1 Introduction
Market equilibria are central objects of study in economic theory [Wal74, AD54, BS00]. However, the notion of an equilibrium deliberately abstracts away how such equilibria are reached or even whether they are reached at all. On this topic, Fisher [Fis83] writes: “Whether or not the actual economy is stable, we largely lack a convincing theory of why that should be so. Lacking such a theory, we do not have an adequate theory of value, and there is an important lacuna in the center of microeconomic theory… To only look at situations where the Invisible Hand has finished its work cannot lead to a real understanding of how that work is accomplished.” This issue is only compounded by the computational complexity results on the hardness of computing various notions of equilibria even in simple models [CDDT09a, NRTV07].
There has, of course, been a vast amount of work done on understanding various market dynamics and whether and how they converge to an equilibrium [Qua96, WZ07, CF08, Zha11, CCD13, MPP15] as well as the computational implications of such dynamics [CMV05, CPV05, CF08, CCD13]. In this paper we are interested in growth in production markets, a scenario whose whole point is that we do not expect (or desire) the economy to approach an equilibrium, but rather to “expand”. Our model is a simple variant of the pioneering model of an expanding economy due to von Neumann [vN46], where the goods are substitutes (the initial model was for perfect complements), represented through additive production functions [Gal89]. The work of von Neumann as well as most of the literature following that focused on analyzing equilibriumlike states where all components of the economy expand by the same constant factor as time progresses. In contrast, we focus on the dynamics themselves, and analyze scenarios that in no sense approach any stable point. We will be interested in understanding which market mechanisms lead to growth in the long run when the players act in a decentralized way, regardless of whether equilibrium states are ever reached or not.
1.1 Model Overview and Example
In our model we have producers, each can only produce one type of (eponymous) good according to his endowed production function, using as inputs various amounts of other goods in the system. In the simplest case, on which we focus, these production functions are simply additive. The main decision that the “economy needs to make” is that of the allocation of resources: for every time unit , how do we split the goods produced at time between the different producers who need to use them to produce goods for time . The model allows for circular dependencies, which are useful for capturing situations in which people would use each other’s products, such as a baker using a computer and the person making computers eating bread. Our goal is growth: having the amounts of resources in the economy increase with time (e.g. more bread, computers, scientific discoveries).
Here is a simple example with two producers:

Every time unit, producer A can make 0.1 units of good A from every unit of good B, and 0.99 units of good A from every unit of good A.^{4}^{4}4I.e., a unit of good A that stays with producer A loses 1% of its quantity every time unit.

Every time unit, producer B can make 10.2 units of good B from every unit of good A, and 0.99 units of good B from every unit of good B.
Looking globally, it is clear that the producers should cooperate in this production market: if each good is allocated back to its own producer (i.e. agent A gets all the existing quantity of good A, and produces from it units of good A, and similarly for B) then the economy will shrink. If on the other hand we allocate the whole quantity of good A each time unit to producer B, and all of good B to producer A, then the economy will grow: units of good A will produce units of good B, which in turn will produce units of good A. The question we ask in this paper is whether and how this growth can be achieved using natural decentralized dynamics. After all, agent B needs significant foresight in order to send essentially everything that he produces to agent A (if he keeps even 2% to himself then the economy will not be able to grow).
It is not hard to globally characterize whether a general market with additive production functions can keep growing in an unbounded way under a globallyoptimal allocation of resources – see Theorem 3.3. We would like to find a “local mechanism” where individual actions by the different producers that use only information they posses can achieve such unbounded growth, whenever growth is possible in the first place. Leaving the space of allowed “mechanisms” intentionally vague, our goal is the following:
Definition: A mechanism is called universal if for any market with additive production functions that can grow (in an unbounded manner) under an optimal allocation of resources, the market also grows in an unbounded manner by following the mechanism.
1.2 Our Results
Here is the basic class of mechanisms that we study. In these, each producer starts with a certain amount of his good that can be used to produce other goods, as well as an amount of “money” (budget) with no intrinsic value but used for trading. At each time unit, every producer splits (all of) his current money into bids that he places on each of the goods. Then each producer sells his good to the producers that bid on it (and collects all the money from these bids).
