CEO-Bench: Can Agents Play the Long Game? — part 5/7

Cleaned-text code points [62420, 77902). Bibliography omitted here; source.md includes it.

On a billing day set by the billing period D bill D_{\mathrm{bill}} , subscription revenue is Y t sub = ∑ i ∈ ℬ t C i , p i , t eff ​ n i , Y_{t}^{\mathrm{sub}}=\sum_{i\in\mathcal{B}_{t}}C_{i,p_{i},t}^{\mathrm{eff}}n_{i}, (18) where ℬ t \mathcal{B}_{t} is the set of subscribers billed on day t t , C i , p i , t eff C_{i,p_{i},t}^{\mathrm{eff}} is the effective price of customer i i ’s active plan, p i p_{i} is the active plan, and n i n_{i} is seats. Seat count multiplies revenue because an enterprise or team subscription pays for more users than an individual account. This simulates recurring subscription billing, where cash arrives in discrete renewal events rather than continuously every day. Before voluntary plan-choice churn, a group-level involuntary churn draw may occur: μ g , m invol = clip ⁡ ( ϵ g , m invol , 0 , 1 ) , ϵ g , m invol ∼ 𝒩 ⁡ ( μ ¯ g invol , σ g invol ) , Z i , t invol ∼ Bernoulli ⁡ ( μ g , m invol ) . \mu_{g,m}^{\mathrm{invol}}=\operatorname{clip}\!\left(\epsilon_{g,m}^{\mathrm{invol}},0,1\right),\qquad\epsilon_{g,m}^{\mathrm{invol}}\sim\mathcal{N}(\bar{\mu}_{g}^{\mathrm{invol}},\sigma_{g}^{\mathrm{invol}}),\qquad Z_{i,t}^{\mathrm{invol}}\sim\mathrm{Bernoulli}(\mu_{g,m}^{\mathrm{invol}}). (19) m m indexes the billing period, ϵ g , m invol \epsilon_{g,m}^{\mathrm{invol}} is the period-specific involuntary churn rate before clipping, μ ¯ g invol \bar{\mu}_{g}^{\mathrm{invol}} and σ g invol \sigma_{g}^{\mathrm{invol}} are group-specific churn parameters, μ g , m invol \mu_{g,m}^{\mathrm{invol}} is the clipped probability used for group g g in period m m , and Z i , t invol Z_{i,t}^{\mathrm{invol}} is the customer-level cancellation draw. This captures background churn such as procurement freezes, budget changes, company shutdowns, or stakeholder turnover that are not caused by the agent. Voluntary churn then follows the participation rule: if no plan clears the customer’s curve, the customer cancels; if a different plan gives higher acceptable surplus, the customer switches. The mechanism simulates the fact that some churn is controllable through product and pricing, while some churn is exogenous noise in the customer base.

#### Support issue generation and resolution.

For a subscriber with no open issue, the issue probability is P ⁡ ( issue i , t ) = clip ⁡ ( ( p 0 + p S ​ ( S ref − S i , t ) + p out ​ 1 ​ { outage t } ) ​ n i , p min , p max ) . P(\mathrm{issue}_{i,t})=\operatorname{clip}\!\left((p_{0}+p_{S}(S^{\mathrm{ref}}-S_{i,t})+p_{\mathrm{out}}\,\mathbb{1}\{\mathrm{outage}_{t}\})n_{i},p_{\min},p_{\max}\right). (20) p 0 p_{0} is the base issue rate, p S p_{S} converts low satisfaction into tickets, S ref S^{\mathrm{ref}} is the reference satisfaction level, p out p_{\mathrm{out}} adds outage-driven issue risk, 𝟙 ​ { outage t } \mathbb{1}\{\mathrm{outage}_{t}\} activates that risk on outage days, n i n_{i} is seats, and p min , p max p_{\min},p_{\max} bound the probability. More seats create more chances for someone to hit a problem, while poor satisfaction and outages make support demand spike. This simulates customer-success queues where large accounts and unhappy users generate more tickets.

