Question 4: Jar Game Consider the following game: Players: 2 - We designate player #1 to...

Question 4: Jar Game Consider the following game: Players: 2 - We designate player #1 to be the one that starts with the jar. Actions: - Each round at the same time both players deposit between 1 to 4 pennies into the jar. - Then both players are able to count the pennies in the jar. - If there are 21 or more pennies, the person with the jar is the winner. - If there are 20 or less pennies, the jar is passed to the other person and another round begins. Do you want to be the first or second player? What is the winning strategy?

Expert Solution

Matching Pennies is a zero-sum game because each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the participants' total gains are added up and their total losses subtracted, the sum will be zero.

The game can be written in a payoff matrix (pictured right - from Even's point of view). Each cell of the matrix shows the two players' payoffs, with Even's payoffs listed first.

Matching pennies is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium.^[1]

This game has no pure strategy Nash equilibrium since there is no pure strategy (heads or tails) that is a best response to a best response. In other words, there is no pair of pure strategies such that neither player would want to switch if told what the other would do. Instead, the unique Nash equilibrium of this game is in mixed strategies: each player chooses heads or tails with equal probability.^[2] In this way, each player makes the other indifferent between choosing heads or tails, so neither player has an incentive to try another strategy.

Variants:

Odds and evens - a game with the same strategic structure, that is played with fingers instead of coins.

From the above observation I will the first player.

ekkarill92 answered 2 years ago

Coin taking game This game is played between 2 players, player 1 and player 2. There...

Coin taking game This game is played between 2 players, player 1 and player 2. There are two piles of coins. The values of a coin can be any integer. Both players know the values of all coins in both piles. Player 1 makes the first move, and play alternates between the players. A move consists of taking a coin from the top of either of the piles (either player can take from either pile). The game ends when both...

Consider the following two-player game, in which Player 1 is the IMF, and Player 2 is...

Consider the following two-player game, in which Player 1 is the IMF, and Player 2 is a debtor country. Reform Waste Aid 3, 2 -2, 3 No Aid -2, 1 0, 0 a) Compute all (pure and mixed) Nash equilibria. b) Do you think that the above game is the case of a resource curse? Interpret the game with a story of a resource curse.

Consider the following game that has two players. Player A has three actions, and player B...

Consider the following game that has two players. Player A has three actions, and player B has three actions. Player A can either play Top, Middle or Bottom, whereas player B can play Left, Middle or Right. The payoffs are shown in the following matrix. Notice that a payoff to player A has been omitted (denoted by x). Player B Left Middle Right Top (-1,1) (0,3) (1,10) Middle (2,0) (-2,-2) (-1,-1) Bottom (x,-1) (1,2) (3,2) (player A) Both players...

Consider the following game. Player 1’s payoffs are listed first: Player 2 X Y Player...

Consider the following game. Player 1’s payoffs are listed first: Player 2 X Y Player 1 A 90 , 1 10 , 0 B 10 , 0 50 , 1 C 100 , 0 80 , 1 Imagine that player 1 makes a decision first and Player 2 makes a decision after observing player 1’s choice. What is the subgame-perfect equilibrium of this game? Imagine that player 2 makes a decision first and Player 1 makes a decision after...

This question is about the sequential moves Stag Hunt game. There are two players. Player 1...

This question is about the sequential moves Stag Hunt game. There are two players. Player 1 moves first, player 2 observes player 1’s move, and then player 2 moves. Players get 10 jollies each if they both choose Stag; 5 jollies if they choose Hare; and 0 jollies if they choose Stag but the other player does not. What is the set of (pure) strategies for player 1? What is the set of (pure) strategies for player 2? Explain why...

Consider the following game: Three cards are labeled $1, $4, and $7. A player pays a...

Consider the following game: Three cards are labeled $1, $4, and $7. A player pays a $9 entry fee, selects 2 cards at random without replacement, and then receives the sum of the winnings indicated on the 2 cards. a) Calculate the expected value and standard deviation of the random variable "net winnings" (that is, winnings minus a $9 entry fee) b) Suppose a 4th card, labelled k, is added to the game but the player still selects two cards...

1. Consider the following game. There are two piles of matches and two players. The game...

1. Consider the following game. There are two piles of matches and two players. The game starts with Player 1 and thereafter the players take turns. When it is a player's turn, she can remove any number of matches from either pile. Each player is required to remove some number of matches if either pile has matches remaining, and can only remove matches from one pile at a time. Whichever player removes the last match wins the game. Winning gives...

A sequential game with two-players 1 and 2, where player 1 has the first move advantage....

A sequential game with two-players 1 and 2, where player 1 has the first move advantage. Each player has two strategies, A and B. If both players choose A, each gets a payoff of 2. Both choose B, each gets a payoff of 3. For player 1 choose A and player 2 choose B, player 1 gets 4, and player 2 gets 1. If player 1 chooses B and player 2 choose A, player 1 gets a payoff of 1,...

Dice Game Rules: 2 - 4 players Each player has 5 Dice. The dice have 6...

Dice Game Rules: 2 - 4 players Each player has 5 Dice. The dice have 6 sides. Each player rolls their dice, and the dice statistics are reported: Sum, number of pairs (and of what), and "straights" - (all dice in order - e.g. 1,2,3,4,5 or 2,3,4,5,6) Player 1 might roll 2,2,3,4,4 so the results would be: Sum: 15, 1 pair (2), 1 pair (4) Player 2 might roll 1, 1, 4, 6, 6 so the results would be: Sum:...

A jar contains 2 red, 3 green, and 6 blue marbles. In a game player closes...

A jar contains 2 red, 3 green, and 6 blue marbles. In a game player closes their eyes, reaches into the jar and randomly chooses two marbles. The player wins the game if at least one of their marbles is red. Suppose it cost $1 to play the game and the winning prize is $3. Mathematically analyze this game and determine if it is in your financial interest to play the game.

Question