ECON 3152/4453/8053 Homework 3

1. Consider repeated prisoners’ dilemma with the following payoff matrix of the stage game: Both players

discount future payoffs by 6 ∈ [0, 1).  Consider the Tit-for-Tat strategy profile: they play (C,C) in the first period; in each Period t + 1 (for t ≥ 0), each player chooses the action played by the other player in Period t. For example, if (C,D) is played in Period 4, then (D,C) is supposed to be played in Period 5.

(a)  [5 marks] Formulate the strategy profile as an automaton by specifying the state space, initial state, policy function and transition rule.

(b)  [5 marks] Compute each player’s value function.

(c)  [10 marks for ECON3152, 5 marks for ECON4453/8053] Is there any 6 for which the strategy profile is a subgame perfect equilibrium? If so, find all such 6 .

2. Ann and Bob currently share the same apartment. Each Saturday, they need to decide whether to spend effort cleaning the apartment.  In this question, a“period” means a week, and their per-period payoff function is as follows: Here the action C is cleaning the apartment and S is shirking, and e ∈ (2, 5) is the

cost of the cleaning effort.  Cleaning is more effective if both flatmates contribute. They both discount future payoffs by 6 ∈ [0, 1). In each period, there is a probability r ∈ (0, 1) that Bob moves out for some reason that he cannot control, after which Ann receives payoff 5 − e in every period. Whether Bob moves out in Period t is thus decided by Nature and assumed to be independent of everything that happened

in the past. Bob receives payoff b ∈ R in each period after he moves out. Technically, this game is not a repeated game as the stage game changes after Bob moves out. However, we try to analyze this game using the tool of automata.

To be specific, consider the following strategy profile:  they play (C,C) until somebody plays S, after which everybody plays S for ever (or until Bob moves out).

(a)  [5 marks] Consider an automaton with three states, Good, Bad and Out.  The first two states are

before Bob moves out and the third state is after Bob moves out. In the Good state, they play (C,C)

and in the Bad state, they play (S,S). There is no strategic choice in the Out state. The initial state is Good.  Find the transition function g  : W × Y → W so that the automaton describes precisely the above strategy profile.

(b)  [5 marks] Computer each player’s value function.

(c)  [10 marks] Find the condition on e, b, δ and r under which the strategy profile is a subgame perfect equilibrium.

(d)  [5 marks] Is there a way to summarize the impact of the two parameters δ  and r  on the value functions by a single combination of them?

(e)  [10 marks for ECON3152, 5 marks for ECON4453/8053] Does the result of the previous part gener- alize to other strategy profiles (automata)?

3. Two software engineers work together to develop a software (or an app). Before they start working, they need to agree on a design decision, such as the relative effort spent between the core functionality and the user interface. For the purpose of this question, this design decision is represented by a number θ ∈ [0, 1]. For example, that θ = 0 represents a software with rich functionality but simple user interface, while that θ = 1 represents a software with simple functionality but beautiful user interface.  Unfortunately, the two engineers have the opposite taste on this issue.  Therefore, if a decision θ  is agreed upon in Period t ≥ 0, Engineer 1 receives payoff δ t (1 − θ2 ) and Engineer 2 receives payoff δ t (1 − (1 − θ)2 ).  If they never reach an agreement, both receive payoff zero. The bargaining protocol is as follows:

• In each even period, Engineer 1 proposes a θ ∈ [0, 1] and Engineer 2 decides whether to accept or not. If the proposal is rejected, the bargaining moves on to the next period.

• In each odd period, Engineer 2 proposes a θ ∈ [0, 1] and Engineer 1 decides whether to accept or not. If the proposal is rejected, the bargaining moves on to the next period.

(a)  [5 marks] Which θ ∈ [0, 1] maximizes the total payoff of the two engineers (assuming that agreement

is reached at Period 0)?

(b)  [5 marks] In what follows, consider the following class of strategy profile.  In every even period, Engineer 1 proposes θ =  − a1  and Engineer 2 accepts a proposal θ if and only if θ ≥  − a1 .  In every odd period, Engineer 2 proposes θ =  + a2  and Engineer 1 accepts a proposal θ if and only if θ ≤  + a2 . Here a1  and a2  are numbers in [0, ] to be determined. Formulate this strategy profile as an automaton.

(c)  [5 marks] Find the value functions of the engineers, which depend on a1  and a2 .

(d)  [5 marks] Is there a subgame perfect equilibrium with a1  = a2  = 0?

(e)  [10 marks for ECON3152, 5 marks for ECON4453/8053] Find conditions on a1  and a2  under which the above strategy profile is a subgame perfect equilibrium.

(f)  [10 marks for ECON3152, 5 marks for ECON4453/8053] Solve the conditions and find the equilibrium values of a1  and a2 .

(g)  [5 marks] What are the limits of a1  and a2  when δ → 1?

4.  (ECON4453/8053 only) Fix a non-empty compact set W  and let C(W) be the set of all continuous  functions on W .  As for compact ordered sets, there are two particularly important cases.  First, W is  a closed and bounded subset of an Euclidean space and C(W) is the set of functions on W that are  continuous in the usual sense.  Secondly, W is a finite set and C(W) is the set of all functions on W .  (Recall that every function on a finite set is continuous.)  For any f, g  ∈ C(W), define the “distance” between f  and g  as d(f, g)  =  maxx∈W  |f (x) − g(x)|.   The maximum exists because the continuous  function f − g has a maximum and a minimum on the non-empty compact set W .

(a)  [5 marks] To be a well-defined “distance” function (or metric), d needs to satisfy three conditions: (i) d(f, f ) ≥ 0 for every f ∈ C(W), with equality only when f (x) = 0 for every x ∈ W .              (ii) d(f, g) = d(g, f ) for all f, g ∈ C(W).

(iii) d(f, g) + d(g, h) ≥ d(f, h) for all f, g, h ∈ C(W).

The first two conditions are obviously satisfied. Verify the third condition.

(b)  [5 marks] Let Y be a non-empty finite set and δ ∈ [0, 1).  Let p be a probability mass function on Y : p : Y → R such that p(y) ≥ 0 for every y  ∈ Y and ∑y∈Y p(y) = 1.  Let u : W × Y → R and g : W × Y → W be functions such that u(w, y) and g(w, y) are continuous in w for every y ∈ Y . For every f ∈ C(W), define a new function Tf : W → R as follows:

(Tf )(w) =  [(1 − δ)u(w, y) + δf (g(w, y))] p(y),  for all w ∈ W.

Show that Tf is a continuous function.