Dynamical Analysis of the EIP-1559 Ethereum Fee Market
Stefanos Leonardos, Barnabé Monnot, Daniël Reijsbergen, Stratis Skoulakis, Georgios Piliouras
DDynamical Analysis of the EIP-1559 Ethereum Fee Market
Stefanos Leonardos ∗ , Barnab´e Monnot ∗ , Daniel Reijsbergen ∗ , Stratis Skoulakis ∗ , andGeorgios Piliouras Singapore University of Technology and Design, { stefanos leonardos,daniel reijsbergen,efstratios,georgios } @sutd.edu.sg Ethereum Foundation, [email protected]
Abstract
Participation in permissionless blockchains results in competition over system resources,which needs to be controlled with fees. Ethereum’s current fee mechanism is implementedvia a first-price auction that results in unpredictable fees as well as other inefficiencies. EIP-1559 is a recent, improved proposal that introduces a number of innovative features such asa dynamically adaptive base fee that is burned, instead of being paid to the miners. Despiteintense interest in understanding its properties, several basic questions such as whether andunder what conditions does this protocol self-stabilize have remained elusive thus far.We perform a thorough analysis of the resulting fee market dynamic mechanism viaa combination of tools from game theory and dynamical systems. We start by providingbounds on the step-size of the base fee update rule that suffice for global convergence toequilibrium via Lyapunov arguments. In the negative direction, we show that for largerstep-sizes instability and even formally chaotic behavior are possible under a wide range ofsettings. We complement these qualitative results with quantitative bounds on the resultingrange of base fees. We conclude our analysis with a thorough experimental case study thatcorroborates our theoretical findings.
The emergence of decentralized, Turing-complete blockchains, such as Ethereum [17], usheredin the possibility of creating alternative economic systems, where traditional institutions (suchas exchanges, banks, e.t.c.) are implemented in open-source code and where the state of thesystem/universal computer is stored in an immutable public blockchain. The extreme versatil-ity of such systems, at least in terms of their fundamental capabilities, naturally raises a lotof critical design considerations as these abstract ideas are fleshed out into concrete implemen-tations. Moreover, as participation in these systems steadily increases over time, these initialdesigns face novel demands and some careful adaptation becomes necessary.Arguably, one of the most critical real-world design decisions in Ethereum, as well as in anyother programmable blockchain, is how the protocol decides on the costs/rewards structure forthe different types of participating entities. The protocol charges users fees for having theirtransactions processed by the network and included in the blockchain. These transactions feesare typically referred to as “gas fees”. These fees are then distributed to the miners rewardingthem for dedicating computational resources to preserving the safety of the blockchain.Ethereum’s current fee system has been recognized as an important design challenge. Theissue primarily lies on the decision to set fees by using a simple first price auction mechanism.Effectively, all users submit their bids in regards to how much they are willing to pay to have ∗ In alphabetical order. a r X i v : . [ c s . G T ] M a r heir transactions included in the blockchain and the miners typically select the highest pricedentries for inclusion given the block capacity constraints. Due to the non-truthful nature of firstprice auctions, choosing an appropriate bidding fee is a non-trivial task and users can end upsignificantly overpaying for system participation.From a traditional mechanism design perspective, the solution to the aforementioned prob-lem seems relatively straightforward: Replace the first price auction with either a Vickrey –Clarke – Groves (VCG) auction [11, 16] or a (generalized) second price auction [10, 5], whichreduce the strategic complexity on the side of the bidders, lead to more efficient outcomesand are known to work well in practice (e.g., internet advertising). Unfortunately, such ap-proaches can be easily exploited and gamed by miners who can artificially increase demandsfor their blocks, increasing the resulting fees while decreasing meaningful system participation.Moreover, such mechanisms are vulnerable to collusion [8, 1].Recently, a new proposal (EIP-1559) has been put forward to address these issues [4]. Akey aspect of this mechanism is the introduction of a base fee that is automatically adjusted bythe protocol depending on the network congestion. This base fee effectively plays the role of areserve price, matching supply and demand. Critically, this base fee is burned, which preventsthe emergence of perverse incentives where miners can extract increased fees from the users byacting dishonestly. Users seeking fast inclusion of their transactions can supplement the basefee with a tip, which is the only fee that is received by the miners. An economic analysis ofEIP-1559 has identified desirable properties, e.g., it is incentive compatible for myopic minersand as well as for users except during time periods with excessively low base fees [14]. Of course,to provide insights about whether such conditions will be satisfied in practice an economicanalysis alone is not sufficient as one needs to explicitly analyze the dynamic evolution of themechanism parameters over time. This raises our driving question: Under which conditionsdo the EIP-1559 dynamics self-stabilize? When these conditions are not satisfied how complex,unpredictable can the resulting behavior be?
Our results.
We perform both theoretical as well as experimental analysis of the dynamicsand the stability properties of the EIP-1559 protocol. In particular, we investigate not onlysufficient conditions for network stability and convergence to equilibrium but, furthermore, weprovide a stress-test type of analysis where we push the system parameters beyond their stablerange and prove the emergence of phase transitions/bifurcations as well as the formal onset ofchaos.Our main observation is that the base fee adjustment parameter (step-size) plays a criticalrole in system stability. In the theoretical part of the paper (Section 3), we provide upper boundsfor the step-size that suffice for system stability (Theorem 3.6). For larger values of the step-size(depending on the other critical parameters of the system, i.e., transaction demand and uservaluations), we show that the base fee dynamics may become formally chaotic (Theorem 3.12).However, even in this unstable regime, the base fees remain within a bounded region and arerelatively well behaved. By contrast, adverse effects are observed in the block occupancies(which may oscillate between their extremes, full to empty and vise versa).On the experimental side we validate our theoretical findings by showcasing high varianceperiods where blocks alternate between full and empty state and base fees fluctuate, using afee market simulation library equipped with agent-based modelling (Section 4). We first lookat the impact of three variables on the prevalence of these high variance periods: the demandvariance, or how “noisy” the demand process is; the initial condition of the demand process,from just enough to fill blocks entirely to twice that demand; and the tolerance of the transactionpool eviction policy, with more tolerant pools keeping transactions even as their fee cap standsat a lower value than the base fee. We find all three variables positively correlate with moreappearances of high variance periods, highlighting the forces inducing variance in the fee market.We additionally find that using stricter pool eviction policies hurts user efficiency and minerrevenue, casting doubt on the incentive compatibility of this strategy to yield more stable base2ee updates.
Our model consists of two parts. The first describes the transaction fee mechanism of EIP-1559with a special focus on the dynamic adjustment of the base fee (Section 2.1) and the seconddescribes the agents’ behavior (Sections 2.2 and 2.3).
