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DECISION DANS L'INCERTAIN ET THEORIE DES JEUX

DECISION DANS L'INCERTAIN ET THEORIE DES JEUX. Stefano MORETTI 1 , David RIOS 2 and Alexis TSOUKIAS 1 1 LAMSADE (CNRS), Paris Dauphine 2 Statistics and Operations Research Department, Rey Juan Carlos University (URJC), Madrid. Tous les mercredis de 13h45 à 17h00 du 20/01 au 24/02.

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DECISION DANS L'INCERTAIN ET THEORIE DES JEUX

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  1. DECISION DANS L'INCERTAIN ET THEORIE DES JEUX Stefano MORETTI1, David RIOS2 and Alexis TSOUKIAS1 1LAMSADE (CNRS), Paris Dauphine 2Statistics and Operations Research Department, Rey Juan Carlos University (URJC), Madrid Tous les mercredis de 13h45 à 17h00 du 20/01 au 24/02

  2. Dominant strategies • Nash eq. (NE) • Subgame perfect NE • NE & refinements • … • Core • Shapley value • Nucleolus • τ-value • PMAS • …. • Nash sol. • Kalai-Smorodinsky • …. • CORE • NTU-value • Compromise value • … No binding agreements No side payments Q: Optimal behaviour in conflict situations binding agreements side payments are possible (sometimes) Q: Reasonable (cost, reward)-sharing

  3. Cooperative games: a simple example Alone, player 1 (singer) and 2 (pianist) can earn 100€ 200€ respect. Together (duo) 700€ How to divide the (extra) earnings? x2 700 I(v) 600 “reasonable” payoff? 400 200 x1 100 700 300 500 x1 +x2=700 Imputation set: I(v)={xIR2|x1100, x2200, x1+x2 =700}

  4. Simple example Alone, player 1 (singer) and 2 (pianist) can earn 100€ 200€ respect. Together (duo) 700€ How to divide the (extra) earnings? x2 In this case I(v) coincideswith the core 700 I(v) 600 “reasonable” payoff (300, 400) =Shapleyvalue = -value = nucleolus 400 200 x1 100 700 300 500 x1 +x2=700 Imputation set: I(v)={xIR2|x1100, x2200, x1+x2 =700}

  5. COOPERATIVE GAME THEORY Games in coalitional form TU-game: (N,v) or v N={1, 2, …, n} set of players SN coalition 2N set of coalitions DEF. v: 2NIR with v()=0 is a Transferable Utility (TU)-game with player set N. NB: (N,v)v NB2: if n=|N|, it is also called n-person TU-game, game in colaitional form, coalitional game, cooperative game with side payments... v(S) is the value (worth) of coalition S Example (Glove game) N=LR, LR=  iL (iR) possesses 1 left (right) hand glove Value of a pair: 1€ v(S)=min{| LS|, |RS|} for each coalition S2N\{} .

  6. Example (Three cooperating communities) 1 N={1,2,3} 100 30 source 3 80 40 90 2 30 v(S)=iSc(i) – c(S)

  7. Example (flow games) capacity owner l1 4,1 N={1,2,3} 10,3 l3 sink source 80 l2 5,2 1€: 1 unit source  sink 30

  8. DEF. (N,v) is a superadditive game iff v(ST)v(S)+v(T) for all S,T with ST= Q.1: which coalitions form? Q.2: If the grand coalition N forms, how to divide v(N)? (how to allocate costs?) Many answers! (solution concepts) One-point concepts: - Shapley value (Shapley 1953) - nucleolus (Schmeidler 1969) - τ-value (Tijs, 1981) … Subset concepts: - Core (Gillies, 1954) - stable sets (von Neumann, Morgenstern, ’44) - kernel (Davis, Maschler) - bargaining set (Aumann, Maschler) …..

  9. Example (Three cooperating communities) 1 N={1,2,3} 100 30 source 3 80 40 90 2 30 v(S)=iSc(i) – c(S) Show that v is superadditive and c is subadditive.

