Game Theory and Game Strategies

Logic puzzles involving game theory, determining optimal strategies for games, or the design of games

Sharing Game Theory Puzzle

In this sharing game theory puzzle, 3 friends take turns taking from a jar of 1000 candies to share. For example, the 1st friend could take 500 candies, then the 2nd friend could take 400, and the 3rd friend would take the remaining 100.

No one wants to be seen as greedy, but no one wants to end up with the fewest candies either. As such, their goals are (in order of preference):

  1. Do not end up with the most candies, nor the fewest candies (a tie for most or fewest also fails this condition)
  2. End up with as many candies as possible

All of them are logical, rational, know each other’s goals, but cannot communicate before or during sharing. How many candies should each friend end up with?

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A truel is a three-way duel. The three participants take turns firing one shot at whichever opponent they choose, until only one is remaining.

Allison, Ben, and Chase are in a truel, and have varying degrees of accuracy: Allison has a 50% chance of hitting her intended target, Ben has a 80% chance, and Chase has a 100% chance. Allison gets to shoot first, then Ben, then Chase, and repeating in that order until only one person is remaining.

Assume that the accuracy of all participants are publicly known, and everyone is trying to maximize their chances of winning. What is Allison’s optimal strategy, and what is her likelihood of winning under that strategy?

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Last Chocolate Game

Two friends are playing a game with two boxes of chocolates. They take turns taking out some number of chocolates from the boxes, and whoever takes the last chocolate wins. Each turn they can take chocolates one of two ways:

  • Take any number of chocolates from a single box
  • Or take an equal number of chocolates from each box

If there are 25 chocolates in one box and 35 in the other, what is the winning strategy?

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Card Pairing Game

You are playing a game with your friend using a standard deck of 52 playing cards. Each round, you flip over two cards:

  • If both cards are black, you keep the cards and earn one point.
  • If both cards are red, your friend keeps the cards and earns one point.
  • If one card is red and one is black, the cards are set aside in a discard pile.

What is the probability that you end up with more points than your friend when all the cards in the deck have been used up?

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Rigging a Vote

There are five candidates participating in an unusual election. There are four rounds of voting, and only two candidates participate in each round, with the winner moving on and the loser dropping out. This means first candidate 1 will run against candidate 2, then the winner will run against candidate 3, and so on, with the winner of the last round being elected.

Polling shows that Stephen Hawking is not very popular. There are five equal-sized groups in the voting population, and polling shows they prefer the candidates in the following order:

  • Group 1: Einstein > Newton > Maxwell > Hawking > Curie
  • Group 2: Newton > Curie > Maxwell > Einstein > Hawking
  • Group 3: Curie > Maxwell > Einstein > Newton > Hawking
  • Group 4: Einstein > Newton > Hawking > Curie > Maxwell
  • Group 5: Newton > Maxwell > Einstein > Hawking > Curie

But Hawking has a secret weapon – he gets to choose the order the candidates participate in each round.

Is there a way for Hawking to organize the rounds such that he wins?

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