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Unique Ergodicity in the Interconnections of Ensembles with Applications to Two-Sided Markets R. Ghosh, W. M. Griggs, J. Marecek,ˇ and R. N. Shorten

Abstract—There has been much recent interest in two-sided reported by the agents without payments, which is important markets and dynamics thereof. In a rather a general discrete- within “artificial intelligence for social good”. While one can time feedback model, we show conditions that ensure that for leverage auditing mechanisms to maximize utility in repeated each agent, there exists the limit of a long-run average allocation of a resource to the agent, which is independent of any initial allocation problems where payments are not possible [9], conditions. We call this property unique ergodicity. our results can be seen as conditions on the information exchange, as well as prices, that allow for certain desirable Our model encompasses two-sided markets and more compli- cated interconnections of workers and customers, such as in a properties. supply chain. It allows for non-linearity of the response functions Our results concern certain desirable properties of such models of market participants. It allows for uncertainty in the response of market participants by considering a set of the possible responses and certain testable conditions assuring these properties [10]. to either price or other signals and a measure to sample from Informally speaking, we call a feedback system uniquely these. Finally, it allows for an arbitrary delay between the arrival ergodic, when for each agent i, there exists the limit of a of incoming data and the clearing of a market. long-run average allocation of a resource to the agent, which is independent of any initial conditions. In turn, the notion of unique ergodicity underlies a natural notion of fairness, distinct I.INTRODUCTION from other popular notions of fairness [11], where the limit Motivated by the success of the business models of Uber coincides across all agents or market participants. A necessary Technologies, Inc. or Upwork Global Inc., and following condition for the feedback system to be uniquely ergodic is Jean Tirole’s pioneering research [1,2,3,4, e.g.], there for the system to be contractive on average, in a sense we has been much recent work on two-sided and multi-sided formalise below. markets [5,6,7,8, e.g.]. One would like to analyse the long- run properties of such markets, including their stability and A. An Application: Two-Sided Markets fairness. Important applications lie within two-sided markets [2,3,4, We consider a discrete-time feedback model of an intercon- 12], which model online labour platforms that enable the nection of ensembles: interaction between the two sides: customers submitting jobs encompassing two-sided markets and more complicated over time, and workers performing jobs. For example, the • interconnections in multi-sided markets, ride-hailing systems of Uber, Lyft, or Didi Chuxing can be allowing for non-linearity of the response functions of modelled as a two-sided market with direct connections. In this • market participants, case, workers would be Uber drivers and jobs can be the rides allowing for uncertainty in the response of market par- of customers. Note that jobs are assumed to be independent • from each other, although one customer might offer multiple arXiv:2104.14858v1 [math.OC] 30 Apr 2021 ticipants by considering a set of the possible responses to either price or other signals and a measure to sample jobs. from these, See Figure1 (right) for an overview of our model of such a allowing for an arbitrary delay between the arrival of 1 1 • two-sided market: controller C suggests prices π (known as incoming data and the clearing of a market. Outside of driver surge pricing [13, 14, 15, 16, 17] in Uber) to customers finite-speed information transmission, this can model the 1 1 Si , whose requests yi (k) for jobs (rides) at time k are based buffering (prior to market clearing by matching) in many 1 on some internal state of each customer i at time k, xi (k), two-sided markets. which are not directly observable. A controller C 2 for the other side of the market matches the jobs (rides) to workers Our model applies even to settings where the organising entity (driver-partners) S 2 whose state x2(k) at time k may be divides resources among agents, based on the information i i partially observable (e.g., availability, position) and partially not observable (e.g., appetite for further work that day). R. Ghosh is at University College Dublin, Dublin, Ireland. W. M. Griggs is at Monash University, Melbourne, Victoria, Australia. Usually [18, 19, 20, 21, 22, e.g.], C 2 is implemented using J. Marecek is at the Czech Technical University, Prague, the Czech 2 Republic. an on-line matching algorithm. Its matches π (k) are pro- 1 R. N. Shorten is at Imperial College London, South Kensington, UK. vided to the workers (drivers), whose acceptance rate yi (k) 2 of the matches is then input into the controller C 1 that and recall the concept of coupling of invariant measures in suggests prices (with driver surge pricing implemented, if order to state a necessary condition for ergodicity, our main many matches go unaccepted, or there simply are not enough results are establised in section III on statistical stability of two drivers). sided market which is modelled by the closed loop system as in figure1, where in Theorem9 we state the result assuming B. Another Application: Multi-sided Platforms that the agents states are evolved as a linear-time-invariant Further important applications lie within multi-sided platforms dynamical system, in Theorem 13 states are governed by [5,6,7] and networked markets [8]. To continue our ride- non-linear iterated function system and in Theorem 16 we hailing example, one could see the ride-hailing systems of assume that the agents state has discrete-range space. In Uber as a multi-sided market, if one also considers “fleet part- SectionIV we extend our results obtained in Section III for the ners” (who are intermediaries for car manufacturers and car interconnection of N number of ensemble of systems, section leasing providers) and taxi operators. Indeed, Uber’s Vehicle V discusses our result adding time delay in our system. And Solutions Program is a platform for fleet partners to offer their finally in Section VIII, we highlights some directions for future vehicles to driver partners. Drivers, in turn, sometimes happen research. to work also as licensed taxi drivers. Similarly, Google’s Android ecosystem and Microsoft’s Win- II.PRELIMINARIES,NOTATION, ANDTHE DESCRIPTION dows ecosystem are sometimes seen [6, 23] as three-sided OFOUR MODEL platforms connecting consumers, software providers, and hard- In this article, we intend to extend and generalize our results ware providers. While some of the incentives offered to from [10] about the closed-loop, discrete time dynamical independent software providers (e.g., free hardware samples, systems of the form depicted in Figure1, consist of a number no-cost licenses of development tools) are not being adjusted N N denote the total number of agents, F denote the ∈ in real-time, others (e.g., promotions for their apps) are. The filter, and a controller C , which act for the central authority details of the control mechanisms have not been made public to produce a signal π(k) at time k. In response, the agents, in this case. modelled by the systems S1, S2,..., SN , modify their use of the resource. C. Contributions and Paper Organization Let (X, B) denote a measurable state-space, where (X, ρ) For these – and many other – applications, we present: is a metric space with ρ a metric on it. In most of our n a novel model of two-sided markets and extensions discussions, X is the Eucleadian space R or a closed subset • towards more general interconnections of ensembles of of it, ρ is the usual Eucleadian metric, and B is a Borel σ- agents, algebra on it. M (X) denote the space of all Borel probability a notion of unique ergodicity, which can be seen as measures on X, and X∞ denotes its associated path space [24, • individual-level predictability of the outcomes in a certain Section 1.1.1], which consists of infinite right-sided sequences closed-loop sense, over X, i.e., a typical element in the path space would conditions ensuring unique ergodicity, look like (x (0) , x (1) , . . . , x (n) , x (n + 1) ,... ): x (i) • { ∈ conditions ruling out unique ergodicity in the same sense. X for all i N . (Ω, B, P) be a probability space where • ∈ } Ω X or X∞ dependening on the context and P is the Our model is based on our earlier work on ensemble control probability≡ on it. [10], which we extend from having a central organising entity and all agents responding to prices it sets or signals it pro- The internal state of the agents in the network is denoted by vides, towards having two-sided markets or more complicated xi(k), i = 1, 2,...,N. In particular, for all i = 1, 2,...,N interconnections. The differences are illustrated in Figure1: xi(k) is a random variable. We model the use of the resource on the left is the model used in our earlier work; on the right, yi(k) of agent i at time k as the output of system Si, which the present model. is again a random variable. In the remainder we will assume yi(k) and π(k) are scalars, but generalizations are easy to Our negative results utilise coupling arguments developed obtain. The randomness can be a result of the inherent ran- by Hairer [24] within the study of stochastic differential domness in the reaction of user i to the control signal π(k), or equations. Our positive results are based on iterated function the response to a control signal that is intentionally randomized systems [25, 26, 27]. Iterated function systems (IFS), albeit [28, 29, 30]. The aggregate resource utilisation not well known in the Economics community, are a convenient N and rich class of Markov processes. As it will be seen in the X y(k) := y (k) (1) sequel, a two-sided markets can be modelled and analysed i i=1 using IFS in a particularly natural way. The use of IFS makes it possible to obtain strong stability guarantees for such at time k are then also random variables. The controller may stochastic systems. not have access to either xi(k), yi(k) or y(k), but only to the error signal e(k), which is the difference of yˆ(k), the output The paper is organised as follows: In SectionII, we present our of a filter F , and r, the desired value of y(k), i.e model and necessary mathematical background from the the- e(k) :=y ˆ(k) r. (2) ory of discrete time Markov chains, iterated function systems − 3