We consider a blockchain-enabled economy in which users make transactions over a distributednetwork. Users submit their transactions to a common pool together with a bid which specifieshow much they are willing to pay for the computational resources that are required for theirtransactions to be processed. The transactions, along with the bids, are viewed by the minerswho select which transactions to include in the blocks they create. In existing mechanisms (in-cluding Ethereum’s current economic model), bids comprise of a single transaction fee. Minerscan sort the transactions and typically select the ones with the highest fees. The miner who willinclude a transaction in a valid block receives the entire fee in a process that closely resemblesa generalized first price auction.According to the proposed reform of EIP-1559, bids comprise two elements ( f, p ): (i) the fee cap , f , which is the maximum amount that the user is willing to pay for their transactionto be processed, and (ii) the premium , p , which is the maximum tip that the user is willing topay to the miner who will eventually process their transaction. In particular, a user who willget their transaction included in the blockchain will never pay more than the fee cap and theminer who will process the transaction will never receive more than the premium.The main element of EIP-1559 and its main difference from existing mechanisms is thestipulation of a dynamically adjusted base fee , b t , t >
0, where t denotes the block height.Every transaction that gets included in a block B t , t > b t , thatis valid at that block. Instead of being transferred from the user to the miner, the base fee is burned , i.e., it is permanently removed from the circulating supply of the native currency (e.g.,ETH). For each included transaction, miners will receive the minimum between the premiumand the difference between the fee cap and the base fee. Specifically, the miner’s tip is definedby miner’s tip := min { f − b t , p } . (1)Blocks have size T and a target block load of T / Let g t denote the number of transactionsthat get included in block B t . Since g t depends on the base fee, b t , we will write g t | b t to denotethe transactions that get included in B t given that the base fee is equal to b t . The base fee isupdated after every block according to the following equation b t +1 = b t (cid:18) d · g t | b t − T / T / (cid:19) , for any t ∈ N . (2)where d denotes an adjustment factor (or step size), currently set at d = 0 .
125 [13]. Equation(2) suggests that the base fee will increase if the load of block B t is larger than the target blockload, i.e., if there is increasing demand or congestion in the system, and will decrease otherwise.The magnitude of the change is regulated by the excess (shortage) of transaction load compared Our analysis is based on the Ethereum blockchain. However, there are other blockchains, such as Filecoin[15], that implement very similar mechanisms and the main ideas of our results (up to technical details) readilyextend to these settings as well. Size is measured in gas, i.e., typically T denotes the gas limit. Here, we express all measurements in units pergas, so under the assumption that all transactions use the same amount of gas, one may think of T as number oftransactions.
3o the target load (i.e.,
T /
2) and parameter d . Our main goal in this paper is to analyze thestability and properties of the dynamical system that is determined by equation (2). In general, we will assume that users (transactions) arrive to the pool according to a randomprocess. We will write N t to denote the random number of transactions that arrive betweentwo consecutive blocks B t , B t +1 for t ≥
0. We assume that N t ∼ P ( λT ) for any t ≥
0, where P ( λ ) denotes the Poisson distribution of parameter λT . To avoid trivial cases, we will assumethat λ > /
2, i.e., that the arrival rate is larger than the target block load. For the theoreticalanalysis, we will assume that users leave the pool if their transaction is not included in the nextblock and return according to the specified arrival process. Whenever necessary, we will indexusers (transactions) with i, j ∈ N .As mentioned above, miners view all transactions in the pool along with their bids, ( f, p ),and decide which transactions to include in the blocks they mine. We assume that miners arewilling to process transactions only if the fees that they receive are at least some commonlyknown (cid:15) >
0. This is due to the intrinsic marginal cost for miners to include the transaction.For instance, each transaction increases the size (in bytes) of the block and its propagationtime over the network of miners, leading to an increase in the risk of producing a stale block(called uncle ). Thus, miners will select a transaction to be included in block B t if f ≥ b t + (cid:15) ,and p ≥ (cid:15) , i.e., if the fee cap is large enough to cover both the base fee and the minimumacceptable premium for miners, and the premium is large enough to satisfy the miner’s tip.These conditions are summarized in the following minimum inclusion requirement miner’s tip = min { f − b t , p } ≥ (cid:15). (3)Finally, each user i ∈ N has a valuation v i which is drawn from some common (for all users)distribution function v ∼ F with strictly positive support S ⊆ R + . For convenience, we assumethroughout that F is continuous and strictly increasing (i.e., non-atomic). We will write F todenote the survival function of F , i.e., F ( x ) = 1 − F ( x ) , for any x ∈ R . For the most part,we will assume that users are non-strategic , which means that they bid their valuations as feecaps, i.e., f i = v i and that they set a premium equal to the miner’s acceptance threshold (cid:15) , i.e., p i = (cid:15) for each user i ∈ N . In short, a non-strategic user with valuation v i , is defined as a userwho bids ( f, p ) = ( v i , (cid:15) ). Based on the above assumptions, the dynamical system that is determined by equation (2)is a discrete time, discrete space stochastic process { b t } t ≥ . The source of randomness is theterm g t | b t , i.e., the number of transactions that get included in block B t given the base fee b t .However, for the most part of the analysis, it will be sufficient to consider a non-atomic (or fluid)approximation of the above system. Accordingly, we will assume that there is a fixed numberof λT arrivals between each two consecutive blocks and that thus the total system demand, ie.,fraction of users who are willing to pay the base fee (plus the miners’ premium), is equal to λT F ( b t + (cid:15) ). Taking into account that a block has a maximum size, T , the above lead to thefollowing discrete time, continuous space deterministic process b t +1 = b t + b t dT / (cid:0) min { T, λT F ( b t + (cid:15) ) } − T / (cid:1) . This assumption only reduces unnecessary complexities in the analysis and is relaxed in the simulationswithout significant effect in the results. It is expected for user wallets to encode this default (cid:15) in their fee estimation strategies, thus supportingcommon knowledge among fee market participants. At the moment, the value of 1 nanoETH (10 − ETH) isrecommended as such a default by the EIP itself [3]. b t +1 = b t + b t d min { , λF ( b t + (cid:15) ) − } . (4)The analysis of the dynamical system { b t } t ≥ defined by equation (4) will be the main subjectof the theoretical part of this paper. In the simulations, we again employ the discrete modeldescribed above. Remark . For practical purposes, the approximation of discrete arrivals by a continuous processis justified by the fact that g t actually denotes gas units , which offer a much larger granularitythan exact numbers of transactions. Moreover, the number of arrivals between consecutiveblocks can be considered fairly constant during stationary periods which are of interest here.If the demand shifts to a new stationary level, then the base fee is also expected to shift toadjust to this new level. From a theoretical perspective, the deterministic dynamical system inequation (4) can be justified as an approximation of the sequence of conditional expectations E [ b t +1 | b t ] as explained in Lemma 3.1 below. Our main task in this section is to analyze the convergence and stability properties of thedynamical system { b t } t ≥ of equation (4). As mentioned in Remark 1, our first observation is that, the deterministic dynamical system inequation (4) can be justified as an approximation of the sequence of conditional expectations E [ b t +1 | b t ]. In particular, the base fee of block t + 1 depends only on the state of block t , whichmeans that the stochastic process { b t } , t ≥ Markov property . This allows us toderive a closed form formula for the conditional expectation E [ b t +1 | b , b , . . . , b t ] as shown inLemma 3.1. Lemma 3.1.
