Perimeter Defense Game

Stackelberg Game · Mixed Strategies

Randomize guard deployment across 4 perimeter sectors to prevent alien infiltration. A pure (deterministic) strategy is exploitable — the attacker simply avoids the guarded sector. Mixed strategies force the attacker to face uncertainty everywhere. This is the key insight of Stackelberg security games.

The Scenario

Guard Allocation

4 perimeter sectors protect the base. You have 2 guards to allocate. Each sector has different payoffs for defender and attacker depending on whether the sector is guarded when attacked. The attacker observes your strategy (coverage probabilities) but not your realization (which guards go where tonight). You must choose a mixed strategy — randomized coverage probabilities — that maximizes your guaranteed worst-case utility.

How this differs from Patrol Route Optimization

The patrol page uses the same SSE framework in a path-based context (ARMOR/IRIS). This page focuses on the pure game-theoretic structure: guard-to-sector assignment, the concept of mixed strategies, and the key insight that randomization is a feature — predictable defense is exploitable. No routes, just allocation probabilities.

Defense Domain	OR Element	Symbol	Example
Wall sector	Target	t	Sector-Beta
Guard unit	Coverage resource	m = 2	2 guards
Guard probability	Coverage	c_t ∈ [0,1]	0.55
Catch infiltrator	Defender covered payoff	U^d_c	5
Breach damage	Defender uncovered payoff	U^d_u	-12
Alien caught	Attacker covered payoff	U^a_c	-7
Successful infiltration	Attacker uncovered payoff	U^a_u	15

Stackelberg Security Game (Tambe 2011): MAXIMIZE d // guaranteed minimum defender utility subject to: d ≤ c_t·U^d_c(t) + (1-c_t)·U^d_u(t) ∀ t ∈ T // worst-case constraint Σ_t c_t ≤ m = 2 // total coverage = 2 guards 0 ≤ c_t ≤ 1 ∀ t ∈ T // Attacker best response (rational adversary): // t* = argmax_t [ c_t · Ua_c(t) + (1-c_t) · Ua_u(t) ] // At Strong Stackelberg Equilibrium (SSE): // attacker is indifferent among targets they might attack // — this is the key game-theoretic property

Interactive Solver

Mixed Strategy Optimizer

★★☆ Heuristic (LP)

4 Sectors · 2 Guards

Sector	U^d_c	U^d_u	U^a_c	U^a_u
Alpha	3	-8	-5	10
Beta	5	-12	-7	15
Gamma	2	-5	-3	6
Delta	4	-10	-6	13

References

Published Tambe, M. (2011). Security Games: Applying Game Theory to Counterterrorism. Cambridge University Press.

Published Korzhyk, D., Conitzer, V., & Parr, R. (2010). “Complexity of computing optimal Stackelberg strategies in security resource allocation games.” AAAI.

Published Kiekintveld, C., Jain, M., Tsai, J., et al. (2009). “Computing optimal randomized resource allocations for massive security games.” AAMAS.

Preparing for First Contact

We do recommend the Hungarian algorithm. It works on any planet.

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Educational Fiction Disclaimer

This is a fictional educational scenario.

All data is entirely fictional
No military applications intended
The author advocates for peace