Are there any simulators for Monopoly that meet my specifications below?

Question

There are simulators for "deterministic" games like chess and Go that will estimate each player's win probabilities (probably using Monte Carlo simulation), and also combat resolution simulators for wargames such as Axis and Allies. I was wondering if there are simulators that will estimate your "win chances" in Monopoly, based on the positions that occur after a given trade.

Example: I won a game after giving an opponent the green monopoly in exchange for the maroons. I won the game because I had $1200 cash (and quickly built three houses on each) while my opponent had only $200 cash. (Consider the remaining properties to be "evenly" distributed, including two railroads and one utility for each person.) I would guess that the outcome might very well have been different if my opponent had the $1200, and I the $200.

This isn't a simulator, but it is a calculator that calculates the theoretical value of properties given various "states" of building development. The main thing that is missing is the role of players' cash positions in win chances, because more cash means that you can develop faster than your opponents.

Is there a simulator that can estimate my win chances given starting monopolies and starting cash, (and assuming that the other properties were evenly divided)?

Answer 1

Using the calculator linked in the question, you can do your own rough simulators. The calculator already calculates the theoretical value of each roll in your making money from each configuration. The condition that "other properties are evenly divided" equates to "outside of the properties in question, the net gain or loss between the players equals zero.

The thing that is missing from the calculator is the cash levels. To make matters simple, you could invest $900 of cash in three houses each on the maroons (which is what I did), as still have $300 left for emergencies. Your opponent probably needed to retain all of his $200 cash for "emergencies." He could not buy as much as one house each for his greens (unless he mortgaged some properties, and he didn't do that in the actual game).

Let's examine the states, $300 for you, three houses on each of the maroons; $200 for him, no houses. According to the calculator, each of his rolls is worth $35 to you. It takes about 5.85 rolls (averaging seven on a 40 square board) each time around, which is about $205 dollars. That is (slightly) greater than his $200 salary, which means he is on the verge of defeat. Add more houses or build hotels, and he should be sunk.

With the money reversed, the story is different. You opponent could build two houses on each of the greens, deplete his cash, and rely on mortgaging property for his turn to turn needs. You could build one house each on the maroons only by mortgaging for $100 above your cash. Each of your rolls is worth $32 each to him (average expectation for the two houses on each of the greens) and each of his rolls is worth $4 to you (one house on each of the maroons). With a net difference of $28, each time around the board, you are on the verge of defeat, and will be sunk when he adds third houses to each green.

Answer 2

You're right that the linked site isn't a simulator, but it may be providing even more valuable information. That site provides calculated probabilities which is generally better than a Monte Carlo simulation, which is a technique used to estimate probabilities. The value of the calculation or simulation also depends on any simplifying assumptions used and how well these assumptions match your situation. Since you wish to only calculate based on Monopoly holdings, you indicate a willingness to use simplifying assumptions. I will also assume that you are playing until bankruptcy, advice may be different for a timed game. I suggest that the typical mid-game trade evaluation situation allows us to simplify away the current token positions of each player, so this makes the provided Markov based calculation tables applicable.

Your key concern seems to be a desire to incorporate each player's cash holdings, not just the properties involved. Post trade cash highly effects the game outcome. For this situation, I think the key metric to calculate is each player's expected-income-per-turn in the long run. This won't give a precise estimate of odds of winning, but it will be close enough for good decision making. To make the trade decision, need to evaluate the pre and post trade game states as outlined below to determine if the trade is in your best interest. Note some other trade might be in your best interest and could also be calculated.

For this evaluation process use the "Expected Income Per Opponent Roll on all Properties Assuming Preferred Long Jail Stay" tables. Using this table assumes you are in the end game state which most games will eventually reach. Each player's expected-income-per-turn = + 28.1144 (from non property squares) - Lookup value from each unowned property + [number of remaining opponents] * Lookup value from each owned property. As your question indicates, since income from unimproved properties is a minor factor, we can assume that this largely cancels out and we only need to do the above calculation for improved properties. Including railroad income for anyone with multiple railroads may also be significant. A key challenge in this process is determining what level of improvement to use when looking up the value of each property. We need to determine where post trade cash will be invested. The best assumption would be that available cash would be invested in improvements with the quickest payback as per the "Expected Number of Opponent Rolls to Recoup Incremental Cost (Short Jail Stay)" table.

Any opponent with a negative expected-income-per-turn will need a good deal of luck to win the game. When multiple players will can have positive short term cash flow from expected-income-per-turn, you may also need to calculate the best case cashflow where properties are fully built up. A typical mutually beneficial trade will result in one player getting the better current expected-income-per-turn but the second player getting the better best case expected-income-per-turn. In this case the second player typically wins IF he can avoid disaster long enough to make the needed improvements (and isn't blocked by a housing shortage strategy) in order to eventually have the highest expected-income-per-turn.