Up Down Up -C, v 0,0 —с, —с Down V, —с
Seagulls love shellfish. In order to break the shell, they need to flyy high
up and drop the shellfish. The problem is the other seagulls on the beach
are kleptoparasites, and they steal the shellfish if they can reach it first.
This question tells the story of two seagulls, named Irene and Jonathan,
who live in a crowded beach where it is impossible to drop the shellfish
and get it before some other gull steals it. The possible dates are t =
0; 1; 2; 3; : : :with no upper bound. Every day, simultaneously Irene and
Jonathan chooses one of the two actions: "Up" or "Down". Up means to
fly high up with the shellfish and drop it next to the other seagulls' nest,
and Down means to stay down in the nest. Up costs c > 0, but if the
other seagull is down, it eats the shellfish, getting payoff v > c. That is,
we consider the infinitely repeated game with the following stage game:
They maximize the discounted sum of their stage payoffs with discount
factor d in (0; 1). For each strategy profile below, find the set of discount
factors under which the strategy profile is a subgame-perfect equilibrium
(a) Irrespective of the history, Irene plays Up in the even dates and Down
in the odd dates; Jonathan plays Up in the odd dates and Down in
the even dates.
(b) Irene plays Up in the even dates and Down in the odd dates while
Jonathan plays the other way around until someone fails to go Up in
a day that he is supposed to do so. They both stay Down thereafter.
(c) For n days Irene goes Up and Jonathan stays Down; in the next n
days Jonathan goes Up and Irene stays Down. This continues back
and forth until someone deviates. They both stay Down thereafter.
d) Irene goes Up on "Sundays", i.e., at t = 0; 7; 14; 21; : : :, and stays
Down on the other days, while Jonathan goes up everyday except
for Sundays, when he rests Down, until someone deviates; they both
stay Down thereafter.
(e) At t = 0, Irene goes Up and Jonathan stays Down, and then they
alternate. If a seagull i fails to go Up at a history when i is supposed
to go Up, then the next day i goes Up and the other seagull stays
Down, and they keep alternating thereafter until someone fails to go
Up when it is supposed to do so. (For example, given the history, if
Irene is supposed to go Up at t but stays Down, then Irene goes Up
at t + 1, Jonathan goes Up at t + 2, and so on. If Irene stays down
again at t + 1, then she is supposed to go up at t + 2, and Jonathan
is supposed to go at t + 3, etc.)
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