A central concept to understand trade in large markets is the notion of competitive or market equilibrium. The computational aspects of competitive equilibria have been a central theme in algorithmic game theory over the last decade, mainly for the prominent class of Fisher markets. In a Fisher market, there are a set of agents or buyers and a set of divisible goods. Each agent brings a budget of money to the market and wants to buy goods, for which she has an increasing and concave utility function. An equilibrium consists of a vector of prices and an allocation of goods and money such that (1) every buyer purchases the most preferred bundle of goods that she can afford, and (2) market clears (supply equals demand).
There are successful approaches based on distributed adaptation processes for converging to market equilibria. For example, tatonnement is governed by the natural intuition that prices for over-demanded goods increase, while under-demanded goods become cheaper. It provides an explanation how decentralized price adjustment can lead a market into an equilibrium state, thereby providing additional justification for the concept. Recently, several works derived a detailed analysis and proved fast convergence of discrete-time tatonnement in markets [13, 14, 11, 10, 8, 9].
It is well-known that network rate control is closely related to Fisher market equilibria [21, 22, 23]. Towards this end, distributed market dynamics called proportional response dynamics (PRD) were proposed and analyzed in the context of peer-to-peer network sharing [29, 24]. These dynamics avoid the usage of prices and work directly on the exchange and allocation of goods. PRD and its generalizations converge toward market equilibria in the full range of CES Fisher markets [31, 5, 12].
While tatonnement and PRD rely on dynamic change of prices and allocation, the existing literature assumes that the market and its properties (agents, budgets, utilities, supplies of goods) remain static and unchanged over time. In fact, to the best of our knowledge, all of the existing work on computation of market equilibrium in algorithmic game theory assumes that the market is essentially a static environment. In contrast, in many (if not all) applications of markets, the market itself is subject to dynamic change, in the sense that supplies of goods changes over time, agents have different budgets at their disposal that they can spend, or the preferences of agents expressed via utility functions evolve over time. Analyzing and quantifying the impact of dynamic change in markets is critical to understand the robustness of market equilibrium in general, and of price adaptation dynamics like tatonnement in particular.
In this paper, we initiate the algorithmic study of dynamic markets in the form of dynamically evolving environments. Our interest lies in the performance of dynamic adaptation processes like tatonnement. We analyze a discrete-time process, where in each round tatonnement provides a price for each good, which is then updated using the excess demand for each good. In each round the excess demand comes from a possibly different, adversarially perturbed market. This dynamic nature of markets gives rise to a number of interesting issues. Notably, even when in each round the market has a unique equilibrium, over time this equilibrium becomes a dynamic object. As such, exact market equilibria can rarely or never be reached. Instead, we consider how tatonnement can trace the equilibrium by maintaining a small distance (in terms of suitably defined notions of distance), which also results in approximate clearing conditions. For PRD, we apply a similar approach based on adaptation of the allocation of goods.
More formally, we consider the prominent class of Fisher markets with agent utilities that exhibit constant elasticity of substitution (CES). In this versatile framework, we analyze the impact of changes in supply of goods, budgets of agents, and their utility parameters. The adaptation approaches equilibrium conditions, but since the equilibrium is moving, prices and allocations follow and chase the equilibrium point over time. Our analysis provides distance bounds, which can be seen as a quantification of the extent of out-of-equilibrium trade due to the interplay of market perturbation and adaptation of agents.
Technically, the majority of our analysis is concerned with quantifying the impact of perturbation in market parameters on several potential functions that guarantee convergence of the dynamics. The results then follow by a combination with the convergence guarantees for static markets. In fact, this approach constitutes a powerful framework to analyze a large variety of protocols and dynamics that are well-understood in static systems, when these systems become subject to dynamic perturbation. In the Appendix C, we discuss further examples – gradient descent for strongly convex functions and diffusion for load balancing in networked systems – where we quantify the performance of natural dynamics in the presence of system perturbations.
Contribution and Outline. After presenting necessary preliminaries in Section 2, we describe in Section 3 the general model for dynamic Fisher markets with CES utilities and a general convergence result. In the subsequent sections, we discuss the insightful case of CES markets with gross-substitute condition. In these markets, the total misspending (absolute excess demand times price) over all goods is a natural parameter to quantify the violation of market clearing conditions. Moreover, one round of tatonnement updates in static markets is known to decrease misspending by a multiplicative factor . In Sections 3.1 and 3.2, we consider markets where the supply of goods, the budgets of agents, and the utility function of the agents are subject to dynamic perturbation, respectively. We quantify the impact of perturbation on the misspending in the market. These bounds reveal that the change is often a rather mild additive change in misspending. Together with the fact that tatonnement decreases the misspending multiplicatively, we see that the price adaptation is indeed able to incorporate and adapt to the changes quickly. Overall, the dynamics can trace the equilibrium point up to a distance that evolves from the change in a small number of recent rounds.