Open issues are resolved by operations pools. For pool P P with spend x P x_{P} , group g g members n g , P n_{g,P} , and pool size | P | |P| , N g , t resolved ∼ Poisson ⁡ ( ( b P + λ g ​ x P ) ​ n g , P | P | ) . N_{g,t}^{\mathrm{resolved}}\sim\mathrm{Poisson}\!\left((b_{P}+\lambda_{g}x_{P})\frac{n_{g,P}}{|P|}\right). (21) b P b_{P} is the pool’s base resolution rate, λ g \lambda_{g} is group-specific operations efficiency, x P x_{P} is spend assigned to support pool P P , n g , P / | P | n_{g,P}/|P| allocates capacity to group g g according to its share of the pool, and N g , t resolved N_{g,t}^{\mathrm{resolved}} is the number resolved. Global operations covers all open issues; targeted operations creates additional pools by group, plan, group-plan pair, or customer. Fast resolutions add relationship boosts, while unresolved issues increase open-issue days and decay relationship. The intuition is queue-based: more operations spend increases throughput, but only for customers covered by that pool. This simulates support staffing and escalation rules, where targeted customer-success effort can protect priority segments but cannot help customers outside the targeted pool.

### A.5 Reputation, Social Media, and Acquisition

#### Reputation impact.

Each active customer contributes a daily reputation delta from satisfaction: δ i , t rep = { ρ + ​ S i , t , S i , t ≥ S 0 , − ρ − ​ | S i , t | , S i , t < S 0 . \delta_{i,t}^{\mathrm{rep}}=\begin{cases}\rho_{+}\,S_{i,t},&S_{i,t}\geq S_{0},\\ -\rho_{-}\,|S_{i,t}|,&S_{i,t}<S_{0}.\end{cases} (22) S 0 S_{0} is neutral satisfaction, ρ + \rho_{+} is the positive-reputation rate, ρ − \rho_{-} is the negative-reputation rate, and δ i , t rep \delta_{i,t}^{\mathrm{rep}} is customer i i ’s daily reputation contribution. Negative satisfaction is allowed to have a different slope from positive satisfaction so that bad experiences can be more reputationally damaging than good experiences are helpful. This simulates word-of-mouth asymmetry in real markets, where angry customers often spread more salient feedback than mildly satisfied customers. For group g g , Δ ​ R g , t = ∑ i ∈ g δ i , t rep max ⁡ ( N g , N min ) ​ log ν N ⁡ ( max ⁡ ( N g , N min ) ) , R g , t + 1 = clip ⁡ ( R g , t + Δ ​ R g , t , R min , R max ) . \Delta R_{g,t}=\frac{\sum_{i\in g}\delta_{i,t}^{\mathrm{rep}}}{\max(N_{g},N_{\min})}\log_{\nu_{N}}(\max(N_{g},N_{\min})),\qquad R_{g,t+1}=\operatorname{clip}(R_{g,t}+\Delta R_{g,t},R_{\min},R_{\max}). (23) N g N_{g} is the active subscriber count, N min N_{\min} is the small-sample normalizer, ν N \nu_{N} controls logarithmic scale, R g , t R_{g,t} is group reputation, and R min , R max R_{\min},R_{\max} bound reputation. The averaging term prevents a single customer from dominating a large segment, while the logarithmic factor lets larger customer bases produce more visible aggregate reputation movement. This simulates customer reviews and public sentiment accumulating within a market segment. Cancellations add event damage D i , t cancel = η D ​ ( β D + ξ t ) ​ ( α D + χ D ​ min ⁡ ( S i , t , S 0 ) 2 ) ​ log ν N ⁡ ( max ⁡ ( N g , N min ) ) max ⁡ ( N g , N min ) , ξ t ∼ Uniform ⁡ ( ξ min , ξ max ) , D_{i,t}^{\mathrm{cancel}}=\eta_{D}(\beta_{D}+\xi_{t})\left(\alpha_{D}+\chi_{D}\min(S_{i,t},S_{0})^{2}\right)\frac{\log_{\nu_{N}}(\max(N_{g},N_{\min}))}{\max(N_{g},N_{\min})},\qquad\xi_{t}\sim\mathrm{Uniform}(\xi_{\min},\xi_{\max}), (24) where η D , β D , α D , χ D \eta_{D},\beta_{D},\alpha_{D},\chi_{D} are configurable damage coefficients, min ⁡ ( S i , t , S 0 ) 2 \min(S_{i,t},S_{0})^{2} makes very negative satisfaction especially costly, ξ t \xi_{t} is event noise, and D i , t cancel D_{i,t}^{\mathrm{cancel}} is the reputation hit from cancellation. This makes visible churn more damaging when the customer was very unhappy. The mechanism simulates public cancellations, angry posts, and negative references that can hurt a brand beyond the lost subscription revenue.

