Explaining the Law of Supply and Demand via Online Learning
Authors: Stratis Skoulakis
Venue: NeurIPS 2025 poster ()
Keywords: Online Learning
Game Theory
No Regret Minimization
Primary area: theory

Abstract: The *law of supply and demand* asserts that in a perfectly competitive market, the price of a good adjusts to a *market clearing price*. In a market clearing price $p^\star$ the number of sellers willing to sell the good at $p^\star$ equals the number of sellers willing to buy the good at price $p^\star$. In this work, we provide a mathematical foundation on the law of supply and demand through the lens of online learning. Specifically, we demonstrate that if each seller employs a no-swap regret algorithm to set their individual selling price—aiming to maximize its individual revenue—the collective pricing dynamics converge to the market-clearing price $p^\star$ . Our findings offer a novel perspective on the law of supply and demand, framing it as the emergent outcome of an adaptive learning processes among sellers.

Explaining the Law of Supply and Demand via Online Learning
Stratis Skoulakis Aarhus University stratisOcs. au.
Abstract
The law of supply and demand asserts that in a perfectly competitive market, the price of good adjusts market clearing price. In market clearing price the number of sellers willing to sell the good at p* equals the number of sellers willing buy the good price p². In this work, we provide mathematical foundation on the law of supply and demand through the lens of online learning. Specifically, we demonstrate that if each seller employs no-swap regret algorithm individual selling price-aiming to maximize its individual revenue- the collective pricing dynamics converge the market-clearing price p" Our findings offer a novel perspective on the law of supply and demand, framing it as the emergent outcome of an adaptive learning processes among sellers
their
Introduction
The law of supply and demand is a fundamental economic principle explaining the price a good in perfectly competitive market (25). A perfectly competitive market consists of a large number of sellers and buyers where each seller is interested in selling one unit of an indistinguishable good and each buyer is interested in buying one unit of the good. The market is thus described by supply S and demand curve D where ((p)/D(p) is the number of sellers/huyers willing to sell/buy the good price p. The law of supply and demand states that the price of the good will converge to the market clearing price p" where S(p") = D(p" )-the number of sellers equals the number of buyers [22 28
The idea behind the law of supply and demand is very intuitive T If the price is higher than p* then more sellers are willing to sell than buyers willing to buy, leading to a market surplus. As a result, some sellers are not able to sell good and will lower their prices to attract buyers. As prices fall, more buyers are willing to buy, and fewer sellers are willing to sell. Conversely, if the price is below p" this will create market shortage that will in turn cause upward trend of the prices. These adjustments continue until all selling prices reach p" where there is neither market surplus or shortage (25 22 28
However, upon closer examination, the explanation above does not rule out the possibility of persistent price fluctuations where a market surplus leads a shortage, which then causes another surplus, and so on-preventing the selling prices from ever converging to the market-clearing price. The latter raises the following fundamental question:
Question 1. Why do selling prices eventually stabilize to the market clearing price pd instead of constantly oxcillating around it?
Surprisingly, despite the law of supply and demand being one of the most fundamental principles in modern economics, this question remains unanswered. In this work, we provide answers to Question through the lens of game theory and online learning.
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39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Our Contribution and Techniques As already discussed, sellers choose their prices to maximize individual revenue- is, to sell their goods at the highest possible price. We model this price competition through a suitable pricing game (see Definition 3). In this game, each seller selects a price for their good, after which buyers arrive sequentially and purchase from the lowest-priced available seller.
The main result of this work consists in establishing that any Correlated Equilibrium (CE) 3 of the pricing game effectively coincides with the market clearing price of the market.
Informal Theorem In any correlated equilibrium, the selling price is the market clearing price p".
To establish this result, we introduce a novel primal-dual technique. Specifically, we construct an appropriate linear program whose optimal value serves as a lower bound on the probability that Correlated Equilibrium (CE) selects the market clearing price. We then show that this probability is at least 1 by demonstrating that the dual program admits feasible solution with objective value To the best of our knowledge, this the first time the dual-fitting technique (2] is applied in the context of establishing convergence properties of learning dynamics. Thus our technique may be of independent interest.
It is well- that in case agents use no-swap regret algorithm 7 to select their actions in repeated game, then the overall joint time-average behavior converges to Correlated Equilibrium [[7 п. We remark that assuming that sellers use no-swap regret learning is natural assumption since these types of algorithms come with strong optimality guarantees no matter the actions of the other sellers. In view of the above the main take-away message of this work is the following:
Tway Message If all sellers use no-swap regret algorithms to select their prices, the selling price of the good will converge to the market clearing price p".
As result, our results offer a novel perspective the law of supply and demand, framing it as the emergent outcome of a learning process among sellers. The latter reinforces the classical understand- ing of market equilibrium but also bridges it with contemporary decision decision-making algorithms.
On the negative side, we show that the weaker solution concept of Coarse Correlated Equilibrium (CCE) does not necessarily align with the market clearing price. In Theorem 3 we construct an instance of the pricing game that admits CCE which does not coincide with the corresponding
market clearing price. This implies that convergence to the market clearing price cannot be guaranteed under general no-regret dynamics, which are known to converge to CCE (20). That being said, in 4 we empirically evaluate both no-regret and no-swap regret algorithms. Our experiments show that both types of learning dynamics converge to the market clearing price, suggesting that the negative result of Theorem 3 may be circumvented by specific classes of no-regret algorithms.
Section
1.1 Related Work
Our work relates with the line of research studying online lcarning dynamics in various market settings. However none of the prior work has not studied perfectly competitive markets and the law of supply and demand.
A significant body of work has investigated Cournot competition under the lens of online learning. Even-Dar et al. [S] were the first to establish that no-regret dynamics converge to the respective Nash Equilibrium in the case of linear Cournot competition. Nadav et al. (26) extended these convergence results in the case of product differentiation and thet also studied Bertrand Duopoly markets. Fiat et al. [T6] study best-response dynamics in variant of Cournot competition where firms aim to either optimize either profit or revenue while Lin et al. [24] establish that best-response dynamics in Cournot competition converge either to a Nash equilibrium or periodic orbit of length two. Immorlica et al. provide bounds on the Price of Anarchy in Cournot competition in case of coalitions among firms while Shit et al. [27] study the convergence properties of no-regret learning dynamics in case of limited information feedback.
Our work also relates with the line of research studying decentralized dynamics in Fisher markets. Bikhchandani et al. [6] showed that proportional response dynamics-an an update rule closely related to online coneeigge to equilibrium in linear Fisher markets. Kolumbus et al. 23 extend these results in case of asynchronous updates. Cheung et al. [II] study the convergence properties of proportional response dynamics in Fisher markets with CES utilities while Branzei et al. [8] study

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proportional dynamics in exchange economies. Finally [3 14
10
9
30
study the convergence
properties of tätonnement-type algorithms in various Fisher markets.
Babaioff et al. 5 consider a pricing game very similar to ours and establish Price of Anarchy bounds the social welfare well identifying structural conditions for pure Nash equilibrium in case of various types of valuations. Golrezaci [18] study the convergence properties of Online Mirror Descent (OMD) case of firn-competition in case of consumer reference prices. Cheung et al. 12 show that in various types of markets, Online Gradient Ascent admits chaotic behavior. Zhu al. 31 study online learning algorithms for betting markets.
2
Preliminaries and Results
Let n denote the number of seller and buyers. We assume & discrete prices [k] = {0, 1, ......... Each seller i [n] can sell 1 unit of good a price smaller than its marginal cost 8; [k.]. Each buyer j E [n]] is interested in buying unit of good at a price at most its valuation bg E [k]. Without loss of
generality we assume that $1 .... I s and b1 2 .... 2 b,.
The supply curve S [k] + [n]] denotes the number of sellers willing to sell the good at given price, ( = (1 E [n] 8; S p) The demand curve [k] + [x] denotes the number buyers willing to buy the good a given price, D(p) E [m] bj 2 p}
Definition 1. A price p E [k] is market clearing price if and only if S(p) =D(p).
We remark market clearing prices are not necessarily unique.
Example 1. Consider n = 3 sellers and buyers. The marginal costs of the sellers are (81, 82, 83) = (0, 5) and the marginal prices of the buyers (s1, $2, 83) (6, 4, 1). All prices (2, 4] are market clearing prices.
Lemma D establishes the fact that under minimal assumption, the set of market clearing prices is not empty and in fact forms consecutive interval.
Lemma 1. In case the marginal costs and prices s;, st,b, E [k] lies in different places. Then the set of market clearing prices is not empty and always an interval (a set of the form (P1.Pr + 1,...,p2 }.
and D(k) S 1 due to the fact no two buyers' valuation can equal k. Similarly S(1) < 1 and D(1)
Proof. Notice that S(k)
Thus, S(k) D(k) > 0 and S(1) - D(1) [ > 0. Since marginal costs and valuations lie in different places, the difference S(p) D(p) can differ by at most for consecutive prices. Thus, there exists p* E [k] such that S(p*) = D(p*). Since S(.)
increasing and D(.) decreasing function, the set of market clearing prices an interval {PI.PI +l.....,P) P2)
Despite the fact that strictly speaking market, clearing prices are not necessarily unique, if k and n are large enough and the marginal costs and prices are not very concentrated, the different market clearing prices will practical correspond to the exact same price.
Definition 2. We denote with p" the highest market clearing price.
With some abuse of terminology in the rest of the paper we refer to p= as the market clearing price. The law of supply and demand sates that the selling price of the good will converge the market clearing price.
2.1 Pricing Games
In this section we introduce the pricing game in order to provide theoretical foundations on the law of supply and demand. Our goal is to establish that the law of supply and demand is the outcome of the strategic pricing of the sellers in their attempt to maximize their revenue.
"With a slight abuse of notation when s; E [k], we actually mean 8, e for an arbitrarily small > 0. This convention is very convenient since it ensures that seller i E [ni] strictly prefer selling the good at price s; rather not selling all.
3

