For the following games, write out the payoff matrix and the reduced matrix after domination.
(a)
Rowan has a 3 of hearts, 4 of spades, 9 of spades and 10 of hearts. Collen has a 5 of spades, 6 of hearts, 7 of hearts and 8 of spades.
Each player selects a card without revealing it, and both players flip their cards over at the same time. If the suits are the same, then Rowan wins the sum of the two card values. Otherwise, Colleen wins that sum.
(b)
The same as (A) but reverse the payoff conditions for Rowan and Colleen.
(c)
Rowan picks an even integer from 1-6, and Colleen picks an odd integer from 1-6. If the difference is less than 3, the player who played the bigger number wins the sum of the two values. Otherwise the player who played the smaller number wins the sum of the two values.
(d)
Rowan picks an integer from 1-3. Colleen picks two guesses which may be the same. Colleen reveals their guesses one at a time. If they guess correctly, they win $ equal to the value of their other guess. If they do not guess correctly, Rowan wins $ equal to the sum of both guesses.
(e)Colonel Blotto.
Colonel Rowan is attacking a town defended by Colonel Colleen. Rowan has three regiments and Colleen has four. There are two routes to the town. Rowan and select any number of regiments to attack along each route, up to a total of three and Colleen can likewise assign her four regiments along either route, neither knowing beforehand the other’s strategy.
Along each route, whatever side has a greater number of regiments wins points equal to the number of regiments sent by the opposing side (as they capture those units). Furthermore, if Rowan wins along either route, he captures the town also worth one point.
(f)Morra.
Rowan and Colleen each simultaneously hold up one or two fingers and shouts a guess for the total number of fingers held. If either Rowan or Colleen guess correctly, then they collect $ from their opponent equal to this number of fingers. If they both guess correctly or both guess incorrectly, then no money changes hands.
2.
For each of the following payoff matrices, determine if there is a value of \(x\) so that the matrix has a saddle point. If so, determine the value(s) of \(x\) and the saddle point(s).
(a)
\(\begin{bmatrix} 1 & 2 \\ x & 3 \end{bmatrix}\)
(b)
\(\begin{bmatrix} 1 & 2 \\ 3 & x \end{bmatrix}\)
(c)
\(\begin{bmatrix} 1 & 3 \\ 2 & x \end{bmatrix}\)
(d)
\(\begin{bmatrix} 3 & x \\ x & 1 \end{bmatrix}\text{.}\)
(e)
\(\begin{bmatrix} 2 & 1 \\ 3 & x \end{bmatrix}\text{.}\)
(f)
\(\begin{bmatrix} 2 & 3 \\ 1 & x \end{bmatrix}\text{.}\)
(g)
\(\begin{bmatrix} x & 1 \\ 2 & 3 \end{bmatrix}\text{.}\)
(h)
\(\begin{bmatrix} x & 2 \\ 1 & 3 \end{bmatrix}\text{.}\)
3.
Write out a reduced payoff matrix with exactly three saddle points.
4.
For each of the following, find the von Neumann value and optimal strategy for the payoff matrix.
For each exercise in Exercise 5.4.1, determine the Von Neumann value and the optimal strategy.
6.
For each of the following games of chance, determine the Von Nuemann value for the game, and optimal strategies for both players.
(a)
Both players secretly flip a coin, they see their own result but not the other. Suppose Heads is greater value than tails.
Rowan then has two choices. He may CALL: both coins are revealed. If Rowan wins, Colleen gives him $2, if Colleen wins, Rowan gives her $4. If both are the same, no money changes hands. He may BID: Colleen then has two choices.
Colleen may FOLD: and Rowan wins $4. Or she may SEE: in which case both coins are revealed, and the winner is awarded $10 from the loser. If there is no winner, no money changes hands.
(b)
Each player adds $2 to the pot. Then they roll a 4 sided dice in secret. Each player knows their own results, but not the other’s.
Rowan two options. He may FOLD: The pot goes to Colleen. He may PLAY: In which case he adds $5 to the pot.
Colleen then has two options. She may FOLD: The pot goes to Rowan. She may PLAY: In which case she adds $5 to the pot.
Then the results are revealed. Whoever wins takes the pot. If they are a tie, then the pot is split between the players.
(c)
There are 6 cards, 2 Jacks, 2 Queens and 2 Kings with Jack < Queen < King. The players each place $1 in the pot. Then, one card each is dealt to each player face down. They may see their own card but not their opponenets.
Rowan now has three choices. He may FOLD: the pot goes to Colleen. He may CHECK: the pot remains as is, or he may RAISE: Rowan adds $2 to the pot.
If Rowan doesn’t FOLD, Colleen also has three choices. She may also FOLD: The pot goes to Rown. She may SEE: Colleen adds money to the pot equal to what Rowan added (if any). She may SEE-RAISE: Colleen adds money to the pot equal to what Rowan added, and then they both add in an additional $2.
If no one has folded at this point, the cards are revealed. The pot goes to the winner. If the cards are a tie, then the pot is split evenly between the players.
7.
Prove that if a reduced payoff matrix has multiple saddle points, they have the same value.
8.
Consider the payoff matrix
\begin{equation*}
\begin{bmatrix} x & 0 \\ 0 & y\end{bmatrix}.
\end{equation*}
(a)
Find neccesary and sufficient conditions for this matrix to be reduced by domination.
(b)
Find the Von Nuemann value and optimal strategies for each player for the game above. There may be multiple cases.
Find the Von Nuemann value and optimal strategies for each player for the game above.
(b)
Suppose \(x\) could be any value, when does this matrix reduce via domination?
(c)
Find the Von Nuemann value and optimal strategies for each player for the game above if \(x\) could be any value.
10.
Consider the payoff matrix
\begin{equation*}
\begin{bmatrix} a & c \\ d & b\end{bmatrix}
\end{equation*}
where \(a < b < c < d\text{.}\)
(a)
Find the optimal solutions and the Von Nuemann value for the above game. (Without loss of generality, we may assume \(a,b,c,d\) are positive, why?)
(b)
Prove that in a game with a two by two payoff matrix where the optimal solution for each player is a pure strategy, then the matrix has a saddle point.
(c)
Prove via contradiction that in a game with a reduced \(n\times n\) payoff matrix where the optimal solution for each player is a pure strategy, then the matrix has a saddle point.
