Pompeii: Omen Distribution

Pompeii is an interesting board game on the Yucata.de website which can be played with 2-4 players. The game has a mechanic which seems to make the 2 player game very swingy. The outcome is sometimes determined not by good play but by who pulls the majority of the omen cards. There was a thread on their message boards a couple months ago complaining about the issue and some people blew it off because they didn't think omen distribution was apt to be terribly skewed. Some of these people also mostly played the game 4 players where the impact is markedly reduced. I decided I wanted to actually quantify how likely it is for omen cards to heavily favour one player or the other.

First off, the basic idea of the game is you spend the first half of them game populating the city of Pompeii and the second half running those people away from an erupting volcano. Winner is whoever saves the most people and while positioning the people near exits and away from initial lava floes is definitely important it's generally favourable to have more people heading into the second half. In the first half on your turn you play a card to add from 1-4 people to the board (generally 2) by playing a card from your hand. Then you draw a new card and end your turn. 25 of the cards from the deck at this point are normal dude adding cards. 7 of them are omen cards. When you draw an omen card you get to immediately kill any person on the board and draw a replacement card. In a 4 player game these will generally target the leader or particularly well placed dudes. In a 2 player game these will always hit your only opponent. So while drawing no omens in a 4 player game is bad it's pretty likely the other players will be hitting each other with them when they draw them anyway. In a 2 player game they're always hitting the same person so if one person draws 5 of them compared to 2 they're just up 3 dudes going into the second half which I've found is a pretty insurmountable lead. But how likely is that to happen?

Your first instinct is to think that the split is going to be relatively even but because you get to immediately draw another card it's not so simple. Because omens don't take up space in your hand you can end up snowballing them. In bridge if someone accidentally dropped 12 cards and they were all spades you could be pretty sure they didn't have the 13th spade since they only have 1 more card left in their hand and there are 39 non-spades to go with the last spade. Here if they drop 6 omens from their 'hand' it just means their hand is 6 cards bigger than their opponent and says nothing about the location of the last omen.

It gets a little more complicated too because the first round ends at an unknown time, with somewhere between 0 and 15 cards left in the deck. So you may have a game with no omens at all!

At any rate, my first thought for working this out was to use Markov chains but when I started planning out what sort of states I'd need I quickly realized there were way too many states to work this out in an evening that way. Then I thought this would be a great use of combinatorial enumeration if only I still remembered how that worked. I really should sit down and pound away at that again some day, but that day is not today. Next up... Just write a program to iterate over every possible configuration. I wrote a VBA script, started it running, and then thought about just how many configurations there are going to be. It turns out 32c7 is actually a really big number. Over 3 times the number of rows in an Excel spreadsheet! I'm letting that run while I make food because I'm curious if my recursive function will eventually crash VBA before hitting the millionth row but it's not going to solve my problem. Time to think things through logically...

First off, let's start with a full deck and no early termination. In fact, let's just pretend there's a huge number of cards and we're just going to go until we get 7 omens. Someone has to get the first omen. At that point each player should have a 50-50 chance of getting the next omen. This can continue on until all 7 are given out, with the following odds:

1/64 - 7-0 split
7/64 - 6-1 split
21/64 - 5-2 split
35/64 - 4-3 split

Now, I'm of the belief that a 5-2 split is swingy enough to pretty much be the determining factor in a game and it (or worse) is going to happen more than 45% of the time with an 'infinite' deck. The person who goes first also has a chance of going last (if there's an odd number of turns then one person gets an extra one) and that person has more chances to proc omens. That you might cut out half the deck with the end condition actually reduces the chance of a 3+ difference by virtue of having fewer omens total. When you're splitting up 6 omens there's a 3+ difference only 22% of the time. With 5 omens there's a 3+ difference only 38% of the time. So you've got one factor pulling one way and another pulling the other. Which way will come out on top? Barring a way to work out the precise numbers I'm not sure. What I do know is that any game which has a nearly guaranteed winner 45% of the time isn't high on my list of things to play. Especially with how many turns actual exist in a game of Pompeii.