Beyond Spot Transactions: Modeling Dynamic Preconfirmation Auctions

Abstract. We investigate a model for an auction for preconfirmations of Ethereum transactions, its relations to existing models modern and classical, and its implications for bidder and provider, in particular block builder, behaviour.

TL;DR

  • An Ethereum execution preconfirmation ("preconf") is an advance commitment to include a specified transaction list into a specified block.

  • The mev-commit protocol and chain provides a new platform for peer-to-peer sales of preconfs through customisable private auctions.

  • Block builders and other providers of execution services may wish to sell preconfs because they fetch a higher price than spot, or to solicit early bids that can be used to update forecasts. If selling preconfs increases block revenue, builders must do it to remain competitive in PBS auctions.

  • The advance guarantee provided by a preconf is valuable to transactors who are more sensitive to fee or censorship risk than to execution outcome risk, especially over a short horizon. It may also be valuable to certain risk-neutral transactors pursuing a multi step strategy.

    Such buyers may be prepared to pay a preconf premium on top of what they would otherwise pay for blockspace at delivery time.

  • To encourage early bids, users need to be convinced that they have some chance to get their bid accepted in a timely manner. The mev-commit protocol addresses this by incentivising early commitments via bids that decay over time.

  • Providers can solicit earlier and higher bids by publicly committing to an auction mechanism to sell off some portion of their blockspace for preconfs. For example, committing to timely accept or reject messages encourages early bidding, while committing to a reservation fee can encourage higher bids.

  • Providers who obtain early bids can also use the resulting price discovery to dynamically adjust their commitment parameters, potentially increasing revenue.

  • In this article we extend the model of Roughgarden (2020) for EIP-1559 gas auctions to study a dynamic preconf auction based on the mev-commit protocol.

Introduction

Execution preconfirmations on Ethereum are a way of issuing an early promise to arrange for execution of specified transactions or bundles.ΒΉ Such promises can be used to mitigate censorship and fee risk in an adversarial MEV environment. With the advent of smart contract chains focused on intents, execution, and block building, they can also be used as the basis for constructing more sophisticated blockspace products.

Definition. A (single transaction/bundle, named block) Ethereum execution preconfirmation is a credible advance commitment to arrange for execution of a specified transaction or bundle in a specified block.Β²

Mev-commit protocol is intended to support more general types of preconfirmations, but this article will focus on the single transaction or bundle, single block case, which for the rest of the article we call simply preconf.

Preconfirmations on mev-commit protocol

The mev-commit protocol comprises a network protocol and programmable blockchain supporting peer-to-peer sales of preconfirmations for Ethereum transactions, with slashing penalties for liveness and safety failures. Bidders establish direct connections to execution service providers and submit their transactions along with confidential bids for preconfirmation thereof. If a provider decides to accept a bid, they may respond directly with a cryptographic commitment that is then registered in a smart contract for later settlement or slashing. Bids can be accepted at any time before the target block, or simply ignored. Accepted bids settle at a fraction of the bid price that decays linearly as a function of the time delay between bid and accept messages. Settlement occurs after the target block time, regardless of when the bid was accepted.

In terms of implementations, all participants run a mev-commit network node.³ Providers typically also run a consensus node for the target chain, or block builder if this role is separated from consensus, which they use to deliver on their commitments.⁴ The blockchain used for registering preconf commitments, slashing, and settlement is called mev-commit chain. At the time of writing, it is running as a POA (Clique) Ethereum testnet sidechain.⁡

Key observations

From a theoretical perspective, the key new ingredient in preconf auctions as compared to spot is their dynamic nature: bids can be accepted at any time, and hence bidder and provider behaviour must be considered in the context of private valuations and strategies that evolve over time.

The model. We begin by adapting the static Roughgarden (2020) model to the case of a single pay-as-bid preconf auction with a new preconf tip bid parameter. We find that the EIP-1559 bid can be bundled into the preconf tip, so the new model is isomorphic to the old one. If, as in op. cit., we assume that the base fee is known ahead of time, the bid space can be reduced to one dimension.⁢

We then discuss how to construct dynamic variants of the model, where bidders and provider must choose which time to submit their messages. With many types of strategy, the bidder will want to wait until the last moment before bidding to fix their transaction parameters before bidding, which makes the "single-mindedness" assumption less realistic in the dynamic setting. On the other side, providers may take actions multiple times, for example holding periodic auctions; among other things, this gives blockspace a dynamic withholding value that the provider must consider when setting up their auction.