Suppose that the number N t of transactions that arrive to the transaction poolbetween consecutive blocks B t , B t +1 follows a Poisson process with rate λT , with λ > / forany t ≥ . Further, suppose that users valuations v i , i ∈ N are independently drawn from acommon distribution v ∼ F for some continuous and strictly increasing distribution function F and that users are nonstrategic, i.e., that their bids satisfy ( f, p ) = ( v i , (cid:15) ) . Then, it holds thatthe stochastic process { b t } t ≥ of equation (2) b t +1 = b t (cid:18) d · g t | b t − T / T / (cid:19) , for any t ∈ N . has the Markov property and E [ b t +1 | b t ] ≤ b t + b t d min { , λF ( b t + (cid:15) ) − } . (5) Proof.
The Markov property is immediate from the definition of b t +1 since b t +1 is fully deter-mined by b t and g t | b t . Thus, E [ b t +1 | b , . . . , b t ] = E [ b t +1 | b t ] for any t ≥
0, with E [ b t +1 | b t ] = E (cid:20) b t (cid:18) d · g t | b t − T / T / (cid:19) (cid:12)(cid:12)(cid:12) b t (cid:21) = b t (cid:18) d · E [ g t | b t ] − T / T / (cid:19) . Formally, a stochastic process X t , t ≥ Markovian , with respect to a filtration F t = σ ( X s | s ≤ t ), if forany fixed time t ≥
0, the future of the process, i.e., X t +1 , is independent of F t given X t .
5o proceed with the calculation of the conditional expectations E [ g t | b t ], we define the randomvariables X i = (cid:40) , if i ’s valuation satisfies the inclusion requirement in block B t , , otherwise.Recall from equation (3), that the minimum inclusion requirement is that min { f − b t , p } ≥ (cid:15) .Since users bid ( f, p ) = ( v i , (cid:15) ) by the assumption that they are nonstrategic, it holds thatmin { f − b t , p } = min { v i − b t , (cid:15) } . Hence, the inclusion requirement is satisfied if and only if v i − b t ≥ (cid:15) which implies that X i = { v i ≥ b t + (cid:15) } . Thus, P ( X i = 1 | b t ) = P ( v i > b t + (cid:15) ) = F ( b t + (cid:15) ) , for any i = 1 , , . . . , N. Thus, conditional on b t , the X i ’s are independent and identically distributed (iid) with distri-bution (denoted by) X | b t ∼ Bernoulli( p = F ( b t + (cid:15) )), so that E [ X | b t ] = F ( b t + (cid:15) ). Since theblock capacity is upper bounded by T , the transactions g t | b t that will get ultimately includedin block B t satisfy the equality g t | b t = min (cid:110) T, (cid:80) N t i =1 X i (cid:111) (cid:12)(cid:12)(cid:12) b t . Putting these together, we cannow upper bound E [ g t | b t ] as follows E [ g t | b t ] = E (cid:104) min (cid:110) T, N t (cid:88) i =1 X i (cid:111) (cid:12)(cid:12)(cid:12) b t (cid:105) ≤ min (cid:110) T, E (cid:104) N t (cid:88) i =1 X i (cid:12)(cid:12)(cid:12) b t (cid:105)(cid:111) = min { T, E [ N t ] E [ X | b t ] } = min { T, λT F ( b t + (cid:15) ) } , (6)where the inequality is due to the interchange of the minimum with the expectation and the(second to last) equality due to Wald’s equation since the random variables X i , i = 1 , . . . , N t are independent of N t . Plugging (6) in the expression for E [ b t +1 | b t ] above, yields E [ b t +1 | b t ] ≤ b t + b t d min { , λF ( b t + (cid:15) ) − } , which is the the inequality in equation (5) as claimed.Next, we show that the dynamical system { b t } t ≥ in (4) has a unique fixed point which is directionally stable . Before proceeding with the formal statement and its proof, we first definethe relevant terms that we will in the subsequent theoretical analysis. Definition 3.2 (Discrete Time Dynamical System) . A one-dimensional discrete time dynamicalsystem , { b t } t ∈ N , is determined by an update rule g : R → R , so that b t +1 := g ( b t ). We will write g t ( b ) := g ◦ g ◦ · · · ◦ g (cid:124) (cid:123)(cid:122) (cid:125) t − times ( x ) , to denote the t -th iteration of the system, i.e., the t -times composition of g with itself (when t = 1, we will simply write g instead of g ). Accordingly, a sequence ( g t ( b )) t ∈ N is a called a trajectory or orbit of the dynamics with b as a starting point. A point b ∗ is called a fixed point of the dynamics if g ( b ∗ ) = b ∗ . A common technique to show that a dynamical system convergesto a fixed point is to construct a function Φ : R → R such that Φ( g ( b ) < Φ( b ) for any b ∈ R unless b is a fixed point of g . We call Φ a Lyapunov or potential function for g . Definition 3.3 (Directionally Stable Fixed Point) . Let { b t } t ≥ be a one-dimensional dynamicalsystem determined by a function g : R → R and let b ∗ be fixed point of g , i.e., g ( b ∗ ) = b ∗ . Then, b ∗ is called directionally stable for { b t } t ≥ if for every t ≥ b t (cid:54) = b ∗ it holds that( g ( b t ) − b t ) / ( b t − b ∗ ) < g ( b t ) = b t +1 for every t ≥ b ∗ is directionally stable for the (non-atomic) base fee dynamics { b t } t ≥ ,let F − ( p ) := inf { x ∈ R : F ( x ) ≥ p } denote the inverse distribution function of F . Since F iscontinuous and strictly increasing by assumption, for any p ∈ [0 ,
1] there exists a unique x ∈ R such that F − ( p ) = x . Moreover, under these conditions, F − is also strictly increasing. Usingthis notation, we can prove Lemma 3.4. Lemma 3.4.
