UNIQUE OPTIMAL TIME MOMENT IN EXPONENTIALLY-CONVEX-REWARD 1-BULLET PROGRESSIVE SILENT DUEL OF ICT INNOVATION LAUNCH

Abstract

Background. In the modern digital economy, the timing of the introduction of new information and telecommunication technologies becomes critical: launching too early may lead to unprepared infrastructure or immature market demand, while delaying too long risks losing the market advantage to competitors. These processes can be interpreted through the lens of progressive silent duels. Each market participant must decide when to act — that is, when to introduce, announce, or deploy a technology — without knowing whether the competitor has already done so. In particular, when the reward of acting grows exponentially with time — representing, for instance, cumulative technological maturity or increasing value of full deployment — yet the risk of being second remains severe, the decision problem aligns with an exponentially-convex-reward duel.

Objective. The paper aims to determine the set of optimal time moments for an exponentially-convex-reward 1-bullet silent duel. From a practical standpoint, the objective of this research is to determine the optimal moment for initiating or announcing a new ICT solution in a competitive environment under the following conditions: readiness and payoff grow progressively over time, information about competitors’ actions is unavailable until after both sides have acted, and only one major strategic action is possible within a given competitive cycle. The goal is to identify a stable and universal decision rule, being an optimal strategy of timing, that maximises expected reward under these uncertainty and competition constraints.

Methods. The finite 1-bullet progressive silent duel is considered, in which each of the two duelists shoots with an exponentially-convex reward. The duel is a symmetric matrix game whose optimal value is 0, and the set of optimal strategies is the same for both duelists, regardless of the duel size and how time is quantised. The duel is silenced because the duelist does not learn about the action of the other duelist until the very end of the duel. The duel time quantisation is such that time progresses by the geometrical progression pattern, according to which every following time moment is the partial sum of the respective geometric series. In this duel, the duelist has a single optimal strategy. It is to shoot always at the third time moment, whichever the number of time moments is. Namely, the unique optimal strategy is to shoot at either the duel end moment in the  duel, or at the three-quarters of the unit time span in bigger duels.

Results. The theoretical finding is that the unique optimal strategy is to act precisely at the third progressive time moment, which is equivalent to around three quarters of the total planning horizon in larger duels. This result suggests that, irrespective of the granularity of internal planning (how finely time is divided into milestones), and regardless of the scale of competition, there exists a universal “sweet spot” for taking action. In real-world terms, this moment corresponds to a late but not final stage of technological preparation — when the solution has reached sufficient maturity and reliability, when market conditions are becoming favourable but not yet saturated, and when the delay is long enough to exploit exponential improvement effects but short enough to avoid being overtaken by a rival.

Conclusions. The “silent” nature of the duel models the real-world asymmetry of competitive information. For enterprises introducing new ICT systems, the third progressive moment represents a strategically balanced readiness threshold. It minimises the risk of premature launch (insufficient maturity) and the threat of excessive delay (competitor’s precedence), producing a dominant timing equilibrium that is independent of specific market size or implementation granularity.