Instead of pay-as-bid, payments may be adjusted by a decreasing function of time between bid creation and acceptance. For simplicity, in this article we mostly confine ourselves to the case of trivial decay; however, nontrivial decay is expected to be used in deployments of mev-commit and this affects several conclusions: dimensional reduction of the bid space is no longer possible, bid timing optimisation is significantly complicated, and providers have the incentive to accept bids sooner rather than later.

It is up to the provider to choose an auction decision rule to optimise their revenue (or other social objective such as OFAC compliance).

Withholding value. The new salient feature of the static preconf model is that the provider's expected payoff for withholding blockspace is nonzero, since it can be reauctioned at a later time closer to the target block.

Because preconf bids may be accepted at any time before the block is committed, with pay-as-bid payment rules the expected revenue of deferring and reauctioning later is always at least as much as the revenue from accepting a bid immediately.⁷ This means that a myopic provider would never accept preconf bids until the last moment unless they were otherwise committed to do so. This gives rise to a theoretical issue which we call the deferring problem: if bids are never accepted early, there is no incentive to bid for a preconf, and the preconf premium vanishes!

The deferring problem does not arise in practice if would-be bidders believe that the provider will issue early "accept" messages under reasonable circumstances. The mev-commit protocol will address this by modifying the payment rules to include a penalty for deferring that increases linearly with time, known as bid decay.

Mechanisms with commitment. A powerful way for a provider to solicit earlier (and in some cases, higher) bids is by making a public commitment β€” whether backed by smart contract or traditional trust-building β€” to auctioning off a fixed-size portion of the blockspace at time tt using a declared credible auction mechanism.⁸ For example, a provider could promise to auction 10% of the block's gas limit with a forward greedy preconf auction in the first 2 seconds of the interblock time. In the context of mev-commit, the mev-commit chain is a natural choice of domain for such smart contract commitments.

As well as soliciting early bids, committing to mechanism parameters β€” for example, a reservation price β€” can increase auction revenue.⁹

Risk management. The main target customers for preconfirmations of blockspace are risk averse. More precisely, they are more sensitive to gas price and censorship risk than to execution outcome risk, the latter arising because the buyer must make an early commitment to transaction parameters. The preconf premium depends on how the market prices the tradeoff between these risks.

Dynamic provider strategies. Apart from capturing risk premia, soliciting early bids holds another potential benefit for providers: they can use the information leaked by the bids of informed users to dynamically update their decision parameters for later preconfs with the same target block. For example, a high bid from a specialist MEV searcher may indicate an imminent fee surge to which the provider may respond by raising reservation fees.

Static model

To illustrate the basic structure of a preconf auction, we study a static model for a preconf auction held tt seconds in advance of the target block.

Bids and mempools

Roughgarden (2020) defines a transaction in terms of three parameters: the gas limit gg, the bidder's private valuation vv for transaction inclusion, and the effective bid per unit gas f=fefff=f_\mathrm{eff}. In implementations, the bidder actually sets the max gas fee and the priority fee, and ff is computed from these and the base fee. Following Roughgarden, we assume the base fee is known ahead of time, so the bidder may deduce these parameters from knowledge of ff.

We enhance the model by adding a new parameter bb, the preconf bid for the transaction as a whole. In implementations, this field goes into a mev-commit preconfirmation bid message that refers by content hash to the underlying EIP-1559 transaction.

A mempool consists of a set Mt={(gi,vi,fi,bi)i∈I} M_t = \{(g_i,v_i,f_i,b_i)_{i\in I}\} of transactions indexed by some auxiliary set II.

Decision problems

Following Roughgarden, we assume for simplicity that bidders are single-minded, that is, each participating bidder is trying to secure inclusion of a single transaction or bundle. Both gg and vv are considered immutable preferences of the bidder, so their bidding action space consists only of setting ff and bb.

The payoff for execution of a winning bid (g,v,f,b)(g,v,f,b) is vβˆ’(gf+b). v - (gf + b). The aggregated fee gf+bgf+b is the effective total fee paid for the transaction, including preconfirmation.