We can provide similar results for a more general approach for CES markets based on a convex potential function . A slight disadvantage is that this potential function does not have an equally intuitive interpretation as the misspending function. On the other hand, it applies to tatonnement in all CES Fisher markets (even without gross-substitute property). The discussion of these results is deferred to Appendix B.
The technique we apply for markets can be executed much more generally for a class of dynamical systems, which we outline in Section 4. These systems have a set of control parameters (e.g., prices in markets, or strategic decisions in games) and system parameters (e.g., supplies or utilities in markets, or payoff values in games). Moreover, these systems admit a Lyapunov function, and a round-based adaptation process for the control parameters (e.g., tatonnement in markets, or best-response dynamics in classes of games) that multiplicatively decreases the Lyapunov function in a single round. Our results provide a bound on the value of the Lyapunov function when the system parameters are subject to dynamic change. We discuss two additional examples of such systems based on minimization of strongly convex functions and network load balancing in Appendix C.
In Section 5 we use a further generalization of the technique based on Bregman divergence to show that proportional response dynamics can successfully trace equilibrium in gross-substitute CES Fisher markets. A more general framework of dynamical systems governed by progress in Bregman divergence is discussed in Appendix D. It seems likely that a similar analysis based on our techniques can be conducted for many more sophisticated systems with significantly more complex dynamics.
Related Work. Competitive equilibrium and tatonnement date back to Walras  in 1874. The existence of equilibrium was established in a non-constructive way for a general setting by Arrow and Debreu  in 1954. Computation of equilibrium has been a central subject in general equilibrium theory. In the past 15 years, there has been impressive progress on devising efficent algorithms for computing equilibria, e.g., using network-flow algorithms [16, 26, 18, 17, 3, 4], the ellipsoid method  or the interior point method .
Decentralized adaptation processes such as tatonnement are important due to their simple nature and plausible applicability in real markets. Tatonnement is broadly defined as a process that increases (resp. decreases) the price of a good if the demand for the good is more (resp. less) than the supply. The price updates are distributed, since the price adjustment for each good is in the direction of its own excess demand and is independent of the demands for other goods.
Arrow, Block and Hurwitz  showed that a continuous version of tatonnement converges to an equilibrium for markets satisfying the weak gross substitutes (WGS) property. The recent algorithmic advances provide new insights in analyzing tatonnement [13, 10]. Cole and Fleischer  proposed the ongoing market model, in which warehouses are introduced to allow out-of-equilibrium trade, and prices are updated in tatonnement-style asynchronously, to provide an in-market process which might capture how real markets work. There has been significant recent interest in further aspects of ongoing markets or asynchronous tatonnement [15, 11, 8, 9].
In contrast, proportional response dynamics are a class of distributed algorithms that originated in the literature on network bandwidth sharing. These dynamics work without prices and come with convergence guarantees in classes of static network exchange, where goods have a uniform value . For the special class of Fisher markets with linear utilities these dynamics can be cast as a form of gradient descent [31, 5, 12].
Notions of games and markets with perturbation and dynamic change are only very recently starting to receive increased interest in algorithmic game theory. For example, recent work has started to quantify the average performance of simple auctions and regret-learning agents in combinatorial auctions with dynamic buyer population [25, 19]. In these scenarios, however, equilibria are probabilistic objects and convergence in the static case can only be shown in terms of regret on average in hindsight. Moreover, the main goal is to bound the price of anarchy.
Fisher Markets. In a Fisher market, there are goods and buyers. Each buyer has an amount of budget. Buyer has a utility function representing her preference. For bundles and , if , then she prefers to . We denote the vector of budgets by and the vector of utility functions by . Let be the total budget in the market.
Given a vector of (per-unit) prices for each good, buyer requests a demand bundle of goods that maximizes her utility function subject to the budget constraint: . In general, the is a set of bundles. In this paper, we concern strictly concave utility function only, for which there is a unique demand bundle.
The sum of amount of good purchased by all buyers is the demand for good , denoted by . The supply of good is , and we set . Let be the vector of excess demand, i.e., demand minus supply: .
A pair is a competitive or market equilibrium if (1) each vector is a demand of agent at prices , (2) for each good with , demand is equal to supply (i.e., ), and (3) for each good with , demand is at most supply (i.e., ). An equilibrium price vector is also called a vector of market clearing prices.
CES Utility Functions. A prominent class of utility functions in markets are utility functions with Constant Elasticity of Substitution (CES). They have the form , where and all .
For and , buyer ’s demand for good is
Dynamic Markets. For CES Fisher markets, tatonnement is known to converge quickly to equilibrium under static market conditions. We here consider a dynamic market where in the beginning of each round our tatonnement dynamics propose a vector of prices . Dynamic market parameters like budgets , supplies and utility functions are manifested, which can be different from their value in previous rounds . Agents request a demand bundle based on the prices and market , which yields a vector of excess demands . Then the system proceeds to the next round .
