PLEASE READ THE SUBMISSION GUIDELINES:

● Use this PDF a template for you answers.

● Write your answers in the designated spots. Otherwise we may not fnd your answer and it will not be graded.

● Use this pdf as a template for your answers. Do not modify it so that they are matched correctly in Gradescope.

●  Fill in the blanks digitally or on the printed out copy.1

●  If you are using paper, scan it (e.g. using a mobile app) to generate a clean PDF.

●  Q2dij requires the knowledge of infnitely repeated games which we will discuss next Tuesday.

1.  Consider a variant of the ultimatum game we studied in class in which players have fairness considerations. he timing of the game is as usual. First, player 1 proposes the split (100 − x, x) of a hundred dollars to player 2, where x ∈ [0, 100]. Player 2 observes the split and decides whether to accept (in which case they receive money according to the proposed split) or reject (in which case they both get 0 dollars).

Now suppose that each player’s utility from an outcome equals the amount of money she gets minus the ”unfairness disutility” which is  proportional to the squared diTerence in the monetary outcomes. hat is, if an oTer of x is accepted by P2, then the ”unfairness disutility” of player i equals βi (x − (100 − x))2 , where βi is a parameters of the game indicating how strongly player i cares about fairness. Note that

if an oTer is rejected, they both get $0 and so the disutility term (and, hence, the fnal utility as well) equals zero. Note also that the case we considered in class corresponds to β1  = β2  = 0.

(a)  [1pt] Represent this game in extensive form.

(b)  [1pt] Let β1  =  , β2  = 0. Which ofers will player 2 defnitely accept? reject? Describe all sequentially rational strategies of player 2.

(c)  [1pt] Let β1  =  , β2  = 0. For each sequentially rational strategy of player 2 you identifed in (c), either describe which proposal maximizes player 1’s continuation value or explain why it does not exist.

(d)  [0.5pt] Let β1  =  , β2  = 0. Describe all SPNE of the game.

(e)  [1pt] Let β1  = 0, β2  =  . Which ofers will player 2 defnitely accept? reject? Describe all sequentially rational strategies of player 2.

(f)  [1pt] Let β1  = 0, β2  =  . For each sequentially rational strategy of player 2 you identifed in (f), either describe which proposal maximizes player 1’s continuation value or explain why it does not exist.

(g)  [0.5pt] Let βJ  = 0, βS  =  . Describe all SPNE of the game.

2. Two producers of the same good repeatedly compete in prices for T periods. hey have the same discount factor 8. In the stage game, each frm chooses between three price levels given below in each part of the problem. he Bertrand stage game profts are given by:

i

Ⅰ(Ⅰ)(JS - pi )(pi - S),     pi  < p-i

Ⅰ

ui (pi , p-i ) =〈Ⅰ  (JS - pi )(pi - S),   pi  = p-i

Ⅰ

Ⅰ(Ⅰ)0,                         pi  > p-i

In each of the following parts you must explain your positive or negative answer. If your answer says that some path can be implemented in an SPNE, then you must list the strategies.

Part I: Suppose the set of available prices is restricted to {S, 寸, J0}.

(a)  [0.5pt] Suppose T = S, and 8 = J. Can the path ((寸, 寸), (寸, 寸)) be implemented in a SPNE?

Hint: Start by writing a ? 根 ? table representing the normal form of the stage game and fnd all stage NE.

(b)  [0.5pt] Suppose T = S, and 8 = J. Can the path ((J0, J0), (J0, J0)) be implemented in a SPNE?

(c)  [0.5pt] Suppose T = S, and 8 = J. Can the path ((J0, J0), (寸, 寸)) be implemented in a SPNE?

(d)  [0.5pt] Suppose T = w, and 8 e (0, J). For which 8 can the path ((S, S), (S, S), …) be implemented in a SPNE using any strategies?