Providers must implement their own decision rules x(M)βŠ†I\mathbf{x}(M)\subseteq I for determining the set of accepted bids. This rule must satisfy the knapsack capacity constraint gx(M):=βˆ‘i∈x(M)gi≀C g_{\mathbf{x}(M)} :=\sum_{i\in \mathbf{x}(M)} g_i \leq C where CC is the block capacity.

The payment rules are pay-your-bid for winning bidders; more precisely,

  • gβ‹…fbaseg\cdot f_\mathrm{base} goes to the burn address.

  • gβ‹…(fβˆ’fbase)g\cdot(f-f_\mathrm{base}) goes to the "fee recipient" target of the block.

  • bb goes to the preconf provider.

Normally, the preconf provider and the fee recipient are the same entity. The provider's immediate payoff for each auction is therefore βˆ‘i∈x(M)[bi+giβ‹…(fiβˆ’fbase)]. \sum_{i\in \mathbf{x}(M)} [b_i + g_i\cdot(f_i-f_\mathrm{base})]. The provider's task is to design an optimal mechanism so as to maximise this payoff.

Example. (Forward greedy first price mechanism). The "standard" 2-parameter knapsack auction implemented as a default in Bitcoin and Ethereum node software is a forward greedy algorithm with capacity CC and reserve fresf_\mathrm{res}. In a spot auction, C=C0C=C_0 is the block size; in a preconf auction, C=Cβˆ’t≀C0C=C_{-t}\leq C_0 is the amount of blockspace the preconf provider allocates to preconfirmations at time βˆ’t-t. See below for some discussion of how reservation fees will be set in the dynamic context.

Dimensional reduction of bid space

Since bidders in this static model have only one parameter β€” their private valuation of inclusion β€” a winning bidder is indifferent to the breakdown of the transaction fee into its three components except inasmuch as it affects their chance of inclusion. When the preconf provider is also the fee recipient, they are also indifferent. Bidders may therefore make the substitution (g,v,f,b)↦(g,v,fbase,b+gβ‹…(fβˆ’fbase)), (g,v,f,b) \mapsto (g,v,f_\mathrm{base},b+g\cdot (f-f_\mathrm{base})), setting the priority fee to zero without affecting any payoffs. This leaves only one degree of freedom to bid with fixed gg and vv.

We find that our static model is trivially isomorphic to Roughgarden's model. In particular, the observations of op. cit. regarding the costliness of including "fake" transactions and OCA-proofness carry over to our setting without modification. Issues relating to ex post efficiency, bidder incentive compatibility, and credibility are isomorphic to the corresponding spot block builder auction.

Dynamic models

The dynamic rules of mev-commit preconf auctions are as follows:

  • The mempool state is now a point process Mtt≀0{M_t}_{t\leq 0} to which transactions are added as they arrive from bidders and are removed as the provider accepts bids.

  • Bidders choose the time they issue their bid. They may choose their transaction parameters ΞΈ\theta at that time. It may be unrealistic to assume that ΞΈ\theta is fixed ahead of time.

  • Bidders' expected valuation vt,ΞΈv_{t,\theta} for the execution of a transaction with parameter ΞΈ\theta evolves over time as a martingale process.

  • Similarly, each bidder's model for the conditional distributions of the other bidders' types evolves over time.

  • The provider may accept any subset xt(Mt)\mathbf{x}_t(M_t) of MtM_t at a finite number of times t1,…,tk{t_1,\ldots,t_k}, removing those bids from the pool. The provider's payoff at delivery time is the sum of the fees received from each of these kk selections.

  • The payment amount for an accepted bid mi=(gi,vi,fi,bi,ti)m_i=(g_i,v_i,f_i,b_i,t_i) may be a decreasing linear function b~(Ξ”ti)\tilde b(\Delta t_i) of the time Ξ”ti=inf⁑{t∣mi∈xt(Mt)}βˆ’ti \Delta t_i=\inf\{t|m_i\in\mathbf{x}_t(M_t)\}-t_i between bid and acceptance, normalised as b~(Ξ”ti)=bi\tilde{b}(\Delta t_i)=b_i.

    For simplicity, in this article we mostly treat the payment rule as if this function were constant, i.e. pay-as-bid.

Bidders may also increment their bids at any time by submitting additional bid messages for the same transaction. The total payment on delivery is then the sum of all the accepted bids.

