1. LA DISCIPLINA EN LA ESFERA MILITAR
1.2. LA DISCIPLINA Y LOS DERECHOS DE LAS PERSONAS
Earlier results allow to compare security investments in a cooperative and a non-cooperative envi- ronment. In this section we focus on this comparison. We intent to determine if under-investments or over-investments prevail in equilibrium (non-cooperative game) relative to social optimum (cooperative game). Also benchmarking results under the random attack are included in the analysis. Table3gives a summary of characteristic equations which uniquely determine security investments.
Case Necessary assumptions Security investment solves
Random attack, Nash equilibrium in proposition5.2 Assumption1
q
νN=
c
0−1(
n1)
Random attack, social optimum in proposition5.3 Assumption1q
νs=
c
0−1(
Dnν)
Strategic attack, Nash equilibrium, Symmetric investments in theorem5.1 Assumption1, assumption 2+and network is vertex-transitive
q
N
=
c
0−1(
1 n+
[n−D]D nψ00(1 n)[1−q
N])
Strategic attack, social optimum in theorem5.2 Assumption1, assumption2 and network is vertex-transitive.q
s
=
c
0−1(
D n)
Table 3 –Summary of characteristic equations and necessary assumptions.
We propose the following definition to adequately compare investment levels.
Definition 5. In the security game in a vertex transitive network wherecsatisfies assumption1andψ satisfies assumption 2+. DefineqN
r as the Nash equilibrium under the random attack. Similarly letqNs be thesymmetricNash equilibrium under the strategic attack. Also setqsas the social optimum investment level37.
The next proposition gives a complete classification of investment levels as introduced in definition5. Theorem 5.3. The security investments outlined in definition5satisfy:
1. for allp∈[0,1]
(a) qN
r ≤qs, (b) qN
r ≤qsN,
2. there exists a uniquep∗such that (a) ifp=p∗thenqN
s =qs, (b) ifp < p∗thenqNs > qs, 37. Note that neitherqN
r orqNs is a vector and consequently notation is slightly abused when saying thatqrNorqNs is a Nash investment level, which has to be a vector. The notation is allowed here however because the Nash equilibrium is symmetric. Similarly, this notation is also applied toqsas opposed to the formally correctqs. Lastly note that the notationqsr andqss are unnecessary as the social optimum investment levels are equivalent under the random and the strategic attack
(c) ifp > p∗thenqNs < qs.
The value ofp∗is the unique solution inside[0,1]of
ψ00(1 n) D−1 D = [n−D][1−c 0−1(D n)]. (5.41)
Proof. The results under1.follow directly by comparing characteristic equations in table3 and using thatcandψare strictly convex and1≤D≤n. The results under2.require more explanation. However whenp= 0: c0(qs) =D n = 1 n < 1 n+ n−1 nψ00(1 n) (1−qN) =c0(qsN),
from which we conclude thatqN
s > qs. Differently whenp= 1, thenD=nand therefore c0(qs) = D n = 1> 1 n =c 0(qN s ),
in turn proving thatqsN < qs. By continuity of bothqsandqNs inpit follows that there necessarily exists an intersection ofqN
s andqs. To prove that this intersection is unique, note that in the intersectionp∗is such that D n = 1 n+ [n−D]D nψ00(1 n) [1−c0−1(D n)].
This expression can subsequently be written as (5.41). Note that the LHS of (5.41) is strictly increasing inp, while - on the other side - the RHS is strictly decreasing inp. By applying lemmaA.1we conclude thatp∗is unique.
Theorem5.3can be seen as the main result in our research as it allows to compare several situations38.
Most striking observation are the over-investments in Nash equilibrium relative to social optimum when pis low (see2.(b)). As discussed earlier, whenpis low, investments in security are strategic comple- mentaries; if one agent invests others will follow (to discourage an attack). Although this may lead to a more social optimal situation in some situations (e.g. following education), in this situation agents invest too much, leading to a less social optimal situation (e.g. an arms race).
The expression in (5.41) characterizes the point where over-investments turn into under-investments. The solutionp∗ is denoted as thetransition point of the security game in this research. Note that this transition point is a function of the number of agentsn, the expected number of documents obtained by each agentD (which in turn depends onpand the network structure) and the cost functions for both the agents and the attacker. In the next section the role of each parameter is analyzed rigorously. First however the following example illustrates the statements in theorem5.3.
Example 5.4. Suppose an information network is complete and holds5agents who mutually share their documents. When agents incur a cost ofc= (1/2)q2 to hold investment levelqand when the attacker
incurs a cost ofψ= (1/2)a2
νfor attacking agentνwith probabilityaν, figure32shows certain investment levels.
Observe that all statements in theorem5.3are illustrated in figure32. Clearly the presence of a strategic attacker makes agents to invest more in security. Although this makes the network more secure, when pis low the network is too secure and benefits of risk-reduction do not compensate the increased costs. Several things strike from the figure which are not mentioned in theorem5.3and in earlier results. First of all, when p = 0the social optimum is the same as the equilibrium investments under the random
38. Prudence is called when applying theorem5.3to real world cases. As the utility of both agents and the attacker are inde- pendent of characteristics of the document,por the network, investment levels can only be compared when the SAME document is considered. For instance one can cannot (directly) conclude from theorem5.3that PIN-codes are better secured than postal codes aspis lower in the first case.
Figure 32 –Security investments inK5 whereψ =a2/2andc =q2/2. As equilibrium investmentsqsN
under the strategic attack are above equilibrium investmentsqrN under the random attack, a strategic
attacker forces agents to invest more in security. Agent may even invest too much in security. Specifically, ifp < p∗thenqN
s features over-investments relative to the social optimumqs. Ifp > p∗thenqsN features
under-investments relative toqs.
attack. This of course follows mathematically but also economically as there is no external effect for an agent whenp = 0. Consequently it is socially optimal to only protect one’s own document against a direct attack; a similar incentive as in the random attack. Second observe that whenp= 1, equilibrium investments under the random attack and the strategic attack are identical. This observation is motivated because whenp= 1an agent shares his document with everybody in the network. This in turn makes it pointless to discourage an attack, which in turn makes incentives under the strategic and the random attack equivalent.
Lastly note that also the statements in theorem 5.1are visible in the figure. For instance qN
s initially increases inpand consequently decreases inp. Also remind thatqN
s attains its global maximum when D =n/2 = 2.5. One can show thatp ≈0.22in this case. Additionally, by working out (5.41) one can show thatqN
s andqsintersect whenD≈3.15. The value ofpis approximately 0.3 in this case.
5.8 Dependency of investments on the network structure, cost functions and the number of