Equilibrium Analysis of the General Model

PART I Toward Efficiency: Security Game 23

III.2 Equilibrium Analysis

III.2.2 Equilibrium Analysis of the General Model

Similarly, we could get the ^v(c−v)_cn -Price of Anarchy as follows, v(c−v)

cn -PoA= cn+c−v

c ,

which is, again, linear inn.

Theorem III.2.4. In theIndependent Multidefender setting, Nash equilibrium among de- fenders (ASE) exists if and only if U^c−U^u ≥kc−^{(n−1)(Ω−U}_n ^c). In this equilibrium all targets are protected with probability 1.

Proof. We firstly claim that Nash equilibrium can appearonly if coverage probabilities of all of targetst_{i j} are identical. Otherwise, there will be a targett_ikwhich has the probability 0 of being attacked, and the defender i has an incentive to decreaseq_ik. To determine a Nash equilibrium, we therefore need only consider scenarios in which all targets have the same coverage probability.

When all targets have the same coverage probability q to be protected, the utility of each defender is

u=(U^c−U^u−nkc)q+U^u+ (nk−1)Ω

n .

Ifq<1, then some defendericould increaseqtoq+δ for all of her targets to ensure none of them are attacked, and obtain utility ofu⁰=kΩ−k(q+δ)c, so that

u⁰−u= (U^c−U^u)(1−q) + (Ω−U^c)−nkcδ

n .

AsU^c≥U^u,Ω≥U^c, andδ can be arbitrarily small,u⁰−u>0 whenq<1, which means that this cannot be a Nash equilibrium. Thus, the only possible equilibrium can beq_{i j} =1 for all targetst_{i j}.

When all targets have the same coverage probabilityq=1, each defender’s utility is

u=U^c−nkc+ (nk−1)Ω

n .

We claim that if a defender i has an incentive to deviate, it is optimal for this defender to use the same coverage probability for all her targets. Otherwise, for some target t_ik which has probability 0 of being attacked, she could decreaseq⁰_ik to obtain higher utility.

If probabilities of targets protected by defenderiare allq⁰(0≤q⁰<1), then her expected

utility isu⁰= (U^c−U^u−c)q⁰+U^u+ (k−1)(Ω−q⁰c), and

u⁰−u= (U^c−U^u−kc)(q⁰−1) +(n−1)(U^c−Ω)

n .

We therefore have two cases:

1) IfU^c−U^u≥kc, thenu⁰−u≤0, andq=1 for all targets is a Nash equilibrium.

2) IfU^c−U^u<kc, the maximal value ofu⁰−ucorresponds toq⁰=0:

0≤qmax⁰<1

u⁰−u=−(U^c−U^u−kc)−(n−1)(Ω−U^c)

n .

Ifkc−^{(n−1)(Ω−U}_n ^c) ≤U^c−U^u<kc,u⁰−u≤0, it is a Nash equilibrium; otherwise, it is not.

To sum up, a Nash equilibrium existsif and only if U^c−U^u≥kc−^{(n−1)(Ω−U}_n ^c), and the equilibrium corresponds to all targets having probability 1 of being protected.

Thus, if a Nash equilibrium does exist, it is unique, with all defenders always protecting their targets. But what if the equilibrium does not exist? Next, we characterize the (unique) ε-equilibrium (ε-ASE) with the minimal ε that arises in such a case. We will use this approximate equilibrium strategy profile as apredictionof the defenders’ strategies.

Theorem III.2.5. InIndependent Multidefender setting, in the optimalε-equilibrium (ε- ASE) all targets are protected with probability^Ω−U_kc ^u. The correspondingεis^{(Ω−Uu)(kc−U}_cnk ^c+Uu). Proof. When all targets have the same coverage probabilityq, the expected utility of each defender is

u=(U^c−U^u−nkc)q+U^u+ (nk−1)Ω

n .

Suppose 0≤q<1. If some defenderiincreasesqtoq+δi j for each of her targett_{i j}, then

she would obtain utilityu⁰=∑^k_j=1Ω−(q+δ_{i j})c, and

u⁰−u= Ω−(U^c−U^u)q−U^u

n −

k

∑

j=1

δ_{i j}c

≤ Ω−(U^c−U^u)q−U^u

n .

(III.1)

Now we consider scenarios in which a defenderi could obtain higher utility by decreas- ing protection probability. We claim that if a defender i has an incentive to deviate, it is optimal for this defender to use the same coverage probability for all her targets. Other- wise, for some targett_ikwhich has probability 0 of being attacked, she could decreaseq⁰_ik to obtain higher utility. Thus, we need only consider cases in which a defender deviates by decreasing coverage probabilities for all her targets to q−δ. Her utility will become u⁰⁰ = (U^c−U^u−kc)(q−δ) +U^u+ (k−1)Ω. SinceU^c−U^u<kc, δ =q (the maximal value ofδ) maximizesu⁰⁰−u:

0<δ≤qmax u⁰⁰−u= Ω−(U^c−U^u)q−U^u

nk +kcq+U^u−Ω. (III.2)

By comparing the value of equation (III.1) and equation (III.2), we get different values of ε forε-equilibrium:

ε=











Ω−(U^c−U^u)q−U^u

n , if 0≤q≤^Ω−U_kc ^u;

Ω−(U^c−U^u)q−U^u

n +kcq+U^u−Ω, if ^Ω−U_kc ^u <q≤1.

