Solutions for Homework 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.
See the page here .
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?
  L    M    R  
U  -1, 1    1, -1    0, 1  
M  -1, 1    0, 0    1, -1  
D  1, -1    1, -1    -1, 1  
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  

Maximin move is "M" for both Ann and Beth, guaranteeing a value of at least 2 for Ann and at least 4 for Beth.

For Ann there is no domination, weak or strong. For Beth, move "M" dominates (strongly, and therefore also weakly) move "L".

After removing option "L" for Beth, suddenly Ann's move "D" is strictly (and therefore also weakly) dominated by both "U" and "M". After eliminating this option as well, we arrive at the following bimatrix:
  M    R  
U  3,4    2,1  
M  2,5    8,1  
Here again Beth has domination---move "M" stricty dominates move "R". After removing move "R", Ann's move "M" strictly dominates move "M". Therefore the IESD and IEWD matrix, the result after iterated eliminating strictly or weakly dominated moves, is
  M  
U  3,4  

The best responses are U-->M, M-->M, and D-->R for Beth, and L-->D, M-->U, and R-->M for Ann.

Therefore the Nash equilibrium is "U" versus "M", since "M" is Beth's best response against Ann's "U", and "U" is Ann's best response against Beth's "M".

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  
Let's do just the first of these six case. Note that the matrix of the first one translates into the following bimatrix:
  L    R  
U  1, -1    2, -2  
D  3, -3    4, -4  
Therefore Ann's maximin move is "D", guaranteeing a payoff of at least 3, and Beth's maximin move is "L", guaranteeing at least a payoff of -3. Ann's move "D" dominates her move "U", and Beth's move "L" dominates her move "R". Therefore "D" versus "L" is the IESD result. This is also the Nash equilibrium.
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.
If in ties the dollar can be split, the payoff bimatrix would look like this. Actually in this case it would be a constant-sum game.
  1    2    4    4  
1  0.5, 0.5    1, 0    1, 0    1, 0  
2  0, 1    0.5, 0.5    1, 0    1, 0  
3  0, 1    0, 1    0.5, 0.5    1, 0  
4  0, 1    0, 1    0, 1    0.5, 0.5  

There is no strict domination, so there is no dominant move equilibrium. However move "1" weakly dominates all others. Therefore the IEWD matrix is
  1  
1  0.5, 0.5  

The pair "1" versus "1" is the only Nash equilibrium of the game.