# Pirates and Paradoxes

## Introduction: The Birthday Problem

### Document It: The Birthday Paradox

- In a group of 30 students what are the chances of two people sharing a birthday?
- Open a text editor and write down your answer with either a fraction or %.

- Watch the video…

- Were you correct?
- What was the correct answer?
- Can you explain it?

### Learn It: The Pirate Game

**The scenario**

- There are 5 rational pirates, A, B, C, D and E.
- They find 100 gold coins.
- They must decide how to distribute them.
- The pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.
- The pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates, including the proposer, then vote on whether to accept this distribution. If the proposed allocation is approved by a majority it happens. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again. Each pirate would prefer to throw another overboard, if all other results would otherwise be equal. Each pirate is very greedy but values his life the most.
- What would be the strategy proposed by A?

**Hints**

Work out the best results for pirate A using Backward induction and start with one pirate then two and keep going until you get a solution.

### Badge It Silver

- Make a presentation to show Pirate A what s/he should do.

### Learn It: Matrix

- A good way to represent Game Theory examples is to use a matrix.
- This is sometimes called the
**normal form**. - Here is one we used for the Prisoner's dilemma:

- Basically the players and their possible actions are on the outside of the table. The outcomes are added in the inside of the table.

### Learn It: Trees

- Another good way to represent Game Theory examples is to use a tree.
- This is sometimes called the
**extensive form**. - Here is one we could have used for the Prisoner's dilemma:

*image courtesy of retroworks.blogspot.co.uk*

- Basically everytime there is a decision you make a branch.

### Research It

- Here is a list of 10 'games':
- Stag Hunt
- Battle of the Sexes
- The Centipede Game
- The Beauty Contest
- Chicken
- Hawk/Dove game
- Diner's dilemma
- Rock, Paper, Scissors
- Dollar Auction
- Traveller's dilemma

**Pick one of them to research for the badge task.**

### Badge It: Gold / Platinum

**You need to explain the game theory principles:**

- number of players
- payoffs
- actions / decisions that are available to the player
- strategy to 'win'
- you may want to show a graphical version of the outcomes as either a matrix or a tree.

The quality of your written answer will determine whether you are awarded Gold or Platinum.