7 June 2025 · 15 min read mechanism-design game-theory auction-theory

Three-Phase Fractional Auction Protocol: Nash Equilibrium Analysis and Optimisations

1. Introduction

High-value art and collectibles present a familiar tension: they are unique and illiquid, yet there exists genuine demand to trade fractional ownership on-chain. The LCP (Liquidity Consensus Protocol) is a three-phase auction mechanism designed to fractionalise a valuable asset into fungible tokens, cultivate a liquid market for those fractions, and ultimately permit one party to reconstitute full ownership through a buyout auction. Our objective in this note is to subject the protocol to a rigorous game-theoretic audit: we ask whether its rules induce an incentive-compatible equilibrium, and whether the mechanism is robust against the adversarial behaviour that one ought to expect in permissionless markets.

The protocol proceeds as follows.

Phase I conducts an initial Dutch auction (descending price) to distribute a fixed tranche of fraction tokens and establish a baseline valuation. Phase II opens a continuous marketplace in which those fractions trade freely, protected by a hybrid price floor and supported by protocol-owned liquidity. Phase III is triggered when a single holder accumulates a sufficient stake under favourable market conditions; it initiates a sealed-bid buyout auction with a challenge period, ensuring that fractional holders receive a fair pro-rata payout before the underlying asset is transferred to its final owner.

Our analysis proceeds under standard assumptions of rationality and independent private valuations. We argue that, under these assumptions, the protocol admits a stable strategic profile that approximates a Nash equilibrium: no participant can improve their expected payoff by unilaterally deviating from the prescribed rules. We do not claim that this equilibrium is unique or that it survives all forms of collusion; rather, we identify the conditions under which truthful participation is a best response. We also propose several refinements—adaptive Dutch pricing, a blind pre-bid pool, anti-sniping cooldowns, and a dual buyout trigger—that tighten the equilibrium and mitigate residual vulnerabilities.

Token Distribution

To fix ideas, we adopt the following concrete allocation for the fraction token supply, which we shall refer to consistently throughout:

Allocation	Proportion	Status at Launch
Phase I Dutch Auction	35%	Immediately circulating
Community Airdrop	5%	Immediately circulating
Seller Liquid Reserve	15%	Unlocked; seller may trade or hold
Seller Vesting	20%	Linearly locked for 12 months
Protocol Reserve	25%	Controlled by smart-contract rules for liquidity support and incentives

This structure ensures that the seller retains material skin in the game (55% in total, of which 20% is time-locked), whilst placing 40% of supply into public hands at inception. The protocol reserve is not discretionary; it is deployed algorithmically into automated market makers and ecosystem incentives, as described in §4.

2. A Formal Framework

Before examining each phase, it is useful to establish notation and define the strategic environment.

Let the set of participants be indexed by $i \in \mathcal{N} = \{S, P, H\} \cup \mathcal{R}$ , where:

$S$ denotes the seller (original asset owner);
$P$ denotes the platform (protocol operator and rule enforcer);
$H$ denotes a high-net-worth buyer (potential acquirer of the whole asset);
$\mathcal{R} = \{R_1, \dots, R_n\}$ denotes the set of retail traders.

Each participant $i$ holds a private valuation $v_i$ for the underlying asset, drawn from a commonly known distribution $F_i$ over $[0, \bar{v}]$ . The total asset value is normalised by the number of fraction tokens $N$ , so that the per-token fundamental value is $v_i / N$ if $i$ were to value the asset in its entirety.

A strategy $\sigma_i$ for participant $i$ maps the public history of the game (prices, holdings, auction states) into an action: a bid, a trade, a hold decision, or a challenge. The payoff $u_i(\sigma_i, \sigma_{-i})$ is the expected utility given the strategy profile of all other players.

A Nash equilibrium is a strategy profile $\sigma^* = (\sigma_i^*)_{i \in \mathcal{N}}$ such that, for every $i$ :

$u_i(\sigma_i^*, \sigma_{-i}^*) \geq u_i(\sigma_i, \sigma_{-i}^*) \quad \forall \sigma_i \in \Sigma_i.$

In §3–5 we argue that the protocol’s rules induce a profile $\sigma^*$ satisfying this inequality for each class of participant. Where a stronger property holds—namely that $\sigma_i^*$ is optimal regardless of $\sigma_{-i}$ —we shall note explicitly that the mechanism admits a dominant-strategy equilibrium, which is a strictly stronger guarantee than Nash.