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In Definition 3 we introduce the pricing game that is an one-shot game capturing the competition between sellers. The strategy each seller is the selling price that the agents selects. When selecting price, an agent needs to balance between high revenue and the risk of not selling its good.
Definition 3. In pricing game, each seller selects price 2 Sy. The payoff Up.. of seller i [n)] is defined as follows:
Let S denote the set of sellers, S =1,.....,n)) All of them initially available.
for each buyer j no
buyer j enters the market and finds the cheapest available seller, How argmin jesP
S bb. Piow
then
1. Buyer j buys the good from flow al price Pisow"
2. Seller how gets utility Un.. (Ps.P=s) = Pinow 8 Show and exits S +- S/filow
A seller E [n]. that did not sell its good, gets utility U.(p..pii)= =
We remark that in the pricing game of Definition 3 only sellers strategic agents. When buyer enters the market, it considers the lowest-priced seller who is still available at that moment. We emphasize that each seller can supply only unit of the good. That is why once seller sells its unit, then it exits the market.
We denote with P; the strategy space of seller i E [n]. P; (s...... k). We also denote with P: = P1 x P2 ... x Pn the set pricing profles. For pricing profile == (Pi....,PM, Pa) P we also the notation p (pt, p-)) denote the price of seller [ni] with the prices selected by the other sellers
2.2 No-Swap Regret Minimization and Correlated Equilibrium
To model the market's behavior over time, we consider sellers repeatedly playing the pricing game of Definition 3 across multiple rounds. We remark that the beginning of each round all sellers admit 1 unit of good regardless of whether they performed a sale in the previous round.
Protocol Pricing Game over time
At each round t = I..... Each seller E [mi]. (secretly) selects price pt E St.
Each seller E [m]] gets utility Uc(p), p",,) (see Definition
Each seller E [n)] learns p", and uses this information to select its next price p. f+1
A seller i E [ri], needs to come up with a pricing strategy that at each round / E [7'] selects a price
of solely based on past prices p_.....,p'_. E [k]. The online learning framework provides such decision-making algorithms that base their decision prior observations (20).
In a pricing games an online learning algorithm A. at round E [T), produces mixed strategy
of A(P:) over set of possible prices P.. The performance of an online learning algorithm A can be quantified through the notion of regret (20 (7].
Definition 4. The regret of an online learning algorithm is defined as
T
(T) =
max EPP
Σ
E Bp.cat
[U.Pi.
Σ Byraat [Ue(P.)L.))
In case of tie. buyer j [n] selects the seller with the highest index.

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The swap regret of an online learning algorithm A is defined as
RA (T) =
max 6:P
Epinat [U:(6(p)), Σ Bpinat (p((..p_)))
t=1
Algorithm is called no-regret if ( o(T)) no matter the prices pl....., p== selected by the other sellers. Respectively if RAA (T) = o(T') the algorithms is called no-swap regret.
no-regret algorithm guarantees that its time-averaged utility converges to the the time- avneeaveedeed utility produced by the best fixed price P E P, regardless the prices selected by the other providers. no-swap regret algorithm provides the stronger optimality guarantees that the time-average payoff' is at least the time-average payoff of the best fixed switching function 8(-)-
It is well known that if all sellers no-regret algorithms to select their prices, the overall behavior converges to a Coarse Correlated Equilibriumm22, Similarly, if sellers use no-swap regret algorithms,
the resulting dynamics converge a Correlated Equilibrium 19 17 formally defined in Definition
Both equilibrium concepts
Definition 5. A probability distribution E ((S) over pricing profiles p == Xie(n] Po is Coarse Correlated Equilibrium for each seller i E [m].
Exr [Us(p:, P-+) 2 max Epon [U:(p',
MEP.
A probability distribution E (SS) over pricing profiles P == XiEln| P.. is Correlated Equilibrium if for each seller i E [[n].
E Eprn UU(pt, 2
E Epun (U:((p)) p-:)).
d.P..rP
2.3 Our Results and Paper Organization
To this end, we present the main contribution of our work: establishing that the law of supply and demand can be viewed as the limiting behavior of no-swap regret algorithms employed by sellers to maximize their individual payoffs.
Theorem 1.
Let pricing game with market clearing price p' and μ E A(P) be Correlated Equilibrium. If (pı,...,Pal Pn) ~ p then with probability 1,
1. all sellers i [n]] with 8; I p", select p; = pp and sell their good.
2. all buyers j E [ni] with by 2 p", buy the good price p"
3. all sellers i E [n] with 8; > pr do not sell anything and all buyers j E [m]] with by < p do not buy anything.
Theorem I establishes that the selling price of the good is the market clearing price p" [k]. This is because executed sales occur at price p². Notice that Theorem I eestablishes that sellers E [n] with marginal costs higher than 8; p" do not sell their good. Similarly, buyers [m] with marginal prices by < p" never buy the good since p" is the lowest price offered in the market.
Combining Theorem I with the known convergence results of no-swap regret dynamics 17 19 we get Corollary [ showing that the law of supply and demand emerges naturally when sellers use no-swap regret algorithms. This is the main takeaway of our work.
Corollary 1. If all sellers in a pricing game use no-swap regret to select their prices, then the price of the good converges to the market-clearing price pz E [k].
In Theorem we show that Coarse Correlated Equilibria (CCE) is not always compatible with the market clearing price. In particular show that there are instances of the pricing game admitting CCES at which the probabily of sellers selecting the market clearing price is arbitrarily close to zero.
Theorem 2. There exists a family of pricing games with = 2 admitting Coarse Correlated Equilibrium μ A(P) such that the probability that any of the sellers plays a market clearing price is at most (1/kk) where [k] is the set of prices.
The proof of Theorem 2 is deferred to Appendix A In Section 3 we present the highlevel ideas behind the proof of Theorem Finally in Section 4 we experimentally evaluate the convergence properties of both no-regret and no-swap regret algorithm in the context of pricing games.
5

