Game theory

Question 1

Which players gains are shown on a payoff matrix

Answer 1

the player on the left / the vertical part

Question 2

There is stable solution if…

Answer 2

the maximum value of the row of minima is the same as the minimum value of the column of maxima

Question 3

playsafe strategy for horizontal player/ player 1

Answer 3

The maximum value of the column of row minima

Question 4

playsafe strategy for vertical player/ player 2

Answer 4

the minimum value of the row of column maxima

Question 5

maximum outcomes when finding playsafe

Answer 5

The maximum outcomes of each column

Question 6

minimum outcomes when finding playsafe

Answer 6

The minimum outcomes of each row

Question 7

Dominance argument for vertical player/ player 1

Answer 7

If the outcomes of row R are always smaller than the outcomes of row P then P dominates R

Question 8

Dominance argument for horizontal player/ player 2

Answer 8

If the outcomes of column R are always smaller than the outcomes of column P then R dominates P

Question 9

Finding the optimal mixed strategy for 2x2 payoff matrix

Answer 9

1. assign probabilities p and 1-p to the two options for player 1
2. find expressions in terms of p for As payoff under each of Bs options
3. Equate these two expressions to find p
4. work out the value of the game for this value of p
5. repeat for player B

Question 10

player 1

Answer 10

The vertical player, the game is from their perspective

Question 11

value of the game

Answer 11

the payoff from the strategies used

Question 12

zero sum

Answer 12

one persons gains = net losses of other components

Question 13

optimal mixed strategy 2xn matrix

Answer 13

1. check for dominance if there is then just do 2x2 method
2. otherwise assign probabilities p and 1-p to player 1s strategies
3. find expressions in terms of p for As expected payoff under each of Bs options
4. plot each of these expressions between 0 and 1 and shade the region under all the lines
5. find the value of p at the highest point of the shaded area
6. calculate the expected payoff for this value of p
7. repeat for player B

Question 14

optimal mixed strategy for nx2 payoff matrix

Answer 14

reflect the matrix in the leading diagonal and swap the signs then follow the 2xn method

Question 15

Finding the nash equilibrium

Answer 15

1. circle the best choice for each player when the other player plays each of their strategies
2. if there is a box which has two circles then that is the nash equilibrium

Question 16

do all games have a nash equilibrium

Answer 16

no

Question 17

weak dominance

Answer 17

if two of the strategies give an equal value

Question 18

non strict nash equilibrium occurs when…

Answer 18

there is weak dominance

Question 19

formulating a game as an LP problem

Answer 19

positive (and non zero) by adding a constant to each value

2. let the new value of this matrix = v, objective function maximise P = v- constant
3. let p, q r equal the probability player 1 chooses each strategy
4. find expressions in terms of p, q, r for As payoff under each of Bs options, these are the conditions (all ≤)
5. solve using simplex

Question 20

objective function from a game theory problem

Answer 20

P = v - constant used to make the whole game positive

Question 21

if you need to solve game theory with LP from other perspective then

Answer 21

reflect matrix in leading diagonal

swap all the signs

Question 22

finding the LP conditions from a game theory problem

Answer 22

• let p, q r equal the probability player 1 chooses each strategy
• find expressions in terms of p, q, r for As payoff under each of Bs options, these are the conditions (all v ≤)

Question 23

LP conditions from game theory

Answer 23

v ≤ ap + bq + cr

Question 24

what time of LP problem is a game theory problem

Answer 24

maximise