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.