3. The citizens of a dictatorship want to overthrow their government. The probability of success depends on how many of them participate in street protests and how many stay home. Also, the citizens differ in their eagerness to protest; we label them by this parameter i. Players: Citizen i E [0, 1] (infinitely, continuously many) Strategies: Each citizen chooses simultaneous whether to stay home (H) or to participate in the protests (P) Payoffs: The utility of staying home is normalized to zero, U:(H; S.;) = 0. The utility of participation U:(P; S-1) = 4x + 3i - 2 depends on the citizens identity i and the mass of participants x in strategy prole s. For example, if i E [0, 0.8] stay home and i E (0.8, 1] participate then the mass of participants equals x = 0.2. Then the utility of participation of citizen i = 0.8, say, equals 4x + 3i - 2 = 4*0.2 + 3*0.8 - 2 = 0.8 +2.4 - 2 = 1.2; thus i = 0.8's best response is to protest. In the same situation, with mass X = 0.2 of citizens protesting, citizen i = 0.3 (who does not enjoy protesting as much) gets only utility 4x + 3i - 2 = 4*0.2 + 3*0.3 - 2 = 0.8 +0.9 - 2 = -0.3 from participation; thus i = 0.3's best response is to stay home. (1) For which citizen i does P strictly dominate H? (2) For which i is BR = {P}? (3) For which i is BR = {P}? (4) Is this game dominance solvable? Why? 3. The citizens of a dictatorship want to overthrow their government. The probability of success depends on how many of them participate in street protests and how many stay home. Also, the citizens differ in their eagerness to protest; we label them by this parameter i. Players: Citizen i E [0, 1] (infinitely, continuously many) Strategies: Each citizen chooses simultaneous whether to stay home (H) or to participate in the protests (P) Payoffs: The utility of staying home is normalized to zero, U:(H; S.;) = 0. The utility of participation U:(P; S-1) = 4x + 3i - 2 depends on the citizens identity i and the mass of participants x in strategy prole s. For example, if i E [0, 0.8] stay home and i E (0.8, 1] participate then the mass of participants equals x = 0.2. Then the utility of participation of citizen i = 0.8, say, equals 4x + 3i - 2 = 4*0.2 + 3*0.8 - 2 = 0.8 +2.4 - 2 = 1.2; thus i = 0.8's best response is to protest. In the same situation, with mass X = 0.2 of citizens protesting, citizen i = 0.3 (who does not enjoy protesting as much) gets only utility 4x + 3i - 2 = 4*0.2 + 3*0.3 - 2 = 0.8 +0.9 - 2 = -0.3 from participation; thus i = 0.3's best response is to stay home. (1) For which citizen i does P strictly dominate H? (2) For which i is BR = {P}? (3) For which i is BR = {P}? (4) Is this game dominance solvable? Why
