SNARE: A TRAP for Rational Players to Solve Byzantine Consensus in the 5f+1 Model

Abstract

The TRAP protocol solves rational agreement by combining accountable consensus with a one-shot BFTCR finalization phase. We present SNARE (Scalable Nash Agreement via Reward and Exclusion), the adaptation of TRAP to $n=5f{+}1$, and prove $ε$-$(k,t)$-robustness for rational agreement tolerating coalitions up to ${\approx}73\%$ with deposits under $0.5\%$ of the gain. A central finding is that appending a single all-to-all broadcast round with the $4f{+}1$ threshold after predecisions yields $ε$-$(k,t)$-robustness for coalitions up to $3f$ (${\approx}60\%$) without any deposit: we need not model or know the utility function of deviating players, only that they participate in the protocol. These players can be \emph{deceitful} (arbitrary unknown utility), not just rational, and the finalization structure prevents disagreement regardless of their motivation. This observation is protocol-agnostic, applies to any $5f{+}1$ protocol at the cost of one message delay that runs concurrently with the next view, and does not require commit-reveal mechanisms. Above $60\%$, the full baiting mechanism with deposits under $0.5\%$ extends tolerance to ${\approx}73\%$. A second finding is that valid-candidacy, the property preventing reward front-running, holds unconditionally regardless of the quorum threshold, removing both the $n>2(k{+}t)$ and $n>\frac{3}{2}k{+}3t$ constraints from the original TRAP. This retroactively extends the $3f{+}1$ bound from $C<n/2$ to $C<5n/9$. The binding constraint in both models is the winner consensus operating on $2f$ residual players after excluding $3f{+}1$ detected equivocators. We explore avenues for relaxing this limit.

Figures (4)

• Figure 1: Regime comparison. Top:SNARE (this work). The no-fork regime ($\mathcal{L}{=}0$) spans $20\%$ to $60\%$; the fork regimes extend to $74\%$ (WC limit). Bottom:$3f{+}1$TRAPTRAP22 with the retroactive valid-candidacy fix (Lemma \ref{['lem:vc']}). No no-fork regime exists (predecision and finalization thresholds coincide at $33\%$); the original bound was $50\%$, extended to $56\%$ by the VC fix. Gray zones require a more fault-tolerant winner selection mechanism.
• Figure 2: Minimum coalition $C_{\textsf{fin}}(a)/n$ for $a$ branches. Dashed: quorum thresholds. Dotted: winner consensus (WC) limits. The $3f{+}1$ WC limit at $5n/9\approx 56\%$ is the retroactive improvement from Lemma \ref{['lem:vc']}; the original bound was $n/2$ (dash-dotted).
• Figure 3: Feasible $(t/n, k/n)$ pairs. The $5f{+}1$ no-fork regime (green, $\mathcal{L}{=}0$) covers $t$ up to $20\%$ for coalitions up to $60\%$. The fork regimes (blue) extend beyond $60\%$ with the same $t\leq 20\%$ Byzantine tolerance as the no-fork regime: the constraint $t<2f/3$ previously stated is redundant because the residual Byzantine count is bounded by $C'<2f/3$ regardless of $t$ (see Lemma \ref{['lem:wc']}). The $3f{+}1$ model (red) tolerates higher $t$ but lower total coalition.
• Figure 4: Required deposit as a fraction of the maximum gain per block for $n{=}101$, $f{=}20$, $t{=}20$ (worst case). The step pattern reflects the discrete baiting threshold $m$. The jump at $C_{\textsf{fin}}(3){=}71$ is due to both the increased gain factor ($2\mathcal{G}$ vs $\mathcal{G}$) and the larger $m$.

Theorems & Definitions (28)

• Definition 1: $\epsilon$-$(k,t)$-robustness ADGH06TRAP22
• Definition 2: Rational agreement TRAP22
• Definition 3: Punishment strategy ADGH06TRAP22
• Definition 4: Baiting strategy TRAP22
• Definition 5: Valid candidate
• Lemma 6: Valid-candidacy
• proof
• Remark 7
• Lemma 8: Winner consensus feasibility
• proof