Introduction to Game Theory and Applications,III

Fioravante PATRONE

DIPTEM, University of Genoa

Luxembourg, June 2010

Summary

Preliminaries and the corePreliminaries: cooperative games with transferable utilityThe core

The Shapley value and some applicationsThe axioms of ShapleyOther axioms for the Shapley valueCost componentsMicroarray gamesCost allocation for connection problems

Conclusions

Bibliography

Basic definitions

Formal definition:

▶ It is given a finite set N of players.

▶ Each group of players, that is, each coalition S ⊆ N, canobtain some amount v(S) ∈ ℝ

▶ So, a cooperative game with transferable utility isv : P(N)→ ℝ (con v(∅) = 0)

Interpretation: how do we interpret v(S)?

▶ Sum of the (transferable) utility values that players of Sobtain.

▶ “Money” that S is able to obtain.

▶ S could be winning or losing. It is the case in whichv(S) ∈ {0, 1}

Basic definitions

▶ Superadditivityv(S ∪ T ) ≥ v(S) + v(T ) if S and T are disjoint.

▶ Allocationsimply one element x of ℝN .

▶ Pre-imputationan allocation x such that

∑i∈N xi = v(N) (for a superadditive

game it is related with efficiency).

▶ Imputationbesides being a pre-imputation, we require also individualrationality: xi ≥ v({i}).

The core

One easily gets the idea of core adding, to conditions for animputation, conditions of “intermediate rationality”:∑

i∈S xi ≥ v(S) for all S .

Example. Simple majority game: N = {1, 2, 3}, v(S) = 1 if#(S) ≥ 2, and 0 otherwise.Easy to see that the core is empty.

Example. Glove game. N is partitioned into “L” and “R” (leftand right glove owners). v(S) = min{S ∩ L,S ∩ R} (unpairedgloves are worthless, pairs of glove have value 1).If #R = #L + 1, there is a unique allocation in the core: “left”owners get 1, “right” owners get 0. This is true for all values of#L, even if #L = 106.

The 4 classical axioms on SG(N).

Given N, SG(N) is the class of all superadditive games whose setof players is N. Or G(N) if we omit superadditivity.We would like to have Φ : SG(N)→ ℝN satisfying:

▶ Efficiency: for all games v , Φ(v) is a pre-imputation. That is:∑i∈N Φi (v) = v(N)

▶ Symmetry: Φi (v) = Φj(v) if, interchanging i with j , the gamev does not change

▶ Null player: Φi (v) = 0 if the contribution of i is null.That is, if:v(S ∪ {i}) = v(S) for all S

▶ Additivity: Φ(v + w) = Φ(v) + Φ(w)

Comments on the axioms.

Efficiency and symmetry are “obvious” conditions.

The “null player” axiom (a special case of the “dummy player”axiom) says that Shapley value distributes the “gains” takingsomehow into account the contributions of the single players.

Additivity: acceptable? Perplexities expressed by Luce and Raiffa(1957)

Mathematically, it works (the reasong can be adapted for SG(N)):

▶ Unanimity games are a basis for G(N).

▶ For the unanimity games the first 3 axioms determine theShapley value.

▶ Using additivity we extend it to all of G(N).

▶ So, the 4 axioms determine a unique solution on G(N).

Formulas for the Shapley value, I

A game v can be expressed as a linear combination of unanimitygames:

v =∑

S⊆N,S ∕=∅ �S(v)uS

So, thanks to linearity, The Shapley value can be calculated usingthe unanimity coefficients (�S(v)), that is:

�i (v) =∑

S⊆N:i∈S

�S(v)

s(1)

for each i ∈ N.Here, s denotes the number of elements of S , i.e., its cardinality.The number �S = �S

s is called Harsanyi dividend, relative to thecoalition S (Harsanyi, 1959).

Formulas for the Shapley value, II

Φi (v) =1

n!

∑�

m�i (v), (2)

where � is a permutation of N, while m�i (v) is the marginal

contribution of player i according to the permutation �, which isdefined as:

v({�(1), �(2), . . . , �(j)})− v({�(1), �(2), . . . , �(j − 1)}),

where j is the unique element of N s.t. i = �(j).Here is a condensed version of the formula:

Φi (v) =∑

S⊆N:i∈S

(s − 1)!(n − s)!

n!(v(S)− v(S ∖ i)) (3)

Some warnings.

▶ The allocation provided could not be stable (outside of thecore: glove game; empty core: simple majority game).

▶ Efficiency without superadditivity is a possibly stupid request.It depends: is one ”obliged” to stay in the ”big coalition”?