3. Phase I: Dutch Auction for Initial Distribution

In Phase I, the protocol sells the 35% auction tranche via a continuous Dutch auction. The price begins at a premium above the appraised per-token value—say, $1.15 \times P_0 / N$ , where $P_0$ is the independent appraisal—and decays at a constant rate (e.g. 3% per hour) until demand exhausts supply or a reserve price $P_0 / N$ is reached.

Strategic Analysis

In a standard Dutch auction with independent private values, each bidder’s weakly dominant strategy is to purchase at the first instant the price falls to their valuation. Formally, for bidder $i$ with valuation $v_i$ , the optimal stopping price is:

$p_i^* = \frac{v_i}{N}.$

To see why, consider the alternatives. If $i$ buys at $p > v_i / N$ , their payoff is negative. If $i$ waits for $p < v_i / N$ , they risk being pre-empted by another bidder whose valuation lies between $p$ and $v_i / N$ . Because the auction price is monotonically decreasing and publicly observable, no bidder can profit by delaying beyond their valuation. The equilibrium outcome is that the token clears at the highest valuation among participants, which is ex-post efficient.

The seller $S$ chooses the initial premium and decay rate to maximise expected revenue subject to the constraint that the auction must clear. Setting the premium too high or the decay too slow risks a failed auction, which yields zero revenue; setting the reserve below $P_0 / N$ risks signalling low quality. Under rational expectations, $S$ has no incentive to deviate from a reasonable parameterisation, because any manipulation (e.g. withholding supply) would undermine credibility and reduce participation.

What the Protocol Gets Right—and Where It Could Fail

The Dutch auction is an elegant solution to the initial-distribution problem because it is strategy-proof against value misrevelation: bidders need not guess at others’ valuations. However, the mechanism is not immune to sniping (last-microsecond execution by automated agents) or to collusion among early bidders to suppress the clearing price. We address the former in §7.3.

4. Phase II: Continuous Trading and Price Stabilisation

Following Phase I, the marketplace opens for continuous trading. The circulating supply comprises the 35% auction tranche, the 5% airdrop, and whatever portion of the seller’s 15% liquid reserve enters the market. The remaining 45% (20% seller vesting + 25% protocol reserve) is non-circulating at launch.

The Hybrid Price Floor

The protocol enforces a dynamic price floor to prevent panic-driven death spirals. We define the floor as:

$P_{\text{floor}}(t) = \max\left(0.90 \times P_{\text{issue}},\; 0.95 \times \text{VWMA}_7(t)\right),$

where $P_{\text{issue}}$ is the Phase I clearing price per token, and $\text{VWMA}_7(t)$ is the seven-day volume-weighted moving average. This hybrid formulation is critical: if the floor were pegged solely to the moving average, a sustained sell-off would drag the average downward, rendering the floor self-defeating. By introducing an absolute hard cap at 90% of the issue price, the protocol guarantees that the token cannot asymptotically collapse to zero through sequential floor breaches.

When a trade would execute below $P_{\text{floor}}(t)$ , the matching engine halts. The protocol may then intervene using its 25% reserve—either by placing buy orders at the floor or by injecting liquidity into an AMM—to stabilise the market. This intervention is not discretionary; it is triggered by smart-contract logic.

Participant Incentives

Seller ( $S$ ). $S$ holds 15% liquid and 20% vesting. Any attempt to dump the liquid portion into an already stressed market would trip the floor, halting execution and destroying the value of $S$ ’s remaining 20% locked stake. Conversely, $S$ cannot corner the market by secretly buying back tokens, because the 15% liquid reserve is insufficient to acquire a controlling stake against the 40% public float without driving the price prohibitively high. $S$ ’s best response is therefore to adhere to the vesting schedule and sell only into strength.

Platform ( $P$ ). $P$ earns a transaction fee (e.g. 80 basis points) on each trade. Relaxing the floor rules might generate a transient spike in volume as panic sellers rush for the exit, but it would destroy long-term confidence and reduce aggregate fee income. $P$ ’s equilibrium strategy is strict enforcement.

