“So the probability we lose is 2/3, multiplied by 5/23, or 10/69. It’s pretty much a 6 in 7 shot.”
I met a friend for dinner and board games a while back, and one of the games we played was Pandemic. Designed by Matt Leacock, the game is themed around managing and discovering cures for diseases before they spread across the world.
Diseases are modelled using cubes, and they spread across a network of cities (which are nodes in a graph). On players’ turns, they perform actions to contain the spread of diseases, research cures for these diseases or share knowledge with other players. After they are done with their actions, players draw cards from an infection deck, where each city (node) is represented once. For each card, players need to add a disease cube of the city’s colour to the city. For example, considering a simplified network:
- If D is drawn, one yellow cube will be added to D. Similarly, if E is drawn, one red cube will be added to E; if F is drawn, one blue cube will be added to F.
- A city is allowed to have a maximum of three disease cubes of each type; if a fourth cube of the same type needs to be placed, an outbreak happens. Instead of placing the fourth cube, all connected cities receive one cube of the relevant colour. For example, if C is drawn, a blue outbreak occurs in C, and one blue cube is added to both B and F.
- The rules on having at most three cubes of each type still apply when placing additional cubes, meaning that outbreaks can cause chain reactions. For example, if A is drawn, a red outbreak occurs in A. This causes one red cube to be added to B, D and E. However, B already has three cubes, meaning that a chain outbreak occurs in B; this causes one red cube to be added to A, C, E and F.
- This looks like it would loop infinitely, but there is some mercy; in a given chain reaction, each city may only outbreak once. Thus, instead of adding a red cube and triggering an outbreak again in A, no red cube is added.
The game ramps up steadily in pace, by increasing the infection rate (number of cards drawn) over time. Players lose the game when the eighth outbreak occurs, when they need to place disease cubes but can’t (these are component-limited), or when the deck of player cards runs out.
Separately, the goal of the game is to research a cure for each of the four diseases in the game, which requires collecting data on the spread of the diseases. There is a deck of player cards which contains twelve city cards associated with each disease; players need to collect five of them (in one player’s hand) and bring them to a research station. This can be tricky, as players draw cards independently, and giving cards is restricted (I can only give you a card if we’re both in the one city that is associated with that card, and if it’s my turn). At the end of each player’s turn, players draw two cards from this deck (if they can’t, the game is lost).
The deck of player cards also contains a few special event cards which help players, and more interestingly several Epidemic cards which are disruptive. The number of Epidemic cards to include is variable (4 are used in an easy game, 6 for a difficult one). When an Epidemic card is drawn, players resolve a three-step process:
- “Increase”: The infection rate marker moves along its track, possibly increasing the infection rate (number of infection cards to draw at the end of each player’s turn). This begins at 2, but eventually becomes 4.
- “Infect”: There is a sudden wave of disease in a new city. Players draw the bottom card of the infection deck, and the city in question receives 3 disease cubes of its colour. The usual Outbreak rules apply, though thankfully only one Outbreak occurs even if the city already has two or three cubes.
- “Intensify”: The infection discard pile is shuffled (including the just-revealed card from the bottom of the infection deck). These cities are likely to suffer from more disease soon, especially since this happens as part of players drawing cards (before the infect phase of the current player’s turn).
Going back to the initial calculation, we were on the second last turn of a game with six Epidemic cards, five of which had already been drawn. There were three cards left in the player deck. We already had seven outbreaks occur, meaning that one more would lose the game. However, we had also cured three diseases, and the Medic which I was playing was sitting in a research station with all of the cards to cure the last disease on the next turn. Furthermore, my friend who was playing the Operations Expert held the One Quiet Night Event card, which allows skipping of the Infection phase.
Thus, the only risk of losing was drawing the last Epidemic card, and then in the Infect step revealing a city on the board which already had disease cubes of its colour (from outbreaks in adjacent cities). These events draw cards from separate decks, so their probability should be independent.
The first term is easy – there were three player cards remaining, one of which was the last Epidemic card, so there would be a 2/3 chance we would draw it. For the second term, we looked through the infection discard pile, which at that time had 24 cards (along with Sydney, which we removed from the deck using a special event card). There were thus 23 unseen infection cards; six of them corresponded to locations on the board which had disease cubes. On my friend’s last turn, we were able to defuse one of these locations, leaving a final loss probability of (2/3) * (5/23).
It turned out that we got lucky as we didn’t draw the Epidemic card, and even then the card at the bottom of the deck would have been safe. I wasn’t initially keen on doing too much of the calculation, figuring that clearing one risky place was easy but two was hard, but my friend (who is a statistician) decided to go ahead with the calculation.