The Rational Pirates
So today while watching videos on YouTube, (which is what I am usually doing almost every second while I am not busy) I came across a brilliant riddle. I clicked on the video and found out that it involved game theory and behavioural economics, and that really fascinated me. I wanted to share the riddle with you, and try to explain and help you better understand the concept of game theory. First of all, you need to know what game theory actually means. Simply kept, game theory is a theoretical structure for understanding social situations among competing players. In some respects, game theory is the science of ‘strategic play’, or at least the optimal decision-making of competing players in a situation.
Now, I would like to share the riddle with you and try to explain the solution and how game theory can help us in various situations:
It’s a good day to be a pirate. Andrew and his 4 mates, Brandon, Christian, David and Edmund have struck gold, a chest with 100 coins. But now, they must divide up the treasure according to the pirate code. The code is as follows:
As the captain, Andrew gets to propose first about how to distribute the coins. Then, each pirate including Andrew himself gets to vote either ‘yes’ or ‘no’. If the vote passes, or if there is a tie, the coins are divided according to the plan proposed by Andrew, but if the majority votes ‘no’, Andrew must walk the plank into the deep sea and Brandon becomes the captain. Then, Brandon gets to propose a new distribution and all remaining pirates would vote again. If his plan is rejected, he walks the plank too and Christian takes his place. This process repeats with the captain’s hat moving to David and then to Edmund until either a proposal is accepted or there is only one pirate left.
Each pirate’s primary objective is to stay alive while getting as much gold as possible. But since they are pirates, none of them trust each other so they can’t really collaborate in advance.
If anyone thinks they will end up with the same amount of gold in two different scenarios, they will vote to make the captain walk the plank just for fun.
Finally, each pirate is excellent at logical thinking and knows that the other pirates are too.
The question here is, what distribution should Andrew propose to make sure he lives and gets as much gold as possible?
If we follow our intuition, it seems like Andrew should try to bribe other pirates with most of the gold to increase the chances of his plan being accepted or maybe to propose a distribution like 40-30-20-10-0 or any other combination for that matter and hope for the best, but it turns out Andrew can do much better than that. Let’s have a look at the solution!
Like you know, the pirates know each other to be top-notch logicians. So, when each one votes, they won’t just be thinking about the current proposal but about all the possible outcomes down the line. Moreover, because the rank order is known in advance, each one of them can accurately predict how others would vote in any situation and adjust their own votes accordingly. Since Edmund is last, he has the most outcomes to consider so let us start by following his thought process. He would reason out by working backward from the last possible scenario with only him and David remaining. David would obviously propose to keep all the gold and Edmund’s one vote would not be enough to override it. So, Edmund wants to avoid this situation at all costs. Now, we move to the previous decision point with three pirates left and Christian making the proposal. Everyone knows that if he is outvoted, the decision shifts to David who would then get all the gold while Edmund gets nothing. So to secure Edmund’s vote, Christian only needs to offer slightly more than nothing, that is, one coin. Since this ensures his support, Christian doesn’t need to offer David anything at all. What if there were 4 pirates? As captain, Brandon would still only need one other vote for his plan to pass. He knows that David wouldn’t want the decision to pass to Christian so he would offer David one coin for his support with nothing for Christian or Edmund. Going forward with other possible scenarios, now we’re back at the initial vote with all 5 pirates standing. Having considered all the other scenarios, Andrew knows that if he goes overboard, the decision comes down to Brandon which would be bad news for Christian and Edmund, so he offers one coin each to them, thus, keeping 98 for himself. Brandon and David vote ‘no’, but Christian and Edmund grudgingly vote ‘yes’, knowing that the alternative would be worse for them.
The pirate game involves some amusing concepts from game theory. One is the concept of ‘common knowledge’ where each person is aware of what the others know and uses this to predict their reasoning and final distribution. Had it been the case where the pirates didn’t know the intellectual capabilities of each other, the situation would flip altogether and there would have been no accurate solution to the riddle but fortunately that wasn’t the case. This is a classic example of ‘Nash-equilibrium’, where each player knows every other player’s strategy and chooses them accordingly. Even though it may lead to a worse outcome for everyone than cooperating would, no individual player can benefit by changing their strategy. So, it seems like Andrew gets to keep most of the gold coins (98 to be precise) and the other pirates might need to find better ways to use those impressive logic skills.
The whole intention behind this article was to explain to you how a simple concept of game theory can help you gain more in life. The example here was not an appropriate one for you to relate to in real life (unless you obviously decide to become a pirate) but there are several other instances in which it might help you. The decision involving whether to go for a movie or a concert with your friends also involves game theory where we know what the other person wants and we try to manipulate the final result so as to do what we want. Also, in financial markets, the decision to invest in a certain bond or a stock after having all the relevant information about the company involves game theory as well. By investing in the company, you become a player and what the company does with that investment determines your gain or loss. These are just some cases where you can utilise game theory, the scope of it is wide-ranging and unexplored, and left for you to discover. As the British writer Charles Lamb rightly said, “Man is a gaming animal. He must always be trying to get better in something or other.”
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