D. Ashlock
New Versions of Risk
1
New Games from
Old: Influence
Maps and New
Versions of Risk
Daniel Ashlock and Colin Lee,
University of Guelph
An influence map is a way of determining how
much one square or tile in a game affects or influences the others. In this column we use influence maps to produce and grade variations of
the popular board game Risk, originally published
by Parker Brothers. In our variation two giant
monoliths appear on the game map that permit
troops to pass between them. The influence map
is used to determine how much a given placement
of the monoliths changes the game.
1 Introduction
isk(tm), a publication of Parker Brother, now
a division of Hasbro, is a simple war game
played on a stylized map of the world. The game
is played with tokens representing armies, five sixsided dice, and a deck of cards emblazoned with
one of the countries and a soldier, cannon, or cavalry soldier. Two jokers have no country but all
three emblems. Players put an initial collection
of armies on countries selected by dealing out the
cards.
The players take turns, gaining new armies
in proportion to number of territories held with
bonuses for owning whole continents. In a turn a
player attacks from a country with at least two of
his armies to an adjacent country held by another
player, as many times as he wishes. Dice rolled by
both players determine armies lost on each side.
If the defender runs out of armies, the attacker
captures the country. In any turn that a country
is taken, the attacker draws a card. At the beginning of the turn triples with the same emblem, or
all different, may be turned in for bonus armies.
The bonus grows as triples are cashed in. The
goal of the game is to take control of the entire
world. A basic version of the Risk board is shown
in Figure 1. The adjacencies of countries are the
wellspring of strategy for Risk.
R
1
Figure 1: The Layout of the Risk Board, courtesy
of Gr0gmint in the Wikimedia commons.
A number of official variations including Risk
2210 and versions based on George Lucas’s Clone
Wars and Tolkien’s Lord of the Rings have been
published. Many different rule variations and
house rules exist. This column will look at a simple modification of the original game which can
be used to disrupt some favorite strategies, such
as holing up in Australia, and which changes the
game in a quantifiable way.
One way of introducing this variation of Risk
can be explained with the following science fiction
metaphor. As the war ramps up, increased scientific research and development causes a breakthrough in remote sensing which leads to the discovery of two huge alien monoliths buried in the
earth. Once activated, the monoliths permits
troops to be transferred between them. This creates an additional link on the Risk board. The
location of the monoliths can be agreed upon by
the players before country selection starts or, if
true chaos is desired, chosen by drawing random
country cards. The second card should be rejected
until it is on a different continent from the first.
This variation on Risk is not a radical change in
game, but a collection of new versions of Risk with
substantially different strategy from the original
game.
The second feature of this publication is an
introduction to the idea of influence maps. A tutorial on influence maps appears in 1 .
Briefly, an influence map is a way of quantifying
how much influence each square, hex, or tile on
a game board has on the others. Computing influence maps requires knowledge of the adjacency
relationship of the regions of the game. For some
games, like chess, this is quite tricky because the
adjacency of squares depends on the identity of
http://aigamedev.com/open/tutorial/influence-map-mechanics/
Game & Puzzle Design, Vol.1, No.1, 2015, pp.1–3.
2
Game & Puzzle Design
the piece occupying the square. In Risk the adjacency relationship, while central to game strategy,
is simple and consists of sharing a border or being explicitly connected by a line printed on the
board. What we are going to do is to abstract
the Risk board as a combinatorial graph, compute a type of influence map for each country on
each other country, and then use a measure of how
much linking two countries with the monolith mechanic changes the overall influence map.
Vol.1, No.1, 2015
{3,4},{2,5},{3,4},{3,5},{4,5}}
A map and its corresponding graph are different
ways of representing the same information. For
the game of Risk, V is the set of countries, and E
is the set of pairs of countries that can attack one
another.
2 The Math
This subsection gently introduces the math we use
to estimate the impact of the placement of different pairs of monoliths on the Risk board.
2.1 A Little Graph Theory
A graph is a set of elements V and collection of
relations between unordered pairs of elements of
V . Formally we can think of a graph as a set V of
elements we call vertices and a set E of elements
we call edges, where each edge is an unordered
distinct pair of vertices. Informally a graph is a
bit easier to understand as a network, where vertices are nodes and edges are arcs connecting two
nodes.
