# REMEMBERING NEUMANN- A BRIEF INTRODUCTION TO GAME THEORY

Game theory is a study of strategic decision making. In a more technical language, as the Wikipedia says- it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers". It has many applications in economics, political science, psychology and even in biology. Not going very deep into its proofs and theorems, which are far beyond the scope of this note, let us look at some of its very interesting applications. We will restrict ourselves to just one example today. PIRATE GAME, IS THE MOST INTERESTING EXAMPLE. PROBLEM: 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? SOLUTION: It might be expected intuitively that Pirate A will have to allocate little if any to himself for fear of being voted off so that there are fewer pirates to share between. However, this is quite far from the theoretical result. We have to work it backwards Case 1: When there is only one pirate left E. It is simple he will keep all the gold to himself Case 2: When D and E, so this is a very Dangerous situation for D, whatever he propose D will vote him out, and then we back to Case1 Case 3: When C, D and E are left, C knows that if D doesn't accept his voting scheme, D will signing his suicide note. So C proposes a distribution scheme of 100, 0, 0 coins Case 4: Now B knows if he gives even one coin to each D and E, he will get their vote. So he propose, 98, 0, 1 ,1 distribution scheme. Case 5: Similarly A will propose scheme 97, 0, 1, 0, 2 or 97, 0, 1, 2, 0. Hence keeps almost all the gold coins.

Note by Kirtan Bhatt
6 years, 7 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

Really NEUMANN Was A legend..... and NASH too!!

- 6 years, 7 months ago

Awesome.. I wasn't aware of the game theory but solved it anyway. I would like to learn more about it

- 4 years, 11 months ago