Diapositiva 1

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A new family of regular semivalues and
applications
Roberto Lucchetti Politecnico di Milano,Italy
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• Main goal
• To rank genes from DNA data provided by
  Microarray Analysis.
• Tools
• Cooperative Game Theory, in particular Power
  indices
• Power indices rank players according to their
  strength in the game.
• In the EU council the strongest states
  (GE,FR,IT,UK) have a some 10 times power w.r.t.
  the weakest state (MT)
• In UN the veto players have a some 100 (10) times
  power w.r.t. non permanent players, according to
  Shapley (Banzhaf).

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A (TU) game is
with N1,,n is the set
of players, v is the characteristic function of
the game. A ? N is called coalition. v(A) is
the utility (or cost) for the coalition A. GN
represents the set of all games having N as set
of players. Remark GN ? R2n-1
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• A Base for GN
• Unanimity games
• Subclass of games
• Simple games.
• Among them the weighted majority games

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Introduction how an array works
A chip can contain millions of DNA probes
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Hybridization helps researchers to identify what
RNA sequences are present in a sample and this
tells them what genes are being expressed by the
organism and how much they are being expressed.
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GeneChip microarrays use the natural chemical
attraction between the RNA target (from the
sample preparation) and the DNA on the array to
determine the expression level of a given gene.
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The RNA extract from a sample is copied in cRNA
(through a process known as PCR). Copying the
RNA allows it to be more easily detected on the
array. At the same time the RNA is copied, a
chemical flourescent molecule called biotin is
attached to the strand. This molecule will show
where the sample RNA has stuck to the DNA probe
on the array.
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If the sample RNA doesnt match it will be
rejected by the probe on the array and when the
laser hits the probe, nothing glows.
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Introduction how a microarray works
The whole point of microarray gene expression
analysis is to compare expression levels among
different samples. Lets simplify the situation
with an example in which we have four genes and
two samples.
Gene1 2RUDE Gene2 2LOUD Gene3 GETOUT Gene4
FATMET
Gene4 is not glowing.
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array 1 array 2 array 3 array 4
gene 1 0,67 0,45 1,32 1,34
gene 2 1,01 1,13 1,54 2,13
gene 3 1,38 1,21 1,23 0,12
gene 4 0,65 0,98 0,54
gene 5 0,17 1,32 2,43

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• The Microarray Game
• An mxn Boolean matrix M such that
• Given the column , supp

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Sample 1 Sample 2 Sample 3
gene1 0.5 0.2 1
gene2 0.4 1 0.3
gene3 0.8 0.4 0.2
Sample1 Sample2 Sample3 Sample 4
gene1 0.7 0.3 1.8 0.8
gene2 0.1 0.2 0.5 0.9
gene3 1 0.6 1.7 0.1
Sample1 Sample2 Sample3 Sample4
gene1 0 0 1 0
gene2 1 1 0 0
gene3 1 0 1 1
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• A power index for the game (N,v) is
  (x1,,xn) such
• that
• xi represents the power of player i in game v.
• weighted voting does not work
• The most famous
• Shapley (?) and Banzhaf (?) .

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Shapley (?) and Banzhaf(?)
the marginal contribution of i to S ? i
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? is a probabilistic value if there is a
probability
on

such that

• Shapley

• Banzhaf

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If pi(S)p(S)gt0, the probabilistic value is
called regular semivalue Examples Banzhaf
Shapley p-binomial
Regular semivalues are points in the simplex
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• Properties for power indices
• Let
• The solution ? has the dummy player (DP)
  property, if for each player such that
• for all coalitions A not containing i,

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Let be a
permutation. Given the game v, denote by
the game and by The solution ? has the
symmetry (S) property if, for each permutation
as above
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The new family of power indices
Let Define on the unanimity game
as and extend it by linearity on a generic
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• Theorem 1
• There exists one and only one value ? fulfilling
  the symmetry, linearity and dummy player
  properties, and assigning aS to all non null
  players in the unanimity game uS , where a11
  and asgt0 for s2,,n.
• ? fulfills the formula

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• Theorem 2
• ? a is a regular semivalue for all agt0.
• ? 2 fulfills the formula

• Corollary
• The family of the weighting coefficients of the
  values ? a, agt0,
• is an open curve in the simplex of the regular
  semivalues, containing the Shapley value.
• The addition of the Banzhaf value to the curve
  provides a one-point compactification of the
  curve.

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• Theorem 3 study of the term
• Key tool
• Let , let
• Then
• Moreover, for all natural l, and positive real
  a,x
• Finally, for each natural m, the following
  formula holds

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Calculating the indices in weighted majority games
Let count in how many ways the sum
of the weights of j players different from i can
give k. Then the following proposition
holds. Let ? be the value defined in the theorem
above. Let qgt0 be a positive integer, and let
w1,,wn be non negative integers. Let
vqw1,,wn be the associated weighted majority
game. Then the following formula holds
An efficient algorithm based on generating
functions and formal series allows for a fast
calculation of the coefficients
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• Applications
• The EU

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• Data from a Colon Rectal Cancer
• 10 Healthy 12 Tumoral tissues
• An extended microarray game considers also how
  much the genes are abnormally expressed w.r.t a
  normality interval.
• Given the normality interval mi,Mi of the gene
  i, si the standard deviation, Nkimi-ksi,miksi,
  assign k to the ij cell of the matrix if value
  of gene i in patient j falls in Nik \ Nik-1
• A weighted Shapley value is used to rank genes.
  This allows better differentiating the genes.
  Taking the first 100 genes in the ranking, the
  game is formed as an average of weighted majority
  games.
• Then we calculate the Shapley, Banzhaf and ?2
  indices

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• Gene expression analysis was performed by using
  Human Genome U133A-Plus 2.0 GeneChip arrays
  (Affymetrix, Inc., Calif).
• The following 7 genes are quoted in medical
  literature as having great importance in the
  onset of the disease
• CYR61, UCHL1, FOS,FOSB, EGR1, VIP, KRT24.
• One of them was ranked around the 100-th position
  by the weighted Shapley value. All other ones are
  among the first 50 and played the subsequent
  game.

S B ?2
FOSB 2 1 1
CYR61 1 2 2
FOS 3 3 3
VIP 5 5 6
EGR1 10 9 9
KRT24 45 35 35
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References

• R.Lucchetti P.Radrizzani, E. Munarini, A new
  family of regular semivalues and applications,
  Int.J.of Game Theory DOI 10.1007/s00182-010-0263-
  5
• R. Lucchetti-S. Moretti-F. Patrone-P. Radrizzani,
  The Shapley and Banzhaf indices in microarray
  games, Computers and Operations Research, 37,
  (2010) p. 1406-1412.
• R. Lucchetti-P.Radrizzani, Microarray Data
  Analysis Via Weighted Indices and Weighted
  Majority Games, Computational Intelligent Methods
  for Bioinformatics and Biostatistics II, Masulli,
  Peterson, Tagliaferri (Eds), Lecture Notes in
  Computer Science, Springer (2010) p.179-190.
• S.Moretti , F.Patrone, S.Bonassi, The class of
  microarray
• games and the relevance index for genes. TOP 15
  (2007), p256-280.
• D. Albino, P. Scaruffi, S. Moretti, S.Coco, C.Di
  Cristofano, A.Cavazzana, M.Truini, S.Stigliani,
  S.Bonassi, G.Ptonini (2008) Stroma poor and
  stroma rich gene signatures show a low
  intratumoral gene expression heterogeneity in
  Neuroblastic tumors. Cancer 113, p. 1412-1422.

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