Catan No Trade Theorem: First Blush

A very common question that comes up when I play with my friends is "Why do people trade in Settlers of Catan? Shouldn't some No Trade Theorem cover it?" (I am sure this says I play with too many Economists...) We tend to puzzle over it for a bit, but never reach a true conclusion about if there is a No Trade Theorem that applies. In this post I will look at why, or why not, a No Trade Theorem applies to the Settlers of Catan universe.

To do this we are going to start at the beginning with the Milgrom-Stokey No Trade Theorem. No I don't expect you to read the paper to understand what follows. Though I will be relying on many concepts in economics and will link the Wikipedia articles on them when they first appear. This theorem has a couple of parts.

1. Utility function depending on the state of the world and one's consumption
2. Agents are risk averse
3. Beliefs are updates from a common prior
4. Initial endowments are Pareto Optimal
5. Agents observe some private information
6. There is a "trade" preferred by all agents after they receive the private info

If all 6 conditions (plus some more technical conditions) hold then the "trade" preferred by all agents provides identical utility to the initial endowment.

The first thing to look at is a technical  issue condition 6, which is why "trade" is in quotes. The reason I didn't say trade is the "trade" in the paper is more complex than those defined by the rules of Settlers. In Settlers a trade is between 2 players and must include each player giving some amount of resources to another. The "trades" in the Milgrom-Stokey are far more general and could involve resources moving between multiple players. This really isn't a huge issue as we could construct a sequence of Settlers trades that result in the Milgrom-Stokey trade, and this could certainly be done to satisfy the constraint that only the active player can trade. I will return to condition 6 later, but need to examine the other conditions first.

Conditions 2, 3, and 4 are clearly satisfied in Settlers. Most people are risk averse, which satisfies condition 2. Condition 3 essentially says that everyone has the same model of how the world works. There is no reason that players should disagree on how things in the game work or the likelihood of events because everything in a board game is completely defined, assuming that people know the rules of Settlers and understand probability. Condition 4 is true because players utilities can be thought of as their probability of winning the game. These probabilities must sum to 1. Hence anytime one player's probability improves another's must necessarily decrease. Therefore every endowment must be Pareto Optimal because any change in endowments that positively effects one agent must negatively effect another.

Condition 1 is the most technically difficult to translate. I think there might be way to apply it to Settlers, but there are 2 big issues with condition 1 relative to the Settlers of Catan universe. First, Settlers is not a one shot game like the Milgrom-Stokey theorem. Players take turns until one wins, so even the length is endogenous! Normally I would just suggest that everything could be considered an Arrow-Debreu securities but the other issue with utilities would prevent this from working and also suggests an alternative way to utilize the Theorem.

Utilities only depend on the state of the world, essentially on how the rolls dole out resources which affects who wins. But the state of the world intimately depends on the resources and buildings each player has. Buildings directly determine most points in the game and alter resource production. Resources change what actions players can take by affecting the available buildings. So there is a feedback loop that depends on how players play the game, which is affected by the trading! Essentially this means that an agents' utility depends on his endowment and the endowments of everyone else. But on any particular player's turn, the other players' can't buy anything so the cards in hand only affect utility through future actions and the buildings on the board make sense as the state of the world. Hence we can apply the theorem to each player's turn individually and simply use backwards induction style argument to calculate utilities from the continuation game. This means that trade should never happen in Settlers... or would mean that if not for issues with the other conditions.

Condition 5 is very odd in the Settlers of Catan world because there is actually very little private information. Any resources rolled are known by all as we can see the numbers and where all the buildings are, even though you may not look at other player's hands. Really only when the Robber moves so a player can steal from another or a Development Card is drawn is there any private info. So most of the time trade occurs in Settlers after not new private info, but new public info! This is even weirder than trade after new private information because every agent already knows how their probabilities of winning have been updated, so there is no information revealing effects of trade to leverage in these cases. Any explanation of trade in Settlers much account for this weird feature.

Condition 6 is where everything breaks down in the Settlers world. We already saw that Milgrom-Stokey "trades" could be reproduced by Settlers' trades. This is not the problem with the condition though. The issue is all players must agree to the trade. In Settlers only the two players on either end of the trade must agree. Hence we could see trades in which the players not party to the trade wishing it didn't happen. Moreover any "trade" that all players agree to must not shift the probabilities of winning because one player would have to be worse off and disagree if they did. This means that the theorem applies, but uses the word "trade" in a different fashion than Settlers does! (Yep, kind of a cop out...)

Overall the Milgrom-Stokey Theorem is very general and flexible. It just takes some work to transform the situation into one that matches the conditions. The truth about Settlers of Catan is the environment does satisfy the conditions of the No Trade Theorem. Unfortunately the theorem does not exclude trades which fewer than the entire set of agents are party to. Thus making it useless for answering the real question of "Why do 2 players ever trade in Settlers? Isn't one of them getting a better deal?" Exploration of this question will have to wait for another post, as this is long enough already. If you have any ideas on how to answer this revised question please let me know in the comments.