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3 Proof of Theorem 1
In this section, present the proof of Theorem I which is based on a dual-fitring argument (29 To the best of our knowledge, this is the first application of such techniques to establish convergence results, and it may be of independent interest.
We first introduce some necessary definitions. We remind that p" denotes the maximum market clearing price, see Definition
Definition 6. We call a pricing profile p (P1...--Pi) PP.) P valid if and only if Pt D for all sellers i E [n]] with s; S p". We also denote with V the set of all valid pricing profiles.
Using D we can restated Theorem I as of the fact that any Correlated Equilibrium μ E ASS). places all of its probability mass on valid pricing profiles. The latter is formally stated in Theorem and consists the main technical contribution of our work.
Theorem 3. Let μ E A(S) be Correlated Equilibrium of a pricing game. Then, p(p) 0 and only if E V.
In the rest of the section we present the proof of Theorem throough our dual fitting technique.
Definition 7. Let the function P + {0, 1) the pricing profiles: r(p) = 1 for all E V and r(p) 0 for any p & V.
Using the reward function defined above, we introduce the following linear program that we will play key role in our proof. Definition 8. Given a pricing game of Definition consider the following linear program,
min Σ
r(p)- m(p)
PEP
s.1.
Σ
p(ps, (U:()) - U.(p. P-i)) S for all i E [n] and ps. p E P:
P-VEP-;
Σ puss) =1
PEP
ppp) 20 EpEP
where (p p-i) the utility of seller E [n]. We also denote with Zip its optimal value.
The idea behind the linear program of Definition 8 is that any Correlated Equilibrium μ E A(P) will satisfies its constraints. One can show that μ. E A(P) satisfies the first constraint by using the fact that Ep- (Pt, p] 2 EEpp (=)))).pp.)) for the switching function with 8(s;) = sy The rest of the constraints are satisfied due to the fact that pu-) is joint probability distribution. The latter is formally stated and established Lemma 6
Lemma 2. Let A(P) be Correlated Equilibrium. Then E A(P) satisfies the constraints of the linear program of Definition 8 and thus ΣpEP p(p) r(p)= 2 Zi.P.
Our cornerstone idea is that the optimal value of the linear program above acts as a lower bound on the probability of sampling a valid pricing profile. The latter is formally stated in Corollary 2 Corollary 2. Let μ. E (PP) be Correlated Equilibrium. Then PÊppul [p is valid) 2 Zi.--
Proof. By Definition 7 r(p) y valid and ( otherwise. Thus, EpprM(D) Epev ((p).
r(p) Pr[p is valid].
To complete our proof we just need to establish that Zip 2 1. We will show that the optimal value of the dual of the linear program of Definition 7 is at least 1. Then the claim follows by weak duality.
Lemma The following LP is the dual of the program in Definition We denote with D* its
optimal value.
max 1
s.t.
+
Σ Σ
Apnn (U((p.p-1) U.(k.. S r(p) for all p P
sE[n] P.EP.
1 x.p.
20 0
Vil E.

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Lemma 4. The optimal value of the dual linear program is at least 1, D 2 1.
Lemma i the main technical contribution of this section. In Section we present sketch of its
proof. in Section 3.1 We complete this section with the proof of Theorem
Proof of Theorem By Corollary we know that Prpsull p is valid] > Zip 2 D* 2 1 where the fact that LLP
D is due to weak duality.
3.1 The Dual-Fitting Argument
In this section we present the proof Lemma stating that the optimal value of the dual program is at least 1, D 2 1.
Our approach is to select assignment of the dual variables (μ, X J that dual feasible and at the same time f 1. Our assignment is presented in Definition that we denote with X se order differentiate it from the variables.
in
Definition 9. For any seller i E [n]] with 5; Sp'
if + p" ifp: = p*
2 piv!
(2nk))anp
For any seller E [n]] with 8; p², Xpxp = O.
Before proceeding, let us provide the high-level intuition behind the assignment of Definition 9 The dual variable P.pp represents deviation of seller i [x] from price p: to price p/.- By setting
X p.rt > only when p; = p' and otherwise, we enforce that all agents E [n] with pt S p* are incentivized to deviate exclusively to price p", and not to any other price.
9 we can select á = 1 and the overall assignment (1, 2) is dual feasible. To establish the latter, for any pricing profile E P we consider
Next, we establish that by selecting the assignment X as of Definition
by r()) Σ Σ
Spp. (Ui(pi,p-.) - -UG(,,--)) PSES:
We will establish that bp 2 1 for all pricing profiles p P. Then it directly follows that (1, Å) is feasible for the dual, and thus D* 2 1, since the dual is maximization linear program. The fact that 6, for all P is respectively established in Lemma
5 and Lemma 6
Lemma 5. Let valid pricing profile E V. Then bp 1.
Proof. Since p E P is a valid pricing profile, r(p) As a result, we only need to establish that
Eiejn) Ey,es, Xp.ny (Uc(p. p-5) - U.(p p-))) = 0. This follows by the fact that X p.P! = 0 for all sellers i E [n] with 8; > p² and for all sellers i [ro] with 5; S pp' p² meaning that
Σ P.S..
di Ppu (U.(p..P.)) U. Ap.p- (U.(p..P..) (p:)) =.
We complete the section with p & The full proof of Lemma 6 presented to Appendix C
the respective claim for non-valid pricing profiles
Lemma 6. Let non valid pricing profile p $ V. Then bn 2
Sketch of Proof. Since the pricing profile & V is not valid, by Definition 7 we get that r(p) = 0. AS result, we need establish that
Σ (E(M) PEES,
Σ
Apa (U.(p). p-1) - U((p..p-))
7

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Since p E [n] is the highest market clearing price, p + 1 is not a market clearing price. Thus, p = by for some buyer j E [n] or p" + 1 s; for some seller i [n]
Let us here consider the case p' = by- To simplify notation let m = I E [x] 8, S p'H and i the highest price, 1" (igmmx((i(w)) a,S,") Pi-
Notice that in case P;- p" then seller i" does not set its good and thus U.- (pe- (p-- ) 0. This because at m buyers are willing to buy the good at price ps+ 2 p" + 1 (recall that p = b,) 1 sellers with lower prices. At the same time, seller i* always sells its
and they are at least good in case ps- p" This is because there are m buyer willing to buy the good at price p". Thus, U. (p", p---) p 84- 1. As result,
Xpp (U .p-i) -
(2nk) 2mp..
Since 1 == (rggax((eEe))) s, P: we know that pi 2 p;- for all sellers i E [n] with p To simplify things here let us assume that > Ps- (the case of ties introduces some additional complications and is presented in the full proof).
Let us now try to identify the worst-case for the quantity Up(p", .p-=) :) - (i(ps.P-s). First notice that U.(P..P.)) S Pt 8; since otherwise (.(p.. p-.) 0 and that U, ,p-.) p" 83- Thus,
Σ %op
U,(", pr-8i
(2nk) 2mpi) (p" -ps)
Spi-8t 91
-n mn(nk))2((p - 1) 1)
> 2 - -(nkk 2n(p. (2)
where the last inequality comes from the fact that ps S p.- and that p* - Pt S k. By Equations and 2 we get that
Σ
Spot (U:(k..P-)) -U:-UU--PHH U.(P.,P-i) 2 (2nk) (22kk) -(2kk) -2n) (2n)) 2 1.
iE|n] P.ES.
4 Experimental Evaluations
In this section we experimentally evaluate the well-known Hedge [0 no-regret algorithm and the no-swap regret algorithm proposed by Blum and Mansour 77.. We consider as the set of prices the [0, 5] interval with 0.2 discretization-thar are the following 30 possible prices {0, 0.2. 4.8, 5).
We first consider the family of instances of the pricing game constructed to establish Theorem (se also Appendix A}. This instance is composed by n sellers and buyers where (s1, 82) (. 0)
and (bz, by) A). As result, the highest market clearing price is p A. Despite the fact that such instances admit CCEs that do not correspond to any market clearing price, our experimental evaluation reveal that the Hedge algorithm always converges to the market clearing price p², see Figure and] 3
Figure 1: X = 0.2
Figure 2: A 1
Figure 3: A = 3
Next we consider more natural set-up with n = 100 sellers and sellers. We consider the linear demand curve D(p) -20p 100 for p [0, 5] and three different supply curves, Stieese (p) ==

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20p, Squad() := p² /0.25 and Stinea( := p0 Ssqet (p) := 100 1p/5 (see Figure 4 Each supply curve intersects with the demand curve different price, resulting different market clearing prices. As TigureSJdepicn if the sellers use the Hedge algorithm, the average selling price converges fast to the respective market clearing price.
Domoad Sanee worei su-40
6
24
70
1008
Tome
Figure 4: Supply and Demand Curves
Figure 5: Average Selling Price
The performance of the no-swap regret algorithm of Blum al. 7 is very similar with the per- formance of Hedge. Since convergence to the market clearing price is ensured by Theorem ! the respective figures are deffered Appendix In Appendix we also present additional experimental evaluations for other supply/demand curves.
Conclusion
In this paper. we establish a rigorous connection between the law of supply and demand and the dynamics of online learning within adequate pricing games. Our main contribution is establishing that any correlated equilibrium (CE) the associated pricing game aligns with the market-clearing price. The latter implies that when all sellers employ no-swap regret algorithms, the price converges the market clearing price. Thus, our results provide interesting theoretical foundation on the law of supply and demand by framing it as the emergent behavior of learning process among the sellers.
While establish that coarse correlated equilibria (CCE) do inherently guarantee convergence the market-clearing price, our experimental evaluations indicate that once sellers use the well-known Hedge algorithm, the resulting price dynamics converge to the market clearing price. As result, is likely that a certain classes of no-regret algorithms, such as mean-based algorithms,, may be able to always converge to the market clearing price. Providing formal theoretical convergence guarantees for specific classes of no-regret algorithms, is very interesting research direction.
Linitations Establishing that general no-regret dynamics converge the market-clearing price would offer a stronger theoretical foundation for the law of supply and demand. However, as Theorem 2 demonstrates. Coarse Correlated Equilibria do not always align with the market-clearing price, such convergence properties cannot hold for all no-regret sequences of play. However our experiments suggest that the Hedge algorithm consistently reaches the market-clearing price. Extending our convergence results to specific clasees of no-regret dynamics remains an open challenge and limitation of this work.
Broader Impact We acknowledge that there are many potential socictal consequences our theoretical results, however none of which feel must be specifically highlighted.
Acknowledgments This project was supported by the Villum Young Investigator Award (Grant no. 72091).