Consider the deterministic dynamical system { b t } t ≥ with b t +1 = b t + b t d min { , λF ( b t + (cid:15) ) − } . Then, b t has a unique stationary point given by b ∗ = F − (1 − / λ ) − (cid:15). (7) Moreover, b ∗ is directionally stable for any initial condition b > and the dynamics { b t } t ≥ converge to a globally attracting db ∗ -neighborhood of b ∗ , i.e., there exists a ¯ t ∈ N , so that b t ∈ [(1 − d ) b ∗ , (1 + d ) b ∗ ] for any t > ¯ t .Proof of Lemma 3.4. Let r t denote the rate of change of b t , i.e., r t := d min { , λF ( b t + (cid:15) ) − } . By definition, r t ∈ [ − d, d ]. The process { b t } t ≥ becomes stationary if only if r t becomes equal to0. Solving the equation r ∗ = 0 for b ∗ under the assumption that F is continuous and increasing(and hence invertible and with an increasing inverse, F − ) yields the unique solution b ∗ = F − (1 − / λ ) − (cid:15), which is the only equilibrium candidate for the deterministic dynamical system { b t } t ≥ . Notethat at b ∗ , it holds that 1 / λF ( b ∗ + (cid:15) ), and hencemin { , λF ( b ∗ + (cid:15) ) − } = 2 λF ( b ∗ + (cid:15) ) − . ( ∗ )To see that the the point b ∗ is directionally stable for { b t } t ≥ , we proceed with a case discrimi-nation on the sign of b t , t ≥
0. Since the dynamical system is one-dimensional, this follows froma sign analysis of r t . • b t < b ∗ . Since F is strictly increasing, it holds that F ( b t + (cid:15) ) < F ( b ∗ + (cid:15) ) for any b t < b ∗ .Hence, r t = d min { , λF ( b t + (cid:15) ) − } > d min { , λF ( b ∗ + (cid:15) ) − } ( ∗ ) = 0 , by definition of b ∗ . Hence, r t >
0, whenever b t > b ∗ . • b t > b ∗ . Similarly, whenever b t > b ∗ , it will be the case that F ( b t + (cid:15) ) > F ( b ∗ + (cid:15) ). Hence, r t = d min { , λF ( b t + (cid:15) ) − } < d min { , λF ( b ∗ + (cid:15) ) − } ( ∗ ) = 0 , where the first equality in the last line follows from the observation that λ (1 − F ( b ∗ + (cid:15) )) < T by definition of b ∗ . 7hus, it remains to show that b t can only have bounded oscillations in a db ∗ neighborhoodaround b ∗ , i.e., that the interval [(1 − d ) b ∗ , (1 + d ) b ∗ ] is globally attracting for the dynamics { b t } t ≥ . Assume that for some t > b t > b ∗ and b t +1 < b ∗ (if b t +1 > b ∗ , then by the definitionof directional stability, b t +1 will be closer to b ∗ than b t and the claim follows). Then, it mustbe the case that, b t +1 = b t (1 + r t ) > b t (1 − d ) > b ∗ (1 − d ) , since r t > − d by definition. Similarly, if b t < b ∗ and b t +1 > b ∗ for some t >
0, then it holds that b t +1 = b t (1 + r t ) < b t (1 + d ) < b ∗ (1 + d ) , since r t < d by definition. Thus, if | b ¯ t − b ∗ | < db ∗ for some ¯ t >
0, it must be that | b t − b ∗ | < db ∗ for any t > ¯ t . This implies that there can only be bounded oscillations around b ∗ within the[(1 − d ) b ∗ , (1 + d ) b ∗ ] intervals as claimed.The next natural step is to determine conditions under which the base fee converges to thiscandidate equilibrium or conditions under which it does not. It is important to understandthat even if the base fee remains in the bounded region specified in Lemma 3.4, it may oscillatethere indefinitely (jumping from above to below the equilibrium value and vice versa) causingsignificant fluctuations in the block load even for stationary demand. Such an instance is givenin Example 3.5. Example 3.5.
Let T = 1000, and assume a fixed number of λT = 3000 arriving transactionsper block with equally spaced valuations in [200 , b = 100, the process { b t } t ≥ has the form that is shown in Figure 1. WhileFigure 1: A case with stationary demand in which the base fee, b t , oscillates perpetually aroundthe equilibrium value b ∗ (right panel). Despite the bounded oscillations in b t , the block loadbounces between its extremes (full to empty and vise versa) (left panel).the base fee converges to the bounded region [(1 − d ) b ∗ , (1+ d ) b ∗ ] as predicted by Lemma 3.4, theblock load bounces between its extremes, i.e., from full to empty (and vise versa). Intuitively,instabilities emerge as the number of arriving transactions with similar valuations increases.If valuations had significant differences, then the base fee would reach a level where only thedesired T / − d ) b ∗ , (1 + d ) b ∗ ])and as it turns out, to extreme (and undesired) oscillations in the block occupancy.Our goal in the subsequent analysis is to formalize the observation in Example 3.5 anddetermine parameter regions for λ and w such that the base fee is provably convergent, oscil-lating or chaotic, leading to (approximately) stable block loads in the former case or significantfluctuations in the other cases. 8 .2 Convergence to Equilibrium For lower step-sizes, we can prove convergence of the base fee dynamics to b ∗ . Here, we providea closed form expression for the threshold under which convergence provably occurs. We remindthat in the non-atomic model the base fee b t is determined by the following dynamics b t +1 = b t (cid:2) d min { , λF ( b t + (cid:15) ) − } (cid:3) . Since the miners’ premium, (cid:15) appears only in the argument of the cumulative distributionfunction, F , we will eliminate it from the following computations without loss of generality(e.g., by appropriately shifting the support of F ). For simplicity, we assume that λ = 1 so thatmin { , λF ( b t ) − } = 2 F ( b t ) − b t >
0. Under these assumptions, b ∗ simplifies to b ∗ = F − (1 / F . Using the above, we can now formulatethe following convergence threshold for the step-size (which holds for arbitrary distributions). Theorem 3.6.
Let b t +1 = b t [1 + d (2 F ( b t ) − , t ≥ denote the non-atomic base fee dynamicswhen λ = 1 . Then, for any initial value b > , and any continuous and strictly increasingdistribution function, F , with support on [ L, U ] with < L < U , b tt ≥ converges to b ∗ = F − (1 / , for any step-size d ∈ (0 , d F ] , where d F = inf b (cid:54) = b ∗ ( b ∗ /b ) − − F ( b ) . Proof.