#### Cross-group reputation spillovers.

Discovered groups receive spillovers from related groups: R h , t + 1 ← clip ⁡ ( R h , t + 1 + ζ R ​ W g , h ​ Δ ​ R g , t , R min , R max ) . R_{h,t+1}\leftarrow\operatorname{clip}\!\left(R_{h,t+1}+\zeta_{R}W_{g,h}\Delta R_{g,t},R_{\min},R_{\max}\right). (25) W g , h W_{g,h} is the influence from group g g to group h h , ζ R \zeta_{R} scales spillover, Δ ​ R g , t \Delta R_{g,t} is the reputation change in the source group, and the clipping bounds keep the recipient group’s reputation in range. The mechanic makes reputation networked: enterprise failures can affect nearby enterprise groups or adjacent market segments. This simulates reference networks, professional communities, and social adjacency, where the experience of one group can change expectations in a related group even before those customers use the product.

#### Customer and agent social media.

Customer social-media candidates are weighted by satisfaction extremity, negative satisfaction, recent satisfaction change, active service events, influencer status, and seat count: w i , t post = n i ​ ω g inf ​ ω i , t new ⋅ ( 1 + α S ​ | S i , t | ) ⋅ ( 1 + α − ​ [ − S i , t ] + 2 ) ⋅ ( 1 + α Δ ​ | Δ ​ S i , t | ) ⋅ ( 1 + α E ​ | ℰ i , t | ) . w_{i,t}^{\mathrm{post}}=n_{i}\,\omega_{g}^{\mathrm{inf}}\omega_{i,t}^{\mathrm{new}}\cdot\left(1+\alpha_{S}|S_{i,t}|\right)\cdot\left(1+\alpha_{-}[-S_{i,t}]_{+}^{2}\right)\cdot\left(1+\alpha_{\Delta}|\Delta S_{i,t}|\right)\cdot\left(1+\alpha_{E}|\mathcal{E}_{i,t}|\right). (26) w i , t post w_{i,t}^{\mathrm{post}} is the post-sampling weight, n i n_{i} is seats, ω g inf \omega_{g}^{\mathrm{inf}} is the group influence multiplier, ω i , t new \omega_{i,t}^{\mathrm{new}} is the new-customer multiplier, α S \alpha_{S} weights satisfaction extremity, α − \alpha_{-} gives extra weight to negative satisfaction, α Δ \alpha_{\Delta} weights recent satisfaction changes, α E \alpha_{E} weights active service events, Δ ​ S i , t \Delta S_{i,t} is the satisfaction change, and ℰ i , t \mathcal{E}_{i,t} is the set of active quality events such as outage, overload, issue, or quota frustration. The simulator samples up to K post K_{\mathrm{post}} posts per day from these candidates, so social media is a noisy but informative public signal rather than a complete survey. This simulates the selection bias of public feedback: large, influential, newly acquired, or upset customers are more likely to be heard than a random satisfied user.

Agent-authored posts are judged per discovered group. For group g g , let e g , t agent ∈ [ − 1 , 1 ] e_{g,t}^{\mathrm{agent}}\in[-1,1] be the judged reaction score. The social multiplier entering acquisition is A g , t = clip ⁡ ( A 0 + α A ​ e g , t agent ​ V g , t , A min , A max ) , A_{g,t}=\operatorname{clip}\!\left(A_{0}+\alpha_{A}e_{g,t}^{\mathrm{agent}}V_{g,t},A_{\min},A_{\max}\right), (27) where e g , t agent e_{g,t}^{\mathrm{agent}} is the judged reaction of group g g to the agent’s post, V g , t V_{g,t} is exposure, A 0 A_{0} is neutral social effect, α A \alpha_{A} converts reaction-weighted exposure into lead impact, and A min , A max A_{\min},A_{\max} bound the multiplier. Public communication can therefore help or hurt growth depending on how each group reacts. This simulates product marketing and public relations: a message that resonates with one segment can accelerate acquisition, while a poorly received message can suppress demand.