  10. Claim 1: (N,v) is superadditive We show that v(ST)v(S)+v(T) for all S,T2N\{} with ST= 60=v(1,2)v(1)+v(2)=0+0 70=v(1,3)v(1)+v(3)=0+0 60=v(2,3)v(2)+v(3)=0+0 60=v(1,2)v(1)+v(2)=0+0 130=v(1,2,3) v(1)+v(2,3)=0+60 130=v(1,2,3) v(2)+v(1,3)=0+70 130=v(1,2,3) v(3)+v(1,2)=0+60 Claim 2: (N,c) is subadditive We show that c(ST)c(S)+c(T) for all S,T2N\{} with ST= 130=c(1,2)  c(1)+c(2)=100+90 110=c(2,3)  c(2)+v(3)=100+80 110=c(1,2)  c(1)+v(2)=90+80 140=c(1,2,3)  c(1)+c(2,3)=100+110 140=c(1,2,3)  c(2)+c(1,3)=90+110 140=c(1,2,3)  c(3)+c(1,2)=80+130

  11. Example • (Glove game) (N,v) such that N=LR, LR=  • v(S)=10 min{| LS|, |RS|} for all S2N\{} • Claim: the glove game is superadditive. • Suppose S,T2N\{} with ST=. Then • v(S)+v(T)= min{| LS|, |RS|} + min{| LT|, |RT|} • =min{| LS|+|LT|,|LS|+|RT|,|RS|+|LT|,|RS|+|RT|} • min{| LS|+|LT|, |RS|+|RT|} since ST= • =min{| L(S  T)|, |R  (S T)|} • =v(S T).

  12. The imputation set DEF. Let (N,v) be a n-persons TU-game. A vector x=(x1, x2, …, xn)IRN is called an imputation iff (1) x is individual rational i.e. xi  v(i) for all iN (2) x is efficient iNxi = v(N) [interpretation xi: payoff to player i] I(v)={xIRN | iNxi = v(N), xi  v(i) for all iN} Set of imputations

  13. Example (N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3, v(1,2,3)=5. x3 (0,0,5) (0,3,2) (x1,x2,x3) I(v) (0,5,0) (2,3,0) x1+x2+x3=5 (5,0,0) x2 I(v)={xIR3 | x1,x30, x23, x1+x2+x3=5} X1

  14. Claim: (N,v) a n-person (n=|N|) TU-game. Then I(v)  v(N)iNv(i) Proof () Suppose xI(v). Then v(N) = iNxi  iNv(i) EFF IR () Suppose v(N)iNv(i). Then the vector (v(1), v(2), …, v(n-1), v(N)- i{1,2, …,n-1}v(i)) is an imputation. v(n)

  15. The core of a game DEF. Let (N,v) be a TU-game. The core C(v) of (N,v) is the set C(v)={xI(v) | iSxi  v(S) for all S2N\{}} stability conditions no coalition S has the incentive to split off if x is proposed Note: x  C(v) iff (1) iNxi = v(N) efficiency (2) iSxi  v(S) for all S2N\{} stability Bad news: C(v) can be empty Good news: many interesting classes of games have a non-empty core.

  16. Core elements satisfy the following conditions: x1,x30, x23, x1+x2+x3=5 x1+x23, x1+x31, x2+x34 We have that 5-x33x32 5-x21x34 5-x14x11 Example (N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3, v(1,2)=3, v(1,3)=1 v(2,3)=4 v(1,2,3)=5. C(v)={xIR3 | 1x10,2x30, 4x23, x1+x2+x3=5}

  17. Example (N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3, v(1,2)=3, v(1,3)=1 v(2,3)=4 v(1,2,3)=5. x3 (0,0,5) (0,5,0) x1+x2+x3=5 (5,0,0) (0,3,2) (0,4,1) C(v) (1,3,1) x2 (1,4,0) (2,3,0) C(v)={xIR3 | 1x10,2x30, 4x23, x1+x2+x3=5} X1

  18. Example (Game of pirates) Three pirates 1,2, and 3. On the other side of the river there is a treasure (10€). At least two pirates are needed to wade the river… (N,v), N={1,2,3}, v(1)=v(2)=v(3)=0, v(1,2)=v(1,3)=v(2,3)=v(1,2,3)=10 Suppose (x1, x2, x3)C(v). Then efficiency x1+ x2+ x3=10 x1+ x2 10 stability x1+ x3 10 x2+ x3 10 20=2(x1+ x2+ x3) 30 Impossible. So C(v)=. Note that (N,v) is superadditive.

