Commitment-Depth Gap in Stackelberg Games

• Commitment–Depth Gap is a measure quantifying the loss in the leader’s expected payoff when followers base responses on N finite observations instead of perfect information.
• It shows that even optimal interior mixed strategies may fall short of achieving full Stackelberg benefits, especially under discontinuous payoff functions.
• Analytical bounds indicate that the gap diminishes polynomially with increasing sample size and can be mitigated via regularized, robust commitment strategies.

The commitment–depth gap quantifies the shortfall in a leader’s expected payoff in Stackelberg (leader–follower) games when the follower can only observe finitely many samples, $N$, of the leader’s mixed strategy, rather than having perfect knowledge. Specifically, for commitment depth $N$, the gap is defined as $\Delta(N) := f^*_\infty - f^*_N$, where $f^*_\infty$ denotes the Stackelberg payoff under perfect commitment (unlimited observations), and $f^*_N$ is the maximal expected payoff achievable when the follower bases their best response on the maximum likelihood estimate (MLE) from $N$ i.i.d. samples. This construct isolates the loss in commitment power arising from partial reputation/incomplete observation, and underpins robust strategy design in security games, network routing, and economic persuasion contexts (Muthukumar et al., 2019).

1. Formal Definition and Interpretation

Let $\pi^*_\infty$ represent the Stackelberg mixed commitment optimizing the leader’s payoff under perfect observation. The leader’s expected payoff with $N$ samples and a follower best–responding to the MLE $\hat\pi_N$ is denoted $f^*_N = \max_{\pi\in\Delta} f_N(\pi)$. The commitment–depth gap is thus

$\Delta(N) := f^*_\infty - f^*_N$

where $\Delta(N)$ encapsulates the degree to which finite observational depth limits the leader’s ability to attain the Stackelberg benefit. In the context of Stackelberg games with discontinuous payoff functions at $\pi^*_\infty$ and interior (mixed) commitments, this gap is strictly positive for all finite $N$.

2. Key Theoretical Bounds

Three principal results delineate the quantitative behavior of $\Delta(N)$:

• Non-robustness of Stackelberg Commitment: For any $2 \times n$ game where $\pi^*_\infty$ is an interior mixed strategy and the leader's payoff is discontinuous at $\pi^*_\infty$, no finite $N$ allows the leader to recover the Stackelberg payoff by simply repeating $\pi^*_\infty$. Formally,

$$f_N(\pi^*_\infty) \leq f^*_\infty - C[\Phi(C'\sqrt{N}) - \tfrac{1}{2} - \tfrac{C'}{\sqrt{N}}] + \exp(-C''N)$$

where $C, C', C''$ depend on the payoff matrices, and $\Phi$ is the standard normal CDF. For all sufficiently large $N$, $f_N(\pi^*_\infty) < f^*_\infty$, and $\lim_{N\to\infty} f_N(\pi^*_\infty) < f^*_\infty$.

• Near–Stackelberg Payoff via Robust Commitments: If the best–response region $K_{j^*}$ is a polytope of dimension $m-1$, then for any $0 < p < \tfrac{1}{2}$ and $N \gtrsim m$,

$$f^*_\infty - f_N(\pi_{N, p}) = \widetilde{O}((m/N)^p + \exp(-\Omega(N^{1-2p})))$$

where $\pi_{N,p}$ is a robust commitment constructed to regularize $\pi^*_\infty$ into the best–response interior, and $\widetilde{O}(\cdot)$ absorbs geometry-dependent constants.

• Upper Bound on True Optimum under $N$ Observations: For all $N$,

$f^*_N \leq f^*_\infty + Cn\sqrt{m/N}$

indicating that even optimal “cheating” commitments cannot surpass the Stackelberg payoff by more than $O(n\sqrt{m/N})$, which vanishes as $N \to \infty$.

3. Construction of Observation–Robust Commitment

To approach the Stackelberg payoff in the partial observation regime, one regularizes the commitment. Let $K_{j^*} = \{\pi : B\pi \leq c\}$ denote the best–response polytope for the follower. By performing an affine change of coordinates,

$$\pi' = T(\pi) \quad \Longleftrightarrow \quad B'\pi' \leq \mathbf{1}$$

the binding constraints at $\pi^*_\infty$ become standard form. For a shrink factor $\delta \in (0,1)$, define

$\pi'_{N,p} := (1-\delta)\,\pi'^*_\infty$

and map back $\pi_{N,p} = T^{-1}(\pi'_{N,p})$. Here,

$$\delta = Z(K_{j^*}, \pi'^*_\infty) \left(\frac{m}{N}\right)^p$$

with $Z(\cdots)$ giving the maximal radius so that the Dikin ellipsoid at $\pi^*_\infty$ remains in $K_{j^*}$.

Pseudocode for Robust Commitment Construction:

Compute affine map T so that B π ≤ c ↦ B'π' ≤ 1 Set δ ← Z⋅(m/N)^p π' ← T(π*∞) π'_N,p ← (1−δ)·π' π_N,p ← T⁻¹(π'_N,p)

Each operation is $O(1)$ or polynomial in $(m, n)$ contingent on prior solution of the Stackelberg LP.

4. Qualitative Characterization of the Gap

The commitment–depth gap is rooted in the geometry of best–response regions in non–zero-sum games. With $\pi^*_\infty$ typically located on the boundary of multiple such regions, sample fluctuations in $\hat\pi_N$ can drive the follower's best response into a different region, resulting in a strictly positive $\Delta(N)$ for any finite $N$. The robust interior strategies $\pi_{N,p}$ trade off bias (moving inland by $\delta$) vs. variance (risk of exiting region) to ensure $\Delta(N) = o(1)$, with the rate essentially polynomial in $1/N$. No alternative commitment can exploit the follower’s learning for more than $O(n\sqrt{m/N})$ gain.

5. Empirical Case Studies

Empirical evaluation includes both toy and practical scenarios:

Game Type	Observed Behavior for $\pi^*_\infty$	Performance of $\pi_{N,p}$
$2 \times 2$	$f_N(\pi^*_\infty) < f^*_\infty$ as $N\to\infty$	Rapid recovery of $f^*_\infty$, empirical gap matches $\widetilde{O}((m/N)^p)$
$2 \times 3$	Permanent shortfall	Near-optimal for all $N$ per brute-force search
Random $5 \times 5$	$f_N(\pi^*_\infty)$ falls below $f^*_\infty$ for $N<10^4$	Robust commitments recover $f^*_\infty$ at rate $N^{-1/2}$

In $2 \times 2$ and $2 \times 3$ discontinuous payoff scenarios, playing $\pi^*_\infty$ alone is inadequate, but the robust strategy $\pi_{N,p}$ quickly approaches optimality. For random zero-sum $5 \times 5$ security games, the Stackelberg commitment exhibits high non–robustness, whereas the robust interior commitments close the gap effectively, as evidenced by log–log and percentage–gap visualizations.

6. Implications and Extensions

Finite commitment depth is a fundamental limitation in leader–follower interaction models where the follower infers the leader’s strategy from finite observations. The commitment–depth gap provides a non–asymptotic framework to analyze strategic reputation in realistic settings. The robust commitment construction advances practical strategy design by exploiting the geometry of best–response regions to offset the payoff loss, with explicit rates for convergence to Stackelberg optimality. A plausible implication is the necessity of regularization in strategic games when observational granularity is limited, which suggests avenues for protocol design in security, routing, and persuasion applications. Future research may refine these bounds for broader classes of games and follower learning processes.