▶ Axioms could not be categorical on subsets of SG(N).Example: additivity for simple games, that is: v(S) ∈ {0, 1}.Replace with transfer:Φi (v ∨ w) + Φi (v ∧ w) = Φi (v) + Φi (w), ∀i ∈ N.

Other axiom systems

▶ Alvin Roth (1977) has taken seriously the idea of expectedvalue.

▶ Peyton Young (1985) emphasizes the marginalistic roots ofthe Shapley value.

▶ Sergiu Hart and Andreu Mas Colell (1987) introduced the ideaof consistency and of potential.

▶ Also Moretti et al. (2007) justify axiomatically the use of theShapley value for the microarray games.

Airport game.

Everyone pays for the components (printers, faxes, copiers, etc.)that he uses.The costs of each component are divided evenly among its users.

Famous case: airport game.Different types of planes need a landing strip of a different length.Each piece of the landing strip is paid equally by all of the planes(landings...) that use it.

Microarray experiment: results and discretization.

Shapley value for microarray games.

▶ Application of Shapley value to find which are the mostrelevant genes.

▶ Why Shapley value?

▶ New axiomatic characterization, which takes into accountspecific characteristics of the situation.

▶ Relevance of GT approach: give due emphasis to theinteractions among genes.

Experimental data.

Selection of microarray data from Alon et al. (1999) on colon cells.

Analyzed 2000 genes, and 62 samples (40 tumoral e 22 normal)

Results on the experimental data.

The five genes with the highest Shapley value:

Name of the gene Shapley value (×10−3)

H.sapiens mRNA for GCAP-II/ 3.83/uroguanylin precursor

Nucleolin 3.56

Gelsolin precursor, Plasma 3.34

DNA-(Apurinic or apirymidinic site) 3.23Lyase

Human vasoactive intestinal peptide (VIP) 3.21

Genes written in blue: it is known that are involved in cancerprocesses.

Minimum cost spanning tree games.

Everyone needs to connect with a source (sink):- irrigation system- electricity provision- sewage facility

Graph with players as nodes; each arc has a cost.v(S) as the cost of the MCST lying inside S.

Best (efficient) solution: minimum cost spanning tree.

An example.

q

1q

3

q2

t

S

2 1

4

12108

v({1}) = 8

v({2}) = 12

v({3}) = 10

v({1, 2}) = 10

v({1, 3}) = 12

v({2, 3}) = 11

v({1, 2, 3}) = 11

Minimum cost spanning tree games: solutions

Question: how to allocate costs?

- Bird rule → a element in the core (extreme element).- use Shapley value...

Conclusions

1. Shapley value is pervasive.

2. Often (not always!) it is easy to calculate.

3. It has unanticipated applications. Not strange, given thesimple and universal conditions it satisfies.

See Moretti and Patrone: Transversality of the Shapley value,about 1) and 3).

▶ Some modest suggestions▶ Integrate various disciplines.▶ Look from different views.

Suggested readings I

A.E. Roth (ed).The Shapley value, essays in honor of Lloyd S. Shapley.Cambridge University Press, Cambridge, 1988.

A.E. Luce, H. Raiffa.Games and Decisions.Wiley, New York, 1957.

S. Moretti, F. Patrone.Transversality of the Shapley valueTop, 16, 1-41, (2008).Invited paper; comments by Fragnelli, Grabisch, Haake,Garcıa-Jurado, Sanchez-Soriano, Tijs; plus rejoinder.

References I

L.S. Shapley:A Value for n-Person Games.in Contributions to the Theory of Games,H.W. Kuhn e A.W. Tucker eds.Annals of Math. Studies, 28Princeton University Press, Princeton (NJ),307–317, 1953.

H. P. Young:Monotonic solutions of cooperative games.Int J Game Theory, 14, 65–72, 1985.

A.E. Roth:The Shapley value as a von Neumann–Morgenstern utility.Econometrica, 45, 657–664, 1977.

References II

S. Hart, A. Mas Colell:Potential, value and consistency.Econometrica, 57, 589–614, 1987.

V. Fragnelli, I. Garcıa-Jurado, H. Norde, F. Patrone, S. Tijs:How to Share Railways Infrastructure Costs?in Game Practice: Contributions from Applied Game Theory,F. Patrone, I. Garcıa-Jurado, S. Tijs (eds.)Kluwer, Dordrecht, 91–101, 1999.

S. Moretti, F. Patrone, S. Bonassi:The class of microarray games and the relevance index forgenes. TOP, 15, 256–280, 2007.