e1(k) π1(k) 1 1 1 u + 1 + C S

yˆ2(k) −

2 1 1 1 z− F SN e1(k) π1(k) 1 1 1 u + 1 + C S

1 1 z− F yˆ1(k)

+ 2 2 + u2 S1 2 2 1 1 π (k) C e (k) 1 z SN −

1 yˆ (k) 1 2 F SN 2

Fig. 1. Left: The feedback model of Fioravanti et al. [10], depicted utilizing our notation. Right: Our feedback model of a two-sided market.

Let Π denotes the set of all admissible broadcast control 1,...,N which are conditioned on x (k) , π(k), but } { i } signals. Assume that the controller has its private state xc(k) stochastically independent. The outputs yi(k) each depend on nc ∈ R and aims to regulate the system1 by providing a signal xi(k) and the signal π(k) only. The behaviour of the overall π(k) Π R at time k. In the simpler static case, the signal system in response to the signal π(k) can be modelled by ∈ ⊆ π(k) is a function of an error signal e(k) and the controller the operator P as P : X Π M (X). In order to reason × → state xc(k), whose range is Π. about the evolution of the state, consider the space X∞ and we introduce the space of probability measures on M (X∞) In the case of purely discrete agents, the non-deterministic with the product σ-algebra. agent-specific response to the feedback signal π(k) Π can ∈ be modelled by agent-specific and signal-specific probability The set-up described in the prequel resembles closely that distributions over certain agent-specific set of actions Ai = of an iterated function system [25, 26, 27]. Iterated function ni ni a1, . . . , aL R , where R can be seen as the space systems are a class of stochastic dynamical systems, for which { } ⊂ of agent’s i private state xi. Assume that the set of possible strong stability and convergence results exist on compact or resource demands of agent i is Di, where in the case that Di complete and separable metric spaces as a system’s state space, is finite, we denote see [26, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. We now introduce some notation and mention some Di := di,1, di,2, . . . , di,mi . (3) { } of the most important results. In the general case, we assume there are wi N state ni ni ∈ Definition 1. A discrete-time, time-homogeneous Markov transition maps Wij : R R , j = 1, . . . , wi for agent → ni chain on is a sequence of -valued random vectors i and hi N output maps Hi` : R Di, ` = 1, . . . , hi for X X ∈ → each agent i. The evolution of the states and the corresponding X(k) k with the Markov property, which is the equality { } ∈N demands then satisfy: of a probability of an event conditioned on past events and probability of the same event conditioned on the current state, x (k + 1) W (x (k)) j = 1, . . . , w , (4) i ∈ { ij i | i} i.e., yi(k) Hi`(xi(k)) ` = 1, . . . , hi , (5) ∈ { | } P (X(k + 1) G X(j) = xj, j = 0, 1, . . . , k) where the choice of agent i’s response at time k is governed ∈ |
= P X(k + 1) G X(k) = xk , (7) by probability functions p :Π [0, 1], j = 1, . . . , w , ∈ | ij → i respectively p0 :Π [0, 1], ` = 1, . . . , h . Specifically, for i` → i m each agent i, we have for all k that Definition 2. Let fi i=1 be continuous self-transformations N m{ } ∈ on X and pi(x) be probability functions on B(X), such
{ }i=1 P xi(k + 1) = Wij(xi(k)) = pij(π(k)), (6a) that,
(6b) m P yi(k) = Hi`(xi(k)) = pi`0 (π(k)). X pi(x): X [0, 1] i [1, m], and pi(x) = 1. Additionally, for all π Π, i = 1,...,N it holds that → ∀ ∈ ∈ i=1 w h Xi Xi The pair of sequences pij(π) = pi`0 (π) = 1. (6c) j=1 `=1 (f1(x), f2(x), . . . , fm(x); p1(x), p2(x), . . . , pm(x)) We assume that the random variables x (k + 1) i = is called an iterated function system (IFS). { i | 4

Any discrete-time Markov chain can be generated by an in distribution. The existence of attractive invariant measures iterated function system with probabilities see Kifer[47, Sec- is intricately linked to the ergodic properties of the system. tion 1.1] or [48, Page 228], although such representation is With this background, our general problem considered in this not unique, see, Stenflo [34]. Informally, the corresponding paper is modelled as a Markov chain on a state space repre- discrete-time Markov process on evolves as follows: choose X senting all system components. Let be the product space an initial point x X . Select an integer from the set [1, m] XS 0 of the state spaces of all agents. The spaces , contain in such a way that∈ the probability of choosing σ is p (x ), XF XC σ 0 the possible internal states for filter and central controller. Our σ [1, m]. When the number σ0 is drawn, define ∈ system thus evolves on the state space X := XS XC XF , × × x1 = fσ0 (x0). as we will see in the following sections.