The basic decisions each producer must make in every time unit are how to split his money into bids and how to allocate his good to the bidders depending on their bids. The mechanism we analyze uses the following rules:

Trading Posts: In each time unit, every good is allocated between the producers that bid on it, in proportion to the bid amount. In our example, if producer A bid 0.2 on good A while producer B bid 0.3 on good A, then producer A gets 40% of the quantity of good A while producer B gets 60%. This rule has been introduced by Shapley and Shubik to explain the formation of prices in the exchange economy [SS77, Shu16].

Proportional Bid Updates: In each time unit, every producer splits his money into bids on the goods in a way that is proportional to the amount he produced from that good in the last round. For example, if in the previous round producer A got 0.5 units of good A, from which it produced units of good A and got 0.3 units of good B from which it produced units of good A, then he will bid a fraction of his money on good A and the rest of on good B.
The combination of these two rules results in rather interesting dynamics, even when we only have two producers as in the example above. Figure 1 presents simulations results of this dynamic of the twoproducer example above (from some simple starting amounts). Note that while the system seems to oscillate in a rather irregular fashion, one can clearly see the quantities produced are growing over time. The budgets of the players also oscillate in an unclear pattern. Figure 3 shows in greater detail the oscillations in a simulation of another twoproducer market where growth is not achieved but instead a repeated oscillation with period 3 (!) is.
It is known that in static models such as Fisher markets, where the players repeatedly come to the market with the same amount of goods and money, the Trading post mechanism converges to market equilibria when the players use proportional bid updates [WZ07, Zha11]. This convergence holds for a large class of valuations known as constant elasticity of substitution. In fact, for additive valuations in the static Fisher model, Trading post with proportional bid updates is equivalent to gradient descent (with respect to the Bregman divergence instead of the Euclidean distance) [BDX11].
Our first main result is that Trading post with proportional bid updates is indeed a universal mechanism: if a production economy can grow unboundedly using an optimal (possibly centralized) allocation rule, the economy also grows unboundedly by following this mechanism.^{5}^{5}5In other words, the performance of a universal mechanism is not too far from that of an optimal centralized allocation rule.
Theorem 1.1 (Universal growth).
The trading posts mechanism with proportional bid updates is universal.
We further study the properties of this mechanism, for example showing that its rate of growth is as fast as possible.
We also focus on the level of inequality between the different producers that develops as the market grows. Figure 3 depicts a heat map of the Gini index (a measure of inequality) reached in a certain two producer economy (after 800 rounds) as a function of the initial bids of the producers. Despite the complexity of the dynamics we are able to show that our mechanism always leads to growing inequality between the producers, in terms of the quantity of goods that they get as time progresses. Specifically,
Theorem 1.2 (Inequality).
The producers get differentiated over time into two classes: the “rich”, who participate in the “most efficient production cycle” versus the “poor”, who do not. While the inequality gaps within the “maxefficiency” class remain bounded by a constant, the inequality gap between this class and the rest grows to infinity.
We obtain the theorem by establishing that as time progresses producers of the “maxefficiency” class trade more and more among themselves, and their production is more efficient than that of any other group of producers.
We find several other phenomena, such as the fact that other producers, who are not part of the “maxefficiency” class, can also grow. In fact, a producer can grow even if he is not part of any efficient production cycle, if he is instead “wellnetworked”. For this we show the existence of phase transitions in the long term quantity of a producer depending on the quality of his connections.
1.3 Open Problems
We believe that we have only started scratching the surface of understanding the dynamics of decentralized models of economic growth and open problems abound. First, there are a bunch of questions that we are not able to answer in our specific model. For example, we would like to characterize the set of players that obtain increasing amount of goods versus those that get vanishing amounts over time. Second, we feel that we have only a very partial understanding of the class of mechanisms that can be utilized. E.g., to what extent are the results that we obtained peculiar to the specific rules that we chose (trading posts and proportional bids)? Which other types of rules will provide similar performance (or better in some sense such as less inequality)? Third, considering additive production functions can obviously only be considered as a first step to understanding wider and more realistic classes of production functions. At a higher level yet, we believe that our whole framework is only a first step, and improved modeling (as well as connection to existing macroeconomic models) is interesting.
