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.
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.
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.
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.
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 dollars from their opponent equal to this number of fingers. If they both guess correctly or both guess incorrectly, then no money changes hands.
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).
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.
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 opponent’s.
If Rowan doesn’t FOLD, Colleen also has three choices. She may also FOLD: The pot goes to Rowan. 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.
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.
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.