Having x1, we select σ1 according to the distribution Notice that for a particular combination of a filter F , con- troller C , and population of agents, the feedback loop can p1(x1), p2(x1), . . . , pm(x1), be modelled by an operator Pk : M (X) M (X) and the → and we define associated dynamical system (Pk)k . Our paper, at some ∈N level, asks what properties of C and F make (Pk)k x = f (x ), ∈N 2 σ1 1 predictable and what properties of C render it impossible and so on. for it to be predictable. Predictability is related to asymptotic convergence in probability in terms of measures over the path Let us denote νn for n = 0, 1, 2,... , the distribution of xn, space, i.e., M (X∞), and will be conveniently characterised in i.e., the next section in terms of the existence of a unique invariant measure, in the language of Markov chains. νn(B) = P(xn B) for some B B. (8) ∈ ∈ Our specific aim is to distribute the resource such that we The above procedure can be formalized for a given x X ∈ achieve the following goals almost surely, i.e., with probability and a Borel subset B B, we may easily show that the 1: transition operator for the∈ given IFS is of the form:

N Definition 6. Given an upper bound r > 0 for the utilization X of the resource, we require for all k N ν(x, B) := 1B (wi(x)) pi(x), (9) ∈ i=0 N X y (k) = y(k) r. (13) ν(x, B) is the transition probability from x to B. where 1B i ≤ denotes the characteristic function of B: i=1 ( More generally, the resource could be time-varying; for the 1 if x B. 1B := ∈ purposes of this paper it will be assumed to be a constant 0 if x Bc. ∈ quantity. Definition 3. The transition operator P of the Markov chain Definition 7 (Unique ergodicity). We call a feedback system uniquely ergodic, when for each agent i, there exists a constant is defined for x Σ, G B by ∈ ∈ ri such that P (x, G) := P(X(k + 1) G X(k) = x). (10) k ∈ | 1 X lim yi(j) = ri, (14) k k + 1 Definition 4 (Invariant probability measure [26, 42]). If the →∞ j=0 initial condition X0 is distributed according to an initial where this limit is independent of the initial conditions. distribution µ, we denote by Pµ the probability measure induced on the path space, i.e., space of sequences with values Further optional requirements may include: fairness, which in Σ. Conditioned on an initial distribution λ, the random could be formulated by saying that all the r coincide, and variable X(k) is distributed according to the measure µ i k optimality so that the vector r = r ... r  is an opti- which is determined inductively by 1 N mum of some associated optimization problem. In addition, it Z is also of interest to achieve the goal after a transient phase, µk+1(G) := P (x, G) µk(dx), (11) i.e., for all k K, where K is a constant. Σ ≥ ? for G B. A measure µ on Σ is called invariant with respect While all of these goals are important from a practical perspec- to the∈Markov process X(k) if it is a fixed point for the { } tive, the principal property of interest in this paper is the goal iteration described by (11), i.e., if of predictability. To analyze this more formally, we consider d P µ? = µ?. (12) an augmented state space X R , which captures the state of the controller, the filter, and⊂ the agents. Definition 5. An invariant probability measure µ is called In our positive results, we consider a class of Markov chains attractive, if for every probability measure ν the sequence that are known as iterated function systems (IFSs). In an λ defined by (11) with initial condition ν converges to µ iterated function system, we are given a set of maps f : { k} { j 5

X X j J , where J is a (finite or countable) index where the error signal depends on the filtered state of the other set.→ Associated| ∈ to} these maps, there are probability functions side of the market: P pj : X [0, 1], where j pj(x) = 1 for all x X. The → ∈J ∈ state X(k + 1) at time k + 1 is then given by fj(X(k)) e1(k) = u1(k) yˆ2(k) (18) with probability pj(X(k)), where X(k) is the state at time − k. Such IFSs have been studied in fractal image compression [49], congestion management in computer networking [50] and The first side has its state filtered too: transportation [51], and to some extent, control theory at large [52, 53, 54]. ( 1 1 1 1 1 xf (k + 1) = Af xf (k) + Bf y (k), The central technique of proving the unique ergodicity of F 1 : (19) yˆ1(k) = C1x1 (k). an IFS is by using several contractivity conditions. Sufficient f f conditions for the existence of a unique attractive invariant measure can be given in terms of “average contractivity”. This The other side of the market is modelled by another population key notion can be traced back to [25, 55, 27]: of participants:

n Theorem 8. [55, Theorem 2.1] Let X R be closed. ( 2 2 2 2 ⊂ xi (k + 1) = Ai xi (k) + bi , Consider an IFS with a finite index set J, and globally 2 : (20) Si 2 2T 2 2 Lipschitz continuous maps fj : X X, j J. Assume that yi (k) = ci xi (k) + di , → ∈ the probability functions pj are globally Lipschitz continuous for i = 1, 2,...,N2, evolving according to its dynamics and and bounded below by η > 0. If there exists a δ > 0 such that the control of a (in general) different controller: for all x, y X, x = y ∈ 6   X fj(x) fj(y) pj(x) log k − k < δ < 0, ( 2 2 2 2 2 x y − 2 xc (k + 1) = Ac xc (k) + Bc e (k), j J k − k C : (21) ∈ π1(k) = C2x2(k) + D2e2(k), then there exists an attractive (and hence unique) invariant c c c probability measure µ for the IFS. where the error signal considers the filtered state of the first side of the two-sided market: We can combine Theorem8 with a result by Elton [25, Corol- lary 1], to obtain that for all (deterministic) initial conditions 2 2 1 x0 X and continuous g : X R, the limit e (k) = u (k) +y ˆ (k) (22) ∈ → k 1 X lim g(X(ν)) = Eµ(g) (15) Likewise, there is the filter for the second side of the two-sided k k + 1 →∞ ν=0 market: exists almost surely (Px0 ) and is independent of x0 X. The ∈ limit is given by the expectation with respect to the invariant ( x2 (k + 1) = A2 x2 (k) + B2y(k), measure µ. 2 f f f f F : 2 2 2 (23) yˆ (k) = Cf xf (k). For more general theorems, the reader is referred to [25, 55, 27, 49, 35, 44, 32, 56, 57] and surveys [31, 58, 37]. Theorem 9. Consider the feedback system depicted in Figure 1 2 1 2 As we have discussed before, this notion precisely guarantees 1, with C , C and F , F given in (17, 21, 19, and 23), respectively. Assume that each agent i 1, ,N has the property of predictability and can be used for an analysis ∈ { ··· } of fairness and optimality. its state xi with dynamics governed by the affine stochastic 1 2 difference equations given in (16 and 20), where Ai ,Ai are 1 2 1 2 Schur matrices and bi , bi and di , di are chosen, at each time 1 2 1 2 ni 1 2 ni III.AN INTERCONNECTION RESULT FOR TWO ENSEMBLE step, from the sets bij, bij R and di`, di` R ac- { } ⊂ { }1 ⊂ 2 SYSTEMS cording to Dini continuous probability functions pij( ), pij( ), 01 02 · · respectively pi` ( ), pi` ( ), that verify (6). Assume furthermore Let us consider a two-sided market, where one side is modelled · ·1 2 01 02 1 that there are scalars δ , δ , δ , δ > 0 such that pij(π) by participants: 0 0 0 0 ≥ δ1 > 0, p2 (π) δ2 > 0, p 1(π) δ 1 > 0, p 2(π) δ 2 > 0 ( ij ij ij x1(k + 1) = A1x1(k) + b1, ≥ 1 1≥ 2 2 ≥ 1 i i i i for all (i, j) and all π Π , π Π . Then, for any S : (16) 1∈ 2 ∈ i 1 1T 1 1 stable linear controllers C , C and any stable linear filters yi (k) = ci xi (k) + di , F 1, F 2 compatible with the system structure, the feedback 2 for i = 1, 2,...,N1, responding to the control signal π loop converges in distribution to a unique invariant measure. produced by

