Rt++ . When each item θτ arrives , OME allocates a fraction x∗iτ of the item to each buyer i and set its price as p ∗ τ . • Suppose the sequentially arriving items are drawn i.i.d . from a known underlying distribution s ∈ L∞+ ( which specifies a random variable θ ∼ s such that P [ θ ∈ A ] = s ( A ) for any measurable set A ⊆ Θ ) and all buyers ’ valuations vi are known . Suppose we have also computed a static ME ( x∗ , p∗ ) ( Definition 2 ) of a market with buyer valuations vi , budgets Bi and item supplies being the distribution s ( the underlying static market ) . Then , when a new item θt ( which is drawn from the distribution s ) arrives at time t , set its price as p∗ ( θt ) and allocate a fraction x∗i ( θt ) /s ( θt ) of it to each buyer i ( assume s ( θt ) > 0 , i.e. , only items with positive supplies can appear ) . Then , the time-averaged utility of each buyer i is 1t ∑t τ=1 vi ( θt ) x ∗ i ( θt ) /s ( θt ) , which converges to Eθ∼s [ vi ( θ ) xi ( θ ) /s ( θ ) ] = ∫ Θ vi ( θ ) x ∗ i ( θ ) dθ = u ∗ i a.s. by to the Strong Law of Large Numbers . Since the online process is carried out using static equilibrium prices and allocations , the static ME properties ( Definition 2 ) ensure the required OME properties hold asymptotically . The above special cases require full knowledge of either the exact future item arrivals or the underlying static market to attain an OME . Next , we propose a simple , distributed dynamics which generates allocations and prices that satisfy the OME conditions asymptotically without requiring such knowledge ( in particular , without knowledge of the distribution s ) .

Summary: This paper investigates the problem of finding market equilibrium in an online Fisher market. Different from the traditional static Fisher market, in the online setting the items arrive one by one following an unknown distribution. The algorithm needs to allocate the items in the same online fashion with the goal that the time-averaged allocation and utilities asymptotically converge to an online market equilibrium. The main result of this paper is a simple PACE algorithm that achieves this goal. The algorithm maintains a pacing variable for each buyer and at each step assigns the item to the buyer with the highest paced value, after which the pacing variables are updated accordingly. In the essence, this PACE algorithm is a version of the dual averaging algorithm applied to the dual of the EG convex program, and therefore enjoys many nice converging properties. The authors also discuss the implication of this PACE algorithm to the online fair division problem and extend it to the Fisher market with a continuum of items. Finally, experiments are conducted to show its quick convergence in practice.