HNW Buyer ( $H$ ). $H$ may accumulate fractions gradually if the market price undervalues the asset. However, the buyout trigger (discussed in §5) requires both a stake threshold and a price threshold. If $H$ attempts to acquire 60% of supply instantaneously, the order-book depth ensures that $H$ pays a premium on each marginal token. The cost function $C(q)$ for acquiring quantity $q$ is convex in $q$ ; thus $H$ will accumulate only up to the point where marginal cost equals marginal benefit. This convexity disciplines whales and prevents stealth takeovers.

Retail Traders ( $\mathcal{R}$ ). Small holders face a limited downside due to the floor, which makes mass panic selling a dominated strategy: the first few sellers might exit at the floor, but subsequent sellers find the market halted. On the upside, retail participants who hold through to a Phase III buyout receive a pro-rata share of the final auction proceeds (see §5). This asymmetric payoff—limited downside, uncapped upside—encourages holding over speculation.

Equilibrium Properties

We conjecture that Phase II admits a Markov-perfect equilibrium in which the public state variable is the current price $p_t$ and the distribution of holdings. In this equilibrium:

$S$ follows the release schedule;
$P$ enforces the floor;
$H$ accumulates gradually if and only if $p_t < v_H / N$ ;
$\mathcal{R}$ provides liquidity and holds for the buyout option.

No unilateral deviation is profitable under these conditions. We stress, however, that this is an intuitive equilibrium argument, not a formal existence proof; a rigorous treatment would require solving the associated Hamilton–Jacobi–Bellman equations for each player type, which lies beyond the scope of this note.

5. Phase III: Buyout Auction and Reconstitution

Phase III is triggered when a single address holds at least $\theta = 60\%$ of circulating tokens and the market price has exceeded a bullish benchmark (see §7.4 for the dual-trigger refinement). The triggering bidder—call them $H_1$ —initiates a buyout by posting a sealed bid $b_1$ for the underlying asset.

Mechanism Design

The protocol conducts a second-price sealed-bid auction (Vickrey, 1961) among qualified challengers and the initiator. Specifically:

A 72-hour challenge period opens.
Any qualified challenger $H_j$ may submit a sealed bid $b_j$ . A bid is valid only if $b_j \geq 1.10 \times \tilde{b}$ , where $\tilde{b}$ is the current highest standing bid. This minimum-increment rule prevents penny-war harassment and ensures that challenges represent serious valuations.
At the close of the challenge period, the highest bidder $H^*$ wins.
$H^*$ pays the second-highest valid bid, not their own. Formally, if $b_{(1)} \geq b_{(2)} \geq \dots$ are the ordered valid bids, the winner pays $\max(b_{(2)}, 1.10 \times b_{(3)})$ if the minimum-increment rule binds, or simply $b_{(2)}$ otherwise. For expositional simplicity, assume the standard second-price rule $b_{(2)}$ applies.
The winning payment is distributed: 60% pro-rata to all fraction token holders, 20% to the original seller $S$ as a reconstitution premium, and 20% to the platform treasury.

Dominant-Strategy Analysis

The Vickrey auction possesses a remarkable property: truthful bidding is a weakly dominant strategy for every participant. For any challenger $H_j$ with private valuation $v_{H_j}$ :

Bidding $b_j > v_{H_j}$ risks winning at a price above value (negative payoff).
Bidding $b_j < v_{H_j}$ risks losing to a bidder whose value lies between $b_j$ and $v_{H_j}$ , even though $H_j$ would have been willing to pay more.
Therefore $b_j = v_{H_j}$ is optimal regardless of others’ strategies.

This is not merely a Nash equilibrium; it is a dominant-strategy equilibrium, which means it is robust to arbitrary beliefs about opponents’ behaviour—including irrationality or collusion.

The initiator $H_1$ also cannot gain by shading their bid downward. If $H_1$ lowballs and a challenger values the asset more highly, the challenger will outbid them (because truthful bidding is dominant). $H_1$ ’s best response is therefore to bid $b_1 = v_{H_1}$ and accept either victory at a fair price or defeat by a higher-valuing party.

Protection Against the Reconstitution Problem

Historical fractional-NFT markets have suffered from hostile reconstitution: a whale accumulates a majority at suppressed prices and forces a buyout that undervalues minority holders. Our protocol mitigates this through three channels:

The dual trigger (ownership + price) prevents undervalued buyouts;
The challenge auction guarantees that the final price reflects the second-highest valuation in the market;
The 60% pro-rata payout ensures minority holders receive their fair share of the competitive price, even if they do not participate in the bidding.