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References
[1] Ioannis Anagnostides, Constantinos Daskalakis, Gabriele Farina, Maxwell Fishelson, Noah Golowich, and Tuomas Sandholm. Near-optimal no-regret learning for correlated equilibria in multi-player general-sum games. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Anmual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 24, 2022, pages 736-749. ACM, 2022.
[2] Ioannis Anagnostides, Gabriele Farina, Christian Kroer, Chung Wei Lec, Haipeng Luo, and Tuomas Sandholm. Uncoupled learning dynamics with Oflog T) swap regret in multiplayer games. In Sanmi Koyejo, S. Mohamed, A. Agarwal, Danielle Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems 35: Annual Conference Neural Information Processing Systems 2022, NeurIPS 2022, 2022.
[3] Robert J. Aumann. Subjectivity and correlation in randomized strategies. Journal of Mathemat- ical Economics, 1(1):67-96, 1974.
[4] Moshe Babaioff, Noam Nisan, and Renato Paes Leme. Price competition in online combinatorial markets. In Chin- Wan Chung, Andrei Z. Broder, Kyuseok Shim, and Torsten Suel, editors, 23rd International World Wide Web Conference, WWW '14, Seoul, Republic of Korea, April 7-11, 2014, pages 711-722. ACM, 2014.
[5] Moshe Babaioff, Renato Paes Leme, and Balasubramanian Sivan. Price competition, fluctuations and welfare guarantees. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC '15, page 759-776. Association for Computing Machinery, 2015.
[6] Benjamin Birnbaum, Nikhil R. Devanur, and Lin Xiao. Distributed algorithms via gradient descent for fisher markets. In Proceedings of the 12th ACM Conference Electronic Commerce, EC '11, page 127-136, New York, NY, USA, 2011. Association for Computing Machinery.
[7] Avrim Blum and Yishay Mansour. From external to internal regret. J. Mach. Learn. Res., 8: 8:1307-1324, December 2007.
[8] Simina Brânzei, Nikhil R. Devanur, and Yuval Rabani. Proportional dynamics in exchange economies. Péter Bíró, Shuchi Chawla, and Federico Echenique, editors, EC '21: The 22nd ACM Conference Economics and Computation, Budapest, Hungary, July 18-23, 2021, pages 180-201. ACM, 2021.
[9] Yun Kuen Cheung, Richard Cole, and Nikhil Devanur. Tatonnement beyond gross substitutes? gradient descent to the rescue. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13 page 191-200. Association for Computing Machinery, 2013.
[10] Yun Kuen Cheung, Richard Cole, and Ashish Rastogi. Tatonnement in ongoing markets of complementary goods. In Proceedings of the 13th ACM Conference Electronic Commerce, EC 12, page 337-354. Association for Computing Machinery, 2012.
[11] Yun Kuen Cheung, Richard Cole, and Yixin Tao. Dynamics of distributed updating in fisher markets. In Proceedings of the 2018 ACM Conference on Economics and Computation, EC 18, page 351-368. Association for Computing Machinery, 2018.
[12] Yun Kuen Cheung, Stefanos Leonardos, and Georgios Piliouras. Learning in markets: Greed leads to chaos but following the price is right. In Zhi-Hua Zhou, editor, Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI 2021, Virtual Event Montreal, Canada, 19-27 August 2021, pages 111-117. ijcai.org, 2021.
[13] Bruno Codenotti, Benton McCune, and Kasturi Varadarqjan. Market equilibrium via the excess demand function. Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, STOC '05, page 74-83, New York, NY, USA, 2005. Association for Computing Machinery.
[14] Richard Cole and Lisa Fleischer. Fast-converging tatonnement algorithms for one-time and ongoing market problems. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC *08. page 315-324, New York, NY, USA, 2008. Association for Computing Machinery.
10

---

[15] Eyal Even-Dar, Yishay Mansour, and Uni Nadav. On the convergence of regret minimization dynamics in concave games. In Michael Mitzenmacher, editor, Proceedings of the 41st Anmual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 June 2, 2009, pages 523-532. ACM, 2009.
[16] Amos Fiat, Elias Koutsoupias, Katrina Ligett, Yishay Mansour, and Svetlana Olonetsky. Beyond myopic best response (in cournot competition). Games and Economic Behavior, 113:38-577 2019.
[17] Dean P. Foster and Rakesh V. Vohra. Calibrated learning and correlated equilibrium. Games and Economic Behavior, 21(1):40-55, 1997.
[18] Negin Golrezaei, Patrick Jaillet, and Jason Cheuk Nam Liang. No-regret learning in price com- petitions under consumer reference effects. In Hugo Larochelle, Marc' Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Anmual Conference on Neural Information Processing Systems 2020, NeurlPS 2020, December 6-12, 2020, virtual, 2020.
[19] Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilib- rium. Econometrica, 68(5):1 2000.
[20] Elad Hazan. Introduction online convex optimization. CORR, abs/1909.05207. 2019.
[21] Nicole Immorlica, Evangelos Markakis, and Georgios Piliouras. Coalition formation and price of anarchy in cournot oligopolies. In Proceedings of the 6th International Conference on Internet and Network Economics, WINE' 10. page 270-281, Berlin, Heidelberg, 2010. Springer-
[22] Sabiou Inoua and Vernon Smith. The classical theory of supply and demand, 03 2020.
[23] Yoav Kolumbus, Menahem Levy, and Noam Nisan. Asynchronous proportional response dynamics: Convergence in markets with adversarial scheduling. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and Levine, editors, Advances in Neural Information Processing Systems, volume 36, pages 25444425444. Curran Associates, Inc., 2023.
[24] Zhengyang Liu, Haolin Lu, Liang Shan, and Zihe Wang. On the oscillations in cournot games with best response strategies, 2024.
[25] Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. Oxford University Press, New York, 1995.
[26] Uri Nadav and Georgios Piliouras. No regret learning in oligopolies: cournot vs. bertrand. In Proceedings of the Third International Conference Algorithmic Game Theory, SAGT" 10, page 300-311, 2010.
[27] Yuanyuan Shi and Baosen Zhang. Multi-agent reinforcement learning in cournot games. In 59th IEEE Conference Decision and Control, CDC 2020, Jeju Island, South Korea, December 14-18, 2020, pages 3561-3566. IEEE, 2020.
[28] Adam Smith. The wealth of nations, 1776).
[29] David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cam- bridge University Press, USA, 1st edition, 2011.
[30] Li Zhang. Proportional response dynamies in the fisher market. Theoretical Computer Science, 412(24))2691-2668 2011.
[31] Haiqing Zhu, Alexander Soen, Yun Kuen Cheung, and Lexing Xie. Online learning in betting markets: Profit versus prediction. In Forty-first International Conference on Machine Learning, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.net, 2024.