We rewrite the base fee dynamics as b t +1 = b t [1 + d (1 − F ( b t )] and define the function g : R + → R + by g ( b ) := b (1 + d − dF ( b )), for any b >
0. We will prove that(ln g ( b ) − ln b ∗ ) − (ln b − ln b ∗ ) < , for any b (cid:54) = b ∗ . Once this is established, the convergence result easily follows since (ln b − ln b ∗ ) acts as apotential function for the dynamics. To proceed, we rewrite the left hand side of the aboveinequality as(ln g ( b ) − ln b ∗ ) − (ln b − ln b ∗ ) = (ln g ( b ) − ln b ) · (ln g ( b ) + ln b − b ∗ )= ln (cid:18) g ( b ) b (cid:19) ln (cid:18) bg ( b )( b ∗ ) (cid:19) = ln [1 + d − dF ( b )] · ln (cid:2) ( b/b ∗ ) · (1 + d − dF ( b )) (cid:3) . Since F ( b ) is a continuous and increasing function by assumption, there are two cases: • b < b ∗ : in this case, it holds that F ( b ) < F ( b ∗ ) = 1 / d − dF ( b ) > d − d/ d − dF ( b )] >
0. Thus, to obtain the desired inequality, weneed to select d > b/b ∗ ) (1 + d − dF ( b )) <
1. Solving for d yieldsthe inequality d ≤ ( b ∗ /b ) − − F ( b ) . Since this inequality must hold for any b < b ∗ , we obtain thethreshold d ≤ inf b b ∗ : in this case, it holds that F ( b ) > F ( b ∗ ) = 1 /
2, which implies that ln[1+ d − dF ( b )] < d > b/b ∗ ) (1 + d − dF ( b )) > . Solving for d yields the same inequality as above (note that now 1 − F ( b ) < b ∗ whenever0 < d ≤ d f , with d F = inf b
Consider the uniform distribution in [0 ,
1] with F ( b ) = b for b ∈ [0 , F ( b ) = 0for b < F ( b ) = 1 for b >
1. Then, b ∗ = 1 / d F is given by d F = inf b (cid:54) =1 / ∈ [0 ,
1] 1 / b − − b .The minimum is obtained for b = 1 which yields the value d F = 3 /
4. This means that in thiscase, the dynamics converge for any d < / d F , consider the parametric case with F ∼ Uniform[
L, U ] with [
L, U ] = [1 − w/ , w/
2] for some w > − w/ >
0. Then, F ( b ) = ( b − (1 − w/ /w for b ∈ [1 − w/ , w/ F ( b ) = 0 for b < − w/ F ( b ) = 1 for b > w/
2. In this case, b ∗ = 1 and d F is the solution of the optimization problem d F = inf d (cid:54) = b ∗ ∈ [ L,U ] (1 + b ) w b , which is obtained from Theorem 3.6 after some trivial algebra. This is decreasing in b whichimplies that the minimum is always attained at the upper bound of the support, b = 1 + w/ d F = w (4+ w )2+ w . Thus, d F is increasing in w which implies that convergenceis easier (harder) as valuations become less (more) concentrated in a specific regime.The last example suggests that for any d >
0, there exists a w > not converge to b ∗ if the valuations are uniformly distributed on aninterval with range w . This raises the question of what happens in the base fee dynamics insuch cases. As we show next, for certain values of w , the dynamics not only fail to converge,but they become provably chaotic. The previous convergence results critically depend on the provided thresholds. If the step-size exceeds these bounds, then the base fee adjustment rule may lead to chaotic updates.As mentioned above, these bounds depend on the number of arrivals, λ , and in the range ofvaluations, w . If λ increases or w decreases, i.e., if the system becomes more congested or if thevaluations become more concentrated around a specific value, then the thresholds go down anda given step-size may not be enough to guarantee convergence. In fact, as we will show, for anystep-size, there exists a (reasonably large) λ and a (reasonably small) w so that the dynamicsbecome chaotic. Formally, we will show that the base fee updates become chaotic in the sense of Li-Yorke [9].If a system is Li-Yorke chaotic, then its trajectories exhibit complex behavior: uncountablymany pairs of trajectories get arbitrary close and move apart infinitely many times as thesystem evolves. Furthermore, the system has periodic orbits of all possible periods. This meansthat different trajectories become indistinguishable and hence, the system cannot be efficientlysimulated or cannot be predicted in practice. The notion of Li-Yorke chaos is a fundamentalnotion of chaos in dynamical systems that is connected to many other definitions of chaos (e.g.,positive topological entropy). For more discussion on these connections, particularly in the caseof game dynamics see [7]. Such chaotic behavior has recently been observed in game theoreticsettings under adaptive agents using different online learning dynamics [12, 7, 6, 2]. To give theformal definition of Li-Yorke chaos (cf. Definition 3.10), we will first introduce some additionalnotation. 10 efinition 3.8 (Periodic Orbits and Points) . A sequence b , b , . . . , b k is called a periodic orbit of length k if b t +1 = g ( b t ) for 1 ≤ i ≤ k − g ( b k ) = b . Each point b , b , . . . , b k is called periodic point of period k . Definition 3.9 (Li-Yorke pair [9]) . Let X = [ L, U ] be a compact interval in R and let g : X → X define a discrete time dynamical system ( x t ) t ∈ N on X , so that x t := g t ( x ) for any x ∈ X . Apair ( x, y ) ∈ X with x (cid:54) = y is called a Li-Yorke pair iflim inf t →∞ | g t ( x ) − g t ( y ) | = 0 < lim sup t →∞ | g t ( x ) − g t ( y ) | . If for any x, y ∈ S with x (cid:54) =, the pair of x, y is a Li-Yorke pair, then S is called a scrambled set.The most classic definition of chaos in the mathematics literature defines chaos as the ex-istence of periodic orbits of all possible periods along with an uncountably large scrambledset. Definition 3.10 (Li-Yorke chaos [9]) . Let X = [ L, U ] be a compact interval in R and let g : X → X define a discrete time dynamical system ( x t ) t ∈ N on X , so that x t := g t ( x ) for any x ∈ X .The dynamical system ( x t ) t ∈ N is called Li-Yorke chaotic if it holds that:1. For every k = 1 , , . . . there is a periodic point in X with period k .2. There is an uncountable set S ⊆ X (containing no periodic points), which satisfies thefollowing conditions: • For every x (cid:54) = y ∈ S ,lim sup t →∞ | g n ( x ) − g n ( y ) | > t →∞ | g n ( x ) − g n ( y ) | = 0 . • For every point x ∈ S and point y ∈ X ,lim sup t →∞ | g n ( x ) − g n ( y ) | > . In particular S is a scrambled set.According to [9], a sufficient condition for a system to be Li-Yorke chaotic is that it has aperiodic orbits of period 3. This will be our main tool to show that the base fee dynamics areLi-Yorke chaotic and is stated next. Theorem 3.11 (Period three implies chaos [9]) . Let X ⊂ R be a compact interval and let g : X → X be a continuous function. Further assume that there exists a point x ∈ X for whichthe points x := g ( x ) , x := g ( x ) = g ( x ) and x := g ( x ) = g ( x ) satisfy x ≤ x < x < x ( or x ≥ x > x > x ) . Then, the system is Li-Yorke chaotic.