1.4 Related Work
Analysis of markets (economies) is central to economics. The growth model due to von Neuman [vN46] for production economies has been extensively studied, e.g., see [KMT56, Pas62, HM65, Lan12], however mainly to understand the growth rate of the economy at equilibrium and under Leontieftype production functions. The classical ArrowDebreu market model which involves both production and consumption has been studied for its equilibrium properties [AD54], and within CS for their computation and complexity [DPSV08, Orl10, Vég12, CSVY06, CDDT09b, EY10, GMVY17].
The question of how the equilibrium prices are reached is analyzed under the natural price adjustment process due to Walras [Wal74], called Tatonnement – increase the prices of over demanded goods and decrease for the under demanded goods. In particular, [AH58, ABH59, AH60] showed that it converges to an equilibrium in markets with valuations restricted to the weakgrosssubstitutes (WGS) property. The work within CS showed fast convergence of specific Tatonmment rules for WGS [CF08, CMV05, CPV05] as well as a class of nonWGS markets [CCD13]. All of these results focus on static markets where goods/money endowments of agents remain fixed. For such a static model, [WZ07, Zha11] studies proportional response dynamics with trading post mechanism, showing fast convergence to a market equilibrium for a large class of valuations known as constant elasticity of substitution. On the contrary, our model is inherently dynamic, where amounts of both money and goods of an agent change based on the trade and production that happened in the previous round. Questions of growth exist in macroeconomics as well [Sol56, S.56, Uri13, Ace09] where the focus is on technical progress, i.e., capital accumulation, population growth, etc., and typically uses CobbDouglas type production functions.
Like in our model, a recent work on trading networks due to [Jag17] also studies trades along edges of an underlying graph, where firms trade contracts consisting of exchange, production and pricing. However, again the focus is on equilibrium existence, which is obtained when the underlying graph is a tree.
There has been extensive work on understanding dynamics in games and auctions under various behavioural model of agents, such as bestresponse, multiplicative weight update, no regret learning (e.g., [FS99, KPT09, DDK15, MPP15, PP16a, RST17, DS16, MT12, HKMN11, BR11, LB10, NSVZ11, BBN17, PP16b, LST16, CD11, DK17, CDE14]). In the former the focus has been on convergence to an equilibrium, preferable Nash, and if not then (coarse) correlated equilibria, and the rate of covergence. In the latter the focus has been on either convergence points and their quality (priceofanarchy), or dynamic mechanism such as ascending price auction to reach a solution (such as the Ausubel auction [Aus04]). Also, the Trading post mechanism has been studied in other scenarios, such as rent seeking, allocating computational resources, matching markets, and fairness [ILWM17, BGM17, FLZ09, Tul80].
Study of evolution naturally involves dynamics at the level of genes or species (e.g., see [Lot10, Vol28, GMM71, Wan78, HS98, LA83, CLPV14, MPP15]). The LotkaVolterra model [Lot10, Vol28, Wan78] studies interdependence of animals and how they “help” or “destroy” each other based on their interactions and reproduction. There seems a highlevel similarity between this model and ours, but the former may be thought of as a mechanism with a fixed splitting rule in our setting.
Organization.
Section 2 formalizes our market model and class of mechanisms. In Section 3 we characterize growth w.r.t. general mechanisms. Trading post (TP) mechanism with proportional bid updates is defined in Section 4, and in Section 5 we analyze the growth under it and show that it is a universal mechanism. Section 6 analyzes inequality under TP, while Sections 7 and 8 further characterizes its various properties. We conclude with Gini index simulations in Section 9.
2 Model
Let be a set of players. Each player can make an eponymous good ^{6}^{6}6In other words, each player makes good ; no other player can make good and player only knows how to make this type of good. using the recipe given by his additive production function described by , where is the amount made by player from one unit of good
. The player knows his own function. A bundle of goods is a vector
, where is the amount of good . Given a bundle , player makes from it an amount of his good.A production economy operates over time . At every time unit , each player produces an amount of his good (from the bundle of ingredients he owns), then trades the good at the market. The bundle obtained by the player from trade at time is the input to his production at time . We will assume that each player starts at by having an initial amount of his good that he directly enters trade with.
