So. Dominion is a game of luck. We all know this - and this luck aspect makes the game appealing for many people. Pulling out a victory from certain defeat is a nice nelson "haha" moment. However, some of us like to believe that through superior playing we can beat even the worst odds. Just how bad can the odds be anyways?
The issue of opening luck i started thinking about early in my dominion playing. When hinterlands game out, as with all other expansions, me and my friends started playing through the recommended sets. We got to the Traveling Circus" set - where for the first time we feature Nomad Camp and Hunting Party. I thought the nomad camps "place on top of deck" was really cool - so I opened with that on a 4/3 split. With a little luck, I ended up with a Nomad Camp, Hunting party opening. After that, I continued to hit 5 coin every turn. If the nomad camp OR hunting party is in your hand, u will probably hit 5, so it's hard to hit repeated 5s with this combo. My poor friend struggled around with some silvers a long time before getting his first 5. By the time he did, I had already scooped up most of the hunting parties. That game didn't go well for him. I declared - huh, I think that with that kind of opening luck, you will lose every time. He didn't believe me. We recreated the openning sequence - and the exact same thing happened. Now I doubt we played that set optimally - would probably play it very different today - but we were both relatively noob. Among us players of roughly equal skill the opening luck This was my first suspicion that maybe there were some boards where opening luck is the dominating factor in victory.
Let's take a look at some basic luck numbers alone for reference.
If you play big money vs big money, going first gives:
61.6% win for BigMoney First Player
31.6% win for BigMoney Second Player
6.8% there is a tie.
So we can roughly quantify a 60/30 split on wins as a one turn advantage. Fortunately, simulators can correct for this bias just by alternating which strategy goes first. In the following simulations, we will swap who goes first.
Now, if I were to tell you that smithy is on the board - clearly buying a single smithy is better than not. You won't always win though:
23.8% win for BigMoney
71.7% win for BigMoneySingleSmithy
4.5% there is a tie.
Even with this clearly superior strategy, you only win in a 3:1 ratio. If you are losing in a 3:1 ratio, you are pretty much losing as poorly has someone who chooses bigmoney vs BigMoneySmithy. None of this is new to any of you I'm sure. Most of you have probably seen the thread discussing the possibility of a strategy that will beat big money all the time. The best we have found is one that wins like 99.98% or something like that. I wanted to rehash some basic numbers to put in some context for the topic I wanted to discuss - Ambassador. I've felt for a while that games involving Ambassador are particularly luck prone. I'm pretty well known among my friends to declare - that's it, game over - after turn 3 or 4. Am I being pessimistic, or can the odds really be that bad. Lets find out.
Lets discuss Ambassador tactics. Always ambassador Estates and coppers. Return as many copies to the supply as possible. Don't return estates when the game is ending please. (I've tried variants where we return fewer copies of cards if it improves the buy - and for the most part that tactic makes things worst - a separate post
)
How does ambassador fair vs BigMoney? Not great - but a little better than statistical noise:
57.9% win for Ambassador
39.7% win for BigMoney
2.4% there is a tie.
So now lets tip the scales. We will play ambassador vs ambassador but force the shuffle luck for the first 3 turns. Each player will always start with a 3/4 split - opening silver + ambassador. On Turn 3 there will be a lucky player, and an unlucky player. For example, if a player has 1 ambassador + 4 copper on turn 3, this will force a double copper return. Lets see some numbers:
1) Ambassador 2 Copper (unlucky) vs Ambassador 1 Estate.
36.7% win for Unlucky
61.3% win for Lucky
2% there is a tie.
2) Ambassador 2 Copper (unlucky) vs Ambassador 2 Estates.
28.1% win for Unlucky
69.8% win for Lucky
2.1% there is a tie.
3) Ambassador 1 Estate (unlucky) vs Ambassador 2 Estates
42.2% win for Unlucky
53.9% win for Lucky
3.9% there is a tie.
So as long as you get to ambassador on turn 3 or turn 4 things aren't bad. If you get to return an estate, there's not much effect. If you have to return a copper though - things start looking little worse. If your opponent gets to ambassador 2 estates and you just get to ambassador 2 coppers - it's a little worse than going second. Not great.
Oh, but it gets better, what happens if you miss your ambassador turn because your ambassador is at card 11 or card 12 in the shuffle? Meanwhile, your opponent gets to ambassador on turn 3 or 4. This will happen to you in about 10% of your ambassador games. After the opening purchase, lets put an estate as card 11 and ambassador as card 12 in the shuffle and see how our luck fares: (we also assume our unlucky player does NOT receive a lucky early gold and is forced into a silver silver purchase).
4) No Ambassador on turn 3/4 (silver silver purchase) vs Ambassador 2 copper
35.1% win for Unlucky
62.4% win for Lucky
2.5% there is a tie.
5) No Ambassador on turn 3/4 (silver silver purchase) vs Ambassador 1 estate
20.6% win for Unlucky
77.3% win for Lucky
2.1% there is a tie.
6) No Ambassador on turn 3/4 (silver silver purchase) vs Ambassador 2 estate
15.6% win for Unlucky
82.7% win for Lucky
1.7% there is a tie.
Whoa. I would put that last one in the solid - you're screwed category. And if we were playing in real life, you'ld probably hear me rant about it the whole game
What's interesting is if you score an early gold, things may not be as bad: Lets assume our unlucky player can buy a gold:
7) No Ambassador on turn 3/4 buys gold (unlucky) vs Ambassador 2 copper
59.2% win for Unlucky
37% win for Lucky
3.8% there is a tie.
8 ) No Ambassador on turn 3/4 buys gold (unlucky) vs Ambassador 1 Estate
48.9% win for Unlucky
48.2% win for Lucky
2.9% there is a tie.