#### Daily new-customer generation.

For discovered target group g g , expected leads are λ g , t = R g , t ​ D g , t ​ C t ​ M g , t ​ A g , t ​ Z t ​ ( ∑ c x c , g , t ​ L c , g , t x ad + ∑ h N h , t ​ W h , g net ) , \lambda_{g,t}=R_{g,t}D_{g,t}C_{t}M_{g,t}A_{g,t}Z_{t}\left(\sum_{c}\frac{x_{c,g,t}L_{c,g,t}}{x_{\mathrm{ad}}}+\sum_{h}N_{h,t}W^{\mathrm{net}}_{h,g}\right), (28) where R g , t R_{g,t} is reputation, D g , t D_{g,t} is market availability, C t C_{t} is the calendar-cycle multiplier, M g , t M_{g,t} is the macro lead multiplier, A g , t A_{g,t} is the agent-social multiplier, Z t Z_{t} is the active demand-surge multiplier, x c , g , t x_{c,g,t} is channel spend, L c , g , t L_{c,g,t} is leads per reference ad spend x ad x_{\mathrm{ad}} , N h , t N_{h,t} is subscribers in group h h , and W h , g net W^{\mathrm{net}}_{h,g} is the referral matrix. The calendar-cycle multiplier makes demand oscillate through recurring seasonal cycles, so otherwise identical ad spend can perform better or worse depending on timing. The macro multiplier captures broad economic expansion or contraction; the social and surge multipliers capture communication effects and temporary external demand spikes; and the referral term captures word of mouth from existing subscribers. Market saturation is D g , t = [ D 0 − ( N g , t cap g , 0 ​ ( α cap + γ g ​ t / Y cap ) ) ν D ] + , n g , t ∼ Poisson ⁡ ( [ λ g , t ] + ) . D_{g,t}=\left[D_{0}-\left(\frac{N_{g,t}}{\mathrm{cap}_{g,0}(\alpha_{\mathrm{cap}}+\gamma_{g}t/Y_{\mathrm{cap}})}\right)^{\nu_{D}}\right]_{+},\qquad n_{g,t}\sim\mathrm{Poisson}([\lambda_{g,t}]_{+}). (29) D 0 D_{0} is baseline availability, N g , t N_{g,t} is the current number of customers in group g g , cap g , 0 \mathrm{cap}_{g,0} is initial market capacity, α cap \alpha_{\mathrm{cap}} is the baseline capacity scale, γ g \gamma_{g} is group capacity growth, Y cap Y_{\mathrm{cap}} is the time scale for capacity growth, ν D \nu_{D} controls saturation curvature, λ g , t \lambda_{g,t} is expected leads, and n g , t n_{g,t} is realized leads. The positive-part operator prevents negative availability, and the Poisson draw converts the expected funnel volume into noisy realized leads. This combines paid acquisition, word of mouth, reputation, macro conditions, seasonal demand, temporary shocks, and finite market size. The mechanism simulates a real go-to-market funnel where the same budget can yield different outcomes depending on brand, timing, segment saturation, and randomness.

#### Demand surges.

External demand surges are temporary acquisition shocks. For each active surge s s , Z t = ∏ s ∈ 𝒮 t z s , z s ∼ 𝒟 s surge , t ∈ [ t s start , t s end ) . Z_{t}=\prod_{s\in\mathcal{S}_{t}}z_{s},\qquad z_{s}\sim\mathcal{D}_{s}^{\mathrm{surge}},\qquad t\in[t_{s}^{\mathrm{start}},t_{s}^{\mathrm{end}}). (30) 𝒮 t \mathcal{S}_{t} is the set of active surges, z s z_{s} is the surge lead multiplier sampled from surge distribution 𝒟 s surge \mathcal{D}_{s}^{\mathrm{surge}} , and t s start , t s end t_{s}^{\mathrm{start}},t_{s}^{\mathrm{end}} are its active days. The product over active surges allows multiple external events to stack. Surges create temporary windows where growth is easier, but the agent must still have pricing, quality, and capacity to retain the acquired customers. This simulates events such as press attention, industry shifts, or sudden demand spikes that increase inbound interest without guaranteeing durable revenue.

### A.6 Market Discovery and Non-Stationarity

#### Market and group research.

Market research reveals new customer groups, while group research increases the information level for a known group after a delay: ℓ g , t + D g , ℓ research info = max ⁡ ( ℓ g , t info , ℓ target ) , D g , ℓ research ∼ 𝒟 g , ℓ research . \ell_{g,t+D_{g,\ell}^{\mathrm{research}}}^{\mathrm{info}}=\max(\ell_{g,t}^{\mathrm{info}},\ell^{\mathrm{target}}),\qquad D_{g,\ell}^{\mathrm{research}}\sim\mathcal{D}_{g,\ell}^{\mathrm{research}}. (31) ℓ g , t info \ell_{g,t}^{\mathrm{info}} is the agent-visible information level for group g g , ℓ target \ell^{\mathrm{target}} is the requested level, D g , ℓ research D_{g,\ell}^{\mathrm{research}} is the configured completion delay, and 𝒟 g , ℓ research \mathcal{D}_{g,\ell}^{\mathrm{research}} is the delay distribution for that group and research depth. The max operator means research can raise the information level but cannot erase already acquired knowledge. Results snapshot current market conditions when the research completes. The purpose is to make information acquisition an operational choice with time cost rather than a free static table. This simulates customer discovery, analyst work, and market research projects that improve visibility only after a delay and may already be slightly stale when delivered.

#### Competitor events.