We will assume the economy (i.e. digraph ) is strongly connected.
2.1 Mechanisms
For the activity in the economy to be completely specified, we must state what mechanism is used for trade.
Definition 1 (Abstract Mechanism).
A market mechanism specifies how trade takes place in each round and is defined as an infinite sequence of “splitting rules” , such that is the fraction received by player from good in round and .
Our interest is in mechanisms where the ’s are determined by player locally from information that he has in time , and such local mechanisms will additionally include a communication protocol specifying the communication between the players. Our main analysis is for a specific mechanism with natural local decisions (trading posts) and very simple natural communication (proportional bids).
Our measure for how well the economy is doing at any point in time will be the total amount of goods in the economy: .
Definition 2 (Growth and decay).
An economy (equipped with some mechanism for trade) grows if and vanishes (or decays) if . Similarly, a player grows if and vanishes if , respectively.
We are also interested in measuring the inequality in the economy, captured by the Gini index.
Definition 3 (Gini index).
Given a vector , its Gini index [Gin18] is:
The Gini index is normalized between and so that means perfect equality and maximum inequality. The latter is achieved when one entry is equal to and all other entries are zero.
Example 1.
Let be a two player economy with initial amounts , i.e. player can make units (of good ) from one unit of good and zero units from one unit of good , while player can make units (of good ) from one unit of good and zero units from one unit of good . If the mechanism used is to give each player of his own good in every round, then player will grow while player will vanish. If on the other hand the goods are split equally in every round, both players will vanish.
The calculations are in Appendix A. The high level idea of the example is that player is “productive” by himself, but not so productive as to support both himself and another player receiving equal shares througout time.
2.2 Cycles
Important objects in our analysis will be simple cycles. In short a cycle will be a set of players , where all the ’s are different. For simplicity we will denote this by and consider them as an ordered set of nodes. Given such a cycle , we will override notation and denote an edge in by , meaning that and is the successor of in the ordering , simply saying that we consider how useful good is for player .
Definition 4 (Good and bad cycle).
A cycle in the digraph is good if the product of weights along the cycle is strictly greater than one (i.e. ) and bad if the product is strictly less than one.
Definition 5 (Best cycle).
A cycle in the digraph is the best cycle
if its geometric mean is the highest among all the cycles.
3 General Mechanisms
We first observe that if all the cycles in the weighted directed graph induced by the values are “good”, then every nonwasteful mechanism will lead to growth, while if all the cycles are “bad”, no mechanism can save the economy from shrinking to zero in the limit. Note an economy is strongly connected if the directed graph with nodes and weights is so.
Theorem 3.1.
An economy vanishes with any mechanism if and only if all the cycles are bad.
A mechanism is nonwasteful if it never throws away goods (i.e. for all ) or allocates goods to players that do not need them (i.e. if then for all ).
Theorem 3.2.
A strongly connected economy grows with any nonwasteful mechanism if and only if it has least one good cycle and each directed cycle is either good or has zero edges all along.
For an economy with additive production to grow, there must exist at least one good cycle, say . Moreover, there exists a mechanism that can grow such an economy by having the players along send their good to their successor in . This gives the next statement.
Theorem 3.3.
A strongly connected economy grows with some nonwasteful mechanism if and only if it has at least one good cycle.
In our model, a universal mechanism will enable growth precisely when the economy has at least one good cycle.
Definition 6.
A mechanism is called universal if for any economy with additive production that has at least one good cycle, the economy grows by using the splitting rule given by .
4 Trading Post
From now on we focus on studying the Trading post mechanism. Each player will be initially endowed with some amount of artificial currency it can use to acquire goods that they like. In a round, every player runs a contest to decide how to allocate its good. The players submit bids on the goods they are interested in buying, after which every good is allocated in fractions proportional to the bids. Thus if the bids in some round are , for all , then player receives the following fraction of good :
Each player collects the money made from selling his good, and this money will be his budget in the next round.
