--- title: Chosen Order Secretary Problem (COSP) url: https://www.emergentmind.com/topics/chosen-order-secretary-problem-cosp type: topic --- # Chosen Order Secretary Problem (COSP) The Chosen Order Secretary Problem (COSP) is a learning-augmented generalization of the classical secretary problem in which the decision-maker possesses machine-learned predictions of candidate values and controls the arrival order of candidates. COSP captures natural scenarios, such as setting interview schedules based on predicted applicant value, where combining predictive information and order selection may yield improved decision quality. The central algorithmic challenge is to maintain consistency—selecting a near-optimal candidate when predictions are accurate—while guaranteeing strong robustness, i.e., a bounded competitive ratio, when predictions are unreliable. The recent work of Karisani et al. systematically develops a framework for COSP, proposes an optimized randomized algorithm, and establishes new competitive-ratio bounds that improve upon previous results for both random and order-controlled settings [2601.07482]. ## 1. Problem Formulation and Model COSP is defined as follows. Given $n$ candidates with unknown true values $v_i \ge 0$, the decision-maker receives before the selection process a (not necessarily reliable) prediction $\hat{v}_i$ for each candidate. Unlike the classical Random Order Secretary Problem (ROSP), the decision-maker in COSP may choose the arrival times $t_i \in [0,1]$, subject to the restriction that the top-predicted candidate $\hat\imath = \arg \max_i \hat v_i$ is scheduled at a specific time $\beta \in [0,1]$, while the remaining arrival times are drawn i.i.d. from $\mathrm{Unif}[0,1]$. Ties in true values or predictions are resolved by infinitesimal perturbations to ensure distinctiveness among all $v_i$ and $\hat v_i$. The online selection proceeds in increasing order of $t_i$; for each candidate, the actual value $v_i$ is revealed and the algorithm must irrevocably choose whether to "hire" or "pass." The process terminates upon hiring, with at most one candidate accepted. ## 2. Prediction Error and Its Role Prediction quality is captured by an $\varepsilon$-multiplicative error parameter, \[ \varepsilon = \max_{i \in [n]} \left|1 - \frac{\hat v_i}{v_i} \right|, \] equivalently, \[ (1 - \varepsilon) v_i \le \hat v_i \le (1 + \varepsilon) v_i \quad \forall i. \] A small $\varepsilon$ implies high predictive accuracy; large $\varepsilon$ corresponds to unreliable model outputs. The competitive performance of the COSP algorithm dynamically interpolates between regimes of low and high $\varepsilon$. ## 3. Randomized Algorithm for COSP The core algorithm uses parameters: error-threshold $\theta \in (0,1)$, time-threshold $\tau \in (0,1)$, arrival time $\beta \in (\tau,1)$ for the top-predicted candidate, and randomized hiring probabilities $\gamma, \delta \in [0,1]$. The continuous-time pseudocode is as follows: ``` 1: Let ĩ ← arg max_i ĥv_i. Fix t_ĩ←β; draw all other t_i∼Unif[0,1]. 2: mode←“Prediction”. (We begin trusting predictions.) 3: For each arriving i in order of t_i: 4: If mode=Prediction and |ĥv_i/v_i −1|>θ then 5: mode←Secretary; t_M←t_i 6: If mode=Prediction: 7: If i=ĩ then hire(i) and STOP 8: Else { mode=Secretary } 9: If t_i>τ and v_i> max {v_j : t_j \theta$. The set $M$ contains such "large-error" candidates; $t_M$ is the arrival time of the first such event. - Line 7: If the top-predicted candidate $\hat\imath$ arrives while still in "Prediction mode," it is immediately hired. - Lines 9–12: In "Secretary mode," the algorithm ignores arrivals before $\tau$, then hires the first "record-breaking" candidate; if this is $\hat\imath$ at $t_M$, acceptance occurs with probability $\gamma$, otherwise with $\delta$. Any other record-breaker is hired unconditionally. ## 4. Competitive Ratio Derivation Sketch Let $v^* = \max_i v_i$. The competitive ratio is \[ \min_{\text{all inputs}} \frac{\mathbb{E}[\text{value