---

A Proof of Theorem 2
Theorem 2. There exists a family of pricing games with 2 admitting Coarse Correlated Equilibrium E (PP) such that the probability that any of the sellers plays a market clearing price most (1/kk) where [k] the set of prices.
Proof. Consider pricing game of Definition with n. = 2 such that 81 82 = 0 and bj = 1 and b2 k. To this end notice that the market clearing prices of the game either 0 or 1. We will show that there exists a Coarse Correlated Equilibrium la E ((P) such that
Pru(s1 E [0, + Pr E (0, 13 s 5 (+)
To simplify notation let : + 1. Up next we define the joint probability distribution E A(P) as follows:
1. randomly select 8 Unif(I,...,
2. then Pr[(s, (s 1) mod k)]
(+1 26+1
and Pr[((s 1) mod k), s) = =
2F+1
Notice that due to the symmetry ya()), Pru[si < $2] = (E+1) and Pr (82 > s1] = E/(22++1) The latter implies that for seller that,
L 1 ΣE
(f+
(s1, 82)]
28 + 1 2 +1
28 +
Similarly for seller 2 we get that
Ei i ale - 1)
E,((U((81,82]]
2E +
2
28 +1
Now let assume that seller 1 deviates to fixed price In this case
E,, (1, s2)] = i. Pr [s2 2 i] (1
2
(e 1)2
(U
28 +1
2
+
Now let assume that seller 2 deviates to fixed price In this case
Fgg [U)( = Pr > = (1 28 +1 =
2 2
E({+2)
S
= F22 (82, 81
2
2 +1
As result, the joint probability distribution μ E is Coarse Correlated Equilibrium (CCE) while probability that the price 0 or 1 (the set of market clearing prices) is playd by either seller 1 or 2 is at most (1/k).
B Omitted Proofs of Section 3
Lemma 2. Let A(P) be Correlated Equilibrium. Then A(P) satisfies the constraints of the linear program of Definition and thus ΣpEp p(p) r(p)= 2 Di.-.
Proof. Since μ E A(P) then Epep plp) = and u(p) 2 0 for all p E P. Notice that by Definition we know that for any seller E [n]] and any switching function P:P: + P:.
ΣΣ u(p.p-.) (U.(P. p-.) -;;(pp P.))) 2 P. P.
Let Pé, Pi,p' p/, P, and consider the switching function 8(p.) p and 6(x) otherwise. Then directly get that
Eu(p.p-)) (U.(p. U.(l,. p))) 20 for all p., P E P. p-
12

---

Lemma 4. The optimal value of the dual linear program is at least 1, D 2 1.
Proof. By taking the Lagragian
: Σ r(p) Σ Nevr
-Vi - tu 1 --(u))
-E
E+
S.Pi.P/
where Xp.D 1 hp 2 O. By rearranging the terms we get that
L:= = u(p) r(p) + Σ p-.) ((pp.pp))) - 4 hp + p E Xosi P.
P
By setting r(p) Eie[ni Eyees, Newps (Ui(pi.p-i) U:(p(.p-i)) 14 kp we get that
1 Σ
Yow (Us(p..-.) U.(().P-i)) S r(p)
vE[n PtE,,
since h 2 O.
C Omitted Proof of Section 3.1
Lemma 5. Let valid pricing profile p E Then bp 1.
Proof. Since p V we know that r(p) 1. Since p is valid pricing profile we know that p; p' for each seller i E S(pp)) with 8; S p". As result,
Σ Ynnt (U.(P..P-)) U.(pl. = Appt (U.(P..P-)) Us(p", .p-i)) (ip = 0 for p\ # p²) MES.
0 since ps = p²
For all the sellers with 8; > p², by Definition 9 we have that Eptes, Appnt since A Xpp 0 for all prices p; and pl--
Lemma 6. Let non valid pricing profile p & V. Then by 2 1.
Proof. Since the pricing profile & V is not valid, by Definition 7 we get that r(p) = 0. As a result,
we need to establish that Eieja) Eptes, X (U, - (. p-:)) )) 2 1. To simplify notation
let S(p*) denote the set of sellers with marginal cost less than p², S(p*) {i E [n] s; S }²}. = 0 for all i & S(p*). To simplify notation we denote m = (S(')) ) Let i* E [r]] be the seller E
Notice that by Definition 9 A e, S(p*) with the highest price pi, it = argmax Pt. In case there
are multiple sellers with price pit, we consider i" to be the one with the lowest index.
Up next show that for any non-valid pricing profile p $ V.
ΣΣ Xos (U4(p).P-)) ((p..p.))) 1.
(E(M) M.E..
We establish the latter claim for the following mutually exclusive cases:
1. Pe S pd
2. P 2 p¹ +
13

---

Let us start with the case p:+ S. p². Since p & V we know that there exists an seller i E [n] such that p; < p" since otherwise p E V. Notice that any seller i S(p") that set p" as its price, is ensured to sell its good. This is because all produce & S(p") must essentially set price p; > p" while there are exactly m buyer willing to pay price p* for the good. As result, for any seller i E [n] we know that
Σ Xiat (U(()),-))) - U.(..p-i) = Xpprr (Ui(P), P-i) - UU(p'' .p-)) PlES.
2 ((p' 8,) (pi si)) =p" - Pi
Since there exists at least one seller E [n] with ps < p" and X = 0 for all $ [n]] we are ensured that
Σ Σ VE(Mi KE(mipt PLES,
X (U1(p) p-1))..-.) (ps...P-)) 2 1.
Let us now proceed with the case where pV. p' + We first argue that Up. (pi ( -p-+) latter is formally established in [
0-the
Claim 1. Let the seller i* = argmaxe (in case of ties in E [n] is the one with lowest index). Then U. [p=,p=))== = O.
Proof. Since p¹ is the maximum market clearing price then p' + is not a market clearing price and thus p- = by for some buyer j E [r] or p* + 8; for some provider i [n]. Since all s; and by lie in different positions the latter two cases are also mutually exclusive.
Let start with the case p' by for some buyer j E [n]. Since p:. 2 p" + we are ensured that there are at most m - 1 buyers willing to buy the good at price p;- E [n]]. However since i" E argmaxlp, and at the same time admits the lowest index, will not sell its good (notice that enter the market in decreasing order with respect to their by and break ties among sellers lexicographically). Thus U+. (pa+, p- 0.
Up next we consider the mutually exclusive cases pd + 1 = bb and pr + 1 = 8; and separately establish that
Σ <E(m) M:S:
Σ
Xex. (U:(p). - Us(p,, 2 1.
We first start with the case p* +1 by for some buyer E [n]. Let E [n]] be the seller E Sip)) )
with the highest price Pt, 8 == argmax Pi. In case there are multiple sellers with price Ps- [k] the it is the one with the lowest index among them.
Let us assume that p:- > p", byClaim I we are ensured that (p--
0. In case seller 1" E [n]] had selected price p² E [n],, it would have sold its good since at price p" E [n]] there are m buyers and m sellers willing to sell. Thus, U,, (p", p_+-) = =p" Str. As result we get that
P (p" (p=.p=...))) U. (p:..pp))))) = (2nk) (mpp) - (p" 8,,) ) 2 (2nk) 2m
Now let a seller i E [n] such that = By definition of i" we get that i 2 1 + 1. Thus, & p.p. (U.(1", .P)) V.(..P-.)) 2
-11- (v* -P--) 2 - (2nk) 2mp, x² 1k
Now let seller i E [n]] such that P: S P 1. In this case
Appp (U'(p", p-s) (. 2 (2nk))n(0rr-1) (p" - Pi. + +1) 2 -(2kk) )(2nk)mmp:- 22n- 22 +)
As result, we overall get that
B, = r(p) Σ Σ pir. (U.(p..p.)) -U: p.))),pp)))
cES(p")p(Es
(2nk) 2mp: -5" n (2nk) 2mps
10:k-n(200)) n2(nk) 2mpp 2n
= (2nkk)
1 1 2
- nk(k2))) -2n-1 21
14

---

We now consider the case where p" + 1 = for some seller i E [n]. To simplify notation we denote this seller as Next. Following the exact same steps as in the previous case, we can prove that
Σ 13f1" PLES,
Pooe (Ui(pi.P-i) (. -n(2nk) 2np - n((nk) 2mp, 2n
Let now consider A p=(p=+11 = (2mk))app where > O is the small positive constant discussed in Section 2
Up next we establish Lemma 6 for the following mutually exclusive cases:
Ps. pp" 1 and Pnext +1
In this case Up (p) 0 since there are sellers before seller i [n]] with higher priority (next has higher index than i*). At the same time U₃- (p" .p-i.) pp 81.. Finally in case next selects p + 1 as its price it gets the good and thus Unext (p" + 1, p* +
Snext = Combining all the above we get that
E iEfir, next) } sES;
Σ Nan, (U.(p. U.(L..-D)) (2nk)
Pr, =p" + 1 and Puent 2 p² +2
In this case Uz- (p) p¹ + at least m buyers willing to buy the good at price p* +1. Similarly U/. (p", -P-i.) pp 8ix. Also notice that next does not sell its good if it sets with price Pnext but sell its if it sets price p" As a result, Unext P_s) O and (p" + 1.pp.) e. Combing all the above we get
since i' E [n] sells its good due to the fact that there are
Σ :Extt,ncxt) &,S.,
Σ
Xmm (U.(p).p-)) U.(p..p.i)
2 -(2nk)amp' (ank))mp*+22-- +2n-1" + (2nk) (2nn))2ppass 2mpaax:-
2 -(2nk) 2mp +2n-i" + (2nk) 2np +4n-next
Pt, 2 p² 2 and Prentt Pi.
In this case notice that Up- (p) since there are sellers with higher priority that seller E [n]]. At the same time U,- (p",p-..) = p* 8i since seller next will never report price p* (Snext p²) thus seller i* always sells its good at price p* E [n]. Similarly as before we get that Unext (p" + [_nxt) (exeeexx =
and Unext (Prext, p-next) S Pnext Snext. Pnext (w" + 1 e). Combining all the above get that
Σ
ExO (U.(1).P)) - U.(p..Pii) 2 (2nk)
(2nk))2mmcalt 2mpmexl -1"
h )2nk) 2mpruxu nexct
(E(t",next) e'ES:
(2nk) 2mppext -1" -i (2nk)
(2nk) 2mpnext
1
2en
1
// (2 for e = 1/n
Ps. 2 p² and Pnext > Pt.
In this case Up- S Pc - 8;- and U,, (p", = 8;., the same time notice that Unexe (p" + 1,,pneext = since seller next. is able to sell its good at price p* +1. While seller next can never sell its good at price Pnext. E [k] since there m other sellers with higher priority. Thus, Upest (Prext, P-next) = O. Combining all the above we get
15

