2 In a game two players are each dealt five cards from a set of ten different cards.
Player 1 is dealt cards A, B, F, G and J.
Player 2 is dealt cards C, D, E, H and I.
Each player chooses a card to play.
The players reveal their choices simultaneously.
The pay-off matrix below shows the points scored by player 1 for each combination of cards.
Pay-off for player 1
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Player 2}
| | C |
| \cline { 3 - 7 }
\multirow{4}{*}{Alayer 1} | D | E | H | I | | |
| \cline { 3 - 7 } | A | 4 | 1 | 3 | 2 | 2 |
| \cline { 3 - 7 } | B | 0 | 2 | 1 | 2 | 1 |
| \cline { 3 - 7 } | F | 0 | 1 | 1 | 2 | 3 |
| \cline { 2 - 7 } | G | 2 | 0 | 3 | 3 | 3 |
| \cline { 3 - 7 } | J | 1 | 2 | 3 | 0 | 2 |
| \cline { 3 - 7 } | | | | | | |
| \cline { 3 - 7 } |
\end{table}
- Determine the play-safe strategy for player 1, ignoring any effect on player 2.
The pay-off matrix below shows the points scored by player 2 for each combination of cards.
Pay-off for player 2
Player 1
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Player 2}
| | C | D | E | H |
| \cline { 2 - 6 }
A | 2 | 2 | 0 | 1 | I |
| \cline { 2 - 6 }
B | 3 | 1 | 2 | 1 | 2 |
| \cline { 2 - 6 }
F | 3 | 2 | 2 | 1 | 0 |
| \cline { 2 - 6 }
G | 1 | 3 | 0 | 0 | 0 |
| \cline { 2 - 6 }
J | 2 | 1 | 0 | 3 | 1 |
| \cline { 2 - 6 } | | | | | |
| \cline { 2 - 6 } |
- Use a dominance argument to delete two columns from the pay-off matrix. You must show all relevant comparisons.
- Explain, with reference to specific combinations of cards, why the game cannot be converted to a zero-sum game.