Seagulls love shellfish. In order to break the shell, they need to fly 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. Everyday, simultaneously Irene and Jonathan choose one of the two actions: "Up" or "Down". Up means to fly high up with the shellfish and drop it next to the other sea gull's 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: Up Down Up --C, -C V, -C Down -C, v 0,0 They maximize the discounted sum of their stage payoffs with discount factor 8 (0.1). For each strategy profile below, find the set of discount factors d under which the strategy profile is a subgame-perfect equilibrium. (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. [10pts) (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.[15pts) Seagulls love shellfish. In order to break the shell, they need to fly 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. Everyday, simultaneously Irene and Jonathan choose one of the two actions: "Up" or "Down". Up means to fly high up with the shellfish and drop it next to the other sea gull's 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: Up Down Up --C, -C V, -C Down -C, v 0,0 They maximize the discounted sum of their stage payoffs with discount factor 8 (0.1). For each strategy profile below, find the set of discount factors d under which the strategy profile is a subgame-perfect equilibrium. (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. [10pts) (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.[15pts)