  19. Example (Glove game with L={1,2}, R={3}) v(1,3)=v(2,3)=v(1,2,3)=1, v(S)=0 otherwise Suppose (x1, x2, x3)C(v). Then x1+ x2+ x3=1 x2=0 x1+x3 1 x1+x3 =1 x20 x2+ x3 1 x1=0 and x3=1 So C(v)={(0,0,1)}. (0,0,1) I(v) (1,0,0) (0,1,0)

  20. Example (flow games) capacity owner l1 4,1 N={1,2,3} 10,3 l3 sink source 80 l2 5,2 1€: 1 unit source  sink 30 Min cut Min cut {l1, l2}. Corresponding core element (4,5,0)

  21. Non-emptiness of the core Notation: Let S2N\{}. 1 if iS eS is a vector with (eS)i= 0 if iS DEF. A collection B2N\{} is a balanced collection if there exist (S)>0 for SB such that: eN= SB(S) eS Example: N={1,2,3}, B={{1,2},{1,3},{2,3}}, (S)= for SB eN=(1,1,1)=1/2 (1,1,0)+1/2 (1,0,1) +1/2 (0,1,1)

  22. Balanced games DEF. (N,v) is a balanced game if for all balanced collections B2N\{}  SB(S) v(S)v(N) Example: N={1,2,3}, v(1,2,3)=10, v(1,2)=v(1,3)=v(2,3)=8 (N,v) is not balanced 1/2v (1,2)+1/2 v(1,3) +1/2 v(2,3)>10=v(N)

  23. Variants of duality theorem • Duality theorem. • min{xTc|xTAb} • | | • max{bTy|Ay=c, y0} • (if both programs feasible) yT x A c bT

  24. Bondareva (1963) | Shapley (1967)Characterization of games with non-empty core Theorem (N,v) is a balanced game  C(v) ProofFirst note that xTeS=iS xi ({1})…….(S)…….… (N) C(v)  v(N)=min{xTeN | xTeSv(S) S2N\{}} duality v(N)=max{vT| S2N\{}(S)eS=eN, 0}  0, S2N\{}(S)eS=eN vT=S2N\{}(S)v(S)v(N)  (N,v) is a balanced game X1 X2 . . . x2 1 1 0 1 . eS . . . . . 0 1 1 1 . . . 1 V(1)………v(S)……..…v(N)

  25. Convex games (1) DEF. An n-persons TU-game (N,v) is convex iff v(S)+v(T)v(ST)+v(ST) for each S,T2N. This condition is also known as submodularity. It can be rewritten as v(T)-v(ST)v(ST)-v(S) for each S,T2N For each S,T2N, let C=(ST)\S. Then we have: v(C(ST))-v(ST)v(CS)-v(S) Interpretation: the marginal contribution of a coalition C to a disjoint coalition S does not decrease if S becomes larger

  26. Convex games (2) • It is easy to show that submodularity is equivalent to v(S{i})-v(S)v(T{i})-v(T) for all iN and all S,T2N such that ST  N\{i} • interpretation: player's marginal contribution to a large coalition is not smaller than her/his marginal contribution to a smaller coalition (which is stronger than superadditivity) • Clearly all convex games are superadditive (ST=…) • A superadditive game can be not convex (try to find one) • An important property of convex games is that they are (totally) balanced, and it is “easy” to determine the core (coincides with the Weber set, i.e. the convex hull of all marginal vectors…)

  27. Example (N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3, v(1,2)=3, v(1,3)=1 v(2,3)=4 v(1,2,3)=5. Check it is convex x3 Marginal vectors 123(0,3,2) 132(0,4,1) 213(0,3,2) 231(1,3,1) 321(1,4,0) 312(1,4,0) (0,0,5) (0,5,0) x1+x2+x3=5 (5,0,0) (0,3,2) (0,4,1) C(v) (1,3,1) x2 (1,4,0) (2,3,0) C(v)={xIR3 | 1x10,2x30, 4x23, x1+x2+x3=5} X1

  28. How to share v(N)… • The Core of a game can be used to exclude those allocations which are not stable. • But the core of a game can be a bit “extreme” (see for instance the glove game) • Sometimes the core is empty (pirates) • And if it is not empty, there can be many allocations in the core (which is the best?)

