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Network Security: an Economic Perspective Marc Lelarge (INRIA-ENS) currently visiting STANFORD TRUST seminar, Berkeley 2011.

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Threats and Vulnerabilities Attacks are exogenous

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Contribution (1) Optimal security investment for a single agent -Gordon and Loeb model, 1/e rule -Monotone comparative statics (2) Optimal security investment for an interconnected agent - Network externalities (3) Equilibrium analysis of the security game - Free-rider problem, Critical mass, PoA

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(1) Single agent Two parameters: – Potential monetary loss: – Probability of security breach without additional security: Agent can invest to reduce the probability of loss to: Optimal investment:

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(1) Gordon and Loeb Class of security breach probability functions: measure of the productivity of security. Gordon and Loeb (2002)

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(1) Gordon and Loeb (cont.) vulnerability Optimal investment (size of potential loss fixed)

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(1) Gordon and Loeb (cont.) Probability of loss for a given investment

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(1) Gordon and Loeb (cont.) High vulnerability Low vulnerability

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(1) Conditions for monotone investment If then is non-decreasing Augmenting return of investment with vulnerability: Extension to submodular functions.

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(1) The 1/e rule If the function is log-convex in x then the optimal security investment is bounded by:,i.e of the expected loss

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Contribution (1) Optimal security investment for a single agent -Gordon and Loeb model, 1/e rule -Monotone comparative statics (2) Optimal security investment for an interconnected agent - Network externalities (3) Equilibrium analysis of the security game - Free-rider problem, Critical mass, PoA

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(2) Effect of the network Agent faces an internal risk and an indirect risk. Information available to the agent: in a poset (partially ordered set). Optimal security investment:

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(2) How to estimate the probability of loss? Epidemic risk model Binary choice for protection Limited information on the network of contagion (physical or not): degree distribution. – Best guess: take a graph uniformly at random. Galeotti et al. (2010)

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Attacker directly infects an agent N with prob. p. Each neighbor is contaminated with prob. q if in S or if in N. (2) Epidemic Model Attacker N S

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(2) Monotone comparative statics If the function is strictly decreasing in for any then the optimal investment is non-decreasing. Equivalent to: Network externalities function is decreasing:

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(2) Strong protection An agent investing in S cannot be harmed by the actions of others:. in previous equation. Decreasing network externalities function.

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(2) Weak protection If, the network externalities function is:

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Contribution (1) Optimal security investment for a single agent -Gordon and Loeb model, 1/e rule -Monotone comparative statics (2) Optimal security investment for an interconnected agent - Network externalities (3) Equilibrium analysis of the security game - Free-rider problem, Critical mass, PoA

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(3) Fulfilled expectations equilibrium Concept introduced by Katz & Shapiro (85) Willingness to pay for the agent of type : multiplicative specification of network externalities, Economides & Himmelberg (95). C.d.f of types: % with Willingness to pay for the ‘last’ agent:

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(3) Fulfilled expectations equilibrium In equilibrium, expectation are fulfilled: The willingness to pay is: Extension of Interdependent Security 2 players game introduced by Kunreuther & Heal (03).

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(3) Critical mass Equilibria given by the fixed point equation

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(3) Critical mass (cont.) Equilibria given by the fixed point equation fraction of population investing in security cost

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(3) Critical mass (cont.) If only one type: willingness to pay = network externalities function. fraction of population investing in security cost

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(3) Price of Anarchy The social welfare function: Because of the public and private externalities, agent under-invest in security (in all cases). Public externalities Private externalities

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Conclusion Simple single agent model: 1/e rule – General conditions for monotone investment Interconnected agents: network externalities function – General conditions to align incentives Equilibrium analysis of the security game – Critical mass, PoA Extensions: In this talk, agent is risk-neutral. What happens if risk-adverse? Insurance?

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Thank you! Feedbacks are welcome: marc.lelarge@ens.fr

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