Cooperative Game Theory

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AACIMP 2011 Summer School. Operational Research Stream. Lecture by Sirma Zeynep Alparslan Gok.

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Cooperative Game Theory

  1. 1. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative Game Theory. Operations Research Games. Applications to Interval Games Lecture 1: Cooperative Game Theory Sırma Zeynep Alparslan G¨k o S¨leyman Demirel University u Faculty of Arts and Sciences Department of Mathematics Isparta, Turkey zeynepalparslan@yahoo.com August 13-16, 2011
  2. 2. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011Outline 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
  3. 3. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction Game theory is a mathematical theory dealing with models of conflict and cooperation. Game Theory has many interactions with economics and with other areas such as Operations Research and social sciences. A young field of study: The start is considered to be the book Theory of Games and Economic Behaviour by von Neumann and Morgernstern. Game theory is divided into two parts: non-cooperative and cooperative.
  4. 4. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction Cooperative game theory deals with coalitions who coordinate their actions and pool their winnings. Natural questions for individuals or businesses when dealing with cooperation are: Which coalitions should form? How to distribute the collective gains (rewards) or costs among the members of the formed coalition?
  5. 5. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction to cooperative game theoryCooperative game theory A cooperative n-person game in coalitional form (TU (transferable utility) game) is an ordered pair < N, v >, where N = {1, 2, ..., n} (the set of players) and v : 2N → R is a map, assigning to each coalition S ∈ 2N a real number, such that v (∅) = 0. v is the characteristic function of the game. v (S) is the worth (or value) of coalition S.
  6. 6. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction to cooperative game theoryExample (Glove game) N = {1, 2, 3}. Players 1 and 2 possess a left-hand glove and the player 3 possesses a right-hand glove. A single glove is worth nothing 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.
  7. 7. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction to cooperative game theoryCooperative game theory G N : the set of coalitional games with player set N G N forms a (2|N| − 1)-dimensional linear space equipped with the usual operators of addition and scalar multiplication of functions. A basis of this space is supplied by the unanimity games uT (or < N, uT >), T ∈ 2N {∅}, which are defined by 1, if T ⊂ S uT (S) := 0, otherwise. The interpretation of the unanimity game uT is that a gain (or cost savings) of 1 can be obtained if and only if all players in coalition T are involved in cooperation.
  8. 8. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction to cooperative game theoryExample Since the unanimity games is the basis of coalitional games, each cooperative 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.
  9. 9. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Basic solution concepts of cooperative game theoryBasic 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 x is individually rational: xi ≥ v ({i}) for all i ∈ N n x is efficient: i=1 xi = v (N) Example (Glove game continues): The imputation set of the glove game 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 system x1 + x2 + x3 = 10, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0).
  10. 10. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Basic solution concepts of cooperative game theoryThe core (Gillies (1959)) The core of a game < N, v > is the set C (v ) = x ∈ I (v )| xi ≥ v (S) for all S ∈ 2N {∅} . i∈S The idea of the core is by giving every coalition S at least their worth v (S) so that no coalition has an incentive to split off. If C (v ) = ∅, then elements of C (v ) can easily be obtained, because the core is defined with the aid of a finite system of linear inequalities (optimization-linear programming (see Dantzig(1963))). The core is a convex set and the core is a polytope (see Rockafellar (1970)).
  11. 11. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Basic solution concepts of cooperative game theoryExample (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.)
  12. 12. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Balanced gamesBalanced game A map λ : 2N {∅} → R+ is called a balanced map if S N S∈2N {∅} λ(S)e = e . Here, e S is the characteristic vector for coaliton S with 1, if i ∈ S eiS := 0, if i ∈ N S. An n-person game < N, v > is called a balanced game if for each balanced map λ : 2N {∅} → R+ we have λ(S)v (S) ≤ v (N). S
  13. 13. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Balanced gamesExample For N = {1, 2, 3}, the set B = {{1, 2} , {1, 3} , {2, 3}} is balanced 1 and corresponds to the balanced map λ with λ(S) = 2 if |S| = 2. That is 1 1 1 v ({1, 2}) + v ({1, 3}) + v ({2, 3}) ≤ v (N) 2 2 2 Let us show it: λ({1, 2})e {1,2} + λ({1, 3})e {1,3} + λ({2, 3})e {2,3} = e N 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 is 1 λ({1, 2}) = λ({1, 3}) = λ({2, 3}) = 2 .
  14. 14. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Balanced gamesBalanced game The importance of a balanced game becomes clear by the following theorem 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 are equivalent: (i) C (v ) = ∅, (ii) < N, v > is a balanced game.
