博弈论问题求助

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 ∈ [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, Ui(H; s-i) = 0. The utility of
participation Ui(P; s-i) = 4x + 3i - 2 depends on the citizens identity i and the mass of
participants x in strategy prole s.
For example, if i ∈ [0, 0.8] stay home and i ∈ (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 BRi2 = {P}?
(3) For which i is BRi3 = {P}?
(4) Is this game dominance solvable? Why?

Consider a first-price sealed bid auction between two bidders, 1 and 2. Assume that they have
different valuations for the object being auctioned: 1’s value is 100 and 2’s value is 90. Further,
assume that ties are broken in favor of bidder 1: if their bids are the same, bidder 1 will get the
object.
(1) What strategies are weakly dominated for bidder 2? For each strategy that you think is
weakly dominated, name a strategy that weakly dominates it.
(2) Show that (90, 90) is a NE.
(3) Give an example of a NE where bidder 2 plays a weakly dominated strategy.

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