In this post I want to formalize a piece of folk knowledge which I think is well-known among many searchers, builders and other participants of the MEV ecosystem: Open bidding auctions with a random deadline and few participants tend to give those participants more payoff than a sealed auction. On a high level, the logic is straightforward. Uncertainty about the auction deadline (in the case of PoS Ethereum through heterogeneous proposer behavior (timing games) and locations) creates for equally fast bidders with equal value a quasi-lottery: Bidders can outbid each other in small increments and whoever manages to place the last bid before the auction ends wins. This is preferable for them to bidding their full value away in a sealed auction (with certain deadline).
The idea dates back to Ethereum PoW times where randomness was inherent in block times. The OG Flashboys 2.0 paper, for example, documents this behaviour, and explains how it can be obtained as a collusive outcome through repeated interactions and threats to deviate.
What I think hasn’t been shown formally is that explicit coordination, collusion or even repetition is unnecessary for this phenomenon. Instead we can get the outcome as an explicit equilibrium of the stage game. This is not to say that coordination and repetition are not helpful to find the equilibrium or implement it in a robust way: as we will see there are many other equilibria of the stage game so that bidders need to coordinate on the equilibrium that gives them the highest rents. However, establishing that this is an equilibrium on its own shows that once bidders have coordinated on it, it is inherently robust. More generally, it gives a rationale for many patterns that might be otherwise attributed to different factors: latency arms races, as documented for example in DataAlways’ work, can to a large extent be driven by “pennying” equilibrium dynamics. Bidders update bids very frequently (and invest in their ability to do so) to increase their chances of winning the quasi-lottery. Bids are often not revised because of new information (e.g. more valuable blocks that can be built), but as a pure artifact of equilibrium play. In particular, I think being aware of this bidding dynamic is informative for discussions around ePBS and sealed bidding, as I will expand on at the very end.
A model
The following model is stylized and there are some obviously unrealistic assumptions, but I think it is the most natural starting point. I will comment on limitations and practically relevant concerns later.
Suppose two risk-neutral bidders have the same, commonly known value v for winning an auction. The auction ends at a random time T, exponentially distributed with rate \mu>0. Thus, conditional on the auction still being open, its remaining duration always has the same distribution. This is the memoryless assumption.
Each bidder gets opportunities to revise their bid according to an independent Poisson process with rate \lambda>0. These processes are also independent of the auction’s ending time. The parameter \lambda captures how quickly bidders can react. At each opportunity, a bidder can submit a bid or do nothing. Submitted bids become visible and effective immediately; the restriction on revision opportunities captures the friction in responding.[1]
The minimum opening bid is r\geq 0, and bids lie on the grid
where \varepsilon>0 is the bid tick and M:=(v-r)/\varepsilon\geq 2 is an integer. A bidder may jump by any number of ticks. Bids cannot be withdrawn or decreased. This last restriction is a simplifying assumption. Submitting and revising bids is costless.
At time T, the highest standing bid wins and pays its bid. The winner’s payoff is v-b, and the loser’s payoff is zero. Equal bids do not displace the incumbent. The auction starts without a bid; if it ends before anyone bids, both bidders receive zero. Both bidders are present and able to participate from the start.
The pennying equilibrium
There is a symmetric stationary Markov perfect equilibrium with the following strategies:
- If no bid has been submitted, bid r at the next revision opportunity.
- If currently winning, keep the standing bid.
- If currently losing and the highest bid is below v, outbid it by exactly \varepsilon at the next revision opportunity.
- Once the highest bid is v, do nothing.
These strategies apply after every history, including histories in which someone has jumped by more than one tick. At a standing bid of v-\varepsilon, the losing bidder is indifferent between bidding v and doing nothing: either action gives them zero. This equilibrium selects bidding v. That boundary choice matters for the other equilibria below.
The important point is that pennying is a best response even though larger jumps are available. To see this, write the current highest bid as b=v-k\varepsilon, and let L_k and F_k be the continuation payoffs of the leader and follower, respectively. Define
Under the proposed strategies, the next event that changes the outcome is either the auction ending or the follower getting a revision opportunity. The latter happens first with probability q. A revision opportunity for the leader changes nothing. Consequently,
and, for k\geq 1,
The first equation says that the leader either wins the auction at the current bid or becomes the follower after being outbid. The second says that the follower either loses when the auction ends or becomes the leader after raising the bid.
Combining the equations gives
The sequence L_k is strictly increasing. Intuitively, being the leader at a lower price is more valuable, even after accounting for all subsequent bidding.
Now consider a follower’s revision opportunity at state k. A bid of v-j\varepsilon, where 0\leq j<k, makes them the leader with continuation payoff L_j. Since L_j is increasing in j, the best bid is the smallest possible increase: j=k-1. Passing instead gives F_k=qL_{k-1}, which is strictly less than L_{k-1} whenever k\geq 2. At k=1, both choices give zero. Similarly, the leader cannot benefit from raising their own bid, since that replaces L_k with a lower L_j.
