Jesse's Pandemonium

—Jesse's Pandemonium is a game of chaos.
By: Jesse Riggs

What Is Jesse's Pandemonium?

There are 105 players in a population. Each player's actions are decided by the roll of a die.

Rules:

  1. Each player is given either two or three coins. Each player must hold their coins until it is their turn. A player who still has their coins is called susceptible.
  2. It becomes a players turn when a coin is transmitted to them by another player, except when the player is chosen to be first ( Player Zero ).
  3. When it is a players turn, they must transmit each of their coins to a different player via connection. A player who is taking their turn is called infectious.
  4. After a player has taken their one and only turn, they are no longer in the game. Any further transmissions must be blocked. A player who is no longer in the game is called immune.
  5. The game is over when the last transmission has been blocked. The winners of the game are those who remain susceptible.

Play

Three buttons control Pandemonium. Full Screen causes Pandemonium to be displayed in full screen mode. Start will start a game, and toggle reset. Opac will change the connection opacity to make players more clear.

image/svg+xml This demonstration requires JavaScript to be enabled.

Effective Reproduction

When a player successfully transmits their coins, they have reproduced that turn — that is, thier turn has been given to at least one other player. The number of coins that a player is given is proportional to the number of times that player can reproduce that turn. The average number of coins that a player has is called the effective reproduction number — or R. The graph below shows the number of infected players, over time, given value R.

image/svg+xml 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Infectious Players Over Time R=2 R=1 R=1/2 Time Infectious Players

As R Shrinks

As the number of immune players grows, the more likely it is that a transmission will fail — the transmission of a coin will be blocked by an immune player. Eventually, the number of susceptible players will drop to a point where it's very difficult to transmit coins and then the number of infected players drops to zero.

We can see in the graph below, that the number of infected players increases for R values above 1. But then levels off when R=1. As R drops below 1, the number of infectious players drops to 0.

image/svg+xml 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 Infectious Players With Varying R R Infectious Time Players

Herd Immunity

If we want to find the reproduction number at any given time, we will need to know what R was in the very beggining, and the proportion of the polulation that is susceptible. The average number of coins that each player has, at the beggining of the game is a special reproduction number called R0

Here, the susceptibility of the population is measured by ( 1 - p ), where p is the proportion of the population that has gained immunity.

R = R0 ( 1 - p )

It would be useful to know when R = 1, because this is when the number of infections level off. This point is known as the herd immunity threshold. If we could find the correct proportion of the population that would need to have immunity to reach this threshold, we could estimate when the infections will taper off. This critical proportion is called pc. We can do a little algebra to find this value.

Pc = 1 - 1 / R

I hope this demonstration has been informative. :)