Notice that if there is a periodic point with period 3, then the hypothesis is satisfied.11 .3.2 Li-Yorke Chaos in the Base Fee Updates
With the above terminology and notation at hand, we now return to the base fee dynamics. Inthe case of the non-atomic approximation (cf. equation (4)), it holds that b t +1 = g ( b t ) with thecontinuous map g defined by g ( b ) = b + bd min { , λF ( b + (cid:15) ) − } . (8)As we showed in Lemma 3.4, the dynamics will ultimately enter the bounded interval X :=[(1 − d ) b ∗ , (1 − d ) b ∗ ]. Moreover, it holds that g ( b ) = b for b ∈ X if and only if b = b ∗ ,i.e., b ∗ is the unique fixed point of function g in X . Thus, according to Definition 3.10 andTheorem 3.11, it suffices to show that the continuous map g : X → X has a periodic point ofperiod 3, i.e., that there exists a point b (cid:48) ∈ X with b (cid:48) (cid:54) = b ∗ , which is a fixed point of g ( b ), i.e., g ( b (cid:48) ) = b (cid:48) , b (cid:48) (cid:54) = b ∗ ∈ X . The two panels in Figure 2 illustrate the two possible cases.Figure 2: Orbits in the base fee dynamics b t +1 = f ( b t ). The left panel shows an instance in whichthe base fee dynamics do not have orbits of period 3 (the graph of g (3) ( b ) does not intersect thediagonal y = b , i.e., g (3) ( b ) does not have fixed points other than the unique fixed point of g ( b )).By contrast, the right panel shows an instance with points of period 3 (multiple intersections of g (3) ( b ) and y = b ). In this case, the dynamics are Li-Yorke chaotic. In both cases, the step sizeis equal to d = 0 .
125 and the valuations are uniformly distributed in [200 , T in the left panel versus 5 T in theright panel).In Theorem 3.12, we invoke Theorem 3.11 and show the more general case, that for any d ,there exists a distribution of valuations so that the system becomes chaotic. Theorem 3.12.
Let g ( b ) = b + bd min { , λF ( b + (cid:15) ) − } denote the non-atomic approximationof the update rule for the base fee dynamics ( b t ) t ∈ N . Then, for any fixed step size d > , thereexists a continuous distribution F of valuations, and a point b ∈ R , so that g ( b ) ≤ b < g ( b ) < g ( b ) , (PO) In particular, for any step-size d , there exists a distribution of valuations F , for which the basefee dynamics become Li-Yorke chaotic.Proof. The proof is constructive and proceeds by creating a specific instance of the uniformdistribution for which condition (PO) is satisfied. Then, the claim that the dynamics are Li-Yorke chaotic follows from Theorem 3.11. To create such an instance, let F ∼ Uniform[ µ − / , µ + 1 /
2] for some µ >
0. Also, let λ = 1 and as above, assume without loss of generalitythat (cid:15) = 0 (e.g., by properly rescaling the distribution F ). Based on these assumptions, it holdsthat 1 > F ( b ) − b > g , of the non-atomic modelbecomes g ( b ) = b (1 + d − dF ( b )) . We will now show that we can construct a sequence of points b , b = g ( b ) , b = g ( b ) and b = g ( b ) with the following properties 12i) b ≤ µ − / b = g ( b ) = µ − δ , for some δ > b = g ( b ) ≥ µ + 1 / b > F ( b ) = 0(since b < µ − / b = g ( b ) = b (1 + d ). Combining this with property (ii), yields the firstnecessary condition, b (1 + d ) = µ − δ , or equivalently b = µ − δ d , for some δ > . ( (cid:63) )Plugging into property (i), this yields the condition µ − δ d ≤ µ − / ⇒ dµ − − d + δ ≥ . (C1)Next, we calculate b = g ( b ) = g ( b ). Since b = µ − δ = b (1 + d ), we can determine g ( b ) asfollows g ( b ) = b (1 + d )(1 + d − dF ( µ − δ ))= b (1 + d ) (cid:18) d − d µ − δ − ( µ − / µ + 1 / − ( µ − / (cid:19) = b (1 + d )(1 + d − d (1 / − δ ))= b (1 + d )(1 + 2 dδ ) = ( µ − δ )(1 + 2 dδ ) , where the last equality follows from ( (cid:63) ). Thus, b = g ( b ) = ( µ − δ )(1 + 2 dδ ) > b = g ( b ).Further, if property (ii) holds, i.e., if b ≥ µ + 1 /
2, or equivalently if( µ − δ )(1 + 2 dδ ) ≥ µ + 1 / , (C2)(which gives a second necessary condition), then it holds that F ( b ) = 1. This allows us tocalculate b = g ( b ) = g ( b ) as follows g ( b ) = b (1 + d − d ·
1) = b (1 − d ) = b (1 + d )(1 + 2 dδ )(1 − d ) . Thus, b = g ( b ) < b and it remains to show that b ≤ b . This yields the third necessarycondition b (1 + d )(1 + 2 dδ )(1 − d ) ≤ b or equivalently (assuming that d < δ ≤ d/ − d ) . (C3)In sum, given d >
0, we need to select µ, δ so that conditions (C1), (C2) and (C3) are satisfiedsimultaneously (note that we already used ( (cid:63) ) in the formulation of (C1)). This gives the system2 dµ − − d + δ ≥ , (C1) dδµ − dδ − δ − / ≥ , (C2) d/ − d ) ≥ δ, (C3)Thus, if we select any δ > µ large enough, conditions (C1) and (C2) are always satisfied (since µ appears only in one termwith positive scalars). Specifically, if we solve (C1) and (C2) for µ , we obtain the (alwaysfeasible) condition µ ≥ max (cid:26) d − δ d , dδ + δ + 1 / δd (cid:27) , (C4)which together with the selected δ , yields an admissible solution of the initial system. In sum,we have shown that if we select a point b ≤ µ − δ where µ, δ satisfy conditions (C3) and (C4),then it holds that g ( b ) ≤ b < g ( b ) < g ( b ), which concludes the proof.13igure 3: Bifurcation diagrams for the input parameters, w (range of valuations) and d (step-size) of the non-atomic base fee dynamics of equation (4). Left panel: route from order to chaosas the step-size increases. Right panel: route from chaos to order as the range of valuationsincreases.Note that the construction in the proof of Theorem 3.11 was based in a favorable scenariofor stability which assumed λ = 1. For higher values of λ , the construction still applies andin fact, chaos obtains for a much wider range of parameters (see Section 3.4). Moreover, theselection of the uniform distribution in the proof is not binding and the proof idea applies forarbitrary distributions. This is illustrated in the next example which concludes this section. Example 3.13 (A Specific Instance with Period 3) . Let b > F ( b ) of the valuations is continuous andsatisfies the following conditions: F (0 . b ) = 6 / F (0 . b ) = 11 / F ( b ) = 11 /
18 and F (1) = 1. Assume that λ = 1 and that F is rescaled so that (cid:15) = 0 (as above). Then, for d = 9 /
10, we have that b = g ( b ) = b (1 + (9 / − / − . b , b = g ( b ) =0 . b (1+(9 / − / −
1) = 0 . b and b = g ( b ) = 0 . b (1+(9 / − / −
1) = b . We, thus, get an example with period 3. The previous paragraphs suggest that there are ranges of parameters for which the base fee dy-namics converge and ranges of parameters for which they become Li-Yorke chaotic. The systemis more prone to chaotic behavior as the step-size, demand (users that submit transactions)or concentration of valuations increase. In this paragraph, we visualize the long-term behaviorof the base fee dynamics and the transitions through the various regimes as the critical inputparameters of the system change. Again, for expositional purposes, we restrict attention touniform distribution of valuations on the interval [
L, U ] = [210 − w/ ,
210 + w/ The resultsare shown in the bifurcation diagrams in Figure 3.The horizontal axis of each diagram corresponds to the varying parameter, w and d re-spectively, with all other parameters being fixed. The vertical axis shows the attractor of thebase fee dynamics (blue dots) for 400 updates (after a burn-in period of 100 updates) and the[(1 − d ) b ∗ , b ∗ , (1 + d ) b ∗ ] bounds. Interestingly, the transition from the stable (convergent) to Simulations with different distributions such as triangle distribution or normal produce qualitatively equiv-alent results which are not presented here. The bifurcation diagram for parameter λ is similar and is not presented here. not occur by a period doubling (as is typical in most game-theoreticapplications of chaos theory) [7, 2]. For practical purposes, the important observation is thatthese phase transitions occur abruptly for small changes in the parameter values. We describe here the main components and results from a simulation environment created toreplicate the Ethereum transaction fee market.
