## 3 The Bike-Share Case
### Evaluation--Causal Mapping  
From the August 2023 data (Figure 3), we observe that, when a station's bike inventory is low (blocking rentals) or high (blocking returns), incentives are often either absent or incorrectly applied, indicating a flaw in the logic behind triggering incentives. In other words, since we have evidence that crowdsourcing as a policy is effective, we can conclude that improving the inventory balancing policy would reduce out-of-stock events (or no-service events). Given that the scope is inventory balancing, Figure 4 illustrates the hypothesised mental model, mapping the causal relationships between relevant variables, with an initial focus on a single bike station for clarity. There are three underlying sets of assumptions in the model: incentives influence user preferences for stations to rent from or return to, perceptions of bike saturation lead decision-makers to employ crowdsourcing to rebalance inventory and reduce stockouts and a station's attractiveness (nearly empty or full) prompts baukling behaviour. These assumptions are grounded in both intuition and empirical studies on Citi Bike and docked bike-sharing systems in general [27; 51; 52]. The _Return_ and _Rental_ rates represent the inflows and outflows of the _Bike stock_, with base rates reflecting demand influenced by weather or other contextual factors. The difference between inflows and outflows, or the net inflow, integrates into the _Bike stock_. Specifically (note that \(s\) represents any point in time between \(t_{0}\) and \(t\)):  
\[\textit{Bikestock}(t)=\int_{t0}^{t}[\text{Return rate}(s)-\text{Rental rate}(s)\bigg{]}ds+\textit{Stock}(t_{0}).\]  
As bike stock increases, _Dock utilisation_ (_Bike saturation_) at the station increases, assuming all else remains equal. This positive relationship (indicated by the plus sign on the arrow) reduces the station's attractiveness for returns (_User perception of return attractiveness_ shown with negative polarity), causing users to avoid full stations and creating some baukling. The balancing loop \"B1\" resists this change, feeding back to stabilise the system. A similar dynamic occurs in the balancing loop \"B2\". As the _Bike utilisation_ or _Dock saturation_ (number of free docks) increases, the attractiveness for rentals (_User perception of rental attractiveness_) decreases, reducing the rental rate and inducing stronger baukling at near-empty stations. The Formulation of Simulation and ML Models section will provide mathematical precision to these qualitative statements. The outer loops \"B3\" and \"B4\" are more consequential to overall system behaviour. The _Perceived bike availability_, a smoothed composite of _Forecasted bike availability_ and real-time _Bike utilisation_, influences the provider's perception of risk. Delays in incorporating real-time data may, however, occur, either due to system limitations or a deliberate \"wait and see\" approach (indicated by dashed arrows) where the provider does not want to react too soon to fluctuations (thus, \"smoothing\") [47]. As the perceived bike availability decreases, the _Perceived risk of bike outage_ increases, which raises the pressure to incentivise returns (_User incentives for returns_). These incentives boost the return rate, increasing bike stock as intended. A similar dynamic plays out in the case of dock outage risk (_Perceived risk of dock outage_) that triggers more rentals (_User incentives for rentals_). It is assumed, both here and in the forthcoming equation formulation, that incentives reroute demand between stations rather than increase overall demand. When coordinated effectively, especially between nearby stations with asymmetric demand, crowdsourced inventory balancing through incentives reduces imbalances across the system. Figure 4: Causal Loop Diagram illustrating the feedback structures involved in bike-share inventory balancing and the hypothesised influences on system performance.