The Patriarch’s Will – a game theory puzzle
Long-time reader Scott found the following puzzle from this site:
Here is a scenario which occurred many millennia ago: The patriarch of a wealthy family was on his deathbed and wanted to divide his gold among his eight sons who were all very, very greedy. Wishing to favor the oldest son (as tradition would have it) but also to reward the more cunning of his progeny, he made the following decree:
The oldest son is to propose a plan for dividing up the gold. The sons are all to vote on this plan, and if it receives at least half of the votes (four or more) then that will be the way the gold is divided. If this plan does not receive half of the votes, the oldest son gets nothing, the next oldest proposes a plan, and there is another vote, now among the remaining seven. Again at least half of the vote (still four or more) is required, and failure removes this son from the process. This is to continue until some son’s plan receives at least half of the votes of the remaining heirs.
Assuming that these sons will do anything to get the most gold possible for themselves, how much (if any) will the oldest son be able to inherit?
This is very much like an extended pirate’s game. Can you figure out the answer?
The solution
Scott also came up with the following well-written solution.
Reasoning Backwards
The last possible outcome is where Son 7 and Son 8 are voting on a plan. In this situation, Son 7 will be proposing the plan. Since only half of the votes are needed, whatever Son 7 proposes will be passed since Son 7 will naturally vote for his own plan. In this situation, Son 7 will end up with all the money and Son 8 will end up with none. Thus, it is in Son 8’s best interest to prevent it from coming to two people.
At three people, we have Son 6 proposing to Sons 7 and 8. Now, Son 6 knows that Son 8 does not want this vote to fail, since that will reduce it to two people, which is against his interests. Son 6 proposes that he gets all but $1, which goes to Son 8. Son 6 votes for this plan, being to his benefit. Son 7 votes against, since he gets nothing (and knows he stands to get everything if this continues). If Son 8 votes for the plan, he gets $1, if he votes against, he will ultimately get nothing. Thus, it is in Son 7’s best interest to prevent it from coming to three people.
At four people, we have Son 5 proposing to Sons 6, 7, and 8. Son 5 proposes that he gets all but $1, which goes to Son 7. Son 5 votes for his own plan. Son 6 votes against. Son 7, not wanting it to go any further, votes for, and Son 8 votes against. The 2-2 vote passes. Thus, Sons 6 and 8 will want to prevent it from coming to four people.
At five people we have Son 4 proposing to Sons 5, 6, 7, 8. He makes $1 offers to Sons 6 and 8, knowing they don’t want it to go any further, with the rest of the money staying with him. The vote ends up 3-2 (Sons 4, 6, and 8 vs. 5 and 7). Thus, Sons 5 and 7 will want to prevent it from coming to five people.
At six people, Son 3 is proposing. He makes $1 offers to Sons 5 and 7. The vote ends 3-3 with sons 4, 6, and 8 getting nothing. Thus, Sons 4, 6, and 8 will want to prevent it from coming to six people.
At seven people, Son 2 is proposing. He makes $1 offers to Sons 4, 6, and 8. The vote ends 4-3 with Sons 3, 5, and 7 getting nothing. Thus, Sons 3, 5, and 7 will want to prevent it from coming to seven people.
Lastly, with all eight sons, the eldest son, Son 1 will be proposing. Knowing all of the above he will propose $1 to Sons 3, 5, and 7. The vote will end 4-4, with Sons 4, 6, and 8 getting nothing.
Generalization
In a Pirate’s Game of N players–and gold pieces outnumber players–if the players are labeled 1, 2 … N, in order of of first to last proposer, player 1 should make a nominal offer to the odd-numbered players 3, 5, 7, etc., and keep the remaining wealth for himself.
It is interesting to note the Pirate’s Game can be extended to arbitrary number of pirates. The situation gets interesting when there are 100 gold pieces and 200 pirate or 500 pirates. The results are astonishing–the weaker pirates manage to survive–and the full write up is in an old issue of Scientific American.
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10 Responses to “The Patriarch’s Will – a game theory puzzle”
It would be interesting to see how this works in reality… where younger brothers can vote in spite (of their $1 interest) just to punish greedy elder brothers
By Nino on Jul 27, 2010
Has play in this game ever been subject to an experiment?
I very much doubt the equilibriums as described are ever reached – being the toughest pirate probably precludes you from being the smartest pirate.
My guess is that this would be fun to run as an experiment after a series of 2 person ultimatum games.
Thanks for reminding me of this game – I do like the conclusion in the SA article about how the meek shall inherit the earth if scarcity obtains.
