HyRiM (version 1.0.3)

mgss: compute a multi-goal security strategy

Description

The function applies fictitious play in a one-against all competition as described by (Sela, 1999) to approximate an equilibrium in the auxiliary game belonging to the input mosg, using the method proposed by (Rass and Rainer, 2014).

Usage

mgss(G, weights, cutOff, ord = 5, eps, T = 1000, points = 512)

Arguments

G

a multi-objective game constructed using mosg

weights

each goal in G can be assigned a weight to reflect its priority. If missing, the weights default to be all equal. The weights do not need to sum up to 1 (and are normalized towards a unit sum otherweise), but need to be all non-negative.

cutOff

(only used for continuous loss distributions) the maximal loss for which no events are expected or otherwise the risk of exceeding cutOff are accepted. If missing, this value defaults to the maximal observation on which the loss distributions were constructed (equivalently, the right end of their common support).

ord

the order up to which the loss distribution shall be approximated. This value may be set to high orders when it is necessary to distinguish distributions that are similar at the tails.

eps

the accuracy at which the fictitious play iteration is being stopped. The error is herein determined as the maximum-norm of the difference between approximations of the equilibrium distributions obtained in the current and the previous iteration of FP.

T

the maximal number of iterations after which FP is stopped (returning whatever approximation has been obtained).

points

the number of points at which the resulting equilibrium loss distributions are approximated.

Value

An object of class mosg.equilibrium, containing the following fields:

optimalDefense

a discrete probability distribution over the action space of player 1 (defender)

optimalAttacks

a discrete probability distribution over the action space of player 2 (attacker)

assurances

a list of loss distributions valid under the assumption that player 1 adheres to the optimalDefense distribution in its randomized action choices. The list can be accessed by the names for each goal as specified through the input mosg object G. Each distribution within assurances is a mixed loss distribution constructed using lossDistribution

%% ~Describe the value returned %% If it is a LIST, use %% \item{comp1 }{Description of 'comp1'} %% \item{comp2 }{Description of 'comp2'} %% ...

Details

The function assumes the payoffs to be Gaussian kernel density estimators (constructed using the function lossDistribution), and computes a Taylor-polynomial approximation at the \(x\) equal to cutOff for each distribution up to order ord. Preferences are decided using the method put forth in (Rass, 2015), using sign-alternating derivatives.

References

S. Rass and B. Rainer. Numerical computation of multi-goal security strategies. In Radha Poovendran and Walid Saad, editors, Decision and Game Theory for Security, LNCS 8840, pages 118-133. Springer, 2014.

Aner Sela. Fictitious play in 'one-against-all' multi-player games. Economic Theory, 14:635-651, 1999.

S. Rass. On Game-Theoretic Risk Management (Part One). Towards a Theory of Games with Payoffs that are Probability-Distributions. ArXiv e-prints, June 2015. http://arxiv.org/abs/1506.07368.

D. Lozovanu, D. Solomon, and A. Zelikovsky. Multiobjective games and determining pareto-nash equilibria. Buletinul Academiei de Stiinte a Republicii Moldova Matematica, 3(49):115-122, 2005. ISSN 1024-7696.

See Also

A brief info on the results can be obtained by print.mosg.equilibrium, and a more detailed summary (showing all loss distributions in detail) is obtained by summary.mosg.equilibrium.

Examples

# NOT RUN {
library(compare)
library(orthopolynom)
## raw data (PURELY ARTIFICIAL, for demo purposes only)
# N=100 observations in each category
obs111<-c(rep(1,40),rep(3,20),rep(5,10),rep(7,20),rep(9,10)); 
obs112<-c(rep(1,50),rep(2,10),rep(4,10),rep(6,20),rep(8,10)); 
obs121<-c(rep(1,20),rep(4,30),rep(6,20),rep(8,10),rep(10,20));
obs122<-c(rep(1,40),rep(2.5,20),rep(5,20),rep(7.5,10),rep(9,10));
obs211<-c(rep(1,30),rep(2,30),rep(5,10),rep(8,10),rep(10,20));
obs212<-c(rep(1,10),rep(2,10),rep(4,20),rep(7,20),rep(10,40));
obs221<-c(rep(1,30),rep(3,30),rep(4,10),rep(7,20),rep(9,10));
obs222<-c(rep(1,10),rep(3,10),rep(5,50),rep(8,20),rep(10,10));
obs311<-c(rep(1,40),rep(2,30),rep(4,10),rep(7,10),rep(9,10));
obs312<-c(rep(1,20),rep(3,20),rep(4,20),rep(7,20),rep(10,20));
obs321<-c(rep(1,10),rep(3,40),rep(4,30),rep(7,10),rep(9,10));
obs322<-c(rep(1,10),rep(4,30),rep(5,30),rep(7,10),rep(10,20));

## compute payoff densities
f111<-lossDistribution(obs111)
f112<-lossDistribution(obs112)
f121<-lossDistribution(obs121)
f122<-lossDistribution(obs122)
f211<-lossDistribution(obs211)
f212<-lossDistribution(obs212)
f221<-lossDistribution(obs221)
f222<-lossDistribution(obs222)
f311<-lossDistribution(obs311)
f312<-lossDistribution(obs312)
f321<-lossDistribution(obs321)
f322<-lossDistribution(obs322)

payoffs<-list(f111,f112,f121, f122,f211,f212,f221,f222, f311,f312,f321,f322)
G <- mosg( n=2,
            m=2,
            payoffs,
            goals=3,
            goalDescriptions=c("g1", "g2", "g3"),
            defensesDescr = c("d1", "d2"),
            attacksDescr = c("a1", "a2"))
eq <- mgss(G,T=1000,weights=c(0.25,0.5,0.25))
print(eq)
summary(eq)
# construct another loss distribution from a given behavior in the game G
suboptimal <- lossDistribution.mosg(G, c(0.1,0.1,0.8), c(0.2,0.3,0.5))
plot(suboptimal)
# }