  29. An axiomatic approach (Shapley (1953) • Similar to the approach of Nash in bargaining: which properties an allocation method should satisfy in order to divide v(N) in a reasonable way? • Given a subset C of GN (class of all TU-games with N as the set of players) a (point map) solution on C is a map Φ:C →IRN. • For a solution Φwe shall be interested in various properties…

  30. Symmetry PROPERTY 1(SYM) For all games vGN, If v(S{i}) = v(S{j}) for all S2N s.t. i,jN\S, then Φi(v) = Φj (v). EXAMPLE We have a TU-game ({1,2,3},v) s.t. v(1) = v(2) = v(3) = 0, v(1, 2) = v(1, 3) = 4, v(2, 3) = 6, v(1, 2, 3) = 20. Players 2 and 3 are symmetric. In fact: v({2})= v({3})=0 and v({1}{2})=v({1}{3})=4 If Φ satisfies SYM, then Φ2(v) = Φ3(v)

  31. Efficiency PROPERTY 2 (EFF) For all games vGN,  iNΦi(v) = v(N), i.e., Φ(v) is a pre-imputation. Null Player Property DEF.Given a game vGN, a player iN s.t. v(Si) = v(S) for all S2N will be said to be a null player. PROPERTY 3 (NPP)For all games v GN,Φi(v) = 0 if i is a null player. EXAMPLE We have a TU-game ({1,2,3},v) such that v(1) =0, v(2) = v(3) = 2, v(1, 2) = v(1, 3) = 2, v(2, 3) = 6, v(1, 2, 3) = 6. Player 1 is null. Then Φ1(v) = 0

  32. EXAMPLE We have a TU-game ({1,2,3},v) such that v(1) =0, v(2) = v(3) = 2, v(1, 2) = v(1, 3) = 2, v(2, 3) = 6, v(1, 2, 3) = 6. On this particular example, if Φ satisfies NPP, SYM and EFF we have that Φ1(v) = 0 by NPP Φ2(v)= Φ3(v) by SYM Φ1(v)+Φ2(v)+Φ3(v)=6 by EFF So Φ=(0,3,3) But our goal is to characterize Φ on GN. One more property is needed.

  33. Additivity PROPERTY 2 (ADD) Given v,w GN, Φ(v)+Φ(w)=Φ(v +w). .EXAMPLE Two TU-games v and w on N={1,2,3} v(1) =3 v(2) =4 v(3) = 1 v(1, 2) =8 v(1, 3) = 4 v(2, 3) = 6 v(1, 2, 3) = 10 Φ w(1) =1 w(2) =0 w(3) = 1 w(1, 2) =2 w(1, 3) = 2 w(2, 3) = 3 w(1, 2, 3) = 4 v+w(1) =4 v+w(2) =4 v+w(3) = 2 v+w(1, 2) =10 v+w(1, 3) = 6 v+w(2, 3) = 9 v+w(1, 2, 3) = 14 Φ Φ = +

  34. Theorem 1(Shapley 1953) There is a unique map  defined on GN that satisfies EFF, SYM, NPP, ADD. Moreover, for any iN we have that Here  is the set of all permutations σ:N →N of N, while mσi(v) is the marginal contribution of player i according to the permutation σ, which is defined as: v({σ(1), σ(2), . . . , σ (j)})− v({σ(1), σ(2), . . . , σ (j −1)}), where j is the unique element of N s.t. i = σ(j).

  35. Unanimity games (1) • DEFLet T2N\{}. Theunanimity game on T is defined as the TU-game (N,uT) such that 1 is TS uT(S)= 0 otherwise • Note that the class GN of all n-person TU-games is a vector space (obvious what we mean for v+w and v for v,wGN and IR). • the dimension of the vector space GN is 2n-1 • {uT|T2N\{}} is an interesting basis for the vector space GN.

  36. Unanimity games (2) • Every coalitional game (N, v) can be written as a linear combination of unanimity games in a unique way, i.e., v =S2N λS(v)uS . • The coefficients λS(v), for each S2N, are called unanimity coefficients of the game (N, v) and are given by the formula: λS(v) = T2S (−1)s−t v(T ).