( 1 1 1 1 1 1 xc (k + 1) = Ac xc (k) + Bc e (k), Proof. Following [55], the proof is centred on the construc- C : 1 1 1 1 1 (17) π (k) = Cc xc (k) + Dc e (k), tion of an iterated function system (IFS) with place-(state- 6

)dependent probabilities that describes the feedback system. bounds [56]. These would complement the present theorem. To this end, consider the augmented state Remark 12. In [10, Theorem 12], we assume that the feed-  1 N 1  (xi )i=1 back loop only consists of stable systems. This assumption 2 (x2)N  may seem quite restrictive, but as shown in the examples, even  i i=1  x1  marginal stability in one of its parts may lead to completely ξ := f 1 2 1 2 1 2 ,  2  XS XS XF XF XC XC unpredictable results. As the result from [55] only requires  xf  ∈ × × × × ×  1  contractivity on average, more general results are possible,  xc  2 but beyond the scope of the present paper. We also point out xc (24) that each agent may have a local stabilising controller, as we whose dynamic behaviour is described by the difference are analysing the feedback loop from a macroscopic level. equation Theorem 13. Consider the feedback system depicted in Figure1. Assume that each agent and ξ(k + 1) = W`(x) := A ξ(k) + β`, (25) i 1, ,N1 i 1, ,N has a state governed∈ by { the··· non-linear} ∈ { ··· 2} where the β` is built from the combinations of the vectors iterated function system b1 , b2 , the scalars d1 , d2 and other signals, and the block- ij ij ij ij 1 1 1 matrix xi (k + 1) = Wij (xi (k)), (27)   Aˆ1 0 0 0 0 0 y1(k) = H 1(x1(k)), (28)  0 Aˆ2 0 0 0 0  i ij i   B11T Cˆ1 0 A1 0 0 0  and A :=  f f   0 B21T Cˆ2 0 A2 0 0   f f  x2(k + 1) = W 2(x2(k)), (29)  0 0 0 B1C2 A1 0  i ij i  − c f c  0 0 B2C1 0 0 A2 c f c y2(k) = H 2(x2(k)), (30) (26) i ij i where 1 1 2 2 are globally Lipschitz-continuous in which the blocks are, Wij , Hij , Wij , Hij 1 01 2 02 functions with global Lipschitz constant lij, lij , lij, lij resp. ˆ1 1 ˆ1 1 A := diag(Ai ), C := diag(ci >), Assume we have Dini continuous probability functions 0 0 2 2 2 2 1 1 2 2 Aˆ := diag(A ), Cˆ := diag(c >). pij, pil , pij, pil so that the probabilistic laws (6) are satisfied. i i 0 0 Assume furthermore that there are scalars δ1, δ 1, δ2, δ 2 > 0 To apply Corollary 2.3 from [55], two observations must be 1 1 01 01 2 2 such that pij(π) δ > 0, pij (π) δ > 0, pij(π) δ > made. First, note that each map is chosen with probability 0 0 ≥ ≥ ≥ W` 0, p 2(π) δ 2 > 0 for all (i, j). Furthermore, assume that p (π) QN δ > 0 and thus these probabilities are bounded ij ` i=1 i the following≥ contractivity condition holds: for all 1 i away from≥ zero. Second, since is lower-triangular block- A N , 1 j J , 1 i N , 1 j J : l1 < 1, l2≤< ≤1. matrix it is easy to notice that the spectrum of A i.e σ (A ) 1 ≤ ≤ 1 ≤ ≤ 2 ≤ ≤ 2 ij ij Then, for every stable linear controller C and C and every can be expressed as follows: 1 2 stable linear filter F1 and F2 compatible with the feedback ˆ1 ˆ2 1 2 1 2 structure1, the feedback loop has a unique attractive invariant σ(A ) = σ(A ) σ(A ) σ(Af ) σ(Af ) σ(Ac ) σ(Ac ) ∪ ∪ ∪ ∪ ∪ measure. In particular, the system is uniquely ergodic. 1 2 and, by hypothesis, for all 1 i N, Ai ,Ai are Schur 1 ≤2 ≤1 2 matrices [59, 60, 61] and Af ,Af ,Ac ,Ac are Schur matrices Proof. Similarly to the proof of [10, Theorem 18] the assump- as well, for any induced matrix norm there exists m N tions on the Lipschitz constants and the the internal asymptotic m ∈ sufficiently large such that A < 1. This provides the stability of controller and filter guarantees that suitable iterates required average contractivityk afterk m steps. The result then of the maps Fm are strict contractions. The result then follows follows from [55]. The proof is complete. again from Theorem 2.1 and Corollary 2.2 of [55].

