mosg.equilibrium {HyRiM}

embodies all information related to a lexicographic Nash equilibrium computed by the function mgss.

Description

The generic functions print and summary provide brief, and detailed information about the lexicographic Nash equilibrium. The generic function plot can be used to visualize the equilibrium.

Usage

## S3 method for class 'mosg.equilibrium'
summary(object, ...)
## S3 method for class 'mosg.equilibrium.summary'
print(x, ...)
## S3 method for class 'mosg.equilibrium'
print(x, extended=FALSE, ...)
## S3 method for class 'mosg.equilibrium'
plot(x, points=100, ...)

Arguments

x

an mgss object as returned by the function mgss.

object

an mgss object as returned by the function mgss.

for print.mosg.equilibrium, the following parameter can be supplied:

extended

if set to TRUE, then the individual assurances are printed as well.

for plot.mosg.equilibrium, the following parameter can be supplied:

points

the number of points to evaluate the density function over its support for plotting

...

further arguments passed to or from other methods.

Value

the result returned by the function summary carries the following fields:

optimalDefense

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

optimalAttacks

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

assurances

an optimal loss distribution valid under the assumption that the defender plays optimalDefense as its mixed strategy. This is a list of mosg.lossdistribution objects, accessible through their assigend names (coming from the underlying game) or by indices.

The action spaces for both players are defined in first place by the game for which the equilibrium was computed (via mgss on a game constructed by mosg).

print gives a shortened output restricted only to displaying the optimal defense for the defender and attack strategies per goal (as defined by the underlying game).

summary returns an object of class summary.mosg.lossdistribution, which has the fields: "range" "mean" "variance" "quantiles" "is.discrete"

range

the minimal and maximal values of the loss (as anticipated by the observations)

mean

the first moment as computed by mean

variance

the variance as computed by variance

quantiles

a 2x5-matrix of quantiles at the 10%,25%,50%,75% and 90% level

is.discrete

a Boolean flag being TRUE if the loss distribution is over categories

plot displays a grid of plots, starting with the optimal defense behavior plotted as a discrete distribution on top of a (m x 2)-matrix of plots. Each line in this grid shows the discrete optimal attack strategy on the right side (as a bar plot), paired with the loss distribution (extracted from x) caused when the defender plays optimalDefense and the attacker plays the respective optimal attack strategy.

Author(s)

Stefan Rass

See Also

print.mosg.equilibrium, mgss, mosg, lossDistribution

Examples

## 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,weights=c(0.25,0.5,0.25))
print(eq)
summary(eq)
plot(eq)

# access the loss distributions computed in the game
summary(eq$assurances$g1)
mean(eq$assurance$g1)    # get the average loss in goal "g1"