Figure 3: The Combinatorial Graph of the Risk
Board.
We will be looking at the underlying graph of
the standard Risk board, shown in Figure 3. We
assign a number between 1 and 42 to each of the
42 different countries on a Risk board and consider these 42 numbers to represent vertices in a
graph. These vertices are connected by an edge
if and only if the corresponding countries are adjacent. Now our question for which pair of countries would adding the monoliths change the game
the most can be formulated as “If we can insert a
single edge into the underlying graph of the Risk
board, which edge would change the connectivity
of the resulting graph the most?”
2.2 Diffusing Gas Analogy
Figure 2: A Drawing of the complete graph on
five vertices.
There are many ways to represent the same
graph. For example a graph can be represented as
a drawing, as in Figure 2, as two sets (V and E),
or in a number of different matrix formats. The
graph shown above is K5 , the complete graph on
five vertices. This graph has five vertices and all
possible edges. The vertex and edge lists for this
graph are:
V ={1,2,3,4,5}
E={{1,2},{1,3},{1,4},{1,5},{2,3},
We now give an analog explanation of the influence map we will be using. Start with some graph
G, suppose we pumped 1 unit of gas into the graph
at a specific vertex at each time step. Let the gas
diffuse across edges, with each vertex absorbing
some constant fraction of gas in each time step.
The gas at a vertex is divided evenly among the
vertex and its neighbors. The rate of absorption is
denoted by 0 < α < 1. The interplay of arithmetic
increase in gas and exponential decay mediated by
α mean that the level of each gas at each vertex
will reach equilibrium. The amount of given gas
at equilibrium at a given vertex x gives us a measure of the connectivity from our starting vertex
D. Ashlock
New Versions of Risk
to the vertex x.
If we calculated the amount of gas present at
each vertex for all possible starting vertices we
have measures of the connectivity of the graph
with respect to all the vertices. Let’s call the
42 × 42 matrix where each column and row corresponds to one of the countries on the Risk board
that has in its (i; j)th entry the amount of gas
present at vertex i after the system comes to equilibrium supposing that gas was pumped in at vertex j, the diffusion character matrix of the graph.
Since the amount of gas in the entire graph does
not depend on the which vertex we pick as our
starting vertex to pump gas into, it follows that
the diffusion character matrix has a constant column sum. So we can normalize this matrix by dividing by the constant column sum to get a matrix
with columns that all sum to 1 and call this a normalized diffusion character matrix. The columns
of the normalized diffusion character matrix can
be thought of as discrete probability distributions,
where the (i; j)th entry corresponds to the probability of ending a random walk along graph edges
at vertex i given that you started at vertex j. The
diffusion character matrix is the influence map,
telling us how much influence each vertex has on
each other. Changes in the connectivity of the
map will causes changes in the influence map.
So one way of measuring the difference in the
connectivity between two graphs (with the same
number of vertices) would be to compare the
normalized diffusion character matrices of those
graphs.
2.3 Looking at Monolith Placements
First we go through and generate all the graphs
that can be obtained by each possible placement
of the monoliths not in already adjacent territories. Realized in the graph representation, this is
all ways of adding one edge to the graph derived
from the Risk game board. For each of these new
graphs we compare it’s normalized diffusion character matrix with the normalized diffusion character matrix of the original graph. Then we note
which new graphs have the largest difference from
the original graph.
Before going on we need to make a quick note
on the difference that choosing two representations make on a graph. If we chose two different
orderings of the countries of the Risk board we
could not simply compare the (i; j)th entries of
the two resulting normalized diffusion character
matrices. We would need to compare the entries
3
corresponding to the same countries. Similarly if
we add an edge to the graph of the Risk board
we cannot simply compare corresponding entries
since the resulting graph may no longer have vertices ordered in the same way, by the test statitic,
as the original graph. In other words, in the new
graph, East Australia may now be strategically
more like Venezuela than East Australia in the
old graph.
