Yesterday I posted about how the Milgrom-Stokey No Trade Theorem applies to the Settlers of Catan game. It was a little technical and very stream of conscious, so I wanted to take another crack at it after thinking about the problem some more and coming up with some interesting implications. Including not only a new No Trade Theorem variation and the result that trade is possible in Settlers, but also trades can occur where everyone knows that one player is getting screwed over! In covering this ground we see that one of the rules of Settlers that looks like it restricts trade is actually key to enabling it.Well done Mr. Teuber, well done!
Though first I have to admit I made a mistake in applying the theorem yesterday. One big difference between the Settlers' universe and the Milgrom-Stokey theorem is player's utility depends on the players' hands and buildings in Settlers, but this is not allowed under Milgrom-Stokey. I gave a convoluted argument about how to fix this in the previous post which is wrong. Luckily one can simply expand the Milgrom-Stokey setup to include utility that depends on the other players' consumption and all the math still works. Overall not too big a deal.
This extension does limit the potency of the resulting theorem though. For example, consider a trade between 2 players that they both agree to. In the Milgrom-Stokey world, the other players are indifferent between this trade happening and not because it doesn't affect their utility. Therefor all players approve of this trade, which means that neither of the two players involved in the trade actually benefit. In the standard setup this result has a lot of bite. Now consider the same 2 player trade, but in a Settlers' world where this can affect the the utility of players not party to the trade. Here we don't have that the other players must be indifferent. Moreover if at least one of the 2 players that are party to the trade is made strictly better off by the trade it must be that one of the players not party to the trade is made strictly worse off. This is a directly implication of the No Trade Theorem in this setting as if some player was not made worse off, all players would agree to the trade and hence no player is made better off.
The insight that both players involved in a trade in Settlers could be made better off, but make some other player worse off is key to understanding how trade could exist in the game. For example consider a trade between players 1 & 2 in Settlers where player 1 gives 2 an Ore in exchange for a Wheat. As a result of this trade Player 1 builds a city on his turn and during Player 2's turn she also builds a city. It is not hard to imagine a board setup where this would put both players far ahead of the remaining player or two not part of this trade. Granted this is a pretty rare occurrence for a trade to so clearly benefit both parties at the expense of the other players. But this partly explains how trade could occur in Settlers even if players are perfectly rational and have perfect understanding of the game.
Personally this is not a fully satisfying answer. Particularly because of how much weaker the no trade theorem is in an environment where utility depends on all players' consumption. A No Trade Theorem which applies to trade that any number of players approve of would be closer to the spirit of the Milgrom-Stokey result. One of my colleagues suggested how to do this. Consider a trade in Settlers that benefits both players and hence harms a player not party to the trade. Assuming he is not budget constrained, this 3rd player would be willing to offer a trade to one of the two players considering trading that would cause him to be hurt less and the other player to gain more than she would under the current trade on the table. This trade then blocks the original trade as one of the players who was part of the original trade prefers it. Clearly if one can construct such a blocking trade for any trade, no one can trade because it will always be blocked. In the language of cooperative game theory the core of this game is empty.
A theorem of this form would prevent trades which don't involve all players as the other players could find a way to block it. Such a theorem would be a true analogue to the Milgrom-Stokey result. At the moment I haven't proved that such a theorem is true, but lets assume that it is and applies to Settlers. Would this theorem actually imply No Trade?
As hinted at in the introduction, such a Theorem would not be able to prevent trade in Settlers. In the motivating example we had the 3rd player offering a different trade to one of the two players involved in the trade on the table. There is nothing that says which of the two players this 3rd player could present an appealing trade to. It could be that the only trade the 3rd player could use to block the trade on the table is with the non-active player. This would violate the rules of Settlers. Only trades in which the active player is one of the parties are allowed. Thus a rule that restricts the possible trades actually breaks a No Trade Theorem, which enables more trades to happen!
The forward thinking reader might object citing the possibility of this blocking trade happening on a latter players' turn. There are two key features which might inhibit the promise of future trades. First and foremost the rules of Settlers explicitly state that agreements are non-enforceable, only spot trades are recognized by the rules. The 3rd player doesn't have to follow through when time for the blocking trade comes around, especially if the blocking trade is harmful to him but was simply less bad than the trade on the table. Secondly, the state of the world has changed by the time the blocking trade is available. Players possibly have new resources and structures. This changes their incentives possibly making the blocking trade unattractive at the future date. Given players are risk averse there are very strong incentives pushing them to accept the trade on the table instead of a future blocking trade.
I promised that I would prove that trades actually happen in Settlers where every player knows one is getting screwed over and can do that now. As motivation, I am sure we have all seen a trade about to happen then some 3rd player makes a better offer out of the blue. For example player 1 is trading two Bricks to player 2 for an Ore and Player 3 offers 1 Ore for 1 Brick. He is not doing this because it a good trade. But he knows that player 1 is going to city and player 2 is going to road and settlement and simply wants to block one so he doesn't fall even further behind. In addition this is a better trade for player 1 because she keeps one brick and doesn't give a road and settlement to player 2. This is exactly the original blocking argument where the 3rd player offers a trade that is harmful to block an even worse one. Therefore by using a one trade to block another, we can actually have trade occur where everyone knows that one of parties to the trade is screwed.
Veterans of Settlers will note that the active player will always benefit from a trade on her turn, assuming perfect playing and knowledge. The only way to get trades which negatively impact one of the parties is when they block another trade. The active player must be party to both the original and blocking trade due to the rules, so must always be made better off by such trades. This means that if you trade when not the active player, odds are you are the sucker. Though if players are budget constrained due to having small hands it is less likely that a block trade could be found, implying that a non-active player could benefit in addition to the active player from a trade.
Between this post and the previous, we have analyzed in detail the classic Milgrom-Stokey No Trade Theorem in the context of Settlers of Catan. With a minor tweak the theorem applies, but doesn't have the same bite as the original. In particular, we don't get that there is no trade, but that any trade which occurs must harm someone, and not necessarily one of the players party to the trade. We also looked at a possible stronger No Trade Theorem for Settlers using a blocking trades argument. Fortunately for Settlers, the rules restricting who can trade cause such a theorem to fail. But using this logic we uncover some interesting implications about trade in Settlers, including the fact that there can be trade where everyone knows the non-active player gets screwed.
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