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Published January 4, 2021 | Version Author Manuscript
Journal article Open

The Price of Defense

  • 1. Department of Computer Science, University of Cyprus, 1678 Nicosia, Cyprus
  • 2. Faculty of Pure and Applied Sciences, Open University of Cyprus, 2220 Nicosia, Cyprus and Research Center on Interactive Media, Smart Systems, and Emerging Technologies, Nicosia, Cyprus
  • 3. Department of Computer Science and Engineering, European University Cyprus, 1516 Nicosia, Cyprus
  • 4. Dipartimento di Scienze Aziendali – Management of Innovative Systems, Università di Salerno, 84081 Fisciano, SA, Italy
  • 5. Department of Computer Science, University of Liverpool, Liverpool L69 3BX, UK, Department of Computer Engineering and Informatics, University of Patras, 26500 Rion, Patras, Greece and Computer Technology Institute and Press "Diophantus", 26500 Rion, Patras, Greece

Description

We consider a strategic game on a graph G(V,E) with two confronting classes of randomized
players: ν attackers who choose vertices and wish to minimize the probability of being caught by
the defender, who chooses edges and gains the expected number of attackers it catches. So, the
defender captures system rationality. In a Nash equilibrium, no single player has an incentive to
unilaterally deviate from its randomized strategy. The Price of Defense is the worst-case ratio, over
all Nash equilibria, of the optimal gain of the defender (which is ν) over the gain of the defender
at a Nash equilibrium. In this work, we provide a comprehensive collection of trade-offs between
the Price of Defense and the computational efficiency of Nash equilibria.
• Through a reduction to a Zero-Sum Two-Players Game, we prove that a general Nash equilibrium
can be computed via Linear Programming in polynomial time. However, the reduction
does not provide any apparent guarantees on the Price of Defense.
• To obtain guarantees on Price of Defense, we analyze several structured Nash equilibria:
– In a Matching Nash Equilibrium, the support of the defender is an Edge Cover of the
graph. We prove that Matching Nash equilibria can still be computed in polynomial
time, and they incur a Price of Defense of α(G), the Independence Number of G.
– In a Perfect Matching Nash Equilibrium, the support of the defender is a Perfect Matching
of the graph. We prove that Perfect Matching Nash Equilibria can be computed in
polynomial time, and they incur a Price of Defense of |V |
2 .
– In a Defender Uniform Nash equilibrium, the defender chooses each edge in its support
with uniform probability. We prove that Defender Uniform Nash equilibria incur a Price
of Defense falling between those for Matching and Perfect Matching Nash Equilibria;
however, it is NP-complete to even decide the existence of a Defender Uniform Nash
equilibrium.
– In an Attacker Symmetric, Uniform Nash equilibrium, all attackers have a common support
on which each uses a uniform probability distribution. We prove that Attacker
Symmetric Uniform Nash equilibria can be computed in polynomial time and incur a
Price of Defense of either
|V |
2 or α(G).
In conclusion, the Perfect Matching Nash Equilibrium both can be computed efficiently and provides
the best (known) Price of Defense, when it exists. Else, Matching Nash equilibria, Defender Uniform
Nash equilibria and Attacker Symmetric Uniform Nash equilibria provide interesting trade-offs
between the Price of Defense and computational efficiency.

Notes

This work has been partly supported by the project that has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 739578 (RISE – Call: H2020-WIDESPREAD-01-2016-2017-TeamingPhase2) and the Republic of Cyprus through the Deputy Ministry of Research, Innovation and Digital Policy.

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Additional details

Funding

RISE – Research Center on Interactive Media, Smart System and Emerging Technologies 739578
European Commission