---

Σ ΣEX (exxt) next] s,ES;
(U:(6):p-)) - U.(pi.pii)
2 - k(2nk) 2n0(praxe 1)-1 + - (2nkk)ampmer 2mphoet -nxtt
= (2nk) 2mpaor ( (-k(2nk) 2n-p" + (2nk)) next
As result, we overall get that
Σ
Σ
Sove (Ui(p),p-i) - (P()) (2nk) (2nk)) 2mpp:
(E next) siES,
At the same time using the exact same arguments as in the case p* = by for some buyer E [n], we establish that
Σ
Σ
X -Ub
p-))) -n(2nk)) 200p -t- -3 n((nk)) 2mp -2n k
.next P/ES;
Putting everything together we get that
Σ
Xpxt (Uc(pt.p-.) Up(p'. 2 (2nk))mpp ((2nk)) 2np.. -2- 2
((2kk2nk)) 2mp. 22-1
<E(M) MEES,
16

---

D Additional Experimental Evaluations
All experiments were conducted in Apple M4 Pro and the Hedge algorithm was run with step-size 0.1.
We also evaluate the Hedge algorithm in the following set-up. We consider the demand curve D(p) 20p - 100 for E [1,5] with a 0.2 discretization. We consider different supply curves parametrized by E {0, 100).
m 100-m 0.2
if S 4.8 otherwise
(p)
5m-480
0.2
In Figure 6 we consider {10, 30, 60. 80}. Each different curve Sm admits different intersection point with the demand curve (see Figure 6). In Figure we see that if all sellers use a no-swap regret algorithm the average selling price converges to the respective market clearing price. The average selling price at each round t, is the average price of the realized trades.
od Supple Cureas
Jeorago
Domend
18
1ine-003
Figure 6: Supply and Demand Curves
Figure 7: Average Selling Price
In all of experimental evaluation we used step-size 0.1 for the Hedge algorithm.
D.1 No-Swap Regret Dynamics
We perform the exact same experimental evaluation for the no-swap regret algorithm Blum et al. [7 In our implementation used the Hedge algorithm with step-size ? 0.1 base no-regret algorithm. all the above experimental evaluation the resulting no-swap regret dynamics converge the market clearing price, something that is to be expected due Theorem
As in Section 4 we first consider the case n = 2 sellers and buyers where (81, 82) 0) and (br, by) = (5, Aj. In this instance the highest market clearing price is p" X. Figures and 10 verify that the resulting no-swap regret dynamics converge to the respective market clearing price of each case.
Figure 2=
Figure ==1
Figure
17

---

Next we consider the case 100 sellers and sellers with demand curve D(p) 20p++0p+-00 100 for e [0, and supply curves, Stincar == Squad := p² /0.25 and Sinear (p) p/5, Saart (p) ==
100 1 p/5. Each supply curve intersects with the demand curve at a different price, resulting in verifies
different market clearing prices (see Figure 12). Figure veriffies that the resulting no-swap regret dynamics converge to the respective market clearing price of each case.
Supply Coreaa
Averac Aica
Sist Buguly
275
Doy Staply
348
30
538
mmP
Time
Figure 11: Supply and Demand Curves
Figure 12: Average Selling Price
Finally consider the demand curve D(p) 20p 100 for p [1, 5] with 0.2 discretization. We consider different supply curves parametrized by m E (10, 30, 60, 80),
m
if S 4.8
(p)
100-m 0.2
p
5m 480
0.2
otherwise
In Figure 14 we see that if all sellers use the no-swap regret algorithm of Blum al. [7], the average price converges to the respective market clearing price.
Dema si and Supely Curaes
Azerage time
Rpbb Convero d0)
Rile
Figure 13: Supply and Demand Curves
Figure 14: Average Selling Price
18

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NeurIPS Paper Checklist
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Question:
Answer: [Yes]
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We described in detail the results of the paper both in abstract and the intro-
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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
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Question: Does the paper provide open access the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
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Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Yes we have included all the necessary details.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [No]
Justification: All of our evaluated online learning dynamics converge to the market clearing price meaning that there is no variance on the final outcome.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification: All of experimental evaluation have been conducted in Mac Pro 4. We have included this information in the experimental section.
19

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9. Code of ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurlPS Code of Ethics
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Justification: All of our research aligns withthe Code of Ethics.
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20


