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Archipelago and game theory

Martin G
United Kingdom
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I've been following an interesting review thread in the Archipelago forum which prompted some thinking about game theory. I should point out that I have not played Archipelago, so I'm not trying to make any points about that game specifically.

The discussion revolves around the way the win conditions in Archipelago work. Basically, if the players collectively do not do enough to prevent a certain condition occurring, then everyone loses. If they do, then everyone wins, and the player with most victory points is declared the 'grand winner'. There's also a traitor mechanic, but that is not important to this discussion.

The problem this is causing people is that if a player knows he can no longer be the 'grand winner' then he may decide to tank the game and make sure everyone loses.

As far as I see it (and as David des Jardins has been arguing in the thread), this situation only arises because players are carrying over their ingrained notions of relative ordering from most other games, and refusing to accept the ordering presented in the rules.

In most games, only relative final placement is important. In such a game, it is natural that if you can't finish first, it is better to place jointly with everyone than behind anyone. But those are not the rules of Archipelago. The designer ranks the placements players are trying to achieve:

1. Grand winner in a winning game
2. Other positions in a winning game
3. Loser in a losing game

If you are not prepared to accept that losing with everyone else is a worse personal outcome than last in a winning game, then you are not playing the game as designed. It's just as though you were playing Power Grid with the objective of having most money at the end of the game instead of the objective of powering most cities.

To make this clearer, it's possible to design equivalent games where the relative ordering is more obvious.

Jim Cote offers the following:

I have $2 and you have $3. If someone doesn't pay $2, we both lose. Otherwise, the most money wins. If I pay, you win. If you pay, I win. This example isn't even a game by my definition.
On the contrary, I think it is a fascinating game, of the sort that is studied in game theory.

I'll reframe it slightly to the following:

It is the last round of a game. I have 300 points and you have 200. The rules of the game state that we must now make a contribution. If together we contribute at least 200 points, the person with most remaining points receives a prize of $100 and the other player receives a prize of $50. However, if we collectively contribute less than 200 points, neither of us gets any prize at all.
It seems pretty clear to me that the rational ordering of desired final placements is:

1. Receive $100
2. Receive $50
3. Receive $0

Let's add another rule: the player with most points has to announce how much they are contributing first. He announces that he is contributing 100 points, exactly half of what is needed. The second player now has a simple choice. Contribute 100 points and receive $50 or contribute less and receive $0. The 'game tankers' in Archipelago are effectively saying they would choose $0, because then at least the other player wouldn't have more than them.

If you're not happy with real money being used to determine the ordering, an alternative is:

1. 'Grand winner' in a winning game
2. Any other placement in a winning game
3. All players are executed

Would you tank the game now?

I then got thinking about an extension this simple game. What if not just the relative positions in a 'winning game' were important, but also the absolute point values?

The game now becomes:

It is the last round of a game. I have $300 and you have $200. The rules of the game state that we must now make a contribution. If together we contribute at least $200, both players keep their remaining money. However, if we collectively contribute less than $200, both players lose all their money.
This opens up a continuum of possibilities, and psychology starts to come in too.

If the first player maintains that they are contributing nothing, the second player gets nothing whatever he does. He can either contribute the full $200 and be left with nothing, or contribute less and have both players left with nothing.

So it seems the first player has to offer something. But how much is enough? If he contributes a penny, the second player has to decide between these splits: $299.99/$0.01 and $0/$0. He might decide that it's worth a penny to teach the first player a lesson for being greedy.

What then is a 'fair' way to decide the contributions? I can think of several possibilities.

1. The 'poll tax'. Each player contributes the same fixed amount, i.e. $100. Final split $200/$100.

2. The 'flat tax'. Each player contributes a fixed percentage of their money, in this case 40%. Final split $180/$120.

3. The 'progressive tax'. Each player contributes 20% of their first $100, 40% of their next $100, and 80% of anything above that. Final split $160/$140.

4. 'Communism'. The players pool their money, pay the contribution, and then redivide the remainder equally. Final split $150/$150.

I seem to have digressed quite far now, so I should probably stop

To round off, I'm not sure I would enjoy playing Archipelago - it sounds like it is probably too complex for my tastes. But I am impressed that the designer is playing around with injecting more complex game theoretic situations into 'our games' and fascinated by the fuss it has caused!
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