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CS711: Introduction to Game Theory and Mechanism Design

CS711: Introduction to Game Theory and Mechanism Design

CS711: Introduction to Game Theory and Mechanism Design Teacher: Swaprava Nath Extensive Form Games Extensive Form Games Chocolate Division Game: Suppose a mother gives his elder son two (indivisible) chocolates to share between him and his younger sister. She also warns that if there is any dispute in the sharing, she will take the chocolates back and nobody will get anything. The brother can propose the following sharing options: (2-0): brother gets two, sister gets nothing, or (1-1): both gets one each, or (0-2): both chocolates to the sister. After the brother proposes the sharing, his sister may \Accept" the division or \Reject" it. Brother 2-0 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 1 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 P (?) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg X (?) = f(2 − 0); (1 − 1); (0 − 2)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Representing the Chocolate Division Game Brother 2-0 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 P (?) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg X (?) = f(2 − 0); (1 − 1); (0 − 2)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother 2-0 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 P (?) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg X (?) = f(2 − 0); (1 − 1); (0 − 2)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g 2-0 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 P (?) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg X (?) = f(2 − 0); (1 − 1); (0 − 2)g Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g 2-0 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 P (?) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g X (?) = f(2 − 0); (12-0− 1); (0 − 2)g 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 P (?) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g X (?) = f(2 − 0); (12-0− 1); (0 − 2)g 1-1 0-2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g X (?) = f(2 − 0); (12-0− 1); (0 − 2)g 1-1 0-2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg Sister P ( ) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 ? ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g X (?) = f(2 − 0); (12-0− 1); (0 − 2)g 1-1 0-2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg Sister P ( ) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 ? ARARAR u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory and Mechanism Design Extensive Form Games Representing the Chocolate Division Game N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Brother H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g X (?) = f(2 − 0); (12-0− 1); (0 − 2)g 1-1 0-2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg Sister P ( ) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 ? ARARAR u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg 2 / 16 Game Theory and Mechanism Design Extensive Form Games For the given example, we can express the utility function as in the following table: B n S AAA AAR ARA ARR RAA RAR RRA RRR 2-0 (2,0) (2,0) (2,0) (2,0) (0,0) (0,0) (0,0) (0,0) 1-1 (1,1) (1,1) (0,0) (0,0) (1,1) (1,1) (0,0) (0,0) 0-2 (0,2) (0,0) (0,2) (0,0) (0,2) (0,0) (0,2) (0,0) Observe that there are many PSNEs in the given game, some of which leads to quite nonintuitive solutions.

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