  15. 15. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Shapley value and Weber setMarginal contribution Let v ∈ G N . For each i ∈ N and for each S ∈ 2N with i ∈ S, the marginal 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 P σ (i) := r ∈ N|σ −1 (r ) < σ −1 (i) consists of all predecessors of i with respect to the permutation σ. Let v ∈ G N and σ ∈ Π(N). The marginal contribution vector mσ (v ) ∈ Rn with respect to σ and v has the i-th coordinate the value miσ (v ) := v (P σ (i) ∪ {i}) − v (P σ (i)) for each i ∈ N.
  16. 16. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Shapley value and Weber setExample Let < N, v > be the three-person game with v ({i}) = 0 for each i ∈ 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, where σ : N → N is identified with (σ(1), σ(2), σ(3)).   σ σ σ σ  m1 (v ) m2 (v ) m3 (v )    (123)   0 3 7   (132)   0 5 5  . (213)   3 0 7   (231)   3 0 7   (312)  5 5 0  (321) 3 7 0
  17. 17. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Shapley value and Weber setThe Shapley value (Shapley (1953)) and the Weber set(Weber (1988)) The Shapley value φ(v ) of a game v ∈ G N is the average of the marginal vectors of the game 1 σ (v ). φ(v ) := n! σ∈Π(N) m This value associates to each n-person game one (payoff) vector in Rn . The Shapley value of the previous example is 1 7 10 13 φ(v ) = (14, 20, 26) = ( , , ). 3! 3 3 3 The Weber set (Weber (1988)) W (v ) of v is defined as the convex hull of the marginal vectors of v . Theorem: Let v ∈ G N . Then C (v ) ⊂ W (v ).
  18. 18. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Shapley value and Weber setExample (LLR game) Marginal vectors can be observed from the following table   σ σ σ σ  m1 (v ) m2 (v ) m3 (v )    (123)   0 0 10   (132)   0 0 10  . (213)   0 0 10   (231)   0 0 10   (312)  10 0 0  (321) 0 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 , 1 , 3 ). 6 6 2 Note that {(0, 0, 10)} = C (v ) ⊂ W (v ) and φ(v ) ∈ W (v ).
  19. 19. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Convex gamesConvex games < N, v > is convex if and only if the supermodularity condition v (S ∪ T ) + v (S ∩ T ) ≥ v (S) + v (T ) for each S, T ∈ 2N holds (desirable for reward games). < N, v > is called concave (or submodular) if and only if v (S ∪ T ) + v (S ∩ T ) ≤ v (S) + v (T ) for all S, T ∈ 2N (desirable for cost games). CG N –The family of all convex games with player set N.
  20. 20. 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 ∈ G N . The following 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} we have v (S1 ∪ {i}) − v (S1 ) ≤ v (S2 ∪ {i}) − v (S2 ). (iv) Each marginal vector mσ (v ) of the game v with respect to the permutation σ belongs to the core C (v ). (v) W (v ) = C (v ).
  21. 21. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Convex games For convex games the gain made when individuals or groups join larger coalitions is higher than when they join smaller coalitions. A convex game is balanced and the core of the convex games is nonempty. The Shapley value is a core element if the game is convex.
  22. 22. 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 ∈ G N and a coalition T ∈ 2N {∅}, the subgame with player set T , (T , vT ), is the game vT defined by vT (S) := v (S) for all S ∈ 2T . A game v ∈ G N is called totally balanced if (the game and) all its subgames are balanced. The class of totally balanced games includes the class of games with a population monotonic allocation scheme (pmas) (Sprumont (1990)).
  23. 23. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Population Monotonic Allocation Schemes (pmas) Let v ∈ G N . A scheme a = (aiS )i∈S,S∈2N {∅} of real numbers is a pmas 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. Interpretation: in larger coalitions, higher rewards (or in larger coalitions lower costs). It is known that for v ∈ CG N the (total) Shapley value generates population monotonic allocation schemes. Further, in a convex game all core elements generate pmas.
  24. 24. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Population Monotonic Allocation Schemes (pmas)Example Let < 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.     1 2 3   N   29 34 39   {1, 2}   20 30 ∗   Φ({1, 3} , v ) → {1, 3}   15 ∗ 35 .  {2, 3}   ∗ 20 30   {1}   10 ∗ ∗   {2}  ∗ 20 ∗  {3} ∗ ∗ 30
  25. 25. 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 Introduction to Game Theory by Tijs and Models in Cooperative Game Theory by Branzei, Dimitrov and Tijs.
  26. 26. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [1]Bondareva O.N., Certain applications of the methods of linear programming to the theory of cooperative games, Problemly Kibernetiki 10 (1963) 119-139 (in Russian). [2]Branzei R., Dimitrov D. and Tijs S., Models in Cooperative Game Theory, Springer (2008). [3]Dantzig G. B., Linear Programming and Extensions, Princeton University 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 of games IV, Annals of Mathematical Studies 40. Princeton University Press, Princeton (1959) pp. 47-85. [6] Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton, (1970).
  27. 27. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [7]Shapley L.S., On balanced sets and cores, Naval Research Logistics Quarterly 14 (1967) 453-460. [8]Shapley L.S., A value for n-person games, Annals of Mathematics Studies, 28 (1953) 307-317. [9]Sprumont Y., Population monotonic allocation schemes for cooperative games with transferable utility, Games and Economic Behavior, 2 (1990) 378-394. [10] Tijs S., Introduction to Game Theory, SIAM, Hindustan Book Agency, India (2003). [11] von Neumann J. and Morgenstern O. , Theory of Games and Economic 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.

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