Game theory

Write down the maximum possible pay-off to Player B if Player A plays option P.

Determine whether or not the game is stable.

Show that there are no Nash Equilibrium points for the game.

The entry in one cell in the zero-sum game above can be increased so that there is a Nash Equilibrium point.

• Identify the cell that should be chosen.

• Write down the smallest value that it could take.

Show that the original game can be reduced to a game with a 3 × 2 pay-off matrix using

dominance.

The entry in cell (Q,Y) in the reduced game is changed to a number less than - 4. This new number is denoted by m. You are given that the value of the reduced game to Player B with this change is 0.4.

Determine the optimal mixed strategy for Player B.

Find the corresponding value of m.

Amir and Beth play a zero‑sum game.

The table shows the pay‑off for Amir for each combination of strategies, where these values are known.

You are given that a < 0 < b < c.

Amir’s play‑safe strategy is R.

Determine the range of possible values of a.

Beth’s play‑safe strategy is Y.

Determine the range of possible values of b.

Determine whether or not the game is stable.

Player 1 and player 2 are playing a two-person zero-sum game.

In each round of the game the players each choose a strategy and simultaneously reveal their choice.

The number of points won in each round by player 1 for each combination of strategies is shown in the table below.

Each player is trying to maximise the number of points that they win.

Determine play-safe strategies for each player.

Show that the game is not stable.

Show that the number of strategies available to player 1 cannot be reduced by dominance.

You must make it clear which values are being compared.

In each turn of a game between two players they simultaneously each choose a strategy and then calculate the points won using the table below. They are each trying to maximise the number of points that they win.

In each cell the first value is the number of points won by player 1 and the second value is the number of points won by player 2.

Find the play‑safe strategy for each player.

Explain why player 2 would never choose strategy Z.

Find the Nash equilibrium solution(s) or show that there is no Nash equilibrium solution.

Player 2 chooses strategy X with probability p and strategy Y with probability 1 - p.

You are given that when player 1 chooses strategy A, the expected number of points won by each player is the same.

Calculate the value of p.

Determine which player expects to win the greater number of points when player 1

chooses strategy B.

Alex and Beth play a zero-sum game. Alex chooses a strategy P, Q or R and Beth chooses a strategy X, Y or Z. The table shows the number of points won by Alex for each combination of strategies. The entry for cell (P, X) is x, where x is an integer.

Suppose that P is a play-safe strategy.

Determine the values of x for which the game is stable.

Determine the values of x for which the game is unstable.

The game can be reduced to a 2 × 3 game using dominance.

Write down the pay-off matrix for the reduced game.

When the game is unstable, Alex plays strategy P with probability p.

Determine, as a function of x, the value of p for the optimal mixed strategy for Alex.

Suppose, instead, that P is not a play-safe strategy and the value of x is - 5.

Show how to set up a linear programming formulation that could be used to find the optimal

mixed strategy for Alex.

Annie and Brett play a two-person, simultaneous play game. The table shows the pay-offs for Annie and Brett in the form (a, b). So, for example, if Annie plays strategy K and Brett plays strategy S, Annie wins 2 points and Brett wins 6 points.

Determine the play-safe strategy for Annie.

Show that the play-safe strategy for Brett is T.

If Annie knows that Brett is planning on playing strategy T, which strategy should Annie

play to maximise her points?

If Brett knows that Annie is planning on playing the strategy identified in part (b)(i),

which strategy should Brett play to maximise his points?

Show that, for Brett, strategy R is weakly dominated.

Using a graphical method, determine the optimal mixed strategy for Brett.

Show that the game has no Nash equilibrium points.