Monopoly Sim

Our family plays Monopoly every Boxing Day and the winner gets to keep our Monopoly trophy for a year with their name inscribed on a new lolly stick. I've never won, I should do something about that!

I wanted to simulate the game of monopoly to find out which locations are most likely. The distribution won't be even because GO TO Jail takes you back to Jail (or Just visiting Jail once you pay or throw a double to get out) and some of the other cards take you to different locations on the board (Go, Pall Mall, Marlebone station, Trafalgar Square or Mayfair.

Naturally, it's not good enough just to simulate the game, the quest is whether it's possible to do that on a 1K ZX81. We can do that by some analysis on what really counts for the simulation.

Firstly, obviously, to find out which locations are most popular, we don't need to draw a simulation of the board, simply listing the popularity of each location as a series of numbers will do. On a ZX81 we'd need to count the memory used by the screen for this. Assuming each location could have a popularity up to 255, then we need 4 locations per location: 160 bytes in total.

Secondly, we don't need to store the names of the properties; the value of the properties or rents since if we simply know the popularities, we can calculate the income we'd get from that and the card values.

Thirdly, we don't need to maintain any money, because we only care about popularity; this means we don't need to simulate the bank or money for passing GO and we don't need to simulate chance and community chest cards that involve money. So, the only cards that matter are those that take you to a new location.

Fourthly, we can note that there are only 2 Community Chest cards that don't involve money, a goto GO and goto Jail, which are also part of the Chance cards, so we can re-use two of the chance cards to simulate Community Chest.

Fifthly, we can assume that a play will get out of jail immediately by paying or by having a card - it makes little difference to distributions (though we could simulate runs where the play prefers to get out of jail using a double, this means they'd only move

This means the simulation is pretty simple:

• We need an array of 40 locations to hold popularities.
• We need to simulate two dice for moving parts: the sum of two random numbers 1..6.
• We need to redirect the locations if we land on one of the 3 Chance or 3 Community Chest locations or Goto Jail.
• The redirect locations are: 0 (GO), 10 (Jail), 11 (Pall Mall), 15 (Marlebone stations), 24 (Trafalgar Square), 39 (Mayfair) or back 3 spaces, but Community Chest is limited to the first two.
• We could calculate going to jail if the last 3 rolls are matching pairs and this would mean an extra jail visit every 1/216 throws on average. Getting out of jail by throwing a matching pair wouldn't affect results because the throw only puts the player into the 'just visiting' location and thus doesn't affect where they land after leaving jail.
• Periodically we should display the current popularities.

With all these rules apart from the even dice jail rule and using the normal ZX81 Basic space-saving rules we can squeeze the code into 1Kb.

Line numbers take up 21*5 = 105b. The main code takes up another 305b. M$ takes about 45 bytes variables take up about 6*4 = 24 bytes. The screen takes 160 bytes, making a total of 639 bytes, which is within the limit of a 1K ZX81.

Results

It's just a list of numbers representing the popularity of every location from GO to Mayfair:

The ZX81 is really slow when run in SLOW mode, taking 0.6 seconds per go, or about 84 seconds per round including the printing. About 4,000 runs were taken which took nearly an hour. We could speed this up by running in fast mode; removing lines 25 and replacing line 130 with GOTO 30. Then it would take 0.15s per run, so 4000 runs will take 10 minutes.

What the results mean:

Site	Pop	Site	Pop
Old Kent	32	Whitechapel	41
King's Cross	29	Angel	40
Euston	43	Pentonville	36
Pall Mall	46	Electric	41
Whitehall	35	Northumberland	40
Marlebone	40	Bow St	47
Marlborough	51	Vine St	44
Strand	47	Fleet St	52
Trafalgar	58	Fenchurch	43
Leicester	44	Coventry	40
Water	47	Piccadilly	49
Regent	46	Oxford	39
Bond	32	Liverpool	45
Park Lane	29	Mayfair	51

In this run there were 4173 throws, so there would have been 19 extra go to jail's due to three successive pairs (boosting Electric to Strand by about 7%).

We can see that the four most popular places are Trafalgar, Fleet St, Marlborough and Mayfair and at the other end of the scale, Park Lane, King's cross, Old Kent Road, and Bond Street are significantly less likely. It's understandable that Marlborough would be likely, but I would have thought Bow Street would have been equally likely (both 5/36 probability), but Trafalgar was unexpected - except that it's an actual card destination. We know that Chance was hit (47+70+54 = 171) times without the player being redirected (11/16), so in total it was hit 248 times and therefore 15 of these involved jumps to Trafalgar (lowering the hit rate to 52). A similar number of hits redirected to Mayfair (51-15 = 45 still leaving it more popular than Park Lane).

The sample size is about 4173 across 40 locations, about 104 hits per location, so the variance will be fairly significant. Therefore I haven't deduced too many details from the results.

Conclusions

The main conclusion is this: it's perfectly possible to solve an interesting question by simulation on an old computer with minimal memory. Even on a 1Kb ZX81 there was spare space. It's worth considering what the minimal computing resources are to solve the same problem. It's certainly under 1Kb, so a basic Jupiter Ace or a 1Kb MSP430 should easily be possible along with a PIC16F54 or even maybe an ancient 1802-based ELF computer with the standard 256b of RAM.