Lookahead Routing Framework

• Lookahead Routing Framework is a dual-paradigm approach where agents predict future outcomes to balance myopic and farsighted decision-making.
• It models congestion games by employing sequential moves and backward induction to achieve equilibrium and improve network efficiency.
• In multi-model LLM systems, it leverages latent response predictions to optimize query routing, yielding notable performance gains over baseline methods.

The Lookahead Routing Framework encompasses two distinct but structurally analogous paradigms—congestion games with limited lookahead and predictive routing for LLMs. In both cases, the core principle is to model agents (players or routers) who “look ahead” into the future consequences of their actions, explicitly forecasting either the strategic decisions of subsequent agents or the latent responses of alternative computational models. This approach interpolates smoothly between myopic (greedy or query-only) and fully farsighted (subgame-perfect or output-aware) decision-making, with profound implications for equilibrium, stability, and efficiency.

1. Formal Models of Lookahead Routing

1.1 Congestion Games with Limited Lookahead

Consider a network congestion game defined by a finite player set $N = \{1, \ldots, n\}$, a directed graph $T = (V, E)$ with origin $o$ and destination $d$, and nondecreasing delay functions $d_e: \mathbb{N} \rightarrow \mathbb{R}_{\geq 0}$ on each edge $e$. Each player chooses a route $A_i \subseteq E$ from $o$ to $d$; congestion on $e$ is $x_e(A) = |\{i : e\in A_i\}|$; player $i$’s cost is

$c_i(A) = \sum_{e \in A_i} d_e(x_e(A)).$

In the $k$-lookahead framework (Groenland et al., 2018), players act sequentially in a fixed order $o$, and each agent, at her move, computes an action optimal under the assumption that the subsequent $k-1$ players will play subgame-perfectly in the $k$-player subgame induced by the current history.

The $k$-lookahead outcome $A = (A_1, \ldots, A_n)$ satisfies, for each agent $i = o^{-1}(j)$,

$$A_i \in \arg\min_{P \in \mathcal{A}_i} V_i^{(j + k - 1)}(H, P),$$

where $H = (A_1, \ldots, A_{j-1})$ and $V_i^{(j+k)}$ recursively denotes subgame value by backward induction.

Special cases:

• For $k=1$, agents simply best-respond to predecessors—classic greedy best-response.
• For $k=n$, agents play to subgame-perfect equilibrium.

1.2 Lookahead Routing in Multi-Model LLM Systems

In the multi-model LLM scenario (Huang et al., 22 Oct 2025), the framework involves a query space $X$, a response space $Y$, and a pool of $T$ candidate LLMs $\mathcal{F} = \{f_1, \ldots, f_T\}$, with each $f_t : X \rightarrow Y$. The routing policy $\pi : X \rightarrow \{1, \ldots, T\}$ determines model choice for each input query $x$ to maximize expected evaluation $s(x, f_{\pi(x)}(x))$. Unlike “query-only” routers, Lookahead explicitly predicts for each $f_t$ a compact proxy latent representation of its likely output without full inference.

This is achieved by training a joint model $F(x, t) \to \tilde{r}_t$ (latent response), and a routing head $$C(x, \tilde{r}_1, \ldots, \tilde{r}_T) \to \hat{c}_{1:T}$$, with joint loss

$$\mathcal{L} = \mathcal{L}_{\text{route}} + \lambda \mathcal{L}_{\text{resp}},$$

where $\mathcal{L}_{\text{resp}}$ encourages $\tilde{r}_t$ to capture semantic properties of the response.

2. Analysis of Lookahead, Equilibrium, and Stability

2.1 Equilibrium Notions in Limited Lookahead Congestion Games

A $k$-lookahead outcome is stable if $A$ forms a Nash equilibrium for the simultaneous-move congestion game:

$$\forall i: \quad c_i(A) \leq c_i(P, A_{-i}) \quad \forall P \in \mathcal{A}_i.$$

Several results hold:

• In generic extension-parallel networks (no ties), for all $k$, the set of $k$-lookahead outcomes coincides with pure Nash equilibria: $k$-LPoA = PoA (Price of Anarchy), independent of $k$.
• In non-generic games (with tie-induced indifferences), full-lookahead (large $k$) can yield unstable and inefficient outcomes not present for myopic ($k=1$) play—known as the “curse of ties.”
• For cost-sharing and consensus games, stability and inefficiency bounds depend on the interplay of lookahead parameter $k$, game structure, and tie-situation.

2.2 Routing Performance and Representation in LLM Lookahead Routing

Lookahead LLM routers achieve higher routing quality by incorporating latent proxies for candidate model outputs. In empirical evaluations:

• The MLM variant produced an average normalized score ($\mu_n$) of 40.8% versus 37.9% for the strongest classifier baseline (RouterDC), a 7.7-point gain (Huang et al., 22 Oct 2025).
• Removing the response modeling loss $\mathcal{L}_{\text{resp}}$ caused substantial drops in normalized scores (6.2–6.8 points depending on variant).
• Joint summarization (MLM) of all candidate responses, rather than isolated or sequential scoring, further improves performance by 3–5 points on open-ended tasks.