Whenq= ^Ω−U_kc ^u, we get the minimalε =^{(Ω−Uu)(kc−U}_cnk ^c+Uu).

We claim that the ^{(Ω−Uu)(kc−U}_cnk ^c+Uu)-equilibrium can appearonly if all targets have the same coverage probability q. We prove this by contradiction. Suppose that targets have different coverage probabilities. This gives rise to two cases: 1) Each defender uses an identical coverage probability for each target she owns (these may differ between defend- ers); and 2)Some defender has different coverage probabilities for her targets. In case 1),

there existβ defenders (1≤β <n) who have the same minimal coverage probabilityq⁰. The expected utility for each defender among theseβ is

u= (U^c−U^u−kβc)q⁰+U^u+ (kβ−1)Ω

β .

When ^Ω−U_kc ^u <q⁰≤1, some defenderiamong theseβ could decrease the coverage proba- bility of all her targets to 0 and obtain the utility ofu₁=U^u+ (k−1)Ω, so that

u₁−u= Ω−(U^c−U^u)q⁰−U^u

β +kcq⁰+U^u−Ω

> Ω−(U^c−U^u)q⁰−U^u

n +kcq⁰+U^u−Ω.

When 0≤q⁰≤ ^Ω−U_kc ^u, some defenderiamong theseβ can increase coverage probabilities of all her targets toq⁰+δ₃to obtain utility ofu₂=kΩ−k(q⁰+δ₃)c, with

u₂−u= Ω−(U^c−U^u)q⁰−U^u−kβcδ₃ β

> Ω−(U^c−U^u)q⁰−U^u

n ,

where the inequality holds because δ₃ can be arbitrarily small. Thus, no profile in case 1) can be a ^{(Ω−Uu)(kc−U}_cnk ^c+Uu)-equilibrium. In case 2), any defender who has different coverage probabilities among her targets can always increase her payoff by decreasing the coverage probabilities of the targets with higher coverage to yield identical coverage for all targets. Consequently, no profile in case 2)can be a ^{(Ω−Uu)(kc−U}_cnk ^c+Uu)-equilibrium.

Armed with a complete characterization of predictions of strategic behavior among the defenders, we can now consider how this behavior is related to socially optimal protection decisions. Since the solutions are unique, there is no distinction between the notions of price of anarchy and price of stability; we term the ratio of socially optimal welfare to welfare in equilibrium as the price of anarchy for convenience.

SW_Ois

SW_O=











U^c−nkc+ (nk−1)Ω, if U^c−U^u≥nkc;

U^u+ (n−1)Ω, if U^c−U^u<nkc.

Proof sketch. First, we claim that we could get optimal social welfare only if all targets have the same coverage probability q. Otherwise, some target j, which is influenced by defenderihas probability 0 of being attacked, and we can decreaseq_i,j to improve social welfare. Consequently, we need only to consider an optimal symmetric coverage prob- ability q to maximize social welfare, which can be done in a relatively straightforward way.

IfU^c−U^u≥kc−^{(n−1)(Ω−U}_n ^c), the Nash equilibrium is unique, with all targets protected with probability 1. The corresponding social welfare is

SW_E =U^c−nkc+ (nk−1)Ω.

So far we have not yet added any constrains to value of Ω, U^c, andU^u (except that Ω≥U^c≥U^u). In order to makePrice of Anarchywell-defined, we need to add constraints that values ofΩ,U^c, andU^u are all non-positive or all non-negative. We add constraints thatU^c,U^uandΩare all non-positive (little changes if all are non-negative).

In the case of a unique Nash equilibrium, the price of anarchy is

PoA=



















1, ifU^c−U^u≥nkc;

U^c−U^u−nkc

U^u+(nk−1)Ω+1, ifkc−^{(n−1)(Ω−U}_n ^c) ≤ U^c−U^u<nkc.

IfU^c−U^u<kc−^{(n−1)(Ω−U}_n ^c), there is no Nash equilibrium. The Social Welfare in the

2 4 6 8 10 12 14 16 18 20 0.8

1 1.2 1.4 1.6 1.8 2

Number of Targets Each Defender Has

(Approximate) Price of Anarchy

n=2 n=3 n=4 n=5 n=6

Figure III.2: (Approximate) Price of Anarchy whenc=1,Ω=−1,U^c=−2 andU^u=−10

optimal approximate equilibrium is

ε-SW_E = (U^c−U^u−nkc)Ω−U^u

kc +U^u+ (nk−1)Ω, and the ^{(Ω−Uu)(kc−U}_cnk ^c+Uu)-Price of Anarchy is ^(U_kcU^c−Uu^u+(nk−1)kcΩ^{−nkc)(Ω−Uu)}+1.

From this result, it is already clear that defenders systematically over-invest in security.

This stems from the fact that the attacker creates anegative externalityof protection: if a defender protects his target with higher probability than others, the attacker will have an incentive to attack another defender. In such a case, we can expect a “dynamic” adjustment process with defenders increasing their security investment well beyond what is socially optimal.

We now analyze the relationship between (ε-)PoA and the values ofnandk. First, we consider (ε-)PoA as the function of n. IfΩ=0, (ε-)PoA linearly increases in n, and is therefore unbounded. However, if Ω6=0, while PoA and ε-PoA are increasing in n, as n→∞, they approach 1− ^c

Ω and 1+^Uu_kΩ^−Ω, respectively. In other words, PoA (exact and approximate) is bounded by a constant, for a constantk.