I have been thinking a bit about good games to introduce people to Eurogames. Settles of Catan was my introduction to them and was my go to for sharing Eurogames with others. Recently, I acquired 7 Wonders and feel I have had more success with it. My friends ask to play 7 Wonders more often than any other game. In thinking over the differences between the games a few things pop out, namely the length of the game and the complexity of the rules. These features end up combining in such that new players grok 7 Wonders just as it ends, so they are ready to play again and use their new found understanding. Whereas in Catan, new players tend to grok the game when it becomes obvious they are going to lose. This usually leaves a good 15-30 minutes of them playing a game where they know they are completely irrelevant, which is a decidedly un-fun experience.This leads me to believe that the best games to teach people about Euorgames are shorter games where the moment of grokking occurs just as the game ends.
I am sure that not everyone is familiar with the word grok. It loosely means "to intuitively understand." That light bulb that just went off in your head where you "got" what grok means is what if feels like to first grok a new concept. It is that experience that helps draw people into a game. When a person learns something new, they immediately want to apply it. The moment when a player sees a new strategy or how the rules interact clicks is one of the times when that player will be most engaged with the game. They will want to both test their knowledge and show off their new understanding.
Capitalizing on the desire to try new knowledge is one way to draw people into a game. One way a game can use this feature of human psychology is through presenting players with same situation repeatedly. This way players can immediately make use of knowledge gained in the previous iteration. Another use, and the point of this post, is to structure the game so new players grok the rules right at the end of the game. This way the players will immediately ask to play again to use the new knowledge from the first play.
7 Wonders does this by having most of the points scored in Age III. Most new players are confused in Ages I & II because they mostly setup Age III. Once the player sees the value of the Age III cards and how expensive they are everything falls into place for them on how the game should flow. The rest of Age III tends to be spend gathering more information by reading up on what different cards do so the new players have a more complete picture of what to anticipate in Age III next game. Luckily each age is fairly quick so new players don't have time to get bored or discouraged about losing as they are still actively learning. Add in that each play of 7 Wonders is about 30 minutes and you have a perfect receipt for how to make new players want to play again.
In contrast, Catan has a much longer end game. It could easily be 15-30 minutes from the point at which a player is inevitably going to win and the game actually finishing. When this point arrives, most new players realize the large mistakes they made in the early game which constrain them from wining now. Moreover, there is little learning for new players at this point as only extremely tight play can save them, which they are not experienced enough to preform. Combine this with an average game of Catan taking a hour or more and it is easy to see how 7 Wonders provides a nicer new players experience.
I think these differences in the moment of grokking a game plays a large role in new players asking to play again. From my limited sample, everyone immediately asks to play 7 Wonders again, whereas I usually have to prod for the second game of Catan. To me this means that 7 Wonders and other short games with late moments of grokking are the best to teach on. These games will end up in more people asking to play again which results in more long term casual gamers. So while I do like Catan, I don't think that it will be my go to game for introducing players to the world of Eurogames.
Monday, December 30, 2013
Friday, December 20, 2013
Catan No Trade Theorem: Take 2
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.
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.
Thursday, December 19, 2013
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.
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.
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.
- Utility function depending on the state of the world and one's consumption
- Agents are risk averse
- Beliefs are updates from a common prior
- Initial endowments are Pareto Optimal
- Agents observe some private information
- There is a "trade" preferred by all agents after they receive the private info
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.
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