By michael webster on Jul 27, 2010
I see a lot of these games come up in game theory which just convince me that game theory has some serious flaws. My prediction for this game is that being offered $1 for your vote is in fact even more insulting than being offered nothing. (Is my vote worth that little to you, screw you! essentially), so if the first son is stupid enough to do that, Son 2 will make an equitable sharing proposal since now the “you get $1″ bluff has been called and he knows making nominal offers will be punished. If Son 1 knows this, he’d start with a mostly equitable share to at least half the brothers. So my prediction if this game were ever played by real people, instead of game theory automatons is that son 1 would get either nothing (if he’s a game theory proponent) or about a quarter.
By Bryan on Jul 27, 2010
@Bryan, you bring up a good point. In fact, in some ultimatum game experiments, people are shown to be unwilling to accept offers of less than 10-15% of the pot. But the key here is that these sons are only motivated by greed, and that works against them. If son 1 agreed to split the inheritance 4 ways with his 3 favorite brothers, he’s guaranteed to win the because he’d have 4 votes he needs. But the other 4 brothers would rather get something over nothing so they’ll offer to take LESS than a quarter each (lets say one-sixth). Now Son 1 would only have to give out a sixth each to three brothers and keep the other half for himself. The 4 brothers left out of this arrangement would then counteroffer lower, and so on and so on, until 3 brothers agreed to accept the lowest possible increment (usually $1 each for formsake). I actually think that birth order would be irrelevant in practice, because even son 2 should see that he’ll either get $1 or nothing in this game.
By Brian on Jul 28, 2010
@Bryan:
Game Theory exercises such as this do depend on certain assumptions. Namely, all participants are capable and willing to reason through the appropriate course of action and, in fact, take the most reasonable course of action. It doesn’t account irrational behavior. Is this a flaw in game theory, or a flaw in people? It’s debatable.
I agree that if you just waltzed in and made the proposed offer, you’d get shot down immediately. You’d have a better shot if you explained the above rationale before making the proposal, but one can’t completely eliminate the chance of spite.
Now, while game theory takes this to the mathematical extreme, maximizing the first proposer’s profit, it’s clear that any proposer has the upper hand and, even if they don’t make nominal offers, stand to profit more than anyone else.
By Scott on Jul 28, 2010
@Bryan:
Additionally, while it may seem that Game Theory misses the mark in *how much* to offer, it is crucial in determining *to whom* to make the offers.
By Scott on Jul 28, 2010
I have a different sense of what would happen in the three person version of this game. My guess is that the strong and weak gang up on the middle and split the loot.
Here is why I think that.
Consider three games, with the three players who have to split 60 pieces of gold
In the first game, the voting rule is that they all have to agree or get nothing. The split is going to be 20,20,20. Unless there is a convincing stubborn holdout.
In the second game, the voting rule is majority wins. There are 3 possible 2 persons coalitions, each with 30,30 as the split. Hard to tell which coalition will form.
Now we get to the three person pirates game. The voting rule is the same, but it is in run-off format. The strongest player proposes and needs 2 votes, if his proposal isn’t accepted the middle player proposes and needs 1 vote, and the weak never gets to vote.
This is very similar to the previous coalition game with majority rule – if the strong can form a coalition with the middle or weak, they will have enough votes. But the problem for the middle is this: no verbal commitment to the strong’s proposal is credible – it would always be better to vote against it, promising the weak a greater share.
If the middle was able to scuttle the strong, then the weak, no matter what was promised, will end up being scuttled by the middle.
This is why the weak has a better chance of a coalition with the strong – both need each other equally.
Just a thought – we love to see how groups might manage this.
By michael webster on Jul 29, 2010
Thanks for the post, Presh! This was a fun puzzle to work out. I proposed this to a friend who also thought a nominal payment to win out votes is insulting and that punishment would be a better strategy for the weaker ones. I agree- it’s just that these pure logical (and probably not very realistic) assumptions are hard to swallow for some people. Still a very interesting scenario!
By Faraz on Jul 31, 2010
I’d like to see the result if you changed the game slightly so that the brother making the offer doesn’t get to vote. At that point, if it got down to the two player scenario, the proposer would be guaranteed to lose, since the last brother only has to vote against him and he inherits everything.
By Bryan on Aug 2, 2010
@Bryan:
In terms of game theory I don’t think this would significantly change things. In that final scenario our observation:
“Thus, it is in Son 8’s best interest to prevent it from coming to two people.”
Becomes:
“Thus, it is in Son 7′s best interest to prevent it from coming to two people.”
Which Son 6 can then use to his advantage making a nominal offer to Son 7, rather than Son 8.
In the end, Son 1 would make nominal offers to 2, 4, 6 and 8 rather than 3, 5 and 7.
By Scott on Aug 2, 2010