Taking risks & driving the P2 seat

This is a guest article by -Stef-, who held the top rank of the Isotropic leaderboard for quite some time.

Suppose we’re playing Borinion, a much more boring game then Dominion. Two players, P1 and P2, take consecutive turns. P1 gets to start, and due to the unfair nature of this game it ends after 9 turns. So P1 always gets to make one extra turn. Both players start at 100 points, and every turn they choose…

option A: 60% chance at +3 points, 40% at -3 points
option B: 50% chance at +5 points, 50% at -5 points
option C: 40% chance at +10 points, 60% at -10 points

If you play this game in isolation, it’s easy: always option A is best. In fact, it’s the only option with a positive mathematical expectation. Simulation over one million games shows that if both players follow this simple minded strategy, P1 will win approximately 555000 vs 445000 (55.5 win% for P1).

However, this game is not about scoring as many points as you can. It’s about scoring more points than your opponent. And thus, plans that might look bad at first glance end up being better after all.

Suppose we leave the game plan of P1 what it was (‘always go A’), but experiment a bit with P2. If we change it to ‘option A when ahead, option B when behind’ the win% for P1 drops to 51%. ‘option A when ahead, option C when behind’ works even better; now P1 only wins 46% of the games. We can improve P2 further by delaying the risky things, and enforcing option A on his first turn (42.5% for P1) or his first two turns (42.1% for P1).

I’m starting to get convinced the optimal strategy is actually quite complex. I think it uses all 3 options, and also includes the ‘turns to go’ and ‘actual point difference’ (not just ‘am I behind or ahead’). But I won’t go into that – I didn’t name it Borinion for nothing.

So what does this game teach us about Dominion? Most of all that the optimal strategy always involves your opponent. Even on kingdoms without any attacks, it matters a lot what he’s doing and how well he’s doing. If you both start out the same, but he has some shuffle luck and you don’t, it’s time for crazy things. I he stumbles where you thrive, try to buy safe cards.

It also suggests it’s good to have options. If you create an option B, C or D for yourself to use later on, that choice itself is already good. Engines give you much more options than BigMoney. So in playing engines right, it’s not just about ‘how can I effectively build this engine in solo play’. It also requires a good feeling/understanding for taking risks. Simulators we use today don’t understand it at all, and that may very well be the reason variants of BigMoney strategies do so well in simulation and so poorly in reality.

Opening two terminals isn’t all that bad just because they might collide. P1 has no real reason to take this risk, but P2 is already behind at the start. It of course depends on the questions ‘how good is it if they don’t collide’ and ‘how bad is it if they do’.

One of my favorites is double Steward, because if they collide I can still get rid of 2 cards. I wasn’t going to do much more on a turn with steward without collision anyway. Don’t get me wrong – I’m not happy with the collision at all – but it’s not a game losing disaster either. And depending on the kingdom, being able to get rid of 4 cards in the first round may very well be winning. As a rule of thumb, opening with two terminals is too soon to take risks though.

It’s not easy to define risky or safe things in general. A safe choice that happens often midgame is adding a little more +actions to your deck than the bare minimum. Another safe choice is to stop playing your engine where you could draw some more cards, just to prevent a reshuffle. Maybe you can put a good card on top for next turn?

Risky things could include adding more cards that require other cards (Baron, Remodel, Forge) or buying slightly too many terminals in general. Village + smithy is more risky than laboratory + laboratory. Swindler has the risk build-in all by himself. So do treasure map and tournament, but they’re not really an addition to a deck – they require building your entire deck around them.

In the endgame it can get quite complex because of the ending conditions of a Dominion game. The most common risky thing is buying the next-to-last province. There is a rule for not doing it, but even if it made you lose, that doesn’t automatically say it was a bad thing to buy it. If your deck is not so good, and you’re losing the long run for sure, try to sneak out a victory now.

To summarize: constantly figure out whether you’re ahead or behind. If you’re ahead consolidate, if you’re behind make a plan to get back. As player 2, you’re behind when you start – do something with it.

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10 Responses to Taking risks & driving the P2 seat

  1. GenericKen says:

    I’ve traditionally known this principle as Classical vs Romantic game theory, but I’m constantly surprised that it’s not more commonly known outside of Magic the gathering.

