I'm asking for a help just to EXPLAIN further and if you can give example basic it's more understandable. Also what does the symbols signify . "Game theory" Thank youuu . Godblesss

1.) Definiton 6.2 so that i can know the difference of Essential and Inessential Games

2. ) THEOREM 6.4.

3. ) THEOREM 6.5.

ASAP.

154 6. N-Person Cooperative Games 6.1.2. Essential and Inessential Games We can single out a class of games which are trivial as far as coalitions are concerned. In fact, they have the property that there is no reason to prefer any one coalition over any other. DEFINITION 6.2. An N-person game v in characteristic function form is said to be inessential if N HDHIM = i=1 (P) = ({P{}). A game which is not inessential is said to be essential. In other words, a game is inessential if the inequality in Corollary 6.3 is actually an equality. In fact, for such a game, a stronger statement is true. THEOREM 6.4. Let S be any coalition of the players in an inessential game. Then v(S) = ({P}). PES PROOF. Suppose not. Then, by Corollary 6.2, v(s) > -({P}), PES and so, by superadditivity, N v(P) 2 v(8) + v(8")> {P}), which contradicts the definition of an inessential game. O Thus, in an inessential game, there is no reason for a coalition actually to form--cooperation does not result in a greater total payoff. The fact that a game is in essential does not make it unimportant. It simply means that there is no reason for a coalition to form. The following theorem shows that there are many inessential games which are, neverthe- less, important. THEOREM 6.5. A two-person game which is zero-sum in its normal form is inessential in its characteristic function form. PROOF. The minimax theorem says that v({Pr}) and 1({B}}) are neg- atives of each other, and thus their sum is zero. But, also, (P) = 0, by THEOREM 6.5. A two-person game which is zero-sum in its normal form is inessential in its characteristic function form. PROOF. The minimax theorem says that v({P.}) and v({P}) are neg- atives of each other, and thus their sum is zero. But, also, (P) = 0, by the zero-sum property. 6.1. Coalitions 155 Thus, (P) = v({P{})+v({P}}). In the context of the previous chapter, an inessential two-person game is one for which there is no advantage to be gained by cooperating. Zero-sum games with more than two players can be essential. See Exercise (5). Exercises (1) Verify the value of the characteristic function for the game shown in Table 154 6. N-Person Cooperative Games 6.1.2. Essential and Inessential Games We can single out a class of games which are trivial as far as coalitions are concerned. In fact, they have the property that there is no reason to prefer any one coalition over any other. DEFINITION 6.2. An N-person game v in characteristic function form is said to be inessential if N HDHIM = i=1 (P) = ({P{}). A game which is not inessential is said to be essential. In other words, a game is inessential if the inequality in Corollary 6.3 is actually an equality. In fact, for such a game, a stronger statement is true. THEOREM 6.4. Let S be any coalition of the players in an inessential game. Then v(S) = ({P}). PES PROOF. Suppose not. Then, by Corollary 6.2, v(s) > -({P}), PES and so, by superadditivity, N v(P) 2 v(8) + v(8")> {P}), which contradicts the definition of an inessential game. O Thus, in an inessential game, there is no reason for a coalition actually to form--cooperation does not result in a greater total payoff. The fact that a game is in essential does not make it unimportant. It simply means that there is no reason for a coalition to form. The following theorem shows that there are many inessential games which are, neverthe- less, important. THEOREM 6.5. A two-person game which is zero-sum in its normal form is inessential in its characteristic function form. PROOF. The minimax theorem says that v({Pr}) and 1({B}}) are neg- atives of each other, and thus their sum is zero. But, also, (P) = 0, by THEOREM 6.5. A two-person game which is zero-sum in its normal form is inessential in its characteristic function form. PROOF. The minimax theorem says that v({P.}) and v({P}) are neg- atives of each other, and thus their sum is zero. But, also, (P) = 0, by the zero-sum property. 6.1. Coalitions 155 Thus, (P) = v({P{})+v({P}}). In the context of the previous chapter, an inessential two-person game is one for which there is no advantage to be gained by cooperating. Zero-sum games with more than two players can be essential. See Exercise (5). Exercises (1) Verify the value of the characteristic function for the game shown in Table