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Prospect Theory for Enhanced Cyber-Physical Security of Drone Delivery Systems: A Network Interdiction Game
Anibal Sanjab, Walid Saad, Tamer Başar
TL;DR
Drone delivery systems face cyber-physical security threats, while prior work gave limited attention to securing delivery operations. This paper models vendor and attacker choices with a network interdiction game and prospect-theoretic behavior; simulations report that subjective decisions can produce delays beyond the vendor’s target delivery time.
Problem
Drone-delivery research largely emphasized efficiency and aviation integration, with little security analysis, while existing UAV threat studies were often qualitative or based on isolated military experiments.
Source: §I · p. 1Method
The paper formulates a zero-sum network interdiction game, solves its Nash equilibrium through linear programming, and incorporates prospect theory to model subjective attack probabilities and delivery-time valuations.
Source: §I · p. 1, §IV · p. 4Results
When γ decreases from 0.9 to 0.1, achieved expected delivery time increases by 11%, and subjective decision-making can yield delivery times exceeding the target.
Source: §V · p. 6, §VI · p. 6Takeaways & Limitations
In the simulations, subjective perceptions lead to risky path choices and delivery delays that can exceed the vendor’s target time.
Source: §VI · p. 6
Abstract
from arXiv · showhide
The use of unmanned aerial vehicles (UAVs) as delivery systems of online goods is rapidly becoming a global norm, as corroborated by Amazon's "Prime Air" and Google's "Project Wing" projects. However, the real-world deployment of such drone delivery systems faces many cyber-physical security challenges. In this paper, a novel mathematical framework for analyzing and enhancing the security of drone delivery systems is introduced. In this regard, a zero-sum network interdiction game is formulated between a vendor, operating a drone delivery system, and a malicious attacker. In this game, the vendor seeks to find the optimal path that its UAV should follow, to deliver a purchase from the vendor's warehouse to a customer location, to minimize the delivery time. Meanwhile, an attacker seeks to choose an optimal location to interdict the potential paths of the UAVs, so as to inflict cyber or physical damage to it, thus, maximizing its delivery time. First, the Nash equilibrium point of this game is characterized. Then, to capture the subjective behavior of both the vendor and attacker, new notions from prospect theory are incorporated into the game. These notions allow capturing the vendor's and attacker's i) subjective perception of attack success probabilities, and ii) their disparate subjective valuations of the achieved delivery times relative to a certain target delivery time. Simulation results have shown that the subjective decision making of the vendor and attacker leads to adopting risky path selection strategies which inflict delays to the delivery, thus, yielding unexpected delivery times which surpass the target delivery time set by the vendor.
I. INTRODUCTION
Drone delivery faces cyber and physical threats, while prior drone-delivery research largely emphasizes efficiency and integration rather than security. The paper formulates a vendor–attacker interdiction game and incorporates prospect theory to model subjective perceptions of attack likelihood and delivery time.
I. INTRODUCTION
- The paper situates its security problem amid physical threats to low-flying UAVs and cyber threats to their communications links and control.The cited examples include civilian hunting rifles and attacks on UAV communications and control.Source: §I · p. 1
- Prior drone-delivery research largely focused on efficiency and aviation integration, leaving cyber-physical security underanalyzed; existing UAV threat studies were often qualitative or based on isolated military experiments.The paper states that, to the authors’ knowledge, no prior work had studied cyber-physical attacks against UAV delivery systems.Source: §I · p. 1
- The paper models delivery security as a zero-sum network interdiction game: the vendor chooses paths to minimize expected delivery time, while the attacker chooses attack locations to maximize it.The game is equivalent to a zero-sum matrix game, and its Nash equilibrium can be found through two linear programs whose values yield the equilibrium expected delivery time.Source: §I · p. 1
- Prospect theory models each player’s subjective perception of attack-success probabilities and assessment of delivery time relative to a reference point, such as the target time.The target matters because the system’s merit is tied to meeting short delivery commitments.Source: §I · p. 1
II. SYSTEM MODEL
The system model represents delivery routes through danger points between the warehouse and customer, reducing continuous travel space to a directed security graph. It uses simple paths through that graph to describe candidate deliveries and attacker locations.
II. SYSTEM MODEL
- Danger points are inevitable locations on possible warehouse-to-customer routes from which an attacker can target a UAV, and the directed graph represents these points and connecting edges.Examples include high hills or buildings that provide line of sight and proximity for physical or cyber attacks.Source: §II · p. 2
- The model reduces infinitely many possible routes to paths through danger points and represents each pair of neighboring points with only its shortest connecting edge.The resulting graph represents the continuous space between origin and destination using danger points and shortest edges.Source: Figure 2 · p. 2, §II · p. 2
- A candidate route is a simple path from O to D, represented by its sequence of nodes; the path-node incidence matrix records whether each node lies on each path.The path distance function gives the time from O to a node along a specified path.Source: §II · p. 2
- The vendor seeks a path strategy that evades attacks and minimizes expected delivery time, while the attacker selects a danger point to maximize that time.Source: §II · p. 2
A. Game Formulation
The game lets the vendor randomize over delivery paths and the attacker randomize over danger points. Their choices determine expected delivery time, which the vendor minimizes against a worst-case attacker response.