Remark 10. Dini’s condition on the probabilities may be Remark 14. Notice that Lipschitz continuity can be rephrased replaced by simpler, more conservative assumptions, such as in many ways. For instance, there is the QUAD condition [62], Lipschitz or Hölder conditions [55]. Also the requirement the sector condition, or, when restricted to convex functions, the bounded subgradient condition. The sector condition re- pij(π) δi > 0 in the theorem statement is not an artefact ≥ quires that there exist constants c1 and c2 such that the vector- of our analysis, as pij(π) = 0 may lead to a non-ergodic behaviour. valued functions W (x) := [Wi(x)] and H (x) := [Hi(x)] T T satisfy W (x) [W (x) c1x] 0 and H (x) [H (x) c2x] Remark 11. As we have stressed that the existence of an at- 0. The bounded subgradient− ≤ condition requires the− existence≤ tractive invariant measure is only a baseline ergodic property. of constants c , c , such that for a given norm , for all 3 4 | · | Under mild but technical additional assumptions, it is possible z, z0 in the subdifferentials of W , H , respectively, at all to prove a geometric rate of convergence [32]. Under further points in the domains of the respective functions, we have assumptions, one can prove results concerning the “shape” of that z c3, z0 c4. Here the functions W (x), H (x) the unique invariant measure, where it exists, such as moment are assumed| | ≤ to| be| ≤ convex with a non-empty subdifferential 7 throughout their domains, but not necessarily differentiable. for all (i, j) and all π. Then, for every stable linear controller The equivalence follows from basic convex analysis, e.g., as a C and every stable linear filter F the following holds: corollary of Lemma 2.6 in [63]. If the graph G = (XS,E) is strongly connected, then there Next, consider the case when the agents’ actions are limited exists an invariant measure for the feedback loop. If in to a finite set. In this case, the Lipschitz conditions cannot be addition, the adjacency matrix of the graph is primitive, then satisfied except in trivial cases. In this case we may use results the invariant measure is attractive and the system is uniquely in [64, 65] to obtain ergodicity results. ergodic.

Let [M] denote the set 1, 2,...,M and suppose, Proof. This is a consequence of [65] and the observation that X ,X ,...,X be a partition{ of a metric} space into a 1 2 M X the necessary contractivity properties follow from the internal non-empty Borel subsets. Consider, for all i [M], let ∈ asymptotic stability of controller and filter.

wi,1, wi,2, . . . , wi,Li : Xi X → We note that a simple condition for the primitivity of the graph be a family of Borel maps such that for each j [Li], there ∈ is that for each agent the graph describing the exists m [M] such that wi,j(Xi) Xm. Finally, for each G = (XS,E) i [M], let∈ ⊆ possible transitions is primitive. ∈

pi,1, pi,2, . . . , pi,Li : Xi (0, 1], → IV. A LARGE-SCALE INTERCONNECTIONFOR ENSEMBLE be a family of Borel measurable probability functions associ- SYSTEMS PLi ated with the maps, and and pi,j = 1 for all x Xi. As hinted at in Section I-B, many markets that are traditionally ∈ j=1 thought of as two-sided are multi-sided platforms [5,6,7]. In Definition 15. [64] We call V = 1, 2,...,M the set of the example of Uber, one can also consider “fleet partners” { } vertices and the subsets X1,...,XM are called vertex sets. (who are intermediaries for car manufacturers and car leasing Further, we call the set providers) and taxi operators. Such interconnections can get rather complicated and we would like to capture these in a E = (i, m ): i [M], m [L ] { i ∈ i ∈ i } matrix, similarly to the adjacency matrix of a graph defining the graph. the set of edges and use the following notations: ˜ pe := pi,m and we := wi,m for (i, m) E . u˜ e˜ π˜ y˜ yˆ ∈ + C P F − The edge is provided with a direction (an arrow) by marking an initial vertex through the map i : E V H → (j, m) j. 7→ Fig. 2. Large-scale interconnection feedback model. The terminal vertex t(j, m) V of an edge (j, m) E is determined by the corresponding∈ map through ∈ Building on the techniques of Section III, consider the in- terconnection of a multi-sided platform as described as fol- t ((j, m)) := s : wj,m(Xj) Xs. lows: ⇐⇒ ⊆ M We call the quadruple G := (V , E , i, t) a directed multigraph p p X q e = u Hpqyˆ , (31) or digraph. Let for each agent i the set Ai is finite. Then XS = − N q=1 Y p Ai is also finite. Consider a directed graph G := (XS,E), where e , for p = 1, 2,...,M, is the input to i=1 ( where there is an arc between vertices representing (x ) xp(k + 1) = Apxp(k) + Bpep(k), i XS C p : c c c c (32) and whenever there is a choice of maps in∈(27) p p p p p (yi) XS Wij π (k) = Cc xc (k) + Dc e (k), and(29). ∈ p u are external inputs, Hpq is a matrix with real and constant Theorem 16. Consider the feedback system depicted in Figure entries, and yˆq, for q = 1, 2,...,M, is the output of 1. Assume that Ai is finite for each i. Assume that each agent ( q q q q q q xf (k + 1) = Af xf (k) + Bf y (k), i 1,...,N1 and i 1,...,N2 has a state governed by F : (33) ∈ { } ∈ { } q q q the non-linear stochastic difference equations. Assume we have yˆ (k) = Cf xf (k). 0 0 Dini continuous probability functions p1 , p1 , p2 , p2 so that ij il ij il We also have, for i = 1, 2,...,Nq, the probabilistic laws (6) are satisfied. Assume furthermore 1 10 2 20 1 ( q q q q that there are scalars δ , δ , δ , δ > 0 such that pij(π) q xi (k + 1) = Ai xi (k) + bi , 0 0 0 0 ≥ S : (34) δ1 > 0, p1 (π) δ1 > 0, p2 (π) δ2 > 0, p2 (π) δ2 > 0 i yq(k) = cqT xq(k) + dq, ij ≥ ij ≥ ij ≥ i i i i 8