Blocks in Ethereum are produced by miners in a random, iterative process. A block builds ona chain of predecessors, such that the chain length always increases in time. The consensusalgorithm ensures all participants (miners and users) agree on the current head of the chain. Inthe following simulations, we adopt the parameter choices of EIP 1559, namely, a gas target of12,500,000, gas limit of 25,000,000 and update rate parameter d = 0 . η = 13 seconds underthe assumption of no latency. User values v are sampled from a fixed distribution F . We assume users all send transactionsconsuming the same amount of gas γ , without loss of generality. The values are expressed asbenefit received per unit of gas, thus if the user’s transaction is included, the user receives γv utility. The parameter value γ is obtained by computing the average gas used in all blocks froma sample period between block 10,900,000 (timestamp Sep-20-2020 03:17:06 PM UTC) to block10,942,000 (timestamp Sep-27-2020 02:40:14 AM UTC). We then take the median of the seriesof average gas used per block, rounded to the nearest multiple of 1,000, to provide a sensible γ estimate. The procedure yields γ = 76 , wallets , which provide fee estimationand generate transaction parameters such as the gas limit and data payload (e.g., inputs andnames of function calls for smart contract interactions). In the current, first-price auction-based fee market, wallets typically provide fee estimation by computing statistics from historicaltransaction inclusion, e.g., the median fee paid by any transaction over the last 200 blocks. Whilewallet behaviors in 1559 are not currently known, we use the following design: • When the previous block was not close to full (less than 90% gas used), the wallet sets thefee cap parameter f to a fixed value derived from the base fee (we use three times the currentbase fee) and the premium p to the commonly agreed (cid:15) miner marginal cost. We guaranteethat fee cap covers at least the premium by setting a lower bound. • When the previous block was close to full (above 90% gas used), the wallet adds an incrementto the average tip recorded in the previous block, inducing competition between users whilebase fee matches the new demand. Average block time chart https://etherscan.io/chart/blocktime .1.3 Demand process For convenience, we introduce two time indices: s, t refer to chain heights (measured in blocks),while k refers to simulation time (measured in seconds). As block inter-arrival times are random,we first generate a demand process ( D k ) k returning for all time indices k an integer-valueddemand volume. D k is interpreted as “users producing transactions between seconds k − k ”. To generate ( D k ) k , we sample Brownian motion (BM) paths with initial condition D , mean0 and variance σ . We obtain demand paths that feature periods of increasing and decreasingvolumes due to the randomness of the BM, yet do not explicitly have a positive or negativetrend. In addition, we simulate random “jumps” where a mass of users is generated at randomintervals, decaying over the next steps at a rate δ , to reproduce instances where an on-chainevent brings a sudden influx of new users (e.g., token sale). Formally, our demand processsatisfies at time k : D k = W k + J k ; J k = (1 − δ ) J k − + M k (cid:88) j =1 ζ j · (cid:98) κ j (cid:99) = k ; J = 0where W k is a discretized Brownian motion (in this case, a random walk with normal incrementsof mean 0 and standard deviation σ ); M k is a Poisson process of rate λ j evaluated at time k ; ζ j is an exponential random variable of mean B , modelling a demand jump; and (cid:98) κ j (cid:99) is thetime index where the j -th jump occurred. Figure 4 depicts the sample paths for one value ofdemand variance and initial condition. Miners run Ethereum nodes exchanging data (including transactions) with other nodes overa peer-to-peer network. Nodes are either run by miners, users or third parties, to relay thisdata in a decentralized fashion. A user either directly sends their transactions from their ownnode or indirectly from a third party node, who receives the transaction from the user via somecommunication protocol.While only miner nodes eventually produce blocks, all nodes feature a transaction poolthat holds pending transactions and continuously receives or sends items to other nodes, asrequested. All nodes are free to decide in practice which transaction pool policies to apply,including the choice of the maximum number of pending transactions held in the transactionpool at any point in time. Geth, Ethereum’s dominant node client as of February 2021, holdsby default a maximum of 4096 transactions in the pool. In our simulations, we abstract this peer-to-peer network of transaction pools with differingpolicies into one logical transaction pool, which instantly receives all user transactions, appliesthe same pool policy at all time steps and is used by all miners to form their blocks. Thepossibility that various pool policies will affect transaction transmission is not considered.