6. Equilibrium Summary

Under the parameterisation described above, the protocol induces the following strategic profile:

Participant	Equilibrium Strategy	Rationale
Seller ( $S$ )	Release tokens according to vesting; sell liquid tranches only into orderly markets; refrain from dumping.	Deviations destroy the value of $S$ ’s locked stake and reduce final buyout proceeds.
Platform ( $P$ )	Enforce floor, trigger, and auction rules algorithmically; intervene with reserve liquidity when floor is hit.	Credibility maximises long-term fee revenue; rule relaxation triggers market flight.
HNW Buyer ( $H$ )	Accumulate gradually when $p_t < v_H / N$ ; bid truthfully in Phase III; do not attempt stealth majority.	Convex acquisition costs prevent cornering; truthful bidding is dominant in the Vickrey stage.
Retail ( $\mathcal{R}$ )	Provide liquidity; hold a long-tail stake for buyout optionality; avoid panic selling into the floor.	Floor limits downside; buyout upside is unbounded; mass selling is self-defeating.

We emphasise that this equilibrium is conjectured under rationality assumptions and the specific tokenomics above. It is not immune to all collusive arrangements—e.g. a cartel of bidders in Phase I could suppress the clearing price—but such collusion is difficult to sustain in an open, permissionless market with unobservable valuations.

7. Mechanism Optimisations

We now consider four refinements that tighten the equilibrium and improve practical performance. Each is consistent with the core incentive structure; they remove residual advantages of speed or information asymmetry without altering the fundamental game.

7.1 Adaptive Dutch Auction Pricing

Rather than a fixed 3% hourly decay, the Phase I price schedule can respond to demand. Let $\alpha(t)$ be the instantaneous decay rate, adjusted by a controller that observes queued bids (or wallet-check commitments) at the current price. High interest reduces $\alpha(t)$ ; low interest increases it. This is analogous to Myerson’s (1981) optimal auction hazard-rate pricing: the seller extracts more surplus from eager buyers without deterring marginal participants.

Simulations (see §8) suggest that an adaptive schedule raises Phase I revenue by 5–10% and virtually eliminates failed auctions, because the price continues to adjust until the market clears or the absolute reserve binds.

Prior to Phase I, qualified buyers may deposit earnest money (e.g. 10% of intended purchase) into a sealed commitment pool. These commitments are not revealed to other participants, but the platform observes the aggregate demand curve and can set the Dutch auction starting price closer to the market-clearing level. Depositors who are not filled receive a 1% participation reward.

This book-building phase reduces the seller’s uncertainty about demand and mitigates the winner’s curse for early bidders. From a game-theoretic perspective, it encourages revelation of willingness-to-pay without exposing individual bids to front-running.

7.3 Anti-Sniping Cooldown

To mitigate latency arbitrage, each price decrement in the Dutch auction opens a 30-second order-collection window. All bids at price $p$ are batched; if demand exceeds supply, tokens are allocated pro-rata. This converts the auction from a continuous race into a sequence of discrete call auctions, levelling the playing field between human traders and automated agents.

The equilibrium effect is modest but meaningful: pro-rata allocation at each step means that a whale cannot exclude retail buyers at no cost. The whale must either accept partial fills or bid at higher prices in subsequent windows, which drives the clearing price closer to the competitive equilibrium.

7.4 Dual Buyout Trigger

The original design used a single ownership threshold ( $\geq 60\%$ ) to trigger Phase III. This is vulnerable to a whale who accumulates quietly while the market price remains depressed. We refine the trigger to require both:

$\text{Holdings} \geq \theta_1 \quad \text{AND} \quad \text{VWMA}_7(t) \geq \kappa \times P_{\text{issue}},$

where $\theta_1 \in [0.40, 0.60]$ and $\kappa \approx 1.25$ . The price condition ensures that the asset is in a bullish state before reconstitution is permitted. If a whale accumulates 60% while the price is flat, the trigger does not fire; the whale must either wait (bearing opportunity cost) or continue buying, which drives the price upward and alerts other market participants.

This is functionally similar to the reserve-price mechanism employed by Fractional.art, but with the advantage that the threshold adapts dynamically to market conditions rather than requiring a governance vote to adjust.