Reviews:
[{"review_id": "Lak8SuAPhD", "reviewer": "Reviewer_CWGT", "summary": "The paper studies convergence of prices in market settings, and specifically showing convergence to the so-called market clearing price, which is where demand meets supply. The game being played is fairly simple and requires sellers to set prices above their willingness to sell according to their strategy, and buyers to accept the price if it is above their willingness to buy. The first buyer gets the cheapest seller, the second the second-cheapest, and so on. Only sellers have strategies in this game, and they play a number of rounds. The aim is to minimise regret. They investigate the general classes of no-regret and no-swap regret algorithms. These are classes of online algorithms, which take past prices as input to produce the next price. They show that, when all sellers use a no-swap regret algorithm, then prices converge to the market clearing price which is a correlated equilibrium. However, they show by example that this does not always hold for coarse correlated equilibria, i.e. for more  no-regret algorithms. The latter is actually quite a suprising result. They then show that, in practice using a specific no-regret algorithm, convergence is still observed. In terms of contribution, existing works have shown converengence of these classes of algorithms, but they have not shown convergence to market clearing prices. At the same time, certain processes such as Walrasian tatonnement have been shown to converge to market clearing prices, but this is not generally shown for more general classes of online learning algorithms. So the contribution is to show the convergence to market clearing prices of a general class of algorithms.", "questions": "1. The sellers and buyers are ordered in terms of their valuation, and the paper states that this is \"without loss of generality\". However, in the definition of the game, i.e. Definition 3, buyers with lower index are able to choose \"first\", i.e. meaning that those buyers with the highest valuation are able to choose first and therefore get the lowest price. This reflects e.g. the double auctions protocol, where sellers with the lowest ask are matched with buyers with the highest bid. Surely, this choice is significant and would affect the results if, say, the buyer order was randomised? A brief note would be useful here.  \n\n2. According to Lemma 1, a market clearing price is (only?) guaranteed if marginal costs and prices \"lies in different places\". First, there is a grammatical error (should be plural i.e. \"lie\" not \"lies\"). Second, the expression is confusing and seems unusual (it might be a formal term I'm just not familiar with). Presumably it means they are all different. Maybe there is a better way (e.g. more formal way even) of defining this? More importantly, is this assumption required throughout, i.e. for all results in the paper? This was not clear. It seems somewhat restrictive.\n\n(if space allows but less important): \n\n3. Def 4 could possibly be standard, but I could not understand why the first term of the first equation is over expectation - i.e. p*_i is not a mixed strategy, and is also not used in the function U so why sample over the mixed strategy? Possibly to be consistent with the second equation?", "limitations": "There are no obvious negative societal impacts from this work.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is very well written, and largely clear. There is novelty and the contribution seem significant and relevant. However, I'm not an expert especially in the methods used for proving the results, and so was unable to check all the technical details. A few questions and minor issues:\n\n- The sellers and buyers are ordered in terms of their valuation, and the paper states that this is \"without loss of generality\". However, in the definition of the game, i.e. Definition 3, buyers with lower index are able to choose \"first\", i.e. meaning that those buyers with the highest valuation are able to choose first and therefore get the lowest price. This reflects e.g. the double auctions protocol, where sellers with the lowest ask are matched with buyers with the highest bid. Surely, this choice is significant and would affect the results if, say, the buyer order was randomised? A brief note would be useful here.  \n\n- According to Lemma 1, a market clearing price is (only?) guaranteed if marginal costs and prices \"lies in different places\". First, there is a grammatical error (should be plural i.e. \"lie\" not \"lies\"). Second, the expression is confusing and seems unusual (it might be a formal term I'm just not familiar with). Presumably it means they are all different. Maybe there is a better way (e.g. more formal way even) of defining this? More importantly, is this assumption required throughout, i.e. for all results in the paper? This was not clear. It seems somewhat restrictive.\n\n- Typo: Below Def 2: \"sates\" -> \"states\"\n\n- L128: \"an one-shot game\" -> \"a one-shot game\" (I know it's confusing but see here for explanation: https://www.reddit.com/r/grammar/comments/8uctlm/is_it_a_one_or_an_one/)\n\n- L156: \"In a pricing games\" -> \"In a pricing game\"\n\n- Def 4 could possibly be standard, but I could not understand why the first term of the first equation is over expectation - i.e. p*_i is not a mixed strategy, and is also not used in the function U so why sample over the mixed strategy? Possibly to be consistent with the second equation?\n\n- In Def 4, o(T) is not defined.", "was_revised": true, "final_justification": "Thanks to the autors for providing the response. I'm satisifed with the answers to my questions and have no further queries. I have also looked at the other reviews and none seem to identify any technical flaws with the paper nor issues around novelty. Hence, I stand by my initial score.", "rebuttal": "Thank you very much for all the very detailed review and positive feedback on our paper. In the revised version of our paper, we will fix all the typos and minor comments.\n\nUp next we address your main questions and concerns.\n\n1. **The sellers...A brief note would be useful here.** You are right and we apologize for the confusion. Wlog was supposed to refer on the ordering of the sellers. We will add a brief note to clarify.\n\n2. **According to Lemma 1...somewhat restrictive.** We believe that our results hold even without this assumption however in such a case one would need to deal with tedious technical details. We chose to use this assumption that greatly simplifies the analysis since in case of sufficiently large discretization, it is natural to assume that each seller admits a slightly different price.\n\n3. **Def 4 could possibly be standard...Possibly to be consistent with the second equation?** You are right that $p_i^\\star$ is a derministic price however observe that all other agents sample their prices $p_{-i}^t$ according to the probablity distribution $x_{-i}^t$ and that is why we use the expectation."}, {"review_id": "0KC8CmR3Du", "reviewer": "Reviewer_XyhP", "summary": "This paper revisits the law of supply and demand through the lens of game theory and online learning. The authors model a pricing game in which sellers independently set prices to maximize their individual revenues. They prove that if sellers adopt no-swap regret algorithms, the resulting price dynamics converge to the market-clearing price. In contrast, they construct a counterexample showing that coarse correlated equilibria (CCE), which arise under general no-regret dynamics, may fail to align with the market-clearing price. Empirical results demonstrate that both no-regret and no-swap regret algorithms typically lead to convergence in practice.", "questions": "1) The analysis assumes a discrete set of prices. Could the framework be extended to continuous prices? How sensitive are the theoretical guarantees to the choice of discretization, and would convergence behavior change in the continuous case?\n2) How would the convergence results be affected by noise in seller feedback or slight uncertainty in buyer valuations?\n3) Theorem 2 presents a negative result where no-regret dynamics may fail to reach the market-clearing price. It would be insightful to include experimental scenarios where this failure is observed, highlighting the contrast with the success of no-swap regret algorithms.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths\n1) The paper provides a novel theoretical foundation for a classical economic principle by framing it through the lens of modern online learning dynamics.\n2) The proposed pricing game, in which sellers act strategically while buyers follow deterministic purchasing rules, is well-motivated and effectively captures essential characteristics of perfectly competitive markets.\n3) A key technical contribution is the use of a dual-fitting technique to show that all correlated equilibria correspond to valid pricing profiles, which is a novel approach and possibly of broader interest.\n4) Simulation results support the theoretical claims demonstrating convergence to the market-clearing price across a variety of market configurations.\n\nWeaknesses\n1) Section 1.1 reviews relevant literature by summarizing key ideas, but lacks a systematic comparison between this work and prior studies.\n2) The model assumes that only sellers are strategic, while buyers arrive sequentially and always purchase from the lowest-priced seller. This assumption simplifies more realistic buyer behaviors, such as strategic decision-making and preference heterogeneity.\n3) The experimental evaluation is mainly illustrative and does not assess performance under large-scale or noisy environments. Moreover, it focuses only on two algorithms without broader comparisons across other online algorithms.", "was_revised": true, "final_justification": "I lean towards acceptance primarily because of the novel perspective it offers on understanding the law of supply. While there are many potential extensions, the paper makes an interesting contribution.", "rebuttal": "Thank you very much for all your work and the positive feedback on our work. Up next we address your main concerns.\n\n\n\n**1.Section 1.1 reviews...this work and prior studies.**\n\nIn the revised version of our work, we will discuss in further detail the related literature.\n\n**2.The model assumes that...preference heterogeneity.**\n\nThe law of Supply and Demand assumes that all provided goods are *indistiguishable*. When buyer preferences vary (preference heterogeneity) the law of Supply and Demand no longer holds. Also notice that in case of indistinguishable goods, strategic buyer behavior reduces to simply purchasing from the cheapest available seller.\n\n**3.The experimental evaluation...online algorithms.**\nThe primary focus of our work is to formally establish the law of supply and demand as an emergent behavior of learning dynamics among sellers. That being said, in the revised version, we will include additional experimental evaluations using a broader set of no-regret algorithms, including FTRL, Regret Matching, Online Mirror Descent, and others.\n\n\nQuestions\n\n1.**The analysis assumes...in the continuous case?**\n\nWe believe that our results and techniques extend to the continuous price model by considering the respective LPs with infinite number of constraints. In the current work we chose the discrete price model since it leads to simpler analysis and better reflects practice where prices are typically discrete.\n\n2.**How would the convergence results be affected by noise in seller feedback or slight uncertainty in buyer valuations?**\n\nThe are not affected at all. There are no-swap regret algorithms that work under noisy perturbations on the payoffs or even bandit feedback (only the revenue of the selected price is observed). If agents adopt such algorithms the overall system will converge to a CE and thus our results apply.\n\n**3.Theorem 2 presents a negative result...no-swap regret algorithms.**\n\nIn the revised version, we will present further experimental evaluations of various no-regret algorithms. However we conjecture that natural no-regret algorithms avoid bad Coarse Correlated Equilibria (CCE) and instead converge to the market-clearing price. Our negative result in Theorem 2 highlights the limitations of reasoning about general CCEs, suggesting that future analysis should focus on specific no-regret algorithms rather than on general CCEs."}, {"review_id": "Adlfp9dtbE", "reviewer": "Reviewer_6iRp", "summary": "The paper proves that in a repeated pricing game, if all sellers use no-swap regret algorithms, the resulting play converges to correlated equilibria that place all probability on profiles where prices equal the market-clearing price. The proof is an interesting application of dual-fitting. The paper also shows that the assumption that the players use no-swap regret algorithms is necessary, since with regular no-regret algorithms, the game converges to a coarse correlated equilibrium, and there are cases where such equilibria does not correspond to a market clearing price. However, such cases are not observed in numerical experiments.", "questions": "How robust are the results to noise, incomplete information, or heterogeneous learning rates?", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* A game dynamics foundation for the law of supply and demand.\n* To the best of my knowledge, the use of dual-fitting to characterize the equilibria is novel.\n\nWeaknesses:\n* While theoretical results are interesting and non-trivial, they might not be at the level of a NeurIPS paper.\n* No-swap regret might be unrealistic in practice.\n* Assumes full information, also somewhat unrealistic.\n* Model is a bit too simplistic.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for all your work and for appreciating the motivation of our work. Up next we adrress your main concerns.\n\n**1.While the theoretical results are interesting and non-trivial, they might not meet the standard of a NeurIPS paper.**\n\nThe Law of Supply and Demand is arguably the most fundamental principle in modern economics. We believe that establishing that it can emerge from a learning process among competing sellers is both a novel and significant contribution. We remark that the latter is not guaranteed—since prices could perpetually oscillate. Our results show that, under mild assumptions, convergence to market equilibrium occurs, which we see as a meaningful insight into market dynamics.\n\n**2.No-swap regret might be unrealistic in practice.**\n\nNo-swap regret algorithms have been extensively studied in the literature as models for the learning behavior of competing agents [1,2]. More recently, several works have shown important advantages of no-swap regret, especially in terms of robustness to exploitation [3,4,5]. The latter suggest that no-swap regret is a compelling modeling choice in strategic multi-agent settings.\n\nReferences:\n\n[1] Hart & Mas-Colell, *A Simple Adaptive Procedure Leading to Correlated Equilibrium*\n\n[2] Foster & Vohra, *Calibrated Learning and Correlated Equilibrium*\n\n[3] Arunachaleswaran et al., *Swap Regret and Correlated Equilibria Beyond Normal-Form Games*\n\n[4] Arunachaleswaran et al., *Pareto-Optimal Algorithms for Learning in Games*\n\n[5] Mansour et al., *Strategizing against Learners in Bayesian Games*\n\n**3. Assumes full information, also somewhat unrealistic.**\n\nOur results do not require any *full information* assumption. If agents use any no-swap learning algorithm with *bandit feedback* (only the revenue of the selected price is learned), they will converge to a CE and our results will apply. The same holds in case of noisy perturbations on the reward. \n\n**4.Model is a bit too simplistic.** Since our model captures the essence of Law of Supply and Demand, we believe that simplicity is a plus.\n\n\n**Questions**\n\n**How robust are the results to noise, incomplete information, or heterogeneous learning rates?**\n\nOur results continue to apply in case of noise / incomplete information / heterogeneous learning rates and even heterogenous algorithms. Our main result establishes any CE is a market clearing price. We remark that convergence to CE is achieved even if agents use different no-swap regret algorithms. At the same time, there are various no-swap regret algorithms that work with noisy perturbations of the reward or even with *bandit feedback* (only the revenue of the selected price is revealed to the seller). Even in such limited information settings, if agents use such no-swap regret algorithms they will converge to CE and thus our results apply.\n\n\n\nWe hope the latter clarify your concerns and encourage a reconsideration of your score. We would be happy to provide further clarifications if needed."}, {"review_id": "4E2bEmBsz4", "reviewer": "Reviewer_7ou3", "summary": "This paper studies the Law of Supply and Demand from a novel perspective, framing it as an outcome of adaptive learning processes among sellers in a perfectly competitive market. The core contribution is that the authors demonstrate how, if each seller employs a no-swap regret algorithm to set their individual selling price, the collective pricing dynamics converge to the market-clearing price. Experimental results are provided to demonstrate the effect.", "questions": "Could you elaborate on what distinguishes these \"specific classes\", in the statement of \"the negative result of Theorem 3 may be circumvented by specific classes of no-regret algorithms\"?", "limitations": "yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "On the positive side, this paper offers a fresh view on a fundamental economic principle by connecting it to the field of online learning and game theory. I found the problem presented by the paper interesting and relevant. In general, this paper proposed an innovative research topic.\n\nOn the other hand, the paper notes that while general no-regret algorithms don't theoretically guarantee convergence to the market-clearing price, the experimental results for Hedge show convergence, suggesting it might be circumvented by \"specific classes of no-regret algorithms.\" This argument is not convincing. Could you elaborate on what distinguishes these \"specific classes\"? For example, including experimental evaluations of other common no-regret algorithms beyond Hedge to further explore this phenomenon. It would be beneficial to see how other common no-regret or online learning algorithms perform, even if they are not expected to converge to CE, and to elaborate on what constitutes these \"specific classes.\"\n\nIn addition, the paper states that \"All of our evaluated online learning dynamics converge to the market clearing price meaning that there is no variance on the final outcome.\" While this might be true for the final outcome, the convergence path and stability during the learning process could still exhibit variance. Including error bars or confidence intervals, especially for the initial phases of learning, would provide a more complete picture of the algorithms' performance.\n\nAnother problem with the paper is its presentation. Unfortunately, the paper doesn’t flow very well. The writing of the paper seems to be a bit rushed. There are typos, and grammatical and formatting errors. For example, in the abstract, “the number of sellers willing to …… equals the number of sellers (the authors meant buyers?) ……”", "was_revised": true, "final_justification": "The main positive result Theorem 1 and negative result Theorem 2 are indeed interesting. My concern is Theorem 2's **contradiction** with the experimental results, which is explained only with the vague term “likely” in the discussion. Just as they provided experimental evidence for Theorem 1, I would expect a similar experimental demonstration for Theorem 2’s non-convergence result, especially since they explicitly state in Line 293: “We first consider the family of instances of the pricing game constructed to **establish** Theorem 2.” Yet the figure’s result directly conflicts with Theorem 2, which is not a tangential point in my opinion, and “likely” is not an acceptable explanation to me for this discrepancy. \n\nThe convergence path and stability during the learning process could still exhibit variance. Including error bars or confidence intervals, especially for the initial phases of learning, would provide a more complete picture of the algorithms' performance. \n\nFinally, including experiments on other common no-regret algorithms beyond Hedge would make the work more convincing. Demonstrating results on just one algorithm is not persuasive enough.  \n \nThat said, I’m willing to raise my rating from 2 to 3, but I will not champion the acceptance of this paper.", "rebuttal": "Thank you very much for all your work and for appreciating the innovative aspects of our work. Up next we address your main concerns.\n\n\n**On the other hand, the paper...to elaborate on what constitutes these \"specific classes.\"**\n\n\nWith specific classes we refer to the *mean-based algorithms*. Notice that we write *\"Ιt is likely that a certain classes of no-regret algorithms, *such as mean-based algorithms*, may be able to always converge to the market clearing price\"*.\n\nMean-based algorithms [4] is a natural class of no-regret algorithms (including Hedge) with the following property: if the cumulative reward of two actions differs by at most $\\mathcal{O}(\\gamma \\cdot T)$ then the respective probabilities differ by at most by $\\mathcal{O}(\\gamma)$. Mean-based nο-regret algorithms are known to avoid degenerate CCE in various games (e.g. auctions, potential games [1,2,3]). As a result, we conjecture that this may be the case for Supply and Demand games. The latter is aligned with our experimental evaluations. \n\nIn the revised version of our work, we will add the above discussion to clarify the situation. We will also add additional experimental evaluations of no-regret algorithms (FTRL, Online Mirror Descent, Regret Matching etc.). We will additionally correct the typos and add error bars.\n\n**Remark:** The main result of our work is formally establishing that *Correlated Equilibrium* (CE) corresponds to the market clearing price in the case of Supply and Demand games. We remark that the discussion on no-regret algorithms does not constitute neither an argument nor a result of the current paper, but rather a conjecture and a suggestion for future research direction.\n\n\nWe hope the latter clarify your concerns and will help reconsider your score.\n\n\n[1] Deng et al. *Nash convergence of mean-based learning algorithms in first price auctions*.\n\n[2] Feng et al. *Convergence Analysis of No-Regret Bidding Algorithms in Repeated Auctions*.\n\n[3]  Heliou et al. *Learning with Bandit Feedback in Potential Games*.\n\n[4] Braverman et al. *Selling to a No-Regret Buyer*."}]