To get around this ordering problem we calculate a measure of the skewness of the probability distribution associated with each column of
a normalized diffusion character matrix and sort
this list and compare differences in the sorted lists
for two graphs. Skewness is a measure of deviation from symmetry - the degree to which the gas
is or is not evenly distributed. We could compare vertices corresponding to countries with the
same name, but when edges are added, the strategic meaning changes. Placing skewness measures
for the vertices in descending order and then using the distance between these vectors provides a
more objective assessment of how much or how
little the board as changed.
It is also worth noting that we do not in fact
have a strong reason to prefer either of our two
skewness measures and we have no reason to suppose that long or short range influence is more
important. For this reason we present six sets
of results - for both skewness measures and three
values of α, which controls the range of influence
considered.
3 Biggest and Smallest Changes
We still need values for α in order to perform our
calculations. Since this parameter controls the degree to which we care about longer or shorter distances, we will calculate diffusion character matrices for three values of α: 1/3, 1/2, and 2/3.
We have chosen two measures of the skewness
of a probability distribution, we calculate these
measures for each column of the normalized diffusion character matrix to get a list of skewness
scores:
Entropy is given by:
−
n
X
pi · log2 (pi )
i=1
where the pi are the n probabilities making up the
probability distribution represented by the entries
in a column of a normalized diffusion character
4
Game & Puzzle Design
matrix.
The column 2-norm is given by:
v
u n
uX
t
p2i ,
Vol.1, No.1, 2015
Largest Entropy List Difference
Score
Pair
0.21447
Peru, New Guinea
0.21447
Peru, Western Australia
0.22802
Brazil, Eastern Australia
0.24523
Argentina, Eastern Australia
0.24991
Peru, Eastern Australia
i=1
although we will actually ignore the square root
and simply take the sum. The following charts
use the Euclidean distance squared between the
sorted lists of skewness measures of the original
Risk board game and a single edge difference Risk
board graph.
Note a high score for both Entropy List Difference and Column 2-norm Difference correspond
to a large difference between the skewnesses of
normalized diffusion character matrices and hence
correspond to large differences in the overall connectivity of the underlying graphs which are surrogates for the degree to which game strategy shifts.
For α = 1/3
Largest Entropy List Difference
Score
Pair
0.063200
South Africa, New Guinea
0.063526
Peru, Eastern Australia
0.064973
Argentina, Western Australia
0.064973
Argentina, New Guinea
0.069028
Argentina, Eastern Australia
Smallest Entropy List Difference
0.0017944
Quebec, Western United States
0.0017297
Ural, India
0.0014588
Western Europe, Scandinavia
0.0014205
Siberia, Kamchatka
0.00056081 Yakutsk, Mongolia
Largest Column 2-norm List Difference
Score
Pair
0.00044022
South Africa, Eastern Australia
0.00044203
Peru, South Africa
0.00045461
Argentina, South Africa
0.00045937
Argentina, Eastern Australia
0.00045971
Peru, Eastern Australia
Smallest Column 2-norm List Difference
0.0000011146
Ural, India
0.0000011106
Egypt, Western Europe
0.0000010968
Western Europe, Ukraine
0.0000010188
Yakutsk, Mongolia
0.00000088760 North Africa, Middle East
For α = 1/2
Smallest Entropy List Difference
0.0020469
Alberta, Greenland
0.0015425
Irkutsk, Japan
0.0015059
Quebec, Western United States
0.00098565 Ural, India
0.00048602 Yakutsk, Mongolia
Largest Column 2-norm List Difference
Score
Pair
0.0016966
Brazil, Eastern Australia
0.0017238
Argentina, New Guinea
0.0017238
Argentina, Western Australia
0.0019256
Peru, Eastern Australia
0.0019491
Argentina, Eastern Australia
Smallest Column 2-norm List Difference
0.00023200 Siberia, Kamchatka
0.00021703 Western Europe, Ukraine
0.00019740 Yakutsk, Mongolia
0.00018912 Western Europe, Scandinavia
0.00014919 North Africa, Middle East
For α = 2/3
Largest Entropy List Difference
Score
Pair
0.67089
North Africa, Eastern Australia
0.67611
Argentina, Eastern Australia
0.71088
Venezuela, Eastern Australia
0.71319
Peru, Eastern Australia
0.74140
Brazil, Eastern Australia