We note the dynamic version of Trading post allows the money to move in the system, from the bidders to the seller, and the quantities to change as the result of production. This is unlike the work on dynamics in static models (e.g. Fisher markets), where the players are endowed with the same budget and bring the same amount of their good in each round.
We will analyze Trading post when the players update their bids in proportion to the contribution of each good in the production from last round.
Definition 7 (Trading post with proportional bid updates).
At each time , given bids and amounts , player receives from trade at time an amount of good , where
After trade, the players produce amounts Each player collects the money made from selling his good: , and updates his bids proportionally to the contribution of each good in production ^{7}^{7}7The bid fractions are unchanged if no production took place. This will turn out to not matter however since we will study nondegenerate starting states, which will imply that throughout time every player will be able to produce a nonzero (but possibly very small) amount.:
For the purpose of our results, we can w.l.o.g. normalize the money in the economy to , so . We will also assume the starting configuration is nondegenerate: and the players bid on goods worth something to them, so .
An example with actual numbers for trading post with proportional updates can be found in Appendix C.
5 Growth
Our first main result is that the trading post mechanism with proportional bid updates leads to growth of the economy (whenever growth is possible). We additionally show that all the players on the best cycle are guaranteed to grow (if that cycle is good), and in fact their growth is within a constant factor of the optimal growth (that could be achieved under some optimal allocation of resources in every round, where the constant may depend on ). Players with good selfloops and players on “good enough” cycles will also grow infinitely.
Theorem 5.1 (Universal growth).
Trading post with proportional updates is a universal mechanism.
Proof.
Let and be any starting quantities and bids with the property that for all and for all where . Let be any good cycle and denote by the product along . Take the product of all the quantities and bids on , given by the function
Since the starting state is nondegenerate, . Rewriting the bids at time as a function of the quantities and bids at time , we obtain
It follows by induction that , for all . Since , we have . The total amount of money is , so for all , thus . Let denote the length of . From the geometricarithmetic mean, we have
Then , so for every good cycle, the sum of quantities of the players along the cycle grows infinitely. ∎
A simulation for two players can be found in Figure 4, showing how the amounts, budgets, fractions invested by the players on each other, and the Gini coefficient evolve over time in an economy where the only good cycle contains both players. In Figure 4(c) it can be seen that each player spends of his money on buying the good of the other player. Figure 4(f) shows the Gini coefficient in terms of the amounts, which oscillates for the whole duration of the time simulated, and Figure 4(g) the Gini coefficient in terms of budgets, which eventually reaches a stable point.
Next we study the question of which players grow in the limit.
Corollary 1.
Each player with a good selfloop grows.
Proof.
Let be any player for which . By Theorem 5.1, . We have and , so as required. ∎
Corollary 2.
In economies where all the cycles are good, every player grows.
Proof.
If all the cycles are good, each selfloop is good. By Corollary 1, each player grows. ∎
Lemma 1.
Let be one of the best cycles. Then there exists a constant such that for each player and time , where is the product along .
Proof.
From Theorem 5.1, we obtain a function with the property: , where since the starting configuration is nondegenerate. Let .
The maximum growth given a number of rounds divisible by can be achieved by passing as much flow as possible through one of the best cycles, so for any player and any ,
Then for any time with , we have the following bound, where is the maximum edge in the graph and is a constant, dependent on the graph and the starting configuration, but independent of time:
This completes the bound. ∎
Lemma 2.
Let be one of the best cycles. Then there exists a constant such that for all , the bids along the cycle are bounded from below by ; that is, for all .
Proof.
By Lemma 1, there exists a constant such that for each player and time , , where and is the length of cycle . Let . Then
(1) 
Combining the expression for with inequality (1) we get: . Recall , for all . Set . Then the product of bids along is . Since for all , the bid of player on good , where , is at least: , for all . Thus the bids along are bounded from below by a constant throughout time as required. ∎
The amounts of the players on the best cycle will turn out to be within a constant factor of the optimal throughout time (where the optimal amounts are those that could be achieved by a central planner by ensuring trade only happens on the best cycle in each round).