A. Game Formulation
- The vendor chooses a probability distribution over candidate paths, and the attacker chooses a probability distribution over possible attack locations.Randomization prevents either player’s choice from being trivially predictable.Source: A. Game Formulation · pp. 2–3
- A successful attack destroys the UAV and triggers a replacement delivery; the model assumes law enforcement clears the initially attacked path before that replacement is sent.Source: A. Game Formulation · p. 3
- The payoff matrix entry for a path and attack location combines whether the route includes that location, attack-success probability, time to reach it, and total path travel time.Expected delivery time is obtained by averaging these terms over the players’ mixed strategies.Source: A. Game Formulation · p. 3
- The vendor’s min-max formulation chooses a path strategy to minimize expected delivery time against the attacker’s worst-case response.The corresponding attacker problem is the max-min counterpart.Source: A. Game Formulation · p. 3
B. Solution Method
The solution method converts the vendor’s and attacker’s security-strategy problems into linear programs whose solutions yield a mixed-strategy Nash equilibrium. Although multiple equilibria may exist, they share the same expected delivery time.
B. Solution Method
- The vendor’s min-max problem is transformed into a linear program using a change of variables, then reduced to standard maximization problem P3.The optimal transformed strategy is used to recover the vendor’s original strategy.Source: B. Solution Method · p. 3
- The attacker’s max-min problem is similarly reduced to standard minimization problem P4, from which its optimal strategy can be recovered.The optimal transformed attacker strategy supports computing the corresponding original strategy.Source: B. Solution Method · p. 3
- The solutions of LP problems P3 and P4 yield mixed-strategy Nash equilibrium strategies and value functions equal to 1/T.The proposition identifies the resulting strategies as an equilibrium of the network interdiction game.Source: B. Solution Method · p. 3
- Multiple Nash equilibria may exist, but all yield the same expected delivery time and are interchangeable across players’ strategies.If two equilibria are (y*, x*) and (y′, x′), the cross-pairs are also equilibria.Source: B. Solution Method · p. 4
IV. PROSPECT-THEORETIC ANALYSIS
The prospect-theoretic analysis models subjective perceptions of attack probabilities and delivery-time outcomes relative to reference points. It applies separate player payoff matrices and worst-case strategy formulations, which do not generally produce a saddle-point equilibrium.
IV. PROSPECT-THEORETIC ANALYSIS
- Prospect theory models subjective weighting of attack-success probabilities and subjective valuation of delivery times relative to a reference point.The reference point can be the vendor’s target delivery time.Source: §IV · p. 4
- The nonlinear weighting function distorts each player’s perceived attack-success probability, while delivery time has attack-success and no-success outcomes.The text reports that players tend to underweight high-probability outcomes and overweight low-probability outcomes.Source: §IV · p. 4
- The vendor’s value function frames delivery times as gains or losses against its reference point and gives losses greater weight than gains.The loss multiplier captures the vendor’s amplified valuation of crossing the promised delivery time.Source: §IV · p. 5
- The model uses distinct prospect-theoretic payoff matrices and worst-case formulations for the players, which do not generally lead to a saddle-point equilibrium.The matrices differ because they represent the vendor’s and attacker’s subjective valuations.Source: §IV · p. 5
V. NUMERICAL RESULTS
The simulations show that prospect-theory parameters change both players’ strategies and the resulting delivery time. Lower rationality and stronger loss aversion are associated with longer expected delivery times, which can exceed the 30-minute target.
V. NUMERICAL RESULTS
- Path 8 is shortest, followed by paths 2 and 14.Fig. 3a presents path lengths; Fig. 3b presents the vendor’s path-selection strategy under CGT and PT.Source: §V · p. 5, Figure 3 · p. 5
- Under CGT the vendor is more likely to choose paths 7 or 9 than shortest path 8, while PT makes the shortest path more likely as perceived attack probabilities flatten.At γ = 0.1, the vendor perceives attack success probabilities as nearly equal across nodes.Source: §V · p. 5, Figure 3 · p. 5
- Under CGT the attacker randomizes among nodes 7, 8, and 9, favoring node 8; under PT the attacker focuses on nodes 5 and 8.The paper notes that nodes 5 and 8 are part of the shortest paths.Source: Figure 4 · p. 5, §V · p. 6
- 11%: lowering γ from 0.9 to 0.1 increases expected delivery time; the simulations use a 30-minute target, which distorted probability perceptions can exceed.The modeled time is expected UAV flight time; re-sending after a successful attack adds processing and handling time.Source: §V · p. 6, Figure 5 · p. 6
- 0.51 to 0.81: as λU rises from 1 to 10, the probability of choosing the shortest path increases, while expected delivery time also rises.The authors attribute this to greater exaggeration of the consequences of missing the target, which makes the vendor more likely to choose shorter, riskier paths.Source: §V · p. 6, Figure 6 · p. 6
VI. CONCLUSION
The paper formulates drone-delivery security as a zero-sum network interdiction game and shows how to solve its Nash equilibrium using linear programming. Adding prospect theory models subjective decisions, which simulations show can produce delivery delays beyond the vendor’s target.
VI. CONCLUSION
- The paper models drone-delivery security against interdiction attacks as a zero-sum game between a vendor and an attacker.Source: §VI · p. 6
- The game’s Nash equilibrium can be obtained by solving standard linear programming problems.Source: §VI · p. 6
- Simulations show that prospect-theory-based subjective decisions by the vendor and attacker can delay delivery beyond the vendor’s target time.Source: §VI · p. 6