Bid timing

Suppose that the bid decay function b~\tilde b is constant in time. Then a single-minded bidder gains the same ex-post payout vβˆ’(gf+b)v-(gf+b) from execution of a given transaction, regardless of when the preconfirmation was accepted. Therefore, the expected payout Ο€(b,t)(vβˆ’(gf+b))\pi(b,t)(v-(gf+b)) depends on bid time only via the probability

Ο€(b,t):=P[miβˆˆβ‹ƒt>tixt(Mt)∣b=bi,t=ti]\pi(b,t) := \mathbb{P}\left[m_i\in\bigcup_{t>t_i}\mathbf{x}_t(M_t)\mid b=b_i,t=t_i\right]

of inclusion. A risk-neutral single-minded bidder will bid bb at time t<0t<0 only if βˆ‚βˆ‚tΟ€(b,t)<0\frac{\partial}{\partial t}\pi(b,t)<0 for some t<0t<0, that is, if the chances of winning with bid bb are decreasing as delivery time approaches. Roughly speaking, this means that the smallest bid needed to secure inclusion is increasing with time, that is, the preconf premium is negative. (In mev-commit protocol, bidders can only express a negative preconf premium by bidding below the market rate for priority fees.)

If b~\tilde{b} is strictly decreasing, the computation is more complex β€” the bidder gains more utility if the bid is accepted late, but it is also less likely to be accepted late than early if no new information reaches the provider. It's an open question whether risk-neutral bidders will sometimes bid early with a positive preconf premium in this setting.

Provider withholding values

The private value of the provider for withholding blockspace in a preconf auction at time tt is, roughly speaking, the expected revenue of selling that same blockspace in some later preconf auction or equivalent spot auction. Subtleties arise in making sense of this quantity because:

  1. Blockspace pricing is not generally uniform;

  2. Selling blockspace early reduces capacity in the spot auction, potentially leading to increased congestion and hence per-unit revenue for late-arriving transactions.

To put it more precisely, let us suppose that the provider holds preconf auctions for a target block at fixed discrete intervals t0,t1,…,tk=0t_0,t_1,\ldots,t_k=0. Suppose also that the payment rule is constant over time. A risk neutral provider will not accept a basket of bids JβŠ†IJ\subseteq I at time t0t_0 if the revenue bJ:=βˆ‘j∈Jbjb_J:=\sum_{j\in J}b_j for accepting them is less than the change Ξ΄J\delta_J in the expected revenue of the time t1t_1 auction under the reduced block capacity Cβˆ’βˆ‘j∈JgjC - \sum_{j\in J}g_j, given knowledge of the preconf mempool Mt0M_{t_0}.

Now, since any bid (g,v,f,b)∈Mt0(g,v,f,b)\in M_{t_0} not accepted at time t0t_0 is still in the mempool and available to accept at time t1t_1, the expected revenue from leaving any subset JβŠ†Mt0J\subseteq M_{t_0} of bids pending until the next stage is at least the revenue for accepting them. That is, bJ≀δJb_J \leq \delta_J always, and our provider will never accept any bids until the last auction tkt_k. We call this the deferring problem.

The mev-commit protocol attempts to address the deferring problem by using a strictly decreasing bid decay function b~\tilde b. The decaying payment rule means that leaving bids pending leads to lower revenue, all other things being equal, so providers do have some incentive to issue early accept messages.

Provider commitments

The deferring problem can also be addressed by providers publishing public commitments to additional rules about how they will conduct the auction. For example here are two types of mechanisms:

  • Committing to a particular auction format with reservation price protects bidders from ex-post increases in the provider's withholding value.

  • Committing to timely issuance of credible "reject" messages gives bids a de facto expiry, eliminating the possibility of deferral. For rejection messages to be credible, the provider must be held accountable for equivocation, that is, issuing both accept and reject messages for the same bid.

Commitment mechanisms can be naturally implemented as smart contracts on mev-commit chain, or more generally any low-latency blockchain endowed with an oracle yielding a commitment to the latest state of the target chain. Alternatively, providers can pursue traditional methods such as announcing their auction parameters on a website and establishing a reputation that makes such an announcement credible.

