cooperative game theory
DESCRIPTION
AACIMP 2011 Summer School. Operational Research Stream. Lecture by Sirma Zeynep Alparslan Gok.TRANSCRIPT
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Cooperative Game Theory. Operations ResearchGames. Applications to Interval Games
Lecture 1: Cooperative Game Theory
Sırma Zeynep Alparslan GokSuleyman Demirel UniversityFaculty of Arts and SciencesDepartment of Mathematics
Isparta, [email protected]
August 13-16, 2011
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Outline
Introduction
Introduction to cooperative game theory
Basic solution concepts of cooperative game theory
Balanced games
Shapley value and Weber set
Convex games
Population Monotonic Allocation Schemes (pmas)
References
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Introduction
Introduction
I Game theory is a mathematical theory dealing with models ofconflict and cooperation.
I Game Theory has many interactions with economics and withother areas such as Operations Research and social sciences.
I A young field of study:The start is considered to be the book Theory of Games andEconomic Behaviour by von Neumann and Morgernstern.
I Game theory is divided into two parts: non-cooperative andcooperative.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Introduction
Introduction
Cooperative game theory deals with coalitions who coordinate theiractions and pool their winnings.Natural questions for individuals or businesses when dealing withcooperation are:
I Which coalitions should form?
I How to distribute the collective gains (rewards) or costsamong the members of the formed coalition?
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Introduction to cooperative game theory
Cooperative game theory
I A cooperative n-person game in coalitional form (TU(transferable utility) game) is an ordered pair < N, v >, whereN = {1, 2, ..., n} (the set of players) and v : 2N → R is amap, assigning to each coalition S ∈ 2N a real number, suchthat v(∅) = 0.
I v is the characteristic function of the game.
I v(S) is the worth (or value) of coalition S .
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Introduction to cooperative game theory
Example (Glove game)
N = {1, 2, 3}. Players 1 and 2 possess a left-hand glove and theplayer 3 possesses a right-hand glove. A single glove is worthnothing and a right-left pair of glove is worth 10 euros.Let us construct the characteristic function v of the game< N, v >.
v(∅) = 0, v({1}) = v({2}) = v({3}) = 0,
v({1, 2}) = 0, v({1, 3}) = v({2, 3}) = 10, v(N) = 10.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Introduction to cooperative game theory
Cooperative game theory
I GN : the set of coalitional games with player set N
I GN forms a (2|N| − 1)-dimensional linear space equipped withthe usual operators of addition and scalar multiplication offunctions.
I A basis of this space is supplied by the unanimity games uT
(or < N, uT >), T ∈ 2N \ {∅}, which are defined by
uT (S) :=
{1, if T ⊂ S0, otherwise.
The interpretation of the unanimity game uT is that a gain (orcost savings) of 1 can be obtained if and only if all players incoalition T are involved in cooperation.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Introduction to cooperative game theory
Example
Since the unanimity games is the basis of coalitional games, eachcooperative game can be written in terms of unanimity games.Consider the game < N, v > with N = {1, 2}, v({1}) = 3,v({2}) = 4 and v(N) = 9.Here v = 3u{1} + 4u{2} + 2u{1,2}.Let us check it
v({1}) = 3u{1}({1}) + 4u{2}({1}) + 2u{1,2}({1}) = 3 + 0 + 0 = 3.
v({2}) = 3u{1}({2}) + 4u{2}({2}) + 2u{1,2}({2}) = 0 + 4 + 0 = 4.
v({1, 2}) = 3u{1}({1, 2})+4u{2}({1, 2})+2u{1,2}({1, 2}) = 3+4+2 = 9.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Basic solution concepts of cooperative game theory
Basic solution concepts of cooperative game theory
A payoff vector x ∈ Rn is called an imputation for the game< N, v > (the set is denoted by I (v)) if
I x is individually rational: xi ≥ v({i}) for all i ∈ N
I x is efficient:∑n
i=1 xi = v(N)
Example (Glove game continues): The imputation set of the glovegame LLR is the triangle with vertices
f 1 = (10, 0, 0), f 2 = (0, 10, 0), f 3 = (0, 0, 10)
I (v) = conv {(10, 0, 0), (0, 10, 0), (0, 0, 10)}
(solution of the linear systemx1 + x2 + x3 = 10, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Basic solution concepts of cooperative game theory
The core (Gillies (1959))
The core of a game < N, v > is the set
C (v) =
{x ∈ I (v)|
∑i∈S
xi ≥ v(S) for all S ∈ 2N \ {∅}
}.
The idea of the core is by giving every coalition S at least theirworth v(S) so that no coalition has an incentive to split off.