Much like the logical transaction pool described above, we also consider a single logical minerrepresenting all miners who produce blocks. [14] provides evidence for the incentive - compati-bility of miner myopic strategies, who maximize greedily the available fees at the time of theirblock production. Thus, we do not consider the possibility of long-range attacks in our simula-tions, where a cartel of miners colludes to lower the base fee to zero to enforce a monopolisticprice of entry. Given the set of pending transactions in the pool, miners order transactions byreceived tip (in decreasing order) and include as many transactions as possible, until the blocklimit is reached or there are no more valid transactions to include. See geth.ethereum.org for defaults, and ethernodes.org for client statistics. .1.6 Simulation steps We provide below a description of a single simulation step, articulating how the various com-ponents are employed.1. The previous block B t − produced at chain height t − η t is sampled from an exponential distribution of mean η = 13, suchthat the block at height t is created at time index θ t = (cid:80) ts =1 η s .3. Given the demand process ( D k ) k , where k is an index over seconds, we obtain N t = (cid:80) θ t k = θ t − D k . N t is the number of users entering the market between blocks t − t ,included at the earliest in block B t .4. The N t users observe the current chain state (e.g., the base fee level) and decide whether ornot to transact. User transactions are formed via wallets which encode shared strategies.5. Transactions are received by miner transaction pools, which hold a set of pending transactionsfrom previous simulation steps. All the while, transaction pools apply eviction policies inorder to manage their limited resources.6. The miner producing B t selects transactions from the pool to maximize their fees. The setof selected transactions must be smaller than the block limit.7. Repeat from step 1. We focus our attention on three independent variables: • The demand path variance σ : We choose σ ∈ { . , . , . } . • Initial condition of the Brownian motion D : Given the mean block inter-arrival time η , weselect D to reproduce conditions of low, medium and high demand. With D set to Tγη , wetarget a user arrival rate that on average is exactly enough to fill the block to its limit (i.e.,to twice its target). Thus we select D ∈ { Tγη , . Tγη , Tγη } . • Transaction pool eviction tolerance τ : size of the eviction band, i.e., evict all transactionswith fee cap smaller than (1 − τ ) times base fee. τ = 0 is the strictest policy (remove alltransactions with fee cap smaller than current base fee), τ = 1 is the most permissive (keepeverything, modulo the pool limit size). We choose τ ∈ { , / , / , } .We sample 20 Brownian motion samples (see Figure 4). Each sample yields nine distinctpaths, one for each value of standard deviation σ and initial condition D , i.e., 180 paths. Foreach path, one simulation is run for each value of τ , yielding 720 sample runs. Each run consistsof 600 blocks, representing approximately half a day of activity on Ethereum. The first 100blocks of each run are discarded from the analysis, as they represent initial conditions wherebase fee has not yet matched the existing demand.Our dependent variable measures the variance of recent realisations of the percentage of gasused by the block. In Section 3, chaotic behavior obtained rapid variations of the gas used,from mostly empty block to mostly full blocks. Experimentally, we measure a moving standarddeviation of the series of gas used, with window size 4. The maximal standard deviation s ∗ isachieved whenever the four values alternate between 0 and 100. We call high variance time stepswhere the moving standard deviation is at least 95% of s ∗ . The percentage of high variancetime steps among all simulation steps is our dependent variable.17 Block height I n c o m i ng u s e r s Figure 4: Demand paths at standard deviation σ = 0 . D = 2 Tγη . We reproduce in Figure 5 the percentage of high variance steps averaged over sample runs foreach single value of treatment variables. We observe consistent increases in high variance stepsas the demand variance increases, the initial demand level increases or the pool eviction policyis more permissive, as evidenced by the two samples presented in Figure 6.The pool eviction has the sharpest contrast between levels of the variable. While highvariance steps almost never occur with the strictest pool policy (never keeping a transactionwith fee cap inferior to the current base fee), even a mild increase of the tolerance to 1/3 inducesa level of high variance steps comparable to any further increase of the tolerance.
Demand standard deviation % H i gh v a r i an c e Initial condition % H i gh v a r i an c e Band width % H i gh v a r i an c e Figure 5: Percentage of high variance simulation steps with changing demand variance, initialcondition and transaction pool eviction tolerance.
In the simulations, user values are randomly sampled from the uniform distribution, with therandomness seeded by the index of the Brownian motion (BM) sample, such that runs withidentical demand paths, demand variance and initial condition generate the same users. Thisallows for comparison of two more dependent variables, user efficiency and miner revenue ,given the band width τ as independent variable and controlling for demand variance and initialconditions. 18 Block height B a s e f ee Block height I n c l uded t r an s a c t i on s Block height P oo l l eng t h Block height B a s e f ee Block height I n c l uded t r an s a c t i on s Block height P oo l l eng t h Figure 6: Two sample runs, one per row, with line plots for base fee, number of includedtransactions and transaction pool length. High variance periods are represented by red bars inthe background. The first sample has D = Tγη , σ = 0 . τ = 0 and features few high varianceperiods. The second sample has D = 2 Tγη , σ = 1, τ = 1 and features more high varianceperiods. Additionally, the transaction pool is continuously full, as the pool eviction policy ismost permissive.User efficiency measures the total benefits received by all users included in the chain. EIP1559 is efficient whenever users with the highest value are included on-chain. Miner revenueconsists of the received tips, either at the marginal cost level 1 Gwei per gas unit or higherwhenever users are competitively bidding.An experiment is represented by a choice of triple (BM index, demand variance, initialcondition), with the band width τ taken as the independent variable. In all experiments,increasing the band width decreases both the user efficiency as well as miner revenue. Thisresult is explained by the dynamics of the pool itself. By keeping transactions that are notcurrently valid for inclusion, miners have “inventory” to spend whenever demand is low and thebase fee has decreased enough to make the transactions valid. This inventory however representsa danger to the stability of the base fee, as a bottleneck of transactions may accumulate in thepool, all becoming valid at the same instant and provoking base fee spikes and instability. Ethereum’s improvement proposal EIP-1559 is aiming to transform the transaction fee mar-ket of the Ethereum blockchain via a dynamic pricing mechanism. The core element of themechanism is a fixed-per-block network fee (termed base fee) that is burned and dynamicallyexpands/contracts block sizes to deal with transient congestion [4]. Our goal in this paper wasto stress-test the base fee both theoretically and experimentally and understand its effects onregulating the transaction fees and block occupancies.A concrete outcome of both our theoretical and experimental analysis is the importance ofthe base fee adjustment parameter on the performance of the mechanism. Our findings provideinsights about the conditions under which the base fee self-stabilizes but also characterize ex-treme operational scenarios under which its dynamics become chaotic. In particular, we showedthat EIP-1559 has promising properties (convergence guarantees under various conditions) toconvey stability to the fee market and identified sources of concern that may destabilize thesystem into regimes of chaotic behavior. Our work develops a systematic framework that com-bines elements from mechanism design, dynamical systems and chaos theory and which aims toaid the ongoing study of transaction fee markets in blockchain-based economies. In this work, we do not consider users with time preferences. cknowledgements Stefanos Leonardos and Stratis Skoulakis acknowledge support from NRF 2018 Fellowship NRF-NRFF2018-07. Barnab´e Monnot acknowledges support from the Ethereum Foundation. DanielReijsbergen acknowledges NRF Award No. NSoE DeST-SCI2019-0009. Georgios Piliouras ac-knowledges support from NRF2019-NRF- ANR095 ALIAS grant, grant PIE-SGP-AI-2018-01,NRF 2018 Fellowship NRF-NRFF2018-07 and the Ethereum Foundation.
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