  37. 1(v) =3 2(v) =4 3(v) = 1 {1,2}(v) =-3-4+8=1 {1,3}(v) = -3-1+4=0 {2,3}(v) = -4-1+6=1 {1,2,3}(v) = -3-4-1+8+4+6-10=0 .EXAMPLE Two TU-games v and w on N={1,2,3} v(1) =3 v(2) =4 v(3) = 1 v(1, 2) =8 v(1, 3) = 4 v(2, 3) = 6 v(1, 2, 3) = 10 v=3u{1}(v)+4u{2}(v)+u{3}(v)+u{1,2}(v)+u{2,3}(v)

  38. Sketch of the Proof of Theorem1 • Shapley value satisfies the four properties (easy). • Properties EFF, SYM, NPP determine  on the class of all games αv, with v a unanimity game and αIR. • Let S2N. The Shapley value of the unanimity game (N,uS) is given by /|S| if iS i(αuS)= 0 otherwise • Since the class of unanimity games is a basis for the vector space, ADD allows to extend  in a unique way to GN.

  39. Probabilistic interpretation:(the “room parable”) • Players gather one by one in a room to create the “grand coalition”, and each one who enters gets his marginal contribution. • Assuming that all the different orders in which they enter are equiprobable, the Shapley value gives to each player her/his expected payoff. Example (N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3, v(1,2)=3, v(1,3)=1, v(2,3)=4, v(1,2,3)=5.

  40. Example (N,v) such that N={1,2,3}, v(1)=v(3)=0, v(2)=3, v(1,2)=3, v(1,3)=1 v(2,3)=4 v(1,2,3)=5. x3 Marginal vectors 123(0,3,2) 132(0,4,1) 213(0,3,2) 231(1,3,1) 321(1,4,0) 312(1,4,0) (0,0,5) (0,5,0) x1+x2+x3=5 (5,0,0) (0,3,2) C(v) (v)=(0.5, 3.5,1) (0,4,1) (1,3,1) x2 (1,4,0) (2,3,0) X1

  41. Example (N,v) such that N={1,2,3}, v(1) =3 v(2) =4 v(3) = 1 v(1, 2) =8 v(1, 3) = 4 v(2, 3) = 6 v(1, 2, 3) = 10

  42. Example (Glove game with L={1,2}, R={3}) v(1,3)=v(2,3)=v(1,2,3)=1, v(S)=0 otherwise C(v) (0,0,1) (v) (1/6,1/6,2/3) I(v) (1,0,0) (0,1,0)

  43. An alternative formulation • Let mσ’i(v)=v({σ’(1),σ’(2),…,σ’(j)})− v({σ’(1),σ’(2),…, σ’(j −1)}), where j is the unique element of N s.t. i = σ’(j). • Let S={σ’(1), σ’(2), . . . , σ’(j)}. • Q: How many other orderings  do we have in which {σ(1), σ(2), . . . , σ (j)}=S and i = σ’(j)? • A: they are precisely (|S|-1)!(|N|-|S|)! • Where (|S|-1)! Is the number of orderings of S\{i} and (|N|-|S|)! Is the number of orderings of N\S • We can rewrite the formula of the Shapley value as the following:

  44. Alternative properties • PROPERTY 5(Anonymity, ANON) Let (N, v)GN σ : N →N be a permutation. Then, Φσ(i)(σ v) = Φi(v) for all iN. • Here σv is the game defined by: σv(S) = v(σ(S)), for all S2N. Interpretation: The meaning of ANON is that whatever a player gets via Φ should depend only on the structure of the game v, not on his “name”, i.e., the way in which he is labelled. • DEF(Dummy Player) Given a game (N, v), a player iN s.t. v(S{i}) =v(S)+v(i) for all S2N will be said to be a dummy player. • PROPERTY 6(Dummy Player Property, DPP) If i is a dummy player, then Φi(v) =v(i). NB: often NPP and SYM are replaced by DPP and ANON, respectively.