I If C (v) 6= ∅, then elements of C (v) can easily be obtained,because the core is defined with the aid of a finite system oflinear inequalities (optimization-linear programming (seeDantzig(1963))).
I The core is a convex set and the core is a polytope (seeRockafellar (1970)).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Basic solution concepts of cooperative game theory
Example (Glove game continues)...
The core of the LLR game consists of one point (0, 0, 10).
C (v) = {(0, 0, 10)}
(solution of the linear system x1 + x2 + x3 = 10,
x1 + x2 ≥ 0, x1 + x3 ≥ 10, x2 + x3 ≥ 10, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.)
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Balanced games
Balanced game
A map λ : 2N \ {∅} → R+ is called a balanced map if∑S∈2N\{∅} λ(S)eS = eN .
Here, eS is the characteristic vector for coaliton S with
eSi :=
{1, if i ∈ S0, if i ∈ N \ S .
An n-person game < N, v > is called a balanced game if for eachbalanced map λ : 2N \ {∅} → R+ we have∑
S
λ(S)v(S) ≤ v(N).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Balanced games
Example
For N = {1, 2, 3}, the set B = {{1, 2} , {1, 3} , {2, 3}} is balancedand corresponds to the balanced map λ with λ(S) = 1
2 if |S | = 2.That is
1
2v({1, 2}) +
1
2v({1, 3}) +
1
2v({2, 3}) ≤ v(N)
Let us show it:
λ({1, 2})e{1,2} + λ({1, 3})e{1,3} + λ({2, 3})e{2,3} = eN
λ({1, 2})(1, 1, 0) + λ({1, 3})(1, 0, 1) + λ({2, 3})(0, 1, 1) = (1, 1, 1).
Solution of the above system isλ({1, 2}) = λ({1, 3}) = λ({2, 3}) = 1
2 .
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Balanced games
Balanced game
The importance of a balanced game becomes clear by the followingtheorem which characterizes games with a non-empty core.
Theorem (Bondareva (1963) and Shapley (1967)): Let < N, v >be an n-person game. Then the following two assertions areequivalent:
(i) C (v) 6= ∅,(ii) < N, v > is a balanced game.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Shapley value and Weber set
Marginal contribution
Let v ∈ GN . For each i ∈ N and for each S ∈ 2N with i ∈ S , themarginal contribution of player i to the coalition S is
Mi (S , v) := v(S)− v(S \ {i}).
Let Π(N) be the set of all permutations σ : N → N of N.The set Pσ(i) :=
{r ∈ N|σ−1(r) < σ−1(i)
}consists of all
predecessors of i with respect to the permutation σ.Let v ∈ GN and σ ∈ Π(N).The marginal contribution vector mσ(v) ∈ Rn with respect to σand v has the i-th coordinate the valuemσ
i (v) := v(Pσ(i) ∪ {i})− v(Pσ(i)) for each i ∈ N.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Shapley value and Weber set
Example
Let < N, v > be the three-person game with v({i}) = 0 for eachi ∈ N, v({1, 2}) = 3, v({1, 3}) = 5, v({2, 3}) = 7, v(N) = 10.Then the marginal vectors are given in the following table, whereσ : N → N is identified with (σ(1), σ(2), σ(3)).
σ(123)(132)(213)(231)(312)(321)
mσ1 (v) mσ
2 (v) mσ3 (v)
0 3 70 5 53 0 73 0 75 5 03 7 0
.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Shapley value and Weber set
The Shapley value (Shapley (1953)) and the Weber set(Weber (1988))
The Shapley value φ(v) of a game v ∈ GN is the average of themarginal vectors of the game
φ(v) := 1n!
∑σ∈Π(N) mσ(v).
This value associates to each n-person game one (payoff) vector inRn.The Shapley value of the previous example is
φ(v) =1
3!(14, 20, 26) = (
7
3,
10
3,
13
3).
The Weber set (Weber (1988)) W (v) of v is defined as the convexhull of the marginal vectors of v .Theorem: Let v ∈ GN . Then C (v) ⊂W (v).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Shapley value and Weber set
Example (LLR game)
Marginal vectors can be observed from the following table
σ(123)(132)(213)(231)(312)(321)
mσ1 (v) mσ
2 (v) mσ3 (v)
0 0 100 0 100 0 100 0 10
10 0 00 10 0
.
The Weber set is W (v) = conv {(0, 0, 10), (10, 0, 0), (0, 10, 0)}.The Shapley value is φ(v) = ( 1
6 ,16 ,
23 ).
Note that {(0, 0, 10)} = C (v) ⊂W (v) and φ(v) ∈W (v).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Convex games
Convex games
I < N, v > is convex if and only if the supermodularitycondition v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T ) for eachS ,T ∈ 2N holds (desirable for reward games).