It remains to check the opening bid of the auction. Let U be each bidder’s continuation payoff before any bid has been submitted. Then
Thus, at an opportunity to open, bidding r is better than passing; opening at a higher price gives a lower L_j. These comparisons verify optimality at every state, including against larger jumps and against passing an opportunity. Since payoffs are bounded and the auction has a strictly positive stopping rate, the corresponding discounted dynamic programming problem also rules out deviations involving an entire alternative sequence of bids. The strategies therefore form a Markov perfect equilibrium.
Equilibrium payoffs
Once the first bid has landed, each subsequent increase flips the identity of the leader. The auction’s random ending time determines which bidder places the last bid.
Conditional on the first bid being submitted before the auction ends, let N be the number of subsequent increases. For n=1,\ldots,M,
Therefore,
The expected winning payment is
and the bidders’ combined expected profit, conditional on a bid being submitted, is
Before either bidder has been selected to open, symmetry gives each bidder expected profit
The factor 2\lambda/(\mu+2\lambda) is the probability that someone submits a bid before the auction ends. Conditional on this event, each bidder’s expected profit, before learning who opens, is S_M/2.
This captures the quasi-lottery particularly clearly when
There can be many changes of leader, while the total increase in price remains small relative to the available surplus. The bidders then retain most of that surplus and split it equally in expectation. No repeated interaction or punishment strategy is needed.
The revision rate is exogenous here. The equilibrium describes bidding by two equally fast bidders; it does not yet solve their incentives to invest in speed. Also, bidders use their revision opportunities immediately. Waiting between opportunities is imposed by the model. Deliberately passing an opportunity despite a positive chance of missing the auction is not a feature of this equilibrium.
An equilibrium with an immediate jump
The pennying equilibrium is not unique. In the same model, suppose
There is another symmetric stationary Markov perfect equilibrium:
- If there is no bid, submit v-\varepsilon at the next opportunity.
- If the highest bid is below v-\varepsilon, submit v-\varepsilon at the next opportunity, even if already leading.
- If the highest bid is at least v-\varepsilon, do nothing.
The last rule selects abstention when the follower could bid v for zero profit. As a result, a bid of v-\varepsilon secures a profit of \varepsilon.
To verify the strategies, let \widetilde L_k and \widetilde F_k denote continuation values in this equilibrium. At a standing bid of v-\varepsilon,
At any lower standing bid, with 2\leq k\leq M, the next relevant event is the auction ending or either bidder getting an opportunity to jump. Hence
Under the stated parameter condition, \widetilde L_k\leq\varepsilon at every such state. Thus, securing \varepsilon by jumping immediately is at least as good as passing or choosing any smaller increase. This holds for both the leader and the follower. Opening at v-\varepsilon is also optimal: any lower opening gives some \widetilde L_k\leq\varepsilon, while passing gives \lambda\varepsilon/(\mu+2\lambda)<\varepsilon. Bidding v gives zero.
Here a large jump is optimal because a smaller bid leaves the bidder exposed to the rival’s next opportunity to submit v-\varepsilon. The strategies also specify credible behavior after a deviation to a lower bid, including the incumbent raising their own bid.
Conditional on a bid being submitted, combined bidder profit in this equilibrium is exactly \varepsilon. Thus, whenever both equilibria exist, pennying gives both bidders higher ex ante profit precisely when S_M>\varepsilon. Small increments need not be bidder-preferred for every parameter choice: with a fixed bid tick and sufficiently rapid revision opportunities, even pennying can dissipate almost all the surplus.
Equilibria with Intermediate increments
More generally, equilibria with larger bid increments exist that still leave room for subsequent competition. Such an equilibrium requires a larger increase to induce a sufficiently more favorable continuation of the auction: saving a few ticks now would expose the bidder to more aggressive subsequent bidding. The symmetric stationary constructions above support pennying and terminal jumps, including paths combining the two, but do not establish repeated intermediate increases. History-dependent continuation rules can support such increases within a single auction, even while one-tick deviations remain available. For example, with \lambda=5\mu and v-r=7\varepsilon, one can sustain the bid path r,\ r+2\varepsilon,\ r+4\varepsilon,\ r+5\varepsilon,\ r+6\varepsilon, after which bidding stops.[2] The distinction is that this construction uses credible responses to deviations, including different choices when a bidder is indifferent about bidding at zero profit. It therefore introduces an additional enforcement mechanism within the auction. This is different from the pennying equilibrium above, whose strategies depend only on the highest bid and who leads.
Sealed bidding and unobservable jumps
The observability of bids is important for the pennying equilibrium. In an open auction, a larger jump induces the rival to respond from a higher price at their next opportunity. It therefore makes subsequent competition more expensive without protecting the bidder against that next response. With fully sealed bids and no feedback about who is winning, the rival cannot condition their bidding on the jump. A bidder can secretly move ahead of a gradual bidding path and remain ahead through several of the rival’s revision opportunities. This changes the incentives underlying the low-price lottery: a larger bid can now buy a more persistent advantage. Uncertainty about the deadline does not remove that incentive. This does not imply that sealed bidding eliminates all rents, since a rival may fail to submit before closure and uncertainty about their standing bid can support bid shading (see also here). It does mean that the observable pennying equilibrium cannot simply be carried over to the sealed setting.