    When behind, up the variance- you need to give yourself a chance to win, even if it’s not classically sound. You may lose epic-ly (see: McCain/Palin) but a loss is a loss. You gave yourself a chance, and players who understand will appreciate your play.

  2. DStu says:

    “Simulators we use today don’t understand it at all, and that may very well be the reason variants of BigMoney strategies do so well in simulation and so poorly in reality.”

    I think the reasons BigMoney strategies do so well in simulation is that in simulations, they only compete against BigMoney strategies usually, and thus naturally one BigMoney strategy wins.

    See Geronimoo’s article on the FirstGame engine for what happens when you really put work into simulating a good engine.

  3. whitecrow says:

    I agree with this article’s point but when I made a quick calculation on excel where both P1 and P2 always play option A and P1’s win probability is 52.8%, not 55.5%. Maybe finding optimal strategy for P2 is far from boring, only had I enough time… =)

  4. onethreeseven says:

    Agreed that 55.5% doesn’t seem to be the right number.

    The optimal strategy can be computed exactly using backward induction: http://en.wikipedia.org/wiki/Backward_induction

  5. When my family and I play Dominion, we have a house rule that makes it so everyone has the same amount of turns. Therefore, if the player who goes first buys out the provinces, the other people get one last turn to try and get a duchy or two to help them have a more even chance. I tend to think this rule is fair because the person who goes first still has more opportunity to buy provinces than every other player, and has a better chance to play the first attack, especially if it’s a discarding attack.
    The only problem is that it can give some strategic edges late game to the people who are going last and know they have another turn, while the person who went first never knows if he’s going to have another turn. I’m not sure how to address this problem though.
    Any ideas or criticisms of this rule?

  6. thorvindr says:

    A lot of games have this rule and a lot of games don’t. I feel like Dominion intentionally doesn’t have this rule because it makes you lazy. If you always know you’re going to get one more turn, you won’t worry so much about how many Provinces are in the pile, or whether you buy the last village or laboratory.

    I think a lot of the strategy (and fun) of Dominion is that it’s not fair, and you need to develop a strategy based on that fact. The deck-building nature of the game limits that a bit, because you essentially decides the odds of everything that might happen on your turn. But it’s definitely true that player one has an advantage over player two. In a six-man game, I feel like it’s not such a huge advantage, but in a two-to-four person game, the difference between player one and player two is significant, while player four is at a distinct disadvantage.

    So, I think your house rule is a good suggestion. It certainly changes the strategy of the game a lot, but I’ve always felt that Dominion is meant to be the ultimate game of “screw your neighbor.” But the bottom line is this: if it makes the game more fun for you, keep doing it.

    The problem will come when you leave the game with that rule and play in a game that doesn’t have it. It’s like using a Designated Hitter in baseball. A pitcher who’s always had one doesn’t train in the batting cage, because there’s no reason for him to know how to hit a baseball: just how to throw one. So when he plays a pickup game with his friends from college, he strikes out every single time because he hasn’t swung a bat since high school.

    The same is true of this rule. It changes the game so much that you’re not really playing “Dominion” any more. You’re playing “Dominion where everybody gets a last turn.” Again, if it’s fun for you, do it. But you’re essentially playing poker with no money.

  7. Anonymous says:

    I have been thinking about Borinon and it and it really is quite interesting. Here is another mini borinon game. Lets say the game ends after two turns but ties go to the first player, who has advantage? Second player! This is quite simple as you can do optimal responses to each move of A, B, C from player 1. If player 1 does C, than 40% of the time he wins outright (as if he gets +10 no move player 2 can make will win), but he also loses outright if he gets -10 as player 2 chooses option A or B and wins, giving option C a 40% win rate overall. if player 1 does B it’s even worse, 50% chance to lose outright as if he getts -5 player 2 does option A and wins, if he gets the +5, player 2 will choose option C, and he still only has a 60% chance of winning giving him a 30% chance of winning overall. Option A is the best option for player 1 but it still doesn’t break 50%, as if he gets +3 player 2 will go option B and have a 50% chance of winning, and if he gets the -3 player 2 will go option A and have a 60% chance of winning, which gives player 2 a 54% chance of winning overall. In conclusion the ability to react to your opponent is fairly huge and finding the optimal strategy for borinon is far from a boring one.

  8. anonymous says:

    re: thorvindr. just wanted to add that i’m pretty sure an mlb pitcher playing pickup baseball still bats about .850…

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