We first provide a basic insight that lies at the core of the analysis and manages to lift convergence results for a class of static markets to a bound for dynamic markets from that class. Formally, assume that the following properties hold:
There is a non-negative potential function , for every market and every price vector . It holds if and only if is a vector of clearing prices for market .
The price dynamics satisfy , for some and every market .
The market dynamics satisfy , for some values and every price vector .
Suppose the price and market dynamics satisfy the Potential, Price-Improvement, and Market-Perturbation properties. Then
Let , then it follows for any
The proof follows by a direct application of the three properties. We prove it for a much more general class of dynamic systems with Lyapunov functions in Section 4.
Consider the three terms in the latter bound for . The first term captures the impact of recent changes to the market. The second term bounds the effect of all older changes. The third term decays exponentially over time. Hence, when the process runs long enough, the potential is only affected by recent changes of the market, while all older changes can be accumulated into a constant term based on and . Intuitively, the price dynamics follows the evolution of the equilibrium up to a “distance” of in the potential function value. Hence, if market perturbation is small and price improvement is large, the process succeeds to maintain market clearing conditions almost exactly.
3 Dynamic Fisher Markets and Misspending
For simplicity, we here describe our techniques for CES markets with gross-substitutes property, i.e., when all buyers have CES utilities with . For a study of general CES Fisher markets see Appendix B below.
The tatonnement process we analyze here updates prices in each round based on the excess demand in the last round, i.e.,
The tatonnement process is known to have the Price-Improvement property based on the misspending potential function in CES markets with . More formally, let , if , then there exists such that . 
3.1 Dynamic Supply and Budgets
Dynamic Supply. Let us first analyze the impact of changing supply on tatonnement dynamics and market clearing conditions. We normalize the initial supply for each good . Suppose that the supplies are then changed additively111We here study additive change for mathematical convenience. The bounds can be adjusted to hold accordingly for multiplicative change. by at time . We parametrize our bounds using the maximum supply change .
Assumption 1. Every price is universally bounded by some time-independent constant , i.e., for any and any time , we have .
Assumption 1 is made for technical reasons, but it is simple to satisfy by constant parameters of the market. For example, if all initial prices are at most , then since Assumption 1 holds with . The main result in this section is as follows.
For any ,
Consider the misspending potential . Tatonnement satisfies the Price-Improvement property. Hence, to show the result, we establish the Market-Perturbation property.
Note that the misspending potential can be given by
Hence, by the triangle inequality and Assumption 1,
Thus, using Proposition 2 with and , the proof follows. ∎
Remark. If the supplies of all goods shrink multiplicatively by the same factor of per time step, then in markets with CES utility functions, it is well-known that the equilibrium price of every good increases by a factor of per time step. However, the tatonnement update rule allows the current price to be increased by a factor of at most per time step. Thus, for plausible tracing of equilibrium, should satisfy .
Dynamic Budgets. We now analyze the impact of changing buyer budgets on tatonnement dynamics and market clearing conditions. Starting from the initial budgets, the budgets are then changed additively by at time . We parametrize our bounds using the maximum budget change . For a proof of the following proposition, see Appendix A.1.
For any ,
3.2 Dynamic Buyer Utility
In this section, we analyze the impact of changing the parameters in the CES utility functions on tatonnement dynamics and market clearing conditions. Starting from the initial utility values, each can in each round be changed by some multiplicative factor . Let and .
For any ,
To show the result, we establish the Market-Perturbation property. Note that the misspending potential can be given by
Using we derive
For the rest of the proof, we construct an upper bound on the difference of two fractions. Fix a buyer . We set and . Moreover, we use and observe . We let
There exists a vector with
For a proof of the lemma, see Appendix A.2. Now let be a vector as in the previous lemma, let and . Using , we obtain
This expression is maximized at and yields an upper bound of . Thus, using Proposition 2 with and , we are done. ∎
4 Parametrized Lyapunov Dynamical Systems
In this section, we prove a general theorem, which includes as special case the bound shown for markets in Proposition 2. Our focus here are dynamical systems, in which time is discrete and represented by non-negative integers. Note, however, that the formulation below can be easily generalized to settings with continous time.
We assume that the dynamical system can be described by two sets of parameters. There is a set of control variables that can be adjusted by an algorithm or a protocol. In addition, there is a set of system parameters that can change in each round in an adversarial way. For example, in our analysis of markets in the previous section, the control variables are prices of goods, whereas system parameters can be supplies of goods, budgets of agents, or utility parameters. As another example, in games the control variables could be the strategy choices of agents, whereas system parameters are utility and payoff values of states. More generally, control variables could also be bird headings in a bird flock, while system parameters are wind direction or velocity, etc.