Author rebuttal:

Meta-review: Two of the reviewers are largely positive about the paper, noting several strengths: new theoretical foundations behind a well-known economic principle through modern online learning; an interesting model of a pricing game; technical contributions that provide useful characterizations of the set of correlated equilibria of the game through a novel approach; and, results are complemented by simulations. Overall, the strengths of the paper are significant. The reviewers praise novelty, clarity, and significance. 

Two of the reviews propose a weak reject. After reading the reviews, and discussion, and taking a quick look at the paper myself, I believe the two weak rejects do not provide sufficient grounds for the paper to be rejected. To elaborate a bit more: 
- one reviewer states that "results (...) might not be at the level of a NeurIPS paper" and "model is a bit too simplitic", without further justfication. I cannot act on these comments, since they do not provide a reason for such evaluation and seem to disagree with the majority of the review team.
- the same reviewer mentions that "No-swap regret might be unrealistic in practice" and "assumes full information, also somewhat unrealistic.", but these assumptions are common in the literature and in theory papers, so I do not see this as an issue. 
- lastly, the second negative reviewer only had negative comments that were mainly disconnected from the main directions of the paper. One complains about proposed future work, claiming that it was not formal enough/in theorem form, but this was not a contribution of the paper so this is not an issue. Another comment was with respect to the neurips checklist, asking for error bars for their convergence guarantees; since said convergence guarantees are in the limit and deterministic, error bars are not needed to evaluate the authors' contributions; what the reviewer is asking is a different contribution, and cannot be construed as a valid criticism of the authors' responses on the checklist. Finally, there seemed to be confusion about the technical results, with Theorem 2, which is a formally proven negative result, which is conflated with a mostly unrelated point in the review. 

Based on this, I recommend the paper for acceptance.
Decision: Accept (poster)