Smallest Entropy List Difference
0.0013345 Irkutsk, Japan
0.0011615 Yakutsk, Japan
0.0010419 Alberta, Greenland
0.0084558 Ural, India
0.0074116 Quebec, Western United States
Largest Column 2-norm List Difference
Score
Pair
0.0051404
Peru, New Guinea
0.0051404
Peru, Western Australia
0.0053178
Brazil, Eastern Australia
0.0055653
Argentina, Eastern Australia
0.0057366
Peru, Eastern Australia
Smallest Column 2-norm List Difference
0.000017741 Ural, India
0.000017652 Yakutsk, Mongolia
0.000017139 Siberia, Kamchatka
0.000016462 Western Europe, Scandinavia
0.000012228 North Africa, Middle East
D. Ashlock
New Versions of Risk
4 Discussion and Conclusions
It is worth noting while in general similar edges
(pairs of countries) appear in the same top 5
blocks for the different values of α, the order in
which they appear varies. The agreement in areas of continents is greater than that of individual
pairs of countries. This is somewhat troublesome
but is still not unexpected as the values of α essentially measure how much we care about long
range connectivity (the smaller the value of α the
shorter the range we care about the most), but we
can none the less get a good idea of which edges
will lead to the greatest difference in the connectivity of the Risk board. These edges are (Peru,
Eastern Australia) and (Argentina, Eastern Australia).
It is reasonable to conclude that there is mathematical justification to believe that either placing a portal between Peru and Eastern Australia,
or between Argentina and Eastern Australia will
result in the largest change in connectivity and
hence largest change in game play. New games
from old would be happy to hear from readers that
try our suggested new versions of Risk.
Those who have played Risk many times before know that there are players that like to hide in
Australia and control Oceania, others that try and
use South America as a springboard to controlling Africa, and occasional overconfident youngsters that try and grab North America right away.
Vizzini’s advice from The Princess Bride, “never
get involved in a land war in Asia” also applies
to all but the ending stages of Risk. Monolith
Risk can change much of this strategic equation.
It would not, however, soften Vizzini’s advice.
This suggests that, in addition to adding links,
one might want to postulate monolith technology
that projected force fields that cut links. These
could also be planned and categorized with diffusion characters.
4.1 Diffusion Characters as
Influence Maps
Computing diffusion characters is equivalent to
inverting a square matrix with as many rows as
there are vertices in the graph. This might seem
a little time consuming. This isn’t the case, however. Computing a diffusion character is an offline
5
activity and, once it is computed, the diffusion
character can be saved and reused for as many
purposes as needed.
The influence of a block of vertices can be computed with a diffusion character - simply add up
the gas from each vertex in the block on any particular vertex and you know the aggregate influence of the vertices in the block on that particular
vertex. It is also possible to modify the computation of the diffusion character itself to achieve
other results. Suppose, instead of adding one unit
of gas to a vertex, you added as many unit of as
gas as there are armies on the country corresponding to the vertex. Then the diffusion character
would measure the armed might you could bring
to bear. How immediate your ability to threaten
a country is can be explored by varying α.
Acknowledgments
The Authors than the Natural Sciences and Engineering Council of Canada (NSERC) and the University of Guelph for their support of this work.
Dr Daniel Ashlock is a Professor of Mathematics
at the University of Guelph in Guelph, Ontario,
Canada. His research interests include applications of computational intelligence to both
mathematical games and games played by people.
Daniel Ashlock, Department of Mathematics and Statistics, University of
Guelph, Guelph, Ontario, Canada.
Email: [email protected]
URL: http://eldar.mathstat.uoguelph.ca/
dashlock/
Dr Colin Lee is a consultant in Guelph, Ontario,
Canada.
Colin Lee, Department of Mathematics and Statistics, University of Guelph,
Guelph, Ontario, Canada, N1G 2W
Email: [email protected]
URL: http://eldar.mathstat.uoguelph.ca/