Theorem 5.2 (Almost optimal rate of growth).
Let be one of the best cycles and its length. Then there exist constants such that for each player and time , where is the product along .
Proof.
Define sequences and containing the maximum and minimum amounts, respectively, on the cycle at each time . By Theorem 5.1, there is a function with the property that . By Lemma 1, there is a constant such that for each and time , , so . Let be the player on the cycle with minimum amount at , breaking ties lexicographically, and a constant. Then
where . Then for any and time , we get that as required. ∎
Corollary 3 (Rate of growth of the economy).
There exists constant (possibly dependent on but independent of time), such that
where is the highest total amount that could be achieved by any (possibly centralized) mechanism at time .
Theorem 5.3 (Growth of players on the best cycles).
Suppose an economy has at least one good cycle. Then for each of the best cycles , all the players in grow.
Proof.
By Theorem 5.2, there exist constants such that for any best cycle and any player , , where . Since , it follows that as required. ∎
The bounds on growth of Theorem 5.2 imply that growth can be achieved also by players outside the best cycle, even if such players don’t have good self loops, as long as they are part of a “good enough” cycle.
Theorem 5.4 (Growth of players on “good enough” cycles).
Suppose an economy has at least one good cycle and let be one of the best cycles. Then all players on any cycle with grow, where and are the product on and , respectively.
Proof.
We leave open the question of understanding more precisely which players grow and, in particular, whether all the players situated on a good cycle are guaranteed to grow.
Open Problem 1.
Are all the players on a good cycle guaranteed to grow?
6 Inequality
In this section we assume the best cycle is unique. We show that the players will get split into two classes: the “rich” (who will turn out to be the players on the best cycle) and the “poor” (everyone else), such that the inequality between these classes will diverge to infinity. The “poor” players will be poor when compared to rich, but some (or even all) of them may grow too, just at a slower rate. The omitted proofs of this section can be found in Appendix D.
En route to proving the inequality theorem we establish several other statements: in the limit the players on the best cycle will bid of their budget on their predecessor on the cycle and in the limit there is no flow of money between the players on the best cycle and rest. We conjecture that in fact the best cycle absorbs all the money in the limit.
We obtain the existence of a limit vector of money (budgets) so that the players on the best cycle rotate these budgets among themselves throughout time. For amounts we get periodicity in a normalized version of the economy, which will imply that the Gini index for amounts cycles with period , where is the length of the best cycle.
Lemma 3.
Consider an economy with a unique best cycle that is run on a sequence of arbitrary splitting rules , such that is the fraction received by player from good in round for each . Suppose and for all other cycles , for some . If there exists an edge , where preceeds , such that player receives infinitely often less than a fraction from the good of player for some , then there is a subsequence of rounds where the amount received by player from goes to zero.
The above lemma is in fact optimal, in the sense that picking a subsequence of rounds is important. Neither the total amount itself nor the quantity received by from need to go to zero as demonstrated by the construction in Example 4 in Appendix D.
Lemma 4 (Normalization).
Consider any economy running trading post with proportional updates. Then there exists such that by dividing each edge of by , we obtain a normalized economy in which any best cycle has product exactly , all the other cycles have product strictly less than , and at any round

the bids of the players in and are identical (i.e. for all ).

the amount of any player in is equal to its amount in divided by (i.e. ).
We refer to the unique value in Lemma 4 as the normalization coefficient of the economy.
Theorem 6.1.
Suppose the best cycle, , is unique. Then in the limit the fraction received by each player in from the good of its predecessor in converges to .
Proof.
By Lemma 4, we can assume that the best cycle, , has product one, while all the other cycles have product less than , for some . Let and denote by the predecessor of player in . Suppose towards a contradiction that the fraction received by player from good does not converge to in the limit. Then there is a constant such that infinitely often, player receives less than a fraction from good . By Lemma 3, it follows that there is a subsequence of rounds for which player receives in the limit of an amount of zero from good .
