Homework 1

1. a) Write down the matrices of the SIMULTANEOUS LEGISLATORS VOTE game in the variant where each of the three voters has also the option to abstain. The raise only passes if more agree than voting against. The loss of face by abstaining is relatively small, only $200.
   b) Solve that game, using the approaches discussed above.
2. Assume a simultaneous two-player game has the best response digraph shown to the right. Display a possible payoff matrix. Can you find a possible zero-sum payoff matrix generating this best responsse digraph?
3. Consider the following two-player game.

   	L	M	R
   U	1,1	3,4	2,1
   M	2,4	2,5	8,1
   D	3,3	0,4	0,9

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the matrix obtained by IESD.
   • Find the matrix obtained by IEWD.
   • Draw the best response digraph.
   • Are there any Nash equilibria?
4. Analyse the following six two-person zero-sum games (Maximin moves, domination, best response digraph, Nash equilibria):

   L     R   U   1     2   D   3     4

   L    R   U   1     2   D   4     3

   L     R   U   1     3   D   2     4

   L     R   U   1     3   D   4     2

   L     R   U   1     4   D   2     3

   L     R   U   1     4   D   3     2

5. Consider the 2 person variant of the GUESS THE AVERAGE game where every player can just choose one of the numbers 1,2,3,4. Create the payoff matrix. Decide whether the game has a dominant move equilibrium, an IEWD equilibrium, an IESD equilibrium, or a Nash equilibrium.