  45. Characterization on a subclass • Shapley and Shubik (1954) proposed to use the Shapley value as a power index, • ADD property does not impose any restriction on a solution map defined on the class of simple games SN, which is the class of games such that v(S) {0,1} (often is added the requirement that v(N) = 1). • Therefore, the classical conditions are not enough to characterize the Shapley–Shubik value on SN. • A condition that resembles ADD and can substitute it to get a characterization of the Shapley–Shubik (Dubey (1975) index on SN: PROPERTY 7(Transfer, TRNSF) For any v,w S(N), it holds: Φ(v  w)+Φ(v  w) = Φ(v)+Φ(w). Here v  w is defined as (v  w)(S) = (v(S)  w(S)) = max{v(S),w(S)}, and v  w is defined as (v  w)(S) = (v(S)  w(S)) = min{v(S),w(S)},

  46.  .EXAMPLE Two TU-games v and w on N={1,2,3} Φ Φ Φ Φ v(1) =0 v(2) =1 v(3) = 0 v(1, 2) =1 v(1, 3) = 1 v(2, 3) = 0 v(1, 2, 3) = 1 w(1) =1 w(2) =0 w(3) = 0 w(1, 2) =1 w(1, 3) = 0 w(2, 3) = 1 w(1, 2, 3) = 1 vw(1) =0 vw(2) =0 vw(3) = 0 vw(1, 2) =1 vw(1, 3) = 0 vw(2, 3) = 0 vw(1, 2, 3) = 1 vw(1) =1 vw(2) =1 vw(3) = 0 vw(1, 2) =1 vw(1, 3) = 1 vw(2, 3) = 1 vw(1, 2, 3) = 1 + = +

  47. Reformulations Other axiomatic approaches have been provided for the Shapley value, of which we shall briefly describe those by Young and by Hart and Mas-Colell. PROPERTY 8(Marginalism, MARG) A map Ψ : GN→IRN satisfies MARG if, given v,wGN, for any player iN s.t. v(S{i}) −v(S) = w(S{i}) −w(S) for each S2N, the following is true: Ψi(v) = Ψi(w). Theorem 2(Young 1988) There is a unique map Ψ defined on G(N) that satisfies EFF, SYM, and MARG. Such a Ψ coincides with the Shapley value.

  48. w({3})- w() = v({3})- v()=1 w({1}{3})- w({1}) = v({1}{3})- v({1})=1 w({2}{3})- w() = v({2}{3})- v()=1 w({1,2}{3})- w({1,2}) = v({1,2}{3})- v({1,2)=1 w(1) =2 w(2) =3 w(3) = 1 w(1, 2) =2 w(1, 3) = 3 w(2, 3) = 5 w(1, 2, 3) = 4 v(1) =3 v(2) =4 v(3) = 1 v(1, 2) =8 v(1, 3) = 4 v(2, 3) = 6 v(1, 2, 3) = 10 .EXAMPLE Two TU-games v and w on N={1,2,3} Ψ3(v) = Ψ3(w).

  49. Potential • A quite different approach was pursued by Hart and Mas-Colell (1987). • To each game (N, v) one can associate a real number P(N,v) (or, simply, P(v)), its potential. • The “partial derivative” of P is defined as Di(P )(N, v) = P(N,v)−P(N\{i},v|N\{i}) Theorem 3(Hart and Mas-Colell 1987) There is a unique map P, defined on the set of all finite games, that satisfies: • P(∅, v0) = 0, • For each iN, DiP(N,v) = v(N). Moreover, Di(P )(N, v) = i(v). [(v) is the Shapley value of v]

  50. Example • there are formulas for the calculation of the potential. • For example, P(N,v)=S2N S/|S|(Harsanyi dividends) 1(v) =3 2(v) =4 3(v) = 1 {1,2}(v) =1 {1,3}(v)=0 {2,3}(v)=1 {1,2,3}(v)=0 v(1) =3 v(2) =4 v(3) = 1 v(1, 2) =8 v(1, 3) = 4 v(2, 3) = 6 v(1, 2, 3) = 10 P({1,2,3},v)=3+4+1+1/2+1/2=9 P({1,2},v)=3+4+1/2=15/2 P({1,3},v)=3+1=4 P({2,3},v)=4+1+1/2=11/2 1(v)=P({1,2,3},v)-P({2,3},v|{2,3})=9-11/2=7/2 2(v)=P({1,2,3},v)-P({1,3},v|{2,3})=9-4=5 3(v)=P({1,2,3},v)-P({1,2},v|{2,3})=9-15/2=3/2

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