3. Treatment of Indifferences, Ties, and Routing Uncertainty

3.1 Congestion Games: Tie-Induced Instability

A congestion game is generic if no two strategies yield precisely the same cost for any player. In non-generic instances, tie-breaking must be imposed, often via lexicographic or preassigned priority (Groenland et al., 2018). The following phenomena arise:

• In generic games, all $k$-lookahead outcomes are stable on simple (extension-parallel) networks.
• In non-generic games, subgame-perfect (large $k$) play can enable instability, where players exploit tie-breaks to their own advantage, yielding non-Nash (unstable) outcomes. In contrast, greedy play ($k=1$) remains stable.
• For consensus games with a shared tie-breaker, all $k$-lookahead outcomes are unanimous and socially optimal for any $k$.
• For cost-sharing games, full lookahead ($k=n$) is required for stability unless all costs are generic.

3.2 LLM Routing: Output Ambiguity and Latent Space “Ties”

In LLM routing, ambiguity and latent overlaps in predicted response features analogously correspond to tie situations in strategy games. Lookahead routing resolves many cases in which query-only classifiers could not differentiate between models, particularly on ambiguous queries, by utilizing richer, predictive latent representations (Huang et al., 22 Oct 2025).

4. Algorithmic Procedures and Complexity

4.1 $k$-Lookahead Computation in Routing Games

The computation of $k$-lookahead outcomes proceeds by limited-depth backward induction for each agent in order:

Input: game G=(N,E,(d_e)), order o, lookahead k Initialize partial profile H ← [] for j = 1 to n do let i = o^{-1}(j) build depth-k game tree rooted at H assign costs to depth-k leaves backward-induction to choose A_i append A_i to H end for return profile A

Each agent's local induction is exponential in $k$ but polynomial for constant $k$ (Groenland et al., 2018).

4.2 Lookahead Routing for LLMs: Instantiations and Pipeline

Causal-LM Variant:

• Employs a small autoregressive backbone (SmolLM2-135M).
• For each candidate $t$, appends a special token $MID_t$, performs a forward pass, extracts the hidden state, and aggregates as $\tilde{r}_t$.
• Routing head merges $\tilde{r}_1, \ldots, \tilde{r}_T$ (optionally with query state) to select optimal model.

Masked-LM Variant:

• Uses a ModernBERT-base backbone.
• Constructs an input with blocks of $MID_t$ tokens, masks full responses.
• Employs curriculum learning to gradually increase masked span.
• Collects joint hidden representations for all candidates, aggregates into the routing prediction.

5. Efficiency, Inefficiency, and Price of Anarchy

5.1 Price of Anarchy in Limited Lookahead Routing

The $k$-Lookahead Price of Anarchy is defined as

$$k\text{-LPoA}(G) = \max_{A \in k\text{-LO}(G)} \frac{\sum_i c_i(A)}{\sum_i c_i(A^*)},$$

where $A^*$ is socially optimal. Key findings:

• On generic extension-parallel networks: $k$-LPoA = PoA, independent of $k$.
• On series-parallel graphs with linear delays: $1$-LPoA $\leq$ PoS = $\rho$ (e.g., $\rho = 4/3$ for affine).
• In generic symmetric singleton cost-sharing games with $d_e(x) = a_ex + b_e$, $k$-LPoA is non-increasing in $k$.
• In consensus games with consistent tie-breaking, $k$-LPoA = 1 for all $k$ (Groenland et al., 2018).

5.2 LLM Routing Efficiency and Representation

Lookahead routing achieves a considerable fraction of ensembling gains at substantially reduced computation. With only 16–18% of training data, response modeling matches full-scale baseline performance (a sixfold data efficiency improvement). Mutual information analysis indicates that the predicted latent features $\tilde{r}_t$ align more closely with oracle model performance than with no-response-modelling baselines (Huang et al., 22 Oct 2025).

6. Empirical Results and Benchmark Comparisons

6.1 Multi-Model LLM Benchmarks

Benchmark evaluation on seven datasets spans instruction following (AlpacaEval-2, Arena-Hard, MT-Bench), mathematical reasoning (GSM8K, MATH), and code generation (HumanEval, MBPP) (Huang et al., 22 Oct 2025). Results are summarized as follows:

Method	Avg. Normalized Score ($\mu_n$)
Random router	0.0%
Oracle router	100.0%
Best ensemble (reward)	48.8%
Similarity-based best	35.4% (SMOOTHIE)
Classifier-based best	37.9% (RouterDC)
Lookahead (CLM)	37.0%
Lookahead (MLM)	40.8%

Key ablation results confirm that removing response modeling and curriculum masking reduces performance, with joint candidate summarization offering further benefits, especially on open-ended tasks.

7. Insights, Limitations, and Implications

The Lookahead framework shows that increasing agent foresight does not universally improve outcomes; in congestion games, non-generic ties can make farsightedness (large $k$) a liability, creating instability or inefficiency, whereas in LLM routing, lightweight predictive “foresight” consistently yields superior empirical routing (Groenland et al., 2018, Huang et al., 22 Oct 2025). In all settings, critical factors include:

• The structure of delay or scoring functions;
• The presence or absence of indifferences/ties;
• The means of representing and aggregating anticipated responses.

Limitations in current LLM applications include lack of cost-awareness during routing, the exclusive use of binary cross-entropy loss, and dependence on potentially biased reward models. Directions for future research include multi-objective routing, alternative loss formulations (contrastive, distributional), reward model ensembling, and dynamic candidate selection mechanisms.

Collectively, these results demonstrate that lookahead, when carefully formulated and implemented, provides a principled mechanism for interpolating between myopic and farsighted decision-making across both strategic routing games and machine learning routing frameworks, with nuanced behaviors dictated by system structure, tie properties, and response modeling.