However by Lemma 2, player bids at least on good in every round and by Theorem 5.2, the amount of good is in an interval bounded away from zero throughout time, that is, there exist constants such that for all times . Thus the amount received by player from good at time remains bounded away from zero throughout time. We obtained a contradiction. Thus the assumption must have been false and the fraction received by player from player converges to as . ∎
Theorem 6.2.
Let be the unique best cycle. Then in the limit there is no money flowing from players in to players in and viceversa, i.e. and for all and .
Proof.
By Theorem 6.1, in the limit each player receives of the good of their predecessor, say , in , which implies that the bids of the other players on good vanish in the limit. In particular, in the limit there is no money flowing from to .
To show that the bids of the players in on the goods in also vanish in the limit, suppose towards a contradiction that they do not. Then there is a constant such that some player sends infinitely often a bid of at least on some player . By Lemma 2, there is a constant such that each player in bids at least on their predecessor in , which implies that for all players and any time . Since the money flowing from to vanishes in the limit, but player sends infinitely often at least to player , it follows that there is a subsequence in the sorted vector of budgets of players in that goes to zero, which is a contradiction. Thus the assumption was false and the money flowing from the players in to also vanishes in the limit. ∎
In the limit, the budgets of the players on the best cycle will converge to some values that get rotated infinitely often along the cycle (from one player in to its successor in ).
Corollary 4.
Suppose is the unique best cycle, where player uses the good of player . There exist values such that for each
Proof.
By Theorem 6.1 and 6.2, we get that in the limit each player in sends all its money to its predecessor in . This means there exist values so that in the long term the budgets of the players are described by these values (with a shift, as the money gets passed around the cycle). This implies the required limit behavior. ∎
Theorem 6.3 (Inequality).
Suppose an economy has a unique best cycle . Then in the limit, the players in are arbitrarily richer than the rest. That is, for all and ,
Proof.
Since the theorem statement requires measuring only the ratios of the amounts, by Lemma 4, we can assume w.l.o.g. that the best cycle, , has product one, while all the other cycles have product less than , for some . Let be any player in and any player in . By Theorem 5.2, there exist constants such that for all . By Theorem 6.1, the bid of player on good vanishes in the limit. We can decompose the good of player at time in two parts: , where is the amount obtained by bidding on the goods of players in and the amount obtained from players in . Let , where and .
From Theorem 6.1 it follows that for each , so . Thus in the limit the players in only receive goods from other players in . But since the subgraph induced by has only cycles that are bad (with product less than ), any amount that completes such a cycle is reduced by at least (multiplicatively), which implies that as well. Thus for each player . On the other hand , so as required. ∎
In our simulation we observed that the best cycle always absorbed all the money in the limit, and if there were multiple best cycles, they absorbed all the money collectively.
Open Problem 2 (Concentration of money).
Do the best cycles absorb all the money in the limit? That is, in an economy with best cycles , is it the case that
A natural property that one may try to use towards settling Open Problem 2 this is that the players outside the best cycle spend a nonnegligible fraction (i.e. bounded from below by a constant ) of their budget on the best cycle throughout time. The simulation for the four player economy , , , shows that this property is likely to be false; see Note 1 in Appendix D. However, it can be observed that the best cycle still absorbs all the money through a different mechanism: player bids a nonnegligible amount of his budget on the best cycle at all times, and player bids in the limit of his budget on player .
We can characterize in some sense the behavior of the amounts in the long run: in the normalized economy the players on the best cycle will “rotate” a vector of amounts among themselves (except the rotation is imperfect, it is multiplied by the prefix given by the position of the current round in the cycle), while the amounts of the players outside the best cycle converge to zero (the latter is immediate from the inequality theorem).
Corollary 5.
Let be any economy with a unique best cycle , where player uses the good of player . Let be the normalization coefficient ^{8}^{8}8Such that by dividing all the edges of by , we obtain an economy in which the best cycle has product and all the other cycles have product less than . of . There exist values such that for each player


for each ,
For each player ,
The next statement follows by Corollary 5.
Corollary 6.
Suppose an economy has a unique best cycle . Then the Gini index for amounts cycles with period .
We conjecture the Gini index for budgets converges as the best cycle will absorb all the money in the limit.
Open Problem 3.