Equilibrium Selection and Preliminary Conclusion
It is important to notice what has and hasn’t been established so far: I have constructed a “pennying” equilibrium that catches, in a stylized way, bidder behaviour familiar from practice. Neither is it the only equilibrium, nor do I claim that it is the bidder-optimal equilibrium (it does generate more payoffs than other easy-to-describe equilibria though). I also did not establish that in a sealed-bid auction with random deadline, bidders wouldn’t be able to achieve similar rents through some cleverly constructed equilibrium. However, if such equilibria exist, they are probably more complicated than the simple pennying dynamic in the candle auction. Thus, the confident preliminary conclusion is that there is an attractive equilibrium to play in a candle auction, and that it is sufficiently straightforward to describe and for bidders to find and play.
Reality
The most stylized feature of the above model is the memoryless closing time. This was a natural approximation in PoW: with roughly fixed hash rate and difficulty, block discovery is approximately exponential, so waiting longer does not bring a bidder closer to the next block (as also assumed in Flash Boys 2.0). In PoS, auction closing times follow distinct patterns driven by proposer behavior, relay policies and network latency. DataAlways’ The Block Auction Infrastructure Race illustrates this: the historical network-wide sample in Figure 9 has an early peak and a later hump, while the bloXroute sample has a sharp cutoff around 800 ms into the slot. The same analysis documents a shift towards later, narrower timing distributions and later bidding. Builders are learning these patterns and can increasingly target the expected deadline. The relevant uncertainty is therefore what remains after conditioning on the particular proposer and relay.
Better deadline prediction can move competition into a shorter window before the anticipated close. Bidding early can raise the price of subsequent competition while providing little chance of actually winning before the rival responds. Waiting avoids some of these counterbids, but risks missing the action altogether. But I do not think better prediction has an obvious monotone effect on bidder rents. Fewer rounds could preserve rents, while a final round in which bids arrive too late to be observed and answered could instead resemble sealed bidding and restore the incentive to jump ahead. Establishing which effect dominates requires solving the time-dependent game, which looks like a daunting task (that real bidders try to approximate each day). What likely remains though is the general pattern: pennying as a reasonable pattern, with more complicated entry and escalation strategies overlayed on top.
Why care?
The equilibrium achieves higher rents for builders (and through pass-through for their upstream suppliers, searchers and orderflow providers) than other equilibria. Depending on who you are this might be good or bad news. As a side effect, it creates an infrastructure arms race which seems obviously wasteful, although a lot of latency optimization on which it might piggyback is due to more reasonable activities, such as faster information aggregation through last ms transaction inclusions and updates.
For a given block value, higher builder rents mean lower proposer payments. If builders manage to coordinate on an equilibrium that leaves them more surplus, that is good for them and bad for the proposer, even when no explicit agreement is needed to sustain it. Frequent bidding and intense competition over latency do not by themselves guarantee that builders are competing away their margins. Making the deadline more predictable is one possible response by proposers: it changes the timing game and may move competition towards a final round. Going sealed is another, since an unobservable jump changes the incentives supporting pennying. Neither is an unconditional revenue guarantee, for the reasons discussed above, but proposers would have a reason to explore both.
This brings us to Glamsterdam. In Trustless payments and relays, we argued that open bidding becomes much more fragile with ePBS: a builder can listen to an open bid feed and then submit its own bid through the (by default) sealed in-protocol channel. A relay trying to maintain an open auction would therefore need to restrict feed access and enforce participation rules that prevent this behavior, or rely on proposers forwarding the bids they receive. The argument here gives a reason why builders might nevertheless want to maintain open bidding, even if doing so becomes more complicated. An accessible equilibrium with high rents is valuable to them, and relays offering services to these builders have an incentive to keep their customers happy. That shared interest does not remove an individual builder’s temptation to listen openly and bid privately, but it helps explain why builders and relays might invest in arrangements that preserve the open auction. Proposers, on the other hand, may want to find a way to keep bids sealed and credibly commit not to reveal them.
The independent Poisson updating times are another strong simplification. In reality, bidders choose when to send updates, subject to the time needed to observe competing bids, prepare a response and have it accepted by the relay. The Poisson clocks are a tractable way to impose a response friction, but they do not model these delays directly and also not realistically: they allow arbitrarily short response times and make opportunities independent across bidders. A more realistic model would let bidders choose submission times subject to latency, with speed depending on infrastructure investment and location. ↩︎
Both bidders strictly prefer the one-tick equilibrium to this two-tick construction. With \lambda=5\mu and v-r=7\varepsilon, each bidder’s ex ante expected profit is approximately 1.543\varepsilon under one-tick bidding, compared with 1.311\varepsilon under the two-tick construction. This comparison concerns these two equilibria at the same parameters; it does not establish that one-tick bidding is optimal among all equilibria. ↩︎