of hired candidate}]}{v^*}. \] For COSP, the parameters \[ \theta = 0.58, \quad \tau = 0.37, \quad \beta = 0.64, \quad \gamma = 0.27, \quad \delta = 0.46 \] yield a provable lower bound: \[ \min_{\text{inputs}} \frac{\mathbb{E}[\text{ALG}]}{v^*} \geq \max \left\{ 0.262, \frac{1-\varepsilon}{1+\varepsilon} \right\}. \] The analysis partitions inputs into seven structural cases depending on whether $\hat\imath$ and/or the maximizing candidate $i^*$ lie in $M$, and arrival-time relations. For each case, the authors derive explicit expressions $C_j(m, k, m_2)$ (using sums and order-statistics integrals) to lower bound the competitive ratio, where $m = |M|$, $k$ is the number of $i$ with $v_i > v_{\hat\imath}$, and $m_2$ is the count of "large-error" candidates with $v_i < v_{\hat\imath}$. Each $C_j$ is reduced, via binomial expansions and integral identities, to computable functions. A regime-by-case evaluation and symbolic bounding for large parameters confirm the claimed universal competitiveness. ## 5. Classical and Prior Results Comparison The following table concisely presents competitive ratios for related secretary-selection models: | Model | Best Proven Competitive Ratio | Reference | |-------------------------------------|------------------------------------------------------|------------------------| | Classical secretary (random arrival)| $1/e \approx 0.368$ | Dynkin (1963) | | ROSP, learning-augmented | $\max\{0.215, (1-\varepsilon)/(1+\varepsilon)\}$ | Fujii–Yoshida (2023) | | Any consistent ROSP | $\leq 0.33$ (robustness upper bound) | Choo–Ling (2024) | | COSP (Karisani et al. algorithm) | $\max\{0.262, (1-\varepsilon)/(1+\varepsilon)\}$ | [2601.07482] | COSP, via the joint use of predictions and deterministic scheduling of the maximally predicted candidate at time $\beta$, surpasses the previous deterministic bound of $0.25$ for learning-augmented order-control protocols [2601.07482]. ## 6. Timeline and Algorithmic Dynamics The execution unfolds along a continuous-time axis $t \in [0,1]$: ``` time t∈[0,1] ─────────────────────────────────────> |---------------|---------------------------| 0 τ=0.37 β=0.64 1 • [0, τ): still in Prediction mode (no hires before τ) • Any large-error candidate in [0,β) → switch to Secretary mode at t_M • If ĩ arrives at t_M or later in Secretary mode → hire with prob. γ or δ • If no error by β → Prediction mode hires ĩ at t=β • After τ in Secretary mode → hire first record-breaker ``` - "Prediction mode" is maintained unless a substantial prediction error is detected or until time $\beta$. - Early errors trigger a fallback to a Dynkin-style “Secretary mode,” giving randomized preference to the top-predicted candidate only if it is also a record-breaker at a critical time. - The mode-switching strategy balances the use of predictive information against adverse instances: *Consistency* is realized when prediction error $\varepsilon \to 0$, while *Robustness* is ensured for large $\varepsilon$, guaranteeing the worst-case lower bound of $0.262$. ## 7. Context and Significance COSP integrates recent advances in learning-augmented online algorithms by harnessing both predictive value estimates and limited order selection in classical decision problems. It serves as a bridge between purely stochastic (random arrival) and prophet-inequality settings (complete order control), marking a substantial improvement in competitive guarantees for secretary-type tasks. The model and analysis of Karisani et al. represent the first provable method to break the deterministic $0.25$ competitive barrier for this setting and approach the classical random-order limit of $1/e$ under high-confidence predictions. Full derivations, including the technical details for each of the seven structural cases and integral computations underlying the competitive-ratio proof, are provided in the appendix of the cited work [2601.07482]. Source: https://www.emergentmind.com/topics/chosen-order-secretary-problem-cosp