where Nq is the total number of systems in the qth ensemble We obtain q q q q denoted by S . Finally, y (k) = y1(k) + y2(k) + +  ˆ  q ··· A 0 0 yN (k). q A := Bf 1ˆCAˆ f 0  , (38) 0 BcHCf Ac Now, by writing − where,  e1   u1   π1   y1   yˆ1    T
2 2 2 2 2 1 := 1 1 1 , 1ˆ := diag 1 ,  e   u   π   y   yˆ  ···         ˜   1 1 2 2 M M
e˜ :=  .  , u˜ :=  .  , π˜ :=  .  , y˜ :=  .  , yˆ :=  . Aˆ := diag A ,...,A ,A ,...,A ,...,A ,...,A ,  .   .   .   .   .  1 N1 1 N2 1 NM M M M M M ˆ diag 1 1 2 2 M M
e u π y yˆ C := c1>, . . . , cN>1 , c1>, . . . , cN>2 , . . . , c1 >, . . . , cNM> , (35) 1 2 M 1 2 M Af := diag(Af ,Af ,...,Af ),Bf := diag(Bf ,Bf ,...,Bf ), the interconnection description may be expressed more com- 1 2 M 1 2 M pactly as Cf := diag(Cf ,Cf ,...,Cf ),Ac := diag(Ac ,Ac ,...,Ac ), 1 2 M e˜ =u ˜ Hy,ˆ˜ (36) Bc := diag(B ,B ,...,B ) − c c c where H is a matrix with block entries Hpq. Let C := The proof then follows in a manner similar to the proof of diag(C 1,..., C M ) such that π˜ = Ce˜, and F := Theorem9. diag(F 1,..., F M ) such that yˆ˜ = F y˜. This set-up is depicted in Figure2. Then, similar to Theorem9, we have the following V. TIME DELAYS result. Consider a scenario where a controller waits for a certain amount of jobs to come in before performing a matching Theorem 17. Consider the feedback system as described of jobs to workers of any sort; that is, some buffering is above and depicted in Figure2. For each ensemble S q, performed. This, indeed, is the only practically relevant sce- q = 1, 2,...,M, assume that each ith system in the ensemble nario: if the controller did not wait, then the matching would q has its state xi with dynamics governed by the affine stochas- be performed on a greedy, first-come, first-served basis. This q tic difference equations given by (34), where Ai are Schur waiting time does, however, introduce a time delay. q q matrices, and bi and di are chosen, at each time step, from sets according to Dini continuous probability functions in the In this section, we provide some mathematical discussion on manner of Theorem9, and that these probability functions are time delays. Specifically, we describe a time delay scenario and offer some analysis on it. It is worth keeping in mind that bounded below by scalars strictly greater than 0. Then, for any there are additional ways in which a time delay scenario could stable linear controllers C 1,..., C M and any stable linear be described. filters F 1,..., F M compatible with the system structure, the feedback loop converges in distribution to a unique invariant Consider the linear discrete-time systems measure. ( xi(k + 1) = Aixi(k) + A˜ixi(k h) + bi, Si : T T − yi(k) = ci xi(k) +c ˜i xi(k h) + di, − (39) for i = 1, 2,...,N, where Ai and A˜i are Schur matrices and Proof. In the spirit of Theorem9, by defining an augmented h is a constant delay. Expressing the first equation in(39), for state example, in a more compact form gives  x˜  ξ := x˜ , x(k + 1) = diag(Ai)x(k) + diag(A˜i)x(k h) + b, (40)  f  − x˜c where     whose dynamic behaviour is described by the difference x1 b1 equation  x2   b2  x :=  .  and b :=  .  .  .   .  ξ(k + 1) = W`(x) := A ξ(k) + β`, (37)     xN bN where Following the technique described in Section 6.1 of [66], we  x1   1   1  f xc x can augment x(k) by its finite history 2 2 2  xf   xc   x  x˜ :=   , x˜ :=   , x˜ :=   ,  x(k)  f  .  c  .   .   .   .   .  x(k 1) M xM xM x (k) :=  −  xf c aug  .   1   2   M   .  x1 x1 x1 1 2 M x(k h)  x2   x2   x2  − 1 :   2 :   M :   x =  .  , x =  .  , . . . , x =  .  . which, from(40), results in an augmented system without  .   .   .  delay: x1 x2 xM N1 N2 NM xaug(k + 1) = Aaugxaug(k) + baug, 9

h i h i ˜ n n where (b) i, diag(Ai) diag(Ai) R × , with 0 α β ∀ α β ∈ ≤ ≤ ≤ diag(A ) 0 diag(A˜ ) b n, are simultaneously symmetrizable (Definition 20), i ··· i  I 0 0  0 A :=  ···  and b :=   . then for any stable linear controllers C 1, C 2 and any stable aug  ......  aug .  . . . .  . linear filters F 1, F 2 compatible with the system structure, we 0 I 0 0 have asymptotic stability of the system with constant delays ··· (39), i.e., have all eigenvalues of Aaug inside the unit circle. Now, let us consider a simple feedback loop as depicted in Fig. 1 of [10]. Let Proof. First, notice that Aaug is a block companion matrix. The proof is based on the work of Martins et al. [72, 70] and the x  aug discrete-time Lyapunov stability theorem [73]. Proposition 2.1 ξ := x aug  f  of [72] (also reprinted as Proposition 2.2 in [70]) shows that x c under the conditions of the present theorem, for any symmetric where xc and xf are as described in (12) and (13), respectively, positive definite matrix W , there exists a unique symmetric of [10]. The dynamic behaviour of ξaug is described by ξaug(k+ solution V of the Lyapunov matrix equation , where 1) = Aaugξaug(k) + β`aug T A V + VAaug = W. (42)   aug − Aaug 0 0 T Subsequently, the discrete-time Lyapunov stability theorem Aaug := Bf 1 [c 0 0c ˜] Af 0  , 0 ··· B C A [74, Theorem 7.3.2] shows the asymptotic stability of − c f c  T   T      c1 c˜1 d1 x (k + 1) = Aaugx (k) , (43) b T T aug c  c˜   d2  T 2 2 which in turn is equivalent [74, Theorem 7.3.1.] to the all the β`aug := Bf 1 d , c :=  .  , c˜ :=  .  , d :=  .   .   .   .  Bcr       eigenvalues of Aaug being inside the unit circle. Consequently, T T cN c˜N dN all the eigenvalues of Aaug are inside the unit circle since the PN other submatrices (Af , Ac, etc) are Schur by the assumption and r is the desired value of y(k) = i=1 yi(k). of stability of the controllers and filters. Definition 18 (Block-companion matrix,[67, 68]). Consider n n EGATIVE ESULTS FOR NTERCONNECTIONSOF the linear operators Bi L (R , R ) for i = 1, 2, . . . , n 1 VI.N R I ∈ − in the space of all linear operators L (Rn, Rn) and let B := MULTIPLE ENSEMBLES (B0,B1,...,Bn 1). Consider the operator polynomial PB as It could be worth investigating in our context to determine the − follows: control strategies that destroy ergodicity of the closed-loop n n 1 feedback system, or in other words, circumstances in which PB(λ) := λ I + λ − Bn 1 + ... + λB1 + B0, λ C. − ∈ an attractive invariant measure fails to exist. In this scenario, B The companion operator C of PB is the n n block operator coupling arguments may be used as they provide criteria for n n × n matrix in the Hilbert space R R R given by the non-existence of a unique invariant measure1. | ⊕ {z⊕ · · · ⊕ } n copies Consider the space of trajectories of a Σ-valued Markov chain   0 IN 0 0 0 0 X(k) k , i.e., the space of all sequences (x(0), x(1), ··· { } ∈N  0 0 IN 0 0 0 0  x(2),...) with x(k) Σ, k N, by Σ∞ (the “path space”).   ∈ ∈  0 0 0 IN 0 0 0  Recall, for example, that Pλ M (Σ∞) is the probability   ∈ B  ......  measure induced on the path space by the initial distribution C =  ......    λ of X(0). A coupling of two probability measures Pµ1 ,Pµ2  0 0 0 0 IN 0  ∈  ···  M (Σ∞) corresponding to two different initial distribution µ1  0 0 0 0 0 0 IN  and µ2, is a probability measure on Σ∞ Σ∞ whose marginals B0 B1 Bn 2 Bn 1 × coincide with Pµ ,Pµ . To be precise, consider a probability − − · · · · · · · · · − − − − (41) 1 2 measure Γ over the product of space M (Σ∞ Σ∞). Γ can be × projected to a measure over one or the other factor space Σ∞; n n Definition 19 ([69], p. 67). A matrix B R × is said to Let us denote this projections by Φ(1)Γ and Φ(2)Γ. The set be symmetrizable if there exists a symmetric-positive∈ definite Ccoup(Pµ ,Pµ ) of couplings of Pµ ,Pµ M (Σ∞) is then T n n T 1 2 1 2 ∈ matrix R = R R × such that B R = RB. defined by ∈ n o Definition 20 ([70, 71]). Two matrices B ,B n n (1) (2) 1 2 R × Ccoup = Γ M (Σ∞ Σ∞):Φ Γ = Pµ , Φ Γ = Pµ . are said to be simultaneously symmetrizable if there∈ exists a ∈ × 1 2 T n n symmetric-positive definite matrix R = R R × such that T T ∈ B1 R = RB1 and B2 R = RB2. Theorem 21. Suppose the following conditions are satisfied: 1Coupling arguments have been used since the theorem of Harris [75, 76], and are hence sometimes known as Harris-type theorems. Generally, they link (a) σ (Aaug) σ ( Aaug) = φ, the existence of a coupling with the forgetfulness of initial conditions. ∩ − 10