9) No Ambassador on turn 3/4 buys gold (unlucky) vs Ambassador 2 Estate
38.1% win for Unlucky
56.8% win for Lucky
5.1% there is a tie.
So if you don't get ambassador, but you get an early gold - you may even come out ahead! However, not even an early gold can not make up for your opponent ambassador 2 estates. But if you do score an early gold - keep your chin up. Worst case scenario, you are down one turn.
While this gives you a pretty good idea of your odds when playing with just big money and ambassador. Lets take a look at a game involving other cards. Based off a game I played recently, consider if the following cards are available:
Ambassador, Caravan, Apprentice and Merchant Guild.
I like this set of test cards, as there is opportunity for purchase at the 3, 4, 5 and 6 level. (Big money biases our results sometimes, as there is no benefit to hitting the 4 or 5 in a big money game - and frankly, how many boards are missing 4s and 5s?!)
So the baseline strategy is open Ambassador + Caravan on a 3/4 split. Purchase up to 5 Caravans, 1 Merchant guild. Get gold whenever possible and silver otherwise. How does this fair vs bigmoney (always a good check):
83.9% win for AmbassadorCaravanMerchantGuild
14.8% win for BigMoney
1.3% there is a tie.
Not too bad.
Now let's do a mirror match and throw a little luck in there. Both players will open ambassador + caravan. One unfortunate player will get the following sequence:
Turn 3) Copper, Copper, Copper, Copper, Estate,
Turn 4) Caravan, Copper, Copper, Copper, Estate, Estate
Turn 5) Ambassador + 4 Coppers + 1 Silver.
Even with the caravan in play (which misses the reshuffle because it's on turn 4), the ambassador is in position 12 - so no play till turn 5 - and even then no estate.
Compare this to the more fortunate player:
Turn 3) Caravan, Copper, Copper, Copper, Estate, Estate,
Turn 4) Estate, Copper, Copper, Copper, Copper Ambassador
What are you the odds of winning this?
23.1% win for Unlucky
74.8% win for Lucky
2.1% there is a tie.
In this real life scenario - the odds of winning with a little misfortune are actually quite similar to the earlier data. (Compare with the scenario: Ambassador 1 estate vs no Ambassador). So I will probably use the above big money data to extrapolate my odds in the future.
What is the value in knowing these numbers? Well - when you are playing an ambassador opening - it's useful to know where you stand. If your odds are not great in an equal strategy, mix it up a little. Play more risky - what have you got to lose? Defer buying that first province, try something you may otherwise not.
You have noticed that in the above strategy, I did not mention apprentice! Maybe our unlucky player can add an apprentice to his deck to make a come back:
12.2% win for Unlucky (if buys apprentice)
86.2% win for Lucky
1.6% there is a tie.
Actually - purchasing apprentice makes things worse. This was a counter intuitive result for myself. In fact, I've tried adding many variants to this strategy - and adding the apprentice seems to almost always makes things worse. In fact, if your unlucky player skips apprentice and the lucky player buys apprentice:
35.8% win for Unlucky (skip apprentice)
61.4% win for Lucky (buys an apprentice)
2.8% there is a tie.
Of course simulators don't play perfect - and probably play apprentice poorly in the late game. The takeaway though is that all it took was for the "lucky" player to make one bad call among otherwise equal strategies - buying the apprentice - and a 2 turn advantage was reduced to a one turn advantage. So With 10 cards on the table - be vigilant. If you make just a couple calls better than your opponent you may be able to make up even the worst of ambassador luck. If you have an unfortunate opening with ambassador, you are like going to be practicing more sportsmanship than dominion skills if your opponent is relatively close to you in skill.
(In case you are curious, here's that last example showing how one sets shuffle luck in Dominulator)
var player1 = Strategies.AmbassadorCaravanApprenticeMerchantGuild.PlayerCustom(playerName: "Unlucky", shouldApprentice:false);
var player2 = Strategies.AmbassadorCaravanApprenticeMerchantGuild.PlayerCustom(playerName: "Lucky", shouldApprentice: true);
var builder = new GameConfigBuilder();
builder.SetShuffleLuckPerPlayer( new CardCountPair[][] {
new CardCountPair[] { new CardCountPair(Cards.Copper, 4), new CardCountPair(Cards.Estate, 1),
new CardCountPair(Cards.Copper, 3), new CardCountPair(Cards.Estate, 2),
new CardCountPair(Cards.Copper, 4), new CardCountPair(Cards.Estate, 1),
new CardCountPair(Cards.Caravan, 1), new CardCountPair(Cards.Copper, 3), new CardCountPair(Cards.Estate, 2),
new CardCountPair(Cards.Ambassador, 1), new CardCountPair(Cards.Silver, 1), new CardCountPair(Cards.Copper, 4),
},
new CardCountPair[] { new CardCountPair(Cards.Copper, 4), new CardCountPair(Cards.Estate, 1),
new CardCountPair(Cards.Copper, 3), new CardCountPair(Cards.Estate, 2),
new CardCountPair(Cards.Caravan, 1), new CardCountPair(Cards.Copper, 3), new CardCountPair(Cards.Estate, 2),
new CardCountPair(Cards.Ambassador, 1), new CardCountPair(Cards.Copper, 4), new CardCountPair(Cards.Estate, 1),
},
});
PlayerAction.SetKingdomCards(builder, player1, player2);
ComparePlayers(
player1,
player2,
builder.ToGameConfig(),
firstPlayerAdvantage:false,
createHtmlReport: true,
numberOfGames: 1000,
shouldParallel: false);