Provider information

As time progresses, the provider has an increasingly informative view of the cumulative preconf mempool ⋃t<TMt\bigcup_{t<T}M_t. Since bidders behave strategically, the bids that arrive in this mempool reflect their private information, whether it be idiosyncratic preferences or specialist insight into opportunities that affect the floor effective gas price of the target block.¹⁰

Providers use this information to incrementally update their forecasts of future bidder behaviour and hence update their decision parameters as time goes on.

This behaviour can be compared to an auctioneer in an ascending bid (English) auction who "lift-lines" reservation prices, successively raising a secret reserve price as new bids come in and reveal more information about the distribution of bidder types.ΒΉΒΉ This bid flow generally consists of noisy and informative components. By analogy with the case of market making in a continuous double auction,ΒΉΒ² we anticipate that to maximize their profit from noise flow, providers may need to accept certain losses to informed agents β€” such as a specialist searcher who expects a gas fee surge and locks in a low gas price early with a preconf β€” so as to incentivise the latter to reveal their private information by bidding early.

Risk

So far, we have discussed matters that affect risk neutral bidders. However, perhaps the major factor influencing preconf prices as a function of spot fees is the demand-side appetite for risk. There are several sources of risk that feed into the computation of a bidder's estimate of the preconf premium.

Inclusion risk

Transactors manage a tradeoff between their preconf bid and the probability of inclusion. Risk-averse bidders may have a preference for high inclusion probability that weighs against EV maximisation. One simple way to incorporate this into a bidding strategy is to optimise with a chance constraint such as P[inclusion]β‰₯0.99\mathbb{P}[\text{inclusion}]\geq 0.99. For bidders who make regular transactions such as rollup operators, this chance constraint translates to a liveness objective: inclusion into at least 0.99n0.99n blocks in each n-block window.

The existence of preconf auctions that consistently sell off blockspace early would, by reducing block capacity, put upward pressure on the minimum bid needed in the spot auction to satisfy the chance constraint. Thus in the presence of such auctions, reliability optimising bidders may be pressured to bid earlier. Meanwhile, bidders unwilling to pay the full preconf premium are left competing for a reduced set of spot opportunities.

Fee risk

That is, risk of the bid amount necessary to secure inclusion.

Fees are positive and their distributions may be highly skewed because of "surge pricing" due to congestion in the target block. To put it glibly, they tend to have more upside than downside. Bidders may therefore wish to take higher moments of the fee distribution into consideration in their risk model to account for this.

Under independent increments models of the fee process, fee variance, and hence the risk-adjusted value of preconfirmation, decreases as the preconfirmation time approaches zero. Hence, it is reasonable to expect fee volatility to correlate positively with the preconf premium. However, because high fees are an indicator of congestion that can lead to an increased backlog and hence further high fees, independent increments may not be a realistic assumption.

Typical examples of signals that contribute to fee risk are the arrivals process in the public mempool, the rate of inclusion into blocks, and the EIP-1559 base fee.

Execution outcome risk

Definition. Execution outcome risk is the risk of adverse market movements impacting the execution value of the target transactions in the period between transaction creation and execution, assuming execution is successful.

This is closely analogous to the notion of β€œexecution risk” used in the context of trading in TradFi; compare, for example, the definition found in the 2023 CFA Program Curriculum (Level III, Vol. 5, p.125). We add the infix β€œoutcome” here to distinguish this concept from the DeFi-specific risk of a transaction failing to execute at all, either because it is not included or because it is included and reverts.

Preconf bidders incur execution outcome risk because they must commit to transaction parameters at the time they bid for the preconfirmation, which may be some time before execution is realised. Under independent increments assumptions, the variance of execution value computed at time tt, and hence the risk incurred by bidding for a preconfirmation, decreases as tβ†’0βˆ’t\rightarrow 0_-. That is, execution outcome risk contributes negatively to the preconf premium.

Tradeoffs

A risk-averse bidder must balance increased execution risk (computed at bid time) with reduced fee risk (computed at acceptance time), possibly with a chance constraint on inclusion probability, when computing optimal bid amounts. This tradeoff looks most attractive when the gap between bid and response time is predictable and short.

The private cost to an individual buyer of these risks depends on their model for the future evolution of the auction, particularly of the behaviour of the auctioneer and other future buyers, their model of the other signals that feed into the realised value of their transaction, and of their approach to risk parameters.