We say that a coupling Γ Ccoup is an asymptotic coupling if where a cascaded PI control has been considered, e.g., for the Γ has full measure on the∈ pairs of convergent sequences. To control of internal combustion engines such that the overall make this precise consider the set H defined by: pollution is regulated, cf. Table 1.1 in [80]. On the other hand,   the situation may be substantially more complicated, when (x (k) , z (k)) Σ∞ Σ∞ : lim x(k) z(k) = 0 . time delays are considered, cf. SectionV. ∈ × k k − k →∞ Remark 25. Notice that the fourth condition can be readily Γ is an asymptotic coupling if Γ(H ) = 1. The following used to reason about, e.g., multiplicative surge pricing at Uber. statement is a specialization of [24, Theorem 1.1] to our While it is not incentive-compatible beyond the single-state situation: model [17], the fourth condition suggests that it is not uniquely Theorem 22 ([24]). Consider an IFS with associated Markov ergodic and hence not fair in the sense that the individuals’ operator P admitting two ergodic invariant measures µ1 and outcomes are independent of the initial state. To see this, µ2. The following are equivalent: consider a linearization of the multiplicative surge pricing, which would not be asymptotically stable. This may have been (i) µ1 = µ2. a motivation beyond the switch to additive surge pricing 2. (ii) There exists an asymptotic coupling of Pµ1 and Pµ2 . VII.RELATED WORK Consequently, if no asymptotic coupling of Pµ1 and Pµ2 exists, then µ1 and µ2 are distinct. Within Economics, Jean Tirole’s pioneering research [1,2, 3,4] on two-sided markets has a substantial following [12]. The existence of asymptotic couplings can then be ruled Especially multi-sided platforms [5,6,7] and networked out with ease for certain controllers. For example, the markets [23] became widespread both in economic theory and Proportional-Integral (PI) controller can, in its simplest form the real world. [77, 78], be cast as: Within Economics and Computation, much attention has been π(k) = π(k 1) + κe(k) αe(k 1), (44) − − − focussed on ride-hailing. When workers cannot reject jobs, where we omit the superscript indices for brevity. Conse- dynamic pricing cannot outperform the optimal static policy in quently, the transfer function from e to π is terms of throughput and revenue. It is known [17] that additive increases provide an incentive-compatible pricing mechanism, 1 π(z) 1 αz− C(z) := = κ − . (45) while certain complications arise [81] from the spatio-temporal e(z) 1 z 1 nature of the problem. Motivated by Uber’s own admission − − This utilizes the well-known Z -transform to convert a [82] that its prices are unfair to female and ethnic minority discrete-time signal into its complex frequency-domain rep- drivers, there has recently been some interest in studying resentation. fairness in related settings [83, 84, 85, 86], often using concept established in fairness [87, 88] in machine learning more Proposition 23. Consider the feedback system depicted in broadly. Figure1 on the right, with C 1, C 2 and F 1, F 2 given in (17, 21, 19, and 23). Assume that each agent i 1, ,N has Related work also appears in the context of multi-agent ∈ { ··· } systems [89]. Multi-agent systems arise, for instance, in the its state xi with dynamics governed by the affine stochastic 1 2 application areas stated above, and many references in the lit- difference equations given in (16 and 20), where Ai ,Ai are 1 2 1 2 erature address these problems using ideas based on consensus Schur matrices and bi , bi and di , di are chosen, at each time 1 2 1 ni 1 2 ni or agreement [90, 91]. The consensus approach is very useful step, from the sets bij R and di`, di` R ac- { } ⊂ { }1 ⊂ 2 in practice, due to its strong links to utility maximization and cording to Dini continuous probability functions pij( ), pij( ), 01 02 · · fairness. The concept of fairness can be related to ergodicity, respectively pi` ( ), pi` ( ), that verify (6). When at least one of the following conditions· · is not satisfied, the system cannot be as an ergodic dynamic behaviour implies several important uniquely ergodic: properties that are necessary for fairness [92]. Finally, we note that our positive results in this paper are based on iterated (1) there are two distinct ergodic measures; function systems [25, 26, 27]. Iterated function systems (IFS), albeit not well known, are a convenient and rich class of (2) there is no asymptotic coupling of Pµ1 and Pµ2 ; (3) one of the controllers C i is proportional–integral (44), Markov processes. The class of stochastic systems arising from while there is no delay; the dynamics of multi-agent interactions can be modelled and (4) one of the controllers C i has a pole on the unit circle, analysed using IFS in a particularly natural way. A wealth of i.e., at z = 1 in (45), while there is no delay; results [25, 55, 27, 49, 35, 44, 32, 56, 57, 31, 58, 37] then applies. Proof. The equivalence of the first two conditions follows from Theorem 22. The third and fourth conditions follow from VIII.CONCLUSIONSAND FURTHER WORK the structure of (26) and Theorem 6 of [10]. A Within feedback systems, the control of ensembles of agents presents a particularly challenging area for further study. Remark 24. Notice that the fourth condition seemingly re- stricts the applications of PI controllers in this space [79], 2https://www.uber.com/blog/your-questions-about-the-new-surge-answered/ 11