I < N, v > is called concave (or submodular) if and only ifv(S ∪ T ) + v(S ∩ T ) ≤ v(S) + v(T ) for all S ,T ∈ 2N
(desirable for cost games).
I CGN–The family of all convex games with player set N.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Convex games
Theorem (characterizations of convex games): Let v ∈ GN . Thefollowing five assertions are equivalent:
(i) < N, v > is convex.
(ii) For all S1,S2,U ∈ 2N with S1 ⊂ S2 ⊂ N \ U we have
v(S1 ∪ U)− v(S1) ≤ v(S2 ∪ U)− v(S2).
(iii) For all S1,S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N \ {i} wehave
v(S1 ∪ {i})− v(S1) ≤ v(S2 ∪ {i})− v(S2).
(iv) Each marginal vector mσ(v) of the game v with respect tothe permutation σ belongs to the core C (v).
(v) W (v) = C (v).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Convex games
I For convex games the gain made when individuals or groupsjoin larger coalitions is higher than when they join smallercoalitions.
I A convex game is balanced and the core of the convex gamesis nonempty.
I The Shapley value is a core element if the game is convex.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Population Monotonic Allocation Schemes (pmas)
Population Monotonic Allocation Schemes (pmas)
For a game v ∈ GN and a coalition T ∈ 2N \ {∅}, the subgamewith player set T , (T , vT ), is the game vT defined byvT (S) := v(S) for all S ∈ 2T .
A game v ∈ GN is called totally balanced if (the game and) all itssubgames are balanced.
The class of totally balanced games includes the class of gameswith a population monotonic allocation scheme (pmas) (Sprumont(1990)).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Population Monotonic Allocation Schemes (pmas)
Let v ∈ GN . A scheme a = (aiS)i∈S,S∈2N\{∅} of real numbers is apmas of v if
(i)∑
i∈S aiS = v(S) for all S ∈ 2N \ {∅},(ii) aiS ≤ aiT for all S ,T ∈ 2N \ {∅} with S ⊂ T and for each
i ∈ S .
I Interpretation: in larger coalitions, higher rewards (or in largercoalitions lower costs).
I It is known that for v ∈ CGN the (total) Shapley valuegenerates population monotonic allocation schemes. Further,in a convex game all core elements generate pmas.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Population Monotonic Allocation Schemes (pmas)
ExampleLet < N, v > be the 3-person game with v({1}) = 10,v({2}) = 20, v({3}) = 30,v({1, 2}) = v({1, 3}) = v({2, 3}) = 50, v(N) = 102.Then a pmas is the (total) Shapley value.
Φ({1, 3} , v)→
N{1, 2}{1, 3}{2, 3}{1}{2}{3}
1 2 329 34 3920 30 ∗15 ∗ 35∗ 20 30
10 ∗ ∗∗ 20 ∗∗ ∗ 30
.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
Population Monotonic Allocation Schemes (pmas)
For detailed information about Cooperative game theory see
I Introduction to Game Theory by Tijs
and
I Models in Cooperative Game Theory by Branzei, Dimitrovand Tijs.
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
References
References
[1]Bondareva O.N., Certain applications of the methods of linearprogramming to the theory of cooperative games, ProblemlyKibernetiki 10 (1963) 119-139 (in Russian).[2]Branzei R., Dimitrov D. and Tijs S., Models in CooperativeGame Theory, Springer (2008).[3]Dantzig G. B., Linear Programming and Extensions, PrincetonUniversity Press (1963).[4]Gillies D. B., Solutions to general non-zero-sum games, In:Tucker, A.W. and Luce, R.D. (Eds.), Contributions to theory ofgames IV, Annals of Mathematical Studies 40. PrincetonUniversity Press, Princeton (1959) pp. 47-85.[6] Rockafellar R.T., Convex Analysis, Princeton University Press,Princeton, (1970).
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011
References
References
[7]Shapley L.S., On balanced sets and cores, Naval ResearchLogistics Quarterly 14 (1967) 453-460.[8]Shapley L.S., A value for n-person games, Annals ofMathematics Studies, 28 (1953) 307-317.[9]Sprumont Y., Population monotonic allocation schemes forcooperative games with transferable utility, Games and EconomicBehavior, 2 (1990) 378-394.[10] Tijs S., Introduction to Game Theory, SIAM, Hindustan BookAgency, India (2003).[11] von Neumann J. and Morgenstern O. , Theory of Games andEconomic Behavior, Princeton Univ. Press, Princeton NJ (1944).[12] Weber R., Probabilistic values for games, in Roth A.E. (ed.),The Shapley Value: Essays in Honour of Lloyd S. Shapley,Cambridge University Press, Cambridge (1988) 101-119.