Conclusion

We discussed phenomena likely to arise in preconf auctions with bidders competing for execution of bundles and single transactions. We found that for preconf auctions to attract early bids, providers may choose to commit to publicly announced auction rules. Moreover, the price discovery entailed by early bids can be used to dynamically update auction parameters to increase expected revenue. Committing to a reserve can also encourage higher bidding, hence improve expected revenue.

The premium paid for early confirmation is a function of opposing fee and censorship versus execution outcome risks, as well as the value of specialist information. In particular, it may be positive or negative depending on the relative weights the market assigns these factors.

We finalize our conclusion here by listing a few interesting avenues for future research:

Dynamic auction decision rules. We discussed a general framework for dynamic preconf but didn't go into detail on any specific auction model. A natural next course for study would be to analyse a simple dynamic mechanism β€” for example, periodic greedy knapsack auctions with reserve β€” and derive results about optimal parameters β€” in this example, time tt capacity and reserve β€” and bidding strategies. The choice of mechanism will naturally be informed by the auction rules we observe preconf providers implementing in practice.

Measuring and interpreting the preconf premium. We have discussed some types of risk that theoretically ought to contribute to the preconf premium. It would be interesting to precisely define empirical measures of the premium and these risks and quantify their contribution in the wild.

Dynamic bidder strategies. In this article we mainly considered single-minded bidders who want a single transaction or bundle included in a target block. But there may be interesting applications of preconfs arise if we drop this requirement and consider multi-stage strategies that require the execution of a sequence of transactions, each preconfirmed separately. If such strategies are deployed in practice, even risk-neutral agents may pay a premium for preconfirmation.

These applications become more powerful if ordering guarantees are added to the preconf protocol, so that, for example, preconfirmed transactions in the same block are executed in the order of the timestamps on their preconfs. Ordering features are expected to be added to mev-commit in the future.

Another strategy available to multi-transaction bidders arises when participating in the preconf auction leaks some information about the types of the other participants. For example, highly sophisticated bidders could try to extract information about the provider's secret reservation price by "dusting" the auction with early bids for small transactions.

Acknowledgments

Thanks to Murat, Evan, Christian, and the rest of the Primev team for comments and feedback on this article. This work was commissioned by Primev.

Bibliography

  • Akbarpour, M., & Li, S. (2019). Credible Mechanisms. https://web.stanford.edu/~mohamwad/CredibleMechanisms.pdf

  • Bagehot, W. (1971). The only game in town. Financial Analysts Journal, 27(2), 12-14.

  • Drake, J. (Nov. 2023). Based preconfirmations. https://ethresear.ch/t/based-preconfirmations/17353

  • Graham, D. A., Marshall, R. C., & Richard, J. F. (1990). Phantom bidding against heterogeneous bidders. Economics Letters, 32(1), 13-17.

  • Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica: Journal of the Econometric Society, 1315-1335.

  • Levin, D., & Smith, J. L. (1996). Optimal reservation prices in auctions. The Economic Journal, 106(438), 1271-1283.

  • Milgrom, P., & Segal, I. (2017). Deferred-acceptance clock auctions and radio spectrum reallocation. Journal of Political Economy.

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  1. (Drake, Nov. 2023).

  2. Compare https://mirror.xyz/preconf.eth/sgcuSbd1jgaRXj9odSJW-_OlWIg6jcDREw1hUJnXtgI.

  3. https://github.com/primevprotocol/mev-commit

  4. Under PBS, block builders may also issue a conditional commitment to include a transaction if they win the auction for the target slot. Under mev-commit protocol, payments are only made if the condition is satisfied, so this should not affect pricing for risk-neutral participants.

  5. https://docs.primev.xyz/concepts/mev-commit-chain/introduction

  6. The same logic applies to the "blob" gas fees introduced in Ethereum's recent "Dencun" update: the max_fee_per_blob_gas bid field can be eliminated from bid parameter space by fixing it equal to the blob base fee, if the latter is known ahead of time.

  7. Because settlement occurs at the same time regardless of when the bid is accepted, time preference effects do not intervene in this comparison.

  8. (Akbarpour and Li, 2018).

  9. This is a well-known principle from mechanism design. See, for example, Levin & Smith (1996).

  10. Compare (Milgrom and Weber, 1982).

  11. (Graham, 1990)

  12. (Bagehot, 1971)

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