Practically important examples of such systems arise in on- [11] M. H. Bateni, Y. Chen, D. F. Ciocan, and V. Mirrokni, line labour platform. Typically, such problems deviate from “Fair resource allocation in a volatile marketplace,” classical control problems in two main ways. First, even in Proceedings of the 2016 ACM Conference on though ensembles are typically too large to allow for a Economics and Computation, ser. EC ’16. New York, microscopic approach, they are not sufficiently large to allow NY, USA: Association for Computing Machinery, 2016, for a meaningful fluid (mean-field) approximation. Second, the p. 819. [Online]. Available: https://doi.org/10.1145/ regulation problem concerns not only the ensemble, but also 2940716.2940763 the individual agents; a certain quality of service should be [12] I. Lobel, “Revenue management and the rise of the provided to each agent. We have formulated this problem as algorithmic economy,” Management Science, 2020. an iterated function system with the objective of designing an [13] M. K. Chen, “Dynamic pricing in a labor market: ergodic control, and demonstrated that controls with poles on Surge pricing and flexible work on the uber platform,” the unit circle (e.g., PI) may destroy ergodicity even for benign in Proceedings of the 2016 ACM Conference on ensembles. Economics and Computation, ser. EC ’16. New York, NY, USA: Association for Computing Machinery, 2016, p. 455. [Online]. Available: https://doi.org/10.1145/ IX.ACKNOWLEDGEMENT 2940716.2940798 The work of Ramen and Bob was in part supported [14] J. C. Castillo, D. Knoepfle, and G. Weyl, “Surge pricing by Science Foundation Ireland grant 16/IA/4610. 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APPENDIX TABLE I: A Table of Notation

Symbol Meaning N the set of natural numbers Z set of all integers Q the set of rational numbers R the set of real numbers 1 a compatible vector of ones c , , c constants used in Remark 14 1 ··· 4 hi the number of output maps Hij lij Lipschitz constant for a transition map lij0 Lipschitz constant for an output map N 1 the number of participants on one side of the market N 2 the number of participants on the other side of the market n a dimension of a generic state space X 1 nc dimension of the state of the controller 1 nf dimension of the state of the filter 2 nc dimension of the state of the controller 2 nf dimension of the state of the filter 1 ni dimension of the state of ith agent’s private state 1 mi the number of possible actions of agent i m an upper bound on the number of possible actions of any agent r1 the reference value, i.e., desired value of y1(k) 1 ri ith agent’s expected share of the resource over the long run wi the number of state transition maps Wij 2 ni dimension of the state of ith agent’s private state 2 mi the number of possible actions of agent i m an upper bound on the number of possible actions of any agent r2 the reference value, i.e., desired value of y1(k) 1 ri ith agent’s expected share of the resource over the long run wi the number of state transition maps Wij z Z -transform variable α a constant used in the PI controller or its lag approximant β a constant used in a lag controller δ a constant in the iterated function system ergodicity used in Theorem8 δ0 a constant in the iterated function system ergodicity used in Theorem8 η a lower bound on the values of probability functions κ a constant used in the PI controller or its lag approximant C 1 controller representing the central authority F 1 filter 1 1 1 S1 , S2 ,..., SN systems modelling agents C 2 controller representing the central authority F 2 filter 2 2 2 S1 , S2 ,..., SN systems modelling agents fj a generic map in a generic iterated function system g a generic function of Hij an output map Hc a map modelling the controller in Hf a map modelling the filter in P (x, G) a generic transition operator P a state-and-signal-to-state transition operator Pk a measure-space-over-states-to-measure-space-over-states operator pj a probability function of a generic iterated function system pij a probability function for the choice of agent i’s transition map continues on next page 15

pi`0 a probability function for the choice of agent i’s output map Wij transition maps Wc maps modelling the controller in Wf maps modelling the filter in Φ(1) projector from a measure over the product of the two path spaces to a single path space Φ(2) projector from a measure over the product of the two path spaces to a single path space Ai a matrix used in the agent dynamics of Theorem 21 1 Ac a matrix used in the controller of Theorem9 2 Ac a matrix used in the controller of Theorem9 1 Bc a matrix used in the controller of Theorem9 2 Af a matrix used in the filter of Theorem9 2 Bc a matrix used in the controller of Theorem9 1 Bf a matrix used in the filter of Theorem9 Bi a matrix used in the agent dynamics of Theorem 1 Cc a matrix used in the controller of Theorem and in Theorem9 2 Bf a matrix used in the filter of Theorem9 Bi a matrix used in the agent dynamics of Theorem 2 Cc a matrix used in the controller of Theorem and in Theorem9. 1 Cf a matrix used in the filter of Theorem9 1 ci a vector used in the agent dynamics of Theorem 1 Dc a matrix used in the controller of Theorem. 2 Cf a matrix used in the filter of Theorem. 2 ci a vector used in the agent dynamics of Theorem 2 Dc a matrix used in the controller of Theorem. 2 di a vector used in the agent dynamics of. X(k) element of a generic state space X(k) k a generic Markov chain { } ∈N A augmented state transition matrix in Aˆ diag(Ai) in β` is built from all bij, dij, and other signals in ˆ T C diag(ci ) in E edges of a graph in the proof of. G a graph in the proof of Theorem. K a constant used in the proof of. Q a scalar or matrix used in the proof. R a marginally Schur matrix used in the proof of. Ai the set of ith agent’s actions Ccoup set of couplings Di set of possible resource demands of agent i B(X) Borel σ-algebra E a real additive group. G a generic event H a set used in the definition of asymptotic couplings J index set of a generic iterated function system M an index set of M (X) a measure-space over X M (X∞) a measure space over the path space OF set of possible output values of the filter F ni Xi a private state space of agent i, often R XS a product space of state spaces of all agents 1 XF a space of internal states of the filter 1 XC a space of internal states of the central controller X a state space of the controller, the filter, and the agents, combined X∞ a path space continues on next page 16

Π the set of admissible broadcast control signals e1(k) the error signal at time k, i.e., yˆ1(k) r π1(k) the signal broadcast at time k − 1 1 xc (k) the internal state of the controller at time k of dimension nc 1 xf resource utilisation of agent i at time k y1(k) the aggregate resource utilisation at time k yˆ1(k) the value of y1(k) filtered by filter F 1 λ initial state (distribution) Pλ a probability measure induced on the path space µ a generic measure, usually on state space Σ Γ a generic measure over the product of the two path spaces, potentially a coupling