The classical theory of dynamical systems often studies the behaviour of systems with static system parameters. However, dynamical systems with varying system parameters often arise in practice (see Appendix C for some examples). Here, we propose a simple framework to analyze Lyapunov dynamical systems with varying system parameters. More formally, the dynamical system is described by an initial control variable vector and an evolution rule , which specifies how the control variables are adjusted. For each time , we have .
The system is called a Lyapunov dynamical system (LDS) if it admits a Lyapunov function such that
for every fixed point (equilibrium) of with it holds ;
for every it holds .
An LDS is called linearly converging (LCLDS) if it further satisfies
there exists a decay parameter such that for any , .
Let be a family of dynamical systems, while each dynamical system is parametrized by a system parameter vector . The family is called a family of parametrized, linearly converging LDS (PLCLDS) if each is an LCLDS and . For each , we denote its evolution rule by and its Lyapunov function by .
In many scenarios, particularly in agent-based dynamical systems, the control variables change by the evolution rule that expresses, e.g., the sequential decisions of the agents, but the system parameters can change in an exogenous (or even adversarial) way. However, in many cases the impact of changes in a single time step is rather mild. The following theorem states our recovery result by relating the Lyapunov value to the magnitude of changes in each step. Intuitively, it characterizes the “distance” that the evolution rule maintains to a fixed point over the course of the dynamics.
Let be a PLCLDS with , let denote the system parameter vectors at times , respectively, and let . Suppose that for every the system parameters invoke a change such that for every , we have The initial control variable vector is denoted by , and the system evolves such that for every we have . Then
Let , then it follows for any
For any time ,
Iterating the above recurrence yields the first result. For the second result, note that
In the scenarios where for small constant , we have the following corollary. In the setting of Theorem 4, if for some constant , then for any constant ,
As , the last two terms of the above inequality diminish. The bound is dominated by the first term, which describes the impact of the changes in the recent steps.
5 Proportional Response Dynamics
In the Fisher market setting, the general protocol of proportional response dynamics (PRD) is as follows. In each round, each buyer splits her budget among the goods according to some rule, and send the bids to the sellers of the corresponding goods. Based on the bids gathered from all buyers, the seller of each good send back (simple) signals to buyers, which are then used by buyers for updating their bids in the next round. We summarize the notation and results we need from Cheung, Cole and Tao  below. When buyer splits her budget among the goods, let denote the spending by her on good . Let denote . Let .
Consider the substitute domain, i.e., when the parameters of all buyers are strictly between and . In each round, the seller of good distributes the good among buyers in proportion to the bids received, and then after receiving the goods, each buyer splits her budget in proportion to the utility generated from the quantity of each good received. More formally, let , then the update rule is
The Kullback-Leibler (KL) divergence is similar to a distance measure. For vectors and such that , the explicit formula is . The above update rule is equivalent to mirror descent w.r.t. the KL divergence (but with different step sizes for different buyers) of the same function:
defined on the domain For our purpose, it suffices to know that any equilibrium of PRD corresponds to a minimum point of . The market potential with proportional response dynamics will be defined as:
Cheung, Cole and Tao  show that for positive constants (which depend on the maximum and minimum values of ) the market potential in a static market is bounded by
In the rest of the section, we analyze the impact of changing utility functions and supplies on the convergence properties of proportional response dynamics. For the varying budgets case, the domain varies too, prohibiting a similar analysis.
Dynamic Buyer Utilities. Starting with the initial utility parameters, suppose that each changes by a factor within . For a given budget allocation , let denote the market potential for the utility of the buyers in round , and the allocation that minimizes .
After rounds, it holds that
and , where .
For any round it holds
where is a constant and .
Consider the function defined above, and let be function in round with utility coefficients . We have
To be able to prove this claim, we need to derive an inequality of the following form:
This inequality is implied by the first one whenever is chosen large enough to satisfy
We choose a value for that satisfies the even larger lower bound of
The first inequality follows since for any , we have and . Finally, we derive an upper bound on the third term in Appendix A.3. This yields the final value of and proves the claim. ∎
Proof of Proposition 5.
Now suppose is given as in the proposition, then with the above claim it follows that:
The potential of the market at round can be bounded by
where the inequalities follow by recursive application of (6). ∎
Dynamic Supplies. It turns out that the case with varying supplies can be reduced to the case with varying utility functions. To see this, note that the function defined in (3) assumes that the supply of each good is normalized to be one unit. When the supply of good is changed from to , by performing a re-normalization of the supply, it is equivalent to changing to .
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Appendix A Missing Proofs
a.1 Proof of Proposition 3.1
To show the result, we establish the Market-Perturbation property. Note that the misspending potential can be given by