What happens with the Gini index for amounts when there are multiple good cycles? Does the Gini index in terms of budgets converge when the best cycle is unique?
7 Phase Transitions
In this section we show the existence of phase transitions, finding that it is possible for a player to grow even if he is not part of any good (simple) cycle, as long as he is networked “wellenough”. Our main theorem investigating this phenomenon is for star networks, where a star is a configuration with players , such that player is the center and trades with everyone else, while the other players only trade with player . Formally, , for all , while all other .
Theorem 7.1 (Phase transitions for stars).
Consider a star economy with players and at least one good cycle, such that player is the center. Let be the product on the cycle between player and player and the product of the best cycle. Then for any player , its amount

grows if .

vanishes if .

stays in a bounded region throughout time if .
The proofs are in Appendix E. We conjecture that for each player with , his amount forms a “pseudocycle” that becomes a perfect cycle in the limit.
We leave open the question of understanding phase transitions more generally. We include several additional examples in Appendix G with phase transitions in twoplayer economies.
8 Fixed Points and Cycles
We briefly study fixed points and cycles. The proofs for this section are in Appendix F.
Proposition 1.
Let be an economy that cycles with period . Then every cycle along which the amounts and bids are nonzero throughout time has a product one.
For fixed points in the bid space, we get the following statement for complete graphs.
Theorem 8.1.
Suppose the economy has a complete graph. If the bids are unchanging throughout time, then

the growth of each player is given by , for all .

for each cycle , the coefficients satisfy the identity: .
Two player economies in which all cycles have product 1 always cycle as long as each player starts by bidding equally on the goods.
Theorem 8.2.
Any two player economy where all cycles have product one ^{9}^{9}9That is, for every , . cycles with period for any initial budgets and amounts, when the players start by splitting their budgets equally on the goods.
Our simulation shows that in fact economies with up to five players cycle for any initial configuration of the amounts and budgets, as long as the players split their budgets equally among the goods, and the period remains .
9 Gini Index Simulation
We studied in simulations the Gini index of multiple economies, with additional simulations in Appendix H. Figure 5 shows a two player economy where the and values are represented on the and axes. All initial amounts are and all initial bids are . The Gini coefficient is computed after 120 iterations, and lighter shades mean higher inequality. It can be observed that there is a threshold at around such that to the right of this value, both on the and axis, the inequality is very high. The reason is that the product of the cycle containing both players and is , with a geometric mean of around . When the selfloop of player becomes higher than this value (at the right of the figure), the inequality becomes very high because player has the best cycle and will grow at a much faster rate than player over time. The figure is symmetric, with the diagonal line showing blue because when both players have the same value for the self loops they will do equally well.
Figure 6 shows the Gini coefficient for the twoplayer economy with initial bids and initial amounts . The only good cycle contains both players. The inequality in this case comes from the differences in the initial bids of the players and from the fact that player sees a “bad” edge (of ) from player compared to his own selfloop, while player sees a very good edge incoming from player (worth ). Thus player is much slower to invest on player than player is on .
At a high level, our findings on the Gini coefficient are related to the discussion on the merits of capitalism versus socialism [Mar06, Pik15, vH44, vM20] and their respective failures. In particular, in recent years a growing concern exists regarding the rising levels of inequality around the world, including countries such as United Kingdom [Wil17] and United States [Gro16].
10 Acknowledgements
We would like to thank Moshe Babaioff, Sergiu Hart, Samuel Kortum, Yuval Peres, Yuval Rabani, and Anthony Smith for useful discussions. This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 740282), and from the ISF grant 1435/14 administered by the Israeli Academy of Sciences and IsraelUSA Binational Science Foundation (BSF) grant 2014389.
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Appendix A Economy with Fixed Splitting Rule
Example 2.
Consider a two player economy with , meaning that player can make units (of good ) from one unit of good and zero units from one unit of good , while player can make units (of good ) from one unit of good and zero units from one unit of good . Let the initial amounts be .
If the mechanism is to give each player of his own good in every round, then , , and generally, . Player receives a zero fraction of good , and he cannot produce anything from his own good, so , and then for all . So player grows while player vanishes.
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