Question

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?

Answer 1

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.