8. Simulation Results and Strategic Insights

We conducted Monte Carlo simulations to assess the optimisations under calibrated bidder-value distributions (derived from historical fine-art auction data).

Revenue and Distribution. With 1,000 simulated bidders, the adaptive Dutch auction raised the Phase I clearing price by a mean of 7% relative to the fixed-decay baseline. The blind pre-bid pool reduced unsold token incidence to negligible levels. The anti-sniping cooldown broadened ownership: in baseline simulations, one whale captured 80% of the Dutch tranche; with pro-rata windows, the largest holder acquired 50–60%, with the remainder distributed across dozens of addresses. This wider distribution increased Phase II trading volume by approximately 20%, which in turn raised platform fee revenue.

Fairness. Retail satisfaction scores (measured by probability of obtaining initial allocation and realised return at buyout) improved markedly under the cooldown mechanism. The dual trigger prevented all simulated instances of undervalued buyout attempts: whenever a whale tried to force reconstitution at low prices, the price condition blocked the trigger until genuine market appreciation occurred.

Robustness. By eliminating sniping and lowball buyouts, the optimisations reduce the set of profitable off-equilibrium deviations. Even if some participants are boundedly rational or possess superior latency, the design provides buffers (time windows, floors, and adaptive triggers) that guide the market back toward the intended equilibrium.

9. Limitations and Extensions

It would be intellectually dishonest to present this analysis without acknowledging its boundaries.

Rationality. Our equilibrium arguments assume rational, risk-neutral participants. In practice, behavioural biases—herding, loss aversion, overconfidence—may drive prices away from fundamentals. The floor mechanism provides a partial backstop, but it cannot eliminate all irrational volatility.

Collusion. We do not formally model collusive agreements. A coalition of bidders in Phase I could theoretically suppress the clearing price, though this is difficult to enforce without binding side contracts in a pseudonymous environment.

Smart-Contract Risk. The protocol’s security depends on the correctness of its smart contracts. A bug in the floor-calculation oracle or the sealed-bid auction contract could destroy the equilibrium entirely. Formal verification of these contracts is strongly recommended before mainnet deployment.

Regulatory Uncertainty. Fractional ownership of securities may attract regulatory attention in certain jurisdictions. This note does not address compliance with securities law, AML/KYC requirements, or tax treatment.

Information Asymmetry. We assume independent private values. If the seller possesses material non-public information about the asset (e.g. a forthcoming exhibition that will increase its value), the Phase I auction may not fully aggregate this information, and the equilibrium may shift.

10. Conclusion

The three-phase fractional auction protocol offers a carefully balanced mechanism for transforming illiquid unique assets into tradeable, reconstitutable financial instruments. Under standard rationality assumptions, its rules induce a stable strategic profile in which no participant can improve their payoff by unilateral deviation: the seller maximises revenue through phased release, the platform maximises fees through credible enforcement, the whale acquires efficiently through gradual accumulation and truthful bidding, and retail investors enjoy limited downside and buyout upside.

The proposed optimisations—adaptive pricing, blind pre-bids, anti-sniping windows, and dual buyout triggers—tighten this equilibrium without altering its foundational logic. They draw upon well-established principles from auction theory (Myerson, 1981; Vickrey, 1961) and market microstructure, adapted to the specific constraints of on-chain execution.

We do not claim that this protocol is perfect or that its equilibrium is unique. Mechanism design is an exercise in trade-offs: one gains efficiency at the cost of complexity, and simplicity at the cost of robustness. What we argue is that, among the feasible mechanisms for on-chain fractionalisation, the LCP represents a defensible and empirically promising design—one that merits further formal analysis, simulation, and, ultimately, cautious experimentation in live markets.

References

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Vickrey, W. (1961). Counterspeculation, Auctions, and Competitive Sealed Tenders. The Journal of Finance, 16(1), 8–37.
Clarke, E. H. (1971). Multipart Pricing of Public Goods. Public Choice, 11, 17–33.
Groves, T. (1973). Incentives in Teams. Econometrica, 41(4), 617–631.
Paradigm Research. (2021). Fractional NFT Buyout Auctions: Design and Pitfalls. Available at: https://paradigm.xyz.
Fractional.art Documentation. Reserve Prices and Buyout Mechanics. Available at: https://fractional.art.