Question

. Dividing Money There is a $10 available prize, and Players 1,2 must try to agree on a di- vision, via the following game: Player 1 writes down a proposed offer x1 to Player 2 (i.e proposes that he, P1, keep 10-x of the prize, giving the remaining xi to Player 2). Player 2 simultaneously writes down a demand for himself, x2: If x1 2x2 (P1's offer weakly exceeds P2's demand), the money is divided according to Pl's suggested split (i.e. P1 gets 10 - xi and P2 gets x1): If xi < x2 (P1 offers less than P2 demands), then both players get $0. (a) If x1x2 are both restricted to the set $0;5;10], write out the payoff matrix, and identify all NE. Now, for parts (b)-(e), allow each player to choose any integer between 0 and 10. (b) Find PI's best responses, (i) for each X2 є {0,1,. . ., 9); and (ii) for X2-10. (c) Find P2's best responses, (i) for each X1 E {1, 2, . . ., 10), and (ii) for x,-0. (d) Using your answers to (b) and (c), find all NE, i.e. all pairs (xi;x2) where both players are best-responding to their opponent. (e) For each of your NE in (d), determine whether either player is using a weakly domi- nated strategy.

Answer 1

aye2 ษุ \四 5, 5 S,5 o, O ouer 0.10 o, 10 action opponent ay Loe e eO Cyrr one one and ax cadh mateH Lo

So, player-2 will be indifferent b/w all three actions he can take, hence what we do is that we tick all the boxes in first row

indicating that if player one playes x1 = 0 then player -2 can play any of the three actions available. We repeat this with every action of player -1 and keep marking the boxes which player -2 will play given player-1's action.

We will get,

Now, repeating the same for player-1,i.e., he will choose his best response associated with each action of player-2 and we will mark those boxes agian. So our box will look like,

au 0 ,10

It is this intersection of marked boxes (i.e, box containing two marks) is the intersection of best responses of two players which gives us the Nash Equilibrium.

To answer Question (b) either draw a big payoff matrix showing all the payoff's or track the pattern by finding best response player-1 by looking at some of the actions of player-2.

au ago tr

When player-2 two choose x2 = 0; the best response of player-1 is to choose x1 = 0

similarly when player-2 chooses x2 =1; the best response of player-1 is choose x1 = 1

i.e every time he will choose x1 = x2 or anything when x2 = 10.

You can verify from the image below,

c.)

Similar arguments can be made to solve this part also that is,

When player-1 chooses x1= 0 the best response of player-2 will be to choose anything because no matter what he chooses he will get payoff = 0.

When player -1 chooses x1 = 1 the best response of player-2 will be to choose any number less than or equal to what player one chooses so that at least he can get what player one has splitted for him. Cause otherwise he will get zero.

Hence every time he will play $\small x_{2}\leq x_{1}$ or anything when x1 = 0.

You can verify this in above image containing payoff matrix.

d.)

We have player-1's best response as

$\small x_{1}^{*}= \left\{\begin{matrix} x_{2} & for&x_{2} \epsilon \left [ 0, \right9 ]\\ & & \\ anything& for & x_{2} = 10 \end{matrix}\right.$

and player-2's best response as,

$\small x_{2}^{*}= \left\{\begin{matrix} \leq x_{1} & for&x_{1} \epsilon \left [ 1, \right10 ]\\ & & \\ anything& for & x_{1} = 0 \end{matrix}\right.$

The intersection of these two best response will give us the Nash Equilibrium.

thus Nash Equilibrium will be all such points for which x1 = x2 and one more pair which will give payoff (0,0) on the top right corner.